Grillage Analogy C.s.surana R.agrawal -for Word p4c4w

Grillage Analogy in Bridge Deck Analysis

C.S. Surana IL Agrawal

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New Delhi Madras Bombay Calcutta London

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C.S. Surana Department of Civil Engineering Indian Institute of Technology, Delhi Nevi Delhi-110 016, India R. Agrawal Department of Civil Engineering Institute of Technology, Banaras Hindu University Varanasi, India .r

Copyright C 1998 Narosa Publishing House NAROSA PUBLISHING HOUSE 6 Community Centre, Panchsheel Park, New Delhi 110 017 35-36 Greams Road, Thousand Lights, Madras 600 006 306 Shiv Centre, D.B.C. Sector 17, K.U. Bazar P.O., New Bombay 400 705 2F-2G Shivam Chambers, 53 Syed Amir Ali Avenue, Calcutta 700 019 3 Henrietta Street, Covent Garden, London WC2E 8LU, UK All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. All export rights for this book vest exclusively with the publisher. Unauthorised export is a violation of Copyright Law and is subject to legal action. ISBN 81-7319-153-0 Published by N.K. Mehra for Narosa Publishing House, 6 Community Centre, Panchsheel Park, New Delhi 110 017, typeset at Innovative Processors, New Delhi-110 002 and printed at Rajkamal Electric Press, Delhi 110 033 (India)

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Preface

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Bridge design and construction all over the 'world has undergone remarkable changes in the past two decades. The increased demand for complex roadway alignments, advances in construction technology and availability of computing power for bridge design, are some of the factors for these developments. Over the years, a number of methods of analysis of highway bridge decks have been- evolved. The methods range from the simplified hand method like Courbon's or graphical methods like Hendry-Jaeger and Morice & Little etc. to highly sophisticated methods like finite element, finite strip etc. The former are conservative and the latter which require fairly complex computer programs and larger computational facilities, are prone to errors of idealization and interpretation of results. Grillage analogy method, which is well-established and computeroriented, bridges the gap between the two. The present book describes bridge deck analysis by grillage analogy. The method is versatile in nature and can be applied to a variety of bridge decks having both simple as well as complex configurations with ease and confidence. Analysis of bridge decks employing grillage analogy is possible on commonly available PCs while retaining the accuracy and versatility of other refined methods that usually require larger computational facilities. A considerable saving in time in the analysis of the bridge is achieved and the method also provides a 'feel' of the bridge, behaviour to the designer. Although, the accuracy of any method of analysis for a particular structure is difficult to predict, the method of grillage analogy is found to be fairly accurate when compared to methods like FEM. The book is mainly intended for professionals and students. Consultants and researchers who are confronted with the problem of analysis for bridge design and those who wish to specialize in the subject, will also find it

viii Preface

The designer and consultant handling a variety of structural problems faces difficulties when confronted with the task of analysis of a bridge deck, whether simple or complex. Ordinarily, one would like to seek solutions in the least possible time and, as far as possible, would like to avoid the cumbersome mathematics involved, without compromising on accuracy. This objective is not easily realised. One may either have to develop his own program or to modify available programs to suit the specific requirements. For this, considerable time, thorough understanding and a lot of confidence is essential. A ready to use computer program of analysis based on grillage analogy has been made available in the book. A discussion on the evolving of the program including the basic assumptions and applications of the concepts used in grillage analogy has also beep provided for the discerning . This endeavour will go a long way towards computerizing the analysis of bridges, in this country and elsewhere. The book begins with an introduction of the recent developments in. the area of bridge analysis, design and construction. Specifications for bridge loadings recommended by Indian Roads Congress (IRC) and adopted in India, are described. The loading standards of some other countries are compared with Indian. Standard Loadings. A brief review of the important methods of analysis of bridges including grillage analogy is undertaken and the applicability of each method to various types of bridges having different plan geometry and. conditions are discussed in Chapter 2. The merits and shortcomings of each method is also dealt with. The procedure for formulation and assembly of matrices using direct stiffness method which is more suitable for mathematical modeling of plane grillage, is illustrated, followed by a simple but generalized computer program in Chapter 3. The listing of this program is given in Appendix I. Chapter 4 idealizes the actual bridge deck into a suitable mathematical model of a grillage. The equivalent elastic properties are evaluated and assigned to the of the grillage. Analysis of the idealized grillage for loading is described in Chapter 5. The interpretation of results obtained and the local effects to be included in the final design are also outlined. Chapter 6 discusses a more elaborate computer program written in FORTRAN based on grillage analogy applied to bridge decks. The Program Manual and s' Manual are provided. These are explained so that any designer, not willing to go through the previous sections of the book and with little exposure to the structural behaviour of bridges, can still prepare the input data and analyze a bridge for different IRC loadings. The program

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Preface ix

is so designed that loads other than Indian standard loadings, can also be easily incorporated. The listing of the program is given in Appendix II. A number of worked out examples of different types of slab, T-beam and box-girder bridges are given in Chapter 7 to explain the use of the program.  The example are chosen from actual life-size bridges and the solutions are obtained for IRC loadings and also for 's specified loadings. Input and selected output modules are given for the convenience of the . Exhaustive and relevant references are included after each chapter for the benefit of the readers. A diskette containing two programs of the Appendices (in a ready to use form) can be ordered from the authors through the publisher. The diskette also contains the input data and exhaustive force responses/output of all the worked out examples given in the book. Although the general methods and concepts postulated by Lightfoot, West, Hambly, Jaeger, Bakht and others are further developed and subsequently expanded, we are deeply indebted to them and to many others whose writings, teachings or personal help have shaped our thinking and approach to the subject matter. The financial assistance and other help rendered by the Curriculum Development Cell of I.I.T. Delhi towards writing the book is gratefully acknowledged. The sabbatical leaves granted to the authors by I.I.T. Delhi and I.T. BHU, Varanasi, respectively for this t venture are also thankfully acknowledged. We are indebted to our research scholars who helped in developing and checking the computer programs and to our numerous students for assistance in worked examples and proof checking. The works of preparing drawings and typing of the manuscript was done by different persons at different times. We acknowledge their help. Any suggestions for further improvement from the readers would be very much appreciated. C.S.SURANA RAMJIAGRAWAL

Contents Preface 1.

Introduction L1 General 1 1.2 Recent Trends in Analysis and Design of Bridges 2 1.2.1 Structural Systems 2 1.2.2 Computer-Aided Methods of Analysis 3 1.23 Design Methodology 4 1.2.4 Modern Construction Techniques 4 1.3 Structural Forms of Bridge Decks 5 1.4 Form of Construction 6 1.4.1 Slab Bridge 6 1.4.2 Slab-on-Girders Bridge 9 1.4.3 Box-Girder Bridge 12 1.5 Plan Geometry or Planforms 15 1.6 Configurations 16 1.7 Bridge Loadings 16 1.7.1 Loading Requirements 18 1.7.2 Dead Loads 18 1.7.3 Live Loads 18 1.7.4 Impact Loads 25 1.7.5 Footway, Kerb, Railing and Parapet Live Loads 26 1.8 Comments on Loading Standards 28 1.9 Organisation of the Text 30 References 33

2.

Methods of Bridge Deck Analysis - 2.1 I ntrod uctio n 35 2.2 Methods of Analysis and their Applicability 35 2.3 Courbon's Method 36

xii Contents

2.4 Orthotropic Plate Theory 38 2.5 Finite Difference Method 40 2.6 Method of Harmonic Analysis 42 2.7 Grillage and Space Frame Analogy 2.8 Folded Plate Analysis 47 2.9 Finite Element Method 48 2.10 Finite Strip Method 50

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References 52

3. Stiffness Method Applied to Grillage Analysis 3.1 Introduction 55 3.2 Matrix Method of Structural Analysis 55 3.3 Degrees of Freedom and Sign Convention 56 3.4 Member Stiffness Matrix 58 3.5 Assembly of Structure Stiffness Matrix 63 3.6 Solution of Simultaneous Equations 65 3.7 Computer Program 66 3.8 Example 69

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References 73

4. Transformation of Bridge Deck into Equivalent Grillage 4.1 Introduction 74 4.2 Idealization of Physical Deck into Equivalent Grillage 75 4.2.1 IdeAlinition of Deck Structure 75 4.2.2 General Guidelines for Grillage Layout 78 4.2.3 Grillage Idealization of Slab Bridge 79 4.2.4 Grillage Idealization of Slab-on-Girders Bridge 90 4.2.5 Grillage Idealization of Box-Girder Bridge 92

4.3 Evaluation of Equivalent Elastic Properties 94 4.3.1 Flexural Moment of Inertia, I 96 4.3.2 Torsional Inertia, .1 96 4.3.3 Flexural and Torsional Inertias of Grillage : Slab Deck 99 4.3.4 Flexural and Torsional Inertias of Grillage : Slab-on-Girders Deck 101 4.3.5 Flexural and Torsional Inertias of Grillage : Box-Girder and Cellular Deck 103 References 109

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Contents xiii

5. Application of Loads, Analysis, Force Responses and their Interpretations 5.1 Introduction 111 5.2 Evaluation and Application of Loads 111 5.3 Identification of s in the Grillage 115 5.4 Transfer of Loads to the Nodes 117

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5.4.1 Transfer i of Dead Loads 118 5.4.2 Transfer of Live Loads 118

5.5 Grillage Analysis and Force Responses 131

5.6

5.5.1 Analysis of Grillage 132 5.5.2 Force Responses 134 5.5.3 Design Envelopes 138 Interpretation of Results 139 5.6.1 Slab Bridges 139 5.6.2 Slab-on-Girders Bridges 143 5.6.3 Box-Girder Bridges 143 References 145

6. Computer Program 6.1 1ntroductiOn 146 6.2 Important Features of the Program 'GABS' 146 6.3 Program Manual for 'GABS' 147

6.4

6.5

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6.3.1 Variables 147 6.3.2 Sign Conventions 150 6.3.3 Main Program and Flow Charts 150 6.3.4 Description of Subroutines 151 Manual for 'GABS' 165 6.4.1 Data Input Module 165 6.4.2 Result Output Module 167 Limitations and. Scope 169 References 169

7. Illustrative Examples 7.1 Introduction 171 7.2 Illustrative Examples 172 7.2.1 Example 1: Right Slab Bridge 172 7.2.2 Example 2: Skew Slab Bridge 179 7.2.3 Example 3: Voided Slab Bridge 183 7.2.4 Example 4: Right T-Beam Bridge 187

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xiv Contents 7.2.5 Example 5: Skew T-Beam Bridge 193 7.2.6 Example 6: Box-Girder Bridge 199 References 203

Appendix I: Listing of Program GRID Appendix II: Listing of Program GABS Index

205 215 251

Chapter 1

Introduction 1.1 , GENERAL Bridge construction has been one of the important engagements of mankind from the earliest days. Bridges are one of the most challenging of all civil engineering works. It has always fired the imagination of people as they seem to lead to hitherto uncharted territory. Bridge construction tqday has achieved a world-wide level of importance. The nuiribers and sizes of bridges have continuously increased in the last fifty years. Man's increasing mobility through railway and motorised transport has caused such complex forms of bridges to be built, which had seemed unrealistic earlier. To cope-up with this demand, tremendous efforts all over the world in the form of active research in analysis, design and construction of bridges is continuing. Over the years, a number of methods of analysis of bridge superstructures have been evolved and are being used. Courbon's method, HendryJaeger method and Morice and Little method are some of the methods which have been in use since long, and, are still popular, as they are found to be easy, amenable to design graphs and also reasonably accurate for bridge decks of simple configurations. But these methods are being gradually replaced where computer facilities are available or more accurate analysis is desired or the cross-section and/or layouts of the bridge decks are complex. Following the advent of digital computers, computer-aided methods like Finite Element, Finite Difference, Finite Strip have been developed and are in use to analyse intricate forms of skew, curved, bifurcated and arbitrary shapes of bridges having usual conditions and cross-sections. But these methods are highly numerical and always carry a heavy cost-penalty_ Grillage Analogy is probably one of the most popular computer-aided methods for analysing bridge decks. The method consists of representing the actual decking system of the bridge by an equivalent grillage of beams. The dispersed bending and torsional stiffnesses of the decking system are assumed, for the purpose of analysis, to be concentrated in these beams. The

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Grillage Analogy in Bridge Deck Analysis

stiffnesses of the beams are chosen so that the prototype bridge deck and the equivalent grillage of beams, are subjected to identical deformations under loading. The actual deck loading is replaced by an equivalent nodal loading. The method is applicable to bridge decks with simple as well as complex configurations with almost the same ease and confidence. The method is easy to comprehend and use. The analysis is relatively inexpensive and has been proved to be reliably accurate for a wide variety of bridges. The grillage representation helps in giving the designer a feel of the structural behaviour of the bridge and the manner in which the loading is distributed and ,eventually taken to the s. The book essentially deals with the Grillage Analogy method and its • applications to variety of bridge decks—both simple and complex. But it is also thought relevant to introduce in brief other existing methods of analysis to the readers (Chapter 2). In order to apprise the readers with the developments in bridge engineering, the present chapter discusses The recent trends in analysis and design of bridge decks. The structural forms of decks based on types of construction, planforms and conditions have been outlined. The loading standards used in India for concrete highway bridges, are given in some detail. Indian Roads Congress (IRC) loadings are compared with the standard loadings adopted by some other developed countries. 1.2 RECENT TRENDS IN ANALYSIS AND DESIGN OF BRIDGES Bridges form vital links in the communication system and the need to build bridges across wide rivers with alluvial and scourable beds, deep gorges, open seas and grade separators on urban highways, calls for the solution of a multitude of engineering problems. The ultimate aim is to evolve the most 'efficient design with the available resources and a technical know-how matching the site conditions. It should emphasise on quality and -the life cycle-cost rather than the minimum material used or minimum cost at the time of construction. Structural engineering and construction technology have undergone a sea change in the last three decades and these have had a great impact on modem bridge engineering. Significant new developments in types of structural system, computer-aided analysis, design methods and modern construction techniques have taken place in recent times. These trends and their scope are briefly outlined in the following sections. 1.2.1 Structural Systems In all recent developments of medium- and long-span concrete bridge construction, prestressing has played a Pentral role and roes* '2rid-cs hui:t today are made of structural concrete using prestressing,. This technology is

Introduction

3

now over fifty years old and has proven its superiority, reliability and economy. The technology of prestressing has undergone a lot of change and the technique of external prestressing was `reinvented'.after more than forty years. External prestressing has a number of advantages like ease of construction and better quality and in the case of corroded or broken tendons, these can be exchanged. Cable stayed bridges present a special case of external prestressing as the stay cables also introduce'prestressing force in the girders of the bridge_ In addition to external prestressing, there has been a trend to reduce the dead load of the superstructure leading to new types of bridge systems known as Alternate Web Systems. Alternate Web System consists of web trusses made of concrete with folded plate webs of corrugated steel sheets. An example of such an innovative design is the bridge at Maupre in [73 The introduction of partial prestressing, which covers the range from zero prestressing to full prestressing, allows a much wider application of prestressing. In.bridge design, partial prestressing leads to a simpler distribution of prestressing cables in the longitudinal direction. Local highly stressed zones are covered by ordinary reinforcement. In the transverse direction, partial prestressing is the only -alternative to ordinary reinforcement because the stresses due to traffic can easily exceed those due to dead load. Furthermore, imposed deformations resulting from temperature, creep, settlement etc. can cause tensile stresses in fully prestressed bridges resulting in cracks of large widths. It can be expected that partial prestressing eventually will gain universal acceptance with full prestressing and reinforced concrete, as the two limiting cases. It appears that prestressing will remain the driving force behind new developments in bridge engineering in the near future. 1.2.2 Computer-Aided Methods of Analysis The introduction and application of computers in planning, design, analysis, construction management and safety control of bridges led to a revolution in bridge construction. In the sixties, the problem of indeterminacy posed serious difficulties to bridge engineers. Today, most engineers and design_offices have easy access to personal computers and also special purpose structural software packages have been developed mainly based on computer-aided methods like finite element, finite difference, finite strip, folded plate and grillage analogy. These packages are selectively used depending-upon the complexity of the problem, both for linear and non-linear analyses. However, the friendliness of most application

4 Grillage Analogy in Bridge Deck Analysis softwares need substantial improvements. The major impact, we can experience today, however, comes from the use of software with enhanced graphics capabilities and CAD/CAM systems. With improved analysis capabilities, structural optimization becomes a reality which in the past had just been a subject of interest to academicians. Expert. Systems -will be the next step towards efficient structural optimization. 1.2.3 Design Methodology In the field of reinforced concrete design, the Limit State Concept is gaining increasing acceptance and this has improved the performance characteristics of structures to be built. Two types of limit states, namely collapse limit state corresponding to maximum load bearing capacity (e.g. bending, shear, torsion, buckling, fatigue etc.) and the serviceability limit state which is related to the criteria governing normal use of the structure (e.g crack width, deflection, vibration, durability etc.) have been incorporated into codes of practice and specifications for bridges of many countries. In India, while concept of limit states have already been introduced for building works, the relevant specifications for bridges are not yet available. Different design formats have been proposed for structural codes in order to for uncertainties both in the relevant load combinations at the ultimate state and in the strength of structural or systems. Both simple as well as more refined methods like partial safety factor method, companion action factor method, load-reduction factor method etc. are being used. Given the wide range of_ design situations, it is evident that great precision of the load combinations can not be expected in the near future. The design of bridges for earthquake and accidental actions, which are capable of producing forces exceeding elastic limit of structural , is gaining importance. Therefore, in order to protect bridges from these actions, it is necessary to absorb the energy .at suitable locations by providing plastic hinges. The inelastic deformations of such hinges are characterised by the ductility ratio. Accordingly, the design for accidental actions calls for both strength of as well as ductility of ts and and needs careful consideration. 1.2.4 Modern Construction Techniques In the last thirty years, construction methods have experienced a rapid development. The method of construction influences the cost of the bridge to a significant extent. It is prudent to conceive the method of construction at the time of selecting the type of bridge for the site and keep this in mind while deg the bridge.

Introduction 5

Concrete bridges are usually constructed on stationary falsework only in the case of a small number of spans and also when the superstructure is located. not at a large height above the ground. For other situations, highly mechanised construction methods are used involving repetitive construction operations. Long span bridges are usually constructed using either launching girders (mechanical formwork) or: segmental cantilever construction. Medium-span bridges are constructed adopting the incremental push launching method, classical balanced cantilever method or cantilever method with launching gantry i.e. segmental construction. The importance of the concept of prefabricating entire cross-sectional blocks and combining these with a span-by-span gantry system is increasing. Prefabrication is very economical for short-span bridges. However, large number of bearings and ts have a negative effect on durability and comfort. Improvements are possible by connecting the girder elements at the to form a continuous beam using post-tensioning. . The objectives of a good construction method should be, to build a bridge at a minimum cost and with a maximum margin of safety during the construction phase. The pre-requisite for efficient construction, especially in the case of long span bridges, is therefore an optimum combination of design and construction methods. How some of the major developments affecting analysis, design and construction of concrete bridges, discussed briefly above, will influence the bridge engineering in future, depends upon the designer, contractor, client, authorities and last, but not the least, on the freedom given by Bridge Codes. 1.3 STRUCTURAL FORMS OF BRIDGE DECKS A bridge may be classified in many ways depending upon its function,. material of construction, form or type of superstructure, plan geometry, conditions or span. It is neither intended to discuss here the choice of a particular type of bridge for a specific situation nor to present a detailed routine bridge classification, which may be available in any textboot on bridge engineering. The present book is concerned essentially with the analysis of highway bridge decks and hence the main factors which govern and influence the choice of analytical techniques, to be discussed, are only identified. They are: I. Form of construction or type of deck 2. Plan-geometry or planform and 3. conditions

6. Grillage Analogy in Bridge Deck Analysis Each of the above parameters is discussed and illustrated. The description will be limited to only those types of bridges which can be gainfully handled by employing the method of grillage analogy. The decks of reinforced concrete, prestressed concrete and composite construction shall form the basis of our discussion in the following sections. 1.4 FORM OF CONSTRUCTION The principal forms of bridge deck construction have been reviewed and categorised in this section. Broadly, the forms of construction can be divided into slabs (solid, voided and pseudo-slabs), slab-on-girders (T-beam and Ibeam) and box-girders (single, multi-cell and multi-spine). Arch, rigid frame, truss, suspension and cable-stayed bridges are not included in our discussions as these are not amenable to Grillage Analogy. 1.4.1 Slab Bridge • Slab bridges are easiest to construct and are frequently used for comparatively .smaller spans. The form is very efficient at distributing point loads because of its two-way spanning ability and high torsional strength. It is relatively easy to construct and this is reflected in its construction cost. The principal disadvantage is its high self-weight which can be counteracted to some extent, by providing suitable. variation in thickness or by providing voids. It may be of reinforced concrete or of prestressed concrete. Solid reinforced concrete slab of constant depth is normally used for spans upto 10 m (Fig. 1.1a). For larger spans, say upto 15 m, haunching or variable depth is adopted to reduce dead load (Fig. 1.1b). A solid slab of uniform depth is. preferred in highly skewed crossings, particularly if significant curvature and variation in width of the deck is involved. Continuous construction can be used with advantage if the possibility of uplift at abutments is expected. Voided slab bridges (Fig. 1.1c) are adopted to reduce the self weight of the bridge. The voids are usually circular or rectangular. The depth of voids is generally restricted ,to sixty per cent of the depth of the slab so that the slab continues to behave like a single plate. If this limit of void-depth is exceeded, the slab may behave more like a cellular deck (discussed later). The voids may either run for the full span length or, alternatively, these may be provided in the central span length only so that solid section is available near the s where shear is large. Voided R.C. slabs with depth upto 100 cm may be adopted for span range of 8 to 15 m. However, for spans between 15 and 30 m, voided pre-

Introduction 7

___________________________ E / a) Solid Slab

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Voided Stab Figure LI Slab Bridges

stressed concrete slabs of depth upto 1.2 m are cheaper. For moderate skew crossings having spans of 15 to 25 m, this type of deck with longitudinal prestressing is useful but for highly skewed crossings, reinforced concrete decks are preferred for ease of construction. If the voided section is found inadequate in shear, it should be kept solid near s (refer voided slab example of Chapter 7). In R.C. slab bridges, span-depth ratio of 15 for simple spans and 20 to 25 for continuous spans are usually adopted for both solid and voided slabs. For cast-in-situ, prestressed concrete voided slab bridges, this ratio is nearly 30. In precast prestressed voided slabs, the ratio ranges between 25 and 30. The deck slab overhang, designated as 'a' in Figs. 1.1b and 1.1c may be provided to produce the desirable aesthetic effect and also to reduce the dead load and the width of sub-structure. In many countries, standard precast prestressed beams are employed for short- and medium-span bridges. These standard beams are closely positioned across the width of the bridge and in-situ concrete is poured to give transverse connection in order to create a slab-type deck. This form of deck is described as Pseudo-slabs. Such type of two-stage casting is also referred to as Contiguous Construction and the standard beams are termed as Contiguous Beams.

8 Grillage Analogy in Bridge Deck Analysis

Many forms of prestressed, precast beams are used in pseudo-slab decks. The Prestressed Concrete Development Group (PCDG) in U.K. produced a series of inverted T-beams, 1-beams and box-beams for various spans (Fig. 1.2). The beams are placed side by side and the gaps are filled In-situ concrete

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with in-situ concrete to form an integral slab. Transverse bond rods are provided just above the bottom flange ing through pre-formed holes in the precast beams. This type of construction can be used upto 20 m span. Another standard group of beams, known as M-Beams have been developed by Cement and Concrete Association (C&CA), U.K., in collaboration with the Ministry of Transport, U.K. (Fig. 1.3) for short- and medium-span bridges. Pre-tensioned cast-in-situ Pseudo-slabs are usually adopted when the erection of formwork presents no difficulty. The advantage of this form of construction is that the structure is monolithic and the stress-distribution in the slab can easily be evaluated. On the other hand, the precast zire.crs v:ith in-situ concrete filled up, are preferred when there is difficulty in ing

Introduction 9

Permanent Pocking a) Pseudo- Box

Beams

b) T-Beams Figure 1.3 C & CA, U.K., Standard M-Beams

the formwork. However, in general, the prestressing of slab, is uneconomical and the formwork is often heavy. The concept of Pseudo-slab bridge construction using pre-tensioned precast beam elements is relatively new to India. 1.4.2 Slab-on-Girders Bridge

Slab-on-girders bridges are by far the most commonly adopted type in the span range of 10 to 50 m. The majority of beam and slab decks have number of beams spanning longitudinally between abutments with a thin slab spanning transversely across the top. T-beam bridges are one of the most common examples under this category and are very popular because of their simple geometry, low-fabrication cost, easy erection or casting and smaller dead loads (Fig. 1.4). Usually I-section or T-section is used for the beam but T-section is found to be more efficient. T-beams are economical where depth of section is not a controlling factor from headroom considerations. The T-beam bridge superstructure may consist of either girders and slab or girders, slab and diaphragms at the s or girders, slab, intermediate cross-beams and

10 Grillage Analog' in Bridge Deck Analysis

end diaphragms. However, T-beam bridge with cross-beams extending into and cast monolithically with the deck slab is found to be more efficient and is recommended for adoption. Sinip1y ed R.C. T-beam is normally adopted for spans upto 25 m. Span-depth ratio is generally kept as 10 for. Services

Precast detachable footpath slab

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h Figure 1.4 T-Beam Bridge

simple spans and 12 to 15 for continuous spans. Higher ratios are possible, but riding qualities are affected by creep characteristics of concrete. The girder spacing 'h' (Fig. 1.4) may vary as justified by comparing the cost of corresponding slab thickness. The usual range of spacing 'h' is between 2 to 3 m for these bridges. The stem-width 'b' is about 300 mm. This stem width is increased to 'B' at the bottom, forming a bulb to accommodate a large number of reinforcement bars there. This '13' may be kept between 500 to 600 mm. The stem width '6' is increased to '13' in the end-region, to take care of large shears occuring there. Slab-on-girders bridge also includes prestressed concrete bridges. Majority of prestressed concrete bridges, constructed in India are of post-tensioned type. The bridge decks, with post-tensioned girders suitable for simply ed construction, may ..have either fully cast-in-situ slab and girders or deck slab with precast prestressed girders alongwith cross-beams, assembled together and transversely prestressed (Fig. 1.5a,b). Such types of construction is convenient for the span range of 20 to 30 m. The span-depth ratio is usually kept as 20 for simple spans and 25 for continuous spans for prestressed concrete girder bridges. The girder spacing 'h' (Fig. 1.5a) is normally kept between 2.0 and 4.5 m. The stem width 'b' should preferably be a minimum of 300 mm to facilitate prestressing of tendons. The deck slab overhang 'a' should be provided as required to produce the desirable aesthetic effect and to reduce transverse moments. Decks with composite construction are also popular for short- to mediumspan bridges. Composite construction refers to the use of structural elements made of two materials in combination in such a way that they act

Introduction 11.

Cross beam

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a) Cast- in- Situ Prestressed Concrete Girder Deck 25mm gap between precast girders

11111111111111Nal111111enIF Precast girder Precast cross- beam b) Tronsversety Precast Prestressed Girder Deck Figure 1.5 Bridge Decks with Post-Tensioned Girders

together. Though, normally, this is understood to refer to the use of rolled steel sections with in-situ concrete slab but it also covers precast prestressed concrete girders with cast-in-situ R.C. slabs and made to act as T-beams. The girders and the floor slabs are connected using shear-connectors by which they work together. The precast prestressed concrete girders and cast-in-place R.C. deck slab (Fig. 1.6a) is suited for medium spans i.e. spans between 30 and 60 m. For such bridges the span-depth ratio of about 18 in case of simple spans and say 20 for continuous spans, are recommended. The composite construction with steel girder (Fig. I.6b) is economical in the short span range of 8 to 15 m. The construction *has the advantage of speedy erection and reduced cost of formwork. The flexural stiffness of a composite beam is about 2 to 4 times that of the corresponding steel beam and this property results in reduced deflections and vibrations. The transition from pseudo-slab to slab-on-girders deck is difficult to define with complete precision. Pseudo-slab is associated with multiple longitudinal beams (five or more for a two-lane bridge) at close spacings whereat, the beams aria sian oriages tall in the category of decks with smaller number of longitudinal beams under similar situation.

12 Grillage Analogy in Bridge Deck Analysis

girder Precast Cast-insitu slab

a)

Precast Prestressed Concrete Girders with' in- situ Stab n-situ R.C. slab

Steel girder

b)

Rolled Steel Girder Sections with Castin- situ R.C. Stab Figure 1.6 Decks with Composite Construction

1.4.3 Box-Girder Bridge

In recent years, single or multi-ell reinforced and prestressed concrete box-girder bridges have been widely used as economic and aesthetic solttions for overcrossing, undercrossing, separation structures and viaducts, found in today's modem highway systems. The main advantage of these type of bridges lies in the high torsional rigidity available because of the closed box-section and convenience in varying the depth along the span. High torsional stiffness gives them better stability and load distribution characteristics and also makes this form particularly suited for Grade Separations, where the alignment of bridges are normally curved in plan. Also, the hollow section may be used to accommodate services such as water mains, telephones, electric cables, sewage pipes etc. and the section has an a d d e d a d va n t 2 o - e of h pi n a B a h r

Introduction 13

The cross-section of a concrete box-girder bridge consists of top and bottom slabs connected monolithically with vertical webs to form a cellular structure. The box-girder may be composed of single cell or multiple cells, with or without side cantilevers. The cells can be either rectangular or trapezoidal, latter being used increasingly in prestressed concrete elevated roadways. Some of the commonly adopted cross sections are shown in Fig. 1.7. A multi-cell deck is distinguished from a voided slab in analysis. In multi-cell decks, the transverse bending of top slab between webs is also considered. Also, the void depth in a multi-cell deck may be larger than the limiting values given earlier for the voided slab, leading to cell-distortions.

p

t w

S

(a) Single -Cell Box-Girder

V./ 4

h ___ I

(b) Multi - Celt Box-Girder

T

5

I.-- a (c) Single-Celt Trapezoidal Box-Girder Figure 1.7 Box-Girder Bridges

Multi-cell box-girder bridges are constructed with or without transverse diaphragms. If diaphragms are provided only at the s, shear deformations in the transverse direction leading to shear-lag and cell-distortions have to be considered in the analysis. But if additional intermediate diaphragms are also provided between the s, the shear deformations and cell-distortions are usually small and mav be ored.

14 Grillage Analogy in Bridge Deck Analysis

In the span range of 20 to 30 m, cast-in-situ multi-cell reinforced concrete box-girder bridges are widely used. For longer spans, say upto 60 m, posttensioned prestressed, cast-in-situ multi-cell box-girders 'have been employed. Two-cell box-girders have been used for span ranging from 30 to 40 m, while single cell trapezoidal box-girders are built with 30 to 50 m span. Post-tensioned precast box-girders prove to be economical for larger spans say upto 100 m. For spans above 60 m and upto 200 m, segmentally erected prestressed concrete box-girder bridges with one, two, or more cells, spaced apart, may be adopted. This form of construction is commonly used in viaducts and is sometimes known as Spine, Spread or Spaced Box-girders (Fig. 1.8). Spine beam bridges, as built in practice, may be defined as structural whose breadth and depth are small in relation to their length and are therefore subjected mainly to longitudinal bending, transverse shear and torsion. Transverse diaphragms are normally provided only over the s. The bridges are generally prestressed longitudinally and reinforced transversely unless they are exceptionally wide and in such a situation they may be prestressed transversely also. These may be continuous or simply ed.

b

SI

—+- b

w

o

-

a —4.— b

Figure 1.8 Multi-Cell Spine Box-Girders

The span-depth ratio for R.C. box-girder bridges are generally adopted as 16 for simple spans and 18 for continuous spans. For prestressed cast-in-situ concrete box girders, this ratio ranges from 20 to 25 depending upon conditions. In case of precast prestressed box-girder bridges, the span-depth ratio is taken between 18 and 20 whereas for spine or spaced box-girders, the ratio lies usually between 18 and 22. In precast, prestressed multi-cell boxbeams, the ratio can be as high as 25 to 30 [9]. The spacing 'h' of box-girders usually lies between 2.0 and 3.5 m. The web of box-girder superstructure should have a minimum thickness `ki: of 200 mm. It is often useful to increase the thickness of webs near s to provide adequate concrete section for shear resistance. Precast box-beams ordinarily have a width '17' of 1.0 to 1.2 m and height 's' in the range of 0.6 to 1.2 m (Figs. 1.7 and 1.8). The bottom slab thickness 'd,' is kept

h

Introduction IS

approximately 1/20 of clear span between webs but it should not be less than 150 mm and may be increased near continuous s. 1.5 PLAN GEOMETRY OR PLANFORMS The horizontal and vertical alignments of a bridge are governed by the geometries of the highway, roadway or channel it crosses. A bridge may either be right or skew, straight or curved or any combination thereof. Typical geometrical planforms of bridge decks are illustrated in Fig. 1.9.

(a) Right Deck

(b) Skew Deck

(c) Symmetrical Curved (d) Unsymmetrical Curved Deck Deck

(e) Arbitrary Shaped Decks

(f) Bifurcated Deck 7,777ned edge — Free edge Figure 1.9 Plan Shapes of Bridge Decks

16 Grillage Analogy in Bridge Deck Analysis The simplest form is the right deck but the demand for skew bridges is increasing because of non-availability of space for traffic schemes. The skew effect becomes more important in design when the skew angle exceeds 15°. The construction of horizontally curved bridges has increased considerably in recent years for highway bridges. The need for smoother dissemination of congested traffic and the limitation of right -of-way alongwith economic and environmental considerations dictate that the bridge alignments meet the overall requirements of the highway system. Also, the current emphasis on aesthetic considerations has motivated increased development in designs • which utilizes curved configurations, either symmetrical or asymmetrical. Due to the geometric complexities, curved girders are subjected to not only flexural stresses but also to very significant torsional stresses. The deflec tions and stresses in such decks are markedly different from those of right bridge decks. There is a large range of arbitrary planforms where the free.edges of the bridge are non-parallel, lines are non-parallel or s are randomly distributed. Bifurcated decks are needed at motor-way exit or entry and may have an arbitrary geometry. 1.6 CONFIGURATIONS Figure 1.10 shows some of the configurations normally used in highway bridges. The simple s are common with slab bridges or with slab-on-girders bridges of smaller spans. Cantilever and Balanced cantilever -bridges are constructed for span range of 35 to 60 m having T-beam or box-girder as their cross-section. Fully continuous bridges are advantageous for spans over 35 m and are suitable with prestressed concrete girders. Further, the bridge may be placed on rigid s or flexible (yielding) s. The conventional plate, rocker or rocker-cum-roller bearings provide rigid s. However, the recent trend is to favour elastomeric bearings. This provides yielding s. These are preferred because of their low height and low cost and require practically no maintenance. Also, they are easy to replace. These bearings can cope up with complex deformations of skew and curved geometry. 1.7 BRIDGE LOADINGS The loading has profound effect upon the design, construction and eventually upon the cost of any bridge of a given span. Besides carrying their own weight, the bridge decks are designed for certain loadings imposed partly by the vehicles and the s and partly by

Introduction 17

(a) Simply ed Arrangement

(b) Cantilever Arrangement Anchor

Canti lever Suspended span

(c) Balance d Cantilever Arrangemen t

ft 11

fi

(d) Continuous Arrangement Figure 1.10 Configurations of Bridge Decks

nature. In order to maintain uniformity in design, loading standards have been laid down for the guidance of engineers. Different countries, including India, have their own loading standards. In India, these standards for Railway bridges are formulated by the Research Design and Standards Organisation (RDSO) of the Indian Railways [17]. For highway bridges, Indian Roads Congress (IRC), a statutory body formed by the Government of India under the Ministry of Surface Transport, prepares the Codes of Practices. These codes are complied faithfully in the design of bridges [18, 19, 20]. The Bureau of Indian Standards (BIS), a body responsible for the "Standardization" in the country, also brings out specifications ror images [2 i But the specifications laid down by IRC supercede those of the BIS, wherever at variance.

-

r 18 Grillage-Analogy in Bridge Deck Analysis 1.7.1 Loading Requirements The deck of the highway bridge has to moving loads in the form of vehicles, men and materials and transmit their effects to the foundation. It has also to and carry the self weight of its various components. The structure is also subjected to vibrations under moving loads giving rise to what is known as impact loading. The details of some other loads and forces such as earthquake,'iwind etc. which also become important in some cases could be referred :from the Codes of Practice [18, 19]. Only the important loads to be used in- the analysis of decks are briefly described here. 1.7.2 Dead Loads The bridge superstructure is to be analysed for its self weight and dead loads imposed on it as well. The dead loads imposed on the bridge consist of permanent stationary load such as that of wearing coat, kerb, parapets etc. In estimating the dead loads, the unit weights of materials specified in reference [19] may be adopted. Dead, loads invariably form a relatively large loading component and result in significant design forces and deformations. It is, however, never a problem to either estimate these loadings accurately or compute their effects on the structure. 1.7.3 Live Loads The main loading on highway bridges is due to the vehicles moving on it, which are transient and hence difficult to estimate accurately. In order to analyse the bridge for these moving loads, IRC Code [19] recommends certain standard hypothetical loading systems. The bridge is then designed for the maximum response values under these standard loads. The live loads usually consist of a set of wheel loads which are patch loads due to tyre area. These patch loads may be treated as point loads acting at the centre of the area. This simplification is found to be acceptable in the analysis. According to Indian Roads Congress classification, the main live loads for road bridges can be put into the following four types [19]: i) !RC Class A Loading—Single Lane and Two Lanes Single lane Class A loading is a train load of eight axles of two wheels each thus having sixteen wheels in total. The total load of the train is 55.4 Ti :Ai Lai: it..ngth of the train is 2U.i in and the distance between the first and the last axle is 18.8 in. The minimum clear longitu .61

Introduction 19

dinal distance between two successive trains is 18.5 m. The minimum centre line distance of the wheel-line from the edge of the Kerb works out to 400 mm. The configuration of the load as well as the position of each wheel is given in Figs. 1.11(a) and 1.11(b).

1200 f--4800

1-- 8300

1200 H-4800 --1

1-/-1.- 3200-F-1- 4300 +3000 +3000+ 3000-1 co. to

6001100 1200 1

c`:

co co  .

1 Co co.

1 I

Axle load (tonnes)

co

cD

Class A

fa irt CO 4:0

(6

-4

Class B

.

(a) Class A and Class B Train of Vehicles

X 4- +

-F. +

+

+

+

Direction of Motion 

1800 + +

(Left Most Front Wheel H81-32004§4-1.300

+3000+3000+3000-

1 (b) Plan

1800

(c) Section on x x (411. Dimension are in mm) Figure 1.11 (a) 1RC Class A and Class B Loadings

Class A-two laneS loading consists of two class A -single lane trains placed side by side at specified minimum clearance. Class A loading is

20 Grillage Analogy in Bridge Deck Analysis

Clear Roadway

.0- •"/".0,0.0.0.0.0 • 7e..0,/,/ —01W

fm.r1_________ Iii),

(d) Cross-Section

Max. w =500 for Class A Max. w :380 for Class B Minimum Carriage way. widih eg, 5500 lo 7500

clearances

Uniformly increasing from 400 to 1200

>7500

1200

ft 1 150 150

(All Dimensions are in mm) Figure 1.11 (b) IRC Class A and Class B Loadings

adopted on all permanent bridges and culverts to be constructed on State and National Highways. ii)

IRC Class B Loading

The Class B loading is identical to Class A loading as far as positions of axles are concerned but the magnitude of axle loads is 60% of the -corresponding loads in Class A vehicle (Figs. 1.11a, b). This loading is intended for temporary structures, timber bridges and bridges in specified areas. iii)

IRC Class AA Loading

This loading is an alternate loading and one train of Class AA vehicle is to be considered for every two lanes of Class A loading. It consists of either a tracked vehicle of 70 tonnes or a two axle wheeled vehicle of 40 tonnes. Detailed dimensions, kerb distances etc. are given Figs. 1.12 (a) and 1.12 (b). Bridges designed for Class AA loading should also be checked for equivalent lanes of Class A loading since under certain conditions. heavier stresses are obtained under such equivalent Class A loading. The nose to tail spacing between two successive vehicles is specified as 90 m.

Introduction

21

3600 _____ 7200 _____ 0) Elevation

4- 1-1-++ ++4- + + -r

Suggested Equivalent

Concentrated Loadings o

un

O

Left Most Front WheeNt- 1-4--++++++1 [4-9@360z3240 ______ (20 toads 3.5t =70t on Two Tracks) (ii) Nan

35t

35t 2900

(iii) Cross-Section (All Dimensions are in mm)

Figure 1.12 (a) IRC Class AA Tracked Vehicle iv) IRC Class 70R Loading This is the revised version of Class AA loading and consists of tracked .and wheel loadings. The loading is detailed in Appendix I of IRC Code [191 The minimum clearance between the road face of the kerb and the outer edge of the track or wheel is same as for Class AA loading. The spacing between successive vehicles is 30 m. This spacing is measured from the rear most point of ground of the leading vehicle to the forward most point of ground of the following vehicle.

22

Grillage Analogy in Bridge Deck Analysis

-43001— -...1 300H-700 —.I 300k- -.I 300 1.-. 1 375t -I+Mos t---

Lett Front Wheel

1 1 ___ I 6.25 t 6.25 t 3.75 t (1) Cross Section + -

10,2

Ill\ +

-I-

+

4-

4.

+ I-

— k 600 —4* 600 --1 __ 1000 2200 ___________ ) Plan

Carriage Way Minimum Width Value of C Single Lane Bridges 3800 and 300 above Multi L ane B ridg es -‹ 5500 600 5500 1200 All Dimensions are in mm Figure 1.12 (b) IRC Class AA Wheeled Vehicle

70R track loading, as before, weighs 70 tonnes (Fig. 1.13). The track dimensions are slightly different than those of Class AA track loading. For design purposes, wherever required, each strip loading could be idealised into a suitable number of point loads say 8 or 10. 70R wheel loading is of two types: (1) 70R Bogie loading weighing 40 tonnes through two axles each weighing 20 tonnes (Fig. 1.14a) and (2) 70R train loading weighing 100 tonnes through seven axles, one axle of 8 tonnes, two axles of 12 tonnes each and four axles of 17 tonnes each (Fig. 1.14b).

Introduction 23

4570 ____ 7920 _____ (a) Elevation t ,+ + + + + + + + -ivSuggested Equivalent I Concentrated Loadings c,

4 + 4 + Left Most Front. Wheel (20 toads 03.5t =70t on Two Tracks) (b) Plan / 1-P-96)457= 4113—.1

35T. 14— 2900

35T

(c) Cross- Section Clearance 4C' same as for Class AA Loading (All Dimensions are in mm) Figure 1.13 IRC Class 70R Track Loading

An axle may have four or eight wheels on it. There are two, four wheel arrangements and one, eight wheel arrangement leading to three alternate wheel arrangements termed as Col. 'I', Col. 'm' and Col. 'n' arrangements [19]. AU axles will have the same arrangement of wheels at a time and all wheels on an axle will have equal loads. The two alternate four wheel arrangements namely Col. '1' and Col. 'm' are given in Figs. 1.14(a, b). The eight wheel arrangement namely Col. 'n' is not found critical and is not given nere. However, if required, details of Col. 'n' arrangement could be obtained from IRC Code [19].

24 Grillage Analogy in Bridge Deck Analysis 4880

I-

20t 20t (i) Elevation " --"F 0

4

+ I

0 JCI

CO

c•i -I-

+t

Left Mos

Front Wheel (ii) Plan Tr," -1`4 W /-"F/

1-e--

n n b

—NI

(iii) Cross-section Note: (i) Min. ro' same as for Class A A Loading (ii) Max. 'w' =410 (iii) Either a=450 & b =1480 (CoI.1') or a=795 & b = 790 (CoL`m1); so that (2a4b) = 2380

Figure 1.14 (a) IRC Class 70R Bogie Loading IRC Code [19] also gives in Appendix I certain other types of track and

wheeled loadings. These are lighter than Class 70R tracked and wheeled loadings discussed above. These are to be adopted if a specifies these for the bridge. For detailed loading Standards and their specifications, the reader is advised to consult the relevant IRC Codes of Practice mentioned under references [18, 19]. The above IRC loads are nlared on the. hriricre clerk anr1 mnveri tong;til_ dinally as well as transversely in small increments to cover the entire bridge

Introduction 25

1

3960 1 t 12t (i)

-

O

T

213 137073050 7'713 — —t 1 t 1 7 1 17t

0 1f

Elevation

44 ++++

00.4co

4 (ii) (Left Most Front Wheel

a

+

P

+. +4-

4

Plan

Note : Cross-Section and distances are as for Bogie Loading shown in Fig. 1.14(a) (All dimension are in mm)

Figure 1.14 (b) IRC Class 70R Wheel Loading deck. One Class A or Class B loading can be put on every lane of the roadway of a bridge. For multi-lane bridges, one lane of Class AA or Class • 70 R per two lanes of the carriageway is allowed as an alternate to Class A loading. 1.7:4' Impact Loads Another major loading on the bridge superstructure is due to the vibrations caused when the vehicle is moving over the bridge. The theoretical estimation of this load is quite complex as it depends upon a variety of factors such as roughness of the surface, spring system of the vehicle, condition of expansion ts at the entry to bridge etc. The IRC Code [l9], however, recommends definite values of impact factors for the vehicles for simplifying the analysis. The value of impact load is expressed as a percentage of the live load, depending upon the material used in the construction of deck of the bridge, ly pc of Luiu!

26

Grillage Analogy in Bridge Deck Analysis

the bridge span. This percentage can be calculated using suitable formulae [19] or could be directly read from Figs. 1.15 (a,b) for both steel and concrete bridges for different types of IRC loadings. 55

54.5

50 74 . Lo

45

L= Span in m

40

35 30

900 13.5+L

44

0

Steel

25 Q. 20 E 15

450 6 +L

Concrete

10

15.4

8.8

I 1 I 5 0 5 10 15 20 25 30 35 40 45 50 Span (m)

Class A and Class B Loadings

Figure 1.15 (a) Impact Percentage Curves

1...7.5 Footway, Kerb, Railing and Parapet Live Loads The following provisions have been made for footpath, kerb, railing and parapet live loadings in IRC:6-1987 [I9]. (i) For all parts of bridge floors accessible only to pedestrians and animals and for all footways, the loading shall be taken as 400 kg/m2. Where crowd loads are likely to occur, such as on bridges located near towns, which are either centres of pilgrimage or where large congregational fairs are held seasonally, the intensity of footway loading be increased from 400 kg/m' to 500 kg/m2. (ii) Kerbs, 0.6 m or more in width, shall be designed for the above loads and for a local lateral force of 750 kg per metre, applied It -,•rizontalt: at the top of the kerb. If the kerb width is less than 0.6 m, no live load

Introduction 27

430 U • 25

.

O

4.  20 0• 15

u10 "" 5 0 5 10 15 20

30 (a)

35

40

45

50

25 Span (m)

Class AA and Class 70R Wheeled Loading .7.30 025C

25

2:20 10

STEEL

40 .7.30

L.= Span in m

025C 4

(

2:20 215

4-

45 50

Concrete & Steel J.

10 CON C. 8.6

U

o. 5 F 0 5 10 15 20 25 .30 35 Span (m) (b) Class AA and Class 70R Tracked Loading Figure 1.15 (b) Impact Percentage Curves

may be necessary in addition to the lateral load specified above. The horizontal force need not be considered in the design of the main structural of the bridge. (iii) In bridges designed for IRC vehicular loadings, the ing the footways shall be designed for the following live load per square metre of footway area, the loaded length of footway taken in each case being such as to produce the worst effects on. the member under consideration: a) For effective span of 7.5 m or less, 400 kg/m2 or 500 kg/m2 as the case may be as per (i) above.

b) For effective spans of over 7.5 in but not exceeding 30 m, the intensity of load shall be determined according to the equation:

28 Grillage Analogy in Bridge Deck Analysis

P =Pi (40L —9 300) c) For effective spans of over 30 m, the intensity of load shall be determined according to the equation: P

=

( p

2 6 0

+ 4800)(16.5 L A 15 )

where P` = 400 kg/m2 or 500 kg/m2 as the case may be based on (i) above. P = the live load in kg/m2L= the effective span of the main girder in m. W = width of the footway in m. • (iv) . Each part of the footway shall be capable of carrying a wheel load of 4 tonnes, which shall be deemed to include impact, distributed over a area. 300 mm in diameter; the permissible working stresses shall be increased by 25 per cent to meet this provision. (v) The railings or parapets shall be designed to resist a lateral horizontal force and a vertical force each of 150 kg/m applied simultaneously at the top of the railing or parapet. These forces need not be considered in the design of the main structural if footpaths are provided. In cases where footpaths are not provided, the effect of these forces shall be considered in the design of the structural system ing the railings and the footpath upto the face of the footpath kerb only. 1.8 COMMENTS ON LOADING STANDARDS Bridge Design Codes of most of the countries prescribe some or the other form of standard live loading to be used in the design. It is observed that there is large variation amongst these live loadings and it is difficult to imagine that the traffic pattern would be differing so widely especially amongst the developed countries. A comparative study of highway bridge loadings of different countries over a span of 100 m for two lane bridges was made by ThomaS [12]. His graphs for maximum bending moments and shear forces are reproduced in Fig. 1.16. It will be seen that the American loading (AASHO loading) gives lowest design values and German loading gives the highest design values over almost the entire span range considered. The existing IRC loadings are complicated in their application, especially if various types of live loadings are to be considered alternately in the

Introduction 29

64 56

E 24 

-43

16 c

gel

8 0

Shear force 00

0 20 40 60 80 100 span(m) (a) Maximum bending moment for two lanes 280

Legend :

240

New zealand —Japan

200 160 120 80 40

--- --.---West --IRC (India) ---A ASHO group -)—BS (HA group) ----Sweden

0 20 40 60 80 100 Span (m) (b) Maximum shear force for two lanes Figure 1.16 Comparison of Highway Bridge Loadings

design to determine the severest effects. Class 70R loading is a newly

30 Grillage Analogy in Bridge Deck Analysis introduced live loading and can be taken as a replacement of IRC Class AA loading. But this loading is also not. simple. The basis for IRC provisions regarding impact is not clear. No systematic study has been made to derive realistic impact factor for road bridges in our country. The impact effect need not be considered for the full length of the load but needs only be applied to the heaviest axle or the pair of adjacent wheels causing the maximum bending moment or shear. The task of the designers will be simplified if some concerted efforts are made to introduce some degree of uniformity into national loading specifications with respect to international loading specifications. Newly formed International Organisation for Standardisation (ISO) has taken up the responsibility of producing a Standard Loading Code and considerable progress has been made in this direction. It is hoped that common internationally acceptable Bridge Specifications including Loading Standards will emerge in the near future. 1.9 ORGANISATION OF THE TEXT Over the years, a number of methods of analysis of highway bridges have been evolved. The method of grillage analogy which is a well-established and computer oriented method, bridges the gap between overly simplified hand computation methods and sophisticated finite element and finite strip methods. To lay the foundation of the book, the present Introductory Chapter is devoted to discuss some of the recent developments in the area of bridge analysis, design and construction that have taken place. Various forms of highway bridge construction, their planforms and configurations are outlined. Loading standards recommended by Indian Roads Congress (IRC) and adopted in India for highway bridges, are described. The details of some of the important loadings for which the analysis of bridge decks are carried out, are also given. The Loading Standards of some of the advanced countries are compared with Indian Standard Loadings. Critical comments on Indian loadings are given and the scope and need for a unified loading is emphasised. In order to make the reader appreciate the relative usefulness and potentialities of grillage analogy, it is considered prudent to discuss various other methods of analysis of bridge decks also. A brief review of the important methods of analysis of highway bridges including grillage analogy is undertaken in Chanter 2. These methods are described in relation to their historical

Introduction 31

background, methodology, applicability, merits and limitations. Further, the applicability of these methods to various types of bridges having different plan geometry and conditions is also presented in a tabular form. It is assumed that the readers have sufficient exposure to matrix methods of structural analysis on which the method of grillage analogy is based. But to maintain a continuity of discussion when the reader is to deal with a computer program in the succeeding chapter of the book using direct stiffness method, a brief outline of the stiffness method and the formulation and assembly of stiffness matrix kw skeletal structures are presented in Chapter 3. Gauss-Elimination Procedure and Cholesky's Factorisation method to solve the large number of resulting simultaneous equations are also discussed. A simple but a general computer program in FORTRAN illustrating different steps involved in matrix formulation of skeletal grid having any planform under externally applied nodal loads, is included and is given in Appendix I. This will provide a background to understand the comprehensive software developed and later presented in Chapter 6. Also, the designer will be able to make changes easily in the program, if needed, as per his requirements. A skew grid is analysed using the program for hypothetical concentrated load at nodes, as an illustration. Chapter 4 idealises the actual bridge structure into a suitable equivalent mathematical model of a grillage consisting of longitudinal and transverse grid lines. The general guidelines for choosing suitable grid layout are given and also illustrated through typical examples of slab, slab-on-girders and boxgirder bridges. The procedure for the evaluation of equivalent elastic properties i.e. flexural moment of inertia and torsional inertia T are also discussed for'the above types of bridge decks. The longitudinal and transverse of the idealised grillage form a mesh having a number of nodes. Grillage analysis requires that the dead, live and impact loads actually acting on bridge decks, are transformed into equivalent loads acting on nodes of the mesh. Chapter 5 discusses different types of loads and identifies the s in which the wheel loads of a vehicular live loading fall. The transfer of loads to nodes of grillage in the form of equivalent nodal loads is also given. The analysis of grillage is then carried out and resulting force responses and design envelopes are discussed. The interpretation of results so obtained and the local effects to be included in the design, are also outlined. Slab, slab-on-girders and boxgirder decks, both of right and skew configuration, are dealt-with.

32 Grillage Analogy in Bridge Deck Analysis Chapter 6 presents a computer program based on grillage analogy developed in FORTRAN for the elastic analysis of bridge decks, covering right and skew layouts. The important features of the program are highlighted. The Program Manual consisting of listing of variables, sign conventions, descriptions of subroutines, flow charts and the s Manual comprising data input and output formats for the force responses, are thoroughly .discussed and explained with illustrations. The designer, not willling to go through the previous sections of the book and with little exposure to the structural behaviour of the bridge, can prepare the input data and can still analyse the bridge for different IRC loadings. The program is designed in such a way that any loading standard other than • Indian loadings can be easily incorporated without involving much changes in the program. The program developed can deal with solid, voided and pseudo-slabs, T-beam and I-beam bridges with or without cross-beams, single and multi-cell box-girder bridges and composite bridges. The bfidges ed on neoprene bearings (yielding s) can also be handled. The complete listing of the- program is given in Appendix II. The limitations and scope of the program are discussed. Theoretical discussion .bnly does not convince a designer of the versatility of a technique unless it is supplemented by its application on a variety of actual problems. To achieve this objective, number of worked out examples of different types of slab, T-beam, and box-girder bridges are included in Chapter 7. The examples chosen are the life size bridges and some of them have already been constructed in India. The bridges are analysed for Indian Standard Loadings and for s' specified loadings also in some cases. To enhance the usefulness of the book, some of the important parameters encountered in practice e.g. yielding of s, effects of variation in numbers .and spacings of longitudinal and transverse grid lines etc. are highlighted through the above worked out examples. Relevant references are added after the end of each chapter to enable the reader to consult further for more details. The book also contains two appendices. Appendix I pertains to the listing of the general program 'GRID' as .discussed in Chapter 3 and Appendix II consists of listing of computer program 'GABS'. Diskette with executable files of these programs, compatible to IBM PC, alongwith input/output modules for the worked examples, illustrated in Chapter 7, has been prepared and can be ordered from the authors through the publishers.

Introduction 33

REFERENCES 1.

2.

3. 4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15. 16.

17.

18.

BAKHT, B.

and JAEGER, L.G., "Bridge Analysis Simplified", McGraw Hill Book Co., New York, 1985. CUSENS, A.R. and PAMA, R.P., "Bridge Deck Analysis", John Wiley and Sons, 1975. HAMBLY, E.C., "Bridge Deck Behaviour", Chapman and Hall, London, 1976. HARVEY, DAVID [., "Recent Trends in Short and Medium Span Highway Bridges in the United Kingdom", Proc. Intl. Conf. on Short and Medium Span Bridges, Vol. I, 1982. KULKA, E and LIN, T.Y., "Comparative Studies of Medium Span Box-Girder Bridges with Other Precast Systems", Proc. Intl. Conf. on Short and Medium Span Bridges, Vol. I, 1982. LEE, D.J., "Progress in Bridge Engineering", Developments in Structural Engineering, Proc. Fourth Rail Bridge Centenary Conference, Vol. I, 1990. MATHIVAT, IE., "Recent Developments in Prestressed Concrete Bridges", FLP Notes, 1988. Pormuswamv, S., "Bridge Engineering", Tata McGraw-Hill Publishing Co. Ltd., New Delhi, 1986. RAINA, V.K., "Concrete Bridge Practice-Construction, Maintenance and Rehabilitation", Tata McGraw-Hill Publishing. Co. Ltd., New Delhi, 1988. ROWE, R.E. and Somenviu.e, G., "Research on Slab Type and Spine Beam Bridge", Proc. Intl. Conf. on Developments in. Bridge Design and Cnnstruction. Crosby Lockwood & Son Ltd., 1971. STEINMAN, D.B. and WATSON, S.R., "Bridges and their Builders", Dover Publications, New York, 1957. THOMAS, P.K., "A Comparative Study of Highway Bridge Loadings in Different Countries", U.K. Transport and Road Research Laboratory-Supplementary Report 135 UC, 1975. VicroR, D.J., "Essentials of Bridge Engineering", Oxford & IBH Publishing Co., New Delhi, 1980. WIELAND, M., "Modern Bridge Engineering in Structural Concrete", Proc. of Asia-Pacific Conference on Road, Highway and Bridge Maintenance and Rehabilitation, Bangkok, 1987. WrreFoirr, H., "Building Bridges", Bentan Verlog GmbH, , 1984. "Specifications for Highway Bridges", Pt. I and Pt. III, Japan Road Association, 1984. "Bridge Rules Specifying the Loads for Deg the Super and Substructures of Bridges and for Assessing the Strength of Existing Bridges", Govt. of. India Publication, Ministry of Railways, New Delhi, 1977. IRC:5-1985, "Standard Specifications and Code of Practice for Road Bridges, Section I-General Features of Design", Indian Roads Congress, New Delhi, I99A.

34 Grillage Analogy in Bridge Deck Analysis 19. IRC:6-I987, "Standard Specifications and Code of Practice for Bridges, Section H—Loads and Stresses", Indian Roads Congress, New Delhi, 1990. 20. IRC:18-1985. "Design Criteria for Prestressed Concrete Road Bridges (Posttensicined Concrete)", Indian Roads Congress. New Delhi, 1990. 21. IS:1343-1981, "Indian Standard Code of Practice for Prestressed Concrete", Bureau of Indian Standards, New. Delhi, 1981.

Chapter 2

Methods of Bridge Deck Analysis 2.1 INTRODUCTION Extensive research into the behaviour of bridge decks under loading had been carried out pver the past five decades and a number of methods of bridge deck analysis were evolved from time to time. Prior to the general use of the computer-aided analysis, hand computation methods and charts based on some approximations and idealizations, provided convenient methods of load distribution. These were reasonably accurate for design purposes. However, with the advent of digital computers, many computer aided methods have been developed and are in use although some of these methods are highly numerical and expensive. The present chapter aims at, giving a broad idea about the various important methods of bridge deck analysis. The applicability of the different methods in relation to bridge type, plan geometry and conditions is also presented in a tabular form. 2.2 METHODS OF ANALYSIS AND THEIR APPLICABILITY Different techniques commonly in use for the analysis of bridge decks of various types and configurations are: 1. Courbon's Method 2. Orthotropic Plate Theory 3. Finite Difference Method 4. Method of Harmonic Analysis 5. Grillage and Space Frame Analogy 6. Folded Plate Analysis 7. Finite Element Method and 8. Finite Strip Method.

36 Grillage Analogy in Bridge Deck Analysis

Each of the above techniques is more suited to a particular type or types of bridge decks depending upon the closeness of the actual structure with the assumptions of the method. The applicability of the methods to various types of bridges is illustrated in Table 2.1. It may be evident that one particular type of bridge deck can be analysed by more than one method and in such cases, the choice rests with the designer depending upon the facility, time available, economy and of course his familiarity with the method. The above methods will be discussed briefly in relation to their historical background, applicability, merits and limitations in succeeding sections. For more details, the references given at the end of the chapter may be useful. 2.3 COURBON'S METHOD Courbon presented the method [10] at the time when other sophisticated and more accurate techniques for bridge deck .analysis were not commonly available. The method makes simplifying assumptions, restricting its applicability to a pertain extent but the method has been very popular because of its simplicity. The method is applicable to inter-connected T-beam-bridges and is still in vogue in India and is recommended-by Indian Roads Congress for live load distribution strictly within its limitations. The method is recommended to be used when the following conditions are satisfied : i) The ratio of span to width is greater than 2, ii) The longitudinal girders are inter-connected by abOut five symmetrically spaced cross-beams, and iii) The cross-beams extend to a depth of at least three-fourth of the depth of the longitudinal girders. These conditions are not difficult to satisfy in majority of actual T-beam bridges. According to Courbon's method, the load R. on any girder i of a bridge consisting of multiple parallel girders (Fig. 2.1) is computed assuming a linear variation of deflection in the transverse direction. The deflection of Inertia I

.0-1 1 ;

Figure 2.1 Deflection Profile in Courbon's Method

TABLE 2.1 Applicability of Analytical Techniques

Type of Deck Method of Analysis

Plan Geometry

Condition

Box- Right Skew 'Curved Arbitrary Beam Girder and > 15" Skew

Slab Pseudo T-

Slab

< Courbon's Method *

*

Finite Difference Method

*

*

*

*

*

Method of Harmonic Analysis

Continuous Arbitrary

51) *

*

*

*

4.

Grillage Analogy Folded Plate Method

*

*

*

* *

* *

*

*

Finite Element Method

*

*

*

*

*

*

Finite Strip Method

*

*

*

*

*

*

* *

* *

*

*

*

*

*

*

*

Methods of Bridge Deck Analysis 37

Orthotropic Plate Theory

*

Simply ed •

38 Grillage Analogy in Bridge Deck Analysis

will be maximum in the exterior girder on the side of the eccentric load (or c.g. of loads, if there is a system of concentrated loads) and minimum on the other exterior girder. The load R1 is given by

PI (PLedEL) R. =L± " EI S E1 , z 4 or, (2.1)

R.=--P4(1±--L1 edij Eli cf where I. = moment of inertia of ith longitudinal girder P = total live load e = eccentricity of live load (or- in case of multiple loads, distance of c.g. of loads from centroid of moment of inertias) d. = distance of ith girder from centroid of moment of inertia For a N-girder bridge with. all girders having same moment of inertia, the above formula reduces to (2.2) R. =

P (i+N edi) 1TT

The above load on the girder is used to determine the bending moment in the girders. For evaluating shear force in the girders, the same load distribution is valid for loads lying beyond 5.5 m from either s. For loads lying at or within 5.5 m of either s, the reactions on the longitudinal girders shall be greater of the results obtained by (i) assuming the deck slab simply ed or continuous, as the case may be, with the s being taken as unyielding and (ii) following Courbon's method [51]. The Courbon's method under-estimates the load on interior girders and generally over-estimates the load on the exterior girders. However, these inaccuracies shall be significantly reduced if the cross-beams are deeper and more in numbers. With the availability of more accurate methods, this procedure may be used for preliminary design of girder-sections only. 2.4 ORTHOTROPIC PLATE THEORY The use of orthotropic plate theory for the approximate analysis of simply ed right concrete bridge decks was pioneered by Guyon [16] and

Methods of Bridge Deck Analysis 39 Massonnet [30]. Design curves were prepared by Morice, Little and Rowe [32] and a summary of the design technique using these curves had been presented by Rowe [39]. Later Bares and Massonnet [2] and Cusens & Pama [11] further developed the method. The design curves are based on the distribution of deflection due to the first harmonic load. The maximum 'calculated longitudinal moments and stresses are increased by 10 per cent to for dropping the higher harmonic loading in the analysis. This approximation is reasonable for longitudinal moments but the transverse moments are highly dependent on the local distribution of harmonic loads and hence superimposition of higher harmonic components becomes essential. The bridge deck is replaced by an equivalent plate with bending and torsional rigidities in two orthogonal directions and the following wellknown partial differential equation, governing the behaviour of the equivalent system, is obtained, 4

D2'—d + 2 H 4

P(x,y) d xw 4

4

+D, d W (2.3) d xd 2 y2 1../ Y d y4

where 21-/ = (Dxy, DD + DI + D2) Dx and D - are the equivalent flexural rigidities and Dxy and D are the equivalent torsional rigidities per unit width in longitudinal and transverse directions respectively. Di and D2 are the coupling rigidities per unit width arising due to Poisson's ratio effects. The flexural and torsional rigidities have significant influence on the load distribution. Their effect is considered through two dimensionless characterising parameters namely flexural parameter 9 and torsional parameter a as given below : b Ds 2a (2.4) Dy. 30.25

and

+ D3w+ Di+ D2 a = _________________________________ (2.5) 2 11 (px where, 2a and 2b represent span and width of the equivalent plate respectively. The coupling rigidities D and D2 are small and can be neglected without introducing any significant error. The dimensionless distribution coefficients, which are given for nine standard reference points and load positions across the bridge width, are plotted

40 Grillage Analogy in Bridge Deck Analysis or tabulated against values of O. The charts or tables are given for two values of a, namely a = 0 and a= 1. Values of the coefficient Ka for any intermediate value of a are obtained by the following interpolation function,

Ka= Ko + (K1— 191Tx

(2.6)

-where Ke and K1 are the corresponding coefficients for a = O. and a = 1.0 respectively. For analysis by this method, the applied loads are converted into equivalent concentrated loads -at the standard locations. The responses corresponding to each load at these locations are then added to for the total effect. Cusens and Pama [11] have improved the distribution coefficient approach by taking seven of the harmonic series and by extending the range of a upto 2.0. Another set of curves are available for transverse moment' coefficients `p' for different values of B and for standard load eccentricities at various prefixed standard stations with reference to the centre line of the bridge deck. :The Poisson's zatio is found to effect the transverse moment coefficients and is assumed as 0.15. The flexural rigidities DF and D of a given slab, T, box or composite section are computed as usual and pose no problem. The torsional rigidity of the section is evaluated based on St. Venant's method by taking the summation of the torsional rigidity of the components forming the section and from this torsional rigidities D and D of the section are computed. However, the orthotropic plate idealization does not always represent the physical behaviour of the bridge decks. In bridge with few girders, say 3, the bending moments obtained are subject to errors, especially if the bridge is wide and load occupies only a fraction of the width. Also, the transverse moments which are complex combination of bending between girders and bending due to non-uniform girder deflections, can not be accurately obtained. Also, the method suffers from the drawback of having to assume the deck to be uniform throughout and also the design curves involve a certain degree of approximation in use. As only the first harmonic component is used in load distribution, the method is not recommended to be used- to estimate shear. Further, the method can not handle skew bridges. y

yx

2.5 FINITE DIFFERENCE METHOD When more complex boundary conditions are encountered in practice, the methnri of iprtheltrnpic plate, discussed earlier, becomes cumbersome and

'-•••••••••••aw

Methods of Bridge Deck Analysis 41

difficult to apply. The finite difference method is the answer for such complex boundary conditions. The method is versatile in nature and has wide applications:The finite difference technique had been used to advantage, first by Neilsen [34], later by Westergaard [49] and applied to bridge decks by Naruoka and Ohmura [33], Heins and Hails [17], Robinson [38] and many others. In this method of analysis, the deck is notionally divided into grids of arbitrary mesh size and the deflection values at the grid points are treated as unknown quantities. The usual governing differential equation of an orthotropic plate is considered in the finite difference method. The differential equation and accompanying boundary conditions are expressed in of these unknown deflections. The resulting sets of linear simultaneous equations are then solved for these unknown deflections. Finally, moments and shear forces are determined from the known deflection pattern. The curved deflection profile of the deck is approximated by a series of straight lines and, naturally, accurate results can be expected only if fine grids are used. Finite difference equations for various boundary conditiOns like simple, fixed, free or a combination of free and simple s can be written down for each case. However, a fixed edge condition is not treated very accurately by this method except when very fine grids are used. The simultaneous equations formulated from these grid points are solved on digital computers which have matrix packages specially designed for these type of problems. The presence of interconnecting beams below the slab, present a special boundary situation. A better representation of slab and beam interaction can be found by treating the two as separate structural elements which are made compatible by satisfying a set of boundary conditions. The interaction of beams and the top deck is based on the assumptions that beam and deck slab have identical deflections, the beams have no torsional rigidity and there is no horizontal force between the beams and the slab. The simplicity of the trigonometric solution for simply ed right bridge decks tends to be lost when extended to the problem of skew decks. However, the finite difference method has been used extensively for skew bridges also where the grid may be taken along the orthogonal coordinates as shown by Jensen [22], Robinson [38] or along skew coordinates as suggested by Naruoka and Ohmura [33]. Most of the solutions and published values of deflections and moments for skew necks roicr oni:yr to isotropic plates; the mesh is court4.... and acc

42 Grillage Analogy in Bridge Deck Analysis racy is doubtful. Naruoka and Ohmura [33] neglected Poisson's ratio. They dealt with torsionless plate (H = 0) as well as isotropic plate (H = Dx= Dy). They had difficulty in setting up satisfactory interpolation equations between these two limiting cases. .favour [21] produced influence surfaces for deflections and moments for a wide range of skew orthotropic slabs for both uniform and concentrated loads making various over-simplifying assumptions. Schleicher- and Wegener [41] published tables of deflections and stresses for continuous isotropic skew slabs under uniform loading. Ghali [13] used finite difference equations at different segments of a_ two-girder skew bridge to determine influence coefficients for deflection; shear and moment assuming that the applied loads act directly on girders leading to erroneous conclusions. Robinson [38] assumed that a concentrated load might reasonably be distributed over one mesh quadrilateral. Paterson [35] adopted this assumption in developing an ALGOL computer program based on standard skew mesh over the skew orthotropic plates. Although, finite difference method, applied to skew bridge decks, has been able to solve a large number of bridge structures but still it has certain inherent difficulties in its use like adoption of fine mesh work, proper assessment of singularities around the obtuse corner, deterioration in convergence with increasing skew 'angles and in some cases, non-compatibility in boundary conditions. The number of parameters involved in skew bridges are such that the preparation of design curves to cover a realistic range of loadings, skew angles and degree of orthotropy, does not appear to be a practical proposition. The finite difference method has also been extended to bridge decks curved in plan by Heins and Hails [17] based on governing equation of orthotropic plate in polar coordinates, neglecting Poisson's ratio. However, the efforts needed to analyse a curved deck by this method is cumbersome and is generally not recommended. .2.6 METHOD OF HARMONIC ANALYSIS In harmonic analysis, the applied load is broken into a number of harmonic components, each consisting of a distributed load parallel to the longitudinal axis of the structure and with intensity varying as pure sine-wave as shown in Fig. 2.2. Mathematically,

P(x) ,t=3

11 It X

a„ sin ___ L

(2.7)

,

Methods of Bridge Deck Analysis

n=1 n=2 n=3 n=4 -+ +

43

+

Figure 2.2 Representation of Load by Harmonic Series

Under the action of each sine-wave load component, every longitudinal. strip of the structure deflects and twists in a pure sine-wave form. Since differential of sine function is a cosine function and vice-versa, the equilibrium equations, which can be thought of as differentials of deflections, can also be expressed as a number of sine or cosine functionS. These resulting equilibrium equations can be solved as conventional simultaneous equations. The general theory of Harmonic Analysis is described in detail in many books of mathematics including Kreyszig [24J. The concept Of harmonics method, applied to bridges, was established by Hendry and Jaeger [18) and their method is referred as Hendry-Jaeger method. In this approach, the actual transverse medium is replaced by a uniform continuous medium of the same total transverse moment of inertia. For torsionally weak bridges, the load distribution defined through distribution coefficient `p' is shown to depend upon dimensionless flexural parameter a, which is defined as 12 (LI3 Eir a = 7r4 71 El

(2.8)

where L = span of the bridge, h = distance between adjacent longitudinals, El = flexural rigidity of a longitudinal girder and Elr= total flexural rigidity of transverse medium. Hendry-Jaeger produced design graphs between a and p for T-beam right bridges having 2, 3, 4, 5 and 6 girders [18]. To cater for a bridge having more than six girders, the bridge is suggested to be converted into an equivalent 6girder bridge. Also, for slab bridges, it is recommended that the deck should be converted into an equivalent 6-girder bridge. To take into the transverse positions of loads, a versus p plotting is done for loads placed on various girders. In order to for higher harmonics, the method suggests modifications in the value of a itself using the same

44 Grillage Analogy in Bridge Deck Analysis

a p plot. As the method can consider higher harmonic components, it can estimate shear also fairly, accurately. Hendry-Jaeger had also given analysis for torsionally stiff girder bridges. The torsional rigidity, CJ of the girder is considered through a dimensionless parameter /3 which is defined as,

'= 2 h CJ

(2.9)

The extreme value of /3 for torsionally stiff bridges has been assumed as infinity. Design graphs between a and p for 13 = cc have also been plotted similar to graphs for torsionally weak bridges namely, 13 = 0. Coefficients for any intermediate value of /3 may be obtained by using following inter' polation function;

11(3+16

____________________________________________________

ffrc7)

(2.10)

A

where po and p„ are the distribution coefficients corresponding to 13 = 0 and p= cc respectively and pp is the distribution coefficient for the desired value of 11 Hendry-Jaeger suggested method to analyse torsionally weak (13 = 0) three and four girder skew bridges. Only the .first harmonic components of the loading and deflection were considered. For more accurate analysis of 3-girder torsionally weak skew bridges, Jaeger, Bakht and Surana [19] extended the method by incorporating second harmonic term also in the analysis. Design graphs were also prepared. Surana [45] developed a general mathematical model for the analysis of skew girder bridges of finite torsional rigidity incorporating first three harmonic components for displacement functions and first two harmonic components for rotation function. To take into the effect of skew angle A, an additional dimensionless skew parameter K = (h/L) tan A., was introduced. The analysis of right bridge could be obtained by putting A = 0. Agrawal [1] and Prasad [37] extended the above method further and gave design tables and charts for use in design offices for three and four girder bridges. Harmonic method has advantage in its inherent capability to completely identify a girder. bridge by a certain combination of non-dimensional structural parameters. These parameters help in visualising the structural

Methods of Bridge Deck Analysis 45

behaviour due to change in bridge dimensions which in turn help in decision-making about the trial sections. The method is best suited -for the analysis of beams and slab right and skew decks with 3 and 4 girders. But for a bridge with more girders, the number of coefficients to be handled for a reasonably accurate result, become rather large. 2.7 GRILLAGE AND SPACE FRAME ANALOGY For any given deck, there will invariably be a choice amongst a number of methods of analysis which will give acceptable results. When the complete field of slab, pseudo-slab and slab-on-girders decks are considered, grillage analogy seems to be completely universal with the exception of Finite Element and Finite Strip methods which will always cany a heavy cost penalty for a structure as simple as a slab bridge. Further, the rigorous methods of analysis like Finite Element Method, even today, are considered too complex by some bridge designers. The grillage analogy method can be applied to the bridge decks exhibiting complicated features such as heavy skew, edge stiffening, deep haunches over s, continuous and isolated s etc., with ease. The method is versatile, in that, the contributions of kerb beams and footpaths and the effect of differential sinking of girder ends over yielding s (such as neoprene bearings) can be taken into . Further, it is easy for an engineer to visualise and prepare the data for a grillage. Also, the grillage analysis programs are more generally available and can be run on personal computers. The method has proved to be reliably accurate for a wide variety of bridge decks. This method of analysis, based on stiffness matrix approach, was made amenable to computer programming by Lightfoot and Sawko [25]. West [47, 48] made recommendations backed by carefully conducted experiments on the use of grillage analogy. He made suggestions towards geometrical layout of grillage beams to simulate a variety of concrete slab and pseudo-slab bridge decks, with illustrations. Gibb [14] developed a general computer program for grillage analysis of bridge decks using direct stiffness approach that takes into the shear deformation also. Martin [28], then followed by Sawko [40] derived stiffness matrix for curved beams and proclaimed a computer program for a grillage for the analysis of decks, curved in plan. The grillage analogy has also been used by Jaeger and Bakht [20] for a variety of bridges. The method consists of converting the structure into a network of skeletal rigidly connected to each other at nodes. The load-deformatiOn relationship at the two ends of a skeletal element with reference to the

1

46 Grillage Analogy in Bridge Deck Analysis member axis is expressed in of its stiffness property. This relationship which is expressed with reference to the member coordinate axis, is then transferred to the structure or global axis using transformation matrix, so that the equilibrium condition that exists at each node in the structure can be satisfied. The bridge structure is very stiff in the horizontal plane due to the presence of decking slab. The translational displacements along the two horizontal axes and rotation about the vertical axis will be negligible and may be ignored • in the analysis. Thus a skeletal structure will have three degrees of freedom at each node i.e. freedom of vertical displacement and 'freedom of rotations about two mutually perpendicular axes in the horizontal plane. In general, a grillage with `71'. nodes will have 3n degrees of freedom. All span loadings are converted into equivalent nodal loads by computing the fixed end forces and transferring them to global axes. A set of simultaneous equations are obtained in the process and their solutions result in the evaluation of the nodal displacements in the structure. The member forces can then be determined by back substitution. Bridges are frequently designed with their decks skew to the s, tapered or curved in plan. The behaviour and rigorous analysis are significantly complicated by the shapes and conditions but their effects on grillage analysis are of inconvenience rather than theoretical complexity. Space Frame idealisation of bridge decks has also found favour with bridge designers. This idealisation is particularly useful for a box-girder structure with variable width or depth where the finite strip and folded plate techniques are inappropriate. However, Scordelis [42] concluded certain disadvantages of.space frame analysis to the extent that the computer-time involved is excessive while the solution is still approximate. Most-road bridges of beams and slab construction can be analysed as three dimensional structure by a space frame analysis which is an extension of grillage analogy. The mesh of the space frame in plan is identical to the grillage, but various transverse and longitudinal are placed coincident with the line of the centroids of the downstand or upstand they represent. For this reason, the space frame is sometimes referred to as "Downstand Grillage" [11]. The longitudinal and transverse are ed by vertical , which, being short, are very stiff in bending. The downstand grillage behaves in a similar fashion as the plane grillage under actions of transverse and longitudinal torsion and bending in a vertical plane and consequently, sectional properties of these are calculated in the

'""--"=• '

.hirhods of Bridge Deck Analysis 47

Space frame programs have been used in the analysis of box-girder bridge decks. However, the simulation of boxes by space frames is not capable of predicting local effects and the method has proved expensive in use. 2.8 FOLDED PLATE ANALYSIS A folded plate is a prismatic shell formed by a series of ading thin plane slabs rigidly connected along their common edges. A box-girder bridge deck may be regarded as a.special type of folded plate structure. in that the plates are arranged so as to form a closed section. Method of analysis originally developed by Goldberg and Leve [15] for folded plate had been adopted for the analysis of box-girders. Scordelis [42] initially applied the method of folded plate to simply ed box-girder bridges and later on extended it to continuous decks [43] also. An approximate method; known as-Finite Segment method, was also used by Sc.onielis [42] and Johnson and Lee [23] in analysing box-girder bridges. The structure consists of a number of rectangular plates connected at longitudinal ts. Each plate is initially assumed to be fixed at the longitudinal ts. Edge forces due to surface loads are determined by plate theory and, for loads in the plane of the plate, by plane stress theory. The stiffness matrix for each plate is then expressed in of the harmonics of a half-range Fourier series. Each t has four degrees of freedom i.e. displacement longitudidally tangential to the t, rotation about an axis tangential to the t and vertical and horizontal displacements. The direct stiffness method is used to analyse the complete structure. The method is applied to simply ed structures with diaphragms. The diaphragms are assumed to be infinitely rigid in their own plane but perfectly flexible in a direction normal to their own plane. The analysis can be extended to include intermediate diaphragms. Harmonic analysis is used to represent applied loads. Folded Plate method is quite suitable for analysis of box-girder bridge having a few number of cells. The method offers a logical approach in the sense that it analyses the structure in its correct form without replacing it by an equivalent structural system. Thus, the field of application of the method is restricted to the right cellular bridge decks of uniform cross-sections having intermediate diaphragms but which must be simply ed at the extreme ends with rigid diaphragms positioned over the s. However, within its field of application, the method is efficient in of computer time, is accurate and yields complete information about the elastic stresses in the structure.

48 Grillage Analogy in Bridge Deck Analysis

2.9 FINITE ELEMENT METHOD During the past two decades, the Finite Element Method (FEM) of analysis has rapidly become a'Arery popular-technique for the computer solution of complex problems in engineering and the method is now wellknown and established. Its early application to problems of plate flexure led to its adoption as a convenient tool in the solution of many bridge deckS where its generality gave it a considerable edge over many other specialised techniques. The method is able to tackle complex planforms, irregularly positioned s, holes in the deck and other anisotropic features. Thus, the Finite Element Method may seem to be very general in application and indeed,• for difficult bridge deck problems, it is sometimes the only valid form of analysis. The technique was pioneered for two dimensional elastic structures by Turner et al [46] and Clough [9] in the latter. half of fifty's. Since then, considerable developments have been made in theory and execution of FEM and it got further impetus due to availability of faster and larger digital computers. The method was fUrther developed by Zienkiewicz [50], Martin and Carey [29], Desai and Abel [121 etc. and was applied to bridge decks by Scordelis [44], Meyer [31], and many others. The current state-of-the-art on the theory of FEM and its application to bridge structure, is available in reference 27.  The FEM consists of solving the mathematical model which is obtained by idealising a structure as an assembly of various discrete two or three dimensional elements connected to each other at their nodal points, possessing an appropriate number of degrees of freedom. The solution by FEM essentially involires four basic steps: 1. Discretisation of the structure into finite elements, 2. Evaluation of element properties, 3. Matrix formulation for element assemblage and its solution, and 4. Interpretation of results. The most important step in the finite element method of analysis is the formulation of a mathematical model of the actual structure which is represented as an assemblage of discrete parts, known as elements. Each element of the model has finite dimensions and properties and in order to perform subsequent analysis, it is necessary to establish the forcedisplacement relationships of each element. The bridge deck is represented as an elastic continuum and the division of the structure in 'elements' can be carried out in many different ways. For

Methods of Bridge Deck Analysis

49

example, in a slab deck; the elements may be taken as triangular or quadrilateral plate elements. The representation may be coarse with a small number of elements or fine using a relatively large number of elements. The actual choice will depend upon the geometry of the structure, on the importance of local features and also on the convergence properties of the element. The usual direct stiffness or diSplacement method of analysis is applied while assembling the elements and solving the bridge deck for stress responses. Equilibrium of the internally and externally applied forces at each node of an element and the compatibility of element deformations are both satisfied. Also, the internal force-displacement relationships must be established with each element as governed by the existing geometry and material property characteristics. The interpretation of stress resultants for design is a cumbersome process and the job becomes still difficult if large number of elements are involved and the conditions and loading systems are complex. The main advantage of the method over other analytical techniques, is its generality. Normally, as was pointed out, it seems possible, by using many elements, to virtually approximate any continuum with complex boundary and loading conditions to such a degree that an accurate analysis can be expected. In practice, however, engineering limitations arise, the most important being the cost of the analysis. As the number of elements increases, the manpower required to prepare the data and interpret the results increases and also a larger amount of computer time is needed for the analysis. Furthermore, the nonavailability of the softwares and large computers may prevent the use of a large number of finite elements to idealise the deck. The method invariably requires a high speed computer and back up storage for solving any real bridge problem. Also, the round-off and truncation errors occuring in the analysis because of finite precision arithmatic are further impediments. It is, therefore, desirable to use only efficient finite element programs which in turn essentially depend upon the use of efficient finite elements, programming methods and the use of appropriate numerical techniques. The application of FEM to bridge problems will need a thorough understanding and knowledge of almost all the facets of advanced structural mechanics and numerical techniques which many a times a design engineer may not have. Also, the softwares based on FEM are not so easily available and also the understanding of its use for the bridge deck analysis is difficult. Versatile as it may be in application, the method can be shown to be uneconomical in analysing bridge decks of regular shape specially right structures with simple end suppbrts. Thus, the method should be reserved for bridge decks which are incapable of solution by any of the simpler and more economical methods.

, 50 Grillage Analogy in Bridge Deck Analysis

2.10 FINITE STRIP METHOD The finite strip approach when first published by Cheung [5] in 1968-69, was recognised as having excellent prospects as a method of analysis for simply ed bridge deck structures M of accuracy and efficiency. Basically, the method is a hybrid procedure which retains advantages of both, the orthotropic plate method and finite element concept. The procedure is applicable to both slab and box-girder bridge decks. The method may be regarded as a special form of the displacement formulation of the finite element procedure. The basic difference between the two methods stems from the assumed displacement patterns. The assumed displacement functions for a finite element in slab or box-girder analysis normally takes the form of a two-way polynomial functions. The displacement functions assumed for the corresponding finite strip are combination of a one-way polynomial function (in the transverse direction) and a harmonic function (in the span-wise direction). The harmonic functions are chosen to satisfy the end conditions. In effect, the finite strip spans -, between two opposite end s. In this method, the structure is assumed to be discretised into a number of strips. Each strip has constant thickness; however, the thickness can vary from strip to strip. The strip is treated as a beam shown in Fig. 2.3. The stiffness matrix of a strip with pre-set end conditions is formulated. The loading can be point load, patch load or line load. Simply ed edge

Auxiliary Nodal Line

Strips

Nodal Lines Figure 2.3 Nodal and Auxiliary Nodal Lines in Finite Strips

The continuum is divided into strips by fictitious lines called Nodal Lines and these strips are assumed to be connected to one another along discrete number of nodal lines which coincide with longitudinal boundaries of the

Methods of Bridge Deck Analysis 51

strips. A displacement function in of the nodal displacement parameter is chosen and then strain and stress fields within each element are obtained. Based on the chosen displacement function, it is possible to obtain stiffness and load matrices which are then assembled to form a set of overall stiffness equations. Since the band-width and the size of the matrix is usually small, the equations can be solved easily by any standard band matrix solution technique to yield the nodal displacement parameters. For more refined solutions of general applicability to both for slab and cellular decks. of uniform cross-sections, the concept of Auxiliary Nodal • Lines (ANL) between the normalstrip boundaries was introduced by Loo and Cusens [26]. In the solution bf the basic force-displacement equation, the unknown amplitudes at auxiliary nodal lines may be; written in of the unknowns at the boundaries Before the matrix equations are assembled from individual strips. This is a useful property of the ANL technique as it permits higher order functions to be incorporated into the. finite strip formulation without affecting the band-width of the overall matrix equation. The resulting computer time far an individual problem is only marginally higher than for the. analysis based on the conventional technique using a third order polynomial function and nodal lines at the boundaries of each strip. The accuracy of a finite strip analysis depends mainly upon the number of finite strips used in representing the actual structure and upon the number . of retained for the Fourier series functions. The finite strip procedure for rectangular slab-type bridge decks had been first suggested by Powell and Ogden [36] and later on, the method was adopted by Scordelis [43] and his collaborators in the University of California at Berkeley, to deal with right and curved box-structures. Research work by Cheung [6] and his co-workers in applying the strip method for the analysis of slab type bridge decks with intermediate column s [4] and simple curved slab and box-bridges [7] gave further impetus to the methodology. Qther studis by Cheung on the analysis of rectangular slabs with end boundary conditions other than ;simple s and with variable cross-sections in the span-wise direction [8], however, have not been convincingly efficient when compared with methods such as finite element analysis. The same criterion applies to the finite strip analysis of skew slab bridges -suggested by Brown and Ghali [3]. Loo and Cusens [26] had also been constantly working on the method and its applications to bridge problems. The method of finite strip, within its field of applications, is well suited for computer use. Any form of loading, including prestressing forces, may be conveniently handled.

52 Grillage Analogy in Bridge Deck Analysis However, the method suffers from the drawback that it is ideally suitable to only prismatic structures with simply ed ends. Apparently, it seems that the method has not been applied to skew girder bridges with diaphragms and only right or circularly curved bridges can be analysed. Further, each finite strip is assumed to have constant geometry and material properties in longitudinal direction. As can be seen froth the discussions of various important methods of bridge deck analysis, practically every method has its merits and limitations and some of them even have limited applicability. However, grillage analogy method seems to be a general, simple, sufficiently accurate, easy to comprehend and convenient to work even on easily available Personal Computers. Therefore, the remaining part of the book deals with different aspects of Grillage-Analogy method in detail. REFERENCES L AGRAWAL, R., "Analysis and Design of Interconnected Skew Girder Bridges", Ph.D. Thesis, Indian Institute of Technology, New Delhi, 1975. 2. BARES, R. and MAssozorET, C., -"Analysis of Beams and Grids and OrthOtropic Plates by the Guyon-Massonnet-Bares Method", Crossby Lockwood, London, 1968. 3. BROWN, T.G. and GuAu, A., "Finite Strip Analysis of Skew Slabs", Proc. McGillEIC Conference in Finite Element Method in Civil Enginering, 1972. 4. CHEUNG, M.S., CHEUNG, Y.K. and CHAU, A., "Analysis of Slab and Girder Bridges by the Finite Strip Method", Building Science, Vol. 5, 1970.5. CHEUNG,IK., "The Finite Strip Method in the Analysis of Elastic Plates with Two Simply ed Ends", Proc. ICE, 40, 1968. 6. CHEUNG, Y.K., "Analysis of Box-Girder Bridges by Finite Strip Method", Proc. 2nd Intl. Symposium on Concrete Bridge Design Chicago, ACI Publications, SP 26, 1969. 7. CHEUNG, Y.K., and CHEUNG, M.S.,,"Analysis of Curved Box-Girder Bridge by Finite Strip Method," Publication, IABSE Vol. 31/1, 1971. 8. CHEUNG, Y.K., "Finite Strip Method in Structural Analysis", Pergamon Press, Oxford, England, 1976. 9. CLOUGH, R.W., "The Finite Element in Plane Stress Analysis", Proc. 2nd ASCE Conf. on Electronic Computation, Pittsburg, Pa., 1960. 10. COURBON, J., "Application de la Resistance des Materiaux au Calculdes Pants", Dunod, Paris, 1950. 11. CUSENS, A.R. and PAMA, R.P., "Bridge Deck Analysis", John Wiley, London, 1975. 12. DESAI, C.S. and ABEL, J.F., "Introduction to Finite Element Method", Von Nostrand Reinhold, New York, 1972.

1

Methods of Bridge Deck Analysis 53 13. GHAI I, A., "Designs of Simply ed Skew Concrete Girder Bridges", Proc. Intl. Symposium on Concrete Bridge Design, SP 26 Toronto, 1969. 14. -Gm A., "Grillage Analysis. Notes for Course on Bridge Deck Analysis", Civil Engg. Dept., University of Dundee, 1972. IS. GOLDBERG, J.E. and LEVE, ELL., "Theory of Prismatic Folded Plate Structures", Publication IABSE, VoI. 17, 1957. 16. GUYON, Y., :`Calcul des Pants Larges a Poutres Multiples Solidarisees Par des Entretoises", Annales des Ponts et Chausees, No. 24, 1946. 17. HEINS, C.P. and HAns, R.L., "Behaviour of Stiffened Curve Plate Model", J. Struct. Div., ASCE, Vol. 95, ST11, 1969. 18. HENDRY, A.W. and JAEGER, L.G., "The Analysis of Grid Frameworks and Related Structures", Prentice-Hall, Englewood Chatto & Windus, London, 1958. 19. JAEGER, L.G., BAiurr, B., and SURANA, C.S., "Application and Analysis of Three-Girder Skew Bridges", Proc. Second Intl. Colloquium on Concrete in Developing Countries, Bombay, 1988. 20. JAEGER, L.G., and &waif, B., "The Grillage Analogy in Bridge Analysis", Canadian Journal of Civil Engineering, 9(2), 1982. 21. JAVOUR, J., Ikew Slabs and Gridwork Bridges", Bratislava Czechoslovakia: Slovenske Uydavataletsvo Technickij Litratury, 1967. 22. JENSEN, V.P., "Analyses of Skew Slabs", Bulletin Series No. 332 University of Illinois, Illinois, 1941. 23. JOHNSON, C.D. and LEE, T., "Long Non-prismatic Folded Plate Structures", J. Struct. Div., ASCE, Vol. 94, 1968. 24. KREYSZIG, E., "Advanced Engineering Mathematics", John Wiley New York, 1962. 25. LIGEar-oar, E. and SAWKO, F., "Structural Frame Analysis by Electronic Computer: Grid Frameworks Resolved by Generalised Slope Deflection," Engineering, 187, 1959. 26. Loo, Y.C. and CUSENS, A.R., "The Finite Strip Method in Bridge Engineering", Viewpoint Publication, Cement and Concrete Association, London, 1978. 27. MAISEL, B.I., "Review of Literature Related to the Analysis and Design of Thin Walled Beams", Technical Report 42:440, Cement and Concrete Association, London, 1970. 28. MARTIN, H.C., "Introduction to Matrix Method of Structural Analysis," McGrawHall Book Co., New York, 1966. 29. MARTIN, H.C. and CAREY, G.F., "Introduction to Finite Element Analysis", McGraw-Hall, New York, 1973. 30. MASSONNET, C., "Methode de Calcul des Ponts a Poutres Multiples Tenant Compte de Leur Resistance a la Torsion", Publication, IABSE, No. 10, 1950. 31. MEYER, C., "Analysis and Design of Curved Box-Girder Bridges", Report SESM 70-22, University of California at Berkeley. 1970. 32. MORICE, P.B., LITTLE, G. and RowE, R.E., "Design Curves for the Effects of Con-Pui-micatIon uri I a, Lcmctit and Concrete Associaton, London, 1956.

54 Grillage Analogy in Bridge Deck Analysis 33.

34.

35.

36.

37.

38.

39. 40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51.

NARUOKA, M.

and OHMURA, I-I,. "On the Analysis of a Skew Girder Bridge by the Theory of Orthotropic Parallelogram Plates", Proc. 1SBSE, No. 19, 1959. NEILSEN, N.J., "Bestemmelese of Spaedinger i Piader Ved Anvendelse of Differensligninger", Doctoral Dissertation, College of Engineering, Copenhagen, 1918. PATERSON, D.K,W., "Load Distribution in Skew Orthotropic Plates", Ph.D. Thesis, University of DUndee, 1970. PoWELL, G.I-1 and OGDEN, D.W., "Analysis of Orthotropic Steel Plate Bridge Decks", Proc. Struct. Div., ASCE, ST5, V.95, 1969. PaAsAn, J., "Modified Hormonics Method for Analysis and Design of Skew Girder Bridges", Ph.D. Thesis, Indian Institute of Technology, New Delhi, 1982. ROBINSON, K.E., "Behaviour of Simply ed Skew Slabs Under Concentrated Loads", Research Report No.8, Cement and Concrete Association, London, 1959. ROWE, R.E., "Concrete Bridge Design", C.R. Books Ltd., London, 1962. Sawx.o, F., "Computer Analysis of Grillages, Curved in Plan", Publication, IABSE, 1967. SCHLEICHER, C. and WEGENER, B., "Continous Skew Slabs: Tables for Statical Analysis", Verlog fur Bauwesen, Berlin, 1968. SCORDELIS, A.C., "Analysis of Simply ed Box-Girder Bridges", Report SESM 66-17, Dept. of Civil Engg., UniVersity of California at Berkeley, 1966. SCORDELIS, A.C., "Analysis of Continous Box-Girder Bridges", Report SESM 6725, University of California at Berkeley, 1967. SCORDELIS, A.C., "Analytical Solutions for Box-Girder Bridges", Proc. Intl. Conf. on Developments in Bridge Design and Construction, Crosby Lockwood, London, 1971. SURANA, C.S., "Interconnected Skew Bridge Girders", Ph.D.Thesis, University of Edinburgh, U.K., 1968. TURNER, M.J., CLOUGH, R.W., MARTIN, H.C. and.Torp, L.J., "Stiffness and Deflection Analysis of Complex Structures", J, Aero Sci., 23, 1956. WEST, R., "The Use of a Grillage Analogy for the Analysis of Slab and PseudoSlab Bridge Decks", Research Report 21, Cement and Concrete Association, London, 1973. WEST, R., "Recommendations on the Use of Grillage Analysis for Slab and Pseudo-Slab Bridge Decks", C&CA/CIRIA. Cement and Concrete Association, London, 1973. WESTERGAARD, H.M., "Formulas for the Design of Rectangular Floor Slabs and ing Girders", Proc. ACI, 22, 1926. ZIENKIEWICZ, 0.C., "The Finite Element Method in Engineering Science", McGraw-Hill, London, 1971. IRC:21-1987, "Standard Specifications and Code of Practice for Road Bridge: Section III-Cement Concrete (Plain and Reinforced)", Indian Roads Congress, New Delhi, 1991.

Chapter 3

Stiffness Method Applied to Grillage Analysis 3.1 INTRODUCTION The bridge deck structure may be considered as an assembly of structural connected together at discrete nodes forming a grid. The deformations and forces at nodes are inter-related by corresponding stiffnesses. In order to satisfy 'the equilibrium and compatibility conditions at each node, a large number- of simultaneous equations will result and the manual solutions of these may be prohibitive. But using the matrix method of structural analysis as a primary approach, it becomes possible to obtain the computer-oriented solution. It is assumed that readers have sufficient exposure to matrix methods of structural analysis. But to maintain the continuity of discussion in succeeding chapters, a brief presentation of stiffness method and its formulation for bridge deck analysis by Grillage Analogy is included. A simple but general computer program, illustrating different steps involved in matrix formulation needed, is also presented. A sample grid is analysed as an example. 3.2 MATRIX METHOD OF STRUCTURAL ANALYSIS For an elastic structure, the actions P (forces and moments) and displacements D (translations and rotations) are directly related as [F] (P}

(3.1)

where [F] is known as Flexibility Matrix of the structure and io defined as the displacement produced by a unit value of action P. Similarly, another way of relating P and D is {P} = [K] {D}

(3.2)

where [K] is the Stiffness Matrix and is defined as the action required to produce a unit displacement.

56 Grillage Analogy in Bridge Deck Analysis

The above two matrix approaches, namely Flexibility Method and Stiffness Method, are commonly employed in the analysis of skeletal structures. In the flexibility method, also known as Force Method, the redundant structure is converted into a 'released' or statically determinate structure by the removal of sufficient internal or external actions. The solution of the problem consists in finding the values of those actions which will restore compatibility of the displacements at the ts and s of the structure. The unknowns are, therefore, static unknowns. In the stiffness method, also known as Displacement Method, the redundant structure is converted into a 'locked up' or fully restrained structure by locking of every t and . The solution of the problem then consists in finding the values of the displacements which must be applied to all ts and s to restore equilibrium conditions at the ts. The unknowns are, therefore, kinematic unknowns. Thus it can be seen that the above two methods approach the problem from two different angle and this represents the essential difference between them. It is generally agreed that the stiffness method is more suitable for the analysis of structures than the flexibility method. One of its advantages over the flexibility method is that it is conducive to computer programming Once the analytical model of a structure has been defined, no further engineering decisions are required in the stiffness method in order to carry out the analysis. In flexibility approach, the reduced structure is- to be solved a number of times as the choice of redundants is not unique. The stiffness method will be used in the analysis of bridge decks by grillage analogy. The method is based on certain assumptions. .The important among these are; (a) Hooke's law applies—leading to the principle of superimposition, (b) small deformation theory holds true and (c) shear deformations can be ignored. The assembly process in stiffness matrix in its original form is computationally inefficient needing more computer time and storage. This inefficiency has been overcome in its improved version called Direct Stiffness Method which is readily programmable on digital computer. Various steps in direct stiffness method are discussed in the following sections. 3.3 DEGREES OF FREEDOM AND SIGN CONVENTION Before we discuss the formulation of stiffness matrix for a member, the concept of Degrees of Freedom (D.O.F.) for a structure is essential. The degrees of freedom for a structure are the independent deformations which

Stiffness Method Applied to Grillage Analysis 57

define the deformed shape of the structure completely. In general, any structure has six degrees of freedom at a node e.g. three components of translation along three orthogonal axes and three components of rotation about them. These degrees of freedom can be written in the form of a vector, known as displacement vector. This vector is generally partitioned into two sub-vectors, namely active displacement vector containing the degrees of freedom along which free displacement is possible and ive or displacemOt vector which corresponds to the restrained degrees of freedom at 's. Depending upon the significance of a deformation, in a particular typeof structure form, some of the above degrees of freedom can be ignored, being insignificant. Then the total number of degrees of freedom get reduced. For example, in a grid structure, the two translations in the plane of grid and a rotation about the axis perpendicular to the plane of grid are insignificant and may be neglected. Thus, the degrees of freedom in the case of grid are reduced to only three at each node. These are translation perpendicular to the plane of grid and rotations about two orthogonal axes lying in the plane of the grid, as shown in Fig. 3.1(a) for a member of the grid in X-Y plane. These are called Global or Structural Degrees of Freedom. For a member, the degrees of freedom are six in number, three at each end of the member (Fig. 3.1b).

83

0

2

(a) '31

L

(b)

Figure 3.1 Global Axes and Degrees of Freedom

Figure 3.2 depicts a member T in conjunction with a set of member axes Xr., Yr and Z.. Member axis (X.) makes an angle y with the global X-axis and the member axis (Z.) is parallel to global Z-axis. The possible displacements of the ends of the member '1' in the directions of member axes are known as Local or Member Degrees of Freedom.

58 Grillage- Analogy in Bridge Deck Analysis

Zm Figure 3.2 Local Axes and Degrees of Freedom

In the above figure, the single headed arrows denote translations and the double headed arrows represent rotations and are taken as positive in the directions shown following right hand screw rules. 3.4 MEMBER STIFFNESS MATRIX Consider a grid member (Fig. 3.3) that is fully restrained at ends j and k. Member stiffnesses consist of reactions exerted at the member ends by the restraints when unit deformations (one translation and two rotations) are imposed at each end of the member in turn. It is assumed that the shear centre and the centroid of the member coincide so that twisting and bending of the member occur independent of one another. Ym 3

6 E1, CJ

Zm

Xm

k 5 (a)

Figure 3.3 Grid Element with End Displacements

The unit displacements are considered to be induced one at a time while all other end displacements are zero. The member stiffnesses corresponding to six possible types of end displacements as shown in Fig. 3.3, are summarized pictorially in Figs_ 3.4 (a) and 3.4 (b). Forces are represented by single headed arrows and moments by rinnhle hpnr.id right hand screw rule as mentioned

L

(b)

r

-r---

Stiffness Method Applied to Grillage Analysis 59



Yrn

1E1 L2

_6E1

pp-XM

-_ 2E1 L3

Zm 12E1 L3 (I) Unit

'Translation

Along Zm at End

Ym G7 k

'Xm

1.0 Zm (ii) Unit Rotation About Xm at End 3

2, L E 1 4E1Z m ' 1 . 0 6E1

r. __ Xm

6E1

L2 (iii) Unit Rotation About Ym at End

Figure 3.4 (a) Determination of Member Stiffnesses Corresponding to Degrees of Freedom at End j earlier. All vectors are drawn in positive senses, but in cases where the restraint actions are actually negative, a minus sign precedes the expression for the stiffness coefficient. In general, it is possible for the member to undergo any one or more of the six displacements shown in Fig. 3.4. All the member stiffnesses shown in the figures are derived by determin ing the values of restraint actions required to hold the distorted member in equilibrium. The stiffness matrix [K] for a grid member as above, therefore, is of the order 6 x 6 and each column in the matrix represents the actions caused by corresponding unit displacement. The member stiffness 'matrix thus obtained is symmetric aria is snown in equation 3.3.

60

Grillage Analogy in Bridge Deck Analysis Ym 6E1

6E1

_ Xm J

k

12(1

0

12E1 L3

(iv) Unit Translation Along Zm at End k vm

GI

G

GAKt

X m

1.0 Zm

(v)

Unit Rotation About Xm at End k Ym 4E1 2E1 17

6E1 L 2

6 E 1 L2

(vi) Unit Rotation About Ym at End k Figure 3.4 (b) Determination of Member Stiffnesses Corresponding to Degrees of Freedom at End k

The stiffness matrix developed above for a grid member is in of local degrees of freedom, which is different for different meeting at a t. Since, the equilibrium at a t is to be satisfied, taking into the end actions of all meeting at that t together with the external forces if any, a common reference is essential. This is provided by global degrees of freedom and a relationship between local and global degrees of freedoms is needed. This relationship is obtained in of a matrix known as Transformation or Rotation Matrix. Consider a grid member 1-2 in Fig. 3.5 with member axes Xm and Yrn. The relation between the deformations at node I in the original direction and in the direction of global axes X and Y is shown. These deformations are related (Eq. 3.4) with the help of a 3 x 3 rotation matrix [A] in of direction cosines.

"V1'

Sti f fness Method Applied to Grillage Analysis 61

12 El

0 GJ

0

L

6E1 [K.]

2

L 12 El

0

L3 0

6E! L2 0

12 El L3 0

4 EI L 6 El

6 El

L2 GJ

6 EI L2

L

2 EI L

11E1 L3 0 6 El L2

0 _GJ

L

6 El 2 L

0 2 El

L

0 GJ

L

6 El. L2

(3.3)

0 4E1

L

Ym Y Y

Xm m5

ems

Figure 3.5 Transformation of Stiffness Matrix l

0 0 = 0 cos y sin y _0 — sin y cosy_

(3.4)

Now, local member deformations can be related to global member deforma-

---411......+nommo••••••••11111111111

62. Grillage Analogy in Bridge Deck Analysis

tions as given in equation 3.5. cosy siny 0 — sin y cos y I I

tyesm3 Wf

0 cosy siny 10

tn4

„,, W m]

0I

I 0 —siny cosy_ O'„,6 or

(3.5) •

{c1} = [T]

where {dm} and {d.'} are local and global member deformation vectors respectively and [T] is a 6 x 6 transformation matrix derived from the rotation matrix fill such that (3.6)

{Pm} = [T] {P:,}

(3.7)

[P.} and {P} are the member end action vectors in local and global degrees of freedom respectively. Since .transformation matrix [T] is an orthogonal matrix, its inverse and transpose should be same. Therefore, we can write, {P',n} = [T]T fP„,1

(3.8)

We may write for actions and displacements referred to local axes X. and Y. as {P.} = [Kin] {din}

(3.9)

and referred to the global axes X and Y as {Pm} =

.1 {d',,}

(3.10)

Substituting for {P.} in equation 3.8 above from equation 3.9, we get, or

(Pim? = [TY{dj T {P„,} = [TI [lc [7] {d:}

(from equation 3.5)

Thus from equation 3.10 above,

[rir,] = [Tr. [Km] [7]

(3.11)

Stiffizess Method Applied to Grillage Analysis

63

Thus, the member stiffness matrix in of the global axes is found by operating on the local member stiffness matrix [IC an] given by equation 3.3 earlier, using the transformation matrix. This takes into the material constitutive laws and orientation of the member. For a grid member, the final global member stiffness matrix is shown in equation 3.12. R

;n:

12E1 L3. 6E1

".n2

34

 Y? mi3

12E1 6E7



GJ C2 4E1 —CS

s

6E1 GJ4E1"L

L'

—

6E1

M"„,5

6E1_

L2

LGJ52+4E1C2 12E1

L3

L2 c

GJ2 L 2E1 2 GJ 2E1 L _ +-S L

Les

M 4 +1 76

6E1

6E1 GI 4E1 2 L2 L 2 L S-

L

2 6E1 GJ 4E1 GJ 2 4E1

a...ILI

2

L

L2

L

L

L

L2 TCS---E—CS TS

+—r-C

(3.12) where

s = sin y, c = cos y

3.5 ASSEMBLY OF STRUCTURE STIFFNESS MATRIX The member stiffness, developed in previous sections, gives the relation between actions and deformations of a single member and satisfies member constitutive laws. But to satisfy equilibrium condition at any t, we have to consider assemblage of all the say 'n', meeting at that t. The structure stiffness matrix element satisfying the t equilibrium can be obtained by an assembly of 'n' member stiffness matrix elements. Symbolically, this assembly process can be represented as [KJ] =

I icno.

(3.13)

i=i

where [K1] is the assembled structure stiffness matrix corresponding to jth degree of freedom and [Kr is the corresponding term of ith member stiffness matrix. This assembled matrix is nothing but the addition of all internal forces which will be subsequently equated to the externally applied forces along the same degree of freedom.

Generally this assembled structure stiffness matrix [K) is quite large in size and hence some technique is to be adopted so that this matrix occupies

64 Grillage Analogy in Bridge Deck Analysis the minimum possible storage space in the computer. One of the basic properties of this matrix is its symmetry and banded nature, advantage of which is taken to reduce the storage memory (RAM) required. The matrix is having non-zero only in a restricted region which can be bounded by two lines parallel to the leading diagonal as shown in Fig. 3.6. This is called band of stiffness matrix and maximum number of non-zero elements in a row is called band-width of the matrix. The number of elements to be Half—Band

Sky line

K1 0 K13 0 Kik K22

K23 •

0

0 K25 ‘K.7)

0 0 0 0 0 0 0 0

K33 K34 K35 K36 )0 0 0 0 •• Kh4 /(45 Kt.6 K47 KI.8

Symmetric

0 0 • K 5 5 K 5 6 K 57 K 5$ ‘ 0 0 • • K66 • 1(67 K68 1(77 1(78K69 K79 • • K88 K89 1(81 •

Figure 3.6 Half-Band and Skyline of a Matrix stored are further reduced by the fact that the lower triangle part of the matrix can be obtained from the upper triangle part and vice-versa. So only half band of the matrix need be stored and rest of the elements can be deduced from this, whenever required. The half band-width of stiffness matrix of a structure can be obtained as Half Band-Width (Degrees of freedom per node) x (maximum difference of numbers of connected nodes f 1) Storage requirements can be further reduced by using Skyline Technique for assembly. As shown in Fig. 3.6, Skyline is an envelope drawn in matrix nose zero Ci erfierit S which do not have any non -zero elements

o

Stiffness Method Applied to Grillage Analysis 65 above them. So the Skyline always remains below the half band boundary line and thus contains smaller number of elements to be stored. The stability of the structure is now considered by introducing the boundary conditions. There are two procedures available to take into the boundary conditions. In one, the s are idealised by a lumped stiffness which can be thought of by proiiding stiff springs along the s. This is achieved by adding a high stiffness term corresponding to the ive degree of freedom. The reactions in that case can be obtained by multiplying the spring stiffness by the corresponding displacement. Another approach is based upon the condition that the displacement along the restricted degrees of freedom i.e. the ive displacement vector, is a null vector. Considering this, structure stiffness matrix is partitioned into four submatrices as given in equation 3.14. (3.14)

[K pp 7, KpRilDp} If}

R

KTpR K RR DR

where [Km] etc. are partitioned submatrices and {pp} and {DR} are free and restrained displacement vectors respectively. (PI is the external load vector and {R} is the reaction vector. From above, we can write [IC p

[Kpp] {Dp}

[KpR] {DR} = {P}

[KPR]T {Dp}

[KRR] {DR} = {R}

and

- (3.15) (3.16)

In case of rigid s,{D R} = 0 and the equation 3.15 reduces to [Kpp] {Dp} = {P}

(3.17)

The solution of the above equation will determine the deformation vector {Dd. This value of {Dp} is substituted in the equation 3.16 to get the reaction, i.e. [IV {Dp} = {R}

(3.18)

The first method can handle flexible as well as rigid s, while the second method is applicable to rigid s only. 3.6 SOLUTION OF SIMULTANEOUS EQUATIONS A large number of simultaneous equations will result depending upon the size of the grid chosen. They are to be solved in an efficient manner utilising itcratvc., the minim'', computer t;.,... .7‘ and Gauss-Seidel Method was one of the most popular techniques to

66

Grillage Analogy in Bridge Deck Analysis

solve the simultaneotis equations. But the method suffers from the disadvantage that the time taken by computer to solve the equations achieving a particular degree of accuracy can be predicted only approximately. Now-a-days, Gauss-Elimination Procedure and Cholesky's Factorization Method [5] are commonly used to solve the simultaneous equations. The Gauss-Elimination Procedure consists of making all the of stiffness matrix, K for j < i, equal to zero i.e., the stiffness matrix is reduced to upper triangle matrix by eliminating all the below the leading diagonal. Then deformations are calculated by back substitution. The Cholesky method [KL] and [Ku] such that

consists

of_factorization_olstiffness-matrix

[K]-into (3.19)

[K] = [KL] [Ku] where

KL = lower triangle matrix Ku = upper triangle matrix Then, from equation 3.2, we can write {P} = [KL] [Ku] {130} Or

{P} = [KL] {Q}

(3.20)

where

{Q} = [Ku] {D}

(3.21)

{Q} can be obtained by Forward substitution from equation 3.20 as [KL] is a lower triangle matrix and {D} can be obtained by Backward substitution from equation 3.21 as [KU] is the upper triangle matrix. The vector {D} thus obtained, gives the nodal deformation of the structure. The member deformations in global as well as local coordinates can be evaluated from the nodal deformations {D} obtained above. From member deformations, member forces can be calculated by multiplying member stiffness matrix with the member deformation vector. An exhaustive treatment of above sections on stiffness matrix and solution of simultaneous equations are available in references given at the end of the chapter and the reader may refer the same for more details. 3.7 COMPUTER PROGRAM For illustrating the basic steps of computer oriented direct stiffness method applied to grids, as discussed above, a simple program 'GRID' is presented in this section. The program will help the reader to understand the approach more clearly. This will also lay a foundation for a more comprehensive computer program to be encountered later in Chapter 6. Figure 3.7 gives the flow chart of the program.

Stffness Method Applied to Grillage Analysis 'V

START

/

I N P U T : G R I D D n o d e s , N o . o f e C o o r d i n a t e s o f M e m b e r p r o p e r t c o n n e c t i v i t i e s ,

E T A I L S N o . o f l e m e n t s , n o d e s , i e s a n d H a l f b a n d - w i d

Generate stiffness matrix and assemble global stiffness matrix for all

i INPUT: Type of , No. of DETAILS s, ed nodes, Spring in case of flexible constant

Modify diagonal of stiffness matrix accordingly

Decompose stiffness matrix

INPUT: LOADING DETAILS No. of loads, Loaded nodes, Value of load

Subroutine "DECOMP"

68 Grillage Analogy in Bridge Deck Analysis

Generate load vector 4r

Solve generated load vector

Subroutine "SOLV"

4.

Print nodal deformations

4. DO 300 Loop for solving for all

• Compute member end forces

Print member end forces

Figure 3.7 Flow Chart of Program 'GRID'

Stiffness Method Applied to Grillage Analysis

69

The computer program described here can be used to determine the deformations and member forces in a skeletal grid having any planform under externally applied nodal loads. The prograril is written in FORTRAN language. In this program, Cholesky method is used for the solution of simultaneous equations. The program can handle both flexible and rigid type of s. In order to reduce the volume of input data, the procedure for automatic coordinate generation has been adopted. The subroutine `DECOMP' has been usOd to decompose the stiffness matrix by Cholesky's method and the solution is obtained by using subroutine `SOLV% For simplicity, the transfer of loads from s of the grid to its nodes has not been dealt with in the program and the forces are assumed to act only at nodes. Many other complicated, but important features, pertaining to bridge deck analysis, have:also not been incorporated in the program and these will be discussed in. Chapter 6. A list of principal variables used in the program and their description is given in Table 3.1 alongwith the symbols. The program listing is given in Appendix I at the end of the book. A diskette containing the program (in a ready to use form) along with the Input and Output files of the example can be ordered from the authors through the publishers. TABLE 3.1 : Variable Description Variable

Symbol in Text

X(1),Y(I) NCN1(I),NCN2(1)

j,k

X1( 1) XJ(I) NNODES NELEMS NDOF NHBAND

I J n m 3*n

G SM(I,J) c,s

G

P(1) D(I) GSM(1,1) TM(I,J)

(P) {DJ [K] [TI

cos y, sin y

Description Coordinate of ith node Node numbers corresponding to the ends of ith member Moment of inertia of ith member Torsional inertia of ith member Total number of nodes Total number of Total number of degrees of freedom Half band width of stiffness matrix Young's modulus of elasticity Shear modulus of elasticity Member stiffness matrix Cosine and sine of the angle y, the member is making with global X-axis Load vector Deformation vector Global (structure) stiffness matrix Transformation matrix

3.8 EXAMPLE 111ar.d nut of inaluies of the above program have been illustrated with the help of a simple example of a skeletal grid shown in Fig. 3.8.

70 Grillage Analogy in Bridge Deck Analysis

Simply ed

Notes: (i) Load of 10,000 kg at node11 (ii)Dimensions in cm and E,Gin kg/cm 2

Figure 3.8 Skew Skeletal Grid—Example It consists of three longitudinals and seven transverse . The longitudinal are simply ed at the ends on rigid s. The transverse are in skew alignments. The grid is to be analysed for a vertical load of 10,000 kg. applied at node 11 located at the center of the grid. The numbers given in boxes in the figure represent the member numbers. The Young's modulus of elasticity E and shear modulus of elasticity G are taken as 142860 kg/cm2 and 62110 kg/cm2 respectively. The input data with description for this example is given in Table 3.2 and the result output of the program is presented in Table 3.3. The moment of inertia I and torsional inertia J for the elements of each group are given as . input. The results are self-explanatory. The deformations at all the nodes and shear force, bending moment and torsional moment for all the of the grid are only given for brevity. However, the program appended also has thc = all thc, in= data, nodal coordinates, moment of inertia I and torsional inertia J of all the . The reactions at

Stiffness Method Applied to Grillage Analysis 71

nodes 1, 2, 3, 19, 20 and 21 are to be obtained by algebraically summing up the shear force in the meeting at the respective nodes. TABLE 3.2: Input Data Description

Input Data

Number of Nodes, Number of Elements Number of groups describing nodal coordinates From node no. to node no., Node number increment, X-coord., Y-coord., X-increment, Y-increment Same for remaining groups

21,32 3

Number of groups describing member sequences From member no. to member no., Member no. increment, Node 1, Node 2, First node increment, Second node increment Same for remaining groups

E, G No. of groups having different member properties I, J, total no. of in group, member numbers of the group Same for remaining groups

Half band width of matrix Type of s (1 for rigid and 0 for flexible) No. of s Node numbers at which s are provided No. of loads Node no. at which load is acting Type of load, magnitude of load Note: Units kg.cm.

1,19,3,0.,0.,200.,0. 2,20,3,250.,250.,200.,0. 3,21,3,500.,500.,200.,0. 5

1,6,1,1,4,3,3 7,12,1,2,5,3,3

13,18,1,3,6,3,3 19,25,1,1,2,3,3 26,32,1,2,3,3,3 142.86E3, 62.11E3 3 370.75E5, 16.5E5,18,1,2,3,4,5,6, 7,8,9,10,11,12,13,14,15,16,17,18 5.36E6, 5.987E6,4,19,25,26,32 6.67E6, 6.72E5,10,20,21,22,23, 24,27,28,29,30,31 12 1 6 1,2,3,19,20,21 1 11 1, —10000.

72 Grillage Analogy in Bridge Dec* Analysis TABLE 33 : Output Data

Total no. of Nodes Total no. of Elements Half Band-Width Total no. bf D.O.F. E = 142860.00 G = 62110.00

-7= = =

21 32 12 63

Deformations

Vertical Deflection -0.166858E-09 -0.250818E-09 -0.189220E-09 - 0.887135E-02 - 0.140837E-01 -0.915430E-02 -0.154419E-01 0.256665E-01 -0.157153E-01 -0.179865E-01 -0.307446E-01 -0.179865E-01 -0.157153E-01 -0.256665E-01 -0.154419E-01 -0.915430E-02 -0.140837E-01 - 0.887135E-02 -0.189220E-09 - 0.250818E-09 -0.166858E-09

Nodb No. I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

X-Rotation Y-Rotation 0.502802E-04 0.465766E-04 0.641898E-04 0.715232E-04 0574660E704 0.484959E-04 0.107143E-04 0.403060E-04 0.641827E-04 0.668466E-04 0.717644E-04 0.410519E-04 -0343813E-04 0.239530E-04 0.431990E-04 0.456668E-04 0.819048E-04 0.230262E-04 -0.721953E-04 0.901100E-06 - 0.472005E-18 - 0.329927E-18 0.721953E-04 -0.9,01100E-06 -0.819048E-04 -0.230262E-04 -0.431990E-04 -0.456668E-04 0.343813E-04 - 0.239530E-04 -0.717644E-04  -0.410519E-04 -0.668466E-04 -0.641827E-04 - 0.107143E-04 - 0.403060E-04 -0.574660E-04 -0.484959E-04 - 0.641898E-04 - 0.715232E-04 -0.465766E-04 -0.502802E-04

Member Forces Member No.

Shear Force

1 2 3 4 5 6 7 8 9

0.145468E+04 0.114928E+04 0.470264E+03 - .466629E+03 -.121693E+04 -.158519E+04 0.196012E+04 0.263379E+04 0.406311E+04

Bending Moment End 2 End 1 0.205935E+05 0.318142E+06 0.563450E+06 0.680321E+06 0.599063E+06 0.355656E+06 - .721619E+05 0.297518E+06 0.80306R.E+k"..•,:.;

- 311530E+06 -.547997E+06 -.657503E+06 -.586995E+06 -.355676E+06 -.386176E+05 -.319862E-F06 - .824277E+06 - .1615.i.;21-07

Torsion 0/02739E+05 0.231074E+05 0.193761E+05 0.497521E+04 - .519601E+04 -.732661E+04 0361352E-1-01 0.107522E+05 .71.7 i :13-; Contd.

Stiffness Method Applied to Grillage Analysis 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

- .406311E+04 -.263379E+04 - .196012E+04 0.158519E+04 0.121693E+04 0.466629E+03 -.470264E+03 - .114928E+04 - .145468E+04 - .943386E+02 0.305410E+03 0.679011E+03 u.y.sotwit+Os 0.750304E+03 0.368260E+03 -.425352E+02 0.425352E+02 - .368260E+03 -.750304E+03 - .936893E+03 - .679011E+03 - .305410E+03 0.943386E+02

0.161569E+07 0.824277E+06 0.319862E+06 0.386176E+05 0.355676E+06 0.586995E+06 0.657503E+06 0.547997E+06 0.311530E+06 -.226007E+03 -.267163E+04 -.135654E-1-05 -.263173E+05. -.157253E+05 .149277E+04 0.324875E+05 0:174490E+05 0.128707E+06 0.249547E+06 0.304925E+06 0.226501E+06 0.105307E+06 -.335797E+05

-.803068E+06 - .297518E+06 0.721619E+05 -.355656E+06 - .599063E46 - .680321E+06 -.563450E+06 - .318142E+06 - .205935E+05 0.335797E+05 -.105307E+06 - .226501E+06 - .304925E+06 -.249547E-1-06 - .128707E+06 - .174490E+05 -.324875E+05 0.149277E+04 0.157253E+05 0.263173E+05 0.135654E+05 0.267163E+04 0.226007E+03

73

0.221355E+05 0.107522E45 0.361352E+01 -.732661E+04 -.519601E-F04 0.497521E+04 0.193761E+05 0231074E+05 0.202739E+05 -.288976E+05 -.667882E+04 -.828866E+04 - .595134E+04 - .134105E44 0.152035E4.04 0.221261E+05 0.221261E+05 0.152035E+04 -.134105E+04 - .595134E+04 - .828866E+04 - .667882E+04 - .288976E+05

REFERENCES 1. BATHE, K.J., "Finite Element Procedures in Engineering Analysis", Prentice-

Hall, USA, 1982. 2. BEAUFAIT, F.W., RowAN, W.H., HOODLEY, P.C. and HACKETT, R.M.,

"Computer Methods of Structural Analysis", Prentice-Hall, USA, 1970. 3. CHAPRA, S.C. and CANALE, R.P., "Numerical Methods for Engineers" McGraw4. 5. 6. 7. 8.

9.

Hill, New York, 1989. CUSENS, A.R. and PAMA, R.P., "Bridge Deck Analysis", John Wiley, 1975. GERE, J.M. and WEAVER, W., "Analysis of Framed Structures", Van Nostrand, 1965. JENKINS, W.M., "Matrix and Digital Computer Methods in Structural Analysis", McGraw-Hill, New York, 1969. RUBINSTEIN, M.F., "Matrix Computer Analysis of Structures", Prentice-Hall, USA, 1966. SURANA, C.S., "Grillage Analogy for Analysis of Super-Structures of Bridges", Proceedings of the Seminar on Modern Trends in Analysis and Design, B.H.U., Varanasi, February 1984. SURANA, C.S. and HUMAR, J.L., "Beam and Slab Bridges with Small Skews", Canadian Journal of Civil Engineering, Vol. 11. No. I, 1984.

Chapter 4

Transformation of Bridge Deck into Equivalent Grillage 4J INTRODUCTION In recent years, the Grillage Analogy Method, which is a computer-oriented technique, is increasingly being used in the analysis and design of bridges. The method is also suitable in cases where bridge exhibits complicating features such as heavy skew, edge stiffening and. isolated s. The use of computer facilitates the investigation of several load cases in shortest possible time. The method is versatile in nature and the contribution of kerb beams and the effect of differential sinking of girder ends over yielding bearings (such as neoprene bearing) can also be taken into and large variety of bridge decks can be analysed with sufficient practical accuracy. Furthermore, the grillage representation is conducive to give the designer a `feel' for the structural behaviour of the bridge and the manner in which the bridge loadings are distributed and eventually taken to the s. The method consists of 'converting' the bridge deck structure into a network of rigidly connected beams at discrete nodes i.e. idealizing the bridge by an equivalent grillage. The deformations at the two ends of a beam element are related to the bending and torsional moments through their bending and torsional stiffnesses. These moments are written in of the end-deformations employing slope-deflection and torsional rotation-moment equations. The shear force in the beam is also related to the bending moment at the two ends of the beam and can again be written in of the end-deformations of the beam. The shear and moment in all the beam elements meeting at a node and fixed end reactions, if any, at the node, are summed-up and three basic statical equilibrium equations at each node namely EF = 0, EMx = 0 and EMy = 0 are satisfied.

Transformation of Bridge Deck into Equivalent Grillage 75 ,

In general, a grid having 'n' nodes will have '3n' nodal deformations and `3n' equilibrium equations relating to these. Back substitution in the slope-deflection and torsional rotation-moment equations • will give the bending and torsional moments at the two ends of each beam element. Shear forces are computed from bending moments and external loads. When a bridge deck is analysed by the method of Grillage Analogy, there are essentially five steps to be followed for obtaining design responses: (i) Idealization Of physical deck into equivalent grillage (ii) Evaluation of equivalent elastic inertias of of grillage (iii) Application and transfer of loads to various nodes of grillage (iv) Determination of force responses and design envelopes and (v) Interpretation of results 'The first two steps of grillage analogy are discussed in this Chapter. The other remaining steps are dealt with in the subsequent chapters. 4.2 IDEALIZATION OF PHYSICAL DECK INTO EQUIVALENT GRILLA(E The method of grillage analysis involves the idealization of the bridge deck as a plane grillage of discrete inter-connected beams. This is the first im- , portant step to be taken by the designer and needs utmost care and understanding of the structural behaviour of the bridge decks. It is difficult to make precise general rules for choosing a grillage mesh and much depends upon the nature of the deck to be analysed, its conditions, accuracy required, quantum of computing facility available etc. and only a set of guidelines can be suggested for setting grid lines. It may be noted that such idealization of the deck is not without pitfalls and the grid lines adopted in one case may not be efficient in another similar case and the experience and judgement of the designer will always play a major role. 4.2.1 Idealization of Deck Structure A rectangular slab-element subjected to loads normal to its plane is equivalent in deformability to an assembly of six beams [81 as shown in Fig. 4.1. The properties of grillage . are given as follows: t3

yL2) x

24 (1 — y2)

Ly ly

r

2

3

2 -j 24 (1 7 )

76 Grillage Analogy in Bridge Deck Analysis

.(E

G

.(E

Ly (1-3y)) G

t3

24(1—y2 )

L, (1.-3y)

= 7(121 +14)13

t3 2 24(1—y ) _________________________________________________ t3/

LyJ24(1_72)

(4.1)

where y= Poisson's ratio and E, G Elastic moduli.

r Ly

X

(a) A Slab Element

J

Y Torsioniess Diagonal (b) Plan of Equivalent Grillage Figure 4.1 Grillage Idealization

For practical purposes, a slab bridge can be regarded as an isotropic plate and can be conceptually divided into a number of rectangles. Each rect angle can then be idealized by the above _mentioned assembly of six beams (Fig. 4.2a). However, the resulting grillage is a complex one due to

Transformation of Bridge Deck into Equivalent Grillage 77

the presence of diagonal . This renders the analysis unsuitable, be cause of time consuming data input requirements and difficulties in th e interpretation of results. The idealization can be made manageable by assuming Poisson's ratio y to be zero, thereby eliminating the need for the torsionless diagonal mem bers. Member prOperties of the resulting assembly of orthogonal beams (Fig. 4.2b) are given by t

Ix =

/

24

E Jx

3

Y

(L24 ),

t3 ) Y

Lx 3 =

24

E (L, t3) G 24

(4.2)

WAMOM MOO_Off MMAM rAM (a) Nan of Idealized Grillage ing for Poisson's Ratio

(b) Nan of Idealized Grillage ignoring Poisson's Ratio Figure 4.2 Plans of Idealized Grillage

In an assembly of orthornal beams, tbR ninment in a beam denends only upon the curvature of the assembly in the direction of beam. The corre sponding moment in a slab depends not only upon the curvature in the

78 Grillage Analogy in Bridge Deck Analysis direction of the moment but also on the curvature in the perpendicular direction. This "slab-action" is represented by the following equations for the moments in x and y directions, M and My respectively, = — D(—d2

c°

y ____2 ) dy

d x2 where

(4.3) M y

D(d2 2)

2

dy d

E t3 D

d

12 (1 y 2 )

Second term in the above equations s for the effect of curvature in the direction perpendicular to that of the moment. It can be seen that the effect of curvature in the perpendicular direction, which is directly related to y, would not effect the idealization which neglects y. 4.2.2 General Guidelines for Grillage Lay-out Because of the enormous variety of deck shapes and conditions, it is difficult to, adopt hard and fast rules for choosing a grillage lay-out of the actual structure. However, some basic guidelines regarding the location, direction, number, spacing etc. of the longitudinal and transverse grid lines forming the idealized grillage mesh, are described here. But each type of deck has its own special features and may need some particular arrangements for setting idealized grid lines and therefore has been discussed separately also. a) Location and Direction of Grid Lines Grid lines are to be adopted along 'Lines of Strength'. In the longitudinal direction, these should be along the centre line of girders, longitudinal webs. or edge beams, wherever these are present. Where isolated bearings are adopted, the grid lines are also to be chosen along the lines ing the centres of bearings. In the transverse direction, the grid lines are to be adopted, one at each end connecting the centres of bearings and along the centre lines of transverse beams, wherever these exist. Ordinarily, the grid lines should coincide with the centre of gravity of the sections but some shift is permissible, if this simplifies the grid lay-out or if it assigns more clearly and easily the sectional properties of the grid in the other direction. b) Number and Spacing of Grid Lines Wherever possible, an odd number of longitudinal and transverse grid lines are to be adopted. The minimum number of longitudinal grid lines may be three and the minimum number of transverse grid lines per span may be live.

Transformation of Bridge Deck into Equivalent Grillage 79

The ratio of spacing of transverse grid lines to those of longitudinal grid lines may be chosen between 1.0 and 2.0. This ratio should also, ordinarily, reflect the span-width ratio of the bridge. Thus, for a short span. and wide bridge, it should be close to 1.0 and for long span and narrow bridge, this ratio may be closer to 2.0. Grid lines are usually uniformly placed, but their spacings can be varied, if the situation so demands. For example, closer transverse grid lines should be adopted near a continuous as the longitudinal moment gradient is steep at such locations. It may be noted that with an increase in number of grid lines, the accuracy of computation increases, but the effort involved is also more and soon it becomes a case of diminishing return. In a contiguous girder bridge, more than one longitudinal physical beam can be represented by one grid line. For slab bridges, the grid lines need not be closer than two to thrrte times the depth of slab. 4.2.3 Grillage Idealization of Slab Bridge Following recommendations are made for setting out grid lines in slab decks with line s at .either end: (i) The direction of longitudinal grid line is ordinarily parallel to the free edge of the deck. (ii) For bridges ed on discrete bearings, longitudinal grid lines are placed along the centre of each bearing. One longitudinal grid line along centre line of each edge beam, if they exist, is also provided. (iii) Total number of longitudinal generally be kept between five and seven (preferably odd number) for two-lane slab • decks without footpath. One additional grid line is provided along the centres of each footpath, if exist. In case of contiguous beam-slab construction (pseudo-slabs), one longitudinal grid line may be provided for two or more physical beams. (iv) The minimum distance between longitudinal grid lines is limited to 2 to 3 times the slab depth and the maximum separation of .lorigitudinal should not be more than one-fourth of the effective span. (v) In general, transverse grillage should be at right angles to longitudinal . But for bridges with skew angle less than or upto 15° or where the transverse directions of strength, such as reinforcement or prestressing, are skew, the transverse grid lines are oriented parallel to the s. (vi) The end transverse grid lines are placed along the centre line of bearings on each side.

80 Grillage Analogy in Bridge Deck Analysis (vii) The spacing of transverse grid lines should be small. Their number depends upon the span of the bridge. Five to seven transverse may be adopted for spans upto 10 m and seven to nine for bridges with span above 10 m. In regions of sudden change such as over intermediate s, a closer spacing is necessary. Ordinarily, one grid line along the centre of bearings at each end and brie at the centre of span are provided initially and then other grid lines are placed in between them. (viii) As far as possible, the spacings of each of longitudinal as well as transverse grid lines, are kept uniform_ (ix) It is important that the idealized grillage is ed at the same positions as the actual deck. Some of the grillage arrangements for right and skew solid slab, voided slab and pseudo slab bridges are discussed with examples. 1. Solid Slab Bridge a) Right bridge with and without footpaths Figures 4.3 and 4.4 show the plans and sections of slab bridges without footpath and with footpath respectively. The longitudinal and transverse grid lines are also shown therein. b) Skew bridge Skewness has considerable effect on the behaviour of the deck and critical design stresses. Skew decks are associated with special characteristics like hogging moment and large reactions near the obtuse corner and small reactions and possible uplift at the acute corner. Moreover there is considerable torsion of deck. Thus special attention is required while laying out the grid lines for a skew bridge. Decks with skew angles less than 15° can usually be handled as right decks. Same guidelines as discussed above, for right bridges, are applicable for such decks with or without footpaths and do not need any further elaboration for setting out the grid lines. However, bridges having skew angles more than 15° pose problems in regards to the positioning and orientation of longitudinal and transverse grid lines and are discussed here. A skew deck can be analysed with grillage having either a parallelogram mesh as shown in Fig. 4.5(a) or orthogonal meshes as in Figs. 4.5(b) and 4 .5(c).

Transformation of Bridge Deck into Equivalent Grillage .81 

-

I

1

r —T

I I —

r

-

r-r-T-7

I I I 1 1 I L --1 A_ i

I I I I _1_11 ! I I

L

- -

1

I I I I I 1 — T ---

I I1 1 11 1 I II I _1. _____ --

I

1-

I

I

I

11_1

I

I

Xm•L

Grid-Lines

i I 1 I I 1  - i-- - -I- --1- - 4-- -4- _I I I ' 1 1 I I- T - TI - i-I- 1 - - - 4 - - 1 I  I

1

I Ii , ! 1 1 I I ,L__4_,_1-1_I_J__ -I I I I I I I- I  ______ 1 I (a) 11Plan 1 i of Solid 11 1 1 1 Slab I

X= 0

L—L __i __..,_ _ _1_ L _ _I -.4 b

1D

I_________________________________________________________________________________1-

(b) Section on A-A

Figure 4.3 Solid Slab Grillage Lay-out without Footpath

While the parallelogram mesh (Fig. 4.5a) is convenient for low skew angles, it is not appropriate for angles of skew greater than 15° because it has -.no close to the direction of dominating structural action. For bridges with larger skew, say. greater than 15°, a parallelogram mesh as Fig. 4.5(a) will result in an over-estimated maximum deflections and moments, the amount increasing with angle of skew. The quantity of reinforcement in such grids is likely to be excessive and uneconomical. An orthogonal grid lay-out as shown in Figs. 4.5(b) or 4.5(c) will be more realistic. In skew bridges, the direction of principal bending moment across the Farancl to skcw span at edge to near normal to

.$ 411

82 Grillage Analogy in Bridge Deck Analysis

rI - — - r- — — - r - - -T - - - 7 - —I-- -1

1 I

1 1 1I

I I

11

i

I

r- --II-

I I 1

r-- - i I l 11 t i

tI

--

I

I

I

I

i

--- 4.

1

i

1 t

ii 1

1 1 I

X•1. t

I I

I

.1

I

t i-

1 I l

I

1 I. 1 t 1 1

.

1

I

t

I I I t I

I

x

1

1

1

I

A

I

1

1

I

I

1 1 I -- --11----F - - H

I 1

1 I

I

- - - 4 - - - -I- - - --I I

-- -1-- --1-

I

tI

I

i

1

I

___I

i I t1 r-- - -I1--.-r - -- -t- --- ir - -- -I- - —1 I I— I I I I I 1 Ii I I 1 t i t__ __ i__ __i_ ___L _ _ _ I _ _ _ _ L _ _ _I

(a) Plan

of Solid Slab

Footpath u

'

( i s

I

1

•,

•

4 .

1 4 1

T

(b) Section on A. A Figure 4.4 Solid Slab Grillage Lay-out with Footpath

abutment in the centre. This is illustrated in Fig_ 4.6. While adopting the orthogonal arrangement of grillage, two different cases arise, depending upon the skew angle and span-width ratio as shown in Fig. 4.7. Case (a) refers to the situation where skew region is small and right Nyc m re-,,r,-.s.rt. this rasp when length AE < EG. The recommended limiting condition is L sin 2, .S B/2 (Fig. 4.7a). E.

1

Transformation of Bridge Deck into Equivalent Grillage 83

Diaphragm Beam

INN AI XIII1111LS I " SiIN►111111

PP-

I

1IP"

(a) Skew or Paretlelogram (b) Mesh Orthogonal to Mesh Span

II II III II II 6.. 1IIIIIIIIIIII

Edge Beam

IIII II I II 111 k

(c) Mesh Orthogonal to , Figure 4.5 Grillages for Skew Decks

1

Figure 4.6 Principal Moment Directions in Skew Slab Decks

Case (b) represents the situation where skew region is large and right region is small i.e. AE > EG. The recommended limiting condition here may be L sin A > B/2 (Fig. 4.7b). The above two cases will decide the orientation of longitudinals with respect to s. In case (a), the main spanning will follow the direction normal to and the longitudinal grid lines will be taken normal to lines and transverse grid lines will be perpendicular to them as shown in Fig. 4.8 (a). As against this, in case (b). the main spanning will fnllnw ctepw rur,:•ction and hence longitudinal grid lines will be taken parallel to bridge axis and

84

Grillage Analogy in Bridge Deck Analysis Right Region

Skew span Skew Region

7: Skew Angie G

A B (a) Small Skew Region (L sin NzS. Bi2) Right Region

L=Skew span

Skew Region

B (b) La rge Skew Region ( L sin h > 6/2 ) Figure 4.7 Cases of Small and Large Skew Regions transverse grid lines will be perpendicular to these as shown in Fig. 4.8(b).

The spacing and number of grid lines in both longitudinal and transverse directions in case (a) and case (b) are discussed below: Case (a) The spacing of longitudinal grid lines may be different in skew region and right region. The spacing of these longitudinal grid lines is to be neither greater than the spacing of transverse grid lines nor greater than three times the thickness of slab. In the right region, .one 11/4-)nezlit4difia1 caLli jakcn along ED and GF and one or more between ED and GF. In skew region,

Transformation of Bridge Deck into Equivalent Grillage 85 C

F

X-L

-Longitudinds _ Transversals

(a) Longitudinal Grid Lines Norm al t o s

14 . / *•„ 4 .

A / / 1'4 /

/

/ • / • /

Transversals Longitudinals

/

A (b) Longitudinal Grid Lines Parallel to Bridge Axis Figure 4.8 Grid Lines in Skew Slab Decks the longitudinal grid lines may originate from the nodes where the transverse lines meet the longitudinal grid lines. However, if the number of longitudinals in skew region become too many, alternate or even lesser longitudinal lines may suffice. As far as possible, the longitudinal grid lines should be equidistant in skew region and right region each. Seven transverse grid lines for skew decks of span upto 10 m and nine grid lines for spans above 10 m may be used. Out of these transverse grid lines, one line each be located at ends of the span ing the centre line of bearings on each abutment and one at the centre of span. The remaining lines can be set in between these. As far as possible, the grid lines be kept equidistant.

86 Grillage Analogy in Bridge Deck Analysis

Case (b) The lay-out of lbrigitudinal grid lines will be similar to as in the case of right bridge discussed earlier. The actual numbers of transverse grid lines will be decided so that the spacing of these lines is not greater than 1.5 to 2 times the spacing of longitudinal grid lines. Referring to Fig. 4.8(b), one grid line each is set along GQ and DR and one at midway between GQ and DR. Others may be set between these if necessary. In triangular portions AGQ and DCR, transverse grid lines may be taken from the end of each longitudinal. However, if the number of transversals in this region becomes too large then alternate or even lesser transverse lines may also suffice. Again, in this case also, the transverse grid lines be placed equidistant, as far as possible. In skew slab deck with foot path, one extra longitudinal grid line is taken for the footpath at its centre at each end of the deck width as in the case of right deck. Sometimes a thick slab deck with thin cantilever and connecting slabs, as shown in Mg./4.9 is also encountered in practice. The thin cantilever slab at the edge, s the kerb, parapet etc. and the middle slab is meant for road divider or verge. The longitudinal grid lines can be placed as in Case (a) or as in Case (b). Location of the longitudinal grid lines as shown, assign the sectional properties of the transverse grid better between 3 and 4 in Fig. 4.9(a) or between 4 and 6 in Fig. 4.9(b). It may be also noted that in Fig. 4.9(b) longitudinal grid lines through 1, 5 and 9 will not have zero end deflections.

Thin Cantilever

(b) Figure 4.9 Thick Slab Deck with Thin Cantilever and Connecting Slab

AM .

Transformation of Bridge Deck into Equivalent Grillage 87

2. Voided Slab Bridge For a voided slab deck as shown in Fig. 4.10, the longitudinal and transverse grid lines are set in a similar fashion as in solid slab except that one longitudinal grid line is adopted covering one or more voids, depending upon their closeness. The grid lines may preferably be taken at the centre of solid portion between voids as shown in Fig. 4.10(b). The edge grid lines may be taken suitably to represent the edge strips of slab. The number of longitudinal lines may be odd or even depending upon the void position and the width of the bridge.

Partipet •

\

m

(a) Section of a Voided Slab •

1 

I

•

I

•

j

•

I

•

1 I

I

- 0

(b) Longitudinal Grid Lines Figure 4.10 Longitudinal Grid Lines for Voided Slab

The number of transverse lines may be located as in solid slab, depending upon span-width ratio and the spacing of longitudinal grid lines. Sometimes, the voids do not run throughout the length of the span but are provided only for about two-third middle segment of the span. In such cases, the longitudinal grid lines run through the entire span as usual but with different inertias for voided and solid end portions. In transverse direction, grid lines are also located at the sections where voided section changes to solid section. 3. Pseudo-Slib Bridge As discussed in Chapter 1, contiguous beams spaced closely with in-situ concrete comes under the category of pseudo-slabs. Usually standard precast beam sections are used in the construction. In such a situation, the number of closely spaced beams are large. A grillage arrangement with longitudinal grid lines coincident with every physical beam will lead to many grid linpc which ,Pc....*.1 arid unmanageable. Therefore, it is proper to represent more than one physical beam by a longitudinal grid line.

88 Grillage Analogy in Bridge Deck Analysis

Sufficient transverse grid lines are to be provided for detailed analysis. Their precise positions are chosen so that they intersect beams at the same points as longitudinal grid lines. For other details regarding the layout of longitudinal and transverse grid lines, recommendations given in Section 4.2.3 can be followed. Figure 4.11 shows the details of a bridge deck having inverted T-beams and in-situ concrete slab forming a Pseudo-slab system. The span and width

8 @ 1= 8 M (a) Cross — Section 3

I

f

- . (b) Longitudi nal Gri d Lines'

.1F

5

1

. I-- - I-- -- - , --4- 1 I 1 1 I I I i I I

----I 1 i I I I I II

I I, I

I 1 I 1 - --- -i Iii;

I 1 I I l- - — +---- i-- - - +

I

I

I

I

i

1 1 1 1 1 1 1 1 ----- 1 -.- --1---- - 1 ----1 1 1 1 I I I I I I 11-1 I -1 -I-- -- --i 1 -I I 1 1 I 1 I I

V O I D S

1---

I -i- - -i E. i

1

k--

r to n

I

--Jk-- - -1

I I .

1 c.4 1e 1I 1 1 I i I 1 I I

Figure 4.11 Grillage Geometry of Inverted T-Beam with in-situ Concrete Slab-Pseudo Slab Bridge

Transformation of Bridge Deck into Equivalent Grillage 89

of the deck are taken as 15 m and 9 m respectively, giving span-width ratio as 1.67. Nine precast beams at 1 m spacing are assumed. Five longitudinal grid lines, along the centres of alternate webs of physical beams are chosen. The edge longitudinal grid line is taken along the centre of edge physical beam as shown. Thus each grid line represents more than one physical beam. Seven transverse grid lines are chosen at the spacing of 2.5 m to represent the transverse medium. If the number of physical beams are in even number, say 10 in the above example, the longitudinal grid lines are taken at the centre of in-situ concrete between the beams as shown in Fig. 4.12. Voids •

1-



•

Figure 4.12 Longitudinal Grid Lines at the Centre of in-situ Concrete Slab-Pseudo Slab Bridge

Another form of pseudo-slab construction consists of standard precast box-beams as shown in Fig. 4.13. In such cases, the longitudinal grid lines are taken along the centre lines of voids. Again, one grid line represents more than one box-beam. The edge grid line may be taken along the centre line of first box-beam. The number of transverse beams are chosen depending upon the span-width ratio and the mesh size. If diaphragms are provided, the transverse lines may be located at each diaphragm first and if necessary, more grid lines may be chosen in between these. EiftiOrD14ffiritrIfffe  (a) Box-Beam Deck



I • I -0 1 •

•

(b) Longitudinat Grid Line Figure 4.13 Longitudinal Grid Lines in Pseudo -Slab (Box-Type)

i

90 Grillage Analogy in Bridge Deck Analysis 4.2.4 Grillage Idealization of Slab-on-Girders Bridge The idealization of beam and slab bridge by an assembly of interconnected beams-seems to confirm more readily to engineering judgement than for slab bridges. The T- and I-beams are by far the most commonly adopted type of bridge decks consisting of longitudinal girders at definite spacing; connected by top slab, with or without transverse cross-beams. Usually, the diaphragms connecting the longitudinal girders, are provided at the s.: The logical choiCe of longitudinal grid lines for T-beam or I-beam decks is to make them coincident with the centre lines of physical girders and these longitudinal are given the properties of the girders plus associated portions of the slab, which they represent. Additional grid lines between physical girders may also be set in order to improve the accuracy of the result. Edge grid lines may be provided at the edges of the deck or at suitable distance from the edge. For bridge with footpaths, one extra longitudinal grid line along the centre-line of each footpath slab is also provided. The above procedure for choosing longitudinal grid lines is applicable to both right and skew decks. When intermediate cross-girders exist in the actual deck, the transverse grid lines represent the properties of cross girders and associated deck slabs. The grid lines are set-in along the centre-Iines of cross-girders. Grid lines are also placed in between these transverse physical cross-girders, if after considering the effective flange widths of these girders, portions of the slab are left out. If after inserting grid lines due to theSe left-over slabs., the spacing of transverse grid lines is still greater than two times the spacing of longitudinal grid lines, the left-over slabs are to be replaced by not one but. two or more grid lines so that the above recommendation for spacing is satisfied. When there is a diaphragm over the in the actual deck, the grid lines coinciding with these diaphragms should also be placed. A typical T-beam bridge with grillage lay-out is shown in Fig. 4.14. When no intermediate diaphragms are provided, the transverse medium i.e. deck slab is conceptually broken into a number of transverse strips and each strip is replaced by a grid line. The spacing of transverse grid lines is somewhat arbitrary but about 1/8 of effective span is generally convenient. As a guideline, it is recommended that the ratio of spacing of transverse and longitudinal grid lines be kept between 1 and 2 and the total number of lines be odd. This spacing ratio may also reflect the span-width ratio of the deck. Therefore, for square and wider decks: the ratio can he kept as I and for long and narrow decks it can approach to 2.

(c) Cross-Section

Transformation of Bridge Deck into Equivalent Grillage 91

(a) Plan

(b) Longitudinal Section

iiiiiiiii 1101111111 11

( d) G ril l a ge Lay - out Figure 4.14 T-Beam Bridge and Grillage Lay-out

The transverse grid lines are also placed at abutments ing the centres of bearings. A minimum of seven transverse grid lines are recommended, including end grid lines. It is advisable to align the transverse grid lines normal to the longitudinal lines wherever cross-girders do not exit. It should also be noted that the transverse grid lines are extended upto the extreme longitudinal grid lines. In skew bridges with small skew angle say less than 15° and with no intermediate diaphragms, the transverse grid lines are kept parallel to the lines as shown in Fig. 4.15(a). Additional transverse grid lines are provided in between these lines in such a way that their spacing does not exceed twice the spacing of longitudinal lines, as in the case of right bridges, discussed above.

IS

92 Grillage Analogy in Bridge Deck Analysis

In skew bridges with higher skew angle, the transverse grid lines are set along abutments PQ and WV and also along PR, ST and UV initially as shown in Fig. 4.15(b). Then extra grid lines are inserted in between PR and UV. When the span-width ratio and the skew angle are such that the skew region QR is larger than the spacing of transverse grid lines in right region VR, additional transverse parallel to interior transverse grid lines starting from the interior nodes are to be placed. FQ

(

Z. Lb ;is./

y,y

,R

>1 7T ,

(a) Deck with Small Skew

(b) Deck with Large Skew A>15 °

Figure 4.15 Grillage Arrangement in Skew T-Beam Bridge without Cross Girders

Sometimes, precast I-sections with in-situ reinforced concrete slab form the deck. In such cases, the grillage is laid out in the same way as the T-beam monolithic deck. The ratio of moduli of elasticity of slab and precast beam materials are to be properly ed for in the analysis, and will be discussed later in the chapter. This is referred to as a twostage construction. 4.2.5 Grillage Idealization of Box-Girder Bridge Idealization of box-girder bridge is in many ways similar to that of slabon-girders construction but there is a behavioural difference betweeen them. The analysis of box-girder bridge is associated with special problems of shear deformations (shear lag) due to usually wide flanges of the deck and distortions of the cells, if sufficient number of intermediate transverse diaphragms are not provided. Although, other rigorous analytical techniques are available for the analysis of box-girder bridges, the grillage analogy has also proved to be sufficiently accurate and can be recommended in many cases. The grillage analogy in this case has the added advantage of being relatively inexpensive in computer time and simple to comprehend.

Transformation of Bridge Deck into Equivalent Grillage 93

The method is to be adopted where the effects of shear deformations and cell-distortions are negligible and could be ignored. The method is most appropriate for multi-cell rectangular box-girder decks (Fig. 4.16). However, it can also be used for decks with one or two cells only. The outer webs may be vertical or inclined, as shown in Fig. 4.17.

I-

•

(a) Deck Section 

t (b) Longitudinal Grid Lines

Figure 4.16 Grid Lines for Multi-Cell Box-Girder Deck (a) Deck Section .0

-

--• ___________ elipt

•

(b) Longitudinal Grid Lines Figure 4.17 Grid Lines for Box-Girder Decks with Inclined Webs

Longitudinal grid lines are usually placed coincident with webs of the actual structure (Fig. 4.16). If the deck has sloping end webs (Fig. 4.17), the grillage simulation is not so precise and engineering judgement must be used to position longitudinal . However, one grid line may be taken at the junction of the inclined web with slab as shown. Normally, additional longitudinal grid lines are located along the edges of the side cantilevers with nominal stiffnesses for the convenience of analysis. Additional longitudinal grid lines are to be adopted for bridges with footpaths at their centres as done in the case of slab and T-beam bridges, discussed earlier. The transverse medium consisting of top and bottom slabs only (with no diaphragms), is represented by equally spaced transverse grid lines along

94 Grillage Analogy in Bridge Deck Analysis the span. The spacing and number of grid lines are similar to as adopted for slab-on-girders bridge. If the deck is having diaphragms, the transverse grid lines are placed along each diaphragm including at s. Additional grid lines representing the top and bottom slabs are placed in between the diaphragms, if needed, to meet the minimum:requirements of transverse grid lines. A closer spacing of transverse grid lines will result in more continuous structural behaviour and will provide greater details of forces etc. For skew box-girder bridges, the procedure to.be followed regarding the setting of longitudinal and transverse grid lines will be the same as in the case of slab-on-girders bridge, discussed earlier. Spaced box-girders, also referred to as Spine box-girders, connected by top slab only, form a distinct class of decks requiring a special approach. Figure 4.18 represents a four-cell spaced box-girder bridge. In deciding the grillage layout for this type of deck, the main problem lies in defining the stiffness in the transverse direction. In the transverse direction, the deck is alternately very stiff over the box-beams and very flexible between the box-beams. It is not possible to replace the transverse media by grillage beams of uniform stiffness, as is usually possible with other types of decks. In this type of deck, each physical beam is replaced by grid lines positioned at the webs as shown in Fig. 4.18(b). This takes into the effect of abrupt change of transverse section more correctly. The other de- tails of layout of transverse grid lines are similar to those of slab-on-girders bridge. The plan of the arrangement of grid lines is shown in Fig. 4.18(c). Sometimes, the width of the cell is small in comparison with their spacing and walls are relatively thick preventing distortion of the cell (Fig. 4.19). For such sections of bridges, the longitudinal grid lines are better placed coincident with centre lines of boxes with additional nominal grid lines running in between the boxes as shown in Fig. 4.19 (b). In the transverse direction, the grid lines will have uniform stiffness corresponding to the deck slab and other details of layout will be similar to those of slab-ongirders bridge, discussed earlier. 4.3 EVALUATION OF EQUIVALENT ELASTIC PROPERTIES After the actual bridge structure is simulated into equivalent grillage, consisting of longitudinal and transverse grid lines meeting at discrete nodes, the second important step in grillage analogy method is to assign appropriate elastic properties i.e. flexural and torsional stiffnesses to each member of the ariliaap cn idpaliceri Thic merle ihP r•nrrInlitt;nr of flexural moment of inertia I and torsional inertia J for the of the grillage ,

FF

Transformation of Bridge Deck into Equivalent Grillage 95

(a) Section of Spaced Box-Girder Deck

(b) Longitudinal Grid

Lines

1- -v-- i-T ---Y-T---Y- T

I

L _L _ _ 1_ I _ _1_4. _ __I I

T

-

I t-

I-

1--

I

1 I

1_1_ --1 1 T -7-

1

I -- - II t - t--- - - F

I

r

I

1 ,

I I

1

I

1 --4----- t — i - -- t - i - -,I 1 T1 1 1

1--7-

I j I I I — I I I 1, 1 I — -11- - --- I - I - -- i - - —I- -1 A--k-A k A i 7 1c —4

— --

4

— ---

I

— i--- 1--

(c) Plan of Grillage Arrangement

Figure 4.18 Grillage Arrangement in Spaced Box-Girder Deck mesh. This is accomplished by considering isolated sections of the deck as if they are individual beams and the inertias are calculated for each section and allotted to the corresponding grillage beams representing that section. The principles involved and the methodology adopted for evaluating the various flexural and torsional inertias, are discussed first. Later on specific bridge decks are considered.

_

4

.

.

4--

.

.

i

96 Grillage Analogy in Bridge Deck Analysis bi

I i

i

er—•, (bi- b2)

-

\,.../ 1-..-b4,-.-1 1-•-b.-,.-1Box-Beams (a) Cross- Section of Deck with Narrow 1

2 •

to 3

(b) Longitudinal Grid Lines (

••• b4/2

• (d) Nominal Stab Grid Line (a) Beam Grid Line Figure 4.19 Grillage Arrangement in Decks with Narrow Box -Beams

4.3.1 Flexural Moment of Inertia, I

The computation of flexural moment of inertia I of different individual geometrical shapes like slab, T or I beams, box-girders etc. is straight forward and needs no elaboration. However, in beams having Tee, Ell or boxsections where slab is cast monolithically with the web of the beam, effective flange-width of the associated slab is to be considered. The Indian Roads Congress (IRC) recommendations [16] for choosing suitable effective flange width of beams are being followed in India for road bridges and will be further discussed elsewhere in this section. 4.3.2 Torsional Inertia, J The torsional inertia J, often referred to as the Saint-Venant torsion constant also, is generally not a simple geometrical property of the cross-section

Transformation of Bridge Deck into Equivalent Grillage 97 as the case with flexuial moment of inertia I and needs careful consideration. There is no accurate analytical procedure for the derivation of J. However, the approximate method for the evaluatidn of J for different cross-sections is based on the elastic theory of torsion of prismatic beams [11, 12] and is discussed here. Saint-Venant [12] derived an approxiMate expression for computing the torsional inertia J, of open sections which is applicable to all crosssectional shapes without having reentrant corners. The expression A4 J= 40 I

(4.4) P

where A is the area of cross-section and I is the polar moment of inertia. . For a rectangle of sides b and d, above expression reduces to, 3 b3 d3 = 10 (b2 + d2 )

(4.5)

In the case of a thin rectangle where b > 5d, the J value is more accurately given by bd3 J= 3

(4.6)

If the cross-section has reentrant corners, J is very much less than that given by equation 4.4 above [13]. In such cases, the value of J is obtained by notionally sub-dividing the section into rectangular shapes without having reentrant corners and summing the values of J of these elements. The value of J of a sub-divided portion with notional cuts on two opposite faces' is to be computed as if sub-division is a part of wide thin strip for which J = bd3I 3. Figure 4.20 shows a T-section with reentrant corners and its sub-division. Thus, if J values of the portions 1, 2, 3 and 4 are designated as J , J , J3 and J4 respectively, then, 1

(1 ' i3 )

2

3)

Jr = - " di

3 b33 J 3 10

(1 4 ±

and for the beam section as a whole,

J 2 =- 1 (b 2

3 J

4

3 b: d34 2 2 10 (b 4 d4)

98

Grillage Analogy in Bridge Deck Analysis

—4-1 Effective Flange Wiciffi---MAE=

4113

11111111a

Yr

i

d41:

1. ______ b1

1I d i

ro2 Figure 4.20 Sub-Division of T-Section with Reentrant Corners -rJ2+J3+J4

(4.7) It may be noted that the value of J of the portion of deck slab forming the flange is' to be halVed to take into its continuity in the other direction [12]. Widths b3 and b4 of segments 3 and 4 are so adjusted •that areas b3xd3 and b4xd4 are same as original areas of the respective segments. While notionally sub-dividing the section, it is worth ing Prandtl's membrane analogy. It is shown that the torsional stiffness of a cross-sectional shape is proportional to the volume under an inflated bubble stretched across a hole of the same shape. Thus, care is to be taken so as to obtain the largest numerical value of J of the section as a whole. This is achieved by choosing the elements so that they maximize the volume under their bubbles [11, 12]. Sub-dividing the cross-section arbitrarily into rectangles and not trying to maximize the. volume under the bubble will result in lower value of J. As an alternative to Equations 4.5 and 4.6 above, expression given by Timoshenko and Goodier [12] can be used for rectangular sections viz. J = yrbd3 where ty depends on the ratio of sides b and d (b > d) of the rectangle. The values of tir for different ratios of b and d are given in Table 4.1_ TABLE 4.1 bld

1

1.2

1.5

2

2.5

3

4

5

10 •

00

.0 .141 0 .166 0 .19 6 0 .229 0 .249 0 .26 3 0 .281 0 .291 0 .312 0333

The coefficient yr for any intermediate value of bid, may be obtained by

Transformation of Bridge Deck into Equivalent Grillage 99

linear interpolation. If the depth or width of an element changes along its length, then the inertias both I and J of the element are to be based upon the mean dimensions. It may also be mentioned here that the load distribution is likely to be more sensitive to the value of flexiaral moment of inertia I rather than to that of torsional inertia J, and as such a small error in the computation of J is not likely to much affect the final results. An incorrect sub-division of a section will invariably lead to an underestimation of J as discussed earlier and this will only lead to a conservative design to a small extent. The computations of I and J for different types of decks are now illustrated in the following sections. 4.3.3 Flexural and Torsional Inertias of Grillage : Slab Deck For solid slab bridges, the computation of flexural inertia is straight forward. The moment of inertia is calculated about the neutral axis of the deck. Thus for an isotropic solid slab of depth d, the flexural inertia (i' per unit width is given by i 3d =

12

If the deck has thin cantilever or intermediate slab strips as in Fig. 4.9, the longitudinal grid lines can be placed as in Fig. 4.9(a) or as in Fig. 4.9(b). In Fig. 4.9(a) the flexural inertias of all are calculated about the deck neutral axis. However if the grid lines are placed as in Fig. 4.9(b), the thin slabs above 1,5 and 9 act primarily as flanges to 2, 4, 6 and 8 respectively. Consequently the inertias of 1, 5 and 9 are calculated about centroid of the thin slab, while for 2, 4, 6 and 8 inertias are calculated with the flanges as in Fig. 4.9(a) but with small inertias of 1, 5 and 9 deducted. In the transverse direction, the thin slab would bend about its own centroid only so that the thin slab depth is used for 1-2, 4-5, 5-6 and 8-9 while the thick slab depth is used for 2-3, 34, 6-7 and 7-8. For a voided slab deck such as in Fig. 4.21, the inertias of longitudinal grid lines are calculated for shaded section about neutral axis. Transversely,

t

fa

Neutral xis Figure 4.21 Voided Slab Section

100 Grillage Analogy in Bridge Deck Analysis

the inertia is generally calculated about the centre line of void. However for void depths less than sixty per cent of the overall depth, the transverse inertia can usually be assumed to be equal to the longitudinal inertia per unit width. Reinforced and prestressed concrete slab bridges usually have similar stiffnesses in longitudinal and transverse directions with the result that sufficient accuracy is obtained by assuming that the full uncracked concrete section is effective, with reinforcing steel ignored. If the percentage of transverse reinforcement is relatively small while longitudinally the bridge is prestressed or heavily reinforced, one should take into flexural cracking, In such cases, the flexural inertias in two directions are calculated separately for different transformed sections. When the torsional inertia j per unit width of the slab is computed, two points are tote noted, namely (a) the deck contributes to torsional stiffness in both the directions and hence the value of f in each direction is to be taken as one-half of the computed value and (b) the lateral dimension is much laiger than the depth. Hence, for the slab, j = 22: 13 d3 I = d3 per unit width

(4.8)

Comparing the values of i and j per unit width for solid slab, it will be seen that j = 2i

(4.9)

There is no simple and rigorous rule for calculating j for voided slabs and the above relationship between i and j can be used with advantage to compute j for voided slabs with sufficient accuracy. Further, the torques in two orthogonal directions of an orthotropio slab are of equal magnitudes. Consequently, the transverse and. longitudinal grillage should have identical torsional inertia per unit width of the deck. It is suggested that, for such slabs, the torsional inertia of the transverse and longitudinal grillage , per unit width, be computed as, j = 2 • ix i

(4.10)

where i and i are the longitudinal and transverse flexural inertias per unit width of slab respectively. Alcn if hac hppn nhcerveri that the. simulation of grillage and physical deck is improved if the width of the edge member is reduced to (b 0.3 D) for calculations of J where D is the thickness of slab and b is the width .

Transformation of Bridge Deck into Equivalent Grillage 101

of. edge member [6]. However, for simplicity, full width of edge member may be used. In pseudo-slab construction, usually precast prestressed concrete beams are closely placed with in-situ reinforced concrete forming the deck. In such cases, the in-situ concrete slab has lower strength and stiffness than. the prestressed concrete beams so that it has a modular ratio m = 0.8, compared to prestressed concrete. Different transformed cracked section inertias are used in the two directions because the transverse reinforcement is usually lighter. It is to be noted that in a skew grillage, the slab width which is to be taken for computing the inertias, is the one normal to the grid line. Referring to Fig. 4.22, the widths of the slab to be considered with the grid lines A and B of the skew system will be U cos a. and B V cos A. respectively. Figure 4.22 Slab widths for inertia in a skew grillage layout

43.4 Flexural and Torsional Inertias of Grillage : Slab-on-Girders Deck Slab-on-girders bridge decks consist of a number of beams spanning longitudinally between abutments with a thin slab spanning transversely across the top. T-beam bridges are the common examples 'tinder this category. The be___ ms may be cast monolithically with the slab or the precast beams with in-situ slab may be used. The decks may be with or without intermediate and/or end diaphragms. The thin slabs can be thought of as flanges of I or T-beams. When such I or T-beams bend, the flanges are subjected to flexural stresses. An element of the flange away from the rib or stem of the beam has less stress than the one directly over the rib due to shearing deformations of the flange. Shear deformation relieves some amount of compressive stress in more distant elements. This phenomenon is known as shear lag. The variation of corn: pressive stress across the width of flange is ed for by considering the effective width of flange. The effective width may be smaller than the actual width and is considered to be uniformly stressed. The effective width of flange is determined based on the concept of constant compressive stress over the entire effective width such that the total compressive force carried by the effective flange is the same as that by the actual flange width with variable compressive stress (Fig. 4.23). Tnis effective width has been found to depend, besides other things, on the span and the relative thickness of slab.

)

102 Grillage Analogy in Bridge Deck Analysis

HEffective

Variation of-' ' Ftexural. S t

widthH

-1_..-- Assumed Uniform ' >--'r----'-'------- Stress over the e s s Effective width

— ----

r

Figure 4.23 Variation of Flexural Stress in the Flange of T-Beam

For the purpose of calculation of flexural and torsional inertias, the effective width of slab, to function as the compression flange of T-beam or L-beam, is needed. A rigorous analysis for its determination is extremely complex and in absence of more accurate procedure for its evaluation, IRC recommendations are followed. IRC : 21-1987 [16] recommends that the effective width of the slab should be the least of the following: 1. In case of T-beams (i) One-fourth the effective span of the beam (ii) The distance between the centres of the ribs of the beams (iii) The breadth of the rib plus twelve times the thickness of the slab 2. In case of L-beams (i). One-tenth the effective span of the beam (ii) The breadth of the rib plus one-half the clear distance between the ribs (iii) The breadth of the rib plus six times the thickness of slab The flexural inertia of each grillage member is calculated about its centroid. Often the centroids of interior and edge member sections are located at different levels. The effect of this is ignored as the error involved is insignificant. Once the effective width of slab acting with the beam is decided, the deck is conceptually divided into number of T or L-beams as the case may be. Some portion of the slab may be left over between the flanges of adjacent beams in either directions. In the longitudinal direction, it is sufficient to consider the effective flange width of T, L or composite sections, in order to for the effects of shear l ag and irrn^r- th 1-ft

Transformation of Bridge Deck into Equivalent Grillage 103

over slab. However, in the transverse direction, the left over slab should be considered by introducing additional grid lines at the centre of each left over slab portion. Sometimes, for the purpose of improving the simulation, it is desirable to place additional grillage of nominal stiffnesses between grid lines representing the beams. The sectional properties of these additional are calculated in a similar manner as outlined in Section 4.3.3 for the deck of Fig. 4.9. The sectional properties of grid lines representing the slab only, are calculated in the usual way i.e. I = bd3/12 and J = bd316. If the construction materials have different properties in the longitudinal and transverse directions, care must be taken to apply correction for this. For example, in a reinforced concrete slab on precast prestressed concrete beams or on steel beams (Fig. 4.24), the inertia of the beam element (1 or J) is multiplied by the ratio of modulus of elasticities of beam Eb and slab E: materials to convert it into the inertia of slab material. For example the total torsional' inertia in of the slab material will be given by Effective Flange Width 1 --*-1 J = Jl + (J2 + J3 + J4) Eb E s _________ b  -d1 d2 J1, J2, J3 and J4 are calculated as below

=12-(13-121 j 2

4)

3b 3 d 3 22

10 (i4 13

- Figure 4.24 Torsional Inertia in Precast Beam with Cast-in-Situ Slab

J3=—3b3d3 J 4

3b3 d3 44

10 (hi + 41)

4.3.5 Flexural and Torsional Inertias of Grillage : BoxGirder and Cellular Deck In the box-girder deck having longitudinal section as shown in Fig: 4.25, the top and bottom slab flex in unison about their common centre of gravity.

104 Grillage Analogy in Bridge Deck Analysis

Following the notations given in Fig. 4.25, the moment of inertia of transverse grillage member per unit width, it, is calculated about the common centroid and is given by =

1

+ t2 d2(4.11) since ti d1 = t2 d2 and (di +

d2) = d, we get 2

i = d t t 2 per unit width t1+ t2

(4.12)

If the transverse grillage also include a diaphragm, as at 'A', (Fig. 4.25) the inertias should be calculated including the diaphragm.

Figure 4.25 Longitudinal Section of Box-Girder Deck

The torsional inertia J of a thick-walled box-section is obtained simply by deducting the value of J of.the inner boundary from that of the whole section. Thus in Fig..4.26, where bi dl < 0.6 bd, 3 b3d3 J= 10 (b2 + d2) (0 +4)

b.?d?

(4.13)

Where continuity in the transverse direction is present, the torsional inertia in each direction is to be taken as half of above value. In thin-walled cellular decks, unlike the flexural inertias, the torsional inertia of grillage can not be calculated by isolating certain portion of the bridge and then asg their properties to grillage beams. The total torsional inertia of a section is distributed amongst the various grillage in proportion to the width that each member represents. For a

Transformation of Bridge Deck into Equivalent Grillage 105

d

Figure 4.26 Thick-Walled Box-Section

longitudinal grillage member representing a thin-walled cellular section, the torsional inertia is given by J =4 A2 f ds t

(4.14)

where 'A' is the area bounded by the centre line of the closed cross-section and 's' and 't' refer to the length and thickness of different units respectively. Referring to the Fig. 4.27, the equation 4.14 can be written as J=

4A2 (E

(Si

4A2 S2 t l

2S 2

(4.15)

3

t

t

3

______________________________________ 4

-7

F - - -

2

Figure 4.27 Thin-Walled Box-Section

106 Grillage Analogy in Bridge Deck Analysis

The above expression is applicable to closed cross-sections with one or two symmetric cells. For multi-cellular sections, as shown in Fig. 4.28, the value of J is obtained on the assumption that-the contribution of the interior webs is negligible. The J value of the section is given by .

I-

(a)

Multi -Celt Cellular Deck

b

t2 (a) Equivalent One -Cell Cellular Deck

Figure 4.28 Computation of J in Multi-cell Cellular Deck

J

4 b2d2

(4.16)

(b b 2d) t2

t3

The term 2d/t3 is small compared to other two in the denominator of equation 4.16 and can be ignored. The equation 4.16 can then be re-written as J

4 bd2ti t 2) If j denotes torsional inertia per unit width in each direction, then,

L

(4.17)

Transformation of Bridge Deck into Equivalent Grillage 107

(b)

= 2d2

r,t2 per unit width of cell (ti +

(4.18)

12)

Comparing equations 4.12 and 4.18, it is seen that] =. 2i. This is the same relationship as was obtained for solid slab bridges earlier and can also be conveniently used for computing the values of J of the transverse of multi-cell boxes of closed section without diaphragms. For obtaining J values of longitudinal , cell-widths in cross-sections are considered and for J values of transverse , spacings of transverse are to be taken. If j is the torsional inertia per unit_width, referring to Fig. 4.29, the values of J for longitudinal 1, 2 and 3, will be given by

(a) Cross-Section of a MuLtf-Cetl. Box-Girder

T a

a

(b) Plan Figure 4.29 Evaluation of J for Longitudinal and Transverse of MultiCell Box-Girder Bridge =

1/2j h and J, = j h

and for transverse 4 and 5. .1 vaiiips will J4=

=

ja

= O M

4 1 6 • 1 1 . 1 1 1 1 . -

108 Grillage Analogy in Bridge Deck Analysis

For transverse. grid of multi-cell box-sections incorporating diaphragms, the value of J is obtained as the larger of the values computed from the following alternatives : (i) treating the transverse (with the diaphragms) as T, L or I-sections and then applying the open section formula given by equation 4.7 and (ii) considering the' top and bottom slabs only and applying the close section formula as in equation 4.12. For precast spaced box-beam bridges (spine bridges) having narrow cells and relatively thick walls preventing cell-distortions (Fig. 4.19), the sec. tional properties of nominal, additional grillage. member 2 is calculated for width of the slab 'as shown. The properties of the grillage 1 and 3, are then evaluated' for the section with flanges including the area in nominal but with previously calculated properties of the nominal deducted from these values. If the boxes are much wider in comparison to the beam spacing as in Fig. 4.30, the values of I and .1 of the longitudinals can each be taken as one-half 1

(a) Cross- Section of Spaced Box-Beam '1

•

2

3 

1•

4 •

(b) Arrangement of Grid Lines

(c) Flanged Box-Section for Computing Land Figure 4.30 Spaced Box-Bridge Having Wide Beams

•

the value of the flanged box section. For the transverse grillage beam between 1-2 (Fig. 4.30b), the value of I will be based on top and

Transformation of Bridge Deck into Equivalent Grillage 109

bottom slabs computed about their combined centroid and the value of I for the beam between 2-3 will be based on top slab about its own centre of gravity. The values of J for beams between 1-2 and 2-3 can be determined from the usual relation i.e. j = 2i. Thus, it is seen that the actual deck of the bridge can be idealised into suitable grillage with appropriate equivalent inertias allotted 'to its each member. Other steps in grillage analogy i.e. the application of loads and its transfer to various nodes of grillage, analysis, stress response, interpretation of results etc. are dealt with in the next chapter. REFERENCES 1.

BAKHT, B. and JAEGER, LG., "Simplified Analysis for Slab-on-Girder Bridges", Bridge and Structural Engineer 12(4), 1982, INDIA. 2. BAKHT, B., JAEGER, L.G. and CHEUNG, M.S., "Cellular and Voided Slab Bridges", Journal of Structural Division, ASCE (ST9), 1981. 3. • BAxnr, B. and JAEGER, L.G., "Bridge Analysis Simplified", McGraw Hill, New York, 1985. 4. BAKHT, B.s JAEGER, L.G., CHEUNG, M.S. and MuFrr, A.A., 'Me State-of-the Art in Analysis of Cellular and Voided Slab Bridges", Canadian Journal of Civil Engineering 8(3), 1981. 5. CUSENS, A.R. and PAMA, R.P., "Bridge Deck Analysis", Wiley, London, 1975. 6. HAMBLY, E.C., "Bridge Deck Behaviour", Chapman and HaIl Ltd., London, 1976. 7. JAEGER, L.G. and BAKHT, B., "Bridge Analysis by Micro-computer", McGrawHill, New York, 1990. 8. JAEGER, L.G. and BAxtrr, B., 'The Grillage Analogy in Bridge Analysis", Canadian Journal of Civil Engineering 9(2), 1982. 9. LtarrFocrr, E. and SAWKO, F., "Structural Frame Analysis by Electronic Computer: Grid Frameworks Resolved by Generalised Slope Deflection", Engineering, 187, 1959. Id. MAISEL, B.I., "Review of Literature Related to the Analysis and Design of Thin Walled Beams", -Technical Report 42.440, Cement and Concrete Association, London, July 1970. 1 I. OMEN, J.T., "Mechanics of Elastic Structures", McGraw Hill, New York, 1967. 12. TIMOSHENKO, S. and GOODIER, J.N., "Theory of Elasticity", McGraw-Hill, New York, 1951. 13. WEST, R., "Recommendations and the Use of Grillage Analysis for Slab and Pseudo-Slab Bridge Decks", Cement and Concrete Association, CIRIA, London; 1973. 14. WEST, R., "The Use of Grillage Analogy for the Analysis of Slab and PseudoSlab Bridge Decks", Research Report 21, Cement and Concrete Association, London, 1973.

110 Grillage Analogy in Bridge De& Analysis 15. YET-IRANI, Al. and HusATN, M.H., "A Gridwork Frame Analogy for Plates in Flexure", Journal of Engineering Mechanics Division, ASCE, June 1965. 16. IRC: 21-1987, "Standard Specifications & Code of Practice for Road Bridges, Cement Concrete (Plain and Reinforced), Section III", The Indian Roads Congress, 1988.

-

Chapter 5

Application Application of Loads, Analysis, Force Responses and Their Interpretations 5.1' INTRODUCTION • The bridge deck has been transformed into an equivalent grillage consisting of longitudinal and transverse grid such that the idealized grillage is very close to the physical deck. Each member of the grillage is allotted flexural and torsional inertias which are equivalent to the corresponding physical properties of the bridge deck. The longitudinal and transverse grid lines form a mesh having number of nodes. The bridge is mainly subjected to vertical loads comprising dead, live and impact loads. Grillage analysis requires that these applied loads be transformed into equivalent loads at nodes. This is done by finding equivalent nodal loads in of either vertical load only or alternately vertical load, bending moment and torsional moment. This Chapter discusses different types of loads, identification of s in which the wheel loads of a vehicular loading system fall and transfer of loads to nodes of grillage in the form of equivalent nodal loads. The analysis of grillage is then carried out and response envelopes and the interpretation of results are discussed. Both right and skew decks have been dealt with. Local effects which are to be ed for in arriving at the design responses, are also discussed. 5.2 EVALUATION AND APPLICATION OF LOADS The loads, consisting of dead, live and impact loads acting on the bridge superstructure, are to be evaluated and nppropriately distributed to the nodes of the grillage. Evaluation of each type of the above loads and their placements on the deck are discussed below.

112 Grillage Analogy in Bridge Deck Analysis

a) Dead Load The deck of a bridge is subjected to dead loads comprising of its self weight and weights due to wearing coat, parapet, kerb etc. which are of permanent stationary nature. The dead loads act on the deck in the form of distributed load. These dead loads are customarily considered to be borne by the longitudinal grid only giving rise to distributed loads on them. This distributed load on a longitudinal grid member is idealised into equivalent nodal loads. This is specially required to be done *hen the distributed load is non-uniform. On the other hand, if the self load is uniform all along the length of the longitudinal grid line then it is not necessary to find the equivalent nodal load and instead it can be handled as a:uniformly distribiited load (u.d.l.) itself. Further, if the dead load is u.d.l. but its centre is non  concident with the longitudinal grid line Then it is substituted by a vertical u.d.l. together with a torsional u.d.l. Figure 5.1 shows a voided slab with voids running only over part length. Longitudinal grid lines will have non-uniform loading, higher intensity of loading in region and lower intensity of loading in central region. In such cases equivalent vertical nodal load is computed from the load on the tributary area. Figure 5.2 shows a solid slab with non-uniform 'spacing

I

I

I

I

I I

1I I 1

'

I

I

I •••Ii•••••

II

I

Longitudinal Grid Line

Transverse Grid Line

2c -L-

Tributary Area

2c

Figure 5.1 Voided Slab Bridge

Application of Loads, Analysis, Force Responses and Interpretations 113 A

Tributary Area

b'p+p 2a pqr,p, 2a pi

Grid Line A: Verticat and Torsions[ u.d.t. Grid Line B: Verticat u.d.[. Figure 5.2 Longitudinal Grid Lines with u.d.l. (b* a)

of longitudinal grid lines. The equivalent vertical load along each grid line is computed in the form of vertical u.d.l. based on its tributary area. For grid . line 'A' the loading is non-central and hence the equivalent load will consists of a vertical -and torsional u.d.l., whereas for grid line `B' the loading is central and hence the equivalent load will be a vertical load only. The self weight of cross-beams and diaphragms needs further considerations. These beams, located at specific intervals, are actually small discrete loads on the longitudinal girders. However, for simplicity of computation, the total weight of all the cross-beams per span should be calculated and equally divided in the form of distributed loads to various longitudinal of the grillage. The dead weight of railings, kerbs, footpaths etc. is lumped on the edge longitudinal grid lines. b) Live Load

The main live loading on highway bridges is of the vehicles moving on it. Indian Roads Congress (IRC) recommends different types of standard

114 Grillage Analogy in Bridge Deck Analysis

hypothetical 'vehicular loading systems, for which a bridge is to be designed. The details of these loadings have been discussed in Section 1.7 of Chapter 1 and are also available in IRC Code of Practice [10]. The vehicular' live loads consist of a set of wheel loads. These are distributed over small areas of s of wheels and form patch loads. These patch loads are treated as concentrated loads acting at the centres of areas. This is a. conservative assumption and is made to facilitate the analysis. The effect of this assumption on the result is very small and does not make any appreciable change in the design. IRC Class A two lane, Class AA Tracked and Wheeled, Class 70R Tracked and Wheeled loads are shown in Figs. 1.11 to 1.14. Three different wheel arrangements for Class 70R Wheeled loads are in existence and are shown in Fig. 1.14. Class 70R Tracked load may be idealised into 20 point loads of 3.5 tonnes each, 10 point loads on each track. The total load of the vehicle, this case is 70 tonnes. . One Class A or Class B loading can be adopted for every lane of the carriageway of the bridge. Thus, for a two-lane bridge, we can have two lanes of Class'A or Class B loading. However, for all other vehicles, only one vehicle loading per two lanes of the carriageway is assumed. Thus, for a twolane bridge, the alternate live loading will be one lane of Class 70R Train or Bogie or Track loading. For a three-lane bridge, three lanes of Class A loading or alternately one lane of Class 70R loading alongwith one lane of Class A loading is usually assumed. It is assumed in the design that the vehicles can not go closer to the kerb by certain . recommended clear distances. These distances are different for different types °flooding and have been given in Section 1.7.3. The wheel loads of the vehicle will be either in the s formed by the longitudinal and transverse grid lines, or on the nodes. The wheel loads falling in the s are to be transferred to the surrounding nodes of the s to facilitate the analysis. The distribution procedure of point wheel loads to nodes of the s will be discussed in the next Section. In order to obtain the maximum response resultants for the design, different positions of each type of loading system are to be tried on the bridge deck. For this purpose, the wheel loads of a vehicular loading system are placed on the bridge and moved longitudinally and transversely in small steps to occupy a large number of different positions on the deck. The largest force response is obtained at each node. c) Impact Load Another major loading on the bridge superstructure is due to the vibrations

. Application of Loads, Analysis, Force Responses and Interpretations 115

caused when the vehicle is moving over the bridge. This is considered through impact loading. IRC gives impact load as a perCentage of live load. As per IRC Code, impact load varies with type of live loading, span length of the bridge and whether it is a steel or a concrete bridge. The impact load can be calculated using formulae or could be directly read from ready-to-use graphical plot (Fig. 1.15). The impact load, so evaluated, is - directly added to the corresponding live load. 5.3 IDENTIFICATION OF S IN THE GRILLAGE When longitudinal and transverse of the grillage cross each other, they form s and the grillage is therefore divided into number of such s. All the wheels of the vehicular loading system may not come directly on the nodes of the grid but usually majority of the wheels fall inside the s. These wheel loads acting on the s are to be transferred to the contiguous nodes forming the , before the grid is analysed by the grillage analogy. Therefore, it is essential to identify the s of the idealised grillage deck in which a particular wheel load is lying. The identification of s involves their types and location in the grillage. The may be rectangular, triangular or in the shape of a parallelogram. Right bridge will have rectangular s only (Fig. 5.3a) while skew bridge with orthogonal arrangements of transversals will have triangular s near the ends and rectangular s in the central region as shown in Fig. 5.3(b). In case of skew deck with small skew angle where the transversals are taken parallel to the s, parallelogram s as shown in Fig. 5.3(c), will form. In case of skew decks with orthogonal arrangements of transversals, the upper and lower triangular segments are formed with polar symmetry, as shown. In skew bridges, it is likely that only one wheel of an axle may lie in the triangular or in the parallelogram , while other wheels are outside the deck, as loads P1 and P5 shown in Fig. 5.3 (b) and Fig. 5.3(c). This is not the case for the right bridge where all the wheels of an axle lie either on the rectangular or outside it (Fig. 5.3a). It is also possible that some of the axles of the train of wheels be out of the span. This happens when the axle load in question has either not entered the span or has already crossed it. In all cases, bridge is analysed only for the wheel loads that lie on or between the lines of the bridge deck. Transversely, the wheels are so positioned that the wheels closest to the kerbs maintain a minimum specified distance from either of the kerbs. When wheels of a vehicular loading system enter the deck, the position of the wheels are considered one by one and the s in which they fall are identified. As the coordinates of the nodes and grillage geometry are

116 Grillage Analogy in Bridge Deck Analysis +P

+P77

+ P6

+ P6

+P5

+P5 t

+P4 +P3

+P .

2•

+121

-1-P4 +P3

Upper Triangular

Ly Lower Triangular s

+P2 Width --1

-1-P1

(a) Rectangular s in Right Bridge

(b) Triangular and Rectangular s in Skew Bridge

Parattelogrum s

&Y

(c) Parattelogrum s in Skew Bridge Figure 5.3 s of Bridge

•

Application of Loads, Analysis, Force Responses and Interpretations 117

known, the type of panetand its location in the grillage are determined. The load in.the is now to be transferred to the contiguous nodes of the in the form of equivalent loads. A may have more than one wheel loads. In such a situation,. each load is transferred to the nodes independently and the effects are summed up algebraically at each node due to all wheel loads falling inside the . Similarly, the nodes will receive loads froti other s around it and also any load which may come on it directly. • 5.4 TRANSFER OF LOADS TO THE NODES The grillage analysis requires that loads be transferred to the corresponding nodes in the form of equivalent-loads. These equivalent nodal loads can be computed using any one Of the following two approaches : i) Simple statical approach where the load is apportioned in the form of -equivalent vertical shear assuming that the between contiguous grillage elements is simply ed along its boundary. ii) Another approach is .where the equivalent load .consists of vertical shear and moments assuming that the between the contiguous grillage elements is clamped at its edges. Although the first approach is simpler, the neglect Of fixed end moments will lead to .some error. The neglect of flied end moments in the longitudinal direction does not usually give rise to any significant error but their neglect in transverse direction can result in some inaccuracy in transverse moments. This is more so when the actual loads are acting on the cantilever portion of the slab or when the spacing between girders is large or when only a few concentrated loads act on the deck. If statical division of load is to be adopted due to its simplicity, then it has to be ensured that the transverse spacing of grillage is small and the loads are well dispersed longitudinally and transversely. Often, this is achieved by placing additional 'nominal' longitudinal grid lines in between or outside the main longitudinal grid lines. These nominal grid lines may be assigned zero inertia values. The second approach, where the loads are distributed in the- form of vertical shear and moments, is more tedious but theoretically superior. As the computer is invariably used in the analysis of grillage, this tediousness may not be considered an impediment to its use. However, both the approaches are in practice and if the grillage mesh size. is small, the results given by both will be close. But if the mesh size is coarse, only the second approach is rp.cnrymnprot.d. Both the approaches for the distribution of dead loads and live loads are discussed.

118 Grillage Analogy in Bridge Deck Analysis

5.4.1 Transfer of Dead Loads The dead load in the form of u.d.l. on the longitudinals, as evaluated earlier, are distributed to the nodes of the grillage. This is done in two ways: i) Assuming the longitudinal grid lines to be simply ed at nodes, the vertical load at each node of the longitudinal grid line due to u.d.l. are obtained statically. The total equivalent vertical load on each node is obtained by summing up the loads coming from the1adjacent . ii) Assuming that the longitudinal grid lines are fixed at nodes, the equivalent nodal loads will consist of a vertical load and a moment. Thus, if W per unit length of u.d.l. due to dead load is acting on any longitudinal grid line as shown in. Fig. 5.4, the vertical load V and moment M at node 2 1is given by V—

w

L

2

)

2

o.

m

(WL

—

WL 12 W/unit Length

(Z) 

nrY•errel

3

1-*-- Li ___________ L }M2 V Figure 5.4 Transfer of Dead Load on the Nodes

Similarly, the torsion T per unit length due to the transverse eccentricity of loads can also be distributed on the nodes treating the concerned longitudinal grid lines as fixed. As a sign convention, downward vertical force has been assumed as positive. For moments, right hand screw rule has been followed. 5.4.2 Transfer of Live LoadsThe live load may either be in the form of u.d.l. on footpaths, if provided, or vehicular loading system moving on the deck. These live loads are to be transferred to the nodes of the grillage suitably for the analysis. a) Live Load on Footpath -

.

.

•

,• .; !IL ithe fOCApath is to be considered. As per IRC:6-1987 [10] the footpath live load

Application of Loads, Analysis, Force. Responses and Interpretations 119

consists of a uniformly distributed load on its area over part or full length. Its magnitude etc. are discussed in Section 1.7 of Chapter 1. The footpath live load may be distributed to the longitudinal grid lines situated in its Vicinity according to the tributary area of each grid line, like dead load, discussed above. The transfer of this live load to the nodes of the grillage will also be done in the same manner as dead load. The bridge structure is analysed for footpath live load on one side of the deck or ontoth sides and over part length or full length and the resulting force responses are combined with force responses due to other live loadings, if the combination results in increase of responses. b) Vehicular Live Load The vertical wheel loads acting inside the s of the grillage are distributed in the form of equivalent loads to the !lodes of the concerned . This is done by adopting any one of the approaches discussed above. Both approaches are dealt with for rectangular, triangular and parallelogram s. The transfer of concentrated wheel load lying in the of the grillage is carried out in two steps. First the load is distributed along the direction parallel to transverse grid lines and then these forces are transferred to the adjacent nodes of the longitudinal grid lines. Case I : Equivalent Vertical Nodal Load The is assumed to be simply ed along the boundary. In a rectangular (Fig. 5.5), the equivalent vertical nodal loads are obtained by simple statical division and are given by ad

P P=

i

ac

P i = Lx L,,

L., L,

P P= be

-

P

bd P 4L,

1,, L,

(5.2)

Ly

For a lower triangular with load P as shown in Fig: 5.6, the equivalent nodal vertical loads are obtained by first distributing thz, load on longitudinal 1-2 and end transversal 1-3 at edges E and F respectively in a direction parallel to the transversal 2-3. The equivalent vertical loads at E and F and on nodes 1, 2 and 3 are given by (s (I) Pr=

P

PF

=s

P

120 Grillage Analogy in Bridge Deck Analysis

Ly

2

X

3 d

P

y

F

I 1

4,

i

a

H—b

Figure 5.5 Statical Distribution of Loads in Rectangular

Ly

d

b

TM)

3 ( C ) X

a

L.

1 (A)

Figure 5.6 Statical Distribution of Load in Triangular

Hence C pE C p

Lx P

P3

=-

=--

6

- 1

L:113 F

L,

=

L„.

(4a — L td) L PE = - p L.,. 4.

La PF =---d , 4. F

(5.3)

Application of Lo'ads, Analysis, Force Responses and Interpretations 121

The above expressions for nodal forces can also be alternately obtained by dividing the load P in ratios of the equivalent corresponding areas of triangles formed and the total area of the triangular element, i.e. P 1=

area of A BPC area ofd ABC area oft APC

P2

area of A ABC

P3 =

area ofd APB

(5.4)

area of A ABC

By substituting in of the dimensions shown in the figure, the nodal loads P1, P2 and P3 obtained from equation 5.4 will be same as given by equation 5.3 above. The equivalent nodal loads in upper triangular elements (Fig. 5.3b) will ,be same in magnitude and direction as in lower triangular element given above. In s having parallelogram shape and containing the wheel load P (Fig. 5.7), the equivalent nodal vertical loads at E and F and on nodes I, 2, 3 and 4 are similarly evaluated and given in equation 5.5. P

E =. 2- P

= ad P

P F

p

1

P

= 3

I

be

p

P

P

Y

ac • _P 2 LXLy

P4 bd 4

L

.

,

p

(5.5)

L y

Case If : Equivalent Nodal Vertical Load and Moments The concentrated wheel load falling in the is distributed to the corresponding nodes of the in the form of vertical shear and moments, treating the between the contiguous grillage elements as fixc:d along the edges. Again, the transfer of wheel load is done in two stages as in Case I, i.e. first the load is distributed on the longitudinals at its edge parallel to the transversals in the form of vertical loads and moments and then these are again transferred to the nodes of the longitudinal as in a fixed beam sub jected to concentrated load and moments on its span. The distribution of loads to the nodes of the rectangular, triangular and parallelogram s is

122 Grillage Analogy in Bridge Deck Analysis

3

X

b

a Ly

Figure 5.7 Statical Distribution of Load in Parallelogram

illustrated and the resulting equations for vertical shear and two moments at the nodes of the s in each case are derived. Vector forces on the diagrams are shown in the positive direction of axes but proper signs are incorporated in the expressions given: Negative sign preceding the equation will indicate that the force is in opposite direction to the vector drawn on the diagrams. Also, the equations pertaining to fixed end distribution of a point load, a bending moment and a torsional moment for a beam are given below which will be useful in subsequent discussions. a) Fixed End Distribution of a Point Load P for a beam AB (Fig. 5.8a) b 2 (3a + b) D I

A

D-

= ab2

M=

a2 (3b + a) P

I 5 a2 b

M=

A L2

L2

P

(5.6)

b) Fixed End Distribution of a Bending Moment M (Fig. 5.8b) 6 ab M

A=-P

P

1,3

M A

b (2a — b) M L2

MB

a (2b — a) M L2

(5.7)

Application of Loads, Analysis, Force Responses and Interpretations 123

c) Fixed End Distribution of a Torsional. Moment T (Fig. 5.8c) TA =1—Lb

Ts= —7:La

(5.8)

P M

A 61

I

1____________ 4 _______ a w1-4

PA i

b

L

1

. OmB .). ,.

P

B

(a) Point Load Distribution

MA 

rm

MB,Ai B b ______.1

t's 11/4 a _________________________________________________ I

-

a

.i

_________________________L Pt:

PB

(b)

Bending Moment Distribution

(c) Torsional Moment Distribution Sign Convention : Moment

Force Torsion

Positive

TA

I•

a

Positive Positive

b

/6- lAl

Figure 5.8 Fired Ended Beam: Distribution of Forces at the Ends (a) Rectangular



Consider a rectangular having nodes 1, 2, 3 and 4 with a wheel load P acting as shown in Fig. 5.9. The load P is first transferred to points E and F lying on the longitudinal 1-2 and 4-3. Treating the edges E and F fixed, the vertical forces and moments about X-axis at E and F are given by, 2 P E —a 3( 3 b + a ) L Dy

b

2

( P3F a 3+ Pb ) Ly =

124 Grillage Analogy in Bridge Deck Analysis M y2

M

Y7

11 MX3

 MX2 11 ;P

A

P

MX E

e )E-•-----ba l

F

4

1 M

t - MXF

___1fhP4 _____________ LY Myi Figure 5.9 Load Distribution in Rectangular

a2 bn

ab2M —=

'14

y

p

2r

AC

L

(5.9)

— L2 4

Again treating the member 1-2 as fixed at its ends, PE will induce at node 1, vertical force P1 and moment and /1//..„x will cause moment Mxi in the directions shown and their values can be written as

P

d2 (3 c + d) = Lg PE x

ar

d2

C

Application of Loads, Analysis, Force Responses and Interpretations 125

Similarly for other nodes of the , the vertical force and the two moments can be evaluated as follows: At node 2 C2

P2

(3d +c) PE Lx

' 3

2

M =—MXE Lx

d

M =-- r

(5.11)

E

L2

At node 3 g (3d+ c) 3

L

=

3

F

c2d

M

r L 2F

(5.12)

ri

13

At node 4 P4

d2 (3c +d) L3

F

c d2

Mx4 = MxF

MY4

LI.

=— L2x. "

(5.13)

(b) Triangular

Figure 5.10 shows a lower triangular of the grid acted upon by a load P. The load P is first distributed to 1-2 and 1-3, parallel to the transversal 2-3 at E and F respectively. The vertical forces PE and PF and associated moments MXE and MXF at E and F are given by PE=(s d) 2 (2d s)

(S M XE

P P

F

—d

2 (3 s — 2 d )

P

s3

— d) 2 d 2

p

S3 d2

M

XF

(s — d)

p

4

(5.14) Now, PE will cause vertical forces P. and P2 and moments M yi and Mn and the moment MXE will induce moments Alm and Mr at nodes 1 and 2

126 Grillage Analogy in Bridge Deck Analysis 6\14'\ -

M Y2

e: \

.4___ Nwr

Mx2 LY _______________________________________________________________________________________

1.R2 C

X

Lx a

1(A) .4,45\

Myl

Figure 5.10 Load Distribution in Lower Triangular

respectively as as shown. Similarly PF, on member 1-3 will lead to vertical forces PHI) and P3(I) and moments M i(1) and M3(1) at nodes 1 and 3 respeclively. The vertical forces and moments due to PEXE and PF. at nodes 1 and 3 are as below At node I due to PE P =C2 (3a + c) L3

E •

c

MX1= MXE— x

At node 2 due to

3

a2 (3 c + a) P

2

or P E t = LS

AlYL =— c2 a P—

(5 .15)

=7J

Application of Loads, Analysis, Force Responses and Interpretations 127

a M o r M X S =LM X E s

2

c

Mr2 or MYB L 2 = a PE

(5.16)

At node 1 due to PF

C2 (3a+ c) Pim= L3 PF

c2 a

M = "I)

sin a PF

(5.17)

At node 3 due to :PF

P3(1) = a2 (3c + a) a2 c PF

M3(1) —LPF (5.18) I sin a

Now MXF acting in the direction of X-axis is resolved into components MXF sin -a and MXF cos a in the directions parallel and perpendicular to member 13 respectively. MXF sin a will cause torsional moments M1(3) and M3(3) at nodes 1 and 3 and MXF cos a will cause bending moments M1(.2) and vertical force Pim at node 1 and bending moment M3(2) and vertical force P3(2) at node 3. The expressions for these force resultants are given below a M3(3) L = MXF sin a

M1(3) = —L MXF sing

P1(2)L 3 = 6 ac MXF sina COS CC c (2a— c) L MXF cosa

M1(2) =

!

P3(2) = 6 acL MXF cosa sina a (2c 1 — a) M

MXF cosa

3(2) = :"1

(5.19)

.

Now at node 1, the total vertical force PA, moment about x-axis MxA and moment about y-axis M„, will be,

128 Grillage Analagy.in Bridge Deck Analysis

PA = PI + P1(1) + PI(2) c2 (3a+ c)

c2 (3a+ c)

L3

L

P+x

L

3 x

6ac

PF

M xF sina cos a

x

MgA = Mio) cos a + M x1 + M1(2) cos a + M1(;) Sin a



c2 a..,, =— .-— MXE cIvi cot c (2a— c) ,, 2 .c . Lx r

Al

m

= M

n

Lx

C2 a n c (2a— c)

L

L

sin2 a

sin a — M1(2) cos a c2 a• c

( 1 )

1 ( 3 )

= - - rPE —- - r + E 2

Lx

L!

— M1

sin a + M

cos2a + — c

Ls2

2F

MXF sin a cosa + —• MXF sin o: cos a Lx (5 . 2 0 )

Similarly at node 3, the total forces Pc, Mxc and M 1 will be,

P = P + PC

3(1)

30.1

a 2 (3c+ a) , bac

rF +— MxF sin a cos a L3x

Mxc = M3(1)cos a + M3(2) cos a +.M3(3)sin a a2 c =— r2 FcoLa

a (2c a)

L!

"L

costa

MxF + —a MxF sine a Lx

M = — M3(1)sin a — M3(2) sin a + M3(3) COS a L2

=—

cr

La(2c—a) +

2 x

M

xF

sin a cos a + —a MxF sin a cos a (5.21)

Similar expressions can be derived for upper triangular also (Fig. 5.11). The distribution of vertical forces, bending and torsional moments on nodes of the are shown. The resultant vertical forces, at ts A, B and C will remain same as in the case of lower triangular . Also, the bending and torsional moments at the three nodes will be the same in magnitude but opposite in direction. (c)

Parallelogram The load in a parallelogram can be distributed to its nodes in the similar manner as in the case of rectangular or triangular s discussed

- - .

a

' Application of Loads, Analysis, Force Responses and Interpretations 129

limxi My1

__ y

41‘;'"s

&-• ty

4.ar Mx

e

MXF

4}h>si> CV') 17,1 1 /4. P



3 `ry 4'62

b

2

4N"' ‘.\ 3(C)

2 (B)

d

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Ly_________________________________________________________________"4

M Y 11M x2

Figure 5.11 Load Distribution in Upper Triangular earlier. Figure 5.12 shows a parallelogram and the distribution of load to its nodes in the form of equivalent vertical loads and moments. The vertical loads and moments at edges E and F are given as

a2 (3 b + a) E

L3

M =—a b P sec a E L2.1. a2 b =

L

PF

P

—

2

tilXE

b 2 (3a+ b) Ls

2

P

MF

MXF = —

b2a 2 — See a L b2 a L2

130 Grillage Analogy in Bridge Deck Analysis

P3 0

mx3 Y4

tlx2

P2

my 2.

MF

MXF

0 YF_

d

PF

Lx PO

ticlx4 P.4

1 01 p H-- b .i*

a —1-1

x

Ly Figure 5.12 Load Distribution in Parallelogram

2b 11.11.E=-- P tan a L2

ab2 m= P tan a YF L

. (5.22)

The forces at E and F are distributed at nodes 1, 2, 3 and 4 and the resultant vertical loads, bending moments and torsional moments at each node are directly written as P

=

3

Lx

d2 (3c +d)D 6cd FE 3 IVIYE

d xk1L E c d2 Lt

PE

d (2 c - d) L;

YE

(5.23)

Application of Loads, Analysis, Force Responses and Interpretations 131 P2 —

c2 (3d + c)

L

3

6cd

E

P E 3+ Y M E 1, x

L a .

M

C 119 XE x ' '

Y

e2 d =L 2x E

2

c (2d — 2 c) mYE L

(5.24)

(3d+c) 6cd 3 I F P3 = L + L3 MYF C2

Lx

Mx3

L i v i x"F x

,Y3

c2 d

c (2d — c) MYF

Lx

= L2 P4 —

r

3

xa

Lx

d2 (3c +d) Lx

M

F

d m

,

F

(5.25)

2 6cd ---

3 MYF

Lx

XF

(5.26) c d2 d (2c — d) frf — r- ________________ MYF L2 F 2 Lx Y4

Thus, it is possible to distribute wheel loads falling in s of different shapes to the contiguous nodes either in the form of vertical forces only by statical distribution or by a combination of vertical loads and moments. The choice rests with the designer. If the computer is used for the transfer of loads, it is prudent to replace the vertical load lying in the by a combination of vertical loads and moments. 5.5 GRILLAGE ANALYSIS AND FORCE RESPONSES After the loads are transferred to the nodes of the grillage in the form of equivalent forces, the grillage may now be analysed to determine nodal deformations and member forces.

132 Grillage Analogy in Bridge Dick Analysis 5.5.1 Analysis of Grillage Direct stiffness method is an effective tool in analysing the grillage on a computer. As mentioned earlier, there are three possible displacements at each t of the grillage. These, for a grid in X-Y plane, are t rotations about X and Y axes and t translation in Z-direction, normal to X-Y plane. The displacements in its own: plane and rotation about Z-axis are small and are ignored. The analysis of grillage by the stiffness method involves the following steps 1. Formulation of Stiffness Matrix The first step in the analysis of grillage involves the generation of the stiffness matrix of the structure corresponding to the appropriate degrees of freedom. A 6 x 6 member stiffness matrix [Km] is generated for each member of the grillage in of its geometric and elastic properties. But the matrix [K.] developed is in of local degrees of freedom which are different for different meeting at a t. Therefore, the matrix [K.3 for a glid member is transformed into global degrees of freedom by using a 6 x6 transformation matrix N. This takes into the orientation of the member. Thus for a grid member, the global member stiffness matrix [K,:] is obtained. Now, to satisfy the equilibrium conditions at a t, the assemblage of all the meeting at that t is considered. The structure stiffness matrix [K] is obtained by assembly of elements of global member stiffness matrix [I C ]. This assembled matrix [K] is nothing but the addition of all internal forces which will be subsequently equated to the externally applied loads along the same degree of freedom. One of the basic properties of the matrix [K] developed is its symmetry and banded nature and the advantage of this fact is taken in storing only the banded upper triangular portion of the matrix. This enables us to analyse bridge with a large number of nodes. 2. Formulation of Load Vectors External equivalent loads applied at the ts (nodes) of the grillage constitute the load vector {B}, having moments about X and Y axes and vertical force along Z axis. The load vectors may either be formed separately for dead load, live load and impact load or their effects are added together and a single vector is formulated. 3. Identification of Conditions The stability of the structure is considered by introducing the boundary

'7417.-.t•-??2;4.7.'

Application of Loads, Analysis, Force Responses and Interpretations 133

conditions. The s of the bridge may be either on compressible (yielding) neoprene type bearings or rigid (non-yielding) steel or concrete bearings. Depending on the type of bearings used, stiffness matrix developed above is to be modified suitably. The neoprene bearings can be assumed to have negligible rotational stiffnesses: The axial stiffness of the neoprene beating is taken and added to the in the stiffness matrix at the position corresponding to the vertical dgfiection of the ed node. Since bearing dimensions are not known in the beginning, a suitable size of the bearing is assumed initially and its axial stiffness is computed and used and finally modified, if reactions obtained so demand, or alternately rigid bearing is assumed initially and based on the reaction and rotations, suitably dimensioned neoprene bearing may be adopted and a revised analysis is carried out. For skew bridges, the stiffness of the bearings has marked effects on. the structural behaviour and it is very important to for their effects properly. Yielding of bearing tends to even out the differences between various reactions. When simple rigid type s are used, the rows and columns corresponding to the vertical deflections are removed from the stiffness matrix. The load vector has also to be modified by removing the rows corresponding to the vertical deflection at the ed nodes. The deformation vector that is obtained by solving the modified stiffness matrix using the modified load vector does not contain the vertical deflections at the ed nodes. The deformation vector is modified to include these. The locations in the grillage is kept at the same place where it actually exist. For obtaining the reactions at the s, the structure stiffness matrix [K] is partitioned into four sub-matrices pertaining to the free and restrained deformation vectors {DP} and {DR} and correspondingly load vectors into external load vector {-P} and reaction vector {R} as discussed in Chapter 3. The reactions can be obtained by using Equation 3.18. 4. Solution of Simultaneous Equations A large number of simultaneous equations will result from the assembly of stiffness matrix considering equilibrium at each t. The number of these equations depends upon the size of the grid. Efficient techniques like Gauss-Elimination and Cholesky Factorisation are available to solve these equa-tions. The resulting deformation vector can be used to compute member forces using basic member properties. The procedure is already dealt in Chapter 3. 5. Determination of Nodal and M h Pin

Pr nOri:ar!r

The solution of simultaneous equations will yield nodal deformations of

134

Grillage Analogy in Bridge Deck Analysis

the structure. The member deformations in global as well as local coordinates can be evaluated by multiplying member stiffness matrix with the member deformation vector. The output or the result consists of vertical deflection and rotations about X and Y axes at each node, shear in each member, bending moments at the two ends of the member and torsional moment in each member. The output also gives reactions at the s.  The formulation of member stiffness matrix, its transformation, assembly, conditions, reactions etc. have been discussed in general in Chapter 3 and may be referred there for detail. Normally the above processes for the analysis of grillage are handled by computer using a suitable program meant for the same. Many compact packages are available for the grid analysis for a set of given loads on the nodes of the grid and may be used. One such computer program in FORTRAN is given at the end of Chapter 3. However, a more general computer program for the analysis of grids subjected to different types of IRC loadings or any other special live loading termed as Specified live loading has been developed and discussed in the next Chapter. The program transfers the load from s to the nodes automatically and analyses the bridge for various positions of loads. 5.5.2 Force Responses As discussed above, the solution of equations yield nodal deformations i.e. deflection, slope and rotation at each end of the member. The shear force for a member, the bending moments at the two ends of the member, the torsional moment in a member and reactions at the ed nodes are the usual output. However, these output can be modified and more details are possible. Ordinarily the output is obtained for various longitudinal and transverse positions of different types of live loading. Invariably the output obtained is very large. Scanning this output, for a grillage of even moderate size, is a problem. Therefore, to reduce the output data, only the critical values of the force responses need to be retained. For the design of any bridge structure we need the envelope diagrams of various responses on it. The envelope diagrams are the response diagrams drawn along the longitudinal grid lines with the largest values of responses picked up under live load. This may be achieved for a particular live load by moving, it over the deck in small increments both longitudinally and transversely and for each of the load positions, the deck is analysed. When the load moves from one position to the next position, the responses are again obtained for this new position of load and these values are comoared with the previous values. The larger values of each force responses like shear force, bending moment and torsional moment for each grid member

Application of Loads, Analysis, Force Responses and Interpretations

135

are retained alongwith the corresponding load position, deleting the smaller values. The process is repeated till the whole length and breadth of the bridge is covered by the live load. The load position for each critical value is given through the coordinates of the left most wheel of the leading axle. This information of load position could be used for positioning the live load on the deck and carrying out a manual check if so desired. The number of movements of loads in longitudinal and transverse directions will depend upon the factors like span, carriageway width, type of live loading, extent of accuracy desired, available computer time, etc. However, as a preliminary guidance, the movements of loads in increments equal to about 1/15th of span length or half the size of the mesh in longitudinal direction is chosen. The movement of loads in transverse direction is very much limited due to the restrictions on wheels from coming closer to the kerb by a specified distance. The loads may be moved transversely in five to seven, equal intervals in a two lane bridge and in steps of about 750 mm in wider bridges. The initial and final positions of the live loading on the deck should be so chosen that no critical response is missed out. The initial and final positions of wheels in longitudinal direction (X min and Xn) and in transverse direction (Ymin and Yn.) are shown in Figs. 5.13a, b. The coordinates of these initial and final positions can be calculated with the help of Table 5.1. TABLE 5.1: Computation of Initial and Final Coordinates of Reference Wheel Loading Reference Number

Type of Loading Class A Two lane Class 70R Train Col. 'I' and Col. 'm' Class 70R Bogie Col.'/' and Col. 'in' Class 70R Track Class A Single lane

Xthin = 0 Ynth, = A + S

2

5300

400

18800

3, 4

2380

1405

13400

5, 6 7 8

2380 2060 1800

1405 1620 400

1220 4113 18800

Xmax =L+U+Btan ) Y =B—A—C-5

where A = Transverse distance of Kerb from exterior grid line (AKERB1* B = Right distance between exterior longitudinal grid lines C = Transverse distance between centres of exterior wheels L = Right span for right bridge and skew span for skew bridge S = Mini mu m tr an s ver se dis t an ce o f c ent r e o f l e ft mos t wh e el fro m Kerb (SCLMIN)* U = Longitudinal length of loading )1. = Skew angle Notations used in the program.

Distances are in mm

_

136 Grillage Analogy in Bridge Deck Analysis

p

Loading Axle

1"1"-- YMAX c

_FH44,R- rncn-751---1

Left Most • Grid Line

XM

B (i) Cross-section ++++

++++ 1

X=L

1 0

B

;41

• Right Most _ Grid Line

0

1

t+ CI)

4T*-

0

(0,0) + 4I+ + + + +

0

0

()Lett Most Wheel of Leading Axle 0 — 0 ©Right Most Wheel of Rear Most Axle 0 — CD (ii) Right Bri dge Plan Figure 5.13 (a) Initial and Final Positions of Loading to Cover Entire Right Deck The total number of positions which a live load may occupy on the deck and for which the bridge will be analysed. are:

Application of Loads, Analysis, Force Responses and Interpretations 137

F"'„

1MAX

c

-0

Loading Axle

1

C

4141-p-n'

Left Most • ___________________________ • Right Most Grid Line 14 _____________ B _____________ Grid Line ( 1 ) Cross—section Y=0

+

OF +

1+

-

02)

X M A X

X=L

XMAX

(0,0)

u I

(I)

Left Most Wheel of Leading AX1e

0-0

Right Most Wheel of Rear Most Axle

(D—

(ii) Skew Bridge Plan Figure 5.13 (b) Initial and Final Positions of Loading to Cover Entire Skew Deck

138 Grillage Analogy in Bridge Deck Analysis (X . — X min +1 ) Y max — Y min +1 I n c r e m e n t ) in X direction Increment in Y direction It is possible to eliminate some of the positions out of the above, based on experience, judgement and a little manual calculations. Alternately, one may take the view that as computer is involved in the analysis, more positions of loads do not matter and carry out the analysis for all the above load positions. 5.5.3 Design Envelopes

.

In order to design a bridge for IRC loading, it is not sufficient to analyse the grillage for any one type of live load only and obtain response envelope diagrams for it. The maximum responses due to one particular type of live load may not be critical at all the points on the deck and it has to be scanned for other types of live loads also to obtain the largest design responses. To achieve this, each live load system is moved longitudinally and transversely in small increments to cover the entire deck. The grillage is analysed for each of these positions and the maximum values of responses are retained alongwith the corresponding load positions; the maximum response results of various types of live loads are compared with each other and the highest values along with their load positions and type of loading are retained giving an overall envelope diagram for each response separately. Normally a one lane bridge is designed for Class A single lane loading. A two-lane bridge is designed for the following alternate live loadings : i) ii) iii) iv)

Two lanes of Class A loading One lane of Class 70R Train loading, One lane of Class 70R Bogie loading, and One Iane of Class 70R Track loading.

Class 70R Train loading and Class 70R Bogie loading each have been considered with two alternate types of wheel arrangements on the axle, Col. `1' and Col. `tn'. Refer section 1.7.3 of Chapter 1. In addition, the program also caters for any other specified live loading. The in such case is required to give full details of the specified live loading. To summarise, following are the loadings together with. their loading ..:.:_:. program.T ne program can move

Application of Loads, Analysis, Force Responses and Interpretations 139

various live loads on the bridge deck through prescribed steps along the span and across the carriageway width. Loading (1) Loading (2) Loading (3) Loading (4) Loading (5) Loading (6) Loading (7) Loading (8) Loading (9)

Dead load Class A Two lane Class 70R Train : Col 1 Class 70R Train : Col m Class 70R Bogie : Col 1 Class 70R Bogie : Col m Class 70R Track Class A Single lane Specified by

Loading (1) represents dead load. Loading (2) to Loading (9) represent various types of live load. It is built-in the program that live load results are given inclusive of dead load results. If only live Ibad results are desired, then in the Input data, dead load can be given as zero. 5.6 INTERPRETATION OF RESULTS The Output or the result obtained from the analysis of grillage consists of vertical deflections and X and Y rotations of each node, shear force and torsional moment of each beam element, bending moments at the two ends of each beam element and reactions at each . The above results are to be judiciously used while deg a bridge deck. Since the deck has been initially idealised as a grillage and the analysis has been performed on the idealised grid, the results may sometimes need modifications and proper interpretations before they are finally used in design. Some of the important interpretations of the Output and its modifications due to the local effects for slab bridges, slab-beam bridges and cellular bridges are discussed below. 5.6.1 Slab Bridges The computer output for deformations like deflection and rotation and force responses like bending moment, shear force and torsional moment are to be thoroughly examined and judiciously interpreted in slab bridges. Modifications in the output results are made, if necessary, due to local effects which are not considered earlier in the grillage analysis and the modified responses are to be used in the design for better accuracy. Some

140 Grillage Analogy in Bridge Deck Analysis

of the significant obseivations pertaining to force responses, for slab bridges are discussed here. The slabs are designed on the basis of per unit force response. The computer output gives response for the width which is represented by a particular grillage member. Hence, these responses should be converted into per unit width befOre these values are considered for design. The. analysis will give positive or negative values for various force responses with respect to each beam element and due care should be taken to adhere to the sign conventions adopted in the computer program. For example, a negative bending moment value at end I of a grillage element will denote a sagging moment while it will indicate a hogging moment at end 2 of the same element. When a grillage member continues across a node, the values of moment at end 2 of one member and end 1 of the adjacent member in continuation will be usually different. This is due to the torsional moments in the framing in other direction. To deal, with such "Stepping" or "Saw -Tooth" in the moment value's along a grid line at nodes, these two moments may be averaged out (Fig. 5.14).

Grid Nodes



•••

.... ..,

_

•l

mertt

\—Griltage Output Figure 5.14 Grillage Output Saw -Tooth Moment Diagram

Only one value of the shear force for a member of the grillage is obtained from the output and the same may be used in design as such. Similarly, maximum reactions printed, are taken as design values for reactions at ed nodes. The torque per unit width in a true orthotropic slab is same in orthogonal directions, however, it is often different when read from 2TilIage output. The torque per unit width at any point should be taken as the average of these two outputs.

Application of Laads, Analysis, Force Responses and Interpretations 141

The design of the section in bending should be based upon the principal moments mi and in,. Referring to Fig. 5.15, the principal moments mi, m2 and their deviation at the nodal point of interest is obtained from the values of bending moments per unit width mx and my, in X and Y directions and the corresponding averaged torsional moment per unit width mx,employing the formulae 2

nix+ my {(mx— my ) m1.2 - +

2

2

2

+m 2 (5.27)

ttan2 0 = m xy mx — my

m1 I

2 My Figure 5.15 Computation of Principal Moments and Their Deviation from X-Y Axes

In reinforced concrete bridges, the direction of reinforcement may not always coincide with the direction of principal moment_ This is more so with skew slab bridges. In such a case, it should be ensured that reinforcement component in the direction of each principal moment is adequate. For a critical load combination, if at a bearing point there is a net negative or downward reaction force as opposed to the usual positive or upward reaction force then it indicates that under that combination of loading there is uplift at the bearing point. This may happen for bridges with large skew angle and there may be one or more such bearing points in the deck.

142 Grillage Analogy in Bridge Deck Analysis

It may be recalled that while obtaining the above results, conditions of zero deflections have been imposed at all the bearing points. Thus the behaviour of the actual structure is not consistent with the assumed condition of the grid-model. In such cases a repeat-analysis is necessary. In the repeat-analysis, the condition of zero deflection is not imposed at the node/nodes expected to be lifting when analysing under loading forming the above critical load combination. Lifting at the bearing is not desirable and should be prevented as far as possible. This could be done by providing a tension bearing at-such bearing point as opposed to the usual compression bearing. In case where a downward reaction force is small, a lateral shift in the position of the bearings can also change it to a net upward reaction force and condition of no lift could be obtained. As pointed out in Chapter 4, a slab bridge, strictly speaking, can be idealised as an assembly of orthogonal beams only if Poisson's ratio v of the slab material is zero. Ignoring Poisson's ratio which is about 0.15 for concrete leads to an under-estimation of moments. This under-estimation is usually negligible for longitudinal bending moments but is not so for transverse bending moments. The error in the prediction of transverse bending moments can be large in somes cases. This is because the curvature along the span is considerably greater than that in the transverse direction. The following relationships can be used to , approximately, for errors resulting from the neglect of Poisson's ratio (M)

70/,)4

(My) = (M)0 + 7(Mx),3

(5.28)

where (M )0and (M)0 are the responses obtained by grillage analysis for which Poisson's ratio = 0 and (Mx) and (My) are the relevant corrected moments for a finite value of Poisson's ratio. Due to dispersion of load, the actual area of application of load is larger than the area of of the wheel with the slab. If this application area is larger than the grillage mesh, the load can be assumed to be sufficiently dispersed for the grillage to reproduce the distribution of moments throughout the slab. No further modification of moments is necessary. On the other hand, if the area of application of the load is small compared to the grillage mesh, no worthwhile information will be obtained about the local high values under the load, though the grillage distributed moment field will simulate that in the deck. The additional moments due to high local curvatures can be obtained for the area of slab within the grillage mesh from influence charts due to Pucher [6].

r

4

Application of Loads, Analysis, Force Responses and Interpretations

143

5.6.2 Slab-on-Girders Bridges In beam and slab decks also, the stepping of moments in on either side of a. node occur. The difference in bending moments in two adjacent meeting at a node will generally be large in exterior girders. Where all the meeting at the node are physical beams, the actual values of bending moment output from the program should be used. If at a node there are no physical beams in the other direction and the grid beain elements represent a slab, thd bending moments on either side of the node should be averaged as there is no real beam of any significant torsional strength. Design shear forces and torsions can be read directly from grillage output without any modifications. In composite construction, where the grillage member stiffnesses are calculated from properties of two dissimilar materials of slab and beam elements, the output force response (torque, bending moment or shear) is attributed to each in proportion to its contribution to the particular stiffness. The local two-dimensional system of moments and torques in the thin slab under a concentrated load are not given by the grillage analysis and can be obtained from Pucher's charts [6]. The local moments obtained from the charts must be added to those in the slab resulting from twisting and relative deflection of the ing beams. If there are no "nominal" grillage between two physical beams and the transverse have not been loaded, these moments can be read directly from grillage output for the local transverse . However, if there is a "nominal" longitudinal member under the load or if transverse have been loaded, the slab moments due to twisting of beams are best calculated from grillage output displacements and rotations of adjacent beams using slope • deflection equations [3]. In case of longitudinal grid lines not physically ed at ends, the load carried by these lines will flow towards nearby s via end transversals. If not taken note of this phenomenon this will give lower value of shear in ed longitudinal grid lines. To for this under-estimation, shear of these beams be added to the shear of adjacent beams which are physically ed. Similarly, in order to avoid under-estimation of bending moment in ed longitudinal beams, the bending moments of uned grid lines also have to be considered for design of ed longitudinal beams. 5.6.3 Box-Girder Bridges drawn from the results of the output for pox-girder or cellular deck again indicate "Saw Tooth" (Fig. 5.14) with

144 Grillage Analogy in Bridge Deck Analysis

large jumps in moments at the nodes. This is because of the transfer of the torsional moment in the transverse at each t to bending moments and shear forces in the longitudinal member [3]. The true design bending moment is obtained by averaging the bending moments on the two sides of each t. The top and bottom slab stresses are calculated from these average moments. The transverse bending moment in the grillage member is equivalent to the opposed transverse compression of the top slab and tension of the bottom slab (or vice-versa) due to transverse flexure without Tlistortion. In narrow decks, it is usually very small compared to the longitudinal bending moment (except in the diaphragms) but in wide decks it can be large specially near skew s. A grillage output transverse moment diagram has a saw-tooth shape like the longitudinal moment diagram and the top and bottom slab stresses are calculated from the average moments. The slab bending moments are derived from the shear force in the transverse grillage . The fraction of this shear force carried by each of the top and bottom slabs of the cell are assumed to be proportional to the flexural stiffnesses of the slabs. Assuming also that points of contraflexure lie midway between the webs, the moment at each end of a slab is simply the shear force it carries multiplied by half the distance between webs. The transverse slab moment in • the cantilever can be taken directly from the grillage output since this member is not representing the cell. Local moments are again obtained using influence charts of Pucher t6]. The total design moment in cellular deck is obtained by adding the slab moments of the cell, cantilever moment and the local moments. The torsion-shear flow in the slabs must be calculated from the average torque per unit width of transverse and longitudinal grillage . The grillage output shear force represents the total shear force in each web of the deck. Grillage analysis ignores the effects of Poisson's ratio on the interaction of longitudinal and transverse moments. This neglect of Poisson's ratio introduces little error in narrow decks while in wide decks with little stiffness against cell-distortion, the calculated transverse moments can be considerably in error, if moments are small and the Poisson's ratio is significant. However, since the concrete has relatively low Poisson's ratio of approximately 0.15, its neglect may not result in any appreciable error. Reviewing the discussions contained in Chapters 4 and 5, taken together, on the grillage analogy method, it can be seen that the procedure is straight forward and is also amenable to an enaine.er who may not he well conversant with higher mathematics and numerical techniques. The setting out of

Application of Loads, Analysis, Force Responses and Interpretations

145

the grid lines and the evaluation of elastic properties of the can be handled manually following the guidelines provided above. The determination of the dead loads etc. is simple. The remaining steps in grillage analogy i.e. transfer of loads to nodes, formulation and inversion of matrix, solution of equations, evaluation of design responses etc. are done on computer using: a suitable software package which are not difficult to find now-a-days. Preparation of input data is also simple (as will be shown later) and can be done mechanically even without understanding the computer program fully However, the interpretations of results and its modifications due to local effects pertinent to a particular type of deck may present some difficulty and will need some understanding of the structural behaviour of the deck 'in question. A general grillage analysis computer program developed for dead load and various types of live loads is discussed in the next chapter. REFERENCES 1.

2.

3. 4.

5.

6.

7.

8.

9.

10.

11.

B. and JAEGER, L.G., "Bridge Analysis Simplified", McGraw Hill, New York, 1985. GERE, M.J. and WEAVER, W., "Matrix Analysis of Framed Struture", D. .Van Nostrand Co., USA, 1965. HAMBLY, E.C., "Bridge Deck Behaviour", Chapman and Hall, London, 1976. JAEGER, L.G. and Bmarr, B., "The Grillage Analogy in Bridge Analysis", Canadian Journal of Civil Engineering, 9(2), 1982. JAEGER, L.G. and BAKHT B., "On the Analysis of Slab Bridges by Grillage Analogy", Proc. Annual Meeting of the Canadian Society for Civil Engineering, Montreal, May 1987. PUCHER. A., "Influence Surfaces of Elastic Plates", Springer Verlag, Wien and New York, 1964. RUSCH. E.H.H. and HERGENROEDER, A., "Influence Surfaces for Moments in Skew Slabs", Werner-Verlag, Dusseldorf, 1969. WEST. R., "Recommendations on the Use of Grillage Analysis for Slab and Pseudo-Slab Bridge Decks". Cement and Concrete Association, London, 1973. WEST. R., "The Use of Grillage Analogy for the Analysis of Slab and Pseudo-Slab Bridge Decks", Research Report 21, Cement and Concrete Association, 1973. IRC 6-1987, "Specifications and Code of Practice for Road Bridges-Loads and Stresses", Section II, Cement Concrete (Plain and Reinforced), Indian Roads Congress. New Delhi - 1990. IRC 21-1987, "Specifications and Code of Practice for Road Bridges". Section III, Cement Concrete (Plain and Reinforced), Indian Roads Congress. New Delhi, 1988. BAKHT,

-- • -

Chapter 6

Computer Program 6.1 INTRODUCTION A Computer Program based on grillage analogy is developed in FORTRAN for the elastic analysis of bridge decks, covering right and skew layouts. The program is compatible with MS-DOS environment in its present form but it can also be compiled to run on mini or main-frame computer. The program is modular and includes different subroutines to execute specific operations. Additional features and modifications can be easily incorporated for a future modular expansion. Sufficient number of explanatory comment statements have been introduced for better and quicker understanding of the program. However, the program is written with the premise that the reader has some prior exposure to computer programming. But if one does not want to go through the drill of understanding the program modules completely, he can still prepare the input data and can analyze the bridge. This Chapter deals with the important features of the program. The Program Manual consisting of listing of variables, sign conventions, main and subroutine segments and flow charts is given. The s Manual comprises data input and output formats. Limitations and scope of the program are also discussed. The listing of the program is given in Appendix II at the end of the book. A diskette with executable file of the program compatible to IBM PC along with input data and result files for all the worked examples illustrated in Chapter 7, is available and can be ordered separately. 6.2 IMPORTANT FEATURES OF THE PROGRAM 'GABS' The program developed can deal with solid, voided and pseudo-slab bridges, T-beam bridges with or without cross-beams, composite bridges, single and multi-cell box-girder bridges and box-beam spine bridges. The bridge may be ed on rigid or flexible bearings. It can handle right bridges and also skew bridges where transversals are either orthogonal or

„

Computer Program 147

skew to the longitudinal girders. Analysis is possible for decks with or without footpaths. The bridges can be analyzed for different classes of Standard IRC live loadings [6] simultaneously. The program can also analyze the bridge for any moving or stationary Specified live loading such as AASHTO [5], OHBD Truck loading [4] etc. The wheel loads moving on the deck are automatically transferred from the s of the grillage to the nodes in the form of equivalent vertical forces and moments. Similarly, the dead load which is given in the form of uniformly distributed load and torsional moment per unit length acting on the longitudinal grid lines, are also transferred to the nodes of the grillage in the form of equivalent vertical forces and moments. The dead and live loads can be dealt with separately and also responses can be obtained when they are combined. The design force responses stored are the maximum values of responses obtained from the analysis of each type of live load after combining it with dead load. Efforts have been made to make the program as general and versatile as possible without making it unduly lengthy and complicated. The program 'GABS' consists of a main program and a number of subroutines. The main program reads the problem details, structural and geometrical parameters, member designations, conditions and loading details. It organizes data and calls different subroutines. The subroutines generate the global axes oriented structure stiffness matrix, set the conditions, decompose the matrix, transfer the loads to the nodes and solve the matrix and give nodal deformations for different load cases after combining with dead load. Member forces and reactions are calculated in the main program itself. TWo versatile subroutines COMPARE 1 and COMPARE 2 are used to produce the output for the maximum live load design responses and these responses are printed through subroutines WRITE 1 and WRITE 2. The main program and subroutines are discussed further in the Program Manual and Manual. 6.3 PROGRAM MANUAL FOR 'GABS' The program manual gives variables used in the program, sign convention adopted, details of main program and flow charts and description of subroutines used. 6.3.1 Variables Some of the variables used in the program for input, output and also those used in more than one subroutines are listed below:

148 Grillage Analogy in Bridge Deck Analysis

AIMP AKERB

Impact factor Right distance of kerb face from the left most longitudinal grid line in transverse direction Skew angle in degrees ANG Array of live load vector (3 x N) B Live load vector due to single wheel BL A character for Yes (Y) or No (N) CH1 CH3 CHOICE A character for STOP (5) or START (0) CORRT A numeral for reading dead load N Overall maximum reaction CORTN Maximum reaction under a live load CORRT Overall minimum reaction CORT Minimum reaction under a live load DD Number of types of groups of elements DLNG Spacing of longitudinal grid lines DLB E Dead load vector of-size (3 x N) and G Young's modulus of elasticity and shear modulus of elasticity of the bridge material El, E2 X-projections of bridge beyond centers of s at the beginning and at the end .FZ Shear at end I of the element PT (i,3) GRIDTYPE A. character representing 'skew' or 'orthogonal' layout of the bridge IK2 A character for dead load case IX Number of ed nodes 1XG, IXL Counters for wheels out of span in longitudinal direction JDD Array of numbers of ed degrees of freedom JRR Type of restraint at JTT Array containing node numbers of ed nodes K(I)Vertical and torsional u.d.l. on longitudinal girders due to dead load L Element length LCASE Load case type LDING A character array containing the names of all types of load cases M Array containing end node numbers of elements M(I,1), M(I,2) Node numbers of element I MM Total number of elements MP I Torsion at end 1 of the element PT (1,4) MN 1 , MN2, MN 1 1 , MN22 Node numbers of the contiguous nodes of the

Computer Program 149

Maximum values of member end deformations Maximum values of member end forces Minimum values of reactions Maximum values of reaction A counter for load placing Moment at end 1 of the element PT (I,5) Moment at end 2 of the element PT (1,6) Total number of nodes Number of longitudinal grid lines • Total number of live load cases desired to be analyzed Number of transverse grid lines Number of wheels, in. the loading Number of wheels per axle identification character (Rectangular '12.' or Triangular 'T') P(I, 1) Length of element in X-direction P(I,2) Length of element in Y-direction P(I,3) Flexural moment of inertia of element P(I,4) Torsional inertia of element POS Wheel position indicative character (YL, YG, XL, XG, etc.) Array containing load on each wheel PW RT Array of reactions S Stiffness matrix Skew angle in radians SANG Span in X-direction• SPAN Minimum distance of the center of left most front wheel SCLMIN from kerb Beating stiffness SUPK T Number of identical elements in a group WW Band width Arrays containing X and Y coordinates of all wheels XC, YC XL, YL X and Y coordinates of center of left most front wheel of loading at start X and Y increments of left most front wheel XINC, YINC XSTOP, YSTOP X and Y coordinates of center of left most front wheel of loading at end Array of cumulative transversal grid line spacings XLLG XLL, YLL X and Y dimensions of XPL, YPL X and Y distances of wheel position with respect to node MNI MAXB MAXPT MAXRT MAXRTN MOVE MQ 1 MQ2 N NLG NOLCASE NTG NW NWPA PATYPE

• •

150 Grillage Analogy in Bridge Deck Analysis

YLLG

Array of cumulative longitudinal grid line spacings (DLNG)

6.3.2 Sign Conventions The coordinate system and the sign conventions adopted in the program are as follows: i) Origin is taken at node on the left most longitudinal grid line as shown in Fig. 6.1(a) and 211 the coordinates are measured from this point. ii) X-axis is along the span length of the bridge and Y-axis is perpendicular to it; the positive directions being as shown in Fig. 6.1. iii) Vertical axis is downward positive and vertical forces acting downward are also taken positive. iv) Rotations about X and Y axes are depicted by the right hand screw rule, the positive directions being as shown in Fig. 6.1(b). v.) The sagging bending moment will be negative at end 1 and positive at end 2 of the element 1-2 according to the above mentioned vector notations. vi) One value of the torsional moment given in the output for a beam element pertains to the value at end 1 of the element. The positive direction is along the member axis as per right hand screw rule. vii) One value of the shear force given by the output corresponds to the shear at end 1 of the beam element. Shear at end 2 will be equal and opposite to shear at end 1. The positive value of shear indicates end 2 up with respect to end 1 and vice-versa. 6.3.3 Main Program and Flow Charts The main program reads all the necessary data required as Input for the analysis and calls various subroutines during the process. The flow charts depict the algorithm followed in the program. The main program does the necessary calculations according to algorithm and prints the input and output values as per given formats. The details of Data Input and Result Output will be dealt with under the Section Manual for 'GABS'. The detailed Micro Flowchart (Fig. 6.3) illustrates the steps involved in the program. Various subroutines of 'GABS' are arranged according to a Macro Flow Chart shown in Fig. 6.2.

Computer Program 151

Left-most Longitudinal Grid line (

2

Mx 'ex

r __ My ,6y

Z (downward)

Z (downward)

(a) Coordinate System (b) Sign Convention Figure 6.1 Coordinate System and Sign Convention

6.3.4 Descriptions of Subroutines Various subroutines are arranged according to the Macro Flowchart given in Fig. 6.2. The functions of each subroutines are briefly described here. 1. Subroutine INODE This subroutine identifies the of the grillage corresponding to end nodes of each member. It stipulates that the node numbers, given as input data, are chosen in such a way that the node number of end 1 of the member is always lesser than the node number of end 2 of the same member.

152 Grillage Analogy in Bridge Deck Analysis Read and Print Input Data. Call "INODE" to Identify Member No. Corresponding to End nodes Call "DEAD LOAD" to Calculate Dead Load Vector Call "STIFF' to Calculate Global Stiffness Matrix [K] If Rigid s

If Flexible s ___ Clan "DECOMP" to Decompose [K]

Call "SORT' and "MODSTIF' to Modify [K]

1 ' Call "IMPACT" to Calculate Impact Factor For Each Live Load and Placing

For Each Wheel

Call "LLOAD" and "WLOAD" to Calculate the Configuration of Vehicular Live Loads Call "PAID!" OR "PAIDRB" to Identify and Wheel Positions in it Call "LDISTT" OR "LDISTR" to get Load Vectors from the Wheel Loads Call "SOLVE" to Analyse for Deformations ,

4

Calculate Member End Forces and Reactions Call "COMPARE!" to Obtain Design Responses for Current Load Type Call "WRITE!" to Print Deformations and Force Responses for the Current Load Type Call "COMPARE2" to Obtain Final Design Responses by Comparing all Load Types Call "WRITE2" to Print Final Design Deformations & Force Responses with Load Types STOP X.4 .L.

1:1317

Computer Program 153

(START) INPUT Grid type, Skew angle, No. of , No. of nodes, No. of types of groups of , E, G, No. of longitudinal grid lines, Spacing of longitudinal grid lines, XProjections of bridge beyond lines, Node nos. of one representative element of the group, Total no. of elements in the group, X and Y lengths of the element, I and J of the element, Nodes nos. of remaining elements of the group.

Identifies member no. corresponding to end nodes.

Call Subroutine "INODE'f

PUT. No. of Is IuN pported nodes, Type of restraint, Node nos. of ed nodes.

154 Grillage Analogy in Bridge Deck Analysis

Bearing Stiffness INPUT

Y-distance of inner edge

/ I N P U T of kerb from left most longitudinal grid line. Call Subroutine "DEAD LOAD" YES INPUT Vertical and Torsional •udl loadings on longitudinal grid lines due to dead loads

INPUT Vertical load at each node due to dead load

Computer Program 155

Assemble global stiffness matrix in half bandwidth form

Call Subroutines "SORT" and "MODSTIFFI

Decomposes the Stiffness matrix into upper & lower triangular matrices

4r--

Sorts out and modifies stiffness matrix by eliminating rows and columns corresponding to restrained degrees of freedom.

Call Subroutine "DECOMP"

/ INPUT "CH3"

YES STOP) NO

156 Grillage Analogy in Bridge Deck Analysis

INPUT TotalTotal no. of live load cases to be analysed DO 1000 Loop for solving all live load cases I N P U T Load case type "LCASE" Dead load Vector is formed

YES

NO INPUT I n i t i a l & f i n a l X a n d c o o r d i n a t e s o f l e f t m o s t f r o w h e e l o f l o a d i n g , I n c r e m e n t s i n a n d Y c o o r d i n a t e

Call Stibroutine "IMPACT"

Place the loading with increments in X and Y coordinates of left front wheel and check for final values.

Calculates impact factor according to type of loading and span.

Y n t X s .

Computer Program 157

Gives no. of wheels; No. of wheels per axle; X and Y coordinates of Wheels for all standard live load —1-Call Subroutine "LLOAD" cases. For 's specified loading, coordinates of wheels and no. of wheels are read in.

Gives the load on each wheel for standard loadings, For 's specified loading min. kerb distance of centre of left most front wheel and load on each wheel are read in.

1Call Subroutine "PAIDRB" A For skew bridge, checks if wheel is outside span or nearer to kerb than allowable limits; Identifies the containing type, its contiguous nodes, dimensions and load positions of the wheel.

For right bridge, checks if wheel is out of span or out of allowable transverse eccentricity; Identifies the containing wheel load by its contiguous nodes, calculates the wheel position on the and the dimensions.

158 Grillage Analogy in Bridge Deck Analysis

r YES :han _____________________________________________

NO Triangular

Rectangular

Using fixed beam analogy, these subroutines calculate the nodal load vector from the wheel load placement on the .

0

4

Next w h e e l t ___ NO

Are all wheel loads analysed and load vectors added

Enhances the final live load vector by impact factor and add dead load vector

Modifies load vector by eliminating values corresponding to restrained degrees of freedom Call Subroutine —)10SOLVE'

Solves for deformations using front and back substitution method on the

decomposed stiffness matrix and load vector

Computer Program 159

Spi Inflates the deformation vector for rigid bearing case with zeros at restrained degrees of freedom.

4, Calculates member end forces using member stiffness matrices and deformation vectors.

4 Calculates reactions by using equilibrium conditions at the ed nodes.

Call Subroutine "COMPARE]."

NO

Are all load positions for urrent load type exhausted YES INPUT IK2

Compares and retains maximum values of member E' end forces, nodal deformations and reactions for the current load type with load position for each maximum value.

41

 160 Grillage Analogy in Bridge Deck Analysis

NO

Is IK2=I

YES

1'

Prints the maximum values of member end forces; deformations, reactions etc. for the current load type.

Call Subroutine "WRITEl"

Next load case

Compares and retains maximum responses after comparison amongst maximum response values for different load types.

NO

Are all load exhaus

Call subroutine "COMPARE2"

Call Sithroutine

Prints the maximum design values of member forces, deformations, reactions due to all types of loads with corresponding. load type and load positions.

Figure 6.3 Micro Flow Chart

L



• 17.. • 7,

,

Computer Program • 161

- 6

LI

2. Subroutine DEAD LOAD This subroutine reads vertical load per unit length and torsional moment per unit length on the longitudinal grid lines due to dead load and transfers this loading to the nodes treating the- grid lines as fixed; alternately,. the subroutine can also take dead load in the form of nodal vertical loads: Loading in-either form is to be calculated manually and then given as input data. Dead load is designated as load type 1, i.e LDING( 1 ). 3. Subroutine STIFF The subroutine assembles the stiffness matrix of the grillage in half band-ividth form using the element stiffness matrix previously calculated. 4. Subroutines SORT and MODSTIF These subroutines sort and modify the stiffness matrix by eliminating rows and columns corresponding to restrained degrees of freedom. If the is flexible, this subroutine adds the bearing stiffness to the appropriate stiffness, matrix . -. 5. : Subroutine DECOMP This subroutine decomposes the stiffness matrix into upper and lower triangular matrices, using Choleskey's [I] Factorisation method. 6. Subroutine IMPACT The subroutine calculates impact factor according to the type of live loading and spans as per provisions of relevant IRC Code of Practices [6]. 7. Subroutine LLOAD For the type of live load specified, this subroutine identifies the total number of wheels and number of wheels per axle. Using the coordinates of left front wheel (given as input), it also calculates the coordinates of all other wheels. Following eight types of live load cases (LDING) are built-in into this subroutine: LDING(2) = Class A Two Lane LDING(3) = Class 701i. Train: Column '1' Wheel Configuration LDING(4) = Class 7CR Train: Column `rd Wheel Configuration LDING(5) = Class 70R Bogie: Column T Wheel Configuration LDING(6) = Class 70R Bogie: Column 'm' Wheel Configuration LDING(7) = Class 70R Track LDING(8) = Class A Single Lane LDING(9) = Specified by

. 1 1 1 1 1 1 . m w

.

•

•

- - s - - - - . : . . . _ , •

-

•

162 . Grillage Analogy in Bridge Deck Analysis

In load case 9, the * specifies the details of loading, namely, the number of wheels and their coordinates. The wheels are numbered beginning from left most wheel of leading axle, successive wheels on the same axle in the positive direction of Y, then the wheels on next following axle and so on. 8. Subroutine WLOAD • This subroUtine describes magnitudes of load on all wheels for each type of live loading. The minimum alliArable distance of center of left most wheel from Kerb is set in. All the wheels are taken as point loads acting at the centers of wheel - area. Class 70R Track loading is described as ten point loads per track, the load on each point being 3.5 tonnes at spacing of 457 mm as shown in Fig. 1.13. For 's Specified loading, the' oad on each wheel inclusive of impact and minimum kerb distance from center of left most front wheel are given as input. 9 . Subroutines PAIDRB and PAID1 When grid type is skew and the transversals are parallel to the s forming parallelogram s or when the grid is right i.e. skew angle is zero, the subroutine PAIDRB is called. On the other hand, when the skew angle is not zero but the transversals are orthogonal to the longitudinals, the program subroutine PAID1 is called. These subroutines use the spacing between longitudinal grid lines and the lengths of elements taken on longitudinals, to calculate the boundaries of the s with respect to node 1 (specified earlier) and then compare the coordinates of wheel load with these boundaries to identify the s containing the wheels. These subroutines are further discussed below. a) Subroutine PAIDRB For each wheel, the Y-coordinate, measured from the origin i.e. from the left most longitudinal grid line, is checked. If the Y-coordinate is less than minimum permissible distance, defined by the distance (Akerb+SCLMIN), the subroutine terminates with position identification character as YL. Also, the subroutine terminates if the Y-coordinate exceeds Y mar (Table 5.1) with position identification character YG. Similarly, if the X-coordinate is such that the wheel is not on the span, the subroutine terminates with position identification character as XL or XG as the case be. The is now identified as 'P.' i.e. rectangular or parallelogram. By using the coordinates of wheels, the containing the wheel is identified by its contiguous nodes MNI,

Computer Program 163

MN2, MN1 1, MN22 as"•shown in Fig. 6.4(a, b). The dimensions of the and the position of the wheel on the are calculated and stored. . b) Subroutine PAID I The checks for X and Y coordinates of wheel are as in the case of the subroutine PAIDRB. If the wheel is on the span with its Y coordinate within allowable limits, then using the coordinates, the containing the wheel is identified as rectangular or parallelogram (`R') or Triangular (T), the triangle being always a right angled one. The contiguous nodes are stored as MN1, MN2, MN11 and MN22 for rectangular and parallelogram s and as MN1, MN2, MN22 for triangulir (Fig. 6.4). MN22

 y

MN11 MN1

MN11 MN1

(a) Rectangular Ranel (b)Parattetogram

MN2

MN22

MN1

MN1.

MN 22

MN2

(c) Lower Triangular (d) Upper Triangular Figure 6.4 Identification in Subroutines `PAIDRB' and `PAIDIP

164 Grillage Analogy in Bridge Deck Analysis

The dimensions of the and the position of the wheel on the with respect to MN1 are calculated and stored. Distinction between lower and upper triangular s as shown in Fig. 6.4(c, d) are also made using position identification characters as 'XG' or 'XL'. If the load is out of the permissible range in Y-direction (nearer to Kerbs than permissible), the program stops. If a particular wheel - load is longitudinally out of span then it is ignored. 10.: Subroutines LDISTR and LDISTT These subroutines distribute the load from the onto the nodes in the form of vertical loads and moments in X and Y directions. The subroutine LDISTR is used for distributing the load to the nodes of rectangular as well as parallelogram s while the. subroutine LDISTT distributes the load to the nodes of triangular s. The final expressions for nodal load vectors for rectangular and parallelogram s are given by equations 5.10 to 5.13 and 5.23 to 5.26 respectively and for triangular s are given by equations 5.16 to 5.21. These have been used in the program with appropriate signs. 11. Subroutine SOLVE This subroutine uses the previously decomposed stiffness matrix and the load vector to solve the set of simultaneous equations by backward and forward substitution procedure and gives the deformations corresponding to three degrees of freedom. If the type is rigid, the deformation vector is 're-inflated' by zeroes at the nodes. This subroutine is called directly if the s are flexible. In case of rigid s, the load vector is first reduced in size by eliminating elements corresponding to degrees of restraints at ed nodes and then the subroutine SOLVE is called. 12. Subroutines COMPARE1 and COMPARE2 These subroutines are used to produce the output for the maximum live load design responses. a) Subroutine COMPAREI The deflections, slopes, moments, shears and torsions at the nodes and the reactions at the nodes are compared for all placings of the same live loading and the critical values of these force responses are retained along with the corresponding load positions. Along with a critical response, wherever relevant, the corresponding values of other related responses, are also retained. For example, along with maximum shear force corresponding torsional moment, along with maximum torsional moment corresponding shear forcc,

Computer Program 165

minimum reaction corresponding X and Y rotations are also stored. All the live loading types are exhausted in a similar manner one by one. b) Subroutine COMPARE2 The critical responses under each type of live loading are compared amongst themselves and the most critical amongst these response values are also stored along with type of loading and load position. 13. Subroutine WRITE1 This subroutine is called to print the maximum deformations and force responses for each load case at various nodes or for various identified by COMPARE1. 14. Subroutine WRITE2 This subroutine is called to pint the final envelope values of deformations and force responses at the nodes or for the as the case be, along with type of live loading and its position on the deck. This subroutine is also called to print the dead load force responses. 6.4 MANUAL FOR 'GABS' The manual of the program comprises of the details of Data Input and Result Output. The final output obtained helps in preparing Envelope diagrams for different deformations and force responses required in the design of the bridge. 6.4.1 Data Input Module

..

The input data -is in tonne-mm unit in the program but. can be in any set of compatible units. The input details are in Free Format. The input data consists of the details of the structure e.g. skew angle, grillage layout, member designations, properties of grillage , conditions etc.; dead load values on each longitudinal grid lines; live load cases and controls and details regarding the movement of vehicular live load on the deck, etc. The description of READ STATEMENTS (data input) is given below in the sequence in which they are needed in the program. 1. Reads a character CH1 which initiates the program. Input 'Y' starts the program while 'N' stops it. 2. Reads type of grid chosen (GRIDTYPE). Grid type can either be `SKEW' (any value other than 90) or 'ORTH' (90). The 'SKEW' grid is designated as where the transversal are not normal to the longitudinal but are parallel to the s. An 'ORTH' i.e. orthogonal (90) grid has its transversal normal to the longitudinal.

168 Grillage Analogy in Bridge Deck Analysis

a check. This is followed by the analysis results. The analysis results consists of force responses and deformation responses. The following is the sequence in which the output result is obtained: I. 2. 3.

Grid details as per input data after suitably titling these. Dead load details as per input data. A list of different types of live loads for which the analysis can be. done and reference numbers of these live loads. 4. Responses under dead load as follows: (a) Shear forces and torsional moments in each beam element. (b) Sagging bending moment at each node on its right and on its left. (c) Hogging bending moment at each node on its right and on its left. (d) Rotations about X and Y axes and vertical deflections at each node. (e) Reactions, rotations about X and Y axes and vertical deflections at each . Deflections at s will occur when grid is ed on yielding s. 5. Responses under individual live load together with impact load and dead load as follows: (a) Span for impact factor calculation and impact factor taken for the live load. (b) Type of live load through its reference number, initial and final positions of reference wheel through its X and Y coordinates and step lengths in X direction and Y direction respectively. Reference wheel is the left most wheel on leading axle of the loading system. (c) Shear force and torsional moment for each beam element (i) maximum shear force, corresponding torsional moment and position of reference wheel; (ii) maximum torsional moment, corresponding shear force and position of reference wheel; (d) Maximum sagging bending moment at each node: (i) at right of the node and corresponding position of reference wheel; (ii) at left of the node and corresponding position of reference wheel; (e) Same as at (d) above but for maximum hogging bending moment. (f) Maximum rotations about X and Y axes and maximum deflection at each node. rom.ar.tinri :et earth cttrwsnrt with Correspondingdeflection, rotations and position of reference wheel.

Computer Program - 169

(h) Same as at (g) above but for minimum reaction. (1) Maximum X-rotation at each with corresponding reaction, deflection, Y-rotation and position of reference wheel. (j) Same as at (i) above but for maximum Y-rotation. 6. Envelope values based on the above live load cases together with dead load and corresponding impact loading. Overall envelope valueS of each of the above responses together with the corresponding live load and location of reference wheel. Thus complete information, regarding the type of load system and the location of wheels on the deck giving the critical or design values.are known to the designer. 6.5 LIMITATIONS AND SCOPE Efforts have been made to keep the program as simple and general as possible without Inaldnpit unduly lengthy and complicated but there is a wide scope for the improvement, additions and modifications. Changes are possible in regard to loading standards, movement of vehicular loads on the deck, output and data generation, etc. The automatic generation of grillage mesh. and the evaluation of elastic properties of grid could also have been in-built in the program as a subroutine instead of doing the same manually as at present. But this has been done with the motive of educating the designer and allow him more flexibility in choosing grid lines depending upon the situations. The program has been primarily developed for Indian Standard Loadings specified by IRC. However, any type of loading can easily be incorporated in the program by modifying the relevant subroutines IMPACT, WLOAD and LLOAD. A minimal change in the main segment of the program will be needed in this process. The program can handle bridges having rigid and yielding simple.s. This can be extended to continuous s also. Curved bridges have not been included in the program but the method of grillage analogy is capable of handling such bridges with a reasonable accuracy and a separate computer program can be written for the same. The analysis of curved bridges, if incorporated in the program 'GABS' will make it too lengthy and cumbersome. REFERENCES 1. GERE. M.J. and WEAveR, W., "Matrix Analysis of Framed Structure', D. Van Nostrand Co., USA, 1965.

170 Grillage Analogy in Bridge Deck Analysis 2. NAYAR, K.K., RAGHUPAIHI, M., SEETHARAMULU, K. and &RANA, C.S., "Computer Aided Bridge Analysis—A Software Based on Grillage Analogy", International Conference on Bridges and Flyovers, Hyderabad, February 1991. 3. SHARMA, K.G., &R ANA, C.S. and AGRAWAL, A.K., "Automated Analysis of T-Beam Bridges Using Grid Method", Proceedings of Second International Conference on Computer Aided Analysis and Design in Civil Engineering, U.O.R., Roorkee, January 1985. 4. "Ontario Highway Bridge Design Code", Ministry of Transportation and Communications, Highway Engineering Design, Downsview, Ontario, Canada, 1983. 5. "Standard Specifications for Highway Bridges", American Association of State Highway and Transportatiion Officials (AASHTO), Washington, D.C., 1977. 6. IRC 6-1987, "Standard Specifications and Code of Practice for Road Bridges Loads and Stresses", Section II, The Indian Roads Congress, New Delhi, 1990.

Chapter 7

Illustrative Examples 7.1 INTRODUCTION This Chapter aims at illustrating the use of the grillage analogy method discussed in earlier chapters through worked examples. A set of six examples have been chosen covering different types oDbridge superstructures. Converting an actual bridge deck to an equivalent grid is not without pitfalls and needs special attention. This conversion involves setting of longitudinal and transverse grid lines and asg flexural and torsional inertia values to various grid . This has been done for all the bridge examples presented in this Chapter along with brief discussions. It is expected that these examples and the detailed recommendations given in Chapter 4 together, will enable a prospective of the grillage method to convert a bridge superstructure to an equivalent grid with requisite accuracy and confidence. The computer program 'GABS' discussed in Chapter 6 is a versatile program. These examples are used to explain the use of the aboVe program when the loading is IRC loading or a -specified loading or a stationary loading. A of the program at some stage would like to know the sensitivity of a solution to the size of the grid mesh, i.e. how will the response result alter, if instead of a normal sized mesh, as recommended in Chapter 4, a finer or coarser sized mesh is adopted. Example 1 of a right slab bridge is devoted to investigate this. End-reactions are non-uniform for a skew bridge even under a uniformly distributed loading on the deck, such as 'self load'. Example 2 is of a skew slab bridge and is chosen to illustrate this variation, when the slab is ed on bearings either rigid or flexible. Computation of Inertia for a voided slab is not straight forward. Example 3 of a voided slab bridge illustrates this computatiori, as well as the handling of 'non-uniform dead loading' along the longitudinal grid

172 Grillage Analogy in Bridge Deck Analysis

In Example 4, a T-beam bridge has been analysed for dead load as .well as live load specified by the Ontario Highway Bridge Design (OHBD) Code of Canada. In example 5, a skew T-beam bridge is analysed for different IRC loadings and the envelope diagrams. for B.M and S.F are presented. A box-girder bridge develops torsional and distorsional Warping under loading. The bending stresses are also affected by shear lag. All these effects are ignored in the program "GAB" presented in this book. In example 6, a box-girder bridge has been analysed by 'GABS' as well as by Finite Element Method and the results are compared. 72 ILLUSTRATIVE' EXAMPLES Examples 1, 2 and 3 pertain to a right slab, skew slab and a voided slab bridge deck respectively. Analysis,of slab-on-girders bridges both right and skew have been illustrated in examples 4 and 5 whereas example 6 discusses a box-girder bridge. • All the bridges are simply ed and have 7500 mm carriageway width. The bridges •are of reinforced concrete. The Young's modulus of elasticity and Shear modulus of elasticity are taken as 2.0 and 0.87 tlmm2, respectively. Wherever dead loads are considered, density of reinforced concrete including density of wearing coat is taken as 2.40 dm'. In the input data, the loading type is designated by a number which refers to the type as given. in Section 5.5.3 of Chapter 5. COmplete Input and Output files of all the six examples of.this Chapter and one example of Chapter 3, with the two programs 'GABS' and 'GRID!, (in a ready to use form) is available on a diskette. The diskette can be ordered from the authors through the publisher. 7.2.1 Example 1: Right Slab Bridge A two-lane right slab bridge (Fig. 7.1) is chosen for this example. The equivalant grid is shown in Fig. 7.2 and is referred to as normal mesh in further discussions. It consists of seven longitudinal and seven transverse grid lines. Recommendations given in Chapter 4 are followed in arriving at the grid pattern. The bridge is analysed for four different types of IRC live loadings alongwith corresponding impact factors. The IRC live loadings chosen are (i) Class A-2 lane loading, (ii) Class 70R Train Col. loading, (iii) Class 70R Bogie Col. '1' loading and (iv) Class 70R Track loading. These loadings are moved on the bridge one at a time in suitably chosen intervaIs—both longitudinally as well as transversely so that the load traverses the entire length and width of the deck. For this example, an

o b .

Illustrative Examples 173

P

—.4600

Wearing Coat 0

470lieroo

7500

L

300

9000

300

Figure 7.1 Right Slab Bridge interval of 500 mm has been chosen for longitudinal movement of all types of loadings. In transverse direction, the intervals are so chosen that the load traverses the full deck width in 5 or 6 steps. The input data is given in Table 7.1 and selective output is given in Table 7.2. It may be noted that maximum longitudinal bending moment is obtained under Class 70R Track loading. The maximum bearing reaction'per unit width for exterior grid lines is under ClassA.-2 Jane loading and for utidr.-.1 701-: Train Col. T loading. This shows that for

te

r

174 Grillage Analogy in Bridge Deck Analysis



7

_0

000

700 7500

H

60 1350 =8100

300 42 7

49

35N,

fi

13

20

27

34

41 -

he

5

12

19

26

33

40

11

18

25

32

39

47 0 o co %0 IL 46 ti+

3

10

17

24

31

38

45

2

9

16

23

30

37

44

1

8 a

15

22

29

36

43

a

Inner Transversal

1-.--1500 -1 T

Inner Longitudinal 1-

I = 4.29 E 10

3.86 700 _t_

.1: 8.58E10 ___________________ j ±700

E101J=7.72 E10

I— 1350 —'-I Outer Transversal N-10504

700

T

I = 3.00 E10 750'1

.1: 6.00 E10

300 's Indicate Locations of arid lines

Outer Longitud

T 1000

inal '1 600 F.—

I

T

6.48E10

700 3,12.97E10

L-4- 6751 300

Figure 7.2 Normal Grid Mesh and Sections of Various Grid

Illustrative Examples 175 f

a particular bridge any one particular type of IRC live loading does not give all the critical responses. TABLE 7.1: Input Data—Right Slab Bridge—Normal Mesh Size

300 300 1 2 12 2 3 3 43 44 44 8 9 3 0 9 10 10, 15 16 16 22 23 23 29 30 30 36 37 37 1 8 1 2 8 15 15 7 14 14 2 9 30 9 16 16 3 10 10 4 11 11 5 12 12 6 13 13 14 3 1 300. 1 0 0 0 1 1 0 0 4 2

1500 4 45

0 4 45

6.48E+10 -12.97E+10 5 5 6 6 46 46 47 47

7 48

48

49

1500 11 17 24 31 38

0 II 17 24 31 38

3.86E+10 12 12 18 .18 25 25 32 32 39 39

14 20 27 34 41

20 27 34 41

21 28 35 42

0 22 21

1350 22 21

29 28

3_0E+10 6.0E+10 29 36 36 43 28 35 35 42

42

49

0 23 17 18 19 20

1350 23 17 18 19 20

30 24 25 26 27

4.29E+10 8.58E+10 30 37 37 44 31 31 24 38 25 32 32 39 33 33 40 26 41 27 34 34

38 39 40 41

45 46 47 48

8

7

0

21 22 28

14 15

0

0

7.72E+10 13 13 19 19 26 26 33 33 40 40

0

0 0

29

0

35 36 42 43 49

0

0

0

0

Y .

Y

90 0 84 49

4; 2.0 0.87

7 1350 1350 1350 1350 1350 1350

0 700. 500. 280. 27800. 2100. 1 0

0 .1705. 500. 385. 22400. 4015. Contd.

.

.`

176 Grillage Analogy in Bridge Deck Analysis 0 5 0 1 0 7 0 1 0 1 0

1705. 500. 385. 10220. 4015.

1920. 500. 440. 13113. 4120.

S

This example is further used to study the effect of size of the mesh formed by grid lines on various force responses. Figure 7.3 shows a coarser mesh for the same bridge where the number of longitudinal grid lines have been reduced from 7 to 5 but the number of transverse grid lines have been kept the same. Figure 7.4 shows a finer mesh for the same bridge where the number of transverse grid lines have been increased from 7 to 11 but the number of longitudinal grid lines are kept same as in normal sized mesh of Fig. 7.2. Both these grids are analysed for the same four types of IRC live loadings as above, keeping the longitudinal and transverse intervals for various IRC loadings same as in the analysis of grid of Fig. 7.2. The input TABLE 7.2: Effect of Mesh Size on Force Responses (a) Maximum longitudinal bending moments and corresponding bending compressive stresses Reference Grid

Normal Coarse Fine

Load Type 7 7 7

Exterior Grid line Central Grid Line B.M. Comp Mean a Load B.M. Comp. Mean a (t m) Sress a (N/mm2) Type (t m) Stress a (N/mm2) (N/mm2) (N/mm2) 29.3 34.1 29.0

2.47 233 2.44

2.48

7 7 7

19.3 28.9 18.8

1.75 1.75 1.70

1.73

(b) Maximum bearing reactions and corresponding bearing reactions per unit width Reference Grid

Normal Coarse Fine

Exterior Grid Line Central Grid Line Load Reaction R per unit Mean Load Reaction R per unit Mean Type R width (r) r Type R width (r) r (t) (t/m) (t/m) (t) (t/m) (t/m) 2 2 2

7.23 9.59 7.32

7.42 7.30 7.51

7.41

3 3 3

17.50 26.02 17.38

13.01 12.tb • i2.91 12.87

Illustrative Examples 177

—4.6001.r1000

700 $

 •

4 @ 2025 — 8100 1

.,,,7

13

6

s' v

Lc, V

20.

27 .

12 :

5

26

 18

11

4

19

J

aS

-r-

34

33

" 25

32

0 0 0 171

0

3

17

"24

9

16

23

30

8

15

22

29

2

1

A

 31 eU 0

10

A

A

A

Inner Longitudinal 1=5.79 E10

P

Outer Longitudina L :FL ___ 1700

2025 --a`Lt.

700 i o _t

00

1'

-i1=7.65 E10 -7( Nate: Inner '4113115 FT J :15.30 E 10 Same as in —04 k- 300 Figure 7.3 CoarseTransversal Grid Mesh and Sections Various Gridare and Outer of Transversal Fig. 1 =11.56 E10

7.2 )

178 Grillage Analogy in Bridge Deck Analysis ^-*1 600h— ........

.

700

• 6e 1 35 0 = 81 0 0 22 10

21

9

20 19

33 32

44 a 55 v

66a 77

43

54

65 .

76

42

53

64

75

30

41

52

63

74

31

7

18

29

40

51

62

73

6

17

28

39

50

61

72

5

16

27

38

49

60

71

15

26

37

48

59

70

3

14

25

36

47

58

69

2.

13

24

35

46

57

68

1

12

23

34

45

56

a

70 X 900..9000

11 v

67

Inner Transversal

Outer Transversal

1 = 2. 5 7 E 10 —. 4 7 50 1. - -

T 1=214E10

700 Jf. 4 29E10

70 0 J = 5 1 5 E 1 0 900 k

-

--pl.

3001.r-( Note: Inner longitudinals and Outer Longitudinals are Same as in Fig 7.2 / Figure 7.4 Fine Grid Mesh and Sections of Various Grid

_______ 4

1151.111.11."m°--

l• - • ..yilt7,- •

Illustrative Examples 179 data have not been given here for brevity. The governing loadings for maximum longitudinal bending stresses and bearing reactions were found to be the same as for normal sized mesh as shown in Table 7.2. The maximum bending compressive stress remained within 2% of its mean value and the maximum bearing reactions per unit width remained within 1.5% of its mean value. This' shows that some variation in fineness or coarseness in mesh pattern can be adopted, if desired,, without affecting the accuracy in any significant manner. 7.2.2 Example 2: Skew Slab Bridge A skew slab bridge having 300 skew angle ed on five isolated bearings at each end is analysed in this example. The example refers to an actual bridge constructed on NH 8 at KM 247/40 near Ajmer, Rajasthan. The bridge was analysed and designed by ,the first author. The cross-section of the bridge, the. positions of the bearings and grid chosen for the analysis are shown in -Fig. 7.5. Preliminary analysis lead to the adoption of neoprene bearings of spring stiffness of 40 t/mm at each point. The flexibility of the is considered in the analysis. 245

10490

7 0 9 24 5 (a) PLan at mid-thickness of Slab and EquIvaLent grid

1775 'a

7500

*linqi

275

.„,

:,, ,rw.c. 75(k.a) \I 1?50 610 --4.1 r• —1-1, rwt 4x2395= 9580 535 /mot

•••••.! •••:.1.1•!t•

—1 610 k-535 • •

octton

Figure 7.5 Skew Slab Bridge ed on Isolated Bearings

180 Grillage Analogy in Bridge Deck Analysis Table 7.3 gives sectional'widths adopted in computing I and J values for grid of various groups. Dead load is assumed to be acting along longitudinal grid lines. The dead load along the two exterior longitudinal grid lines inclusive of weight of railing @ 0.15 Um is found to be 4.0 t/rn each. The dead load along interior longitudinal grid lines is taken as 4.74 Tim each. The types of liVe load considered on the carriageway are the sarn4 as in example I, namely, Class A-2 lane loading, Class 70R Train Col. '1' loading, Class 70R Bogie Col. '1' loading and Class 70R Track loading. Table 7.4 presents grid input data assuming flexible bearings at s. The bridge is also alternately analysed for rigid bearings for the sake of comparison. For brevity, only bearing reactions are presented and discussed. Bearing reactions under dead load are given in Fig. 7.6. Also the reactions are tabulated under dead load and under dead and live load combine in Table 7.5. The curve of Fig..-7.6 of bearing reactions under dead load for rigid bearings gives the shape of a saddle. The reaction is very large at the obtuse angled corner. This is followed by a smaller reaction at the second bearing. The curve then rises again and finally diminishes at the acute angled corner. The curve corresponding to flexible bearings has more even distribution of reactions. The same trend is followed by bearing reactions under dead and live load combine. The minimum bearing reactions under dead and live load combine, at times, is smaller than its value under dead load alone. This indicates that for certain positions of live load on the deck, the bearing

50 —. 40 oc

Q.. Rigid Bearings

30

Flexible

20 cc 10

0

9

18

27

36

45

Bearing Node Number

Figure 7.6 Bearing Reactions Under Dead Load

MOM

Illustrative Examples 181 reaction due to live load alone is negative. However, special attention is required to be paid if the minimum bearing reaction under dead and live load combine becomes negative. This will indicate an uplift at the bearing. TABLE 7.3: Group 1&2 3&4 5 6 7

.

8

Widths Adopted in Computing I and J Values for. Grid of Various Groups Width '6' Typical Remarks (mm) 1-2, 5-6 I0-H, 13-14 1-10 6-14 2-10

1807 2395 245 1645 1383

5-13

1514

Along exterior longitudinal grid lines Along interior longitudinal grid lines Along skew transverse lines Along interior transverse lines Along exterior transverse lines, not covering full grid width Along exterior transverse lines covering full grid width

Notes: 1. Constant thickness of 750 mm is assumed for all . T hus flexural mo me nt of inertial I for a me mber of width ` If mm is (750)3 b/12 nun'. 2. Torsional Inertia J of the me mber is taken as t wice its flexural mo ment of Inertia, I. TABLE 7.4: Input Data-Skew Slab Bridge on Flexible Bearings Y

90 30 80 285 1 2

4

45 8 285 2 8 3 3

2:0 .87 5 2395.

2395. 2395. 2395.

4382.5 0. 6.35E+10 12.71E+10 4 4 5 41 42 42 43 43 44

8 1645. 0. 7 8 8

44 45

5 6

6 7

6.35E+10 12.71E+10 9 37 38 38 39 39 40 40

10 II 33

11 12 1382.5 0. 8.42E+10 16.84E+10 12 12 13 17 18 19 20 20 21 25 26 26 34 34 35 35 36

27 28 29

13 14 29

14 12 1645. 0. 8.42E+10 16.84E+10 15 15 16 16 17 21 22 22 23 23 24 24 30 30 31 31 32 32 33

25

1 9

10 8 1382.5 2395. 0.86E+10 1.72E+10 18 10 19 18 27 19 28 27 36 28 37 36

6 7 31

14 12 0. 2395. 5.78E+10 11.57E+10 15 8 16 14 22 15 23 16 24 22 30 23 39 32 40

41

45 31

24

1/

30

JV

Contd.

182 Grillage Analogy in Bridge Deck Analysis 2 10 12 0. 2395. 4.86E+10 9.72E+10 20 18 26 20 3 11 4 12 11 19 12 35 43 36 44 5 13 8 0. 2395. 5.32E+10 10.64E+10 29 25 33 29 9 17 13 21 17 25 21 10 1 1 9 10 18 19 - 27 28 36 37 40 1040. 1 4.74E-3 4.00E-3 .967 4.74E-3 0 4.74E-3 0 0 1 1 1 0 4 2  280 36500 2840 500 0 1440 1 0 3 385 31000 4755 500 0 2445 1 0 5 385 19000 4755 500 0 2445 1. 0 7 440. 22000 4860 500 0 2660 1

28 26 34 27

35 34 42

37 33 41 45

0 4.00E-3 —.967

L

0 S

TABLE 7.5: Bearing Reactions Under Dead Load and Dead & Live Loads Combine Bearing Node

9 or 37 18 or 28 27 or 19 36 or 10 45 or 1

Dead Load Alone Reaction (t) Flexible Rigid Bearing Bearing 44 31 28 24

13

51 23 29 27 11

Dead and Live Loads Combine Min. Reaction (t) Max. Reaction (t) Flexible Rigid Flexible Rigid Bearing Bearing Bearing Bearing 44 31 28 24 11

49 22 29 27 9

68 55 52 42 20

74 55 61 47 16

Illustrative Examples

183

7.2.3 Example 3: Voided Slab Bridge A voided slab bridge is chosen for this example. The voids are running through the central part of the span. In regions where shear is large, solid section has been provided. Figure 7.7 shows the bridge details and the proposed grid arrangement is shown in Fig. 7.8. The example is chosen to explain (a) how inertia values are to be computed for such voided sections and (b) how non-uniform dead load along the longitudinal grid line is to be dealt with. The inertia values are computed on the principle that slab has same Liu8 , 8 est 4 1••••••

•Mis.mmiM

§ 14441 § §. I

l

- , ._,

!---. 7 '

in Fin r :

ri I il I II I III 1ii

11 I I I

1 1 I 1 iI I1 1 I I II : I I I 1 I II

1 I

I

1I

iI1

I II :I I

I I

, , _11.11i iilriii Hi!!I 1 11 1;111 i I II Ii ' 1 I 1 1 II I i. I II I Ill!! 1 I 1 1 LJLJLJULJLJUII 1

I

I

C

r

•

I

Ig 1 ec. r

I I

§ N.

0 -•

Q

1 8 N, it

I I I .. co o ' 0

90 7500

9C11,_IS‘uf:port _

F()11(3 01 0 010 0 0 H- 7 x 960 u 6720 —.4 1-.

480 

• • 90 1830 1920

480 • • 1920 1830 90

Figure 7.7 Voided Slab Bridge

0

184 7

Grillage Analogy in Bridge Deck Analysis

v5

V

14

13

5

12

21

20

19

28

6

3 0 0 - 2 0

27

26

•

0 N 11

0

N

(0

4

1

. M . 1 1 1 1 . •

11

18

25

10

17

24

9

16

23

8 15 A 2 9 i+-1830-441— 2 x 19 2 0 = 3840

3

.

1

30

22 1830-44

Figure 7.8 Grid Relating to Voided Slab Bridge

inertia per unit width in the two mutually perpendicular directions. This is true only if void depth is upto 60 per cent of the overall depth. For voided section, the moment of inertia per unit width of the slab is computed from the section showing the void/voids and the same is assumed per unit width in the orthogonal direction. Torsional inertia .1 of the section is computed from moment of inertia 1 adopting the formula .1 = 21 for all slab sections. Table 7.6 gives the sections and their I values. Moment of Inertia values of belonging to groups 3 to 10 have been calculated on the basis of I per unit width (i) of the of groups I and 2. The dead load alone the longitudinal grid line is non-uniform as in the end regions the section consists of solid slab and in the central region the section consists of voided slab. Such non-uniform loading is dealt in by



1..„-:-,,''''''"Ff7e•-.72.14.-,1=.0M-•;.1';'.•:'•-•,•

i,---7411111111°.11c12— Illustrative Examples 185 TABLE 7.6: Sections and Moment of Inertias Section (D = IOW)

x 10 - 4

--I 160 800 '(i = 0.8333) --04 960

p

736 (i 17...7671)

.8333 x 1920 = 1600 1 9 2 0

.7671 x 1920 = 1473 1600 kg 6001 1400

.8333 x 1400 = 1167

I.- 2000 66001

_L

0

(.8333 + .7671) 1000 = 1600

1.10001.-10001

2000 -.4 6110

Typical member

Group Member Location •

-•

1

Exterior long.  grid line

2

—do--

2-3

1-2

.7671 x 2000 = 1534

3

Interior long. grid line

4

8-9

9-10

5,6

transverse grid line

1-8 8-15

7,8

First Interior transverse grid line

2-9 9-16

Interior transverse grid lines

3-10 10-17

9,10

co mp uting gravity lo ad s at the vario us nodes based o n the load o f the tributary area or contributary area of each node. Table 7.7 gives details for e. T— •L ,y;ven in the same sequence in which node numbering is done. r•-trin +1.•Itn

AIM

186 Grillage Analogy in Bridge Deck Analysis TABLE 7.7: Computation of Nodal Loads Section*

Typical Node

Area (m2)

1

Length (m)

As for member 1-2 Mean.of 1-2 and 2-3 As for member 2-3 As for member 8-9 Mean of 8-9- and 9-10 As for member 9-10

2 3 8 9 10

Load

0.960

1.40

3.23

0.824

2.0

3.93

0.677

2.0

3.25

1.92

1.4

6.45

1.648

2.0

7.86

1.354

2.0

6.50

1

* Refer Table 7.6 for Sections. Table 7.8 gives the input data required for dead load analysis. Data relating to live load analysis is not given for brevity though it can be given in the same manner as for earlier examples. TABLE. 7.8:

Input Data-Voided Slab Bridge

Y

90 0 58 35 10 2.0 .87 5 1830. 1920. 1920. 1830. 400. 400. 1 6

2 7

2 3

3 4

8 9 13 14

4 2000. 0. 8.0+E10 16.0+E10 29 30 34 35 8 4 5

2000. 0. 7.36E+10 14.72E+10 5 6 30 31 31 32 32 33 33 34

6 2000. 0. 16.E+10 32.E+10 15 16 20 21 22 23 27 28

9 10 12 2000. 0. 14.73E+10 29.46E+10 10 11 11 12 12 13 16 17 17 18 IS 19 19 20 23 24 24 25 25 26 26 27 1 7

8 14

4 0. 1830. 11.67E+10 22 29 28 35

23.33E+10

8 15 14 21

4 0. 1920. 11.67E+10 15 22 21 28

23.33E+10

9

4 0. 1830.

16.E+10

32.E+10 Contd.

Illustrative Examples 6

187

13 23 30 27 34

9 16 4 0. 1920. 16.E+10 32.E+10 13 20 16 23 20 27 3 10 6 0. 1830. 15.34E+10 30.68E+10 4 11 5 12 24 31 25 32 26 33 10 17 6 0. 1920. 15.34E+10 30.68E+10 11 18 12 19 17 24 18 25 19 26 10 3 1 7 8 14 15 21 0 100 3.23 3.93. 3.25 3.25 6.45 7.86 6.50 6.50 6.45 7.86 6.50 6.50 6.50 7.86 6.50 6.45 3.23 3.93 3.25 3.25 1

22 28 29 35 3.25 6.50 6.50 6.50 3.25

3.93 7.86 7.86 7.86 3.93

3.23 6.45 6.45 6.45 3.23

1 S

7.2.4 Example 4: Right T-Beam Bridge A right T-beam bridge having three longitudinals and five transverse beams as shown in Fig. 7.9, has been analysed. in this example. The beams will have T-sections because of monolithic construction. The flange widths is computed following IRC 21-1987 Clause 305.12.2[3]. The three longitudinal beams between them will cover the entire deck width as the flange width with the middle beam covering central 3 m width as its flange. To apply the above clause for asg the flange widths of transverse beams, their effective spans are required to be estimated. The effective span length may be defined as the distance between points of zero bending moment and its evaluation initially is not straight forward. It is easy to visualize that under 'loading, an interior transverse beam can develop sagging bending moment at a section where it crosses the central longitudinal beam. The longitudinal beams usually have small torsional inertia and hence transverse beams at sections where they end into exterior longitudinal beams, will develop small bending moments only. It is, therefore, suggested that the span length of the interior transverse beams for applying the above IRC Clause be taken as centre to centre distance between exterior longitudinal beams, which in this case is 6 m. The case of end transverse beam is however different. The end transverse beam under loading can develop

l•Im.M10.*

188 Grillage Analogy in Bridge Deck Analysis 7500 210 70mm Wearing coat

 •,1

-

150 -C1-1-1-*--3000 ± 1- 600 1 1"3000

r100.3

51

C

A B

D

Ci

e t A'

—0.11950 -1..s.-- 1. @1500: 6000 ----1.-11050 I•-(a) Cross- section at Mid-span 5 _______ 5250 3601 1 <150 x 300

1.1

•

210

I 10500 E

•

G (b) Longitudinal section Figure 7.9 Right T-Beam Bridge

hogging bending moment over interior . The span length of exterior transverse beam will be less than centre to centre distance between adjacent bearings. It may be assumed as 0.8 times the distance between centres of adjacent bearings, that is 2.40 m. The flange widths of individual transverse beams applying the relevant Clause will be then as shown in Fig. 7.10. We now proceed to idealize the bridge deck into grid structure. Let us take transverse grid lines first. Referring to Fig. 7.9 (b), adopt grid lines along centres of transverse beams. These are then at E, G and J. The effective 'slab widths covered by these transverse T-beams still leave out large widths of slabs between them. To take into these remaining

•

Illustrative Examples 189 -i-

--0

— dm.

O O

cV



0 1• • •

MO ,

o +m,

OFR

O I -

•••••••1

0 on

.0-o

• (12

Figure 7.10 Plan of Bridge Showing Widths of Deck Slabs Assigned to Various Transverse

slab strips adopt transverse grid lines between them at F and H giving five transverse grid lines in half length of the bridge or nine transverse grid lines in full length of the bridge. The widths of the slab strips represented by these are shown in Fig. 7.10. The assumed locations of all the nine grid lines are shown in Fig. 7.11. Now, I and T values to these grid corresponding to their sections are to be assigned. In longitudinal direction, referring to Fig. 7.9 (a), grid lines along the centres of three longitudinal beams are adopted. These are at B, D and B'. I and J values based on corresponding T-sections of these beams are assigned to these grid lines. The spacing of longitudinal beams is 3000 mm

N/RiNsam.,_

MUM

190 Grillage Analogy in Bridge Deck Analysis 18

27

36

45

63

62

61

6 0

•a 0 V II

in N

50

59 _ii

eV co

58

57

56

11

10

55 401500 = WOO I

Figure 7.11 Equivalent Grid

which is more than spacing of transverse grid lines which is 2625 MILL But the spacing of longitudinal grid lines should be less than spacing of transverse grid lines for the width to span plan dimension of this bridge. Hence two additional grid lines at C and C' between the longitudinals are introduced. The live load wheel while traversing over the carriageway may come on the cantilever slab i.e. on the left of B or right of B' and hence additional longitudinal grid lines, one each, say at centres of each of the kerb slab at A and A' be adopted. The inertia value of entire transverse section is covered by die loligatudinal beams at B, D and B' and hence the grid lines at locations A, C, C' and A' may be assigned zero inertia values or alternately

Illustrative Examples 191

these grid lines could be assigned small nominal inertia values and correspondingly inertia values of beams at B, D and B' be reduced. The former option is adopted herein. The location of all seven longitudinal grid lines is .shown in Fig. 7.11. The bridge is analysed for dead load and live load. The dead load is due to self load and is assumed to be acting along the seven longitudinal grid lines as per corresponding tributary widths. The live load is .due; to OHBD Truck load and is chosen to demonstrate the use of the program! for a specified live loading. The OHBD truck loading is used for the design of highway bridges in Ontario State of Canada and has been taken from Ontario Highway Bridge Design Code [2]. The impact factor for the loading is taken as 0.3. The minimum kerb distance for the loading is taken as 600 mm. The truck loading is given in Fig 7.12. The truck is made to traverse the length and breadth of the entire carriageway. The lengths of the steps in longitudinal and transverse directions are kept as 1500 mm and 750 mm. respectively. The input data is given in Table 7.9. The longitudinal design bending moment along exterior and interior beams at one-eighth span point along 6 3

14 14 7 710 I

20 16 Axle load 8 Wheel load

(a) Elevation Showing Magnitude of Axle and Wheel Loads

250

. 4

250 .0

I . -

0

0

0 0 GO 0

2SCRE3—

st

U 7 L

00

Wheel Reference

1200

)7/2fC2

____ M *Ti 81

1-

3600 1-

T 44 6000 180Q0

•-1 7200 11 i

.. I

(b) Plan Showing Dimensions of Truck Loading Figure 7.12 OHBD Truck Load

192 Grillage Analogy in Bridge Deck Analysis with corresponding positions of the truck is given in Table 7.10. The truck position is indicated through X-Y coordinates of reference wheel, which is the left most wheel of the leading axle. The X -Y coordinates are given with respect to node 1 (Fig. 7.11). TABLE 7.9: Input Data-Right T-Beam Bridge Y

90 0 110 63 112.0 .87 7 1050. 1500 1500. 1500. 1500. 500. 500.

1050.

1 2 32 2625. 0. 0. 0.. 2 3 3 445 5 667 7 8 8 9 19 20 20 21 21 22 22 23 23 24 41 41 42 42 43 43 44 44 45 55 56 24 25 25 26 26 27 37 38 38 39 39 40 40 63 56 57 57 58 58 59 59 60 60 61 61 62 62 10 11 16 2625. 0. 53.93E+10 2.78E+10 11 12 12 13 13 14 14 15 15 16 16 17 17 49 50 50 51 51 52 52 53 53 54

18 46 47 47 48 48 49

28 29 8 2625. O. 45.65E+10 2.78E+10 29 30 30 31 31 32 32 33 33 34 34 35 35

36

10 19 8 0. 1500. 9.94E+10 .91E+10 19 28 28 37 37 46 18 27 27 36 36 45 45

54

11 20 8 0. 1500..32E+10 .65E+10 20 29 29 38 38 47 17 26 26 35 35 44 44

53

12 21 12 0. 1500. 13.77E+10 .97E+10 21 30 30 39 39 48 14 23 23 32 32 41 41 43 52 13 22 8 0. 1500. .29E+10 .58E+10 22 31 31 40 40 49 15 24 24 33 33 42 42 1 10 4 0. 1050. 1.27E+10 2.25E+10 46 55 9 18 54 63 2 11 4 0. 1050. .89E+10 1.78E+10 47 56 8 17 53 62 3 12 6 0. 1050..32E+10 .64E-F10 48 57 5 14 50 59 7 16 52 61 4 13 4 0. 1050. .80E+10 1.59E+10 49 58 6 15 51 60 6 3 10 18 28 36 46 54 300.

50 16 25 25 34 34 43

51

Illustrative Examples

193

1 5.2E-04 8.79E-02 2.38E-03 9.64E-02 1.01E-03 0. 2.53E-03 0. 1.01E-03 0. 2.38E-03 -9.64E-02 5.2E-04 -8.79E-02 0 1 1 1 0 1 9 0 900 1500 750 39000 5400 10 1800 -3600 0 -3600 1800 0 -4800 '0 -4800 1800 -10800 -10800 1800 -18000 0 -18000 1800 600 3.9 3.9 9.1 9.1 9.1 9.1 13. 13.10.4 10.4 1 0 S

TABLE 7.10: Design Bending Moment Along Longitudinal Reams

Location along span 0.125L 0.25L 0.375L 0.500L

Exterior Beam Coordinates of Bending Reference wheel Moment (t m) (m) 177 297 361 379

Interior Beam Bending Coordinates of Moment Reference wheel m) (m)

22.5, 0.9 19.5, 0.9 18.0, 0.9 15.0, 0.9

136 202 255 256

22.5, 3.15 19.5, 3.15 18.0, 3.15 15.0, 3.15

7.2.5 Example 5: Skew T-Beam Bridge A skew T-beam reinforced concrete two-lane simply ed bridge with 40° skew angle is chosen for this example. The example refers to an actual bridge constructed on NH-46 in Tamil Nadu at KM 40/2. The bridge was analysed and designed by the authors for the Ministry of Surface Transport. The bridge is shown in Fig. 7.13. The longitudinal and transverse beams are cast monolithically with deck slab. The longitudinal and transverse beams are taken as T-beams as in example 4. For end diagonal beams, the effective span has been taken as 0.8 times the skew distance between centres of bearings for asg the flange

.01MEMil

194 Grillage Analogy in Bridge Deck Analysis

KERB 3@ 5500- 16500 Skew span =2070 (a) Plan

11

111 1

I-

450

li

w e k rs a cross beam

60  ••• A0C0 H 1475 hi-441250r

475

Hnoo I, 2500 .1. 2500 1:1700.1 (b) Right cross - section at mid span Figure 7.13 Skew T-Beam Bridge

Flange width of beam Figure 7.14 Beams exist along lines and 0 Additional transverse grid tines are put along tines 0 0 g0 Width of deck stab contributing to I , J Values to grid tine , say 3,is (b + c )

Transverse Grid Lines in End Region

Illustrative Examples 195

width to these beams. Seven longitudinal grid lines as in Example 4 are adopted. The mathematical model of this bridge requires comparatively a larger number of transverse grid lines. Figure 7.14 shows the transverse grid lines in the end region and Fig. 7.15 shows the equivalent grillage for the bridge. A closely spaced grid pattern in the end region will provide more accurate force responses. This adoption also ensures equal number of nodes along each longitudinal grid line, which is a pre-requisite of .the program 'GABS'. The analysis for only IRC live loads has been carried out. The live loads considered are same as for Example 1. Table 7.11 gives the input data. The 91 78

13

z

1238 _ 30

65 77 64 52 76 89 9 51 63 75 88 26 38 50 62 74 87 25 37 61 86

in0§0

rc

*4'

2750

85 . 12

24 36 48 6072

11

2 3 3 5 47 59 71

10 9

8 7 6

22 34 46 58 7 0

21 33 45 576 9

X 0 3 2 44 566 8

_ 84

83

0 0 0

82

II

0

81

U) (--

csa x 4

80

19 31 43 55 _____y1 6 .79 18 30 42 17 29

1512 1238

4

3 2 (0,0) 1

1 0*

. 1--,-4x1250= -`.1 1-.1475 5000 1475 • Figure 7.15 Equivalent Grillage

196 Grillage Analogy in Bridge Deck Analysis

200 '150 100 .50

co 0

Bending Moment Envelope for Exterior Beam

7 no 100 oi

50 0 Bending Moment Envelope for Central Beam 30 • 10 0

Shear Force Envelope for Exterior Beam 50 30 In 10

— Shear Shear Force Envelope for Interior Beam 0246

8 10 12 14 16 18 20' Span (m)

Figure 7.16 Bending Moment and Shear Force Envelopes for Beams

overall envelope values for bending moments and shear forces under live loadings for exterior and interior beams are presented in Fig. 7.16. In preparing these envelope diagrams the response values at polar symmetric points have been compared and wherever these were different, the larger one has been adopted. The shear force envelope diagrams show sudden change in values at locations of cross beams. The envelope diagrams for shear forces also show maximum reaction values at the s. The reactions at the two ends of the exterior beam differ considerably. The reaction is minimum at the acute angled corner and is maximum at the obtuse angled corner. This is expected in a skew bridge.

•

Illustrative Examples TABLE 7.11: Input Data-Skew T-Beam Bridge Y

(

90 40 168 91 26 2.0 .87 7 1475. 1250. 1250. 1250. 1250. 1475. 450. 450. 2 14 2 0. 1475. 0.12E+10 0.24E+10 78 90 3 15.4 0. 1475. 0.11E+10 0.22E+10 4 16 76 88 77 89 5 17 2 0. 1475. 0.10E+10 0.2E+10 75 87 6 18 6 0. 1475. 0.13E+10 0.26E+10 9 21 11 23 69 81 71 83 74 86 7 19 4 0. 1475. 0.144E+10 0.288E+10 13 25 73 85 67 79 8 20 4 0. 1475. 0.304E+10 0.607E+10 12 24 68 80 72 84 10 22 2 0..1475. 0.45E+10 0.9E+10 70 82 1 2 8 1238. 0. 0. 0. 6 7 30 31 37 38 54 55 61 62 85 86 90 91 2 3 16 1050. 0. 0. 0. 3 44 5 5 6 86 87 87 88 88 89 89 90 27 28 28 29 29 30 38 39 53 54 62 63 63 64 64 65 7 8 8 1512. Q. 0. 0. 12 13 31 32 36 37 55 56 60 61 79 80 84 85 8 9 16 2750. 0. 0. 0. 58 59 9 10 10 11 11 12 32 33 33 34 34 35 35 36 56 57 57 58 59 60 80 81 81 82 82 83 83 84 14 15 8 1050. 0. 44.90E+10 2.23E+10 15 16 16 17 17 18 74 75 75 76 76 77 77 78 18 19 4 1238. 0. 44.90E+10 2.23E+10 25 26 66 67 73 74 19 20 4 1512. 0. 44.90E+10 2.23E+10 24 25 67 68 72 73 20 21 8 2750. 0. 44.90E+10 2.23E+10 21 22 22 23 23 24 68 69 69 70 70 71 71 72 40 41 4 1050. 0. 38.0E-F10 1.66E+10 41 42 50 51 51 52 42 43 2 1238. 0. 38.0E+10 1.66E+10 49 50 43 44 2 1512. 0. 38.0E+10 1.66E+10 48 49

197

198 Grillage Analogy in Bridge Deck Analysis 44 45 4 2750. 0. 38.0E+10 1.66E+10 45 46 46 47 47 48 1 1412 1238. 1475. 3368E+10 7.98E+10 14 27 27 40 40 53 53 66 66 79 13 26 26 52 65 65 78 78 91 15 27 60. 1250A.07E+10 0.14E+10 16 28 28 40 52 64 64 76 65 77 17 29 6 0. 1250. 0.063E+10 0.12E+10 29 41 41 53 39 51 51 63 63 75 18 30 16 0. 1250. 4.64E+10 0.64E+10 30 42 42 54 54 66 26 38 38 50 50 62 62 45 57 57 69 23 35 35 47 47 59 59 71 19 31 8 0. 1250. 0.091E+10 0.18E+10 31 43 43 55 55 67 25 37 37 49 49 61 61 20 32 8 0. 1250. 0.192E+10 0.384E+10 32 44 44 56 56 68 24 36 36 48 48 60 60 22 34 4 0. 1250. 0.283E+10 0.567E+10 34 46 46 58 58 70 6 3 14 26 40 52 66 78 225. 1 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 1 1 0 4 2 0 625 1 0 3 0 1630 1 0 5 0 1630 1 0 7 0 1845 1 0 1

1000 280 47000 2025

1000 385 42000 3940

1000 385 29000 3940

1000 440 32000 4045

39 39 52

74 21 33 33 45 73 72

0.

Illustrative Examples 199

7.2.6 Example 6: Box-Girder Bridge A single cell box of trapezoidal cross-section with 30 m span is considered for this example. The bridge is shown in Fig. 7.17. Converting a bridge of this type to an equivalent grid and asg inertia values to various needs special care. Referring to Fig. 7.17(b) six longitudinal grid lines have been assumed; namely two at C and C' in alignment with locations of centre of bearings, two at B and B' at locations where inclined webs intersect the deck slab and two at A and A' where the decking slab ends. The end ones at A and A' are znecessiated because the cantilever projections are large and wheels of live load could go on the cantilever

3 5 0 0 1.•— 4176 --oi (a) Cross Section



• 1075 • 3500 . 1075 *

2725

A



BC

2725

C B'

0,

A'

i

(b) Location of Longitudinot Grid Lines

30 0 4-

est

7

)35 225 30000

50 (c) Longitudinal Section

450 

•

41 ____________

•

•

•

•

•

8x 3750 c 30000 ________

(d) Location of Transverse Grid Lines

Figure 7.17 Box-Girder Bridge

•

-

5

200 Grillage Analogy in Bridge Deck Analysis

slabs beyond B and B'. The moment of inertia of the cross section of the bridge about a common axis is computed and this is divided equally among the two longitudinals at B and B'. Similarly torsional inertia of the closed trapezoidal section is computed using equation 4.15 and one half of this is assigned to each of the longitudinals at B and B'. Thus it is assumed that in longitudinal direction, the entire inertia is Concentrated along grid lines at B and B'. The remaining four longitudinals which are located at A, A', C and C' each are assigned zero inertia values. Referring to Fig. 7.17 (c, d), nine transverse grid lines as shown have been assumed. The flexural and torsional inertia values of these transverse are computed employing equation 4.12. The end transverse also have diaphragms and hence moment of inertia of diaphragms are added while computing moment of inertia of end transverse . The bridge is analysed for a stationery position of Class 70R Col. 'm' train loading. The equivalent grid and position of live load on it is shown in Fig. 7.18. Table 7.12 gives the grid input data. The maximum longitudinal bending moment at mid-span was found to be 350.0 t.m giving a bending compressive stress of 1.76 MPa and bending tensile stress of 3.36 MPa. To compare the results of aboye grillage analysis, the bridge is alternately analysed by Finite Element Method (FEM). Standard package SAP IV is used for the analysis. Rectangular plate element with six degrees of freedom at each node is chosen. Boundary elements are introduced at locations for reactions. Figure 7.19 gives the_ structural idealisation chosen for the FEM analysis. The maximum bending compressive and tensile stresses at mid span were found to be 1.78 MPa and 3.20 MPa respectively. The comparison is quite close and acceptable. For more detailed information on the comparison, readers may refer to reference [1]. In the above examples, wherever necessary, the sections adopted for the grillage .elements are given. The inertia values of these sections can be taken from the respective grid Input data given for each example. TABLE 7.12:

Input Data—Box-Girder Bridge

Y 90

0 93 54 8 1 2

2.0 0.87 6 2725.0 737.0 4176.0 737.0 2725.0 450 450

2 32 3750. 0. 0. 0. 3 3 4 4 5 56 67

7 8

8 9 Contd.

'

19 20 20 21 21 22 22 23 23 24 24 25 25 26 28 29 29 30 30 31 31 32 32 33 33 34 34 .35 46 47 47 48 48 49 49 50 50 51 51 52 52 53

26 27 35 36 53 54

10 11 16 3750.0 0.0 136.76E+10 203.4E+10 11 12 12 13 13 14 14 15 15 16 16 17 17 18 3.7 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45

t

10 4 0.0 2725.0 0.38E+10 0 .76E+10 37 46 9 18 45 54 4 10 19  0.0 737.0 92.86E+10 185.72E+10 28 37 18 27 36 45 19 28 27 36

2 0.0 4176.0 92.86E-F10 185.72E+10

2 11 14 0.0 2725.0 0.62E+10 1.23E+10 38 47 3 12-39 48 4 13 40 49 5 14 41 50 6 15 42 51 7 16 43 52 8 ,17 4453 11 20 14 0.0 737.0 135.04E+10 270.08E+10 29 38 12 21 30 39 13 22 31 40 14 23 32 41 15 24 33 42 16 25 34 43 17 26 35 44 20 29 7 0.0 4176.0 135.04E+10 270.08E+10 21 30 22 31 23 32 24 33 25 34 26 35 4 3 19 27 28 36 1800. 1 0. 0. 0. 0.•0. 0. 0. 0. 0. 0. 0. 0. 0 I 1 0 0 1 3 23980. 3205. 1. 1. 23980. 3205. 1 0 S

Illustrative Examples

201

202 Grillage Analogy in Bridge Deck Analysis

795 790 f•3205

e

795

-0

positionsof 7 2

9 3 6

1

w h els I

53

-

I

11101111 43

2

[email protected]

51

MINI

St

42

F cc;

14

Yir

4

171 17t -

17± 17t

40

ao

12t 12t

N 3

12

30

121 21 9

39 1•7

2 0

11725.4 I.-3500 --I 27251 1075 1075 Figure 7.18 Equivalent Grid with Position of Live Load

•

Illustrative Eramples 203

. c• H In

0

on te r

0

01

71:7 0 .111

co

0

0

IA 0 to  -•grati

-t

O

U 1 5

In O

g 2 0 0 0 = 3 0 0 0 0

O

I0 w

-1

%.t w

tp

1 2 725 1180188411E183 2725 I 1917

O c-I

11917) 184.1. ["1- 1144 1146

Figure 7.19 Box-Girder Bridge-Structural Idealisation for F.E.M. Analysis

REFERENCES 1. ALOK

2. 3.

13uowt.ucK, "To study the Behaviour of Box-Girder Bridges under Live Load by Analysing the Bridge Using Several Methods and Comparing the Results", An M.Tech. Project, Civil Engineering Department, I.I.T. Delhi. 1990. IRC 21-1987, "Standard Specifications and Code of Practice for Road Bridges", Section III, The Indian Roads Congress, New Delhi. 1991. "Ontario Highway Bridge Design Code", Ministry of Transportation and Communications, Highway Engineering Design. Downsview. Ontario, Canada, 1983.

9

r t

Appendix I

Listing of Programs Grid'*

* This program can analyse a grid under nodal loading.

206 Grillage Analogy in Bridge Deck Analysis C C C C

PROGRAM 'GRID' THIS PROGRAM CAN ANALYZE A GRID UNDER APPLIED NODAL LOADS. THE CONFIGURATION OF THE GRID COULD BE RIGHT, SKEW OR QUADILATERAL. IMPLICIT DOUBLEPRECISION (A-H2O-Z) COM MON/BX I /X(100), Y(I 00), XI(I00), XJ(100), SM (6,6), NCD (6), NN(50) COMMON/BX2/GSM(200,50) COMMON/BX3/NCN1(200),NCN2(200),NSN(20),SUPSTF(20) COMMON/BX4/P(300),D(300) COMMON/BX5/GDM(6),DM(6),PM(6),TM(6,6) COMMON/BX6/LNODE(200),NLT(200),XLOAD(200) OPEN(UN1T=3,FILE='GRID.IN') OPEN(UNIT=4,FILE='GRID.OUT)

C C

READING INPUT DATA INITIALISATION OF MATRICES DO 10 1=1,6 DO 5 J=1,6 SM(I,J)=0.0 TM(I,J) =0.0 5 CONTINUE 10 CONTINUE C

GEOMETRICAL PROPERTIES OF THE GRID READ (3,*) NNODES, NELEMS

C

COORDINATES OF NODES READ (3,*)NGROUP DO 20 I=I, NGROUP READ (3 ,*)N1 ,N2,NINC, AX,AY,XINC,YINC DO 15 11=N1,N2,NINC X(I1)7AX Y(I1) =AY AX=AX+XINC AY =AY+Y1NC 15 CONTINUE 20 CONTINUE C

MEMBER CONNECTIVITIES READ (3,*)NGROUP DO 30 1=1, NGROUP READ(3,*)N1,N2,N1NC,NC I ,NC2,NC11NC, NC21NC DO 25 11=NI,N2,NINC NCN1(I1)=NCI NCN2(I1)=NC2 NC 1 =NCI +NC IINC NC2=NC2+NC2INC 25 CONTINUE 30 CONTINUE C

MEMBER PROPERTIES READ (3,*)E,G READ (3,*)NGROUP

Appendix I 207 DO 40 I=1,NGROUP READ(_S,*)ALAJ,N,(NN(I1),I1 =1,N) DO 35 I1=1,N 12=NN(11) XI(12)=AI XJ(12)=AJ 35 CO N TI NU E 4 0 C O N TI N U E C

HALF BANDWIDTH OF STIFFNESS MATRIX . READ (3,*)NIIBAND NDOF=3*NNODES

C

INITIALIZE THE STIFFNESS MATRIX DO 45 IA=I,NDOF DO 45 IB=1,NHBAND 45 GSM(IA,IB)=0.0 C

PRINTING INPUT DATA _WRITE(4,*)'TOTAL NO. OF NODES=',NNODES WRITE(4,*).TOTAL NO. OF ELEMENTS=',NELEMS WRITE(4,*)'HALF BAND WIDTH=',NHBAND WRITE(4,*)'TOTAL NO. OF DOF.=',NDOF WRITE(4,50) 50 FORMAT (4X,'NODE NO.',9X,'X-COORD.',7X,'Y-COORD.') DO 60 I=1,NNODES WRITE (4,55)I,X(I),Y(I) 55 FORMAT(6X,I3,9X,F10.4,6X, F10.4) 60 CO N TI NU E WRITE(4,*) WRITE(4,*)'E=',E,' G=',G WRITE(4,*) WRITE(4,65) 65 F013.MAT(4X,'MEMBERNO.',8X,'M.I.',14X,T,8X,'NODE1',4X,'NODE2') DO 75 I=1, NELEMS WRITE (4,70)LXI(1),XJ(I),NCN1(1),NCN2(1) 70 FORMAT (8X,I3,4X,E14.6,4X,E14.6,3X,13,6X,13) 75 CONTINUE C MEMBER STIFFNESS MATRIX DO 90 I=1,NELEMS C DETERMINATION OF ANGLE OF TRANSFORMATION II =NCN1(I) 12=NCN2(1) X L = DS QRT((X(12)-X(I1))**2 + (Y(I2)-Y(I1))**2) C =(X(12)-X(I1))/XL S=(Y(I2)-Y(I1))/XL X1 =12.*E*XI(1)/(XL**3) X2 =6.*E*XI(I)/(XL**2) X3 =4.*E*XI(I)/XL X4 = G*XJ(I)/XL SM(1,1)=XI SM(2,1)=X2*S SM(2,2)=X3*S*S+X4*C* C SM(3,1)= -X2*C

208

80

Grillage Analogy in Bridge Deck Analysis SM(3,2)=-X3*C*S+X4*C*S SM(3,3)=X3*C*C-I-X4*S*S SM(4,1)=-X1 SM(4,2)=-X2*S SM(4,3)=X2*C SM(4,4)=X1 SM(5,1)=X2*S SM(5,2)=X3/2.*S*S-X4*C*C SM (5 ,3) =-X3/2.*C*S-X4*C*S SM(5,4)=-X2*S SM(5,5)=X3*S*S-i-X4*C*C SM(6,1)=:-X2*C SM(6,2)=-X3/2.*C*S-X4*C*S SM(6,3)=X.3/2.*C*C-X4*S*S SM(6,4) =X2*C SM(6,5)=-X3*C*S+X4*C*S SM(6,6)---X3*C*C+X4*S*S DO 80 J=1,5 DO 80 K=(.I+1),6 SM(J,K)=SM(K,J)

C. " ASSEMBLY OF STIFFNESS MATRIX NCD(1)=1.1*3-2 NCD(2)=NCD(1)+1 NCD(3)=NCD(2)+1 NCD(4)=12*3-2 NCD(5)=NCD(4)+1 NCD(6)=NCD(5)+1 DO 85 IA=1,6 NI =NCD(IA) DO 85 IB=1,6 N2=NCD(IB)-NI+1 IF(N2.LT.1) GO TO 85 GSM (N1,N2)=GSM(N1,N2)+ SM(IA,113) 85 CO N TI NU E 90 CONTINUE C ' CONDITION C READ TYPES- 0 FOR FLEXIBLE, 1 FOR RIGID READ (3,*)NST IF(NST.EQ.0) GC) TO 100 READ (3,*)NSUP READ(3,*)(NSN(I),I = 1 ,NSUP) DO 95 I=1,NSUP IDOF = 3*(NSN(I)-1)+ 1 GSM(IDOF,1)=GSM(IDOF, 1)*1.0E + 6 95 CONTINUE GO TO 110 100 READ (3,*)NSUP READ(3,*) (NSN(1),SUPSTF(0,1=1,NSUP) DO 105 I=1,NSUP IDOF=3*(I-1)+1 ncivr(1nrw i)=GSM(1DOF,I)+SUPSTF(I) 105 CONTINUE

Appendix 1 209 C 110

DECOMPOSITION OF STIFFNESS MATRIX CALL DECOMP (NDOF, NHBAND, EXIT)

C

INIALIZATION OF LOAD VECTOR DO 115 1=1,NDOF P(I)=0.0 115 CONTINUE C

LOAD VECTOR

. READ (3,1)NLOADS DO 120 I=1,NLOADS READ(3,*)LNODE(I),NLT(I),XLOAD(I) . 120 CONTINUE DO 125 I=1,NLOADS IDOF=3*(LNODE(1)-1)+NLT(1) P(IDOF) =XLOAD(I) 125 CONTINUE C

SOLUTION OF STIFFNESS MATRIX CALL SOLV (NDOF, NHBAND) WRITE(4,*) WRITE(4,*)'DEFORMATIONS' WRITE(4,130) 130 FORMAT(4X,'NODE NO.',4X,'VERT.DEP.',6X,'X-ROT.',10X,'Y-ROT.') DO 140 IA=1, NNODES 11=3*IA-2 I2=I1 +I 13=12+1 WRITE(4,135)IA,D(I1),D(I2),D(13) 135 FORMAT(4X,I3,4X,3(E14.6,2X)) 140 CONTINUE C

MEMBER FORCES WRITE(4,*) WRITE(4,I'MEMBER FORCES' WRITE(4;145) 145 FORMAT(1X,'MEM.NO.',IX,'S.FORCE',7X,'B.MOMENT',12X,'TORSION') WRITE(4,150) 150 FORMAT(26X,'END 1',9X,'END 2') DO 195 I= LNELEMS C

DETERMINATION OF ANGLE OF ROTATION I1=NCNI(I) 12=NCN2(I) XLz---DSQRTaX(I2)-X(11))**2+(Y(12)-Y(I1)j**2) C=(X(12)-X(II))/XL S=(Y(12)-Y(I1))/XL DO 155 IA=1.6 DO 155 113=1,6 155 TM(IA,IB)=0.0 TM(1,1)=1.0 TM(2,2)=C TM(2,3)=S

210 Grillage Analogy in Bridge Deck Analysis TM(3,2)=-S TM(3,3)=C DO 160 IA=4,6 II =1A-3 DO 160 IB=-4,6 11=1B-3 160 TM(IA,113)=TM(11,1.1)

1: 1.

1 1

C

GLOBAL MEMBER END DEFORMATIONS GDM(I)=1)(11*3-2) GDM(2) = D (1 1 *3-1) GDM(3)=D(11*3) GDM(4)=D(12*3-2) GDM(5)=D(1249-1) GDM(6)=D(12*3) DO 170 I=1,6 SUM =0.0 DO 165 K=1,6 SUM=SUMA-TM(1,10*GDM(K) 165 CONTINUE DM(J)=SUM 170 CONTINUE X I =12.*E*X1(1)/(XL**3) X2=6.*E*XI(I)/(XL**2) X3=4.*E*X1(I)/XL X4 =G*XJ(I)/XL SM(1,1)=XI SM(2,1)=0 SM(2,2) =X4 SM(3,1)=-X2 SM(3,2)=0 SM(3,3)=X3 SM(4,1)=-X1 SM(4,2)=0 SM(4,3)=X2 SM(4,4)= X1 SM(5,1)=0 SM(5,2) =-X4 SM(5,3)=0 SM(5,4)=0 SM(5,5)=X4 SM(6,1)=—X2 SM(6,2) =0 SM(6,3)= X3/2. SM(6,4) =X2 SM(6,5)=0 SM(6,6)=X3 DO 175 J=1,5 DO 175 K= (J+1),6 175 SM(J,K)=SM(K,J) DO 185 2=1,6 SUM =0.0 DO 180 K =1,6 SUM =SUM

C C C

Appendix I 211 180 CONTINUE PM(J) =SUM 185 CONTINUE WRITE(4,190)I,PM(1),PM(3),PM(6),PM(2) 190 FORMAT(4X,I3,2X,4(E12.6,2X)) 195. CONTINUE STOP END C C C C C C C

SUBROUTINE DECOMP: DECOMPOSITION.OF STIFFNESS MATRIX SUING CHOLESICEY'S METHOD. N ---->TOTAL NO. OF EQUATIONS IBW = >HALF BAND OF THE MATRIX EXIT = >INDICATING VARIABLE S = <STIFFNESS MATRIX TO BE DECOMPOSED

wrai

SUBROUTINE DECOMP(N,IBW,EXIT) IMPLICIT DOUBLEPRECISION (A-H2O-Z) COMMON/BX2/S(200,50) WRITE(*,*) DECOMPOSITION STARTS' DO 260 I=1,N. WR1TE(s,*)I IP=N-I+ 1 IF (IBW.LE.IP) GOTO 200 GOTO 205 200 IP=IBW 205 DO 260 LI P SUM =S(I,J) IQ=IBW-J IF ((I-1).LE.IQ)GOTO 210 GOTO 215 210 IQ =I-1 215 IF (IQ.EQ.0) GOTO 230 GOTO 220 220 DO 225 K =1,IQ 225 SUM =SUM-S(I-K,1+K)*S(I-K,J+K) 230 IF (I.NE.1) GOTO 235 GOTO 240 235 S(I,J) =SUM/TEMP GOTO 260 C37 IF (SUM.LE.0.0) GOTO 245 240 GOTO 250 245 EXIT=0.0 RETURN 250 EXIT =1.0 255 TEMP=DSQRT(SUM) S(I,J) =TEMP 260 CONTINUE RETURN END C C

SUBROUTINE SOLD': SOLUTION OF STIFFNESS MATRIX DECOMPOSED BY

212 Grillage Analogy 0: Bridge Deck Analysis C

CHOLESKEY 'S METHOD FOR GIVEN LOAD 5 D7 = -Pl*Pl*Pa- Fr2*P2*P3 D8 =-P1*P2*(PS -P4) D9= P I *P 1 *P3 -In *P2*F4 M1=3*M(I,1) M2 =3*M(I,2) M3 =M1-1 M4 =M1-2 M5=M2-1 M6 =M2-2 MXI = D l*B(Mrh) +D2*B(M3) + D3*(B(M1)-B(M2)) + D7*B(M6) + D8*B(M5) MY1=D2*B(146.4)+D4*B(M3)+D5*(B(M1)-B(M2))+D8*B(N16)+D9*B(M5) MX2 =D7*B(M4) + D8*B(M3) + D3*(B(M1)-B(M2)) +D1*B(1V16) +D2*B(M5) MY2 =D8*B(145.4) +D9*B(M3)+D5*(B(M1)-B(M2))+D2*B(M6)+D4*B(M5) FZ =D3*(B(M--4)+B(M6))+D5*(B(M3)+B(M5))+D6*(B(M1)-B(M2)) FZ1 =F5 J=I-IBW+ I IF ((I +1).LE.JBW) GOTO 265 GOTO 265 265 J=1 DO 270 K=1,1-1 270 SUM = SUM-S(K, (I-K+ 1))*D(K) 275 D (I) = SU/s4/S(I ,I) 280 CONTINUE DO 305 I =-• I ,N J=N-I+IBW IF (J. GT. GOTO 285 GOTO 290 285 J=N 290 SUM =D(N-1+ I) IF (I, EQ.1) GOTO 300 DO 295 K=(N-I+2),1 '295 SUM =SUM-S(N-I+1,K-N +D*D(K) 300 D(N-I+1)=SUM/S(N-I+1,1) 305 CONTINUE RETURN END

Appendix II

Listing of Program Gabs*

* This program can analyse a grid under generalised deck loading.

214 Grillage Analogy in Bridge Deck Analysis PROGRAM 'GABS' THIS PROGRAM CAN ANALYSE A GRID UNDER GENERALISED DECK LOADING.THE CONFIGURATION OF THE GRID COULD BE RIGHT , SKEW OR TRIANGULAR. IMPLICIT real*8"(A-H2O-Z) INTEGER GG,DD,WW,R,T,EE,EE1,EE2,T1,R3,PATYPE,GRIDTYPE real*8 L,MP1:,MP2,MQ1,MQ2,MX1,MX2,MY1,MY2,E1,E2, +MAXPT(6,4;500),MAXB(4,500),MAXRT(4, 140),MAXPT2(6,2:4,500), +MAXB2(2:4;500),MAXRT2(2:4,100),MAXRTN(4,100),CORSPT(2,500), +MAXR2N(2:4,100),CORSPTI(2,500),CORSPT2(2,500),CORK(100), +CORX1(100),CORY1(100),CORIX(100),COR1Y(160),COR2X(100), +COR1X1(100),COR1Y1(100),COR2X1(100),COR2Y1(100), + coRRx(2,500), CORRY(2 ,500),CORRX1 (2,500), CORRX2(2,500), + CORY(100);COR2Y(100), CORRY1(2,500),CORRY2(2,500), .+CORD(100),CORD1(100),CORD2(100) CHARACTER*1 CH1,CH3 CHARACTER*2 POS, INPFILE*12, OUTFILE*12 COMMON/BLOCK2/S(1,50000),U(50000),P(1000,7),PT(1000,7),W(100), +D(100),B(1000),BD(1000),RT(100),FEL(500,6),RTN(100) COMMON/BLO CK3/XC (100), YC(100),P W(100),PI ,SANG,PT1 (1000 ,2), +DLB(1000),XLLG(0:100),YLLG(0:100) DIMENSION M(1000,2), ITT(100),JDD(100),JDD1(100), +PM(1000,1) DIMENSION DLNG(100),BL(1000) COMMON/BLOCK6/ SCLMIN,XLL,YLL,AKERB COMMON/BLOCK8/0MAXPT(6,4,1000), oMAX13(4,1000), OMAXRT(4,100), +0MAXRN(4,100) COMMON/XX1/AIMP CHARACTER*20 FILE1, FILE2 WRITE(*,*) 'Enter the file name for Input ' READ(*,5) FILEI 5 FORMAT(A20) WRITE(*,*) 'Enter the file name for Output READ(*,5) FILE2 • OPEN(UNIT=3,FILE:=F1LE1) OPEN(UNIT =4 ,FILE =FILE2) C OPEN(UNIT=3,FILE='*SRC.EX2') C OPEN(UNITL-4,FILE='*SRC.GRID3OU') MOVE=O XL1 =0.0 YLI =0.0 XINT=0.0 YINT=0.0 DO 10 i=1.6 DO 10 J=1,4 C C C

Appendix II 215

),

DO 10 K=1,500 10 MAXPT(I,J,K)=0.0 DO 15 1=1,4 DO 15 3=1,500 15 MAXB(I,J)=0.0 DO 20 I=1,4 DO 20 3=1,100 MAXRTN(I,J)=0 20 MAXRT(I,J)=0 P1=4*ATAN(1.0) 25 READ(3;230) CHI IF (CHI.EQ.'N') GO TO 315 READ(3,*) GRIDTYPE C GRIDTYPE CAN EITHER BE SKEW(ANY VALUE OTHER. THAN 90) OR C ORT H.(90) READ (3,*) ANG READ (3,*) MM,N,DD,E,G,NLG,(DLNG(I),I=1,NLG-1) READ (3,*) E1,E2 SANG = ANG*PI/180 WRITE (4,235) WRITE (4,240) ANG WRITE (4,245) MM,N,DD,E,G WRITE(4,290)NLG,(DLNG(I),I=1,NLG-1) WRITE(4,295) WW =0 1=0 DO 45 GG=1, DD I= I+ 1 READ (3,*) M(I,1),M (1,2),T,P(I,1),P(1,2),P(1,3),P(I,4) WRITE (4,250) GG WRITE (4,255)T,(P(1,1C2),K2=1,4) PM(I, 1) = P(I , 1). L=SQRT(P(I,1)**2+P(I,2)**2) P(I,1) =P(I, p(1,2)=P(1,2)/L. P(I,3)=2*E*P(1,3)/L H =P(1,3) P(I,4)=G*P(I,4)/L P(I,5) = 6*H/(L**2) P(1,6)=2*H P(I,7) =3 *HiL EE=I EE1=EE+1 EE2=EE+T-1 DO 30 II I =EE1,EE2 30 PM(II I, I) = PM(EE,1) READ (3,*) (M(IAX, 1), M(IAX ,2),IAX =EE1,EE2) Nil —1+T 1

216

Grillage Analogy in Bridge Deck Analysis

WRITE (4,260) (M(IAX1,1), M(IAX1,2),IAX1=1,N11) T1=T-1 DO 40 R=1,T1 I=R+EE DO 35 .1=1,7 35 P(I,J)=P(I-1,J) 40 CONTINUE 45 CONTINUE NTG=N/NLG WW =3*NTG +3 YLLG(0) DO 50 I=1,NLG-1 YLLG(1) = YLLG(1- 1.)+ DLNG(1) 50 CONTINUE. YSPAN =YLLG(NLG-1) XLLG(0) =0 DO 55 I=1,NTG-1 CALL INODE(M,I,I+1,L1) XLLG(I) = XLLG(I-1) + PM(L1, 1) 55 CONTINUE SPAN=XLLG(NTG-1) N3=N*3 NROWS=N3 NROWS1=NROWS N8 = N3*WW DO 601=1, 15 60 JTT (I)=0 READ (3,*) IX,JRR, (JTT(1), I=1,IX) IF (JRR.EQ.1)THEN READ(3,*) SUPK ENDIF WRITE (4,265) WRITE (4,270) IX,(JTT a1m1=1,1X) IF (JRR.NE.3) WRITE(4,280) (JRR.EQ.3) WRITE(4,275) IF (JRR.EQ.1)THEN WRITE (4,285)SUPK ENDIF READ (3,*) AKERB AKERB = AKERS - 0.5 NNODES =N CALL DEAD LOAD (DLB,N,NLG,M,PM,E1,E2) C THE FOLLOWING DO LOOP STORES IN JDD THE VERT DEFN. NOS C CORRESPONDING TO THE ED NODES. DO 65 I=1,IX 65 JDD(I)=3*JTT(I) CALL STIFF(N,MM,WW,P,M,S) IF (JRR.EQ.3) THEN

-•-

Appendix II 217

70

C 75

80

85

CALL SORT(IX,JDD,JDD1) CALL MODSTIF(WW,IX,JDD1,S,NROWS) ELSE DO 70 1=1,1X NE=WW*(JDD(I)-1)+ I S(1,NE)=S(1,NE)-1-SUFK CONTINUE ENDIF CALL DECOMP(NROWS,WW,S,U) COMPUTATIONS OF LOAD MATRIX STARTS READ (3,230) CH3 IF (CH3.EQ.'S') GO TO 315 NL=N N9=3*N+WW READ (3,*) 'NOLCASE WRITE(4,80) FORMAWING(1) = DEAD LOAD ONLY'/ + ' LDING(2) = CLASS A - TWO LANE'/ + ' LDING(3) = CLASS 70-R TRAIN:COL -L'/ + ' LDING(4) = CLASS 70-R TRAIN:COL -M'/ LDING(5) = CLASS 70-R BOGIE:COL -1.1/ LDING(6) = CLASS 70-R-BOGIE:COL -M'/ LDING(7) = CLASS 70-R TRACK'/ + ' LDING(8) = CLASS A - SINGLE LANE'/ + ' LDING(9) = SPECIFIED BY '/) DO 225 IMM1=1,NOLCASE READ(3,*)LCASE IF(LCASE.EQ.I)THEN WRITE(*,*)'DEAD LOAD ONLY' WRITE(4,*)'DEAD LOAD RESULTS' DO 85 JA3=1,3*NL B(JA3)=DLB(JA3) GOTO 135 ENDIF READ(3,*)XL,YL,XINC,YINC,XSTOP,YSTOP MODIN1=0 MODIN2=0 XFIN = XL + REAL(INT((XSTOP-XL)/XINC))*XINC YFIN = YL REAL(INT((YSTOP-YL)/YINC))*YINC WRITE(4:*)' WRITE(4,310)LCASE WRITE(4,*)' -----------------------CALL IMPACT (LCASE, SPAN,SANG) WRITE(4,'(10X,A)TTHE RESPONSE IS INCLUSIVE OF DEAD LOAD RESULTS' WRITE (4,300)XL, YL, XINC, YINC,XFIN, YFIN,AIMP MOVE =0 IM1 =0 YL1 =YL-YINC

218 :Grillage Analogy in Bridge Deck Analysis 90

XL1 =XL-XINC YL1 =YL1+YINC 95 XL1 =XLI + XINC IF (YLLGT.YSTOP)THEN GO TO 200 ENDIF IF (XLLGT.XSTOP)THEN GO TO 90 ENDIF IF(MODIN1. EQ.1) THEN XL1 =XL1-1.0 MODIN1 =0 ENDIF IF(XLI.EQ.0) THEN XL1=1.0 MODINI =1 ENDIF IF(MODIN2.EQ.1) THEN XL1 =XL1 +1.0 MODIN2=0 ENDIF IF(XLLEQ.SPAN) THEN XL1=XL1-1.0 MODIN2=.1 ENDIF C WRITE(*,*)'LOAD CASE ',LCASE,' LEFT FRONT WHEEL AT ',XL1, C +' AND ',YL1 IM1=IMI+1 CALL LLOAD (LCASE, XLI, YL1, XC, YC, NW; NWPA,IM1) CALL WLOAD (LCASE, PW,SCLMIN, NW, IM1) IXL=0 IXG=0 DO 100 JA2=1,3*NL 100 B(JA2)=0.0 DO 110 JA2=1,MM DO 105 JA3=1,6 105 FEL(IA2,1A3)=0.0 110 CONTINUE DO 120 I5P=1,NW IF ((GRIDTYPE.NE.90).bR.(SANG.EQ.0.0)) THEN CALL PAIDRB(XLLG,YLLG,XC(15P),YC(I5P),SANG,IXL,IXG,A10ERB , SCLMIN, SPAN, POS, PATYPE, MN1, MN 2, MN11 , MN22, X LL, YLL, XPL, YPL + ,NTG,NLG) ELSE CALL PAID1(M,PM,N,NLG,DLNG,MN1,MN2,MN11.MN22,PATYPE, + XC(I5P),YC(15P),XLL,YLL,XPL,YPL,SANG,IXG,IXL,POS, + AKERB,SCLMIN) ENDIF

Appendix II 219 IF(POS.EQ.'YOGO TO 90 IF(POS.EQ.'YG')GO TO 90 IF(PATYPE.EQ.0)G0 TO 120 IF(PATYPE.EQ.3)THEN CALL LDISTT (MN1,MN2,MN22,PW(15P),XPL,YPL,XLL,YLL,BL,POS,NL) ELSEIF(PATYPE.EQ.4)THEN IF (GRIDTYPE.EQ.90)THEN SANG! =0.0 CALL LDISTR (MN1,MN2,MN1I,MN22,PW(I5P),SANG1,13L,XPL,YPL,XLL, + Yi,L,POS,NL) ELSE • SANGI=SANG CALL LDISTR (MN1,MN2,MN11, + MN22,PW(15P),SANG1,BL,XPL,YPL,XLL,YLL,POS,NL) ENDIF ENDIF DO 115 JOG=I,3*NL 115 B(JOG)=B(JOG)+BL(JOG) 120 CONTINUE IF(IXG.EQ.NW)GO TO 90 IF(LXL.EQ.NW)G0 TO 95 MOVE=MOVE+1 DO 125 IFI=1,3*N 125 B(IFI)=B(IFI)*AIMP DO 130 JA3=1,3*NL 130 B(JA3)=B(JA3)+DLB(IA3) C COMPUTATION OF LOAD MATRIX ENDS 135 DO 140 I=1,NROWS1 BD(I)=-B(1) 140 CONTINUE IF (JRR.EQ.3) THEN DO 150 1=1X,1,-1 J=JDDI(I) DO 145 IP= J,NROWS1 B(IP)=B(IP+1) 145 CONTINUE 150 CONTINUE CALL SOLVE (NROWS,WW,U,B,B) DO 155 I=NROWS+1,NROWSI 155 B(I)=0 DO 165 I=1,IX J=.113D1 (I) DO 160 IP =NROWSI,J,-1 B(IP + I) =B(IP) 160 CONTINUE B(IP +1)=0 165 CONTINUE LT C."0

220

Grillage Analogy in Bridge Deck Analysis

CALL SOLVE(NROW S1 , WW , U, B , B) ENDIF C COMPUTATIONS OF ELEMENT FORCES STARTS DO 170 I =1,MM P1 =P(I,1) P2 =P(I,2) P3 =P(I ,3) P4 =P(1,4) P5 =P(1,5) P6 = P(I, 6) P7 =P(I,7) D1 =-P1*P1*P4 +P2*P2*P6 D2 =P1*P2*(P4-P6) D3 = P2*P7 D4 =P2*P2*P4+P 1 *Pl*P6 D5 = -P 1 *P7 D6 = P5 D7 = -Pl*P 1*P4 +P2*P2*P3 D8 =-Pl*P2*(P3 +P4) D9 =P1*P1*133-P2*P2*P4 M =3*M(I, I) M2 =3*M(I,2) M3 =M1-1 M4 =M1-2 M5 =M2-1 M6 = M2-2 MX1 =D1*B(M4)+D2*B(M3)+D3*(B(M1)-B(M2))+D7*B(M6)+D8*B(M5) MY1 = D2*B(M4) + D4*B(M3) + D5*(B (M1)-B (M2)) + D8*B(M6) + D9*B (M5) MX2 = D7493(M4) + D 8*B (M3) + D3*(13(M1)-B(M2)) + D 1*B (M6) + D2*B(M5) MY2 = D8*B (M4) + D9*13(M3) + D5*(B (M1)-B(M2)) + D2*B(M6) + D4*B (M5) FZ = D3*(B (M4) + B(M6)) + D5*(B (M3) + B (M5 ))+136*(B(M1)-B (M2)) FZ1= FZ MQ 1= (-MX1)*P2 +(MY1)*P1 MQ2 = (-MX2)*P2 + (MY2)*P1 MP1 = (MX1)*P1 +(MY1)*P2 MP2 = (MX2)*P1 +(MY2)*P2 PT(I,3) =FZ1 PT(I,4) = MP 1 PT(L5) = MQ I PT(I, 6) =MQ2 170 CONTINUE C COMPUTATIONS OF ELEMENT FORCES ENDS DO 175 II =1,1X RT(11 ) =0 175 CONTINUE DO 185 11=1,1X DO 180 12=1,MM IFWTT(11 ) . EQ. Ivi(12, 1)).OR.(11-f(11).EQ.M(12,2))) THEN

Appendix 11 221 IF (JTT(II).EQ.M(I2,I)) THEN RT(I1) =RT(I1) +PT(12,3) ELSE RT(I1) =RT(I1)-PT(I2,3) ENDIF ENDIF 180 CONTINUE 185 CONTINUE DO 190 II =1,IX NP1=3*JTT(II) RT(I1)=RT(11)-BD(NP1) 190 CONTINUE DO 195 JP8=1,2 DO 195 IP8=1,MM KP8 =JP8 +4 195 PTI(IP8,11)8)=PT(IP8,KP8) CALL COMPARE1 (PT,B,RT,MAXPT,MAXB,MAXRT,MAXRTN,MM,N,IX, +MOVE,PT1,XL1,YLI,LCASE,CORSPT,CORX,CORY,CORXI,CORY1, + CORRX, CORRY, CORD I ,JTT) IF(LCASE.EQ.1) GOTO 200 GO TO 95 200 READ (3,*)IIC2 IF (IK2.EQ.I)CALL WRITE1(B,MAX7F,MAXB,MAXRT,MAXRTN,MM, +N,IX,MOVE,M,JTT,LCASE,CORSPT,CORX,CORY,CORX1,CORY1, +CORRX,CORRY,CORDI) IF(NOLCASE.EQ.1) GO TO 75 DO 205 IL1=1,4 IL2 =11,1+2 DO 205 IL3 =1,MM PT (IL3,112)=MAXPT(IL1,1,IL3) DO 205 1=2,4 MAXPT2(IL I ,I ,IL3) =MAXPT(IL1,I,IL3) CORSPT2(1,IL3)= CORSPT(1,IL3) CORSPT2(2,IL3) =CORSPT(2,IL3) 205 CONTINUE DO 210 1=1,N DO 210 .1=3*1-2,3*1 B(J)=MAX13(1,J) DO 210 K=2,4 MAXBAK,J)=MAXB(K,J) CORRX2(1,I)=CORRX(1,I) CORRY2(1,I)=CORRY(1,I) CORRX2(2,I)=CORRX(2,I) CORRY2(2,I) =CORRY(2,I) 210 CONTINUE DO 215 1=1,1X RT(I) =MAXRT(I ,I) RTN(I)=MAXICIN ,

222 Grillage Analogy in Bridge Deck Analysis DO 215 J=2,4 MAXRT2(J,I) =MAXRT(J,1) MAXR2N(.1,1)=MAXRTN(J,I) COR2X(D=CORX(I) COR2Y(I)=CORY(I) COR2X1(I)= CORX I (I) COR2Y1 (I) = CORY1 (I) CORD2(I)=CORD1(I) 215 CONTINUE DO 220 JP8=1,2 KP8=JP8+4 DO 220 108=1,MM PTI(IP8,JP8) = MAXPT(I8, 1, IP8) DO 220 1=2,4 MAXPT2(KP8,I,IP8)=MAXPT(I8,LIP8) 220 CONTINUE CALL COMPARE2(PT,B,RT,RTN,OMAXPT,OMAXB,OMAXRT,OMAXRN,MM, +N,IX,IMM1,PT1,MAXPT2, +MAXB2,MAXRT2,MAXR2N,JTT,COR2X,COR2Y,COR2X1,COR2Y1, +COR1X,CORIY,COR1X1,CORIYI,CORSPT1,CORSPT2, +CORRX1,CORRX2,CORRY1,CORRY2,CORD,CORD2) 225 CONTINUE WRITE(4,80) LCASE =0 READ(3,*) 11C3 IF (IK3.NE.1) GO TO 75 CALL WRITE2(0MAXPT,OMAX13,0MAXRT,OMAXRN,MM,N,IX,MOVE,M,JTT, + LCASE,CORIX,CORlY,CORIX1 , CORlY I ,CORSPT1,CORRX1,CORRYI,CORD) GO TO 75 230 FORMAT (1A1) 235 FORMAT(2(/),30X,'BRIDGE DETAILS'/29X,16('*')//) 240 FORMAT(10X,'SKEW ANGLE = ',F9.3, ' DEGREES') 245 FORMAT(10X,'NO. OF ELEMENTS= ',13/,10X, 'NO. OF NODES=', + 13/,10X,'NO. OF ELEMENT GR.OUPS-=',12/, + 10X, 'YOUNGS MODULUS OF ELASTICITY=', + E14.7, ' T/SQ.MMVIOX,'SHEAR MODULUS OF ELASTICITY = ',E14.7, + ' T/SQ.MMV) 250 FORMAT(//15X,'GROUP',I2/15X,7('-')) 255 FORMAT(20X,'NO. OF ELEMENTS IN THIS GROUP = ',12/20X, + 'LENGTH OF ELEMENT IN X-DIRECTION =',F7.0,' MM'/20X, + 'LENGTH OF ELEMENT IN Y-DIRECTION =',F7.0,' MM'/20X, + 'MOMENT OF INERTIA OF ELEMENT =',E14.7,' MM4'/20X, + 'TORSIONAL INERTIA OF ELEMENT =', E14.7,' MM4') 260 FORMAT(/20X,'ELEMENTS IN THIS GROUP:'//7(20X,5(12,'-', + I2,3X)/)) 265 FORMAT (/15X,'BOUNDARY CONDITIONS- NODES' + /15X, 33 ('-')) 270 FORMAT (20X,'NO. OF NODES 12120X,

2' 2 2 2

Appendix Il 223 + 'NODE NO. OF S:', 10(12,2X)) 275 FORMAT (20X,'NON YIELDING S') 280 FORMAT (20X,'YIELDING S') 285 FORMAT(20X,'STIFFNESS OF BEARINGS',2X,E9.3,'N/MM') 290 FORMAT(/5X,'NO. OF LONGITUDNAL GRID LINES = ',14//5X, + 'SPACINGS (IN MM) = ',(F8.0,5X)) 295 FORMAT(/3X,'DETAILS OF ELEMENTS IN EACH GROUP') 300 FORMAT(/15X,'INITIAL COORDINATE OF ' +,'LEADING LEFTMOST WHEEL IS','(',F8.1,',', F8.1,')',/15X, + 'INCREMENT ALONG-X-AXIS=', F7.0,14M',/15X,'INCREMENT ALONG ' +,'Y-AXIS=', F7.0,'MM' +,/15X,'FINAL CO-ORDINATE OF LEADING LEFTMOST WHEEL IS',

a

+,/15X,'IMPACT FACTOR= ',F8.4//) 310 FORMAT(' ENVELOPE VALUES UNDER LIVE LOAD ',I3) 315 STOP END c*********************************************************************** SUBROUTINE STIFF(N,MM,WW,P,M,S) IMPLICIT real*8(A-H2O-Z) DIMENSION S(1,50000),P(1000,7),M(1000,2) INTEGER U,WW,R1,R2,R3,R N3 =N*3 N8 =N3*WW DO 320 II=1,N8 320 S(1,II)=0.0 DO 325 I=1,MM P1=P(I,1) P2=P(I,2) P3 =P(I,3) P4 =P(I,4) P 5 = P ( I , 5 P 6 = P ( I , 6 ) = P ( I M 1 = 3 * M ( I M1 = (MI-1)*WW M2 =(MI-2)*WW M3 =(MI-3)*WV/ S(1,M3 +1) =S(1,M3 + 1) + P 1 *P I *P4 +P2*P2*P6 S(1,M3+2)=S(1,M3+2)+P1*P2*(P4P6) S( I ,M3 +3) =S(1,M3 +3) + P2*P7 S(I,M2+1)=S(I,M2+1)+P2*P2*P4+P1iP1*P6 S(1 , M2 +2) =S(1 ,M2 +2)-P I *P7 S(1,M1+1)=S(1,M1+1)+P5 MI=3*M(I,2) MI = (MI-I )*WW M2 =(MI-2)*WW M3 =(MI-3)*WW

) , ,

P 7 1

. 7 ) )

224 • Grillage Analogy in Bridge Deck Analysis S(1,M3+1)=S(1,M3+1)+Pl*Pl*P4+P2*P2*P6 S(1 ,M3 +2) =S(1,M3 + 2) +P I *P2*(P4-P6) S(1,M3 +3) =S( I ,M3 +3)-P2*P7 S(1,M2+1)=S(I,M2+1)+P2*P2*P4+Pl*P1*P6 S(1,M2+2)=S(1,M2+2)+P1*P7 S(I,M1+1)=S(1,M1+1)+P5 C ABOVE STATMENTS SET UP DIAGONAL ELEMENTS U=3*IABS(M(I,2)-M(I,1)) R=3*M(I,1) RI =(R-1)*'WW+U R2 =(R-2)*W1V+U R3 =(R-3)*WW +U S(1,R3+1)=S(1,R3+1)-Pl*P1*P4+P2*P2*P3 S(1,R3+2)=-S(1,R3+2)-Pl*P2*(P4+P3) S(1 ,R3 +3) = S(1,R3 +3)-P2*P7 S(1,R2)=S(1,R2)-P1*P2*(P4+P3) S(1,R2 + I) = S(1,R2+ 1)-P2*P2*P4 +Pl*Pl*P3 S(1,R2+2)=S(1,R2+2)+P1*P7 S(1,R1-1)=S(I,R1-1)+P2*P7 S(1,R1)=S(1,R1)-PI*P7 S(1,R1+ I)=S(1,R1+1)-P5 325 CONTINUE C ABOVE STATMENT SET UP OFF DIAGONAL ELEMENTS RETURN END • c*********************************************************************** SUBROUTINE INODE (M,I1,I2,16) IMPLICIT real*8(A-H2O-Z) DIMENSION M(1000,2) IF (12.LT.I1) THEN WRITE(*,*)*ERROR' READ(*,*)0 ELSE K1=0 330 K1 =K1+1 IF (M(K1,1).NE.II.OR.M(K1,2).NE.I2) GO TO 330 I6=K1 ENDIF RETURN END c*********************************************************************** SUBROUTINE PAID1(M,PM,N,NLG,DLNG,MN1,Mi',12,MN11,MN22, + PT,XL,YL,XLL,YLL,XPL,YPL,SANG,IXG,IXL,POS,AKERB,SCLMIN) IMPLICIT real*8(A-H2O-Z) INTEGER PT CHARACTER*2 POS DIMENSION M(1000,2),PM(1000,1),DLNG(100),XLLG(100),SLNG(I00) POS='NN'

Appendix-11 225 X=0 DO 335 I=1,NLG-1 SLNG(I)=0 X=X+DLNG(I) 335 SLNG(1)=X SPAN =0 DO 340 !IFF =1 ,(N/NLG-1) XLLG(IFF) =0 CALL INODE(M,IFF,IFF+1,IN1) SPAN=SPAN+PM(IN1,1) XLLG(IFF)=SPAN 340 CONTINUE IF (YL.LT.(AKERB+SCLMIN))GO TO 470 IF(YL.GT.(SLNG(NLG-I)-(AKERE+SCLMIN)))GO TO 475 IF((XL.LT.(SLNG(NLG-I)*TAN(SANG))).AND.(YL.GT.XL/ + TAN(SANG))) GO TO 480 IF ((XL.GT.SPAN).AND.(YL.LT.((XL-SPAN)/TAN(SANG)))) + GO TO 485 XLEN=0.0 DO 345 II =1,NLG-1 CALL INODE(M,I1,II + I ,LI) 345 XLEN=XLEN+PM(L1,1) IF ((XL.GT.XLEN).AND.(XL.LE.SPAN))GO TO 350 IF (XL.LT.XLEN) GO TO 380 IF (XL.GT.SPAN) GO TO 425 3 5 0 PT = 4 DO 355 II =NLG,N/NLG-I CALL INODE (M,I1J1+1,L1) XLEN=XLEN+PM(L1,1) IF (XL.LE.XLEN) GO TO 360 355 CONTINUE 360 IF(YL.GT.0)GOTO 365 MN 1=11 MN 2=11+1 MN 11 = II +(N/NLG-1) MN 22=11+N/NLG XLL=XL-(XLEN-PM(LI,1)) YLL=0.0 XPL=PM(L1,1) YPL=DLNG(1) RETURN 365 DO 370 12=1,NLG-1 IF ((SLNG(12)-YL).GE.0.0)G0 TO 375 370 CONTINUE 375 MN 1=11+ (I2-1)*(N/NLG- ) MN 2=iviNi+1 MN 1I=MN1+(N/NLG-1) MN 22=MN11+1

226 Grillage Analogy in Bridge Deck Analysis XLL=XL-(XLEN-PM(L1,1)) IF(12.NE.1)YLL=YL-SLNG(I2-1) IF(12.EQ.1)YLL=YL XPL=PM(L1,1) YPL=DLNG(12) RETURN 380 POS = 'XL' DO 390 13= 1,NLG-1 IF(13.NE.1)GOTO 385 IFaL.LE.XLLG(1)).AND.(YL.LE.XL/TAN(SANG))) GOTO 395 GOTO 390 385 IF ((XL.LE.XLLG(I3)).AND.(YL.LE.XL/TAN(SANG)).AND.(YL.GE.SLNG + (13-1))) GO TO 395 390 CONTINUE GO TO 400 395 PT=3 MN 1 =1 ,+(I3-1)*N/NLG MN 2=MN1+I MN 22=MN1+N/NLG MN 11=0 CALL INODE(M,I3,13 +1,131) XPL=PM(I3I,1) YPL=DLNG(13) IF(13.NE.1)XLL=XL-XLLG(I3-1) IF(I3.NE. DYLL = YL,SLNG(13-1) IF(I3.EQ.1)XLL=XL IF(I3.EQ.1)YLL=YL RETURN 400 PT =4 DO 405 14 =2,NLG-1 IF (XL.LE.XLLG(I4)) GO TO 410 405 CONTINUE 410 DO 415 15=1,NLG-1 IF ((YL-SLNG(15)).LE.0) GO TO 420 415 CONTINUE 420 MN I =14+(15-1)*(N/NLG-1) MN 2=MN1+1 MN 11 =MN1 +N/NLG- I MN 22 = MNI 1 + 1 CALL INODE(M,I4,14+1,I3I) XPL=PM(I3I,I) YPL=DLNG(I5) XLL = X L-XL.TL-G(I4-1) IF(15.NE.1)YLL =YL-SLNG(15-1) IF(I5.EQ.1)YLL=YL RETURN 425 POS = 'XG' XXL= (SPAN +XLLG(NLG-1))-XL

Appendix II 227 YYL=SLNG(NLG-1)-YL DO 435 16=1,NLG-1 IF(I6.NE.1)GOTO 430 IF((XXL.LEALLG(1)).AND.(YYLLE.XXL/TAN(SANG)))GOTO 440 GOTO 435 430 IF ((XXL.LE.XLLG(16)).AND.(YYLLEJO(L/TAN(SANG)). + AND.(YYLGE.SLNG(16-1))) GO TO 440 435 CONTINUE GO TO 445 440 PT =3 MN I =(146)*N/NLG+N MN 2=MN1-1 MN 22=MN1-NINLG MN 11=0 CALL INODE(M,I6,16+1,131) XPL = PM(I3I, 1) YPL=DLNG(NLG-16) IF(I6.NE.1)XL,L=XXL-XLLG(16-1) IF(I6 .NE. DYLL =YYL-SLNG(I6-1) IE(16.EQ.1)XLL=XXL IF(I6. EQ. 1)YLL=YYL RETURN 445 PT=4 DO 450 17=-2,NLG-1 IF (XXL.LEILLG(17)) GO TO 455 450 CONTINUE 455 DO 460 I8=1,NLG-1 IF ((YYL-SLNG(18)).LE.0) GO TO 465 460 CONTINUE 465 MN22= N-(18-1)*N/NLG-(I7-18) MN11=MN22-1 MN2=MN22-(N/NLG-1) MN1=MN2-1 CALL INODE(M,I7,17+1,131) XPL=PM(I3I,1) YPL=DLNG(I8) XLL=XLLG(I7)-XXL IF(18.NE.1)YLL =SLNG(18)-YYL RETURN 470 POS='YL' RETURN 475 POS='YG' RETURN 480 IXL =Del,4-1 GO TO 490 485 IXG =IXG+ I 490 PT =0 RETURN

228 Grillage Analogy in Bridge Deck Analysis END c********************************************************************** SUBROUTINE PAIDRB(XLLG,YLLG,XL,YL,SANG,IXL,IXG,AKERB,SCLMIN, SPAN,POS,PT,MN1,MN2,MN11,MN22,XLL,YLL,XPL,YPL,NTG,NLG) IMPLICIT real*8(A-H2O-Z) DIMENSION YLLG(0:100),XLLG(0:100) INTEGER PT CHARACTER*2 POS POS= PT=0 • IF (YL.GT.(AKERB+SCLMIN)).THEN IF (YL.LT.(YLLG(NLG-1)-(AKERB+SCLMIN))) THEN IF (XL.GT.(YL*TAN(SANG))) THEN IF X(XL-SPAN).LT.(YL*TAN(SANG))) THEN PT =4 IF (YL.LT.0.0) THEN 11=1 495 IF (XL.LT.XLLG(I1)) THEN  MN1=I1 MN2=I1 +1 MN11=MN1+NTG MN22=MN11+1 XLL=XL-XLLG(I1-1) YLL = 0 XPL =XLLG(I1)-XLLG(11-1) YPL=YLLG(1) ELSE II =I1 +1 GO TO 495 ENDIF ELSE IF (YL.GT.YLLG(NLG-1)) THEN 11=1 G=XL-YLLG(NLG-1)*TAN(SANG) 500 IF (G.LT.XLLG(I1)) THEN MN II = II +NTG*(NLG-1) MN22 =MN11 +1 MN1=MN11-NTG MN2 =MN1 +1 XLL=G-XLLG(I1-1) YLL=YLLG(NLG-1)-YLLG(NLG-2) XPL=XLLG(I1)-XLLG(I1-1) YPL=YLL ELSE I1 =II +1 GO TO 500 ENDIF ELSE 11=1

Appendix 11 229 505 IF (YL.LT.YLLG(I1)) THEN N2=11 ELSE I1=I1+1 GO TO 505 ENDIF II =1 G=(XL-YL*(TAN(SANG))) 510 IF (G.LT.XLLG(I1))THEN N1 =I1 ELSE 11=I1+1 GO TO 510 ENDIF MNI =N1 +(N2-1)*NTG MN2=MN1+1 MN11=MN1+NTG MN22=MN11 +1 XLL=XL-XLLG(N11)-YL*TAN(SANG) YLL=YL-YLLG(N2-1) XPL=XLLG(N1)-XLLG(N1-1) YPL=YLLG(N2)-YLLG(N2-1) ENDIF ELSE IXG PT =0 ENDIF ELSE IXL=IXL +1 PT=0 ENDIF ELSE POS = 'YG' ENDIF ELSE POS ='YL' ENDIF RETURN END C **************************************************** SUBROUTINE WLOAD (LCASE,PW.SCLMIN,NW,MOVE) IMPLICIT real*8 (A-H2O-Z) DIMENSION PW(100),PWI(100) DO 515 11=1,28 515 PW (11)=0.0 GO TO(10,1,3,3,5,5,7,1,9),LCASE IF AIL= 1 RETURN

230 Grillage Analogy in Bridge Deck Analysis 1 SCLMIN=400 PW(I)=1.35 PW(2)=1.35 PW(3)=135 PW(4)=1.35 PW(5)=5.70 PW(8)=5.70 PW(6)=5.70 PW(7)=5.70 PW(9)=3.40 PW(10)=3.40 PW(11)=3.40 PW(12)=3.40 PW(13)=3.40 PW(14)=3.40 PW(15)=3.40 PW(16)=3.40 IF (LCASE.EQ.2)GO TO 2 RETURN 2 DO 520 1=17,32 520 PW (I)=PW(I-16) RETURN . 3 SCLMIN=1405 DO 525 1=1,4 PW (1)=2.0 PW (I+4)=3.0 525 PW (1+8)=3.0 DO 530 1=13,28 530 PW (I)=4.25 RETURN 5 SCLMIN=1405 DO 535 1=1,8 535 PW (I)=5.0 RETURN 7 SCLMIN=1620 DO 540 1=1,20 540 PW (1)=3.50 RETURN 9 IF (MOVE.NE.1) GO TO 550 READ (3,*) SCLMIN1 READ (3,*)(PW1(K),K=1,NW) WRITE(4,545)(PW1(K),K=1,NW) 545 FORMAT('LOAD ON WHEEL' ,10E10.2) 550 SCLMIN =SCLMINI DO 555 K=1,NW PW (K) =PWI(K) 555 CONTINUE 10 RETURN

Appendix 11 231

r.

END c********************************************************i************** SUBROUTINE LLOAD (LCASE,XT,YT,XW,YW,NW,NWPA,MOVE) IMPLICIT reals8(A-H2O-Z) DIMENSION XW(100),YIN(100),XW1(100),YW1(100) DO 565 11=1,28 XW (II)=0.0 565 YW (11)=0.0 X=XT Y=YT GO TO (10,1,3,4,5,6,7,1,9),LCASE IF AIL=1 • RETURN 1 NW =16 NWPA =2 DO 570 11 =1,8' YW (I1*2-1)=Y 570 YW (11 41)=Y+1800 XW (1)=X XW (2)=XW(1) XW (3) =XW(2)-1100 XW (4)=XW(3) XW (5) =XW(4)-3200 XW (6)=XW(5) XW (7)=XW(6)-1200. XW (8)=XW(7) XW (9)=XW(8)-4300 XW (10)=XW(9) XW (11)=XW(10)-3000 XW (12)=XW(11) XW (13)=XW(12)-3000 XW (14)=XW(13) XV(I5)=XW(14)-3000 XV, (16)=XW(15) IF (LCASE.EQ.2) GO TO 2 RETURN 2 NW =32 NWPA =4 DO 575 1=17,32 575 XW (1)=XW(I-16) DO 580 1=9,16 YW (1*2-1)=YW((I-8)*2)+1700 580 YW (I*2)=YW(I*2-1)+1800 RETURN 3 NW=28 NWPA =4 K-7 585 DO 590 1=1,K -

232 Grillage Analogy in Bridge Deck Analysis YW(I*4-3)=Y YW(I*4-2) =Y +450 YW(I*4-1)--= Y+ 1930 590 YW(I*4) =Y +2380 IF (LCASE.EQ.5)GO TO 615 595 DO 600 I=1,4 XW(I) =X XW(I+4) =XW(I)-3960 XW(I + 8) = XW(I +4)-1520 VW (I+ 12) =XW(I +8)-2130 XW(I+16)=XW(I+12)-1370 XW(I+20)=XW(I+16)-3050 600 XW(I +24) = XW(I +20)-1370 RETURN 4 NW=28 NWPA=4 L=7 605 DO 610 I=1,1, YW(I*4-3)=Y YW(I*4-2)=Y+795 YW(I*4-1) = Y + 1585 YW(I*4)=Y +2380 610 CONTINUE IF (LCASE.EQ.6)GO TO 615 GO TO 595 5 NW =8 NWPA=4 K=2 GO TO 585 615 DO 620 I=1,4 XW(I) =X 620 XW(I+4)=X-1220 RETURN 6 NW=8 NWPA=4 L =.2 GO TO 605 7 NW=20 NWPA=2 DO 625 1=1,10 YW(I*2-1)=Y 625 YVV(I *2) =Y +2060 DO 630 1=1,10 XW(I*2-1)=X-((I-1)*457) 630 XW(I*2)=XW(I*2-1) RETURN !F.

N!'r 1 )r:Ct TO Fall

READ (3,*)NW I

Appendix 1.1 233 IF(NW1.EQ.1)G0 TO 640 DO 635 12P=2,NW1+1 IF (12P.EQ.NW1+1)GOTO 635 READ (3,*)XW1(I2P),YW1(I2P) 635 CONTINUE 640 NW=NW1 XW(I)=XT YW(1)=YT DO 645 I1P=2,NW XW(I1P) =X1V(1)+ XW 1(I1P) YW(IIP)=YW(1)+YWI(I1P) 645 CONTINUE I0 RETURN END SUBROUTINE DECOMP(NROWS,IWW,A,U) IMPLICIT real*8(A-H,G-Z) DIMENSION A(1,50000), U(50000) DO 650 I=1,NROWS*IWW 650 U(I)=0.0 DO 670 I=I,NROWS IP = (I-1)*IWW+ 1 SUM=0.0 DO 655 K=1, I-1 IF ((I+1).GT.(K+IWW)) GO TO 655 SUM= SUM + (U((K-1)*IWW+ I-K+1))**2655 CONTINUE U(IP)=SQRT(A(1,IP)- SUM) DO 665 J= 2,IWW SUM=0.0 DO 660 K= 1,1-1 IF ((I+J).GT.(K+IWW)) GO TO 660 SUM= SUM + (U((K-1)*IWW +/-K +I))*(U((K-1)*IWW+I-K +1)) 660 CONTINUE U(IP+.1-1)=(A(1,IP+J-1)-SUM)/U(IP) 665 CONTINUE 670 CONTINUE RETURN END c*********************************************************************** SUBROUTINE SOLVE(NROWS,IWW,U,F,D) IMPLICIT real*8(A-H2O-Z) DIMENSION U(50000),F(1000),D(I000),X(1000) X(1)=F(1)/U(1) DO 680 I =2,NROWS SUM=0 DO-675 J=1, I-I IF ((I-J).GE.IWW) GO "ID t,-75

234 Grillage Analogy in Bridge Deck Analysis SUM= SUM + U((J-1)*IWW + I + 1-1)*X(J) 675 CONTINUE X(I)=(F(I)-SUM)/U(1+(I-1)*IWW) 680 CONTINUE D(NROWS)=X(NROWS)/U((NROWS-1)*IWW+1) DO 690 I=NROWS-1,1,-1 SUM =0 DO 685 J=2 ,IWW SUM=SUM + U(IWW*(I-1)+J)*D(I+J-1) 685 CONTINUE D(1)=(X(1)-SUM)/U((I-1)*IWW +1) 690 CONTINUE RETURN END c*********************************************************************** SUBROUTINE MODSTIF(IWW,IX,JDD,S,NROWS) IMPLICIT real*8(A-H2O-Z) DIMENSION S(1,12000),JDD(100) DO 705 I=IX,1,-1 J=JDD(I) DO 700 IP = 1,J-1 IF((J-IP +1).GT.IWW) GO TO 700 IA=(IP-1)*IWW+(J-Ip+1) I B =a p - i r r ww+ r ww DO 695 1Q-=-IAJB-1 S(1,IQ)=S(1,IQ+1) 695 CONTINUE S(1,IQ)=0 700 CONTINUE 705 CONTINUE DO 715 I=IX,1,-1 J =JDD(I) IA= (J-1)*IWW +1 IB=(NROWS-1 )*IWW DO 710 IP =IA,IB S(1,IP)=S(1,1P+IWW) 710 CONTINUE NROWS=NROWS-1 715 CONTINUE RETURN END SUBROUTINE SORT(IX,JDD,JDD1) IMPLICIT real*8(A-H2O-Z) DIMENSION JDD(100),JDD1(100) DO 720 I=1 ,IX 720 JDD1(I) =JDD(I) IP=IX+1

72

7:

Appendix II 235 DO 730 1=1,a-1 IP=IP-1 IK = 1 DO 725 J=1,IP IF (JDD1(IK).GT.JDD1(J)) GO TO 725 IK 725 CONTINUE EMP =.113D1(IP) IDD1(IP)=./DD1(1K) IDD1(1K) =EMP 730 CONTINUE RETURN END c***************m******************************************************* SUBROUTINE LDISTR (MN1,M N2,MN11, M N22, PLOAD , SANG, + BL,XPL,DLNG,XLL,YLL,POS,NL) IMPLICIT real*8(A-H, 0-Z) DIMENSION BL(I000)CHARACTER*2 POS DO 735 I=1 ,3* NL 735 BL(I)= 0 C =XLL B =YLL/COS(SANG) D =XPL-XLL A =(DLNG-YLL)/COS(SANG) X =XPL Y =DLNG/COS(SANG) PE =PLOAD*A*A*(3*B +A)/Y**3 PF =PLOAD*B*B*(3*A+B)/Y**3 BMF =-PLOAD*A*B*B/Y/Y*COS(SANG) BME=PLOAD*A*A*B/Y/Y*COS(SANG) BMFI =-PLOAD*A*B*B/Y/Y*SIN(SANG) BME1 = PLOAD *A*A*B/Y/Y*SIN(SANG) BL(3*MN I )=PE*D*D*(3*C + D)/X**3-6*BMEI*D*C/X**3 BL(3*M N2) = PE*C*C*(3*D + C)/X**3 +6*BMEI*D*ca**3 BL(3*MN11)= PF*D*D*(3*C + D)/X**3-6*BMF1*D*C/X**3 BL(3*MN22)=PF*C*C*(3*D +C)/X**3 +6*BMF1*D*C/X**3 BL(3*MN1-1)=-(PE*C*D*D/X/X-BME1*D*(2*C-D)/X**2) BL(3*MN2-1)= -(-PE*C*C*D/X/X-BMEl*C*(2*D-C)/X**2) BL(3*MN11-1) --(PF*C*D*D/X/X-BMFI*D*(2*C-D)/X**2) BL(3*MN22-1) =-(-PF*C*C*D/X/X-BMFI*C*(2*D-C)/X**2) BL(3*MNI-2) =BME*D/X BL(3*MN2-2) =BME*CIX BL(3*MN I1 -2) =BMF*D/X BL(3*M N22-2) = Bne*ca RETURN END c***********************************************************************

236 Grillage Analogy in Bridge Deck Anidysis SUBROUTINE LDISTT (MN1,MN2,MN22,PLOAD,XPL,DLNG,XLL,YLL,BL,POS,NL) IMPLICIT real*8(A-H2O-Z) DIMENSION BL(1000) CHARACTER*2 POS DO 740 I=1,3*NL 740 BL(I)=0 A =XLL D =YLL C =XPL-XLL B=DLNG-YLL X=XPL Y=DLNG SINT=X/SQRT(X*X+Y*Y) TANT=X/Y COST=SINT/TANT EF=A/TANT PE=PLOAD*(EF-D)**2*(2*D+EF)/EF**3 PF=PLOAD*(3*EF-2*D)*D*D/EF**3 BMF=PLOAD*D*D*(EF-D)/EF/EF BME=PLOAD*D*(EF-D)**2/EF/EF BL(3*MN1)=(PE+PF)*C*C*(3*A+C)/X**3+6*A*C*BMF/X**3*SINT*COST BL(3*MN2)=PE'iA*A*(3*C+A)/X**3 BL(3*MN22)=PF*A*A*(3*C+A)/X**3-6*A*C*BMF*SINT*COST/X**3 BL(3*MNI-1)=-(BMF*C/X*SINT*COST*(1+(2*A-C)/X)+PF*A*C*C/X/X + +PE*C**2*A/X**2) BL(3*MN2-1)=PE*A*A*C/X/X BL(3*MN22-1)=-BMF*A/X*SINT*COST*(1+(2*C-A)/X)+PF*C*A*AfX./X BL(3*MN1-2)=C/X*(PF*A*C/XJTANT+BME+BMF*((2*A-C)/X*COST*COST + -SINT*SINT)) BL(3*MN2-2)=BME*A/X 13L(3*MN22-2) =-PF*C*A*A/X/X/TANT-BMF*A/X*(SINT*SINT-(2*C-A)/X + *COST*COST) IF(POS.NE.'XG')GOTO 750 DO 745 1=1,2 BI.,(3*M N1-1) = -BL(3*MN1-1) BL(3*MN24)=-BL(3*MN2-I) 745 BL(3*MN22-1)=-BL(3*MN22-I) 750 RETURN END C ***************************************,ie************* SUBROUTINE IMPACT(LCASE,SPAN,SANG) IMPLICIT REAL*8(A-H2OZ)' COMMON/XXUAIMP real*8 L • WRITE(*, *)'SPAN = ',SPAN WRITE(*,*)'ANGLE=',SANG L=SPAN*COS(SANG)/1000 WRITE(*,*)'SPAN FOR IMPACT FACTOR CALC. =' ,L

'IL)

O

Appendix II 237 WRITE(4,755)L 755 FORMAT(10X,'SPAN FOR IMPACT FACTOR CALCULATION= ',F6.2) GO TO (1,1,3,3,3,3,1,1,9),LCASE 1 IF (L.LE.3.0)AIMP=1.5 IF (L.GE.45.0)AIMP=1.0875 IF (L.GT.3.0.AND.L.LT.45.0)AIMP —1.0+(4.5/(6.0+L)) RETURN 3 IF (LLE.12.0)AIMP=1.25 IF (L.GE.45.0)AIMP=1.0875 IF (L.LT.45.0.AND.L.GT.12.0)AIMP=1.0 +(4.5/(6.0+L)) RETURN 7 IF (L.LE.5.0)AIMP=1.25 IF (L.GT.39.0)GO TO 1 IF (L.GE.9.0)AIMP=1.10 IF (L.LT.9.0.AND.L.GT.5.0)AIMP=1.25-0.15*(L-5.0)/4.0 RETURN 9 A IM P = 1 . 0 RETURN END c***************************************************#******************* SUBROUTINE DEAD LOAD (B,N,NLG,M,PM,E1,E2) IMPLICIT REAL*8(A-H2O-Z) real*8 K(100),B(1000),E1,E2 DIMENSION M(1000,2),PM(1000,1) DO 760 I=1,3*N 760 B(I)=0.0 READ(3,*)CHOICE IF(CHOICE.EQ.1)THEN READ (3,*)(1C(1),I=1,2*NLG) WRITE(4,795) WRITE(4,800)(I,K(I*2-1),K(2*1),I=1,NLG) WRITE(4,805)E1,E2 J=0 DO 765 I=1,N,NINLG• J=1+2 CALL INODE(M,I,I+1,I1) AL=PM(I1,1)/2. B(3*I)=B(3*I)+K(J-1)*AL B(3*I-1)=B(3*I-1)-K(J-1)*AL*AL/3. B(3*I-2)=B(3*12)+K(J)*AL JI=I+N/NLG-1 CALL INODE (M,II-1,II,I1) AL=PM(I1,1)/2. B(3*II)=B(3*II)+K(J-1)*AL B(3*II-1)=B(3*II-1)+K(J-1)*AL*AL/3. B(3*II-2)=B(3*II-2)+K(.1)*AL 765 CONTINUE DO 785 i=2.INI

238 Grillage Analogy in Bridge Deck Analysis DO 770 J=1,NLG IF (I.EQ.N/NLG*J)G0 TO 785 IF (I.EQ.N/NLG*J+1)G0 TO 785 770 CONTINUE DO 775 J=1,NLG IF (I.LT.N/NLG*J)G0 TO 780 775 CONTINUE 780 CALL INODE(M,I-1,I,I1) CALL INODE(M,I,I+1,12) AL= (PM(I1, 1) +PM(I2, 1))/2. ALl = PM(I1,1)/2.0 AL2 = PM(I2,1)/2.0 B(3*1)=B(3*I) +K(J*2-1)*AL B(3*I-1)=B(3*I-1)-K(J*2-1)*(-PM(I1,1)**2+PM(12,I)**2)/12 B(3*I-2)=B(3*I-2)+K(J*2)*AL 785 CONTINUE DO 790 I=1,NLG B(3*((1-1)*N/NLG+I))=El*K(2*I-1)+B(3*((I-1)*N/NLG+1)) 790 B(3*N*IINLG)=E2*K(2*I-I)+B(3*N*I/NLG) ELSE READ(3,*)(B(3*I),I=1,N) WRITE(4,8I0) WRITE(4,815)(I,B(3*1),I=1,N) ENDIF 795 FORMAT(/5X,'DEAD LOAD ALONG LONGITUDINAL GRID LINESV5X,39c*'), + I/5X,'LONG. GRID',5X,'VERTICAL LOAD',5X,'TORS1ONAL MOMENT'/5X, + 'LINE NO.',13X,'TIMM',14X,'T MMTMW/60('*')/) 800 FORIvIAT(8X,I3,9X,E11A,9X,E11.4) 805 FORMAT(!/'THE END PROJECTIONS OVER LINES ARE ',E1I.4, +' & A,' MM'/' RESPECTIVELY.'//) 810 FORMAT(/5X,'DEAD LOAD AS SUPPLIED,DIRECTLY ON NODES,VERTICAL' + ,'. LOAD ONLY',/5X,30('*')//5X,'NODE NO.' ,10X,'LOAD(T)' + /30('*'),/) 815 FORMAT((6X,I3,I3X,E11.4)) RETURN END c*********************************************************************** SUBROUTINE COMPARE1(PT,B,RT,MAXPT,MAX13,MAX.RT,MAX RTN,MM,N + ,IX,IMI,PT1,X,Y + ,LCASE,CORSPT,CORX,CORY,CORX1,CORY1,CORRX,CORRY,CORD1 + ,JTT) IMPLICIT REAL*8(A-F1,0-Z) DIMENSION PT(1000,7),B(1000),RT(100),PT1(1000,2),JTT(100) REAL *8 MAXPT(6,4,500),MAXB(4,500),MAXRT(4,100),MAXRTN(4,100) + ,CORSPT(2,500),CORX(100),CORY(100),CORX1(100),CORY1(100) + C ORRX(2,500). CORRY(2 .500) CORD I (100) IF (1M1.GT.1)G0 TO 850

Appendix II 239

),

DO 820 I=1,4 1 =I +2 DO 820 K=I,MM MAXPT(J, 1,K) = PT(K, J) MAXPT(I,2,K)=LCASE MAXPT(L3,K) = X MAXPT(I,4,K)=Y CORSPT(I,K) =PT(K,4) CORSPT(2,K)=PT(K,3) 820 CONTINUE KL DO 825 I=1,N N11 =3*I-2 N13 =3*I . CORRX(1,I) -=13(N11 +1) CORRX(2,I)=B(N13) CORRY(1,I) =B(N 11) CORRY(2,I)=B(N13) DO 825 J=N11,N13 MAXB(1,J)=B(J) MAXB(2,J)=LCASE MAXB(3 ,J) =X MAX13(4,J)=Y 825 CONTINUE DO 830 I=1,IX MAXRT(I ,I) =RT(I) MAXRT(2,I) =LCASE MAXRT(3,I) =X MAXRT(4,I) =Y CORXI (I) = B(JTT(I)*3 -2) CORYI (I) =B(JTT(I)*3-1) CORD1(1) = B(ITT(I)*3) MAXRTN(1,I) =RT(I) MAXRTN(2,I)=LCASE MAXRTN(3,I) =X MAXRTN(4,I) =Y WRITE(*,*)LJTT(I),B(JTT(I)*3-2),B(JTT(1)*3-1) CORX(I)=B(JTT(1)*3-2) CORY(I)=B(JTT(I)*3-1) 830 CONTINUE DO 835 1=5,6 DO 835 K=1,MM J=1-4 MAXPT(I, 1,K) = PT1(K,J) MAXPT(1, 2,K) = LCASE MAXPT(1,3,K)= X MAXPT(I,4,K)=Y 835 CONTINUE

240 Grillage Analogy in Bridge Deck Analysis DO 845 1=3,6 DO 845 K=1,MM IF (1.GT.4)G0 TO 840 IF (MAXPT(I,1,K).LT.0.0)THEN MAXPT(I,1,K)=0 MAXPT(I,2,K)=0 MAXPT(I,3,K)=0 MAXPT(I,4,K) =0 ENDIF GO TO 845 840 IF (MAXPT(I,1,K).GT.0.0)THEN MAXPT(L1,K)=0 MAXPT(I,2,K)=0 MAXPT(I,3,K)=0 MAXPT(I,4,K) =0 ENDIF 845 CONTINUE RETURN 850 DO 855 1=1,2 J=I+2 DO 855 K=1,MM IF (ABS(MAXPT(I,1,K)).GE.ABS(PT(K,J)))GO TO 855 MAXPT(I, 1,K) = PT(K,J) MAXPT(I,2,K) =LCASE MAXPT(1,3,K)=X MAXPT(I,4,K) =Y CORSPT(I,K)=PT(K,54) 855 CONTINUE DO 860 1=3,4 J=1+2 DO 860 K=1,MM IF (MAXPT(I,1,K).GE.PT(K,J))GO TO 860 MAXPT(1,1,K)=PT(K,J) MAXPT(I,2,K) =LCASE MAXPT(1,3,K)=X MAXPT(1,4,K)=Y 860 CONTINUE DO 865 1=5,6 DO 865 K=1,MM J=I-4 IF (MAXPT(1,1,K).LT.PT1(K,J))G0 TO 865 M A'XPT(I , 1 ,K) = PT1(K,J) MAXPT(1,2,K)=LCASE MAXPT(1,3,K)=X MAXPT(1,4,K)=Y 865 CONTINUE KL1 = I DO 870 I=1,N

Appendix II 241 N 1 =3*1-2 IF (ABS(MAX13(1,N11)).GE.ABS(B(N11)))G0 TO 870 MAXI3(1 ,N 1) =B(NI1) MAXI3(2,N11)=LCASE MAX13(3,N11)=X MAX13(4,N11)=Y CORRX(1 ,I) = B(N1 I +1) CORRX(2,I)=B(N11+2) 870 CONTINUE KL1 =1 DO 875 I=1,N N12=3*1-1 IF (ABS(MAXB(1,N12)).GE.ABS(B(N12)))G0 TO 875 MAXB(1,N12) =-13(N12) MAX13(2,N12)=LCASE MAXB(3,N12)=X MAX13(4,N12)=Y CORRY(1,I)=B(N12-1.) CORRY(2,1)=B(N12+1) 875 CONTINUE DO 880 I=1,N N13 =3*I IF (ABS(MAX13(1,N13)).GE.ABS(B(N12)))G0 TO 880 MAXB(1,N13)=B(N13) MAXB(2,N13)=LCASE MAXB(3,N13)=X MAXB(4,N13)=Y 880 CONTINUE DO 885 I=1,IX IF(MAXRT(1,1).GT.RT(I))G0 TO 885 MAXRT(1,I)=RT(I) MAXRT(2,I) =LCASE MAXRT(3 ,I) =X MAXRT(4,I)=Y CORX1(I) = B (JTT(I)*3 -2) CORY1(I) = B (JTT(I) *3-1) CORD1(I)=B(ITT(I)*3) 885 CONTINUE DO 890 I=1,IX IF (MAXRTN(1,I).LT.RT(I)) GO TO 890 MAXRTN(1,I)=RT(I) MAXRTN(2,I)=LCASE MAXRTN(3,I)=X MAXRTN(4,I)=Y WRIT E(*,*)i,yrixo, B(JTT(1)*3-2),B(JTT(1)*3-1) CORX(I)=B(ITT(I)*3-2) CORY(I)=B(JTT(1)*3-1) 890 CONTINUE

242 Grillage Analogy in Bridge Deck Analysis RETURN END c*********************************************************************** SUBROUTINE COMPARE2(PT,B,RT,RTN,MAXPT,MAXB,MAXRT,MAXRTN, + MM,N,IX,IM1,PT1, + MAX PT2 ,MAXB2,MAXRT2;MAXR2N,JTT ,COR2X,COR2Y,C OR2XI ,COR2Y1 , + COR1X COR I Y, COR1XI ,CORlYI,CORSPT1,CORSPT2, + CORRRX I , CORRX2 ,CORRY I ,CORRY2 , CORD,CORD2) IMPLICIT REAL*8(A-H2O-Z) DIMENSION PT(1000,7),B(1 000), RT(100),RTN(100),PT I (1000, 2), JTT(100) REAL*8 MAXPT(6,4,500),MAXB(4,500),MAXRT(4,100) + ,MAXPT2(6,2:4,500) + ,MAXB2(2:4,500),MAXRT2(2:4,100),MAXRTN(4,100),MAXR2N(2:4,100) + ,COR2X(100), COR2Y(100),COR2X1(100) ,COR2Y1( I00) ,CORSPT1 (2 ,500) + ,CORSPT2 (2,500),COR1X( I00),CORI Y(100),COR1X1(100),COR1Y1 (100) +, CORRX1(2,500), CORRX2(2,500) ,CORRY1(2,500) ,CORRY2 (2,500) +, CORD( I00),CORD2(100) IF (Im1.NE.1)G0 TO 925 DO 895 I =1,4 J=I+2 DO 895 K = 1,MM MAXPT(I, I ,K) = PT(K,J) MAXPT(I,2,K)=MAXPT2(1,2,K) MAXPT(I,3,K) = M AXPT2 (I ,3,K) MAXPT(I,4,K) =MAXPT2(I,4,K) CORSPT1 (1 ,K) = CORSPT2 (1,K) CORSPT I (2 ,K) = CORSPT2(2,K) 895 CONTINUE DO 900 I=1,N NI I =3*I-2 N13 =3*1 DO 900 J=N11,N13 MAXB(1,J) = B(J) MAXB(2,J) = M AXB2(2 ,J) MAXB(3,J) MAXB2(3 ,J) MAX B(4, J) = M AXB2 (4, J) CORRX1(1 ,I) =CORRX2(1,I) CORRY I (1 ,I) =CORRY2(I,I) C ORRX I (2 , I) = CORRX2 (2,I) C ORR Y I (2,D= CORRY2(2,I) 900 CONTINUE DO 905 I = 1,IX MAXRT(1,1) = RT (I) M AXRT(2 , I) = M AXRT2 (2,1) M AXRT (3 , I) = MAXRT2 (3,1) M AXRT (4, I) = MAXRT2(4,I) COR1XI(I)=COR2X1(I)

90

9

Appendix II 243 CORI Y f(I) =COR2Y1(1) CORD(I)=CORD2(I) MAXRTN(1,I) =RTN(I) MAXRTN(2,I) =MAXR2N(2, I) MAXRTN(3,I)=MAXR2N(3,I) MAXRTN(4,I)=MAXR2N(4,1) COR1X(I)=COR2X(I) CORIY(I)=COR2Y(I) 905 CONTINUE DO 910 1=5,6 DO 910 K=1,MM J=I-4 MAXPT(L1,K)=PT1(K,J) MAXPT(I,2,K) =MAXPT2(I,2,K) MAXPT(1,3,K)=MAXPT2(1,3,K) MAXPT(L4,K)=MAXPT2(I,4,K) 910 CONTINUE DO 920 I=3,6 DO 920 K=1,MM IF (I.GT.4)G0 TO 915 IF (MAXPT(I,1,K).LT.0.0)THEN MAXPT(I,1,K)=0 MAXPT(I,2,K)=0 MAXPT(I,3,K)=0 MAXPT(1,4,K)=0 ENDIF GO TO 920 915 IF (MAXPT(1,1,K).GT.0.0)THEN MAXPT(I,1,K)=0 MAXPT(I,2,K) =0 MAXPT(1,3,K)=0 MAXPT(I,4,K)=0 ENDIF 920 CONTINUE RETURN 925 DO 930 I=1,2 .1= I + 2 DO 930 K=I,MM IF (ABS(MAXPT(I,I,K)).GT.ABS(PT(K,J)))GO TO 930 MAXPT(I,1,K)=PT(K,J) MAXPT(1,2,10=MAXPT2(1,2,K) MAXPT(I,3,K)=MAXPT2(I,3,K) MAXPT(I,4,K)=MAXPT2(I,4,K) CORSPT1(I,K)= CORSPT2(I,K) 930 CONTINUE DO 935 1=3,4 J=1+2 DO 935 K=i,iviNi

244 Grillage Analogy in Bridge Deck Analysis4 IF (MAXPT(I,1,K).GT.PT(K,D)G0 TO 935 MAXPT(I,1,K)=PT(K,J) MAXPT(I,2,IC) =MAXPT2 (I, 2, K) MAXPT(1,3,K)=MAXPT2(I,3,K) MAXPT(I,4,K)=MAXPT2(I,4,K) 935 CONTINUE DO 940 1=5,6 DO 940 K=1,MM J=I4 IF (MAXPT(I,1,K).LT.PT1(K,i))GO TO 940 MAXPT(I, I ,K) =PT1(K,J) MAXPT(1,2,K)=MAXPT2(1,2,K) MAXPT(1,3,K) =MAXPT2(1,3 ,K) MAXPT(1,4,K)=MAXPT2(I,4,K) 940 CONTINUE KL1=1 DO 945 I=1,N • N11=3*1-2 IF (ABS(MAXB(1,N11)).GT.ABS(B(N11)))G0 TO 945 MAX'S(1 ,N11)= B(N I 1) MAXB(2, N11) = MAXB2(2, N11) MAXB(3,N11)=MAXB2(3,N11) MAXB(4,N1I)=MAX132(4,N11) CORRX1(1,1)=CORRX2(1,I) CORRX1 (2 ,I) CORRX2(2 ,I) 945 CONTINUE DO 950 1=1,N N12=3*1-1 IF (ABS(MAXB(1,NI2)).GT.ABS(B(N12)))GOTO 950 MAXB(1,N12)=B(N12) MAXB (2, N12) = MAXB2(2 , N 12) MAXB(3,N12)=MAX132(3,N12) MAXB(4,N12) = MAXB2(4,N 12) CORRY I (1 ,1) = CORRY2(1, I) CORRY I (2,I) = CORRY2(2, I) 950 CONTINUE DO 955 I=1,N N13 =3*I IF (ABS(MAXB(1,N13)).GT.ABS(B(N13)))GOTO 955 MAXB(1,N13)=B(N13) MAXB(2,N13)=MAXB2(2,N13) IVIAXB(3 , N13) =MAXB2(3 , N13) MAXB (4 , N13) = MAXB2(4, N13) 955 CONTINUE DO 960 I=1,IX IF(MAXRT(1,I).GT.RT(I))GO TO 960 MAXIIT(11)=RT(T MAXRT(2,I)=MAX—RT2.(2,I) 1

Appendix11 245 MAXRT(3,I)=MAXRT2(3,I) MAXRT(4,I)=MAXRT2(4,I) COR1XI(I)=COR2X1(1) COR1Y1(I)=COR2Y1(1) CORD(I) =CORD2(I) 960 CONTINUE DO 965 I=1,1X IF(MAXRTN(1,11.1-T.RTN(1))00, TO 965 MAXRTN(1,I)=RTN(I) MAXRTN(2,I)=MAXR2N(2,I) MAXRTN(3,I) =MAXR2N(3,I) MAXRTN(4,I)=MAXR2N(4,i) COR1X(I)=COR2X(I) COR1Y(1)=COR2Y(1) 965 CONTINUE ' RETURN END c***************************************************************i******* SUBROUTINE WRITE2 (MAXPT,MAXB,MAXRT,MAXRTN,MM,N,IX,MOVE,M, + JTT,LCASE,COR1X,CORlY,CORIX1,COR1Y1,CORSPT1,CORRX1,CORRY1 + ,CORD) IMPLICIT REAL*8(A-H2O-Z) DIMENSION M(1000,2);JTT(100) real*8 MAXPT(6,4,500),MAXB(4,500),MAXRT(4,100),MAXRTN(4,100) + ,CORIX(100),COR1Y(100),COR1X1(100),CORIY1(100),CORSPTI(2,500) +,CORRXI(2,500),CORRY1(2,500),CORD(100) WRITE (4,1025) DO 970 I=1,MM WRITE (4,1030)I,M(L1),M(L2),(MAXPT(K,1,0,CORSPT1(K,1), +(MAXPT(K,J,1),J= 2,4),K = 1 ,2) 970 CONTINUE WRITE (4,1015) WRITE (4,*)'MAX SAGGING MOMENT ON THE ELEMENTS' WRITE (4,1040) DO 975 I=1,MM 975 WRITE (4,1035)I,M(I,I),M(I,2),(MAXPT(5,7,1),J= I,4),(MAXPT(4,5,1) + ,J=1,4) WRITE (4,1015) WRITE (4,*)'MAX HOGGING MOMENT ON THE ELEMENTS' WRITE (4,1040) DO 980 I= I,MM 980 WRITE .(4,1035)I,M(I,1),M(1,2),(MAXPT(3,J,I),J = 1,4), (MAXPT(6,1,1) + ,J = 1,4) READ (3,*) IK4 IF (IK4.NE.1) GO TO 990 WRITE (4,11345)LNAME,LNAME,LNAME DO 985 I=1,N N11=3*1-2

246 Grillage Analogy in Bridge Deck Analysis N12 =31-1 N13 = 3*I WRITE (4,1050)1,1, (MAX.13(J,N11),J =1,2),(MAX13(1,NI2),J= 1,2), + (MAXB(J,N13),J = 1,2) 985 CONTINUE 990 CONTINUE WRITE (4,1060) DO 995 I=1,LX WRITE (4,1065)LITT(1),-MAX.RTN(1,I),COR1X(I),CORlY(I) + ,MAXB(I,3*JTT(I)),(MAXRTN(J,I),J =2;4) 995 CONTINUE WRITE (4,1020) WRITE (4,1055) DO 1000 I=1,IX WRITE (4,1065) LITT(1),-MAXRT(1,1),COR1X1(1),CORIY1(1),CORD(1) + ,(MAXRT(I,I),J=2,4) 1000 CONTINUE WRITE (4,1075) DO 1005 I=1,IX N1I=3*.ITT(1)-2 WRITE (4,1085)LITT(1),MAXB(1,N11),CORRX1(1,JTT(1)), + C ORRX1(2, ITT(I)),(MAXB(J, N11), I =2,4) 1005 CONTINUE WRITE (4,1080) DO 1010 I=1,IX N12=34TIT(1)-1 WRITE (4,1085)I,JTT(I),MAXB(1,N12),CORRY1(1,ITT(I)), +CORRY1(2,Ms(I)),(MAXB(J,N12),J=2,4) 1010 CONTINUE WRITE (4,1020) 1015 FORMAT(/80('*')) 1020 FORMAT(/80('*')) 1025 FORMAT(//,'MAX. SHEAR AND TORSION ON THE ELEMENTS'/ +110('*')/'SNO ELEMENT MAX SHEAR TORSION' +,' LCASE X Y ' + ,' MAX TORSION SHEAR', + ' LCASE X Y'/19X,'T TMM',12X,'MM MM',5X, + TMM T',14X,'MM MM'/110('*')///) 1030 FORMAT(I3,3X,I2, I2,2(2X, E11.4, IX, E11.4, 1X +,F3.0,2X,F6.0,2X,F6.0)) 1035 FORMAT(13,3X,I2,'-',12,2(2X,E11.4,1X,F3.0,2X,F6.0,2X,F6.0)) 1040 FORMAT(80('*')PSNO ELEMENT AT END1 LCASE X Y ' + ,' AT END2 LCASE X Y'/18X,'TMM',13X,'MM MM',5X, + 'TMM',14X,'MM MM'/80('*')/) 1045 FORMAT(99('*')/'SNO',2X,'T',3X, + 'MAX X-ROTATION',1X,1A8,2X,'MAX Y-ROTATION',1X,1A8,2X, + 'MAX Z-DEFLECTION',1X,1A8,/22X,'RAD'.24X,'RAD',27X,  'MM',/99('*')/)

Appendix II 247 1050 FORMAT(I3.4X,13,2(4X,E13.6,5X,F6.0),5X,E13.6,6X. +F6.0) 1055 FORMAT(/,'MINIMUM REACTION ON SUPERSTRUCTURE/K('*') +PSNO T REACTN COR ROT-X COR ROT-Y + CO R D EFL LC A S E X r/ + 80('*')/) 1060 FORMAT(//,`MAXIMUM REACTION ON SUPERSTRUCTURE/80(n +PSNO T REACTN COR ROT-X COR ROT-Y COR DEFL , ' LC A S E X Y '/ + 80('*')/) • 1065 FORMAT(I3,2X,I3,2X,4(E11A,1X),F3.0,2X,F6.0,2X,F6.0) 1070 FORMAT(13,2X,13,2X,3(E11.4,1X),F3.0,2X,F6,0,2X.F6.0) 1075 FORMAT(80(w),/,' SNO T MAX ROT X COR ROT Y ' + COR DEFL LCASE X Y 'PUNITS IN T AND mm'/ + 80(P')/) 1080 FORMAT(80C*),/,' SNO T MAX ROT Y COR ROT X ' +,' COR DEFL LCASE X Y '/ + 80('*')/) 1085 FORIvIAT(I3,2X,I3,3(3X,E11.4),2X,F3.0,2X,F6.0,2X,F6.0) 1090 FORMAT(I3,2X,I3,2X,E11.4,6X,F3.0,2X,F6.0,2X,F6.0) RETURN END c********************************************************************** SUBROUTINE WRITE1(13,MAXPT,MAXB,IvIAXRT,MAXRTN,MM,N,IX,MOVE,M, TIT,LCASE,CORSPT,CORX,CORY,CORX1,CORYLCORRX,CORRY,CORD1) IMPLICIT REAL *8(A-H2O-2;) DIMENSION M(1000,2),JTT(100),B(1000) real*8 MAXPT(6,4,500),MAXB(4,500),MAXRT(4,100),MAXRTN(4,100) +,CORSPT(2,500),CORX(100),CORY(100),CORX1(100),CORY1(100) +,CORRX(2,500),CORRY(2,500),CORDI(100) IF (LCASE.NE.1) GO TO 1115 WRITE(4,1245) DO 1095 I=1,MM WRITE(4,1260)1,M(I, 1),M(1,2),(MAXPT(K,1,I),K= I ,2),LCASE 1095 CONTINUE WRTTE(4,1165) WRITE(4.1250) DO 1100 I=1,MM WRITE(4,1265)I,M(1,1),M(1,2),MAXPT(5, 1 , I),MAXPT(4, 1,D,LCASE 1100 CONTINUE WRITE(4,1165) WRITE(4,1255) DO 1105 I= LIN.EvI WRITE(4,1265)I,M(1,1),M(1,2),MAXPT(3,1,D,MAXY19(6,1,I),LCASE 1105 CONTINUE WRITE(4,1165) WRITE(4,1240) DO 1110 1=1,1X WRITE(4,1270)LJTT(I),-MAXRTN(I,I),(MAXB(I.J). + J=3*-TIT(I)-2,3*JTT(0),LCASE 1110 CONTINUE WRITE(4,1165) GO TO 1275

248 Grillage Analogy in Bridge Deck Analysis Ills WRITE (4,1175) DO 1120 I=1,MM WRITE (4,1180)I,M(I,1),M(I,2),(MAXPT(K,1,1),CORSPT(K,I), +(MAXPT(K,J,1),J=2,4),K=1,2) 1120 CONTINUE WRITE (4,1165) WRITE (4,*)IMAX SAGGING MOMENT ON THE Fr PMENTS' WRITE (4,1190) DO 1125 I=1,MM 1125 WRITE (4,1185)1,M(I,1),M(1,2),(MAXPT(5,J,I),J=1,4),(MAXPT(4,J,I) +,J=1,4) WRITE (4,1165) WRITE (4,*)'MAX HOGGING MOMENT ON THE ELEMENTS' WRITE (4,1190) DO 1130 I=1,MM 1130 WRITE (4,1185)1,M(1,1),M(1,2),(MAXPT(3,J,I),J=1,4),(MAXPT(6,J,1) +,J=1,4) READ(3,*) IKI IF (1K1.NE.1) GO TO 1140 WRITE (4,1195)LNAME,LNAME,LNAME DO 1135 I=1,N N11=3*1-2 N12=3*1-1 N13=3*I WRITE (4,1200)I,I, (MAX13(J,N I 1),J=1,2),(MAXE (J,N12),J=1,2), +(MAX13(1,N13),J=1,2) 1135 CONTINUE 1140 CONTINUE WRITE (4,1210) DO 1145 I=1,IX WR1TE(4,12 I5)I,M(1),-MAXEITN(1,1),CORX(1),CORY(1) ±,MMCB(1,3*JTT(1)),(MAXRTN(J,1),J=2,4) 1145 CONTINUE WRITE (4,1170) WRITE(4,1205) DO 1150 I=1,1X WRITE (4,1215) 1,JTT(I),-MAXRT(1,1),CORX1(1),CORY1(1).CORD1(I) + ,(MAXRT(J,I),J=2,4) 1150 CONTINUE WRITE (4,1220) DO 1155 I=I,DC JI =3*.ITT(1)-2 WRITE (4,1230)I,ITT(1),MAXB(1,11),CORRX(I,ITT(I)),CORRX(2,TIT(1)) t(MAXE3(J,J1),J=2,4) 1155 CONTINUE WRITE (4,1225) DO 11601=1,1X J13491T(1)-1 WRITE (4,1230)1,JTT(I),MAXB(I,J1),CORRY(1,JTr(1)),CORRY(2,TrraD +,(MAXB(J,J1),J=2,4) 11Ni CONTINUE

WRITE (4.1170) 1165 FORMAT(/100(*)) 1170 FORMAT(/80('4'))

Appendix II 249

7

1175 FORMAT(//,'MAX. SHEAR AND MAX. TORSION ON THE ELEMENTS'/ + 1 IO('*')PSNO ELEMENT MAX SHEAR TORSION' +,' LCASE X Y ' + ,' MAX TORSION SHEAR'. +' LCASE X Y719X,'T 'FM1v1',12X,'W MM',5X, +' TMM T,14X,'MM MKT/110( 1 *V) 1180 FOFtMAT(L3,3X,I2,'-',I2,2(2X,E11.4,1X,E11.4,1X, + F3.0.2X,F6.0,2X,F6.0)) 1185 FORMAT(13,3X,I2,'-',12,2(2X,E11.4,1X,F3.0,2X,F6.0,2X,F6.0)) 1190 FORMAT(SOMPSNO ELEMENT AT ENDI LCASE X Y ' +,' AT END2 LCASE X Y'/18X.,'IMM 1 ,13X,MM MM',5X, + 'TMM,14X,' MM MM1/80('*')/) 1195 FORMAT(99(n/2X,'SNO',2X,IT,3X, + 'MAX X-ROTAIION;IX,1A8,2X,'MAX Y-ROTATION',1X,1A8,2X, + 'MAX Z-DEFLECTION',1X,1A8,/22X;RAD',24X,RAD',27X, + TM3/1.199C*)/) 1200 FORMAT(13,4X,13,2(4X,E13.6,5X,F6.0),5X,E13.6,6X, '+ F6.0) 1205 FORMAT(/,'MINIMUM REACTION ON SUPERSTRUCTURE' 4/80('*')PSNO T REACTN COR ROT-X COR ROT-Y' +,' COR DEFL LCASE X +,11X,RAD',9X,RAD',9X.,'nme,11X,Irtute,6X,'mne/ 80(1*')//) 1210 FORMAT(//,'MA)IMUM REACTION ON SUPERSTRUCTURE' +/80(41PSNO T REACTN COR ROT-X COR ROT-Y ' + COR DEFL LCASE X Y'/' T' +111X,'RAD',9X,TAD',9X;rarre,11X,'Enmi,6X,'mm? 80('*')/) 1215 FORMAT(13,2X,13,2X,4(E11.4,1X),F3.0,2X,F6.0,2X,F6.0) 1220 FORMAT(80C*),/,' SNO T MAX ROT X COR ROT Y ' + COR DEFL LCASE X Y '/ +16X,'RAD`,12X,RAD',11X,'nuns,13X,smm',5X,'mm./ 80(41//) 1225 FORMAT(80(n,/,' SNO T- MAX ROT Y COR ROT X ' + COR DEFL LCASE • X Y '/ +16X;RAD',12X,RAD',11X.Anm1,13X,'ma,5X,'nuni 80 C nil) 1230 FORMAT(13,2X,I3,3(3X,E11.4),2X,F3.0,2X,F6.0,2X.F6.0) 1235 FORNLAT(13,2X,13,2X,E11.4,6X,F3.0,2X,F6.0,2X,F6.0) 1240 FORMAT(//,'REACTION ON SUPERSTRUCTURE AT S' +/80('*')PSNO T REACTN ROT - X ROT - Y ' + ,' DEFLECTION LCASE/17X.7,8X,'RAD',10X,'RAD' +,9X'nulf/80('*')/) 1245 FORMAT(1/,'SHEAR AND TORSION ON ELEMENTS'/ + 80(*VSNO ELEMENT SHEAR TORSION LCASE'/ + 19XT,10X,I'mm1/80(1*')/) 1250 FORMAT(//,'SAGGING MOMENT ON ELEMENTS' +/80('*')/SNO ELEMENT AT ENDI AT END2 LCASE'I + 18X,Tmm.',8X,'Tnine/80('*')/) 1255 FORMAT(//,'HOGGING MOMENT ON ELEMENTS' +/80('*')PSNO ELEMENT AT END1 AT END2 LCASE'/ + 18X,Trran',8X,Tmrn'/80('*')/) 1260 FORMAT(13,3X,I2,'-',I2,2X,E I X.I3 ) 1265 FORMAT(13,3X,I2,'-',12,2X,E1 L 4.1X,E11.4.2X.13 ) 1270 FORMA.T(3,2X,13,3X4(Eii.4, X), iX,I3) 1275 CONTINUE RETURN END

F.

Subject Index Analysis, 35, 111, 131139 Arbitrary planforms, 16 Assumptions, 36, 56 Axes, 42, 46 • Axle loading, 19

0

Band-width, 51, 64 Beam contiguous beam, 7, 87 effective flange width, 96, 101 equilibrium equations, 74 spaced beam, 14 spine beam, 14 Bearing stiffness, 133 Bearings neoprene, 16, 32, 133 rocker, 16 roller, 16 Bending Moment diagram, 140, 143 Box-girder bridges, 6, 12-15, 92-94, 103-109, 143-146 Bridge classification, 5 Bridge codes, 5, 17, 18 Bridge deck analysis (see analysis) Bureau of Indian Standards (BIS), 17 Cable stayed bridges, 3, 6 CAD/CAM systems, 4 Cantilever bridges, 16 Cantilever construction, 5 Cell distortion, 13, 93, 108, 144 Cellular deck, 6, 51, 103 Cement and concrete association (C & CA), S Cholesky's factorisation method. 31, 69, 133

Clearance, 20, 22-24 Codes of practices (see bridge codes) Composite construction, 10 Coarse mesh, 41, 171 Contributory area (see tributary area) Computer aided methods, 1, 3 Computer programs GRID, 32, 66-69, 205-212 GABS, 32, 146-1,69, 171203, 213-250 Contiguous beams (see beams) Contiguous nodes, 115, 117 Courbon's method, 1, 36-38 Curved deck, 42 Curved geometry, 16 Data input, 150, 165 output, 150, 165 Dead load, 18, 112, 118 Degrees of freedom, 46, 56-58 Design curves, 1, 39, 43, 44 Design envelopes, 31, 138 Diaphragms, 10. 13 Direct stiffness method, 31, 55-73, 132 Displacement method, 56 Distortion, 13, 93, 108. 144 Ductility-ratio, 4 Edge grid lines, 87, 89 Effective width of flange, 96, 101 Elastic properties, 94-109 Envelope diagrams, 134, 138, 196 Equiiihrium equations. 74 Equivalent loads (see transfer of loads)

252 Subject Index Examples, 69-73, 171-203 Expert system, 4 External prestressing, 3 Fine mesh, 42, 171 Finite difference method, 1, 40-42 Finite element Method (FEM), 1, 48, 49, 200 . Finite strip method, 1, 50-52 Flexibility matrix,•55 Flexible , 16 Flexural moment of inertia, 31, 38, 94-109 Flexural parameter, 39, 43 Flowcharts, 66-68, 150-160 Folded plate analysis, 47 Footpaths, 28, 113, 118 Force method,. 56 Force responses, 31, 32, 111, 134 FORTRAN, 31, 32., 69, 146 Fourier analysis (see harmonic analysis) Gauss-Elimination procedure, 31, 66, 133 Gauss-Seidel method, 65 Grade separation, 12 Grillage examples box-girder deck, 143, 199-203 grid deck, 69-73. skew deck, 69-73, 179-182 slab deck, 172-187 slab-on-girders deck, 187-198 voided slab deck, 183-187 Grillage section properties box-girder-deck, 103-109 cellular deck (see box-girder deck) skew deck, 101 slab deck, 99-101 slab-on-girders deck, 101-103 Harmonic components, 39, 42, 44 Hendry-Jaegar method, 42-45 design graphs, 43-44 interpolation function, 44 Highway bridge decks, 5, 18 Hypothetical loading system, 18, 144

Idealization bridge deck, 74-94 loading, 117-131 Impact load, 25-27, 114 Inclined webs, 93 Indian Railways, 17 Indian Roads Congress (IRC), 2, 17, 30, 36, 96, 113 Influence surfaces, 42 Interpretation of results, 139-145 Isotropic decks, 41-76 Kerb, 18, 19, 26, 28, 113, 162 Launching, 5 Limit state concept, 4 Lines of strength, 78 Live load AASHO, 28, 147 IRC, 18-25, 28-30 OHBD, 147, 191 Loading (see dead load, live load, impact load) Load distribution, 35, 43, 99 Loading standards, 17, 24, 28, 30 Local effects, 111 Longitudinal grid lines, 31, 78 Matrix method of analysis, 31, 55 Mbeam, 8 Method of harmonic analysis (see Hendry-Jaeger method) Ministry of surface transport, 17, 193 Ministry of transport, UK, 8 Moment of inertia (see flexural moment .of inertia) MS-DOS, 146 Multi-cellular decks, 13, 106 Neutral axis, 99 Nodal deformations, 66, 131, 133 Nodal loads, 31, 69, 117 Ontario highway bridge design code, 147, 172. 191 Orthotropic plate theory, .3`6-4.,)

Subject Index 253 coupling rigidities, 39 equations, 39 flexural parameter, 39 torsional parameter, 39 Orthotropic slab, 38-40, 100,140 s parallelogram, 115, 121, 128-131 rectangular, 115, 119, 123-125 triangular, 115, 119, .121, 125-129 Parapets, 18, 26 Partial prestressing, 3 Patch loads, 18, 50, 114 Physical deck, 74, 75, 99, 111 Plan geometry, 5, 15, 31, 35 Planforms, 2, 5, 15, 69 Plastic hinges, 4 Poisson's ratio, 40, 42, 77, 144 Post tension, 10, 14, 142 Prandtl's membrane analogy, 98 Prerast prestressed concrete girders, 11 Prestressed concrete bridges, 11, 12, 16 Principal moments, 141 Program manual, 32, 147 • Pseudo-slab, 7, 8, 11, 87, 101, 146 Pucher's chart. 142 Push launching, 5 Re-entrant corners, 97 Responses, 31, 32, 111, 119, 134, 147 Result output, 150, 165 Rigid frame bridges, 6 Rigid s, 16, 65, 133 Saint-Venant, 40, 96 Segmental construction, 5 Shear deformation, 13, 92, 101 Shear lag, 13, 92, 101, 172 Shear modulus, 70, 76, 166 Sign convention, 56, 150 Skeletal structures, 46, 56 Skew angle, 80 Skew decks finite difference method, 41, 42 finite element method, 45 orthotropic plate theory, 40

Skew grid, 31, 42, 69-73, 80 Skyline technique, 64 Slab bridges, 6-9, 79-89, 99-101, 139-142 Slab-on-girders bridges, 6, 9-12, 9092, 101-103, 143 • Slope deflection, 74, '75 Solid slab, 79, 80, 90, 99, 146 Spaced box-girders, 94 Space frame analysis, 46 Spine beam bridges, 14 Spine boxgirders, 14, 94, 146 Stiffness method (see direct stiffness method) Subroutines (see computer programs) conditions, 2, 5, 31, 35, 132 s, 6, 13, 15, 69 T-beam bridges (see slab-on-girders bridges) Torsional inertia, 31, 94-109 Torsional moment, 111 Torsional parameter, 39, 44 Torsional rigidity, 39, 44 Transfer of loads, 117-131 Transformation matrix, 46, 60 Transverse grid lines, 31, 78 Trapezoidal cell, 14 Tributary area, 112, 113 Trigonometric solution, 41 Tyre area, 18 s manual (see computer programs) s specified loading, 32, 134, 147, 162 Vehicular live loading (see also live load), 27, 31, 111, 115, 119 Voided slab (see also slab bridges), 6, 13, 87, 99, 146, 182 Warping, 172 Young's modulus. 70, 76, 166

c),

ti

Subject Index Analysis, 35, 111, 131-139 Arbitrary planforms, 16 Assumptions, 36, 56 Axes, 42, 46 Axle loading, 19

C

V

Band-width, 51, 64 Beam contiguous beam, 7, 87 effective flange width, 96, 101 equilibrium equations, 74 spaced beam, 14 spine beam, 14 Bearing stiffness, 133 Bearings neoprene, 16, 32, 133 rocker, 16 roller, 16 Bending moment diagram, 140, 143 Box-girder bridges, 6, 12-15, 92-94, 103-109, 143-146 Bridge classification, 5 Bridge codes, 5, 17, 18 Bridge deck analysis (see analysis) Bureau of Indian Standards (BIS), 17 Cable stayed bridges. 3, 6 CAD/CAM systems, 4 Cantilever bridges, 16 Cantilever construction, 5 Cell distortion, 13, 93, 108, 144 Cellular deck, 6, 51, 103 (`am not , Anri nr.rvisrdaCci no,

(.1

CA), 8 Cho lesky' s factorisation method, 31, 69, 133

Clearance, 20, 22-24 Codes of practices (see bridge codes) Composite construction, 10 Coarse mesh, 41, 171 Contributory area (see tributory area) Computer aided methods, 1, 3 Computer programs GRID, 32, 66-69, 205-212 GABS, 32, 146-169, 171-203, 213-250 Contiguous beams (see beams) Contiguous nodes, 115, 117 Courbon's method, 1, 36-38 Curved deck, 42 Curved geometry, 16 Data input, 150, 165 output, 150, 165 Dead Ioad, 18, 112, 118 Degrees of freedom, 46, 56-58 Design curves, 1, 39, 43, 44 Design envelopes, 31, 138 Diaphragms, 10, 13 Direct stiffness method, 31, 55-73, 132._.Displacement method, 56 Distortion, 13, 93, 108, 144 Ductility-ratio, 4 Edge grid lines, 87, 89 Effective width of flange, 96, 101 Elastic properties, 94-109 Envelope diagrams, 134, 138, 196 Equilibrium equations, 74 Equivalent loads (see transfer of loads)

252 Subject Index Examples, 69-73, 171-203 Expert system, 4 External prestressing, 3 Fine mesh, 42, 171 Finite difference method, 1, 40-42 Finite element method (FEM), 1, 48, 49, 200 Finite strip method, 1, 50-52 Flexibility matrix, 55 Flexible , 16 Flexural moment of inertia, 31, 38, 94-109 Flexural parameter, 39, 43 Flowcharts, 66-68, 150-160 Folded plate analysis, 47 Footpaths, 28, 113, 118 Force method,.56 Force responses, 31, 32, 111, 134 FORTRAN, 31, 32, 69, 146 Fourier analysis (see harmonic analysis) Gauss-Elimination procedure, 31, 66, 133 Gauss-Seidel method, 65 Grade separation, 12 Grillage examples box-girder deck, 143, 199-203 grid deck, 69-73 skew deck, 69-73, 179-182 slab deck, 172-187 slab-on-girders deck, 187-198 voided slab deck, 183-187 Grillage section properties box-girder-deck, 103-109 cellular deck (see box-girder deck) skew deck, 101 slab deck, 99-101 slab-on-girders deck, 101-103 Harmonic components, 39, 42, 44 Hendrv-Jaear method, 42-45 design graphs, 43-44 interpolation function, 44 Highway bridge decks, 5, 18 Hypothetical loading system, 18, 144

Idealization bridge deck, 74-94 loading, 117-131 Impact load, 25-27, 114 Inclined webs, 93 Indian Railways, 17 Indian Roads Congress (IRC), 2, 17, 30, 36, 96, 113 'Influence surfaces, 42 Interpretation of results, 139-145 Isotropic decks, 41-76 Kerb, 18, 19, 26, 28, 113, 162 Launching, 5 Limit state concept, 4 Lines of strength, 78 Live load AASHO, 28, 147 IRC, 18-25, 28-30 OHBD, 147, 191 Loading (see dead load, live load, impact load) Load distribution, 35, 43, 99 Loading standards, 17, 24, 28, 30 Local effects, 111 Longitudinal grid lines, 31, 78 Matrix method of analysis, 31, 55 Mbeam, 8 Method of harmonic analysis (see Hendry-Jaeger method) Ministry of surface transport, 17, 193 Ministry of transport, UK, 8 Moment of inertia (see flexural moment of inertia) MS-DOS, 146 Multi-cellular decks, 13, 106 Neutral axis. 99 Nodal deformations, 66, 131, 133 Nodal loads, 31, 69, 117 Ontario highway bridge design code, 147, 172, 191 Orthotropic plate theory, 38-40

coupling rigidities, 39 equations, 39 flexural parameter, 39 torsional parameter, 39 Orthotropic slab, 38-40, 100,140 is s parallelogram, 115, 121, 128-131 rectangular, 115, 119, 123-125 triangular, 115, 119,.121, 125-129 Parapets, 18, 26 Partial prestressing, 3 Patch loads, 18, 50, 114 Physical deck, 74, 75, 99, 111 Plan geometry, 5, 15, 31, 35 Planforms, 2, 5, 15, 69 Plastic binges, 4 Poisson's ratio, 40, 42, 77, 144 Post tension, 10, 14, 142 Prandtl's membrane analogy, 98 Precast prestressed concrete girders, 11 Prestressed concrete bridges, 11, 12, 16 Principal moments, 141 Program manual, 32, 147 Pseudo-slab, 7, 8, 11, 87, 101, 146 Pucher's chart, 142 Push launching, 5 Re-entrant corners, 97 Responses, 31, 32, 111, 119, 134, 147 Result output, 150, 165 Rigid frame bridges, 6 Rigid s, 16, 65, 133 Saint-Venant, 40, 96 Segmental construction, 5 Shear deformation, 13, 92, 101 Shear lag, 13, 92, 101, 172 Shear modulus, 70, 76, 166 Sign convention, 56, 150 Skeletal structures, 46, 56 Skew angle, 80 Skew decks finite difference method, 41, 42 finite element method, 45 thccry, '!"

Vehicular live loading (see also live load), 27, 31, 111, 115, 119 Voided slab (see also slab bridges), 6, 13, 87, 99, 146, 182 Warping, 172 Young's modulus, 70, 76, 166

Subject Index 253 Skew grid, 31, 42, 69-73, 80 Skyline technique, 64 Slab bridges, 6-9; 79-89, 99-101, 139142 Slab-on-girders bridges, 6, 9-12, 9092, 101-103, 143 Slope deflection, 74, 75 Solid slab, 79, 80, 90, 99, 146 Spaced box-girders, 94 Space frame analysis, 46 Spine beam bridges, 14 Spine boxgirders, 14, 94, 146 Stiffness method (see direct stiffness method) Subroutines (see computer programs) conditions, 2, 5, 31, 35, 132 s, 6, 13, 15, 69 T-beam bridges (see slab-on-girders bridges) Torsional inertia, 31, 94-109 Torsional moment, 111 Torsional parameter, 39, 44 Torsional rigidity, 39, 44 Transfer of loads, 117-131 Transformation matrix, 46, 60 Transverse grid lines, 31, 78 Trapezoidal cell, 14 Tributary area, 112, 113 Trigonometric solution, 41 Tyre area, 18 s manual (see computer programs) s specified loading, 32, 134, 147, 162

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