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THEORY1
Structural Theory 1
Chapter 1
Introduction; Engineering Structures; - the word structure has various meanings. By an engineering structure, that means roughly something constructed or built. The principal structures of concern to civil engineers are bridges, buildings, walls, dams towers and shell structures. Structures as such are composed of one or more solid elements so arranged that the whole structures as well as their components are capable of holding themselves without appreciable geometric change during loading and unloading. To design a structure involves many considerations, among which are two major objectives that must be satisfied; -
the structure must meet the performance requirement
-
the structure must carry the loads safely
Consider, for example, the roof trusses resting on columns shown in figure below. The purposes of the roof truss and of the columns are, on the hand, to hold in equilibrium their own weights, the load of roof covering the wind; and, on the other hand, to provide shelter for housing a family, a manufacturing plant, or for other uses.
the one and for
Theory of Structures Defined; - the complete design of a structure is outlined in the following stages;
-
Developing a General Layout; - the general layout of a structure is selected from any possible alternatives. The primary concern is the purpose for which the structure is to be built. This stage involves the structural type, the selection of material and a tentative estimation of cost based on a reasonable analysis of a preliminary structural design. It may also involve selecting the best location or adapting the structure to a site that has not been predetermined. Proper attention should also be given to the aesthetic value of the structure since the requirement of good appearance can reasonably be met without much increase in cost. There are many other considerations, including the legal, financial, sociological, cultural, and environmental aspects. It is clear that this stage of design calls for an engineer with a high order of experience, skill, general knowledge and imagination.
-
Investigating the Loads; - before a refined structural analysis, it is necessary to determine the loads for which a given structure should be designed. General information about the loads imposed on a structure is usually given in the specifications and codes. However, it is part of the designer’s responsibility to specify the load conditions and to take care of exceptional cases. The weight of the structure itself together with the material permanently attached to it is called the dead loads and is regarded as fixed in magnitude and location. Since the dead load must be approximated before the
Engr. I.R. Bonzon
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Chapter 1
structure is designed, the preliminary data are only tentative. Revision must be made if the initial estimation is not satisfactory. All loads, other than the dead loads, may be called live loads. Live loads are generally classified as movable loads and moving loads. Movable loads are loads that may be transported from one location to another on a structure without dynamic impact; for example, people, furniture and goods on a building floor or rainwater on a roof. Moving loads are loads that move continuously over the structure, such as railway trains or tracks on a bridge, wind on a roof or wall or hydrostatic pressure on an abutment. Moving loads may also be applied suddenly to the structure, for example, the centrifugal and longitudinal forces induced by the acceleration of vehicles and the dynamic forces generated by earthquakes. In an ordinary structural design, all loads are treated as static loads in order to simplify the analysis. In this way the impact due to a moving live load is expressed as a percentage of the live load and the earthquake force is commonly considered to be a horizontal force equal to a fraction of the weight of a structure. Other load considerations may include thermal effects and resistance to bomb blasting.
-
Stress Analysis; - once the basic form of the structure and the external loads are defined, a structural analysis can be made to determine the internal forces in various of the structure and the displacements at some controlling points. When live loads are involved, it is important to determine the maximum possible stresses in each member being considered. The principles governing this phase of the design is usually discuss in the theory of structures.
-
Selection of Elements; - the selection of suitable sizes and shapes of and their connections depends on the results of the stress analysis together with the design provisions of the specifications or codes. A trial and error approach may be used in the search for a proportioning of elements that will be both economical and adequate. A sound knowledge of strength of material and process of fabrication is also essential.
-
Drawing and Detailing; - once the makeup of each part of the structure has been determined, the last stage of the design can begin. This final stage includes the preparation of contract drawing, detailing, job specification and the final cost; this information is necessary for construction to proceed.
-
these five stages are interrelated and may be subdivided and modified. In many cases, they must be carried out more or less simultaneously. The subject matter of the theory of structures is stress analysis with occasional reference to loading. The emphasis of structural theory is usually on the fundamentals rather than on the details of design.
Theory of Structures Classified; - structural theory may be classified from various points of view. convenience, structural theory is characterize by the following aspects;
-
For
Statics versus Dynamics; - ordinary structures are usually designed under static loads. Dead loads are static loads that cause no dynamic effect on structures. Some live loads, such as, trucks and locomotives moving on bridges, are also assumed as
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concentrated static load systems. They do cause impact on structures; however, the dynamic effects are treated as a fraction of the moving loads to simplify the design. The particular, specialized branch that deals with the dynamic effects of structures of accelerated moving loads, earthquake loads, wind gusts or bomb blasts is structural dynamics.
-
Plane versus Space; - no structure is really planar. However, structural analysis for beams, trussed bridges or rigid frame buildings are usually treated as plane problems, although they are never two-dimensional in themselves. On the other hand, in some structures, such as towers and framing for domes, the stresses are interrelated between the not lying in a plane in such a way that the analysis cannot be simplified on the basis of component planar structures. Such structures must be considered as space frameworks under noncoplanar force system.
-
Linear versus Nonlinear Structures; - linear structures mean that a linear relationship is assumed to exist between the applied loads and the resulting displacements in a structure. This assumption is based on the following conditions; −
the material of the structure is elastic and obey the Hooke’s Law at all points and throughout the range of loading considered
−
the changes in the geometry of the structure are so small that they can be neglected when the stresses are calculated
Note that if the principle of superposition is to apply, a linear relationship must exist, or be assumed to exist, between loads and displacements. The principle of superposition states that the total effect at some point in a structure due to a number of causes (forces or displacements) acting simultaneously is equal to the sum of the effects for the causes acting separately. A nonlinear relationship between the applied actions and the resulting displacements exists under of two conditions; − −
the material of the structure is inelastic the material is within elastic range, but the geometry of the structure changes significantly during the application of the loads
The study of nonlinear behaviour of structures analysis of structures and buckling of structures.
-
includes
plastic
Statically Determinate versus Statically Indeterminate Structures; - the term statically determinate structure means that the structural analysis can be carried out by statics alone. If this is not so, the structure is statically indeterminate. A statically indeterminate structure is solved by the equations of statics together with the equations furnished by the geometry of the elastic curve of the structure in linear analysis. It can be noted that the elastic deformations of the structure are not only associated with the applied loads on the structure but are also affected by the material properties (e.g., the modulus of elasticity) and by the geometric properties of the member section (e.g., the cross-sectional area or the moment of inertia). Thus, loads,
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material properties and geometric properties are all involved in solving a statically indeterminate structure, while load factor alone dominates in a statically determinate problem.
-
Force versus Displacement; - the force and displacement are two categories of event occurring to a structure. The objective of a structural analysis is to determine forces and displacements pertaining to the structure and their relationships specified by the geometric and material properties of structural elements. Structural analysis can be divided into two categories; the force and the displacement methods. In the force method, the forces are treated as the basic unknowns and the displacements are expressed in of forces; whereas, in the displacement method, it is the displacements that are regarded as the fundamental unknowns and the forces are expressed in of displacements. In matrix analysis of linear structures, the force method is often regarded to as the flexibility method, and the displacement method is called the stiffness method.
Actual and Ideal Structures; - all analyses are based on some assumptions not quite in accordance with the facts. It is impossible for an actual structure to correspond fully to the idealized structure on which the analysis is based. The materials of which the structure is built do not follow exactly the assumed properties and the dimensions of the actual structure do not coincide with their theoretical values. To illustrate, let us take a simple example. In deg a reinforced concrete beam of rectangular section, the values of the modulus of elasticity and the moment of inertia are usually assumed to be constant. However, the amount of reinforcing steel placed in the beam varies with the stresses; therefore, the values of the modulus of elasticity and the moment of inertia are not constant throughout the span. Besides, there is a great uncertainty involved even in choosing a constant value of the modulus of elasticity and the moment of inertia. Even without considering other factors, such as the s, the connections and the working dimensions of the structure, it can be noted that the behaviour of an actual structure often deviates from that of an idealized structure by a considerable amount. However, it does not follow from this that the results of analysis are not useful for practical purposes. Therefore, the process of structural solution is setting an idealized model in order to carry out practical analysis and from the practical analyses, the idealization can be made more and more consistent with actuality. Scope of Theory of Structures; - three major types of basic structures are thoroughly considered in structural theory;
-
Rigid Frame; - composed of connected by rigid ts capable of resisting moments. of a rigid frame, in general, are subjected to bending moment, shear and axial forces.
-
Truss; - composed of connected by frictionless hinges or pins. The loads on a truss are assumed to be concentrated at the ts. Each member of a truss is considered as a two-force member subjected to axial force only.
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-
Structural Theory 1
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Beam; - a straight member subjected only to transverse loads. A beam is completely analyzed when the values of bending moment and shear are determined.
Stability and Determinacy of Structures; Equations of Equilibrium for a Coplanar Force System; - the first and major function of a structure is to carry loads without showing appreciable distortions. In structural statics, all force systems are assumed to act on rigid bodies. Actually, there are always some small deformations that may cause some small change of dimension in structure and a shifting of the action lines of the forces. However, such deviations are neglected in stress analysis. A structure is said to be in equilibrium if, under the action of external forces, it remains at rest relative to the earth. Also, each part of the structure, if taken, as a free body isolated from the whole, must be at rest relative to the earth under the action of the internal forces at the cut sections and of the external forces thereabout. If such is the case, the force system is balanced, or in equilibrium, which implies that the resultant of the force system (either a resultant force or a resultant couple) imposed on the structure, or segment thereof, must be zero. The generally balanced coplanar force system must then satisfy the following three simultaneous equations;
where:
ΣFx
=
0
ΣFx
-
summation of x component of each force in the system
ΣFy
-
summation of y component of each force in the system
ΣM a -
ΣFy
=
0
ΣM a =
0
summation of moment about any point a in the plane due to each force
The subscripts x and y indicate two mutually perpendicular directions in the Cartesian coordinate system.
-
Special Cases of the Coplanar Force System in Equilibrium; -
Concurrent Forces; − if a system of coplanar, concurrent forces is in equilibrium, forces of the system must satisfy the following equations;
ΣFx
=
0
ΣFy
=
then the
0
Another set of independent equations necessary and sufficient for the equilibrium of the forces of a coplanar, concurrent force system, provided that point a is not on the line through the concurrent point of forces and perpendicular to y axis, is; ΣFy
=
0
ΣM a =
0
A third set of equations of equilibrium for a coplanar, concurrent force system, provided that line through a and b does not through the concurrent point of the forces, is;
ΣM a = Engr. I.R. Bonzon
0
ΣM b =
0
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THEORY1
-
Structural Theory 1
Chapter 1
Parallel Forces; − if a coplanar, parallel force system is in equilibrium, the forces of the system must satisfy the equations; ΣFy
=
0
ΣM a =
0
Another set of independent equations of equilibrium for a system of a coplanar, parallel forces, provided that line through a and b is not parallel to the forces of the system, may be given as;
ΣM a =
-
0
ΣM b =
0
Simple, Special Cases of Equilibrium; -
Two-Force Member; − in the figure shown below, a body is subjected to two external forces applied at a and b. If the body is in equilibrium, then the two forces cannot be in random orientation, but must be directed along ab. Furthermore, they must be equal in magnitude and opposite in direction.
-
Three-Force Member; − in the figure shown below, a body is subjected to the action of three external forces applied at a, b, and c. If the body is in equilibrium, then the three forces cannot be in random orientation, but must be concurrent at a common point O; otherwise the total moment about the intersection of any two forces would not vanish. A limiting case occurs when point O moves off at an infinite distance from a, b, and c in which case the three forces will be parallel.
Reactions; - structures are either partially or completely restrained so that they cannot move freely in space. Such restraints are provided by s that connect the structure to some stationary body, such as the ground or another structure. The first step in structural analysis is to take the structure without the s and calculate the forces, known as reactions, exerted on the structure by the s. The reactions are considered part of the external forces other than the loads on the structure and are to balance the other external loads in a state of equilibrium. The distribution of the reactive forces of a may be represented by a single force completely specified by three elements-the point of application, the direction and the magnitude.
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-
Hinge ; - a hinge can resist a general force, P, in any direction but cannot resist the moment of the force about the connecting point. The reaction of a hinge is assumed to be through the center of the connecting pin; its magnitude and slope of action line is yet to be determined. It is therefore a reaction with two unknown elements, which could equivalently represented by unknown magnitudes of its horizontal and vertical components both acting through the center of the hinge pin.
-
Roller ; - the mechanism used is such that the reaction acts normal to the ing surface through the center of the connecting pin. The reaction may be either away from or toward the ing surface. As such, the roller is incapable of resisting moment and lateral force along the surface of the . A roller supplies a reactive force, fixed at a known point and in a known direction, the magnitude of which is unknown. It is therefore a reaction with one unknown element.
-
Fixed ; - a fixed is capable of resisting force in any direction and the moment of force about the connecting end, thus preventing the end of the member from both translation and rotation. The reaction supplied by a fixed may be represented by the unknown magnitudes of a moment, a horizontal force and a vertical force acting through the centroid of the end cross section O. M
Rx
Rx
R
Ry hinge
R roller
Ry fixed
Internal Forces at a Cut Section of a Structure; - a truss structure is composed of pin-connected and is assumed to be pin-loaded. If any one of the is taken from its connecting pins as a free body, the forces exerted on the member must be concentrated at the two ends of the member through the center of the pins. Furthermore, these two systems of concurrent forces can be combined into two resultant forces that must be equal, opposite and collinear. In other words, each member of a truss is a two-force member. Hence, the internal forces existing in any cut section of a truss member (assumed straight and uniform) must be a pair of equal and opposite axial forces to balance the axial forces exerted on the ends.
of structures, such as beams and rigid frames, are acted on by more than two forces. It is obvious that forces of internal constraint must exist between these two portions in order to hold them together. Such internal Engr. I.R. Bonzon
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forces, of course, always occur in pairs of equal and opposite directions. To maintain equilibrium of the free body, however, the internal forces must be statically equal and opposite to the system of forces acting externally on the portion considered and the internal forces can always be represented by a force applied at the centroid of the cross section together with a couple of moment.
Equations of Condition or Construction; - structures such as truss, beam and rigid frame may sometimes be considered to be one rigid body sustained in space by a number of s. Out of several such rigid bodies, a compound form of structure may be built by means of connecting devices, such as hinges, links or rollers and mounted on a number of s. In either the simple or the compound type of structure, the external force system of the entire structure, consisting of the loads on the structure and the reactions, must satisfy the equations of equilibrium if the structure is to remain at rest. However, in the compound type of structure, the connecting devices enforce further restrictions on the force system acting on the structure, thus providing additional equations of statics to supplement the equations of equilibrium. Equations supplied by the method of special construction (other than external s) are called equations of condition or construction. Stability and Determinacy of a Structure with Respect to s when the Structure is considered a Monolithic Rigid Body; - when one considers the design of a structure, careful thought must be given to the number and arrangement of the s directly related to the statical stability and determinacy of the structure. If a structure is treated as one monolithic rigid body mounted on a number of s, there will be no internal condition involved and the stability and determinacy of the structure will be judged solely by the stability and determinacy of s. -
two elements of reaction supplied by s, such as two forces each with a definite point of application and direction, are not sufficient to ensure the stability of a rigid body, because the two are either collinear, parallel or concurrent. In this case, the condition of equilibrium is violated, not because of the lack of strength of s, but because of the insufficient number of elements. This is referred to as statical instability.
-
at least three elements of reaction are necessary to restrain a body in stable equilibrium. In a stable equilibrium condition, the rigid body is
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subjected to restraints by three elements of reactions and the restraints can be solved by the three available equilibrium equations. The satisfaction of all the three equilibrium equations, for loads and reactions acting on a body guarantees, respectively, that the body will neither move horizontally or vertically nor rotate. The system is said to be statically stable and determinate.
-
-
if there are more than three elements of reaction the body is necessarily more stable because of the additional restraints. Since the number of unknown elements of reaction is more than the number of equations for static equilibrium, the system is said to be statically indeterminate with regard to the reactions of .
-
the number of elements of reaction should be at least three is a necessity but not a sufficient condition for an externally stable structure. There are many cases that are obviously not stable with respect to the system even though three or more than three elements of reaction are supplied. When, for example, the lines of reaction are all parallel, the body is unstable because it is vulnerable to lateral sliding. Another case is when the lines of three reaction elements are originally concurrent at a point. The system is also unstable because, even though complete collapse probably will not result, a small initial rotation about the point because of the moment caused by any force not ing through the point will certainly occur until the three reaction lines form the triangle. This instability, which results from the inadequacy of arrangement of s, is referred to as external geometric instability.
-
a monolithic rigid body is rigid by definition; hence, it will have no problem of internal instability. Furthermore, at any cut section of a monolithic rigid body, the elements of the internal force, which are no more than three in number, can always be determined by the equations of equilibrium, once the reactions are completely defined. Therefore, the stability and determinacy of the entire system are solely determined by the stability and determinacy of s and reactions.
summing up the above-mentioned discussion, the following conclusions can be derived;
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-
if the number of unknown elements of reaction is fewer than three, the equations of equilibrium are generally not satisfied and the system is said to be unstable - if the number of unknown elements of reaction is equal to three and if no external geometric instability is involved, then the system is statically stable and determinate -
if the number of unknown elements of reaction is more than three, then the system is statically indeterminate; it is stable provided no external geometric instability is involved. The excess number, n, of unknown elements designates the nth degree of indeterminacy.
General Stability and Determinacy of Structures; - structural stability and determinacy must be judged by the number and arrangement of the s as well as the number and arrangement of the and the connections of the structure. They are determined by inspection or by formula.
-
General Stability and Determinacy of Beams; - if a beam is built up without any internal connections (internal hinge, roller or link), the entire beam may be considered as one piece of monolithic rigid body placed on a number of s and the question of the stability and of the determinacy of the beam is settled solely by the number and arrangement of s. Geometric instability is most likely to occur whenever internal connections are inserted into an originally stable structure. The beam will obviously become unstable under general loading as the result of a relative rotation on both sides of the beam because the hinge has no capacity to resist moment and this will constitute a restriction on the external forces acting on the structure.
Also by investigation, there are three elements of reaction supplied by the s, whereas, there are four conditions of statics to resist the external forces-three from equilibrium plus one from construction. This means that the number of unknown elements of reaction is one fewer than the independent equations of statics available for their solution. Therefore, the equations of statics for the force system are generally not satisfied. The beam is unstable unless at least one additional element of reaction, such as a roller , is provided which makes the total number of unknown elements of reaction equal to the total number of independent equations of statics needed to determine the elements. If this is done, the beam will be restored to a statically stable and determinate state. From the above discussion, a criterion can be established for stability and determinacy of beams.
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if r < (c + 3), the beam is unstable
−
if r = (c + 3), the beam is statically determinate provided that no geometric instability (external or internal) is involved
−
if r > (c + 3), the beam is statically indeterminate r c
-
the number of reaction elements the number of equations of conditions
= = =
1.0 2.0 0
-
for hinge for roller for beam without internal connection
General Stability and Determinacy of Truss; - a truss is composed of a number of bars connected at their ends by a number of pinned ts so as to form a network, usually a series of triangles, and mounted on a number of s. Each bar of a truss is a two-force member; hence, each represents one unknown element of internal force. The total number of unknown elements of the entire system is counted by the number of bars (internal) plus the number of independent reaction elements (external). If the truss is in equilibrium, every isolated portion of the truss must likewise be in equilibrium. For a truss having a certain number of ts, the entire system may be separated into free bodies, the number of which is the same as the number of ts, in which each t yields two equilibrium equations. Therefore, a criterion can be established for the stability and determinacy of trusses. −
if (b + r) < 2j,
system is unstable
−
if (b + r) = 2j,
system is statically determinate provided that it is also stable
−
if (b + r) > 2j,
system is statically indeterminate
where:
-
Chapter 1
−
where:
-
Structural Theory 1
b r j
-
the number of bars the number of reaction components the number of ts
General Stability and Determinacy of Rigid Frames; - a rigid frame is built of beams and columns connected rigidly. The stability and determinacy of a rigid frame may also be investigated by comparing the number of unknowns (internal unknowns and reaction unknowns) with the number of equations of statics available for their solution. Like a truss, a rigid frame may be separated into a number of free bodies of ts, which requires that every member of the frame be taken apart. There are usually three unknown magnitudes existing in a cut section of a member. However, if these quantities are known at one section of a member, similar quantities for any other section of the same member can be determined. Hence, there are only three independent, internal, unknown elements for each member in a frame. The criteria for the stability and determinacy of a rigid frame are established by comparing the number of unknowns with the number of independent equations. −
if (3b + r) < (3j + c),
Engr. I.R. Bonzon
the frame is unstable
11
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−
if (3b + r) = (3j + c),
the frame is statically determinate provided that it is also stable
−
if (3b + r) > (3j + c),
the frame is statically indeterminate
where:
b r j c
-
the the the the
number number number number
of of of of
reaction components ts equations of conditions
EX. Classify each of the beam shown.
(A)
(B)
(C)
(D)
(E)
(F)
r
c
c + 3
Condition
A
3
0
3
statically determinate
B
4
0
3
statically indeterminate (first degree)
C
6
1
4
statically indeterminate (second degree)
D
4
2
5
statically unstable
E
2
0
3
statically unstable
F
6
0
3
statically indeterminate (third degree)
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EX. Classify each of the trusses shown.
(A)
(B)
(C)
(D)
(E)
(F)
(G)
(H)
r
b
j
b + r
2j
condition
A
3
17
10
20
20
statically determinate
B
2
17
10
19
20
unstable
C
3
21
10
24
20
D
3
16
10
19
20
unstable
E
4
26
15
20
30
statically determinate
F
3
10
7
13
14
unstable
G
3
11
7
14
14
statically determinate
H
4
10
7
14
14
statically determinate
Engr. I.R. Bonzon
statically indeterminate (fourth degree)
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EX. Classify each of the plane frame shown.
(A)
(B)
(C)
r
b
j
c
3b + r
3j + c
condition
A
8
5
6
0
23
18
statically indeterminate (fifth degree)
B
3
4
4
0
15
12
statically indeterminate (third degree)
C
9
10
9
0
39
27
statically indeterminate (twelfth degree)
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Introduction; Engineering Structures; - the word structure has various meanings. By an engineering structure, that means roughly something constructed or built. The principal structures of concern to civil engineers are bridges, buildings, walls, dams towers and shell structures. Structures as such are composed of one or more solid elements so arranged that the whole structures as well as their components are capable of holding themselves without appreciable geometric change during loading and unloading. To design a structure involves many considerations, among which are two major objectives that must be satisfied; -
the structure must meet the performance requirement
-
the structure must carry the loads safely
Consider, for example, the roof trusses resting on columns shown in figure below. The purposes of the roof truss and of the columns are, on the hand, to hold in equilibrium their own weights, the load of roof covering the wind; and, on the other hand, to provide shelter for housing a family, a manufacturing plant, or for other uses.
the one and for
Theory of Structures Defined; - the complete design of a structure is outlined in the following stages;
-
Developing a General Layout; - the general layout of a structure is selected from any possible alternatives. The primary concern is the purpose for which the structure is to be built. This stage involves the structural type, the selection of material and a tentative estimation of cost based on a reasonable analysis of a preliminary structural design. It may also involve selecting the best location or adapting the structure to a site that has not been predetermined. Proper attention should also be given to the aesthetic value of the structure since the requirement of good appearance can reasonably be met without much increase in cost. There are many other considerations, including the legal, financial, sociological, cultural, and environmental aspects. It is clear that this stage of design calls for an engineer with a high order of experience, skill, general knowledge and imagination.
-
Investigating the Loads; - before a refined structural analysis, it is necessary to determine the loads for which a given structure should be designed. General information about the loads imposed on a structure is usually given in the specifications and codes. However, it is part of the designer’s responsibility to specify the load conditions and to take care of exceptional cases. The weight of the structure itself together with the material permanently attached to it is called the dead loads and is regarded as fixed in magnitude and location. Since the dead load must be approximated before the
Engr. I.R. Bonzon
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structure is designed, the preliminary data are only tentative. Revision must be made if the initial estimation is not satisfactory. All loads, other than the dead loads, may be called live loads. Live loads are generally classified as movable loads and moving loads. Movable loads are loads that may be transported from one location to another on a structure without dynamic impact; for example, people, furniture and goods on a building floor or rainwater on a roof. Moving loads are loads that move continuously over the structure, such as railway trains or tracks on a bridge, wind on a roof or wall or hydrostatic pressure on an abutment. Moving loads may also be applied suddenly to the structure, for example, the centrifugal and longitudinal forces induced by the acceleration of vehicles and the dynamic forces generated by earthquakes. In an ordinary structural design, all loads are treated as static loads in order to simplify the analysis. In this way the impact due to a moving live load is expressed as a percentage of the live load and the earthquake force is commonly considered to be a horizontal force equal to a fraction of the weight of a structure. Other load considerations may include thermal effects and resistance to bomb blasting.
-
Stress Analysis; - once the basic form of the structure and the external loads are defined, a structural analysis can be made to determine the internal forces in various of the structure and the displacements at some controlling points. When live loads are involved, it is important to determine the maximum possible stresses in each member being considered. The principles governing this phase of the design is usually discuss in the theory of structures.
-
Selection of Elements; - the selection of suitable sizes and shapes of and their connections depends on the results of the stress analysis together with the design provisions of the specifications or codes. A trial and error approach may be used in the search for a proportioning of elements that will be both economical and adequate. A sound knowledge of strength of material and process of fabrication is also essential.
-
Drawing and Detailing; - once the makeup of each part of the structure has been determined, the last stage of the design can begin. This final stage includes the preparation of contract drawing, detailing, job specification and the final cost; this information is necessary for construction to proceed.
-
these five stages are interrelated and may be subdivided and modified. In many cases, they must be carried out more or less simultaneously. The subject matter of the theory of structures is stress analysis with occasional reference to loading. The emphasis of structural theory is usually on the fundamentals rather than on the details of design.
Theory of Structures Classified; - structural theory may be classified from various points of view. convenience, structural theory is characterize by the following aspects;
-
For
Statics versus Dynamics; - ordinary structures are usually designed under static loads. Dead loads are static loads that cause no dynamic effect on structures. Some live loads, such as, trucks and locomotives moving on bridges, are also assumed as
Engr. I.R. Bonzon
2
THEORY1
Structural Theory 1
Chapter 1
concentrated static load systems. They do cause impact on structures; however, the dynamic effects are treated as a fraction of the moving loads to simplify the design. The particular, specialized branch that deals with the dynamic effects of structures of accelerated moving loads, earthquake loads, wind gusts or bomb blasts is structural dynamics.
-
Plane versus Space; - no structure is really planar. However, structural analysis for beams, trussed bridges or rigid frame buildings are usually treated as plane problems, although they are never two-dimensional in themselves. On the other hand, in some structures, such as towers and framing for domes, the stresses are interrelated between the not lying in a plane in such a way that the analysis cannot be simplified on the basis of component planar structures. Such structures must be considered as space frameworks under noncoplanar force system.
-
Linear versus Nonlinear Structures; - linear structures mean that a linear relationship is assumed to exist between the applied loads and the resulting displacements in a structure. This assumption is based on the following conditions; −
the material of the structure is elastic and obey the Hooke’s Law at all points and throughout the range of loading considered
−
the changes in the geometry of the structure are so small that they can be neglected when the stresses are calculated
Note that if the principle of superposition is to apply, a linear relationship must exist, or be assumed to exist, between loads and displacements. The principle of superposition states that the total effect at some point in a structure due to a number of causes (forces or displacements) acting simultaneously is equal to the sum of the effects for the causes acting separately. A nonlinear relationship between the applied actions and the resulting displacements exists under of two conditions; − −
the material of the structure is inelastic the material is within elastic range, but the geometry of the structure changes significantly during the application of the loads
The study of nonlinear behaviour of structures analysis of structures and buckling of structures.
-
includes
plastic
Statically Determinate versus Statically Indeterminate Structures; - the term statically determinate structure means that the structural analysis can be carried out by statics alone. If this is not so, the structure is statically indeterminate. A statically indeterminate structure is solved by the equations of statics together with the equations furnished by the geometry of the elastic curve of the structure in linear analysis. It can be noted that the elastic deformations of the structure are not only associated with the applied loads on the structure but are also affected by the material properties (e.g., the modulus of elasticity) and by the geometric properties of the member section (e.g., the cross-sectional area or the moment of inertia). Thus, loads,
Engr. I.R. Bonzon
3
THEORY1
Structural Theory 1
Chapter 1
material properties and geometric properties are all involved in solving a statically indeterminate structure, while load factor alone dominates in a statically determinate problem.
-
Force versus Displacement; - the force and displacement are two categories of event occurring to a structure. The objective of a structural analysis is to determine forces and displacements pertaining to the structure and their relationships specified by the geometric and material properties of structural elements. Structural analysis can be divided into two categories; the force and the displacement methods. In the force method, the forces are treated as the basic unknowns and the displacements are expressed in of forces; whereas, in the displacement method, it is the displacements that are regarded as the fundamental unknowns and the forces are expressed in of displacements. In matrix analysis of linear structures, the force method is often regarded to as the flexibility method, and the displacement method is called the stiffness method.
Actual and Ideal Structures; - all analyses are based on some assumptions not quite in accordance with the facts. It is impossible for an actual structure to correspond fully to the idealized structure on which the analysis is based. The materials of which the structure is built do not follow exactly the assumed properties and the dimensions of the actual structure do not coincide with their theoretical values. To illustrate, let us take a simple example. In deg a reinforced concrete beam of rectangular section, the values of the modulus of elasticity and the moment of inertia are usually assumed to be constant. However, the amount of reinforcing steel placed in the beam varies with the stresses; therefore, the values of the modulus of elasticity and the moment of inertia are not constant throughout the span. Besides, there is a great uncertainty involved even in choosing a constant value of the modulus of elasticity and the moment of inertia. Even without considering other factors, such as the s, the connections and the working dimensions of the structure, it can be noted that the behaviour of an actual structure often deviates from that of an idealized structure by a considerable amount. However, it does not follow from this that the results of analysis are not useful for practical purposes. Therefore, the process of structural solution is setting an idealized model in order to carry out practical analysis and from the practical analyses, the idealization can be made more and more consistent with actuality. Scope of Theory of Structures; - three major types of basic structures are thoroughly considered in structural theory;
-
Rigid Frame; - composed of connected by rigid ts capable of resisting moments. of a rigid frame, in general, are subjected to bending moment, shear and axial forces.
-
Truss; - composed of connected by frictionless hinges or pins. The loads on a truss are assumed to be concentrated at the ts. Each member of a truss is considered as a two-force member subjected to axial force only.
Engr. I.R. Bonzon
4
THEORY1
-
Structural Theory 1
Chapter 1
Beam; - a straight member subjected only to transverse loads. A beam is completely analyzed when the values of bending moment and shear are determined.
Stability and Determinacy of Structures; Equations of Equilibrium for a Coplanar Force System; - the first and major function of a structure is to carry loads without showing appreciable distortions. In structural statics, all force systems are assumed to act on rigid bodies. Actually, there are always some small deformations that may cause some small change of dimension in structure and a shifting of the action lines of the forces. However, such deviations are neglected in stress analysis. A structure is said to be in equilibrium if, under the action of external forces, it remains at rest relative to the earth. Also, each part of the structure, if taken, as a free body isolated from the whole, must be at rest relative to the earth under the action of the internal forces at the cut sections and of the external forces thereabout. If such is the case, the force system is balanced, or in equilibrium, which implies that the resultant of the force system (either a resultant force or a resultant couple) imposed on the structure, or segment thereof, must be zero. The generally balanced coplanar force system must then satisfy the following three simultaneous equations;
where:
ΣFx
=
0
ΣFx
-
summation of x component of each force in the system
ΣFy
-
summation of y component of each force in the system
ΣM a -
ΣFy
=
0
ΣM a =
0
summation of moment about any point a in the plane due to each force
The subscripts x and y indicate two mutually perpendicular directions in the Cartesian coordinate system.
-
Special Cases of the Coplanar Force System in Equilibrium; -
Concurrent Forces; − if a system of coplanar, concurrent forces is in equilibrium, forces of the system must satisfy the following equations;
ΣFx
=
0
ΣFy
=
then the
0
Another set of independent equations necessary and sufficient for the equilibrium of the forces of a coplanar, concurrent force system, provided that point a is not on the line through the concurrent point of forces and perpendicular to y axis, is; ΣFy
=
0
ΣM a =
0
A third set of equations of equilibrium for a coplanar, concurrent force system, provided that line through a and b does not through the concurrent point of the forces, is;
ΣM a = Engr. I.R. Bonzon
0
ΣM b =
0
5
THEORY1
-
Structural Theory 1
Chapter 1
Parallel Forces; − if a coplanar, parallel force system is in equilibrium, the forces of the system must satisfy the equations; ΣFy
=
0
ΣM a =
0
Another set of independent equations of equilibrium for a system of a coplanar, parallel forces, provided that line through a and b is not parallel to the forces of the system, may be given as;
ΣM a =
-
0
ΣM b =
0
Simple, Special Cases of Equilibrium; -
Two-Force Member; − in the figure shown below, a body is subjected to two external forces applied at a and b. If the body is in equilibrium, then the two forces cannot be in random orientation, but must be directed along ab. Furthermore, they must be equal in magnitude and opposite in direction.
-
Three-Force Member; − in the figure shown below, a body is subjected to the action of three external forces applied at a, b, and c. If the body is in equilibrium, then the three forces cannot be in random orientation, but must be concurrent at a common point O; otherwise the total moment about the intersection of any two forces would not vanish. A limiting case occurs when point O moves off at an infinite distance from a, b, and c in which case the three forces will be parallel.
Reactions; - structures are either partially or completely restrained so that they cannot move freely in space. Such restraints are provided by s that connect the structure to some stationary body, such as the ground or another structure. The first step in structural analysis is to take the structure without the s and calculate the forces, known as reactions, exerted on the structure by the s. The reactions are considered part of the external forces other than the loads on the structure and are to balance the other external loads in a state of equilibrium. The distribution of the reactive forces of a may be represented by a single force completely specified by three elements-the point of application, the direction and the magnitude.
Engr. I.R. Bonzon
6
THEORY1
Structural Theory 1
Chapter 1
-
Hinge ; - a hinge can resist a general force, P, in any direction but cannot resist the moment of the force about the connecting point. The reaction of a hinge is assumed to be through the center of the connecting pin; its magnitude and slope of action line is yet to be determined. It is therefore a reaction with two unknown elements, which could equivalently represented by unknown magnitudes of its horizontal and vertical components both acting through the center of the hinge pin.
-
Roller ; - the mechanism used is such that the reaction acts normal to the ing surface through the center of the connecting pin. The reaction may be either away from or toward the ing surface. As such, the roller is incapable of resisting moment and lateral force along the surface of the . A roller supplies a reactive force, fixed at a known point and in a known direction, the magnitude of which is unknown. It is therefore a reaction with one unknown element.
-
Fixed ; - a fixed is capable of resisting force in any direction and the moment of force about the connecting end, thus preventing the end of the member from both translation and rotation. The reaction supplied by a fixed may be represented by the unknown magnitudes of a moment, a horizontal force and a vertical force acting through the centroid of the end cross section O. M
Rx
Rx
R
Ry hinge
R roller
Ry fixed
Internal Forces at a Cut Section of a Structure; - a truss structure is composed of pin-connected and is assumed to be pin-loaded. If any one of the is taken from its connecting pins as a free body, the forces exerted on the member must be concentrated at the two ends of the member through the center of the pins. Furthermore, these two systems of concurrent forces can be combined into two resultant forces that must be equal, opposite and collinear. In other words, each member of a truss is a two-force member. Hence, the internal forces existing in any cut section of a truss member (assumed straight and uniform) must be a pair of equal and opposite axial forces to balance the axial forces exerted on the ends.
of structures, such as beams and rigid frames, are acted on by more than two forces. It is obvious that forces of internal constraint must exist between these two portions in order to hold them together. Such internal Engr. I.R. Bonzon
7
THEORY1
Structural Theory 1
Chapter 1
forces, of course, always occur in pairs of equal and opposite directions. To maintain equilibrium of the free body, however, the internal forces must be statically equal and opposite to the system of forces acting externally on the portion considered and the internal forces can always be represented by a force applied at the centroid of the cross section together with a couple of moment.
Equations of Condition or Construction; - structures such as truss, beam and rigid frame may sometimes be considered to be one rigid body sustained in space by a number of s. Out of several such rigid bodies, a compound form of structure may be built by means of connecting devices, such as hinges, links or rollers and mounted on a number of s. In either the simple or the compound type of structure, the external force system of the entire structure, consisting of the loads on the structure and the reactions, must satisfy the equations of equilibrium if the structure is to remain at rest. However, in the compound type of structure, the connecting devices enforce further restrictions on the force system acting on the structure, thus providing additional equations of statics to supplement the equations of equilibrium. Equations supplied by the method of special construction (other than external s) are called equations of condition or construction. Stability and Determinacy of a Structure with Respect to s when the Structure is considered a Monolithic Rigid Body; - when one considers the design of a structure, careful thought must be given to the number and arrangement of the s directly related to the statical stability and determinacy of the structure. If a structure is treated as one monolithic rigid body mounted on a number of s, there will be no internal condition involved and the stability and determinacy of the structure will be judged solely by the stability and determinacy of s. -
two elements of reaction supplied by s, such as two forces each with a definite point of application and direction, are not sufficient to ensure the stability of a rigid body, because the two are either collinear, parallel or concurrent. In this case, the condition of equilibrium is violated, not because of the lack of strength of s, but because of the insufficient number of elements. This is referred to as statical instability.
-
at least three elements of reaction are necessary to restrain a body in stable equilibrium. In a stable equilibrium condition, the rigid body is
Engr. I.R. Bonzon
8
THEORY1
Structural Theory 1
Chapter 1
subjected to restraints by three elements of reactions and the restraints can be solved by the three available equilibrium equations. The satisfaction of all the three equilibrium equations, for loads and reactions acting on a body guarantees, respectively, that the body will neither move horizontally or vertically nor rotate. The system is said to be statically stable and determinate.
-
-
if there are more than three elements of reaction the body is necessarily more stable because of the additional restraints. Since the number of unknown elements of reaction is more than the number of equations for static equilibrium, the system is said to be statically indeterminate with regard to the reactions of .
-
the number of elements of reaction should be at least three is a necessity but not a sufficient condition for an externally stable structure. There are many cases that are obviously not stable with respect to the system even though three or more than three elements of reaction are supplied. When, for example, the lines of reaction are all parallel, the body is unstable because it is vulnerable to lateral sliding. Another case is when the lines of three reaction elements are originally concurrent at a point. The system is also unstable because, even though complete collapse probably will not result, a small initial rotation about the point because of the moment caused by any force not ing through the point will certainly occur until the three reaction lines form the triangle. This instability, which results from the inadequacy of arrangement of s, is referred to as external geometric instability.
-
a monolithic rigid body is rigid by definition; hence, it will have no problem of internal instability. Furthermore, at any cut section of a monolithic rigid body, the elements of the internal force, which are no more than three in number, can always be determined by the equations of equilibrium, once the reactions are completely defined. Therefore, the stability and determinacy of the entire system are solely determined by the stability and determinacy of s and reactions.
summing up the above-mentioned discussion, the following conclusions can be derived;
Engr. I.R. Bonzon
9
THEORY1
Structural Theory 1
Chapter 1
-
if the number of unknown elements of reaction is fewer than three, the equations of equilibrium are generally not satisfied and the system is said to be unstable - if the number of unknown elements of reaction is equal to three and if no external geometric instability is involved, then the system is statically stable and determinate -
if the number of unknown elements of reaction is more than three, then the system is statically indeterminate; it is stable provided no external geometric instability is involved. The excess number, n, of unknown elements designates the nth degree of indeterminacy.
General Stability and Determinacy of Structures; - structural stability and determinacy must be judged by the number and arrangement of the s as well as the number and arrangement of the and the connections of the structure. They are determined by inspection or by formula.
-
General Stability and Determinacy of Beams; - if a beam is built up without any internal connections (internal hinge, roller or link), the entire beam may be considered as one piece of monolithic rigid body placed on a number of s and the question of the stability and of the determinacy of the beam is settled solely by the number and arrangement of s. Geometric instability is most likely to occur whenever internal connections are inserted into an originally stable structure. The beam will obviously become unstable under general loading as the result of a relative rotation on both sides of the beam because the hinge has no capacity to resist moment and this will constitute a restriction on the external forces acting on the structure.
Also by investigation, there are three elements of reaction supplied by the s, whereas, there are four conditions of statics to resist the external forces-three from equilibrium plus one from construction. This means that the number of unknown elements of reaction is one fewer than the independent equations of statics available for their solution. Therefore, the equations of statics for the force system are generally not satisfied. The beam is unstable unless at least one additional element of reaction, such as a roller , is provided which makes the total number of unknown elements of reaction equal to the total number of independent equations of statics needed to determine the elements. If this is done, the beam will be restored to a statically stable and determinate state. From the above discussion, a criterion can be established for stability and determinacy of beams.
Engr. I.R. Bonzon
10
THEORY1
if r < (c + 3), the beam is unstable
−
if r = (c + 3), the beam is statically determinate provided that no geometric instability (external or internal) is involved
−
if r > (c + 3), the beam is statically indeterminate r c
-
the number of reaction elements the number of equations of conditions
= = =
1.0 2.0 0
-
for hinge for roller for beam without internal connection
General Stability and Determinacy of Truss; - a truss is composed of a number of bars connected at their ends by a number of pinned ts so as to form a network, usually a series of triangles, and mounted on a number of s. Each bar of a truss is a two-force member; hence, each represents one unknown element of internal force. The total number of unknown elements of the entire system is counted by the number of bars (internal) plus the number of independent reaction elements (external). If the truss is in equilibrium, every isolated portion of the truss must likewise be in equilibrium. For a truss having a certain number of ts, the entire system may be separated into free bodies, the number of which is the same as the number of ts, in which each t yields two equilibrium equations. Therefore, a criterion can be established for the stability and determinacy of trusses. −
if (b + r) < 2j,
system is unstable
−
if (b + r) = 2j,
system is statically determinate provided that it is also stable
−
if (b + r) > 2j,
system is statically indeterminate
where:
-
Chapter 1
−
where:
-
Structural Theory 1
b r j
-
the number of bars the number of reaction components the number of ts
General Stability and Determinacy of Rigid Frames; - a rigid frame is built of beams and columns connected rigidly. The stability and determinacy of a rigid frame may also be investigated by comparing the number of unknowns (internal unknowns and reaction unknowns) with the number of equations of statics available for their solution. Like a truss, a rigid frame may be separated into a number of free bodies of ts, which requires that every member of the frame be taken apart. There are usually three unknown magnitudes existing in a cut section of a member. However, if these quantities are known at one section of a member, similar quantities for any other section of the same member can be determined. Hence, there are only three independent, internal, unknown elements for each member in a frame. The criteria for the stability and determinacy of a rigid frame are established by comparing the number of unknowns with the number of independent equations. −
if (3b + r) < (3j + c),
Engr. I.R. Bonzon
the frame is unstable
11
THEORY1
Structural Theory 1
Chapter 1
−
if (3b + r) = (3j + c),
the frame is statically determinate provided that it is also stable
−
if (3b + r) > (3j + c),
the frame is statically indeterminate
where:
b r j c
-
the the the the
number number number number
of of of of
reaction components ts equations of conditions
EX. Classify each of the beam shown.
(A)
(B)
(C)
(D)
(E)
(F)
r
c
c + 3
Condition
A
3
0
3
statically determinate
B
4
0
3
statically indeterminate (first degree)
C
6
1
4
statically indeterminate (second degree)
D
4
2
5
statically unstable
E
2
0
3
statically unstable
F
6
0
3
statically indeterminate (third degree)
Engr. I.R. Bonzon
12
THEORY1
Structural Theory 1
Chapter 1
EX. Classify each of the trusses shown.
(A)
(B)
(C)
(D)
(E)
(F)
(G)
(H)
r
b
j
b + r
2j
condition
A
3
17
10
20
20
statically determinate
B
2
17
10
19
20
unstable
C
3
21
10
24
20
D
3
16
10
19
20
unstable
E
4
26
15
20
30
statically determinate
F
3
10
7
13
14
unstable
G
3
11
7
14
14
statically determinate
H
4
10
7
14
14
statically determinate
Engr. I.R. Bonzon
statically indeterminate (fourth degree)
13
THEORY1
Structural Theory 1
Chapter 1
EX. Classify each of the plane frame shown.
(A)
(B)
(C)
r
b
j
c
3b + r
3j + c
condition
A
8
5
6
0
23
18
statically indeterminate (fifth degree)
B
3
4
4
0
15
12
statically indeterminate (third degree)
C
9
10
9
0
39
27
statically indeterminate (twelfth degree)
Engr. I.R. Bonzon
14
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