Vierendeel Girders 1b5t6y

Vierendeel Bridge Grammene Belgium

Vierendeel girder and frame Vierendeel structures

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Arthur Vierendeel (1852–1940) born in Leuven, Belgium was a university professor and civil engineer. The Vierendeel structure he developed was named after him. His work, Cours de stabilité des constructions (1889) was an important reference during more than half a century. His first bridge was built 1902 in Avelgen, crossing the Scheldt river

Vierendeel structures

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Berlin Pedestrian Bridge

Vierendeel structures

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Berlin HBF:

Vierendeel structures

Vierendeel frame

Vierendeel elevator shaft

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Vierendeel detail

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Vierendeel girder and frame Named after 19th century Belgian inventor, Vierendeel girders and frames are bending resistant 1 2 3 4

1-bay girder Gravity load Lateral load Articulated Inflection points

Vierendeel structures

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3-bay girder Gravity load Lateral load Articulated Inflection points

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Base girder Global shear Global moment Bending  Chord forces

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Pin ts Strong web Strong chord Shear  Chord shear

One-way girders 1 Plain girder 2 Prismatic girder 3 Prismatic girder

Space frames 4 2-way 5 3-way 6 3-D

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Salk Institute, La Jolla Architect: Louis Kahn Engineer: Komendant and Dubin

Viernedeel girders of 65’ span, provide adaptable interstitial space for evolving research needs

Perspective section and photo, courtesy Salk Institute Vierendeel structures

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Yale University Library Architect/Engineer: SOM

Vierendeel structures

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Vierendeel facade Vierendeel elements Cross section

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The library features five-story Vierndeel frames

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Four concrete corner columns the frames

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Length direction span: 131 feet Width direction span: 80 feet

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Façades are assembled from prefab steel crosses welded together at inflection points

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The tapered crosses visualize inflection points

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Commerzbank, Frankfurt Architect: Norman Foster Engineer: Ove Arup Floors between sky gardens are ed by eight-story high Vierendeel frames which also resist lateral load

Vierendeel structures

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Commerzbank, Frankfurt Architect: Norman Foster Engineer: Ove Arup Vierendeel elevation / plan

Vierendeel / floor girder

Vierendeel / floor girder t detail

Vierendeel structures

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Hong Kong Shanghai Bank Architect: Norman Foster Engineer: Ove Arup Gravity / lateral load : • Hanger / belt truss • Vierendeel towers

Vierendeel structures

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Vierendeel steel girder Assume: 10” tubing, allowable bending stress Fb = 0.6x46 ksi Girder depth d = 6’, span 10 e = 10x10’ DL= 18 psf LL = 12 psf  = 30 psf Uniform load w = 30 psf x 20’ / 1000 t load P = 0.6 x 10’ Max shear V = 9 P/2 = 9 x 6/2 CHORD BARS Shear (2 chords) Vc = V/2 = 27/2 Chord bending (k’) Mc = Vc e/2 = 13.5x5 Chord bending (k”) Mc = 67.5 k’ x12” Moment of Inertia I = Mc c/Fb = 810 k” x 5”/27.6 ksi 2nd bay chord shear Vc = (V–P)/2 = (27-6)/2 2nd chord bending Mc = Vc e/2 = 10.5 x 5 2nd chord bending Mc = 52.5 k’ x 12” WEB BAR (2nd web resists bending of 2 chords) Web bar bending Mw = Mc end bay + Mc 2nd bay Mw = 810 + 630 Moment of Inertia I = Mw c/Fb = 1440 k” x 5”/27.6 ksi Vierendeel structures

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Fb= 27.6 ksi L = 100’

w = 0.6 klf P= 6 k V = 27 k Vc = 13.5 k Mc = 67.5 k’ Mc = 810 k” I = 147 in4 Vc = 10.5 k Mc = 52.5 k’ Mc = 630 k” Mw=1,440 k” I = 261 in4 11

Compare Multiframe vs. schematic results

MF load

MF shear

MF bending

Schematic results

A

20’

B

6x10 = 60’

Vierendeel with overhangs MAL= (2x10/2)(5’) = 50 k’ MAR= (5x10/2)(5’)-50 = 75 k’ Bars @ B are symmetrical

Vierendeel structures

20’

(Multiframe MAL = 61.3 (Multiframe MAR = 73.63

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Vierendeel structures

Chord bars Moment of Inertia required Use ST10x10x5/16

I= 147 in4 I= 183>147

Web bars Moment of Inertia required Use ST10x10x1/2

I= 261 in4 I= 271>261

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Sport Center, University of California Davis Architect: Perkins & Will Engineer: Leon Riesemberg Given the residential neighborhood, a major objective was to minimize the building height by several means: • The main level is 10’ below grade • Landscaped berms reduce the visual façade height • Along the edge the roof is attached to bottom chords to articulates the façade and reduce bulk Assume Bar cross sections 16”x16” tubing, 3/16” to 5/8” thick Frame depth d = 14’ (max. allowed for transport) Module size: 21 x 21 x 14 ft Width/length: 252 x 315 ft Fb = 27.6 ksi Structural tubing Fb = 0.6 Fy = 0.6x46 ksi DL = 22 psf LL = 12 psf (60% of 20 psf for tributary area > 600 ft2)  = 34 psf Note: two-way frame carries load inverse to deflection ratio: r = L14/(L14+L24) = 3154/(3154+2524) r = 0.71 Uniform load per bay w = 0.71 x 34 psf x 21’/1000 w = 0.5 klf Vierendeel structures

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Modules: 21x21x14’

Vierendeel structures

Design end chords t load P = w x 21’ = 0.5klf x 21’ P = 10.5 k Max. shear V = 11 P /2 = 11 x 10.5 / 2 V = 58 k Chord shear (2 chords) Vc = V/2 = 58 k / 2 Vc = 29 k Chord bending Mc = Vc e/2 = 29x 21’x12”/2 Mc= 3654 k” Moment of Inertia required I = Mc c /Fb = 3654 x 8”/27.6 ksi I = 1059 in4 Check mid-span compression Global moment M = 3969 k’ M = w L2/8 = 0.5 x 2522/8 Compression (d’=14’–16”=12.67’) C = M/d’= 3969 k’/ 12.67 C = 313 k

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Chord bars Moment of Inertia required Use ST16x16x1/2 Check mid-span chord stress Compression Allowable compression

I= 1059 in4 I= 1200

C = 313 k Pall = 728 k 313 <<728

Note: End-bay bending governs

Vierendeel structures

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Commerzbank, Frankfurt Design edge girder Assume: Tributary area End bay width Loads: 70 psf DL+ 30 psf LL Allowable stress Fb =0.6 x36 Girder shear V = 60’x20’x 100 psf/1000 Bending moment M = V e/2 = 120x20/2 Required section modulus S = M/Fb = 1200 k’ x 12”/ 21.6 ksi Use W40x192

60’x20’ e = 20’ ∑=100 psf Fb = 21.6 ksi V = 120 k M = 1200 k’ S = 667 in3 S = 706 in3

Note: check also lateral load Variable bay widths equalize bending stress Load at corners increases stability

Vierendeel structures

Prof Schierle

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Vierendeel steel girder Assume: 10” tubing, allowable bending stress Fb = 0.6x46 ksi Girder depth d = 6’, span 10 e = 10x10’ DL= 18 psf LL = 12 psf  = 30 psf

Fb= 27.6 ksi L = 100’

Uniform load w = 30 psf x 20’ / 1000 w = 0.6 klf t load P = 0.6 x 10’ P= 6 k Max shear V = 9 P/2 = 9 x 6/2 V = 27 k CHORD BARS Vc = 13.5 k Shear (2 chords) Vc = V/2 = 27/2 Chord bending Mc = Vc e/2 = 13.5 x (10’x12”)/ 2 Mc = 810 k” Moment of Inertia I = Mc c/Fb = 810 k” x 5”/27.6 ksi I = 147 in4 2nd bay chord shear Vc = (V–P)/2 = (27-6)/2 Vc = 10.5 k 2nd chord bending Mc = Vc e/2 = 10.5 x 120”/2 Mc = 630 k” WEB BAR (2nd web resists bending of 2 chords) Web bar bending Mw = Mc end bay + Mc 2nd bay Mw = 810 + 630 Mw=1,440 k” Moment of Inertia I = Mw c/Fb = 1440 k” x 5”/27.6 ksi I = 261 in4 Vierendeel structures

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Scheepsdale Revolving Bridge Bruges, Belgium 1933 Vierendeel structures

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Railroad Bridge Vierendeel structures

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Dallvazza Bridge Swiss, 1925

Gellik Railroad Bridge Belgium

Anderlecht Railroad Bridge Belgium

Osera de Ebro Bridge, Zaragoza, Spain, 2002

Pedestrian Bridge Vierendeel structures

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Vierendeel girder and frame endure Vierendeel structures

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