Vierendeel structures Copyright Prof Schierle 2011 1

Vierendeel girder and frame

Vierendeel Bridge Grammene Belgium

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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 Copyright Prof Schierle 2011 3

Berlin Pedestrian Bridge

Vierendeel structures Copyright Prof Schierle 2011 4

Berlin HBF: Vierendeel frame Vierendeel elevator shaft Vierendeel detail

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1 Base girder2 Global shear 3 Global moment4 Bending 5 Chord forces

6 Pin joints7 Strong web 8 Strong chord9 Shear 10 Chord shear

1 1-bay girder2 Gravity load 3 Lateral load4 Articulated

Inflection points

5 3-bay girder6 Gravity load 7 Lateral load8 Articulated

Inflection points

One-way girders1 Plain girder2 Prismatic girder 3 Prismatic girder

Space frames4 2-way5 3-way6 3-D

Vierendeel girder and frameNamed after 19th century Belgian inventor, Vierendeel girders and frames are bending resistant

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Salk Institute, La JollaArchitect: Louis KahnEngineer: Komendant and Dubin

Perspective section and photo, courtesy Salk Institute

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

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Yale University LibraryArchitect/Engineer: SOM

1 Vierendeel facade2 Vierendeel elements3 Cross section

• The library features five-story Vierndeel frames

• Four concrete corner columns support the frames

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

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

• The tapered crosses visualize inflection points

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Commerzbank, FrankfurtArchitect: Norman FosterEngineer: Ove Arup

Floors between sky gardens aresupported by eight-story highVierendeel frames which also resist lateral load

Vierendeel structures Copyright Prof Schierle 2011 9

Commerzbank, FrankfurtArchitect: Norman FosterEngineer: Ove Arup

Vierendeel elevation / plan

Vierendeel / floor girderjoint detail

Vierendeel / floor girder

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Vierendeel steel girderAssume: 10” tubing, allowable bending stress Fb = 0.6x46 ksi Fb= 27.6 ksiGirder depth d = 6’, span 10 e = 10x10’ L = 100’DL= 18 psfLL = 12 psf = 30 psfUniform load w = 30 psf x 20’ / 1000 w = 0.6 klfJoint load P = 0.6 x 10’ P= 6 kMax shear V = 9 P/2 = 9 x 6/2 V = 27 kCHORD BARSShear (2 chords) Vc = V/2 = 27/2 Vc = 13.5 kChord bending (k’) Mc = Vc e/2 = 13.5x5 Mc = 67.5 k’ Chord bending (k”) Mc = 67.5 k’ x12” 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 k2nd chord bending Mc = Vc e/2 = 10.5 x 5 Mc = 52.5 k’2nd chord bending Mc = 52.5 k’ x 12” 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

Load

Shear

Bending

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Chord barsMoment of Inertia required I= 147 in4

Use ST10x10x5/16 I= 183>147

Web barsMoment of Inertia required I= 261 in4

Use ST10x10x1/2 I= 271>261

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Sport Center, University of California DavisArchitect: Perkins & Will Engineer: Leon Riesemberg

Given the residential neighborhood, a major objective was tominimize 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 bulkAssumeBar cross sections 16”x16” tubing, 3/16” to 5/8” thickFrame depth d = 14’ (max. allowed for transport)Module size: 21 x 21 x 14 ftWidth/length: 252 x 315 ftStructural tubing Fb = 0.6 Fy = 0.6x46 ksi Fb = 27.6 ksiDL = 22 psfLL = 12 psf (60% of 20 psf for tributary area > 600 ft2) = 34 psfNote: two-way frame carries load inverse to deflection ratio:r = L14/(L14+L24) = 3154/(3154+2524) r = 0.71Uniform load per bayw = 0.71 x 34 psf x 21’/1000 w = 0.5 klf

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Design end chordsJoint loadP = w x 21’ = 0.5klf x 21’ P = 10.5 k Max. shearV = 11 P /2 = 11 x 10.5 / 2 V = 58 kChord shear (2 chords)Vc = V/2 = 58 k / 2 Vc = 29 kChord bendingMc = 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 compressionGlobal momentM = w L2/8 = 0.5 x 2522/8 M = 3969 k’Compression (d’=14’–16”=12.67’) C = M/d’= 3969 k’/ 12.67 C = 313 k

Modules:21x21x14’

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Chord barsMoment of Inertia required I= 1059 in4

Use ST16x16x1/2 I= 1200

Check mid-span chord stressCompression C = 313 kAllowable compression Pall = 728 k

313 <<728Note:End-bay bending governs

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Commerzbank, FrankfurtDesign edge girderAssume:Tributary area 60’x20’End bay width e = 20’Loads: 70 psf DL+ 30 psf LL ∑=100 psfAllowable stress Fb =0.6 x36 Fb = 21.6 ksi

Girder shearV = 60’x20’x 100 psf/1000 V = 120 kBending momentM = V e/2 = 120x20/2 M = 1200 k’Required section modulusS = M/Fb = 1200 k’ x 12”/ 21.6 ksi S = 667 in3

Use W40x192 S = 706 in3

Note: check also lateral loadVariable bay widths equalize bending stressLoad at corners increases stability

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Vierendeel steel girderAssume: 10” tubing, allowable bending stress Fb = 0.6x46 ksi Fb= 27.6 ksiGirder depth d = 6’, span 10 e = 10x10’ L = 100’DL= 18 psfLL = 12 psf = 30 psf

Uniform load w = 30 psf x 20’ / 1000 w = 0.6 klfJoint load P = 0.6 x 10’ P= 6 kMax shear V = 9 P/2 = 9 x 6/2 V = 27 kCHORD BARSShear (2 chords) Vc = V/2 = 27/2 Vc = 13.5 kChord 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 k2nd 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

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Commerzbank, FrankfurtDesign edge girderAssume:Tributary area 60’x20’End bay width e = 20’Loads: 70 psf DL+ 30 psf LL ∑=100 psfAllowable stress Fb =0.6 x36 Fb = 21.6 ksi

Girder shearV = 60’x20’x 100 psf/1000 V = 120 kBending momentM = V e/2 = 120x20/2 M = 1200 k’Required section modulusS = M/Fb = 1200 k’ x 12”/ 21.6 ksi S = 667 in3

Use W40x192 S = 706 in3

Note: check also lateral loadVariable bay widths equalize bending stressLoad at corners increases stability

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Scheepsdale Revolving Bridge Bruges, Belgium 1933

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Railroad Bridge

Dallvazza Bridge Swiss, 1925

Gellik Railroad Bridge Belgium

Anderlecht Railroad Bridge Belgium

Osera de Ebro Bridge, Zaragoza, Spain, 2002

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Pedestrian Bridge

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Vierendeel Space Frame

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Vierendeel girder and frame endure