vierendeel girders

Vierendeel structures Copyright Prof Schierle 2011 1

Vierendeel girder and frame

Vierendeel Bridge Grammene Belgium


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


Berlin Pedestrian Bridge


Berlin HBF: Vierendeel frame Vierendeel elevator shaft Vierendeel detail


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


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


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


Commerzbank, FrankfurtArchitect: Norman FosterEngineer: Ove Arup

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


Commerzbank, FrankfurtArchitect: Norman FosterEngineer: Ove Arup

Vierendeel elevation / plan

Vierendeel / floor girderjoint detail

Vierendeel / floor girder


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


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


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


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’


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


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


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


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


Scheepsdale Revolving Bridge Bruges, Belgium 1933


Railroad Bridge

Dallvazza Bridge Swiss, 1925

Gellik Railroad Bridge Belgium

Anderlecht Railroad Bridge Belgium

Osera de Ebro Bridge, Zaragoza, Spain, 2002


Pedestrian Bridge


Vierendeel Space Frame


Vierendeel girder and frame endure

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