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Vierendeel girder and frame

Vierendeel Bridge Grammene Belgium

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Vierendeel structures Copyright Prof Schierle 2011 2

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

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Vierendeel structures Copyright Prof Schierle 2011 4

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1 Base girder

2 Global shear

3 Global moment

4 Bending 

5 Chord forces

6 Pin joints

7 Strong web

8 Strong chord

9 Shear 

10 Chord shear

1 1-bay girder

2 Gravity load

3 Lateral load

4 Articulated

Inflection points

5 3-bay girder

6 Gravity load

7 Lateral load

8 Articulated

Inflection points

One-way girders

1 Plain girder

2 Prismatic girder

3 Prismatic girder

Space frames

4 2-way

5 3-way

6 3-D

Vierendeel girder and frame

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Vierendeel structures Copyright Prof Schierle 2011 6

Salk Institute, La Jolla

Architect: Louis Kahn

Engineer: Komendant and Dubin

Perspective section and photo, courtesy Salk Institute

Viernedeel girders of 65’ span, provide adaptable

interstitial space for evolving research needs

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Yale University Library

Architect/Engineer: SOM

1

Vierendeel facade

2

Vierendeel elements

3

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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Vierendeel structures Copyright Prof Schierle 2011 8

Commerzbank, Frankfurt

Architect: Norman Foster

Engineer: Ove Arup

Floors between sky gardens are

supported by eight-story high

Vierendeel frames which also

resist lateral load

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Commerzbank, Frankfurt

Architect: Norman Foster

Engineer: Ove Arup

Vierendeel elevation / plan

Vierendeel / floor girder

joint detail

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Vierendeel steel girder

Assume:

10” tubing, allowable bending stress F

b

= 0.6x46 ksi

F

b

= 27.6 ksi

Girder depth d = 6’, span 10 e = 10x10’

L = 100’

DL=

18 psf

LL =

12 psf

 =

30 psf

Uniform load

w = 30 psf x 20’ / 1000

w = 0.6 klf

Joint load

P = 0.6 x 10’

P= 6 k

Max shear

V = 9 P/2 = 9 x 6/2

V = 27 k

CHORD BARS

Shear (2 chords)

V

c

= V/2 = 27/2

V

c

= 13.5 k

Chord bending (k’)

M

c

= V

c

e/2 = 13.5x5

M

c

= 67.5 k’

Chord bending (k”) M

c

= 67.5 k’ x12”

M

c

= 810 k”

Moment of Inertia

I = M

c

c/F

b

= 810 k” x 5”/27.6 ksi

I = 147 in

4

2nd bay chord shear V

c

= (V–P)/2 = (27-6)/2

V

c

= 10.5 k

2nd chord bending M

c

= V

c

e/2 = 10.5 x 5

M

c

= 52.5 k’

2nd chord bending

M

c

= 52.5 k’ x 12”

M

c

= 630 k”

WEB BAR (2nd web resists bending of 2 chords)

Web bar bending

M

w

= M

c

end bay + M

c

2nd bay

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Load

Shear

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Vierendeel structures Copyright Prof Schierle 2011 14

Chord bars

Moment of Inertia required I= 147 in

4

Use ST10x10x5/16

I= 183>147

Web bars

Moment of Inertia required I= 261 in

4

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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

Structural tubing F

b

= 0.6 Fy = 0.6x46 ksi

F

b

= 27.6 ksi

DL = 22 psf

LL = 12 psf (60% of 20 psf for tributary area > 600 ft

2

)

 = 34 psf

Note: two-way frame carries load inverse to deflection ratio:

r = L1

4

/(L1

4

+L2

4

) = 315

4

/(315

4

+252

4

) r = 0.71

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Vierendeel structures Copyright Prof Schierle 2011 16

Design end chords

Joint 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 /F

b

= 3654 x 8”/27.6 ksi I = 1059 in

4

Check mid-span compression

Global moment

M = w L

2

/8 = 0.5 x 252

2

/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 bars

Moment of Inertia required

I= 1059 in

4

Use ST16x16x1/2

I= 1200

Check mid-span chord stress

Compression

C = 313 k

Allowable compression

P

all

= 728 k

313 <<728

Note:

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Vierendeel structures Copyright Prof Schierle 2011 18

Commerzbank, Frankfurt

Design edge girder

Assume:

Tributary area

60’x20’

End bay width

e = 20’

Loads: 70 psf DL+ 30 psf LL

∑=100 psf

Allowable stress F

b

=0.6 x36

F

b

= 21.6 ksi

Girder shear

V = 60’x20’x 100 psf/1000

V = 120 k

Bending moment

M = V e/2 = 120x20/2

M = 1200 k’

Required section modulus

S = M/F

b

= 1200 k’ x 12”/ 21.6 ksi

S = 667 in

3

Use W40x192

S = 706 in

3

Note: check also lateral load

Variable bay widths equalize bending stress

Load at corners increases stability

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Vierendeel steel girder

Assume:

10” tubing, allowable bending stress F

b

= 0.6x46 ksi

F

b

= 27.6 ksi

Girder depth d = 6’, span 10 e = 10x10’

L = 100’

DL=

18 psf

LL =

12 psf

 =

30 psf

Uniform load

w = 30 psf x 20’ / 1000

w = 0.6 klf

Joint load

P = 0.6 x 10’

P= 6 k

Max shear

V = 9 P/2 = 9 x 6/2

V = 27 k

CHORD BARS

Shear (2 chords)

V

c

= V/2 = 27/2

V

c

= 13.5 k

Chord bending

M

c

= V

c

e/2 = 13.5 x (10’x12”)/ 2 M

c

= 810 k”

Moment of Inertia

I = M

c

c/F

b

= 810 k” x 5”/27.6 ksi

I = 147 in

4

2nd bay chord shear V

c

= (V–P)/2 = (27-6)/2

V

c

= 10.5 k

2nd chord bending

M

c

= V

c

e/2 = 10.5 x 120”/2

M

c

= 630 k”

WEB BAR (2nd web resists bending of 2 chords)

Web bar bending

M

w

= M

c

end bay + M

c

2nd bay

M

w

= 810 + 630

M

w

=1,440 k”

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Vierendeel structures Copyright Prof Schierle 2011 20

Commerzbank, Frankfurt

Design edge girder

Assume:

Tributary area

60’x20’

End bay width

e = 20’

Loads: 70 psf DL+ 30 psf LL

∑=100 psf

Allowable stress F

b

=0.6 x36

F

b

= 21.6 ksi

Girder shear

V = 60’x20’x 100 psf/1000

V = 120 k

Bending moment

M = V e/2 = 120x20/2

M = 1200 k’

Required section modulus

S = M/F

b

= 1200 k’ x 12”/ 21.6 ksi

S = 667 in

3

Use W40x192

S = 706 in

3

Note: check also lateral load

Variable bay widths equalize bending stress

Load at corners increases stability

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Vierendeel structures Copyright Prof Schierle 2011 22

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Vierendeel structures Copyright Prof Schierle 2011 28

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References

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