Vierendeel girders resist load in combed beam action and frame action as shown on the left and right diagrams, respectively. Load causes global shear and bending which elongates the bottom in tension and shortens the top in compression. The internal reaction to global shear and bending is different in a Vierendeel compared to a beam. A beam's bending stress varies gradually over the cross-section, but global bending in a Vierendeel causes concentrated tension and compression forces in the chords. By visual inspection we can derive simple formulas for approximate axial and shear forces and bending moments. Respective stresses are found using formulas for axial, shear and bending stress and superimposing them. Chord tension T and compression C are computed, dividing the respective global moment M by frame depth D (distance between centers of chords).
Bending of individual struts can be visualized too. In a structure where moment resistant strut/chord connections are replaced with hinges, chords would deleci as independent beams6. Assuming flexible chords and stiff webs, vertioa' «hear would aefcrm chords to S-shapes with inflection point. Assuming flexib'e webs and siifi ch Drds: horizontal shear, caused by a compressive force pushing outvie' on Lop anu a tensile force pulling inward on the bottom, would deform webs to S-shapes with inflection point. The combined effect of these two idealized cases imparls S-snaped deformation and inflection points in both chord and we!; strut?. The deformation yields strut bending moments which vary from positive to negative -siong each strut. Top and bottom chords carry each about half the iota! shear V. Assuming inflection points at midpoints of chords, the local chord moment M is half the shear V multiplied by half the chord length.
The moment M is maximum at supports where shear is greatest and equal to support reactions. For equilibrium, webs have to balance chord moments at each joint. Their moment equals the difference of adjacent chord moments.
1 Gravity load on a Vierendeel
2 Global shear (in overall system rather than individual members)
3 Global bending (in overall system rather than individual members)
4 Compression and tension in top and bottom chord, respectively
5 Free-body visualizes derivation of chord tension T and compression C
6 Global shear deformation
7 Chord bending, assuming flexible chords and stiff webs
8 Web bending, assuming flexible webs and stiff chords
9 Combined chord and web bending under actual condition
10 Free-bodies visualize derivation of chord bending moment M
Vierendeels may have various configurations, including one-way and two-way spans.
One-way girders may be simply supported or continuous over more than two supports. They may be planar or prismatic with triangular or square profile for improved lateral load resistance. Some highway pedestrian bridges are of the latter type. A triangular cross-section has added stability, inherent in triangular geometry. It could be integrated with bands of skylights on top of girders.
When supports are provided on all sides, Vierendeel frames of two-way or three-way spans are possible options. They require less depth, can carry more load, have less deflection, and resist lateral load as well as gravity load. The two-way option is well suited for orthogonal plans; the three-way option adapts better to plans based on triangles, hexagons, or free-form variations thereof.
One-way planar Vierendeel g One-way prismatic Vierende» One-way prismatic Vierende» Two-way Vierendeel space frame Three-way Vierendeel space frame Multi-story Vierendeel space frame
Salk Institute, La Jolla, California (1966)
Architect: Louis I. Kahn
Engineer: August Komendant and Fred Dubin Associates
The Salk Institute in Lajolla houses two research laboratories, flanked by service towers facing the outside, and study cells for scientists facing a central courtyard with view to the Pacific Ocean. North being oriented to the left on the drawing, offices are located on both west ends of the two lab wings. Mechanical rooms are located to the east. The courtyard concept was inspired by Dr. Salk's memory of the monastery of St. Francis of Assisi which he visited in the 1950's. This concept and the sparse use of material, reinforced concrete, accented by the study cell's teak wood finish, give the Salk Institute a serene beauty.
To avoid interior columns for greater flexibility, 9 ft (2.7m)-deep Vierendeel girders of prestressed concrete span across the width of the labs. These girders are located at interstitial spaces between ceiling and floor to provide space for duct,5 ar.d pipes, allowing access to change them as required by evolving research need?. The Vierardeel struts vary in thickness and shape to reflect variable stress pattern«. Top- and bottom chords are tapered for increased depth toward the supports v/heie global shear, which generates chord bending stress, is giea'.est. Vertical struts, too, vary from minimum thickness at ¡nid-span to maximum iowaid the supports, since their bending is the sum of Ihe bending o' all chords connected to them. However, the greatest bending in vertical struts occur« not over the supports but at struts next to them, since they absorb bending irom chords on both sides; yet end struts transfer bending to only one set of chords.
Perspective section, courtesy Salk Institute
Perspective section, courtesy Salk Institute
Beinecke library, Yale University, New Haven, Connecticut (1963)
Architect and Engineer: Skidmore, Owings and Merrill
The Beinecke library of Yale University for rare books has a 5-level central book tower, freestanding within a single story donut-shaped hall that extends over the full height of the tower. The tower holds 180,000 books and is climatically separated from the surrounding hall by a glass curtain wall.
The library's five-story open space is framed by a unique structural concept. Four Vierendeel steel frames, 50 ft (15 m) high, support the roof and wall load and span 131 and 88 ft (40 and 27m) in length and width, respectively. The frames are supported by a reinforced concrete plate that transfers the load via steel pin joints to four reinforced concrete pylons. The Vierendeel frames consist of 8-8" (2.6m) prefabricated steel crosses, welded together during erection. The cladding of the crosses express pin joints at mid-points of chord and web struts, where inflection points of zero henrfing occur. The pin joints express two-way hinges by the tapering. This does not represent actual construction of the steel frame behind the cladding, it the vranr.e nad in fasv two-way hinges, out-of-plane instability would result. This is a visual weakness of an otherwise compelling tectonic expref,:
Was this article helpful?