Structural Engineering & Statics
where the load goes
Concept

Vierendeel — where it appears

A rectangular frame with rigid corners and no diagonals, which carries shear by bending its chords rather than by axial force. Its panels are open, which is what it is for, and the price is a deflection an order of magnitude above the triangulated equivalent.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

Cross the hangers and the chords stop bending. The same tied arch, the same sixteen hangers, the same load on half the span — hung vertically and hung as a network. Vertical hangers make the two chords a Vierendeel frame, which has no truss action at all, so a partial load is carried by bending: 3316 kNm in the tie and 6234 in the arch. Inclined hangers can carry the shear between the chords axially, and the same load gives 686 and 831 — factors of 4.8 and 7.5. The thrust is identical in both, because that is decided by the span and the rise and nothing else.

Cross the hangers and the bending goes

A tied arch with vertical hangers is a Vierendeel frame with a curved top chord — it has no truss action at all, so a load on half the span is carried by bending. Incline the hangers so they cross and the same two chords become a truss.

structures · Network arch The shear goes round the corner instead of across it. A 6-panel Vierendeel girder, 21 m by 6000 mm, under 300 kN at mid-span. There is no diagonal in it, so each panel's 150 kN of shear is carried as bending in the chords: the curves drawn along them are the chord moments, and every one passes through zero at the middle of its own panel. The local moment is the panel shear times the panel length over four, 131.3 kNm, and it adds to an axial force of 263 kN from the global moment at the same point. The girder deflects 109.09 mm against 6.53 mm for the same members triangulated — 16.70 times — and 98% of that movement is chord bending that a diagonal would have removed entirely.

The frame is a girder stood on end

Every unbraced building frame is a Vierendeel girder turned through ninety degrees, and the identification is not an analogy — it is the same equations with the axes swapped. Which means the frame inherits results that read as absurd for a building — more bays is stiffer, a wider building is not, and doubling one section property halves the sway.

structures · Vierendeel However heavy the chord, ordinary posts cap its share. The heavy chord's share of the shear in a panel of uniform shear in a Vierendeel girder of 2 m panels, 1.5 m deep, against how many times stiffer it is than the light chord, from the one-panel model (lines) and from a whole girder (dots). Splitting by second moment (dashed) runs from 0.5 towards 1. With posts as stiff as the light chord the share is 0.54 at a ratio of 4 and still only 0.55 at 30; posts 4, 16 and 100 times stiffer reach 0.65, 0.80 and 0.93 at 30. The posts, not the chord, set the ceiling. The dots are a twenty-panel girder at its sixth panel, far from both support and load: with ordinary posts 0.50, 0.53, 0.56, 0.60, 0.65, 0.69 at ratios of 1, 2, 4, 10, 20, 30. They follow the model until the heavy chord's own second moment becomes a noticeable part of the girder's — 31 per cent at a ratio of 30 — and it starts to carry the global moment by bending as a beam.

The heavy chord the posts will not let work

A Vierendeel girder carries its shear as bending in its two chords, and when one chord is four times as stiff as the other the obvious rule gives it four fifths of the shear. With posts as stiff as the light chord it takes 0.56. The heavy chord's end moments have to be carried round each joint by a post, the post bends, and the heavy chord turns with its joints instead of bending between them. No amount of extra steel in that chord takes its share past 0.55 in the middle of a long girder; only the posts can, and they have to be about 150 times as stiff as the light chord to make the rule come true.

structures · Vierendeel A load between the posts racks the panel it stands in. Elevation of a Vierendeel girder of six 2 m panels, 1.5 m deep, its deck chord as stiff as its other chord and its posts, the load standing in the middle of its second panel; the racked shape exaggerated, the unloaded girder faint. The panel the load stands in carries most of its shear, and the deck chord there bends both as a Vierendeel chord and as a beam between its posts: 38 kN·m against 27 with the load at mid-span. The posts either side carry 56 and 11 kN·m, and the heaviest post 56.

The load that visits every panel

A Vierendeel girder checked for a load at mid-span is checked for one position of a load that, on a bridge, stands everywhere in turn. Crossing it, a 100 kN load puts 1.7 times the mid-span check's moment into the end panels' chords, loads the middle post that a central load leaves idle, and makes the deck chord a beam between its posts in every panel where its share of the shear is small. Equal chords stay the stiffest arrangement; a vehicle whose axles span a panel cannot stand in the middle of one.

structures · Vierendeel A haunch moves the worst stress range to its toe. The stress range up the height of the first interior post of a Vierendeel girder of six 2 m panels, 1.5 m deep, every member of the same section (I = 117 × 10⁶ mm⁴, W = 780 × 10³ mm³), crossed by a 100 kN load, over one crossing: the plain post (dashed), and the post with 250 mm haunches 2.00 times as deep at both corners (solid); with the constant-amplitude limits 52.3 MPa for detail category 71 (dotted). Plain, the range is 78.3 MPa at the corner. Haunched, it is 43.3 at the corner and jumps to 61.2 at the toe, 250 mm along the post — the largest anywhere in it, 61.2.

The corner that counts its crossings

As a load crosses a Vierendeel girder, the moment at a post's corner changes sign when the load passes that post. A static check reads the peak, and the peak says the posts beside the supports work twice as hard as the middle one. Fatigue reads the range, and the range says every interior post works the same. A haunch at the corner lowers the corner's range and moves the worst of it to the haunch's toe — so a deeper haunch soon buys nothing, a longer one keeps paying, and a toe two detail categories worse than the corner cancels the haunch outright.

structures · Vierendeel Lightened where it reverses, and the range follows. The stress range each post's corner sweeps over one crossing of a Vierendeel girder of six 2 m panels, 1.5 m deep, its chords all of one section (I = 117 × 10⁶ mm⁴, W = 780 × 10³ mm³), crossed by a 100 kN load: uniform posts, posts sized to their peaks, and posts sized to both their peaks and their ranges. Uniform, every interior corner ranges 76.1 to 78.3 MPa. Sized to peaks, the middle post's range is 150.8 MPa and its neighbours' 101.6, against the uniform girder's worst of 78.3. Sized to both, no corner exceeds 78.3 MPa.

The post sized by what it must survive

A Vierendeel girder's posts are usually all one section, and a crossing load gives them peaks that differ by a factor of two and ranges that barely differ at all. Size each post to its peak and the steel falls by a quarter — and the middle post, lightened to two-fifths of the chords, sweeps twice the stress range the whole girder used to, and lasts a seventh as long. Size each post to its range instead and the girder needs less steel than the uniform one, not more, because the posts that can be lightened are the ones at the ends.

structures · Vierendeel

Named alongside it

The objects these essays reach for when they reach for this one.

Influence lineLoad sharingStiffnessContraflexureDetail categoryFatigueMoving loadRigid jointStress rangeBending stiffnessChordDrift

All concepts