Structural Engineering & Statics
where the load goes
Structural form

Depth is the cheapest strength there is

Doubling the depth of a truss halves its chord forces without adding a gram of material to the chords. Nothing else in structural design is that cheap, and almost every structure has already spent it.
9 min read 7 figures Geometry beats materialScale changes everything

There are three ways to make a structure carry more. Use more material, use better material, or move the material further apart. The third is free, and it is the one every efficient structure has already exploited to the limit of what something non-structural would allow.

Chord force against truss depthThe force in a truss chord for a fixed bending moment, against the depth of the truss. The relationship is a reciprocal: the chords form a couple whose lever arm is the depth, so a shallow truss pays for it steeply.0.511.52050100150200250300depth of the truss2501671251007150the same moment, resisted by a longer lever arm
Fig. 1 Chord force against truss depth for a fixed bending moment. The relationship is a reciprocal — halving the depth doubles the force in the chords, and nothing about the chords themselves has changed.

The couple, and its arm

A truss resists a bending moment with a pair of forces: the top chord pushing and the bottom chord pulling. Two equal and opposite forces separated by a distance form a couple, and the moment of a couple is the force times the separation.

M=F×d⟹F=Md.M = F \times d \quad\Longrightarrow\quad F = \frac{M}{d}.

The moment is set by the loads and the span, and nothing about the truss changes it. So the chord force is the moment divided by the depth, and depth is the only term available to reduce it.

That is the whole argument, and its economics are unusual. Doubling the depth halves the chord force, which halves the chord area required, which halves the chord material — while the web members get longer and there are the same number of them. The saving is real and one-sided.

A Pratt truss of 6 panelsA Pratt truss under equal panel-point loads. The joint equilibrium equations were assembled and solved; 10 members came out in tension, 9 in compression and 2 carrying nothing.tensioncompression2 carrying nothing
Fig. 2 A shallow truss. The chords are working hard because the lever arm between them is short, which the solver shows as heavy line weights along the top and bottom.
A Pratt truss of 6 panelsA Pratt truss under equal panel-point loads. The joint equilibrium equations were assembled and solved; 10 members came out in tension, 9 in compression and 2 carrying nothing.tensioncompression2 carrying nothing
Fig. 3 The same span and the same loads at three times the depth. The chord forces have fallen to a third, and the diagonals have grown longer and slightly harder-worked — which is the whole of what depth costs.

Comparing the two figures is the argument made visible. The load, the span, the panel count and the material are all identical. Only the depth differs, and the chord forces differ by a factor of three.

For a beam it is squared

A solid beam plays the same game continuously rather than at two chords, and the payoff is steeper.

Every strip counts by the square of its distanceA rectangular section divided into equal strips, with each strip's contribution to the second moment of area drawn beside it. The strips are identical in size; only their distance from the neutral axis differs.neutral axiscontribution of each striptotal I = 79.86 × 10⁶the outer strips do almost all of the work
Fig. 4 A rectangular section divided into strips, with each strip’s contribution to the second moment of area drawn beside it. The strips are identical; only their distance from the neutral axis differs.

Each strip of material contributes in proportion to the square of its distance from the neutral axis, so the second moment of area of a rectangle goes as the cube of its depth, and the section modulus — which is what resists moment — goes as the square.

Doubling the depth of a beam therefore quadruples its bending strength, at twice the material. Efficiency per unit of material doubles. Turning a joist on edge rather than laying it flat is the same trick applied by ninety degrees, and it is the reason floor joists are deep and narrow.

Moving the flanges apartThe second moment of area of an I-section against its depth, with the flange and web areas held constant. The growth is close to quadratic, because the parallel-axis term dominates everything the flanges contribute about their own centres.1001502002503000M10M20M30M40M50M60Moverall depth1.0×2.6×4.9×7.9×13.8×21.3×same steel, moved apart
Fig. 5 The second moment of area of an I-section against its depth, with the flange and web areas held constant. The growth is close to quadratic, and none of it is bought with extra steel.

The I-section is the logical endpoint. Since the outer material does nearly all the work, the middle can be reduced to whatever is needed to hold the flanges apart and carry the shear. An I-beam is a truss with a solid web, and the flanges are its chords.

The limits, and none of them are structural

If depth is nearly free, the obvious question is why every structure is not enormously deep. The answers are almost all about something other than statics.

Headroom. A floor structure competes with ceiling height. Every extra millimetre of beam depth is a millimetre of storey height, multiplied by the number of storeys, multiplied by the cost of façade.

Approach gradients. A deep bridge either sits high, needing long ramps, or sits low, fouling what passes beneath.

Wind. A deep structure catches more of it, and for a long-span roof or a tall building the lateral load can grow faster than the bending capacity gained.

Stability. A deep, thin member is more likely to buckle sideways before it fails in bending. The compression chord of a deep truss is a long column, and lateral-torsional buckling of a deep beam is a real limit on how far this argument can be pushed.

Transport and fabrication. A section deeper than about four metres cannot be moved by road.

The practical result is a set of span-to-depth ratios that have barely moved in a century: about 20 for a steel beam, 25 to 30 for a concrete slab, 10 to 15 for a roof truss, and around 8 for a deep bridge truss. Those numbers are not physics. They are the point at which the free structural gain stops being worth the non-structural cost.

What happens when depth is denied

A structure forced to be shallow pays in a way that is easy to trace.

The chord force goes up in inverse proportion, so the chord area goes up with it. That extra material weighs more, which increases the load, which increases the moment, which increases the chord force again. The loop converges for ordinary spans and does not for long ones — at some span, a structure of a given depth cannot carry its own weight, and the only escape is more depth.

That is the structural expression of the square-cube law: the load a beam must carry grows with its volume while its capacity grows with its section, so scaling everything up uniformly makes a structure weaker relative to its own weight. A big structure is not a small one enlarged, and the difference shows up first as depth.

Long-span roofs are the clearest illustration. Past about sixty metres, solid beams stop being possible at any depth and the structure has to become a truss, a cable or an arch — three different ways of getting the material further apart than a rolled section allows.

Where the gain runs out

Depth is nearly free and the freedom ends somewhere, and the place it ends is a stability limit rather than a strength one.

The column curveFailure load against slenderness, as a fraction of the squash load. A stocky column crushes; a slender one buckles at the Euler load; the crossover is where the two curves meet, and real columns fall below both near it.5010015020000.20.40.60.811.2slenderness (effective length ÷ radius of gyration)they cross at λ = 75squashingEuler bucklingreal columns, which are neither
Fig. 6 Failure load against slenderness. A deep truss has a long compression chord, and a long compression member is judged on this curve rather than on its material strength.

The compression chord of a deep truss is a column, and its capacity falls as the inverse square of its unrestrained length. Making the truss deeper reduces the force in that chord and does nothing about its length, so beyond some depth the extra bracing needed costs more than the chord material saved.

The same material, four waysFour cross-sections of identical area, so identical weight and cost, with the second moment of area computed from each profile's own geometry. Only the arrangement differs, and the stiffest is many times the flattest.the same, laid flatI = 0.06 × 10⁶1.0× the firstsquareI = 0.75 × 10⁶13.3× the firsttall rectangleI = 10.00 × 10⁶177.8× the firstI-sectionI = 24.29 × 10⁶431.8× the firstevery section here has an area of 3000 — only the shape differsthe bar is the second moment of area, to scale
Fig. 7 Four sections of equal area. The same argument at the scale of a cross-section — get the material away from the middle, and stop when the plates become too thin to stay flat.

The section-level version of the argument has exactly the same shape and exactly the same limit, which is a good indication that the principle is about geometry rather than about trusses.

Where the model stops

A constant moment. The reciprocal curve at the top assumes the bending moment is fixed while the depth varies. In reality a deeper truss weighs more, which raises the moment slightly, so the gain is a little less than the curve promises.

Chords carry all the bending. In a real truss the web members carry some, and in a beam the web carries a small share too. The couple model is an idealisation, though a close one for a parallel-chord truss.

Compression is not free. The reciprocal curve treats both chords alike, and only one of them can buckle. The compression chord’s capacity depends on its unrestrained length, so a deeper truss with the same bracing spacing does not gain quite what the arithmetic suggests.

Shear does not scale the same way. Depth reduces the chord forces and leaves the shear unchanged, so a very deep, very shallow-loaded truss ends up governed by its web members and its connections rather than by its chords.

The figures have a limitation worth stating. Each truss is drawn to a fixed canvas, so a deep truss and a shallow one appear at different scales rather than side by side at the same one. The chord forces are computed and comparable; the appearance of relative depth is not, and comparing the two drawings by eye will mislead about how much deeper the second one actually is.

The ladder from here

Later rungs: the section modulus and its square law. Span-to-depth ratios across materials and forms. Self-weight and the span at which a form runs out. The square-cube law in structures. Lateral-torsional buckling as the limit on depth. Web design in deep girders. Vierendeel frames, which give up diagonals to keep depth usable. Castellated and cellular beams, which buy depth from the same steel by cutting and re-welding it. And the long-span forms — cable, arch, shell — which are what happens when depth alone runs out.

The span-to-depth ratio of a Roman timber roof truss and of a modern steel one differ by less than a factor of two. Materials have improved by an order of magnitude; the geometry has not needed to change.

Where this goes next

tensioncompression2 carrying nothing The triangle that cannot fold, and everything built out of it A square of pinned bars is a mechanism. A triangle is not, and that single fact is the reason trusses exist and the reason they look the way they do. neutral axiscontribution of each striptotal I = 79.86 × 10⁶the outer strips do almost all of the work The material far from the middle does nearly all the work A strip of steel contributes to bending stiffness in proportion to the square of its distance from the centre. Move the same steel outward and the section gets stiffer for nothing. 11.522.533.54050100150200250300span, relative to the first1×5×16×81×256×moment: the squareload: the first powerdeflection: the fourth Span to the fourth, which is why spans are short Doubling a span multiplies its deflection by sixteen. No other relationship in ordinary structural work is that steep, and it is the reason long spans are always a different kind of structure.