Depth is the cheapest strength there is

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.

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.

$$M = F \times d \quad\Longrightarrow\quad F = \frac{M}{d}.$$

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.

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 exactly the factor of three the depths do. The web members do not: one of them did not change at all and the other changed by half.

For a beam it is squared #

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

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.

Hold the flange and web areas of an I-section constant and let only its depth grow, and its second moment grows close to quadratically with no extra steel bought at all — which is the same reciprocal argument the truss makes, run continuously instead of at two chords.

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 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.

Push the same truss to five times its original depth and something more specific happens, which the reciprocal on its own conceals.

That is where the free gain stops, and it stops for a reason that is about which member governs rather than about how much steel is in the chords. The section-level version of the argument has exactly the same shape and exactly the same limit — get the material away from the middle, and stop when the plates become too thin to stay flat — which is a good indication that the principle is about geometry rather than about trusses.

Which free body gives $M/d$ #

The expression $F = M/d$ is quoted so often that it can start to look like a definition. It is not: it is the result of one cut and one moment equation, and the cut is worth naming.

Take a parallel-chord truss and cut it clean through a panel, severing exactly three members — the top chord, the bottom chord and the diagonal between them. Discard everything to the right and keep the left-hand piece, which now has on it the reaction at the support, whatever loads have been applied between the support and the cut, and three unknown axial forces on the cut faces.

Now take moments about the joint where the bottom chord meets the diagonal. Two of the three unknowns pass through that point and contribute nothing, which leaves one equation in one unknown:

$$F_{\text{top}} \times d = \Sigma(\text{moments of the loads and the reaction about that joint}).$$

The right-hand side is the bending moment at the cut, computed on a free body that never mentioned the truss. So the top chord force is that moment divided by the depth, and the depth appearing in the answer is the perpendicular distance from the chosen joint to the line of the chord — which for a parallel-chord truss is the truss depth and for a curved-chord truss is not.

Two things follow that the formula on its own hides. The first is that the answer required no knowledge of the diagonals at all — the method of sections answers one question without solving the structure, which for a fifty-member truss where only the worst chord matters is an enormous saving. The second is that the moment on the right-hand side is a property of the span and the loading only. Nothing a designer does to the truss changes it, which is precisely why the depth in the denominator is the only handle there is.

What the web pays for it #

Depth is free for the chords. It is not free for everything, and the place it is paid for is the web.

A diagonal in a panel of width $p$ and depth $d$ has a length $\sqrt{p^2+d^2}$, so deepening the truss lengthens every diagonal. Worse, each of those diagonals is carrying roughly the panel shear divided by the sine of its inclination, and while a steeper diagonal carries a smaller force, it does so over a greater length — and a longer compression diagonal is judged on slenderness rather than on strength, so the section cannot be reduced in proportion.

The result is a total-weight curve with a genuine minimum. Chord weight falls as $1/d$; web weight rises roughly linearly with $d$; the sum has a bottom. For a uniformly loaded parallel-chord truss the bottom sits somewhere around a span-to-depth ratio of six to ten, which is deeper than almost anything built.

The same web arrangement, at nearly twice that depth, separates the two behaviours as cleanly as anything on this page.

Which member of the web pays it is a decision rather than a fact, and the decision is made by pointing the diagonals.

A Pratt truss puts the long members in tension and the short ones in compression, and a Howe does the reverse, which is why the Pratt is the steel arrangement and the Howe the timber one — a long compression member is judged on its slenderness and a long tie is not. The couple is the same job in both, so the entire difference is a choice about which web members take the compression.

The useful part of that result is not where the minimum is but how flat it is. A sum of a reciprocal and a linear term is very insensitive near its bottom: moving twenty or thirty per cent away from the optimum depth typically costs a few per cent of weight. Which means the structural optimum is worth almost nothing and the non-structural constraints are worth a great deal, and the designer who fights the architect for another two hundred millimetres of depth is arguing over a rounding error while the cladding it costs is not one.

That flatness is the quiet reason the span-to-depth ratios listed above are so stable across a century. They are not sitting at a structural optimum. They are sitting where a very forgiving structural curve meets a very unforgiving set of everything else.

The ratio is a deflection, and it says the beam is at half strength #

The span-to-depth ratios were explained above by a flat weight optimum meeting a set of non-structural costs. That is true and it is not what actually fixes them. The binding constraint is stiffness, and the arithmetic says so in one line.

Take a simply supported beam at its full bending stress. Then $I = wL^2d/16\sigma$, and substituting into $\delta = 5wL^4/384EI$ makes the load and the section cancel:

$$\frac{\delta}{L} = \frac{5}{24}\cdot\frac{\sigma}{E}\cdot\frac{L}{d}$$

The span-to-depth ratio and the deflection ratio are the same statement. No load, no second moment, no breadth — only a material’s working strain and a shape.

Put steel in it. At $\sigma = 275$ and $E = 210{,}000$, a beam at $L/d = 20$ deflects

$$\frac{5}{24} \times 1.31\times10^{-3} \times 20 = \frac{1}{183}$$

which fails every deflection limit anybody writes. To reach $L/360$ at that depth the stress has to come down to 140 N/mm² — so a steel beam at the conventional $L/20$ is working at 51 per cent of its yield stress, and half of it is there for the deflection check.

That is the honest reading of the ratios in the list above. They are not a weight optimum and they are not an architectural compromise; they are a stiffness limit divided by a material’s working strain, and the material’s strength enters only by deciding how much of the section is wasted.

Which explains the essay’s own closing observation better than the essay does. $\sigma/E$ is the quantity, and it barely moves between structural materials: 1.31 × 10⁻³ for steel at yield, 1.0 for timber at its working stress, 0.86 for aluminium. Materials have improved by an order of magnitude in strength and by nothing at all in working strain, so the span-to-depth ratio a Roman carpenter arrived at is the ratio the arithmetic still gives. Strength is what changed; stiffness is what the ratio was always about.

The depth that is already in the building #

There is a further move available, which is to stop paying for depth and start finding it.

Composite action. A steel beam bolted to the concrete slab it supports, with shear studs to make the two act as one, has a neutral axis up in the slab and an effective depth far larger than the beam alone. The slab was going to be there regardless; the studs cost a few per cent; the stiffness gain is routinely fifty per cent or more. The whole of it comes from making a lever arm out of two things that were already in the building.

Service integration. Much of a floor’s depth is not structure but the zone below it holding ducts, pipes and cable trays. A cellular beam with holes cut through its web lets the services pass through the structural depth rather than beneath it, so the floor-to-floor dimension buys both at once. A castellated beam goes further: the web is cut on a zig-zag, the two halves offset and welded, and a beam roughly half again as deep emerges from the same rolled section with no extra steel at all.

Vierendeel panels. Where a diagonal would block an opening, it can be removed and the joints made rigid instead. That is a genuinely expensive substitution — bending in the chords replaces axial force in a diagonal, and bending is the inefficient way to carry anything — but it buys usable depth in a wall that has to have a door in it.

Each of these is the same observation from a different angle: depth is cheap, so anything that produces it as a by-product is worth more than it looks, and anything that consumes it without carrying load is worth less.

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.

Where the ratio was found #

The argument was arrived at experimentally before it was arrived at algebraically. Robert Stephenson’s Britannia Bridge of 1850 had to carry a railway two spans of 140 metres each across the Menai Strait, and no formula then available could say what section would do it. William Fairbairn and Eaton Hodgkinson built and tested a long series of scale tubes instead — round, elliptical and finally rectangular — loading each to destruction and recording where it gave way.

What the tests showed was that depth was worth more than anything else available, and that the failures were not tensile ruptures but buckles in the compression top. The bridge that resulted was a wrought-iron box deep enough for the trains to run inside it, with a cellular top flange stiffened against exactly the failure the tests had produced. The design is a direct transcription of a set of experiments, and it contains both halves of this essay: take all the depth available, then discover that the limit on doing so is stability rather than strength.

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.