The same span, four ways

Four structures cross the same gap under the same load. A beam bends. A truss turns the bending into a couple of axial forces. An arch turns it into a compression along a line. A cable turns it into a tension along the same line, upside down.

The usual account of the choice between them is a list of considerations, and the list is not wrong. What it hides is that the four differ in exponent, not in coefficient — so there is no span at which they are all within a factor of two, and the ordering changes as spans change rather than the gaps between them narrowing.

Which free body produced the number #

Cut each of the four at midspan and look at what appears on the cut.

A beam shows a bending moment resisted by a stress distribution across its depth, with a lever arm of a fraction of the depth. Capacity is $\sigma Z$, and $Z$ contains the depth squared and the breadth once.

A truss shows the same bending moment resisted by two axial forces at the full depth apart. Capacity is $\sigma A d$ with the whole area available at the extreme fibres, so a truss is a beam whose section modulus has been made as large as its depth permits.

An arch and a cable show no bending moment at all if their shape is right, because the internal force follows a line whose eccentricity from the axis is zero everywhere. Capacity is $\sigma A$ with no depth in it, and the depth reappears as the sag, which decides the magnitude of the force rather than its capacity.

Four cuts, four different quantities on the cut face. That is the whole taxonomy, and everything else in this essay follows from it — the exponents, the crossovers, and the fact that the four cannot be compared by any single number.

The exponent, made visible #

The clearest demonstration is not a span at all. It is a pressure vessel against a flat plate.

The membrane thickness is $t = pR/\sigma$ — linear in the pressure. The bending thickness is $t = \sqrt{3p R^2/\sigma}$ — a square root of it. Their ratio is $\sqrt{3\sigma/p}$, which for this case is 23.7 and which grows as the pressure falls.

That last clause is what makes it an exponent argument rather than a factor. The advantage of the curved form is not a constant 24 to be traded against buildability; it is unbounded as the load gets lighter, which is why a lightly loaded roof is a shell or a fabric and a heavily loaded floor is a slab. The surface that carries by being curved is the essay about the mechanism; this is what it is worth.

The shape that has to be found #

Both the arch and the cable only carry axially if their shape is the funicular of the load, and the funicular is a solved quantity rather than a chosen curve.

The catch is that a funicular is the funicular of one load case. Change the load and the shape is wrong, bending appears, and the form loses precisely the advantage it was chosen for. So an arch that has to carry a variable load is either given a depth to resist the bending — at which point it is a curved truss — or given hinges so the bending cannot develop.

That is why the line that must stay inside is the governing check for a masonry arch and why steel arches are usually stiffened. It is also the honest limit of the whole family: the axial forms are the efficient ones for the load they were shaped for and no other.

What the thrust costs #

An arch delivers a horizontal thrust at its springings and a cable delivers a horizontal pull at its anchorages, and both have to go somewhere.

The thrust is $wL^2/8r$ — inversely proportional to the rise — so a shallow arch is not merely less efficient, it is expensive at both ends. Halving the rise doubles the thrust and doubles whatever has to resist it, which may be a rock face, a tie across the span, or a very large foundation.

That is the arch’s characteristic trade: it converts a bending problem into an axial one and a foundation problem. Where the foundation cannot take a thrust, the thrust that never reaches the ground is the alternative, and the tie it needs is a member in tension the length of the span.

Stiffness, which is a different ranking #

Everything so far has been about strength, and the four forms rank differently on stiffness.

A cable has no stiffness at all until it has tension in it, which is the state a slack tie is in, and the tension it has is a design decision rather than a property. So a cable structure’s stiffness is bought with pretension and with sag, and neither is free — the pretension has to be resisted by the anchorages and the sag has to be accommodated by whatever the cable is holding up. The stiffness that comes from the shape is the essay about the mechanism.

A beam’s stiffness, by contrast, is a property of the section, and it goes as the fourth power of the span.

That fourth power is the reason a beam runs out of usefulness at a span long before it runs out of strength, and it is the single strongest argument for changing form as spans grow: the strength penalty for a beam is a square and the stiffness penalty is a fourth power, so a form that reduces the bending reduces the more expensive of the two.

Three ways to do one job #

The abstraction is easiest to check on a problem with only one answer required.

The truss is a ninth of the beam’s weight and four times as stiff, because it has the whole storey depth as a lever arm rather than a metre of concrete. The wall is three times the beam’s weight and thirty-five times as stiff — and 36% of its deflection is shear rather than bending, which is what a member as deep as it is long always does and is why beam theory does not describe one.

No consideration of taste separates those three. They are the three mechanisms of this essay applied to one job, and the numbers are as far apart as the mechanisms are.

The material question rides on top #

Having chosen a mechanism, the ranking of materials for it is decided by an exponent too, and it is not the same exponent for each.

For a tie, where the shape is fixed and only the area is free, the index is $E/\rho$. For a beam of free depth it is $E^{1/2}/\rho$; for a plate of free thickness, $E^{1/3}/\rho$. Three lines through the same point with three slopes, and three different winners — which is the ranking belonging to the load case rather than to the material.

So form and material cannot be chosen separately. Timber beats steel eight to one as a beam and loses to it as a tie, so a decision to use a truss rather than a beam is partly a decision about which materials are in the running.

The truss, which is a beam with the middle taken out #

The truss deserves a closer look than the exponent argument gives it, because it is the one form in the family whose advantage is a pure geometric ratio.

A beam of depth $d$ has a section modulus $Z = \alpha A d$ with $\alpha$ about 0.4 for a rolled shape and a sixth for a rectangle. A truss of the same depth has an effective $\alpha$ of a half, because all of the material is at the extreme fibres. That alone is a factor of between one and three — worth having and not decisive.

What makes the truss decisive is that its depth is not limited by the same things. A beam’s depth is limited by what can be rolled, cast or transported; a truss’s is limited by the storey height, which is an order of magnitude larger. So the truss’s advantage is not its better $\alpha$ but the depth its openness allows, and the reciprocal in the figure at the top of this essay is what turns that depth into a force.

It buys the depth with connections. A truss has two or three joints per panel, each of which is a place where the idealisation fails — the joints are not pins, the members are eccentric to their working points, and the whole thing is a set of compression members whose effective lengths are decided by out-of-plane bracing nobody has drawn yet. Its efficiency in a calculation and its efficiency in a fabrication shop are quite different numbers.

The Vierendeel is the same trade taken to its end: remove the diagonals for the sake of the opening they block, and the panel has to carry its shear in bending — which puts the mechanism back where it started and costs, on a typical geometry, a factor of several in material.

Where the crossovers actually are #

It is tempting to want a table of spans at which each form takes over. The exponents make it possible in principle and the answer is disappointingly wide, for a reason worth stating.

The crossover between a beam and a truss depends on the depth available, which is a client’s decision about floor-to-floor height. The crossover between a truss and an arch depends on whether the ground can take a thrust. The crossover between an arch and a cable depends on whether the load can reverse, because a cable cannot take compression and an arch under an uplift is an arch in the wrong direction.

Every one of those is a boundary condition rather than a structural quantity, which is why the same span is crossed by different forms in different places, and why the four have coexisted for two centuries rather than one of them winning.

What the exponents do settle is the direction: as spans grow, forms move from bending toward axial action, and never the other way. That trend is not a fashion. It is the observation that a bending capacity contains a depth that cannot grow with the span and an axial capacity does not.

Where the model stops #

Self weight has been left out. Every comparison above is at a fixed applied load, and at long spans the structure’s own weight is most of what it carries — which shifts the ranking further toward the axial forms and introduces a fixed point that none of the arithmetic here contains.

Fabrication is not in the exponents. A truss is a ninth of the weight of a beam and several times its cost per tonne, because it is made of many small pieces with connections at each. Below some span, the beam wins on money while losing on every quantity in this essay — and where that span falls is a property of a labour market rather than of a material.

Buckling has been assumed away. Every capacity quoted above is a stress times an area, and the compression members in three of the four forms are governed by something that falls as the inverse square of a length instead. An arch has the whole of its length in compression and is the form this bites hardest.

A form is not chosen once. A real structure is a hierarchy — a deck on beams on trusses on arches — and each level answers a different span with a different mechanism. The interesting choices are usually about where to put the boundaries between levels rather than about which single form to use.

What the picture cannot show #

Every figure here compares forms at one span, and the comparison a designer actually faces is a comparison of systems including their supports, their stability, their erection sequence and their tolerance to a load nobody predicted.

The last of those is the one the exponents are silent about and it decides many real projects. A beam is robust to almost anything: change the load, move it, add a hole, and it goes on being a beam. An arch shaped for a load case is not a structure once the load case changes, a cable cannot be pushed, and a shell is a membrane only while its edges are held.

Efficiency and tolerance run in opposite directions here, and the drawings show only the first.

The generalisation #

The habit worth carrying is to ask what exponent a design decision moves.

Choosing a bigger section changes a coefficient. Choosing a deeper truss changes a coefficient. Choosing to carry a load axially rather than in bending changes an exponent, and an exponent beats a coefficient at every scale far enough from where they were compared.

That is the sharpest form of a thread that runs through this whole collection: geometry beats material, and it beats it by more the further the problem is pushed. The reason is always the same — a material property enters a capacity linearly and a geometric one enters it squared, cubed, or as a square root of the load — and the reason it is so often ignored is that the material property is the one on the order and the geometry is the one on the drawing.