The member that is not worth stiffening

The deflection of a truss is a sum with one term per member:

$$\delta = \sum \frac{F f L}{EA}$$

with $F$ the force from the real load and $f$ the force from a unit load applied where the answer is wanted. A total conceals the terms, and the terms are where the design decisions are.

A term is a product, so either factor can kill it #

The most useful consequence of the sum’s form is that a member’s contribution depends on two forces, and a zero in either one makes the term vanish.

That member is fully loaded, entirely necessary and completely irrelevant to the deflection being asked about. Doubling its area changes 2,019.41 units of movement by nothing at all.

Two members can carry identical forces and be worth completely different amounts, and which is which is decided by the virtual force rather than by the real one. That is not a subtlety — it is the reason a deflection calculation and a strength calculation rank the members differently, and why a truss optimised for stiffness looks different from one optimised for strength.

Which free body produced the number #

Two free bodies, applied to the same structure, and the second is the one that carries the information.

The real system is the truss under its actual loads, and its member forces $F$ come from ordinary joint equilibrium.

The virtual system is the same truss under a single unit load applied at the point, and in the direction, the deflection is wanted — and nowhere else. Its member forces $f$ come from the same joint equilibrium on a different load case.

Virtual work then equates the external work of the unit load moving through the real deflection, $1 \times \delta$, with the internal work of the virtual member forces moving through the real member extensions, $\sum f \cdot (FL/EA)$.

The virtual system is a question rather than a load case. Its unit load is not something the structure ever carries; it is the way of asking “deflection where, and in which direction”. Move the question and every $f$ changes while every $F$ stays put — which is exactly why the same truss has a different ranking of members for every point on it.

There is a third kind of zero worth separating from the other two, because it is the one that catches people out.

A member with no force under the real load contributes nothing and is easy to spot: the members carrying nothing fall out of the joint equilibrium and everybody notices them.

A member with no force under the virtual load contributes nothing and is invisible, because it looks fully loaded on the drawing that matters.

And a member with both forces non-zero but of opposite signs contributes a negative term — it is being stretched by one system and compressed by the other, and it is reducing the deflection rather than adding to it. Enlarging such a member makes the truss deflect more, which is a genuinely counter-intuitive result and is real.

Negative terms are rare on a simple truss and common on a cantilever or a truss with an overhang, where the virtual load’s force pattern differs from the real one over part of the structure. Their existence is the sharpest possible argument for reading the sum term by term rather than trusting the total.

What the shares actually are #

The category totals in the first figure are worth reading, because they are not what the eye expects from looking at a truss.

The chords supply 80 per cent — 49.4 from the top and 30.8 from the bottom — on this truss at this depth. The web supplies 20.

That is the opposite of the impression a truss gives, which is of a lattice of busy diagonals with two long members along the outside. The reason is the lever arm: a chord force is $M/d$ and grows with the span, while a diagonal’s is $V/\sin\theta$ and does not; and a chord is long while a diagonal is short.

The two trusses differ by 3 per cent in deflection and by rather more in how it is made up, which is a fair summary of the difference between named truss types: they are much more alike than their drawings suggest.

How it scales, and why the exponent is not four #

A beam’s deflection goes as $wL^4/EI$. A truss’s does not, and the difference is instructive.

Two effects are mixed in that exponent and separating them explains it.

The load grows with the span. Each new panel adds a node and a 15 kN load, so the total load is proportional to the span rather than fixed. A beam under a fixed total load goes as $L^3$; under a fixed intensity it goes as $L^4$; this truss is the second case.

And the composition changes. The chords’ share rises from 55 per cent to 92, because chord forces grow with the span and web forces do not. A long truss is very nearly a beam made of two chords, and its web contributes almost nothing to its movement.

The result is an exponent between three and four, closer to four, and not a constant — it drifts as the composition drifts, so quoting a power law for a truss is quoting a local slope.

Depth is worth less than the inverse square a chord-force argument alone would suggest, and the reason is on the second axis: as the truss deepens the chords get cheaper and the web gets longer, so the web’s share rises and the benefit saturates.

Past a depth of about a fifth of the span the web is supplying half the movement, and further depth is buying less than it looks. That is a serviceability argument for a proportion, and it lands close to the span-to-depth ratios trusses are actually built at.

The same sum, read as energy #

There is a second reading of the same terms that makes the ranking feel less arbitrary.

Each term $FfL/EA$ is, when $f$ happens to equal $F$, twice the strain energy stored in that member. So a truss’s deflection under its own load is a derivative of its strain energy, and the per-member shares are shares of that energy.

A member’s share of the deflection is its share of the derivative of the energy, which is why the two figures rank members the same way and why neither ranks them by force. That connection is Castigliano’s theorem, and it is the reason the unit-load method and the energy method are not two techniques but one.

What to do with the ranking #

The per-member sum is a design tool rather than a curiosity, and it is used in three ways.

To choose where to add material. The worst member on the Pratt above supplies 11.9 per cent of the movement. Doubling its area removes half of that — 5.9 per cent of the deflection — for a small amount of steel in one place. Doubling the area of the whole truss removes 50 per cent for a great deal.

To know what not to bother with. Seven verticals supply 3.0 per cent between them. Nothing done to any of them will register.

And to find the member that governs neither check. A member with a large $F$ and a small $f$ is sized by strength and is irrelevant to stiffness; one with a small $F$ and a large $f$ is the reverse. The two lists are different and both are needed, which is the practical content of the whole page.

The same reading works for a camber. A truss cambered by shortening its bottom chord is being cambered by the members with the largest $f$, and the amount is $\sum f \times$ (the shortening imposed) — the same sum with the extensions chosen rather than computed.

Why the composition matters more than the total #

A designer given a truss that deflects too much has four moves, and the shares say which is worth making.

Deepen it. Worth the 1.56 power, and worth less the deeper the truss already is, because the web’s share is rising.

Lengthen the chords’ area. Worth up to 80 per cent of the deflection on a shallow truss and under 50 on a deep one. The chords are also the members that are cheap to enlarge, since they are continuous and prismatic.

Enlarge the web. Worth 20 per cent on a shallow truss and over 50 on a deep one, and it is spread over sixteen or twenty small members rather than concentrated in four large ones — so it costs more in fabrication for the same tonnage.

Or camber it. Worth all of it, costs nothing structurally, and does not change the deflection under live load at all — which is the distinction that decides whether camber is an answer.

The first three swap places as the proportions change, which is why a rule of thumb about trusses is a rule about a proportion rather than about trusses. On a shallow long-span truss the chords are the whole answer; on a deep short-span one they are half of it.

An example worth doing, in numbers #

Taking the eight-panel Pratt and asking what a fixed budget of extra steel buys makes the point better than the shares do.

Suppose 10 per cent more steel is available. Three ways of spending it:

Spread evenly over every member. Every area rises by 10 per cent, every term falls by the same fraction, and the deflection falls by 9.1 per cent — from 2,019 to 1,835.

All of it in the six top chords, which are 49.4 per cent of the movement. Six members out of twenty-nine, so their areas rise by roughly half again, and their terms fall by a third: the deflection falls by about 16 per cent.

All of it in the seven verticals, which are 3.0 per cent of the movement. Their terms fall by a third and the deflection falls by 1 per cent — a tenth of what the even spread bought, for the same steel.

A factor of sixteen between the best and the worst use of the same material, decided by a ranking that costs one extra analysis to obtain. That is the whole argument for computing the terms rather than the total, and it is why a stiffness-governed truss is a genuinely different optimisation problem from a strength-governed one.

The corollary is worth stating because it is what a designer actually does. A truss governed by deflection is designed at its chords and detailed at its web; one governed by strength is designed at whichever member has the largest force over its own capacity, which is usually an end diagonal. Two different members, two different drawings, one truss.

The two-way reading #

There is a symmetry in the sum that is worth noticing because it connects this page to the last one.

The term $Ff L/EA$ is symmetric in $F$ and $f$. Swap them — apply the “real” load as a unit load and vice versa — and every term is unchanged. That is Maxwell’s theorem appearing inside the summation rather than as a separate statement, and it is why the influence line for a truss’s deflection is itself a deflected shape.

It also gives a check nobody uses and everybody could: compute the deflection at A from a load at B and at B from a load at A. The two sums share no terms in common — a different set of real forces $F$, a different set of virtual forces $f$ — and they must agree. On this truss they do to a part in 10¹⁴.

What this is a special case of #

The per-member sum is the discrete version of something continuous, and seeing the connection makes both easier to use.

For a beam the equivalent is $\int Mm/EI \, dx$, and its integrand is a density along the member — a map of where the deflection came from — rather than a list of members. The two are the same statement, with a sum over discrete elements in one case and an integral over a continuum in the other.

That correspondence carries the design reading across. On a beam, the density says which part of the span is worth stiffening; on a truss, the share says which member is. In both cases the answer is not where the internal force is largest, because the product of two forces is what matters and only one of them is the one a designer has been looking at.

It also carries the trap across. A beam’s density is largest near mid-span for a mid-span deflection, and a truss’s shares are largest in the chords near mid-span for a mid-span deflection — and in both cases moving the question to a quarter point moves the map completely. There is no single diagram of “where a structure is flexible”; there is one per question asked, and the question is the unit load’s position.

Where the model stops #

Only axial deformation is counted. The members are treated as pin-ended bars, so the sum omits the bending in the chords that continuous chords actually carry and the secondary bending from rigid joints. Both are small for deflection and neither is zero.

The joints are perfect. A bolted truss with clearance holes takes up its slack before it deflects elastically, and on a lightly loaded truss that take-up can exceed the calculated deflection.

Every member is prismatic and elastic. A member that has yielded, buckled or gone slack has left the sum, and a truss with tension-only members has a different sum in each load case.

The geometry is the undeformed one. For an ordinary truss that is excellent; for a very slender one carrying large chord forces the second-order effect is real.

And the sum answers one question at a time. Each $f$ belongs to one point and one direction. A truss’s whole deflected shape needs one virtual system per point, which is where a stiffness solution stops being the slower method.

What to carry away #

Compute the terms, not the total. A truss’s deflection is a sum of one term per member, and the ranking of the terms is not the ranking of the forces.

A term is a product and either factor can kill it. A fully loaded member can contribute nothing; a lightly loaded one can contribute a great deal; and a member with opposite signs in the two systems contributes a negative amount.

The composition moves with the proportions. Chords supply 80 per cent of a shallow truss’s movement and under half of a deep one’s, and 55 per cent of a short span’s against 92 of a long one’s.

And the map belongs to the question. One virtual load, one point, one direction, one ranking — and moving any of them produces a different list.

The ranking this produces is the useful output, and it is the same ranking which member moved the roof computes from the other direction: a member’s contribution to a deflection, rather than its force. Both come out of the deflection read as a derivative of the strain energy, and both are instances of the distinction stiffness is not strength is built on — that the member which is working hardest and the member which is moving the structure are usually not the same member.

The ladder from here #

Later rungs on this anchor: camber computed as an imposed set of member extensions rather than as a deflection. Lack of fit and the self-stress a misfitting member produces, which is the same sum with no external load in it. Deflection of a truss under a temperature change, where the extensions are imposed and the forces are the unknowns. The Williot-Mohr diagram, which is this calculation done graphically and gives the whole deflected shape at once. Shear deformation in a deep truss, where the panel is short enough that the web’s contribution is the whole answer. And the optimisation this page is one step from: the fully stressed design against the minimum-deflection design, which choose different members and are both defensible.

The unit-load method for trusses is Maxwell’s, from 1864, and it is one of the very few analytical results whose presentation has been improved by computers rather than made obsolete by them. The sum was always available; drawing every member at the width of its own share was not, and a table of two hundred terms is not the same object as a picture of them.