Where a deflection comes from

The unit-load method says that to find how far one point of a structure moves, put an imaginary unit force there, multiply two moment diagrams together, and integrate:

$$\delta = \int \frac{M\,m}{EI}\,dx$$

Every use of that expression in this collection has treated it as a machine for producing a number. It is also a machine for producing a map, and nobody looks at the map.

The integrand is a density. It says how much of the answer each millimetre of the member contributed, and the distribution is nothing like uniform.

Which free body produced the number #

The cantilever case has a closed form and no arithmetic in it worth hiding.

A tip load $P$ gives $M(x) = -P(L-x)$, measuring $x$ from the root. The unit load at the tip gives $m(x) = -(L-x)$. Their product is $P(L-x)^2$, so the density goes as the square of the distance from the tip, and the share supplied by the outer portion of length $a$ is $(a/L)^3$.

The half nearest the tip therefore supplies $(1/2)^3 = 1/8$, and the half nearest the root supplies 7/8. No length, no load, no material, no section: seven eighths, for every tip-loaded cantilever there has ever been.

Under a uniform load the real moment goes as $(L-x)^2$ while the virtual one is still linear, so the density goes as the cube and the root half supplies $1 - (1/2)^4 = 15/16$.

The simply supported case, which is the one everybody has #

For a mid-span deflection under a uniform load the two diagrams are different shapes — a parabola and a triangle — and the density is their product, $wx^2(L-x)/4$ on the left half.

Integrating it returns $5wL^4/384EI$, which is the answer everybody knows, and integrating it piecewise returns something nobody quotes: the middle half of the span supplies 67/80 of the deflection, or 83.75 per cent, and the two outer quarters supply the remaining sixth between them.

Which is the whole argument for a haunch, and against one #

A haunch is material added where the moment is largest, and it is usually justified on strength: the section is deepened where it has to carry the most.

The map says something stronger and more specific. Deepening the busy region buys stiffness at five times the rate that deepening the quiet region does — so a haunch is not a marginally better place to put material, it is very nearly the only place worth putting it.

And the same map says where not to bother. The outer quarters of a simply supported span contribute a sixth of the deflection between them; a designer worrying about the depth available at the bearing is worrying about a region that could be made half as stiff for an eight per cent penalty.

This is where the argument meets the tapered member from the other side. That essay asks where a tapered beam is checked — a strength question, whose answer is the station where the demand-to-capacity ratio peaks. This one asks where a tapered beam is stiff, and the two stations are not the same. For a tip-loaded cantilever the governing section is where the depth has doubled, and the stiffness is coming from the root.

Two load cases on one cantilever, and the sharper map pays better. Against the 5.2 to one the simply supported span offers, a uniformly loaded cantilever returns fifteen to one on the same decision — so the case for deepening a member where it meets its support is stronger than the case for deepening it at mid-span, and it gets stronger as the load spreads out along the member.

The map depends on which deflection is asked for #

The density contains the virtual moment diagram, and the virtual diagram depends entirely on where the deflection is being asked about.

Ask for the mid-span deflection of a uniformly loaded span and the virtual diagram is a triangle peaking at mid-span, so the map is concentrated there. Ask for the deflection at the quarter point and the virtual diagram peaks at the quarter point, and the map moves — the region supplying the answer shifts towards the end of the beam even though the real moment diagram has not changed at all.

Half of the density belongs to the question rather than to the structure. That is easy to say and easy to forget, and it is why “where the beam is working hardest” is not a well-formed phrase: the beam is working hardest at mid-span for every question about strength and for only some questions about movement.

The clean statement of it is reciprocity: the influence of a load at $A$ on the deflection at $B$ equals the influence of a load at $B$ on the deflection at $A$. The density is that theorem written out as a distribution rather than as a pair of numbers.

In an indeterminate structure part of the map is negative #

Everything so far has had both diagrams of the same sign over the whole member, so every millimetre contributed positively and the running total climbed. That is a property of determinate structures with a single load, not of structures in general.

Take a propped cantilever under a uniform load. Its real moment diagram is hogging over the prop and sagging in the span, so it changes sign; the virtual diagram for a deflection somewhere in the span changes sign too, at a different place. Their product is therefore negative over part of the member, and that part of the beam is reducing the deflection being asked about.

Two consequences follow, and the second is the useful one.

The running total is no longer monotonic, so “the middle half supplies eighty per cent” stops being a well-formed sentence — there are regions supplying more than a hundred per cent and regions supplying less than nothing.

And stiffening a region with a negative density makes the structure deflect more. That is not a paradox: stiffening it attracts moment to it, which is a redistribution, and the redistribution moves the answer the other way. It is the clean statement of something every designer of continuous structures learns by accident — that adding material somewhere can be counterproductive — and the density says exactly where the boundary is.

The shear map is the other half of the beam #

Shear deflection has a density of its own, $V\,v/GA_v$, and it is the product of two shear diagrams rather than two moment diagrams.

For a uniformly loaded simple span the real shear is largest at the supports and zero at mid-span; the virtual shear for a mid-span deflection is a constant $\pm\tfrac12$. So the shear density is largest at the supports and smallest in the middle — exactly the opposite of the bending map.

On a rolled steel beam that hardly matters, because the shear term is 1.4 per cent of the total. On a sandwich panel it is 9 per cent at ordinary proportions and 60 at short ones, on a timber joist it is several times a steel beam’s, and on a deep beam it is most of the answer.

For those members the two maps have to be read together, and they point in opposite directions: the bending stiffness is wanted in the middle and the shear stiffness at the ends. A haunched sandwich panel would be a strange object; a panel with a stiffer core at the supports is exactly what a well-detailed one has.

And a truss has the same map, one dimension down #

A truss’s deflection is a sum with one term per member rather than an integral, and the terms are $F f L/EA$ — the same product of a real quantity and a virtual one, weighted by a flexibility.

The map is therefore a ranking of members rather than a distribution along a length, and it says the same kind of thing: a handful of members supply most of the deflection, and they are not the members carrying the largest forces. A member with a large force and a small virtual force contributes nothing, and the classic case is a member carrying a full panel load in a truss where the deflection being asked about puts no virtual force through it at all.

The continuum and the discrete versions are the same theorem, and reading them together is what makes the density worth naming: the question “which part of this structure produced the movement” always has an answer, it is always available from a calculation already being done, and it is almost never plotted.

The map is also a sensitivity, which is what makes it a tool #

There is a second reading of the density that is more useful than the first, and it follows from reciprocity.

The contribution of a length $dx$ to the deflection is $M m\,dx/EI$, so the sensitivity of the deflection to a change in $EI$ there is that contribution divided by $EI$ — the density again. A designer asking how much the answer moves when the stiffness at a station changes is asking for exactly the curve that has been plotted.

That converts a plot into a procedure. Compute the deflection once, plot the density, and read off where a change is worth making — no second analysis, no trial section, and no iteration. It is the same trick strain energy differentiated with respect to a load plays for a different variable, and it is available from the first analysis rather than from a sweep.

And it is the honest reply to a common request. Asked to make a beam stiffer, the reflex is to make the whole beam deeper. The density says which part of it is doing the work, and on an ordinary simply supported span the answer — five to one in favour of the middle half — is large enough to change the detail rather than merely to reassure.

And it says where a splice may go #

The map has a use nobody teaches, because it answers a question that is usually settled by convention: where to put a joint.

A splice is a discontinuity in stiffness — a bolted cover plate is stiffer than the section, a site weld with a backing bar is not quite, and a bolted end plate under-develops the flange until it is fully tightened. Wherever it goes it perturbs $EI$ locally, and the density says what that perturbation costs: nothing at all in the outer quarters of a simply supported span, and several per cent of the deflection at mid-span.

The received rule is to splice near a point of contraflexure, which is a strength rule about putting the joint where the moment is small. The density says the same thing for a different reason and does not always agree: for a deflection asked at mid-span of a continuous beam, the contraflexure point is where the real moment vanishes and the product need not, because the virtual diagram is not zero there. The two rules coincide often enough that nobody has noticed they are different rules.

Finding the station itself is ordinary work — moment distribution, or any other continuous-beam calculation, puts the zero of the real moment diagram wherever the support moment and the span moment cancel. What none of those calculations produces is the zero of the product, and the two are in different places for the same reason everything else in this essay is: one diagram belongs to the load and the other to the question.

Where to put the hole #

The map answers a question that arrives on every project and is almost never asked of the structural engineer at all: where along the beam should the services pass through it?

A web opening removes stiffness over a short length. What it costs is the density at that station times the length of the opening — so the same hole is worth quite different amounts depending only on where it is cut.

For a uniformly loaded simple span the density goes as $x^2(L-x)$ on the left half, so relative to a hole at mid-span:

A hole at mid-span costs 2.7 times one at the quarter point and 14 times one a tenth of the span in. Nothing about the hole changed; only where somebody drew it.

The same arithmetic runs for a row of openings, which is the usual case. A cellular beam has holes at a constant pitch along its whole length, so what it loses is the integral of the density over all of them — which is why a cellular beam’s deflection is computed with a reduced effective stiffness rather than hole by hole, and why the reduction is nothing like proportional to the fraction of web removed.

That is a large enough factor to be worth a conversation, and the conversation almost never happens, because a services route is set out on a coordination drawing to suit a duct run and the structural engineer is asked afterwards whether the hole is acceptable. The honest answer is that acceptability is not a property of the hole.

The two maps then pull against each other, which is what makes the question interesting rather than merely a preference. The shear density is the reverse of the bending one — largest at the supports, zero at mid-span — so a hole near the support is cheap for deflection and expensive for shear, and a hole at mid-span is the other way round. There is no position that is good for both, and the least-bad one depends on which of the two the member is nearer its limit in.

For an ordinary rolled beam, governed by deflection and with shear to spare, the map says put the holes in the outer quarters and accept a few per cent. For a deep short member, or a plate girder with a slender web, it says the opposite. And for a member where both matter — a sandwich panel or a timber joist — the answer is that a large opening should not be there at all, which is a conclusion worth reaching before the ductwork is ordered rather than after.

Where the model stops #

The density assumes linear elastic behaviour and superposition. A member that has yielded, cracked or slipped has a curvature that is not proportional to its moment, and the integrand is then $M\kappa$ rather than $M m/EI$ — still a density, no longer a product of two diagrams, and no longer symmetric under reciprocity.

The comparison between stiffening the middle and stiffening the ends holds $EI$ constant elsewhere. A real haunch changes the moment diagram too, in an indeterminate structure, by attracting moment towards the stiffened region — so the gain is smaller than the map predicts and the redistribution is another calculation.

Nothing here is about axial deformation. A frame’s sway comes partly from its columns shortening, and that density is a third map with its own shape.

And the map is drawn for one load case. A structure is designed for several, and the region that supplies the deflection under a uniform load is not the region that supplies it under the pattern loads that govern a continuous beam.

What the pictures cannot show #

The density is normalised to its own peak in every figure here, so nothing on these axes says how large the deflection is. Two beams with identical maps can differ by a factor of a hundred in how far they move, and the map has no opinion about it.

Nor can it show the practical constraint that decides most real profiles, which is that a beam has to be made. A haunch is a fabrication operation, a tapered member is a cutting and welding sequence, and a variable-depth precast unit needs a variable mould. The map says where the material is worth putting; the price list says whether the saving covers the cost of putting it there, and for spans under about fifteen metres it usually does not.

The assumption the figure rests on #

The whole essay assumes the deflection worth knowing is the one at a single named point.

Serviceability is not usually written that way, and where it is, the point is chosen by convention rather than by consequence. What a cladding panel cares about is the differential movement between its two fixings; what a partition cares about is the change in deflection after it was installed; what a crane rail cares about is the slope. Each of those is a different virtual load — a pair of opposed unit forces, a unit force applied only to the loads that arrive later, a unit moment — and each therefore has a different density.

Which means the map is not a property of the beam at all. It belongs to the pair of a structure and a question, and changing the question moves it. That is a discipline rather than a difficulty: it says that “stiffen the middle” is an answer to a specific question, and that the specific question is the one worth writing down first.

There is an older reading of the same integral worth naming beside it. The area-moment theorems say that the area of a curvature diagram is a rotation and that the first moment of that area is a deflection — which is this essay’s density with the weighting supplied by geometry rather than by a virtual load. The two are the same statement made twice: a deflection is an accumulation along a member rather than a property of a section, and what it is accumulated against is chosen by whoever asked the question.