The theorem that swaps the question round

Load a beam at one point and measure how far a second point moves. Then move the load to the second point and measure the first. The two numbers are the same.

Not approximately, and not because the beam is symmetric — the two deflected shapes in the figure below have nothing in common. The equality holds for any elastic structure of any shape, and it is one of the few results in the subject that is genuinely surprising the first time it is met and completely unremarkable once its proof is seen.

Why it has to be true #

The proof takes two paragraphs and turns on the fact that the final state of an elastic structure does not remember how it got there.

Apply load $P_A$ at point A, then load $P_B$ at point B. The work done is: $P_A$ moving through its own deflection as it is applied, then $P_B$ moving through its own, and — the extra term — $P_A$ moving through whatever additional deflection $P_B$ causes at A, at full value, since $P_A$ is already fully applied by then.

Now apply them in the other order. The same three kinds of term appear, with the cross term being $P_B$ moving through the deflection $P_A$ causes at B.

The stored energy at the end is the same either way, because an elastic structure’s strain energy is a function of its final state and not of the loading history. So the two cross terms must be equal:

$$P_A\,\delta_{AB} = P_B\,\delta_{BA}.$$

Set both loads to one and the theorem falls out: $\delta_{AB} = \delta_{BA}$. The deflection at A due to a unit load at B equals the deflection at B due to a unit load at A.

The same argument in the language of the unit-load method is even shorter. Both quantities are the integral $\int M_A M_B / EI\,dx$ — one calling the first diagram real and the second virtual, the other calling them the other way round — and multiplication is commutative. The theorem is the symmetry of a product.

What counts as a displacement #

The theorem generalises further than it first appears, because “load” and “displacement” can be any pair whose product is work.

A force pairs with a displacement. A couple pairs with a rotation. So the rotation at B caused by a unit force at A equals the deflection at A caused by a unit couple at B — a statement relating two quantities with different units and different dimensions, which is nevertheless exactly true.

That version is the useful one in practice. It says that a quantity which is awkward to measure can be swapped for one that is easy, provided the pairing is respected. Measuring the rotation of a bridge deck under a vehicle is difficult; measuring the deflection under an applied couple is not much easier; but the swap gives a designer a choice about which experiment to do, and in a computer model it gives a choice about which analysis to run.

The formal statement of all this is that the flexibility matrix is symmetric. Collect the deflections at every point due to unit loads at every point into a matrix, and $f_{ij} = f_{ji}$. Its inverse, the stiffness matrix, is symmetric for the same reason — which is why every structural analysis program stores only half of it, and why a stiffness matrix that comes out unsymmetric is a sign of a modelling error rather than of an unusual structure.

Influence lines, obtained by pushing #

The most valuable consequence is a method for finding the worst position of a moving load, and it converts a hard question into an easy one.

An influence line plots one quantity at one station against the position of a unit load. Constructing it directly means re-solving the structure for every position of the load — which is what the generator behind those figures does, several hundred times.

Müller-Breslau’s principle says the shape of that influence line is the deflected shape the structure takes when the restraint corresponding to the quantity is released and a unit displacement imposed in its place.

For the reaction at a support: lift the support by one unit and draw the shape. That shape is the influence line for the reaction.

For the bending moment at a station: cut the beam there, insert a hinge, and rotate the two faces apart by one unit. The resulting shape is the influence line for the moment.

For the shear: cut and slide one face past the other by one unit.

The reason this works is reciprocity. The influence line’s ordinate at $x$ is a quantity at the station due to a load at $x$; the principle swaps it for a displacement at $x$ due to an action at the station. The theorem says the swap is free.

The practical value was enormous before computers and is still real. A physical model of a bridge, pushed at one point, displays the influence line for the whole span directly — and celluloid models were used exactly this way for decades. In a computer model the same trick answers in one analysis what would otherwise take hundreds.

Not the same thing as the diagram #

The commonest confusion in this area is worth heading off, because the two objects look alike and are drawn alike and mean opposite things.

A moment diagram has position along the beam on its horizontal axis, and its ordinate is the moment at that position, for one fixed set of loads.

An influence line has the position of a moving load on its horizontal axis, and its ordinate is a quantity at one fixed station, for a load standing at that position.

Two curves that both run the length of the beam, both peak somewhere, and answer questions with almost nothing in common. A moment diagram says which part of the beam is worst loaded; an influence line says where to stand to make one particular part worst.

The reaction influence line above is the easiest case to keep straight, because its answer is known in advance: a unit load standing directly over a support delivers all of itself to that support and nothing to the other, so the line must run from one to zero. Any influence line that fails an equivalent sanity check at its ends has been drawn as a diagram by mistake.

There is a further distinction that matters for design. A moment diagram is superposable — two load cases add — while the envelope of influence-line results is not, because the worst arrangement for one quantity is not the worst for another. A structure is checked against an envelope assembled from many separate worst cases, and no single load arrangement produces the whole of it.

Measuring it by pushing #

Reciprocity is not only a computational convenience. It licenses an experiment, and the experiment has been done at every scale from a laboratory model to a completed bridge.

The trick is that a quantity awkward to load for can be swapped for one easy to displace. Rather than run a vehicle across a bridge model and record the moment at a section for every position, insert a hinge at that section, rotate its faces apart by a measured amount, and photograph the resulting shape. The photograph is the influence line for the whole span.

Celluloid and perspex models were used exactly this way from the 1920s through the 1960s, with the deflected shape traced directly off the model. Beggs deformeters — small mechanical devices that impose a controlled displacement or rotation at a point — were made commercially for it, and a large indeterminate bridge could be analysed with a set of them and a good camera.

The modern descendant is dynamic rather than static and rests on the same symmetry. In modal testing, a structure is struck at one point and the response measured at another; reciprocity says the transfer function between the two is unchanged if the roles are swapped, which is what allows a whole structure’s dynamic behaviour to be mapped by hammering a single accessible point and moving the sensors around. Structural health monitoring uses it in the same way, and so does acoustics, and the reason is the one at the end of the previous section — the theorem belongs to conservative systems rather than to structures.

The residue for anyone reading a load test report is worth carrying: a measured influence line is a real measurement of a real structure, including all the stiffnesses nobody could estimate — partial fixity at the bearings, participation from parapets and surfacing, the true condition of the deck. It is one of the very few structural quantities that can be measured directly rather than computed, which is why bridge assessment leans on it so heavily — and why a measured stiffness beats an assumed one wherever one can be got.

What the shape says about where to stand #

Once an influence line exists, reading it is straightforward and the answers are frequently counter-intuitive.

A load does most damage where the ordinate is largest. So for the moment at a quarter point of a simple span, the worst place for a single load is at that quarter point, and the peak ordinate is $ab/L$ — for the span of ten with its station at three drawn below, exactly $2.1$.

A spread load should stand wherever the ordinate has the sign being maximised, and nowhere else. That is why the shaded region in the figures matters: a uniform load covering only that region produces a larger effect than the same load covering the whole span, because the rest of the span would be pushing the other way.

For a continuous beam the answers become genuinely unguessable. The worst hogging over a support comes from loading the two adjacent spans and leaving the next ones empty, because the influence line alternates in sign span by span. Pattern loading is that observation turned into a set of load cases, and nobody arrives at it by thinking about the structure directly.

The reading also has to respect what the ordinate is a coefficient of. An influence line for a moment has units of length, because it is a moment per unit load; multiplying its ordinate by an actual load gives the moment that load produces. So the effect of several loads standing at once is the sum of ordinate times load over all of them, and the effect of a spread load is the area under the influence line over the loaded length times the intensity.

Where reciprocity fails #

The theorem is exact within its assumptions and the assumptions are worth knowing, because the places it fails are all places where a structure has a memory.

Nonlinear material. The proof required the strain energy to be a function of the final state alone. A structure that has yielded does not satisfy that — its state depends on the order in which the loads arrived, and the two cross terms differ.

Nonlinear geometry. A structure whose stiffness changes as it deforms fails the same test, since loading in one order leaves it at a different geometry from the other. Reciprocity is a first-order property.

Anything with friction, slip or gapping. A bolted connection that slips into bearing, a support that lifts off, a crack that opens and closes — each of these is a system whose response depends on history, and each breaks the theorem.

Non-conservative loading. A load whose direction follows the structure as it deforms — wind on a moving surface, a follower force — does work that depends on the path, and the energy argument collapses.

The interesting entry on that list is the first, because it is the one that reveals what the theorem is really about. Reciprocity is not a fact about structures; it is a fact about conservative systems, and the same symmetry appears in electrical networks, in optics, in acoustics and in heat conduction for exactly the same reason. Maxwell, who proved this version in 1864, proved several of the others too.

Where the model stops #

Elastic, linear and conservative. As above.

Small displacements. The deflections being compared are measured on the undeformed geometry, and the theorem’s proof assumes the loads do not move as the structure deflects.

One structure. The two experiments must be on the same structure with the same supports. Comparing a beam with one support arrangement to the same beam with another is not an instance of the theorem, however similar the two look.

Müller-Breslau needs a determinate release. For a redundant structure the principle still holds and the released shape is harder to obtain, because releasing a restraint on an indeterminate structure leaves something that still requires analysis.

The figures have one distortion worth naming, and it is the standard one on this site. The deflected shapes are drawn at an exaggeration of a hundred or more; at true scale the two panels would be indistinguishable straight lines, and the equality being demonstrated would be an equality between two invisible quantities. The exaggeration is what makes the claim visible, and it also makes the two shapes look more different than they are, which in this case helps — the point is that shapes with nothing in common produce identical readings.

The ladder from here #

Later rungs on this anchor: Betti’s theorem, of which Maxwell’s is the special case with two unit loads. The symmetry of the flexibility and stiffness matrices. Müller-Breslau’s principle proved. Influence lines for determinate and indeterminate structures. Influence surfaces for slabs. Reciprocity in the force method. Reciprocal theorems in dynamics, where the same symmetry appears between force and response at two frequencies. And the limits of reciprocity in nonlinear and non-conservative systems, which is where the interesting modern work is.

Maxwell published the theorem in 1864 in the same paper that contains the counting rule and the theory of reciprocal figures. Betti generalised it eight years later, Müller-Breslau turned it into influence lines in 1886, and the result was that a structure could be interrogated about a moving load by pushing it once in the right place.