An influence line is a deflected shape

Maxwell’s theorem swaps the question round: a load at A causing a deflection at B equals the same load at B causing the deflection at A.

That is a curiosity until it is used, and the use is one of the most economical results in the subject.

The question it answers #

Finding the worst place for a load to stand means asking, for a fixed quantity — the moment at one section, say — how it varies as a unit load walks across the span.

Three hundred solves for one curve, and a structure has as many such curves as it has sections worth checking. That is the brute-force route and it is what a computer does.

What reciprocity does to it #

Write the influence line’s ordinate as a function of the load’s position $x$: it is the moment at the fixed station $a$ due to a unit load at $x$, call it $M_a(x)$.

Now use the theorem on the quantity itself. The moment at $a$ due to a unit load at $x$ equals the displacement at $x$ due to a unit release at $a$ — a unit relative rotation imposed across a hinge cut at that section.

The right-hand side is a single deflected shape. It has an ordinate at every $x$ without anybody having to move a load anywhere, because the load has been replaced by a displacement and the displacement is imposed once.

One analysis replaces three hundred, and the answer is not an approximation to them: it is the same numbers, obtained by asking the reciprocal question.

Which free body produced the number #

The free body is the whole structure, twice, and the theorem is a statement about the two together.

Take system 1: the structure carrying a load $P_A$ at A, in equilibrium, with displacements $\delta_{BA}$ at B. Take system 2: the same structure carrying $P_B$ at B, with displacements $\delta_{AB}$ at A.

Now compute the work done by system 1’s forces moving through system 2’s displacements, and system 2’s forces moving through system 1’s. For a linear elastic structure both equal the same expression — the integral of one system’s internal forces against the other’s internal deformations — so

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

which is Betti’s theorem, and Maxwell’s is the case with both loads equal to one.

Two things about that derivation decide where the result applies. It is an energy statement, so it needs the material to be elastic and the structure to be linear — nothing about determinacy or redundancy appears anywhere. And it is symmetric in the two systems, which is exactly the property that makes the flexibility matrix symmetric and the stiffness matrix with it.

What the released shape looks like #

The principle is easier to trust once the shapes are recognisable, and for a simply supported beam they are.

Cut a hinge at $x = 4$ on a 10 m span and open it by a unit rotation. The beam becomes two rigid links, each pinned at its own support and hinged to the other, so the deflected shape is two straight lines meeting in a kink. The kink’s depth is fixed by the geometry: a unit rotation across the hinge, with lever arms of 4 and 6 m, gives an ordinate of $4 \times 6 / 10 = 2.4$ m.

That is exactly the 2.400 the three-hundred-solve calculation returned. The influence line for a moment on a simple span is $ab/L$ at the section and straight lines either side, and the whole of that result is the geometry of two rigid links.

The same construction on a continuous beam gives a shape that is not straight, because the released structure is still redundant and bends. That is the case where the shortcut is worth most: three hundred solves become one, and the answer is a curve nobody could have written down.

A determinate structure’s influence lines are straight because its released structures are mechanisms, and an indeterminate one’s are curved because its released structures are still structures. That is a one-line test for which kind of answer to expect.

The generalisation, which is where the practical value is #

Müller-Breslau’s principle is Betti’s theorem applied with one of the two “loads” being a released internal action, and it works for any quantity a structure has.

A reaction. Release the support, jack it up by a unit, and the deflected shape is the influence line for that reaction.

A bending moment. Insert a hinge at the section, impose a unit relative rotation across it, and the shape is the influence line for the moment.

A shear force. Insert a shear release — a cut that allows relative vertical movement but not relative rotation — impose a unit relative displacement, and the shape is the influence line for the shear.

The step in the shear influence line is the released displacement itself, which is a satisfying check on the whole construction: the discontinuity in the diagram is the discontinuity that was imposed.

Why it can be measured rather than computed #

The most useful consequence is that a deflected shape can be produced physically.

Cut a model of the structure at the section of interest, impose the release, and photograph the shape. That is Beggs’s method, it was standard practice for indeterminate bridges between the 1920s and the 1950s, and it produced influence lines for structures nobody could analyse.

The reason it works is entirely reciprocity: the model does not need to be at the same scale, made of the same material or loaded to the same level, because the influence line is a shape and shapes are dimensionless once normalised. Only the linearity has to be shared.

There is a modern version that matters more. A real bridge can be loaded with a known vehicle at a series of positions and its response measured — which produces a measured influence line for the instrumented section, describing the structure that exists rather than the one that was drawn. Reciprocity says that measurement is equivalent to a much easier one: push the bridge at one point and measure the shape it takes.

Where the answer comes from along the member #

Reciprocity has a companion result that says not only what the deflection is but where it came from.

That figure is the reason the unit-load method survives its own obsolescence. A computer returns a deflection; the integrand returns a map of where the deflection was produced, and the map is not the map of where the moment is largest. Anybody stiffening a member is choosing where to put material, and this is the diagram that says where.

Two diagrams that are constantly confused #

An influence line and a bending moment diagram are both curves drawn along a member and they answer opposite questions, which is worth stating because the confusion is nearly universal on first meeting.

A bending moment diagram has position along the member on its horizontal axis and the moment at that position on its vertical. The load is fixed; the section moves.

An influence line has the position of the load on its horizontal axis and the value of a quantity at one fixed section on its vertical. The section is fixed; the load moves.

The two are the same shape only by coincidence, and for a simple span under a unit load they are — which is precisely the coincidence that makes the confusion so durable. On a continuous beam they look nothing like each other: the moment diagram has a hogging peak over each support and the influence line for a mid-span moment has small negative lobes in the adjacent spans.

The negative lobes are the entire practical content. They say that loading the adjacent span reduces the moment being watched, which is why pattern loading is a search over subsets rather than a matter of loading everything — and a moment diagram never contains that information for any section but its own.

The energy reading #

There is a third statement of the same fact, and it is the one that connects to everything else.

Reciprocity, virtual work and Castigliano’s theorem are three faces of one statement: for a linear elastic structure, the strain energy is a quadratic form in the loads, and a quadratic form has a symmetric matrix. Maxwell’s theorem is that symmetry, the unit-load method is the derivative of the form, and the influence line is a row of it.

That is worth having because it says what breaks all three at once, which is the next section.

The theorem’s other job #

Reciprocity earns its keep twice, and the second job is quieter than the first.

Every matrix method in structural analysis rests on the symmetry of the stiffness matrix. That symmetry is not an accident of assembly: it is Maxwell’s theorem, expressed as $k_{ij} = k_{ji}$ — the force at freedom $i$ from a unit displacement at $j$ equals the force at $j$ from a unit displacement at $i$.

Three practical things follow from it and none is obvious.

A solver stores half the matrix. Symmetry halves the memory and roughly halves the factorisation, which on a large model is the difference between an analysis that runs and one that does not.

Cholesky factorisation is available, which is about twice as fast as the general method and is numerically better behaved — and it requires symmetry and positive definiteness, both of which the theorem supplies for a properly restrained elastic structure.

And a lost symmetry is a diagnosable error. A stiffness matrix that comes out unsymmetric has something non-conservative in it — a follower force, a friction element, a one-way spring — and the asymmetry is a signal rather than a nuisance. A follower force is exactly such a case, and its matrix is unsymmetric for a real reason.

So a theorem published in 1864 about two loads on a beam is why a modern solver is fast, and the connection is a good deal more direct than it looks.

Where reciprocity fails #

Every application above needs three hypotheses, and each of them fails somewhere real.

Linearity. A structure with a tension-only member, a gap, a contact or a friction interface is not linear, and its flexibility matrix is not symmetric. A crossed pair of diagonals with one member slack is the ordinary case: the influence line for a diagonal’s force has a kink where the other one goes slack, and no single deflected shape produces it.

Elasticity. Once anything has yielded, the energy is no longer a quadratic form and there is no symmetric matrix to be the theorem about. That is why a plastic influence line — the load position that produces a mechanism — is a different construction with a different theorem behind it.

And small displacements. A cable, a membrane or any structure whose stiffness depends on its own deflection has a flexibility that changes with the load, so the “same” structure is not the same in the two systems the theorem compares.

The first of those is the one that catches people. A frame with tension-only bracing is linear under any one load case and not linear across load cases, so its influence lines are correct as long as nothing changes state — and useless the moment something does.

Where it is used and where it is not #

Given how strong the result is, it is worth asking where it earns its place now that a computer can do the three hundred solves without complaint.

It is used for pattern loading. The shaded region on the influence lines above is where a spread load has to stand to make the quantity worst, and that region is read off the shape’s sign. On a continuous beam that is the whole of why an envelope needs several load arrangements, and the arrangements are chosen from the influence lines rather than enumerated blindly.

It is used for bridge assessment. A measured influence line is a description of an existing structure, and reciprocity is what makes it cheap to obtain.

It is used to know what to expect. An engineer who knows the shape of an influence line can tell at a glance whether a computed one is plausible — a moment influence line that is negative where it should be positive is a modelling error, and the shape is the test.

And it is not used to save arithmetic any more. The three hundred solves take milliseconds. What survives is everything the shortcut was about rather than the saving it produced, which is a common fate for a good result and is not a diminished one.

What to carry away #

An influence line is a deflected shape. Release the quantity, impose a unit displacement, and read the ordinates. One analysis, not three hundred.

The magnitude comes with it, provided the release is a unit one. The shape is not merely proportional to the influence line; it is the influence line.

It survives redundancy and it is the same theorem that makes a flexibility matrix symmetric. Nothing about the derivation mentions determinacy.

And it needs linearity, elasticity and small displacements. A tension-only member breaks the first, a hinge breaks the second, and a cable breaks the third.

Where the model stops #

Only flexural deformation is counted. The integrands above are $Mm/EI$, and a real influence line includes shear, axial and torsional terms that are small in a beam and not in a truss or a deep member.

The release has to be exact. A “unit rotation across a hinge” is a mathematical operation; producing it in a physical model needs a mechanism that imposes it without adding restraint anywhere else, which is what made Beggs’s apparatus a specialist instrument.

The structure is assumed prismatic where the shape is read. A member whose section changes has a deflected shape with kinks in it, which is correct and is easy to mistake for an error.

Nothing here handles a moving load’s dynamics. The influence line is a static object, and a vehicle crossing at speed produces a response that is the influence line convolved with the structure’s own dynamics.

The unit release is assumed not to change the structure. For an elastic analysis it does not. For a real bridge being tested it does: cutting a hinge into an existing member is not an option, so the measured version pushes rather than releases and relies on the reciprocal form.

And the measured version measures the structure that is there. That is the point of it and it is also its limitation: a measured influence line includes every unintended restraint, every bearing that has seized and every crack, and cannot be extrapolated to the structure after a repair.

The line is only worth drawing because of what is done with it. The worst place to stand is the question it answers directly, a train worse than its heaviest axle is the question it answers that no single load case can, and one deflection without solving everything is the same reciprocity used the other way round — a single answer extracted without the whole structure being solved.

The ladder from here #

Later rungs on this anchor: Betti’s theorem stated and proved in its general form, of which Maxwell’s is the two-unit-load case. The symmetry of the flexibility and stiffness matrices, and what it costs a solver when it is exploited. Influence lines for indeterminate structures, where the released shape is a curve rather than a pair of straight lines. Influence surfaces for slabs, where the release is imposed on a plate and the shape is a surface. Reciprocity in the force method, where the symmetry of the flexibility matrix halves the arithmetic. Reciprocal theorems in dynamics, where the same symmetry appears between force and response at two frequencies. And the limits of reciprocity in non-linear and non-conservative systems, which is where the interesting modern work is.

Müller-Breslau published the principle in 1886, twenty-two years after Maxwell’s theorem and twelve after Mohr’s independent statement of it. What he added was not the mechanics but the recognition of what the mechanics was for — and for the seventy years before computers it was the only practical way to find where a train should stand on a bridge nobody could analyse.