The worst place to stand

A floor beam carries a load that sits still. A bridge carries one that moves, and the difference is not a detail — it changes what a design case even is.

For a fixed load there is one moment diagram and the beam is sized against its peak. For a moving load there is a different diagram for every position, and the question becomes: for each station along the beam, where should the load stand to make things worst there?

A different axis entirely #

The object above looks like a moment diagram and is not one, and keeping the two apart is the whole beginning of the subject.

A moment diagram answers: for this fixed load, what is the moment at each station? Horizontal axis, station. One load case.

An influence line answers: for this fixed station, what is the moment when the load stands at each position? Horizontal axis, load position. Every load case at once, for one station.

So an influence line is built by moving the load, not by moving the cut. The generator behind these figures does exactly that, literally: it places a unit load at each of three hundred positions, solves the beam from scratch each time, and records one number. Nothing about the method is clever — the definition says to do this, and doing it is cheap. Every ordinate is a free body summed at one station, so the curve is a summary of several hundred of them, one per position of the load.

The simplest one there is makes the definition concrete, because its answer is known in advance and can be checked against the picture.

Nothing in that figure needed an influence line to find, which is exactly why it is worth drawing: it is the case where the definition can be checked against something already known. The curve is a straight line from one to zero, the peak is at the support, and both facts fall out of one moment equation. Every other curve on this page is produced by the identical procedure on a quantity where the answer is not obvious.

The reward is that the resulting curve answers a question no single analysis can. Once it exists, the effect of any load anywhere is a multiplication: a load $P$ standing at position $x$ produces $P$ times the ordinate at $x$. Several loads at once produce the sum. A spread load produces the intensity times the area under the curve over the loaded length.

Reading it for a real load #

Three rules follow, and each is worth stating because each is used constantly.

A single load does most damage at the peak. For the moment at a station $a$ from one end of a simple span, the influence line is two straight lines meeting under that station, with a peak of $ab/L$. So the worst place for a single wheel is directly over the station being checked — which is intuitive, and it is the only one of the three that is.

A spread load should cover the regions of one sign only. Where the ordinate has the sign being maximised, load; where it does not, leave empty. The shaded band in these figures marks exactly that region, and for a simple-span moment it is the whole span, while for a shear influence line it is half of it.

A train of axles has to be searched. Several loads at fixed spacings produce a total that depends on where the group is placed, and the maximum is at some position that no rule of thumb gives. The standard method is to try every position where an axle sits over a peak of the influence line, which is a small finite set — the maximum of a sum of piecewise-linear functions occurs at a break point of one of them.

The shear case is where instinct fails first. A uniform load over the whole span produces less shear at that station than the same load over part of it, because the two lobes work against each other. Anyone sizing a web from a fully loaded span has used a load case that is not the worst one.

In a truss, the load arrives at the joints #

A beam takes load anywhere along it. A truss does not — the deck delivers to the panel points, and between them the load reaches the frame through the deck rather than through the truss.

That changes the shape of the influence line in a specific and useful way. Between two adjacent panel points the ordinate varies linearly, whatever the influence line would have done for a beam, because a load between the joints is shared between them in inverse proportion to its distance from each. So a truss influence line is a polygon with vertices at the panel points, and the vertices are the only values that have to be computed.

That is a considerable saving and it is also a source of error. The peak of a truss influence line is at a panel point, and interpolating a beam’s smooth curve to find a worst position gives an answer between the joints that the truss cannot experience.

The diagonals are where this matters most, and the beam’s own shear line at the same position is the shape they inherit. A diagonal carries the panel shear resolved along its length, so its influence line is the shear influence line for that panel with a constant factor on it — and near a support that line has two lobes.

Reading the two signs is the design consequence. A load anywhere in the long part of the span puts the diagonal one way and a load in the short part puts it the other, so it is in tension for a load in one part of the span and compression for a load in another — which means the member has to be designed for both, and for a slender diagonal that were only ever expected to carry tension, the compression case can govern. Trusses carrying moving loads frequently have counter-diagonals in the middle panels for exactly this reason: a second diagonal leaning the other way, present only because the load reverses.

The general observation is worth keeping. Which members pull and which push is a property of a load case, and for a moving load there is no single answer — a member’s sign is a function of where the load stands, and the influence line is the object that says so.

The envelope, which is what gets designed against #

Since each station has its own worst arrangement, no single loading produces the worst case everywhere. What a bridge is designed against is the envelope: the outer boundary of every diagram from every position of the load.

The envelope is not a moment diagram either. It is not achievable by any one arrangement, and it does not satisfy the differential relations — differentiating it does not give a shear envelope, because the two envelopes come from different load positions at every station.

For a simple span under a single moving load the envelope of maximum moment is a parabola with a peak of $PL/4$, which is a useful case because it looks exactly like a moment diagram and is not one. The moment diagram for a load at mid-span is two straight lines. The envelope is the curve traced by the peaks as the load walks across, and every point on it belongs to a different load position.

Where the envelope and the influence line touch is the one station at which the two objects can be compared directly.

The relationship between the two is worth stating exactly, because it is nearly a coincidence and is not one. The envelope’s ordinate at any station is the largest moment that station ever sees, which for a single load is that station’s own influence-line peak — $ab/L$. So the envelope is the locus of the peaks of all the influence lines, and not any one of them. At three metres the envelope reads 2.100, which is the peak of the opening figure; at five metres it reads 2.500, which is the peak of this one; and neither curve is the other.

That is also where the identification stops. It holds only while there is one load, because a peak is then a single ordinate. Put two axles on the beam and the largest moment at a station is a sum of two ordinates at a fixed spacing, the maximum moves off the station, and the envelope is no longer the locus of anything the influence line contains by itself.

The practical consequence is that a bridge’s reinforcement or its flange curtailment follows a curve that never occurs. That is not an inefficiency but a statement about what the structure has to be ready for, and the same logic runs through every design against a variable action.

Continuous spans, where nothing is guessable #

For a simple span the answers are unsurprising. For a continuous beam they are not, and this is where influence lines stop being a formalism and start being necessary.

A continuous beam’s influence lines alternate in sign span by span. Loading one span produces sagging in it and hogging in its neighbours and sagging again beyond them, with the effect dying away as it goes. So the influence line for hogging over a support is positive in the two adjacent spans and negative in the next ones out.

The worst hogging therefore comes from loading the two adjacent spans and leaving the next ones empty — an arrangement nobody proposes by instinct, since it means deliberately not loading part of a bridge to make things worse.

That is what pattern loading is: the influence-line answer, precomputed as a set of load arrangements so that a designer does not have to derive it each time. The alternation also explains something that puzzles people about long viaducts — that a vehicle five spans away has a computable, non-zero and occasionally relieving effect on a section here.

The axle train, searched #

The rule that a train of axles has to be searched deserves the mechanics, because it is the part of the technique that a computer does and a person still has to understand.

Take an influence line made of straight segments meeting at break points, and a set of loads at fixed spacings sitting on it. The total effect is the sum of each load times the ordinate beneath it. As the train advances, each load slides along its segment, so the total is a piecewise-linear function of the train’s position — and a piecewise-linear function attains its maximum at a break point.

The break points occur where any axle passes a peak of the influence line. So the search is finite and short: for each axle in turn, place it over each peak, compute the total, and take the largest. A five-axle vehicle on a line with two peaks needs ten evaluations, not a continuum.

Two features of the result are worth knowing before it is computed. The heaviest axle does not always want to be at the peak — a cluster of two moderate axles sitting either side of it can beat one heavy axle on it, depending on the spacings and the line’s slopes. And the governing arrangement usually differs between stations, so a single “worst vehicle position” for a whole bridge does not exist.

That is the reason codes replace real vehicles with idealised load models: a uniformly distributed load plus a tandem of point loads, calibrated so that the effects they produce envelope the effects of the real traffic. The model is not a vehicle anybody has seen. It is a device for making the search unnecessary, and it exists because the search’s answer differs for every station and every quantity.

Where they came from, and what they cost #

Influence lines are one of the few structural techniques invented for a specific commercial reason and still in use for it.

The reason was railways, and specifically the question of what a bridge’s supports and members would see under an engine nobody had designed for. A locomotive is a set of heavy axles at fixed spacings, moving, and by the 1860s bridges were being asked to carry engines heavier than anything the original designers had contemplated. The question “will this bridge take that engine” is not answerable from a moment diagram; it needs the influence line and the axle search, and Winkler’s introduction of the technique in 1867 is contemporary with exactly that pressure.

The cost was arithmetic, and it was severe. Each station needed its own influence line, each influence line needed the axle train searched across it, and a bridge of any size meant hundreds of such operations. Müller-Breslau’s principle — that the influence line is a deflected shape — was what made it survivable, because a deflected shape can be drawn or modelled rather than computed point by point.

Today the arithmetic is free and the technique survives for a different reason: it is the only representation that makes the moving-load question visible. A table of maximum moments hides where they came from; an influence line shows immediately which parts of a structure a load has to avoid, which spans matter, and how much of the answer comes from the axle nearest the station.

The line also counts the cycles #

Everything above uses the influence line to find a maximum. It contains something the maximum throws away, and for a bridge it is the more expensive quantity.

One vehicle crossing traces the whole influence line as a stress history at that station. The ordinate under each axle, summed, plotted against time as the train advances — that is exactly what a strain gauge at the station would record, and it is exactly what a fatigue check needs, which is a history rather than a peak.

So the shape of the line decides how many cycles a crossing costs, and the shapes differ sharply. Compare the two lines drawn above, both at the station three metres along a ten metre span.

The moment line rises from zero to $ab/L = 2.1$ m and falls back. A single axle crossing gives one cycle, of range $2.1P$, entirely one-signed.

The shear line has two lobes. Its ordinate runs from zero down to $-0.3$, jumps to $+0.7$ as the axle passes the station, and falls to zero. The same axle, the same crossing, produces a range of $1.0P$ through zero — a full reversal, where the moment detail saw none.

That is why the number of fatigue cycles a bridge accumulates is not the number of vehicles that crossed it. It is the number of vehicles times the number of peaks the relevant influence line has, and a detail on a shear connection or on a truss diagonal near a support is counting reversals while a detail at mid-span counts gentle one-signed excursions of comparable range.

The counter-diagonals mentioned above are the same fact seen at ultimate load rather than at working load. A member whose influence line changes sign is a member the traffic works in both directions, and the design consequence is a compression check at the extreme and a fatigue check at the ordinary — two different limits, both traceable to a single feature of one curve.

Where the model stops #

Static loads, moved slowly. An influence line is a sequence of static analyses. A vehicle crossing at speed applies a dynamic amplification, handled in practice by a factor on the static result and in reality by an analysis that includes the mass of both structure and vehicle.

The load path is assumed. An influence line for a beam in a floor assumes a decision about how the slab above it delivers load, and that decision is a stiffness question rather than a geometric one. A wheel between two beams does not split between them in proportion to distance unless the deck is behaving as the model says.

One influence line per quantity per station. The technique answers about one thing at one place, so a full design needs many — and combining them is not automatic, because the arrangement worst for one is not worst for another. A section critical in combined bending and shear has to be checked under arrangements that are worst for neither individually.

Linear elastic behaviour. Everything here assumes superposition: the effect of several loads is the sum of their separate effects. Once anything is nonlinear that fails, and the influence line stops being a multiplier.

Fixed geometry. The line is computed on the undeformed structure — the first-order assumption again — and for a suspension bridge — whose geometry changes materially under load — the concept needs care.

Determinate lines are straight; indeterminate ones are not. For a determinate structure every influence line is made of straight segments, which is why hand construction was feasible. For a redundant one the segments curve, because the load’s effect depends on stiffness, and the whole line has to be computed rather than drawn from two ordinates.

The figures share a distortion worth stating. The shaded band marking where a spread load should stand is drawn as a rectangle over the whole height of the plot, which suggests the load has an intensity that varies with the ordinate. It does not — the band marks a region of the horizontal axis and nothing else, and the ordinate is what converts the load into an effect.

The ladder from here #

Later rungs on this anchor: influence lines for reactions, shear and moment in determinate beams. Müller-Breslau’s principle. Influence lines for trusses and for continuous beams. Axle trains and the search for the worst position. Moment and shear envelopes. Influence surfaces for slabs and for bridge decks. Dynamic amplification of moving loads. Load models in bridge codes and where they came from. And influence lines used in assessment, where a measured one describes the structure that exists rather than the one that was drawn.

Winkler introduced the technique in 1867 and Müller-Breslau gave it its geometric interpretation in 1886. Between them they turned the question “where should the train stand” from an exhaustive search into a shape that could be drawn once and read for ever after.