The train that arrives in time with itself

The worst position for a train on a bridge is a static question with a static answer: roll the axle pattern along the span, evaluate the moment at every station for every position, and take the envelope. Nothing about how fast the train is moving appears in it.

That is right for most bridges and it stopped being right for high-speed railways, and the reason is not that trains got heavier. It is that a train’s axles are regularly spaced, so a train passing at speed $v$ is not a moving load — it is a forcing function with a frequency $v/d$, where $d$ is the distance between the repeating groups. Line a multiple of that up with the bridge’s own frequency and every coach arrives exactly in step with the motion the last one left behind.

Which free body produced the number #

Take the bridge’s first mode alone. For a simply supported span the shape is a half sine, and projecting the equation of motion onto it turns a beam into a single oscillator:

$$m^* \ddot{q} + 2\zeta\omega m^*\dot{q} + \omega^2 m^* q = \sum_j P \sin\frac{\pi x_j(t)}{L}$$

with $m^* = \bar{m}L/2$ the modal mass, $\omega = (\pi/L)^2\sqrt{EI/\bar{m}}$ the first frequency, and $x_j = vt - (j-1)d$ the position of the $j$-th axle. Each axle contributes to the generalised force only while it is on the span, and it contributes in proportion to the mode shape where it stands.

That right-hand side is the whole of the problem. It is a sum of terms that switch on and off as axles arrive and leave, and its dominant period is $d/v$ — the time between successive coaches — because each coach delivers the same pattern of force as the last.

Resonance is then the ordinary condition. If

$$\frac{v}{d} = \frac{f_1}{k}, \qquad\text{that is}\qquad v_{res} = \frac{d f_1}{k}$$

for integer $k$, each coach’s contribution arrives in phase with the free vibration left by the one before, and the response builds up coach by coach. The formula has nothing in it about how heavy the axles are or how strong the bridge is — only a length and a frequency.

For a 25-metre span at 4 Hz under 18-metre coaches, that gives 72 m/s, 36 and 24: 259, 130 and 86 km/h. All three are operating speeds and the first of them is a design speed for a high-speed line.

The response between the spikes #

The other thing the figure shows is how narrow the spikes are. Away from a resonant speed the response is a fraction of the peak, and the amplification of a single load crossing at any speed is modest — around 1.7 at worst, at a crossing time comparable with the natural period.

So a bridge on a high-speed line is not generally in trouble — it is in trouble at three or four specific speeds, and the design question is whether any of them falls inside the operating range with the margin the code demands. That is a very different design activity from checking a load case: it is a sweep, and it produces a chart rather than a number.

The narrowness is itself a consequence of damping. With one per cent damping the amplification at exact resonance is bounded by the number of coaches rather than by 1/2ζ, because the train runs out before the response does — a fifty-fold amplification needs a hundred coaches to build. That is the same finite-duration argument a suddenly applied load makes at the other extreme, and it is why train length is a design variable on this problem.

It also explains something counter-intuitive about operating rules. If a bridge is critical at 259 km/h and the line runs at 300, the bridge is fine — but the train has to be prevented from dwelling at 259, which is exactly what happens when it is held at a signal approach or run at reduced speed for maintenance. A speed restriction can put a train onto a resonance rather than away from one.

What actually fails, which is not a stress #

The quantity that governs a high-speed railway bridge is the deck’s vertical acceleration, and this is the part of the subject with no counterpart anywhere else on this site.

Ballasted track holds the rails in place through the interlock of the stone. Shake the deck hard enough and the ballast fluidises — the stones lose contact with one another, the track’s lateral resistance falls away, and the alignment can be lost. The threshold is somewhere around 0.7 g measured on the ballast, and design codes work to half of it: 3.5 m/s² on the deck for ballasted track, 5 m/s² for direct fastening.

That limit is the dashed line across every sweep on this page, and it is the only acceptance criterion in this collection that is neither a stress nor a deflection. A floor and a railway deck are the two places the site meets one: both are governed by how hard they shake rather than by what they can carry.

Strength appears nowhere in that criterion, which is why the second refutation above is a flat “false” rather than a qualification. A stronger deck of the same mass and stiffness accelerates exactly as much. It is the same relationship a vibrating floor has with its own limit, one field along and with ballast in place of people.

The stress check does not disappear; it becomes a fatigue check, because a resonance at 4 Hz for the twenty seconds a train takes to cross is eighty cycles, and a line carrying two hundred trains a day accumulates them quickly. But the ultimate strength check is almost never what sizes the span.

What the sweep costs, and why it is done at all #

A speed sweep is not free. Each speed is a time integration through the crossing and the free decay after it; a hundred and sixty speeds against ten train types is sixteen hundred analyses, and for a continuous deck with several modes each one is a solve rather than a formula.

The reason it is nonetheless routine is that the integration is over a modal model rather than a finite-element one. Extract the modes once — which is one eigenvalue problem — and every speed is then an ordinary differential equation in a handful of coordinates, integrated in milliseconds. The expensive part is done once and the cheap part is done sixteen hundred times.

That is a general pattern in dynamics and it is why modal analysis survives in an era of large computers. It is not an approximation adopted to save work; it is a change of coordinates in which the work becomes separable, and the modal mass is the quantity that says how many coordinates are needed.

What is being sampled sixteen hundred times is one oscillator’s amplification against forcing frequency, and the speed axis of every chart here is that axis rescaled by the axle spacing. A sweep is expensive because the rescaling is not the whole story — the buildup, the cancellation and the finite train length all live in the time integration and none of them is visible on the amplification curve.

The sweep also has a property worth exploiting: it is embarrassingly parallel, and the answer wanted from it is a maximum. So the practical procedure is a coarse sweep to find the neighbourhoods, then a fine one round each peak — which is the same two-stage search a collapse mechanism’s hinge position is found by, in a completely different subject.

The three things that can be changed #

The equation of motion has three parameters and each of them does something different.

Mass. Raising the mass lowers $f_1$ as its square root, which moves every resonant speed down. It also raises $m^*$, which reduces the response at any given forcing — so adding mass helps twice, and it is why heavy concrete decks behave better than light steel ones on this problem. It is also why the ballast, which is dead weight to every other calculation, is a structural benefit here — a reversal of the usual relationship between mass and performance.

Stiffness. Raising $EI$ raises $f_1$ and moves the resonant speeds up. Whether that helps depends entirely on where they land: pushing the first resonance from 259 to 320 km/h takes it out of a 300 km/h operating range, and pushing it from 200 to 259 takes it in. Stiffening a bridge is not a safe direction on this problem, which is the reverse of the intuition the rest of this collection builds.

Damping. Raising $\zeta$ reduces the peak and does nothing to its location. It is the only one of the three that is unambiguously good, and it is also the one nobody controls: the damping of a bridge is measured, not designed, and the only thing that stops a resonance is a quantity nobody specifies: the code values used for a short concrete span are around 1 to 2 per cent with an addition for spans under 20 metres that reflects what has been measured rather than any mechanism.

Which is why retrofit solutions for this problem are damping devices rather than strengthening: tuned mass dampers on the deck, viscous dampers between deck and pier, or simply a heavier parapet.

The cancellation nobody designs for #

There is a second condition hidden in the sum, and it is the reason the peaks are not all the same height.

Each axle leaves the span having imparted some free vibration, and the next axle arrives with the span already moving. But a single load crossing a span also produces a free vibration whose amplitude depends on the crossing time — and at certain crossing times that amplitude is zero, because the load leaves the span at the instant the mode has returned to rest. Those are the cancellation speeds, at $v = 2Lf_1/(2i-1)$, and a resonance that coincides with one of them does not build up at all.

That is why a bridge’s worst resonance is often the second or third rather than the first, and why the response chart has to be computed rather than reasoned about. The interference between the buildup and the cancellation is the whole shape of the curve, and neither effect alone predicts it.

It also means the span length matters in a way that has nothing to do with strength: a span whose length puts a cancellation on top of the resonance is quiet, and one a metre different is not. On lines with many identical short spans, choosing the span to avoid a resonance is a real and available design move.

Why the criterion picks out short spans #

The choice of an acceleration limit rather than a stress one does something to the design population that is worth making explicit, because it selects a completely different set of bridges.

The modal mass of a simply supported span is $\bar{m}L/2$, and the generalised force from an axle is the axle load times the mode shape where it stands. So the acceleration at resonance goes as force over modal mass, which for a given axle load and a given track construction means

$$a \;\propto\; \frac{1}{L}$$

A short span accelerates more for the same train. Half the span, twice the acceleration, with nothing about the section or the strength involved.

A stress criterion would have said the opposite. Bending moment goes up with span, so a stress check picks out long spans and a short one passes trivially — which is exactly why a hundred years of railway bridges built to a stress check with an impact factor never met this problem, and why it arrived on the shortest spans of a high-speed line rather than on its viaducts.

There is a competing effect that softens it and does not reverse it. Since $f_1 \propto 1/L^2$, a short span’s fundamental is high, so its first resonance sits at a speed far above anything a train does; what lands in the operating range is a higher-order one, at $k = 3$ or 4, and the coach-passing force has less content at those harmonics than at the first. So a short span has weaker resonances and a much smaller modal mass, and the second wins.

Which explains a fact about the modern rules that otherwise looks arbitrary. The dynamic check is demanded for spans below about 40 metres and is often waived above them; the acceleration limit is the criterion; and the bridges that fail it are the little ones nobody would have looked at twice. The check was written for the population a stress check cannot see.

The lines that found it #

This is one of very few problems in structural engineering that arrived as a surprise on a working railway rather than in a laboratory, and the history is worth having because it explains why the rules look the way they do.

Conventional railways never met it. At 160 km/h with 20-metre coaches the forcing frequency is 2.2 Hz, and a short span’s first mode is well above that; the resonance would need a bridge softer than anybody was building. The dynamic allowance in the older codes is an impact factor — a single number multiplying the static answer, calibrated on measurements and rising with span — and it works because there is nothing resonant happening.

The French high-speed lines changed the arithmetic in the 1980s and 1990s by raising $v$ by a factor of two, which raised the forcing frequency into the range short spans actually occupy. Measurements on the Paris–Lyon line and its successors found accelerations well above anything the impact factor implied, on spans that were nowhere near their strength limits, and the response was traced to exactly the condition above.

The reason the check is written against a set of trains rather than a train is visible on the same bridge as soon as the spacing changes.

What followed is the reason a modern railway bridge code looks so different from a highway one. The dynamic check became a speed sweep rather than a factor; a set of ten fictitious articulated trains, the HSLM family, was defined with spacings chosen to envelope every real train likely to run; and the acceptance criterion became a deck acceleration. All three of those are unusual in structural codes, and all three follow from the mechanism rather than from caution.

A single number cannot describe a resonance. That is the whole content of the change, and it is why an impact factor calibrated on a hundred years of measurement stopped being adequate the moment a length scale in the problem changed.

Where the model stops #

One mode is not always enough. The displacement of a simply supported span is well described by its first mode — the one-mode static deflection is within 1.5 per cent of the exact — but acceleration is not, because differentiating twice multiplies each mode by the square of its frequency and the second mode is nine times more important in acceleration than in displacement. Codes require modes up to 30 Hz for exactly that reason, and the single-mode model in this essay under-predicts the acceleration it is being used to check.

The train is not a set of forces. Its axles sit on suspensions, and the sprung mass interacts with the deck rather than merely pressing on it. That vehicle–bridge interaction adds damping to the coupled system and usually reduces the response, so the moving-force model is conservative — which is the reason it is allowed for design and the reason a bridge that fails the check is then reanalysed with a vehicle model rather than strengthened.

And the span is not simply supported. Continuity over piers changes both the mode shapes and the way a load entering one span affects the next, and a continuous deck has more modes in the range of interest than a simple one. Almost every result here is exact for the simplest case and indicative for the real one.

And the instrument this replaced is still correct. An influence line answers where a load should stand, and the envelope in the second figure is that answer for a whole axle pattern. Neither has any opinion at all about how fast the train arrives, which is not an error in them — it is the question they were built to answer, asked of a railway where it was the only question there was.

The generalisation #

The idea worth carrying is that a regularly repeating load is a frequency, and this collection has been treating such loads as sequences of positions.

A row of columns being craned into place, a line of vehicles on a bridge deck, a crowd walking in step, a machine’s rotor passing an unbalanced position once per revolution, a wave train arriving at a jetty — every one of them has a spacing and a speed, and the ratio is a forcing frequency that can be compared with a natural one.

The comparison costs nothing and it is not made by any static analysis, however careful. That is the failure this whole field exists to catch: a load case that is correct about magnitude and silent about timing, evaluated by a method that has no way of noticing.

The instrument is always the same and it is one division. Find the repeating length, divide by the speed, and compare the result with the structure’s own period. If they are within a factor of about two of each other, the static answer is not the answer — and if they are not, it is. That comparison takes seconds and it is the only thing standing between a correct envelope and a bridge that shakes its ballast loose.

One qualification belongs on that division, and it is what modal mass is for: the natural period to compare against is the period of a mode that carries some of the mass. A forcing frequency landing on a mode with nothing in it excites nothing. The comparison is with the modes that would move if the structure were pushed, which on a simple span is the first one and on almost anything else is a short list rather than a single number.