Girder Bridge and Moving Truck (2D): Deflection, Moment Diagram and Resonance

What you see

A steel plate-girder road bridge over a river valley with a lorry (or a convoy) driving across. The span is a simply supported beam: a concrete deck on 3 to 6 steel I-girders. The picture draws the bending of the span much larger than real so you can see it; every number is the true value. Girder colours show the bending stress (blue compression on top, red tension in the bottom flange).

The model

The whole span is one Euler-Bernoulli beam of stiffness E I (the sum of all girders; the slab is ignored, which is conservative) and mass per metre m (girders plus 15 % for bracing, slab and asphalt). Its mode shapes are sin(n pi x / L) with frequencies w_n = (n pi / L)^2 sqrt(E I / m), so f1 = (pi / 2 L^2) sqrt(E I / m). The lorry is a set of moving point loads (front axle 20 %, two rear axles 40 % each of the weight, spaced 4.8 m and 1.3 m). Each mode obeys q'' + 2 zeta w q' + w^2 q = (2 / (m L)) sum P_k sin(n pi x_k / L), advanced with the exact solution of a damped oscillator over each step. Eight modes are carried, and the mode-acceleration method adds the exact static response of the loads at their present positions, so the deflection and the bending moment (which converges slowly mode by mode) are accurate: y = y_static + sum (q_n - q_n,static) sin(n pi x / L).

What to look for

• Static envelope. The worst bending moment is not under the middle: by Barre's theorem the heaviest axle and the resultant of the loads sit symmetrically about midspan. The dashed curve in the second graph is the maximum over every truck position.
• Dynamic amplification. A single truck at normal speed adds a few per cent to the static deflection because the load changes slowly compared with the bridge's own period.
• Resonance. A convoy repeats its load every gap / v seconds; when that equals 1 / f1, that is v = f1 x gap, every truck pushes the bridge in step and the amplitude builds up, limited only by damping. Use the preset "Convoy at the resonant speed" and watch the dynamic factor in the third graph, the sweep over speeds.
• Depth and span. Halving the girder depth cuts I by about 7 and lowers f1 by a factor of 2.6, while deflection grows with L^3: the L / 800 limit is the usual live-load serviceability check.

Honest simplifications

• One span, simply supported, no skew, no continuity over piers, and no composite action between slab and girders (real bridges are stiffer); the deck is a rigid transverse distributor so every girder carries an equal share.
• The lorry is a set of moving forces: there are no tyre-spring suspension dynamics, no road roughness and no braking, all of which add to the real dynamic factor (design codes use factors around 1.1 to 1.4).
• Damping is viscous and the same ratio for every mode; real bridges are between 0.5 % and 5 %.
• Only the live-load response is shown; the dead-load deflection is assumed to be removed by camber. Stresses are for live load only, steel grade S355 (355 MPa yield), no fatigue, no buckling, no shear check.
• Axle weights, spacings and the 15 % bracing allowance are typical round numbers for an illustrative 4-axle-equivalent lorry, not a design code vehicle. Educational physics only; it must not be used to assess a real bridge.

Checks

Five boxes under the controls: the I-section second moment of area by two methods together with the modal series for a midspan load against P L^3 / (48 E I); the free-vibration period of the numerical oscillator against 2 pi / w1 and a Rayleigh estimate; the moving-force response of one mode against Fryba's closed-form solution; the maximum bending moment of the lorry by brute force against Barre's rule; and the slow-crossing limit (no amplification) together with the 90 km/h dynamic factor.