BridgeBeat
scroll ↓
VEHICLE-BRIDGE INTERACTION AND WEIGH-IN-MOTION

A bridge bends under traffic. Run the physics backwards and the same response gives the weight of the trucks and the condition of the bridge.

Everything here is computed from the equations of structural dynamics, not drawn by hand. A truck crosses, a finite-element model of the deck computes how it bends, and inverting that bending recovers the truck's weight (Bridge Weigh-in-Motion). The same vibration carries information about damage, and on a real bridge (KW51, Belgium) it tracks sixteen months of condition. The five sections below walk through this. The Math switch, top right, shows the equations and live numbers behind each step.

What the model computes
FORWARD · Euler–Bernoulli beam, 2-node elements (2 DOF/node), consistent mass M & stiffness K M ü + C u̇ + K u = F(t)
LOAD · moving wheel placed by cubic Hermite shape functions; marched in time F(t) = N(x(t))·P  (RK4 & implicit Newmark-β)
INVERSE · fit the measured moment to the section influence line (Moses), regularized + Bayesian M(t)=Σₖ Pₖ·IL(xₖ) → P = (CᵀC)⁻¹CᵀM  (±95% CI)
VALIDATE · real modal frequencies (KW51) · fatigue economics · multiple-presence gate fₙ=(nπ/L)²√(EI/m̄)/2π · damage∝(W/W_ref)ᵐ · weigh iff 1 vehicle on span

Tip: flip Math (top-right) at any point to see the equations behind that step.

scroll to begin ↓
01 · WEIGHING A MOVING TRUCK

Weighing a truck from how the deck bends

Drag the truck across. The deck dips by less than a millimetre, but that dip, measured at a single point, is enough to recover the truck's weight while it is still moving.

this truck weighs
— tonnes
measured from the bridge's dip alone; the truck is never put on a scale.
B-WIM (Moses) recovery · true — t · error — · DAF —
how much the deck dips right now
0.0 mm
peak — mm ≈ the thickness of a 1-cent coin. Shape drawn ×—; data magnification ×—.
truck speed— km/h
faster trucks bounce the bridge more, so the weight gets a little harder to read.
higher speed → larger DAF → larger dynamic B-WIM error (see error above).
Mechanism
equation of motion (beam FEM)M ü + C u̇ + K u = F(t)
marched by numerical integration (RK4), step by step in timeu̇ = v,  v̇ = M⁻¹(F−Ku−Cu̇)
weigh by least-squares signal inversion (Moses)P = (CᵀC)⁻¹CᵀM → —
dynamic amplificationDAF = max|w|/wstatic = —
02 · DETECTING DAMAGE

A loss of stiffness lowers the natural frequency

Damage softens the bridge, so it vibrates a little slower. Drag the damage up and the peak frequency moves left. A passing vehicle can pick this up without any sensors on the bridge itself.

verdict
HEALTHY
damage at mid-span0 %
the bridge's note: — Hz (—)
fdetected = — Hz, — below healthy · peak from vehicle FFT
honest limit
frequency says something changed, not where. A single global mode cannot localise damage. Confirmed on real data in Act 3.
Mechanism
beam natural frequencyf₁ = (π/L)²√(EI/m̄) / 2π
damage softens the sectionEI → (1−d)·EI ⇒ f₁ ↓ —
read from the vehicle's acceleration spectrum (FFT)f̂ = argmax |FFT(a(t))|
03 · REAL DATA

Sixteen months of a real bridge's natural frequency

Everything so far is simulated. This is not. It is the measured natural frequency of the KW51 bridge, recorded daily for sixteen months and coloured by deck temperature.

Mechanism
remove the temperature trend (fit on the healthy period)f = a·T + b  → residual
a reduced beam tuned to f₁ matches it, but not the higher modesfₙ = n²·f₁  (anchor behaviour, not geometry)
MEASURED DATA
KW51 · Leuven, BE
Zenodo 3745914
deck temperature
→
The retrofit is visible in the data.
DATE
—
FREQUENCY
— Hz
04 · WHY IT MATTERS

Weight, damage, and cost

Heavy trucks do not cause a little more damage, they cause a great deal more. A small number of overloaded trucks accounts for most of the wear, and the same method that weighs them also identifies them and estimates the cost.

a 2× overloaded truck does
0×
the road damage of a legal one. 8× for a steel detail (m=3); exact from the power law.
overloaded 20% of trucks cause
0%
of the bridge's entire wear bill, and B-WIM catches exactly those trucks.
one heavy truck eats
€0
of bridge life in a single crossing. vs €0.80 legal; asset-depreciation estimate.
Mechanism
S–N (Wöhler) fatigue law → damage per passN(Δσ) = N_C(Δσ_C/Δσ)ᵐ ⇒ d ∝ (W/W_ref)ᵐ  (m=3 steel, 4 pavement)
Miner's rule sums it; cost = depreciationD = Σ nᵢ/Nᵢ  ·  € = d × replacement

Verified against closed-form physics (Frýba, modal frequencies) · validated on the real KW51 bridge · ratios exact per Eurocode EN 1993-1-9 / AASHTO · euro figures are an explicit asset-depreciation illustration. See REPORT.pdf.

Built on real physics, checked against textbook formulas and a real bridge. Flip the Math switch (top-right) to see the equations.

05 · A VIADUCT UNDER TRAFFIC

A 500-metre viaduct carrying a traffic stream

A half-kilometre viaduct on its piers, carrying a full stream of cars and trucks with dozens on the deck at once, seen from above: counting the trucks, flagging the overloaded ones, and totalling the wear they leave behind.

on the deck now
0 vehicles
0 t of live load
vehicles passed
0
0 trucks · rest cars
overloaded caught
0
trucks over the legal limit
bridge life consumed
€0.00
fatigue wear from this traffic
What is being computed, live
every vehicle presses on the deck; we solve the bridge's deflection K u = F(t)—
the colour is the bending the deck feels, reddest where it works hardest M = −EI·w″—
each truck spends bridge life by the cube of its weight d ∝ (W/W_ref)³—
a continuous viaduct on piers, not one long beam (a multi-span modal forest) w = 0 at every pier · 10×50 mf₁ = 1.66 Hz