M023 · Structural mechanics

Truss Bridge Force Explorer

What is the least material a truss bridge needs to carry a load anywhere on its deck? Just enough that, with the load in its worst position, no member exceeds its tension strength or buckles in compression. Each member's worst position is different, so move the load across the whole deck instead of checking only the middle.

Build a Warren truss or a reinforced planar truss, move the load along it and measure the displacement. Compare tension, compression and buckling of the pin-ended members as the load crosses the whole deck.

Model domain $\text{—}$

Left vertical reaction $\text{—}$

Right vertical reaction $\text{—}$

Largest displacement magnitude $\text{—}$

Probe vertical displacement $\text{—}$

Maximum axial stress magnitude $\text{—}$

Largest ideal utilization $\text{—}$

Elastic strain energy $\text{—}$

Cross-section area $\text{—}$

Section second moment $\text{—}$

Actual displacement display gain $\text{—}$

Acquired load stations $\text{—}$

Modulus fitted from raw data $\text{—}$

Fit residual RMS $\text{—}$

Inspected member number $\text{—}$

Selected axial force: tension positive $\text{—}$

Selected unloaded length $\text{—}$

Selected axial stress $\text{—}$

Selected ideal pinned buckling load $\text{—}$

Horizontal pin reaction $\text{—}$

Free-node equilibrium residual $\text{—}$

Minimum equal rod area $\text{—}$

Minimum rod material volume $\text{—}$

Verified deck-node load cases $\text{—}$

Design a structure, then load and measure it

What is the least material a truss bridge needs to carry a load anywhere on its deck?

Applied loads and support reactionsEach configuration keeps one fixed force scale.Span

L

10.00\,m

Truss depth

H

2.50\,m

Deck panels

n

4.00\,

Bracing: Warren / reinforced

b

0.00\,

Equal rod area

A

25.00\,cm^2

Young modulus

E

200.00\,GPa

Ideal yield strength

\sigma_y

250.00\,MPa

Moving point load

P

80.00\,kN

Load position / span

a/L

0.50\,

Displacement probe / span

x_p/L

0.50\,

Displacement noise bound

\epsilon

0.00\,mm

Requested displacement gain

G

100.00\,

The truss is a single flat frame with pinned joints, held by a pin on the left and a roller on the right. All members are solid round rods with the same area and modulus. A massless rigid floor beam passes the moving load to the neighbouring deck joints, and added bracing changes the paths the load takes. The weight of rods and fixtures, bending at the joints, fatigue, sideways stability and post-buckling behaviour are left out. Orange means tension and blue means compression.

The minimum-area task keeps the layout, geometry and load fixed, puts the load at every deck joint in turn and assumes all rods are equal and solid. Yield and Euler buckling of a pinned rod each get a safety factor of 1.5, and the deflection is separately limited to one three-hundred-sixtieth of the span. The task is an idealized estimate of material volume. A red limit means the linear solution is being extrapolated; it does not simulate a collapse.

The raw points are simulated gauge readings, each with bounded, reproducible noise. A load sweep keeps only the loads that stay elastic. A least-squares line of displacement against force gives the compliance, and the known geometry turns it into a modulus; the fit never reads the reference modulus. Changing the load or display gain keeps the readings, while changing the geometry, material, probe or noise clears them. Measuring the same station again replaces the old reading; it is not an independent measurement. The CSV leaves out the reference modulus.

The model is linear elastic and quasi-static. Displacements are drawn magnified, but the readouts show true physical values. Simulated sensor readings are kept separate from the model's reference values. The strength limits are an idealized first check, not a certified design.

Start with the essentials

Focus question: What is the least material that can carry the load at every position?
One-sentence intuition: The geometry decides which members carry the load; the cross-section and the modulus decide how far the truss gives under it.

Core mathematical model

Joint equilibrium

\mathbf K_{ff}\mathbf u_f=\mathbf f_f

Only the free joint motions are solved for. The left pin blocks motion in both directions; the right roller blocks only vertical motion.

Axial rod law

N_e=\frac{EA}{\ell_e}\Delta\ell_e,\qquad\sigma_e=\frac{N_e}{A}

A rod stretches by the difference of its two end displacements, projected onto the unloaded rod direction. A positive force means tension.

Two separate limits: yield and buckling

|N_e|\leq\frac{A\sigma_y}{s},\qquad -N_e\leq\frac{\pi^2EI}{s\ell_e^2}

The second condition applies only to rods in compression, treated as ideal straight rods with pinned ends. The design safety factor is 1.5.

Material volume of solid round rods

I=\frac{A^2}{4\pi},\qquad\mathcal V=A\sum_e\ell_e

The smallest common rod area must meet the strength and displacement limits with the load at every deck joint. A load between two joints gives a convex combination of those two cases, so the joint cases cover it.

Independent energy check

U=\sum_e\frac{N_e^2\ell_e}{2EA}=\frac12\mathbf f^{\mathsf T}\mathbf u

When a linear elastic truss is loaded gradually, the axial strain energy in the rods must equal half the work of the external forces through their displacements.

Common difficulties

The drawn bend is magnified

Typical misconceptionHow far the bridge bends on screen is how far it really moves.

Better mental modelRead the displacement from the readouts or the plot. The 3D display gain only changes how the deformation is drawn.

Past a limit, the model does not show collapse

Typical misconceptionA rod or beam pushed past an ideal limit still behaves as this linear model says.

Better mental modelRead a red solution as an extrapolation. The model leaves out plastic flow and what happens after buckling.

The modulus comes from the data

Typical misconceptionThe fitted modulus just copies the reference slider.

Better mental modelDelete the recorded data and the estimate disappears. Record several loads and change the noise to see how well the modulus is recovered.

Run the experiment

01
Change the load path
Compare the Warren truss and Reinforced web presets.
What to observe: The support reactions and the member forces change together.
02
Move the load and probe
Drag the two colored station handles below the bridge.
What to observe: The probe reads the physical deflection. Moving it clears earlier measurements that no longer match the new position.
03
Measure the material
Use Acquire nine load steps, add bounded sensor noise, then Export raw observations.
What to observe: The fitted slope gives the modulus, and the residuals show how much the sensor readings scatter.
04
Find the minimum rod area
Use Check all deck load stations, then Use minimum equal rod section.
What to observe: The minimum-area search checks both strength and the deflection allowed in service.