M023 · Structural mechanics

Truss Bridge Force Explorer

Build a Warren or reinforced planar truss, move its load and measure displacement. Compare tension, compression and pinned-rod buckling across 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 / LOAD / MEASURE

How little material survives every load position?

Applied loads and support reactionsFixed force scale per configurationSpan

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\,

A single planar pin-jointed truss has a left pin and a right roller. Solid circular rods share one area and modulus. A massless rigid floor-beam transfers a moving load to adjacent deck joints; reinforcement changes the load paths. Rod and fixture self-weight, joint bending, fatigue, lateral stability and post-buckling are omitted. Orange means tension, blue compression.

The minimum-area task holds topology, geometry and load fixed, uses all deck-node positions, and assumes equal solid rods. Yield and pinned Euler buckling have a safety factor of 1.5; the separate displacement limit is one three-hundred-sixtieth of the span. This is an ideal material-volume screening task. A red limit means the linear solution is an extrapolation, not a simulated collapse.

Raw points are simulated gauge readings with reproducible bounded station noise. Load sweeps retain only elastic stations. Least squares fits displacement against force, then known geometry converts compliance into modulus; the fit never reads the reference modulus. Different loads and display gain preserve observations. Geometry, material, probe or noise edits clear them. Repeating a station replaces it rather than creating an independent measurement. CSV excludes the reference modulus.

Linear elastic, quasi-static model. Displacements are magnified; readouts remain physical. Simulated sensor observations are separate from model references. Strength limits are ideal screening, not a certified design.

Start with the essentials

Focus question: How little material survives every load position?
One-sentence intuition: Geometry fixes the load paths; cross-section and modulus fix compliance.

Core mathematical model

Joint equilibrium

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

Only free degrees of freedom are solved. The left pin fixes both translations; the right roller fixes vertical translation.

Axial rod law

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

Extension is the projection of joint displacement difference on the unloaded rod direction. Positive force is tension.

Two separate strength limits

|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 compression. Ideal straight rods have pinned ends. The design safety factor is 1.5.

Solid circular material budget

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

The minimum equal area must satisfy strength and displacement limits at every deck-node load station. Intermediate load positions are convex combinations.

Independent energy check

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

Axial strain energy must equal half of the external force–displacement work in a linear elastic load ramp.

Common difficulties

Magnification is not deformation

Typical misconceptionThe visibly bent apparatus gives the physical displacement directly.

Better mental modelUse the readout or quantitative plot; the 3D gain changes only presentation.

Limits do not predict collapse

Typical misconceptionA rod or beam beyond an ideal limit still obeys this linear model.

Better mental modelTreat the red solution as extrapolation; plastic flow and post-buckling are omitted.

Raw data come first

Typical misconceptionThe fitted modulus simply repeats the reference slider.

Better mental modelClear observations: the estimate disappears. Acquire several loads and change noise to inspect inference quality.

Run the experiment

01
Change the load path
Compare Warren and reinforced webs.
What to observe: Reactions and internal forces change together.
02
Move the load and probe
Drag the two colored station handles below the apparatus.
What to observe: The probe follows physical deflection; moving it clears incompatible measurements.
03
Measure the material
Acquire nine load steps, add bounded sensor noise and export raw readings.
What to observe: Slope estimates modulus; residuals reveal sensor scatter.
04
Audit the boundary
Check all deck positions and use the minimum area.
What to observe: The material task checks strength and service deflection.