---
title: "Truss Bridge Force Explorer Interactive Lab | Sandbox Physics"
description: "What is the least material a truss bridge needs to carry a load anywhere on its deck?"
canonical_url: "https://sandboxphysics.com/labs/truss-bridge/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/truss-bridge/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## Follow the load path, then measure its compliance

**Background** The load is applied slowly, so the truss stays in equilibrium and inertia can be ignored.

**Why it matters** MIT notes on mechanics of materials derive joint equilibrium, axial elasticity and Euler–Bernoulli bending from the same force balance and energy argument.

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 misconception** How far the bridge bends on screen is how far it really moves.

**Better mental model** Read 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 misconception** A rod or beam pushed past an ideal limit still behaves as this linear model says.

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

#### The modulus comes from the data

**Typical misconception** The fitted modulus just copies the reference slider.

**Better mental model** Delete 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

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

Trust a plotted linear solution only inside the range where the model says it holds.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)