---
title: "Beam Bending Laboratory Interactive Lab | Sandbox Physics"
description: "Can measured bending reveal material stiffness? Yes, within the linear range."
canonical_url: "https://sandboxphysics.com/labs/beam-bending/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/beam-bending/"
last_modified: "2026-10-27"
---

M025 · Structural mechanics

# Beam Bending Laboratory

**Can measured bending reveal material stiffness?** Yes, within the linear range. Deflection then depends on the supports, the cross-section, the load and Young's modulus. With the supports and cross-section known, plot deflection against load: the slope gives the modulus.

Change the supports and switch between solid and hollow rectangular sections. Read off the shear force, bending moment, curvature and fiber stress, then work out Young's modulus from the measured deflections.

> 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

## A support changes the entire bending field

**Background** The load is applied slowly, so inertia is left out.

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

Start with the essentials

- **Focus question**: Can measured bending reveal material stiffness?
- **One-sentence intuition**: Geometry decides how the load travels through the structure; the cross-section and modulus decide how much it deflects under that load.

### Core mathematical model

#### Moment and curvature

$$
EI\,v^{\prime\prime}(x)=M(x)
$$

Upward displacement is positive. A positive, sagging moment compresses the upper fiber and stretches the lower fiber.

#### Second moment of area and fiber stress

$$
I=\frac{bh^3-b_i h_i^3}{12},\qquad\sigma_x=-\frac{My}{I}
$$

For a solid section the inner rectangle is zero. Height runs vertically and width runs in depth.

#### Distributed load and the jump at a point load

$$
M^{\prime}=V,\quad V^{\prime}=-q,\quad V(a^+)-V(a^-)=-P
$$

A downward point load makes the shear force jump, while moment, slope and displacement stay continuous.

#### Boundary conditions matter

$$
v(0)=v(L)=0\quad\text{or}\quad v(0)=v^{\prime}(0)=0
$$

The first pair describes a simply supported beam, the second a cantilever clamped at the left end. At the free tip, moment and shear are zero apart from any load applied there.

#### Modulus from the slope of measured deflection

$$
v_p=\alpha P+\beta,\qquad\widehat E=\frac{C_{\mathrm{geometry}}}{\widehat\alpha}
$$

Fit the raw simulated deflections. The known geometry gives the compliance for a unit modulus, and a free intercept absorbs the fixed offset from the distributed load.

### Common difficulties

#### The drawing exaggerates the bend

**Typical misconception** You can read the real displacement straight off the bent apparatus.

**Better mental model** Use the readouts or the plotted curves. The 3D gain changes only how the bend is drawn.

#### A red limit is not a collapse prediction

**Typical misconception** A rod or beam past an ideal limit still follows this linear model.

**Better mental model** Read the red solution as an extrapolation. Plastic flow and post-buckling behavior are not modeled.

#### The modulus comes from the data

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

**Better mental model** Clear the observations and the estimate disappears. Record several loads and change the noise to see how good the estimate is.

### Run the experiment

1. 01

   #### Change the load path

   Compare the Simply supported and Cantilever tip load presets.

   **What to observe:** The reactions and the internal forces change together.

2. 02

   #### Move the load and probe

   Drag the two colored station handles below the apparatus.

   **What to observe:** The probe follows the real deflection. Moving it clears measurements that no longer match.

3. 03

   #### Measure the material

   Press Acquire nine load steps, add bounded sensor noise and export the raw readings.

   **What to observe:** The slope gives the modulus, and the residuals show the sensor scatter.

4. 04

   #### Test the model's limits

   Try Rectangular tube, then Short-beam warning.

   **What to observe:** A low stress does not clear the warning about shear deformation.

Physical conclusion

Trust a plotted linear solution only within the range where its assumptions hold.

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)