Beam deformation
Euler-Bernoulli Beam Deflection
This interactive visualization demonstrates the deflection \(w(x)\) of beams under concentrated loads, computed using Euler-Bernoulli beam theory.
How to use
- Select a support configuration from the dropdown menu
- Enter a position along the beam (0 to 10)
- Enter the force magnitude (positive = downward)
- Click “Add force” to apply the load
- Observe how the deflection curve changes
Available support configurations
| Configuration | Left End | Right End | Description |
|---|---|---|---|
| Cantilever | Clamped | Free | Fixed at one end, free at the other |
| Simply supported | Hinged | Hinged | Supported at both ends, free to rotate |
| Fixed-fixed | Clamped | Clamped | Both ends fully restrained |
| Propped cantilever | Clamped | Hinged | Fixed at one end, simply supported at the other |
Theory
The Euler-Bernoulli beam equation relates the deflection \(w(x)\) to the bending moment \(M(x)\):
\[ EI \frac{\mathrm{d}^2 w}{\mathrm{d} x^2} = -M(x) \]
where:
- \(E\) is Young’s modulus
- \(I\) is the second moment of area
- \(EI\) is the bending stiffness
Boundary conditions
The different support types impose the following boundary conditions:
Clamped end: \[ w = 0 \quad \text{and} \quad \frac{\mathrm{d}w}{\mathrm{d}x} = 0 \]
Hinged end: \[ w = 0 \quad \text{and} \quad M = EI\frac{\mathrm{d}^2 w}{\mathrm{d}x^2} = 0 \]
Free end: \[ M = 0 \quad \text{and} \quad V = -EI\frac{\mathrm{d}^3 w}{\mathrm{d}x^3} = 0 \]
Example: Cantilever with end load
For a cantilever beam of length \(L\) with a concentrated force \(F\) at the free end:
\[ w(x) = \frac{F x^2}{6EI}(3L - x) \]
The maximum deflection occurs at the free end:
\[ w_{\max} = \frac{F L^3}{3EI} \]