Cantilever Beam Tip Deflection Formula
Calculates the maximum tip deflection and rotation slope of a rigid cantilever beam fixed at one end and loaded with a concentrated point load at the free tip.
\[ \delta_{tip} = \frac{P \cdot L^3}{3 \cdot E \cdot I} \quad \text{and} \quad \theta_{tip} = \frac{P \cdot L^2}{2 \cdot E \cdot I} \]
Plain Text Formula:
delta_tip = (P * L^3) / (3 * E * I) and theta_tip = (P * L^2) / (2 * E * I)
📐 Variable Definitions & Parameters
| Symbol | Variable Name | Standard Unit | Physical Definition |
|---|---|---|---|
| \[ \delta_{tip} \] | Tip Deflection | mm or m | Downward displacement at free overhang end |
| \[ P \] | Point Load | Newtons (N) or kN | Concentrated force at tip |
| \[ L \] | Cantilever Overhang Length | meters (m) | Length from fixed wall support to point load |
| \[ E \] | Young's Modulus | Pa | Flexural elastic modulus |
| \[ I \] | Moment of Inertia | m⁴ | Second moment of area |
💡 Worked Engineering Example
Problem Statement:
A 2.5 m cantilever steel beam carries a 10 kN tip load. If E*I = 5,000 kN·m², find tip deflection.
Step-by-Step Solution:
- P = 10 kN, L = 2.5 m, L^3 = 15.625 m³
- Numerator: 10 * 15.625 = 156.25
- Denominator: 3 * 5,000 = 15,000
- Deflection = 156.25 / 15,000 = 0.01042 m = 10.42 mm
🎯 Result: The cantilever tip deflects downward by 10.42 mm.