Beam Deflection: Uniform Distributed Load
Calculates the maximum center-span elastic deflection of a simply supported structural beam subjected to a uniform distributed load (UDL).
\[ \delta_{max} = \frac{5 \cdot w \cdot L^4}{384 \cdot E \cdot I} \]
Plain Text Formula:
delta_max = (5 * w * L^4) / (384 * E * I)
📐 Variable Definitions & Parameters
| Symbol | Variable Name | Standard Unit | Physical Definition |
|---|---|---|---|
| \[ \delta_{max} \] | Mid-Span Deflection | meters (m) or mm | Maximum downward vertical displacement at x = L/2 |
| \[ w \] | Uniform Distributed Load | N/m or kN/m | Total continuous load per unit length |
| \[ L \] | Clear Span Length | meters (m) | Distance between support pin and roller |
| \[ E \] | Young's Modulus | Pascals (Pa) | Elastic modulus of beam material |
| \[ I \] | Moment of Inertia | m⁴ | Moment of inertia about the horizontal bending axis |
💡 Worked Engineering Example
Problem Statement:
A steel beam (E=200 GPa, I=8,000 cm⁴) spans 6 meters under a load w=15 kN/m. Find center deflection.
Step-by-Step Solution:
- Convert I: 8,000 cm⁴ = 8e-5 m⁴
- w = 15,000 N/m, L = 6 m
- Numerator: 5 * 15,000 * 6^4 = 5 * 15,000 * 1,296 = 97,200,000
- Denominator: 384 * 200e9 * 8e-5 = 6,144,000,000
- Deflection = 97,200,000 / 6,144,000,000 = 0.01582 m = 15.82 mm
🎯 Result: Mid-span deflection is 15.82 mm, which satisfies the standard L/360 deflection limit (6,000 / 360 = 16.67 mm).