Beam Stress and Deflection Formulas

Overview

This page provides engineering reference data and calculation tools for Beam Stress and Deflection Formulas in statics.

Calculator

Key Formulas

The fundamental relationships for beam stress and deflection formulas are described by:

$Q = U \cdot A \cdot \Delta T$

Where:

• $Q$ = heat transfer rate (W)
• $U$ = overall heat transfer coefficient (W/m²·K)
• $A$ = heat transfer area (m²)
• $\Delta T$ = temperature difference (K)

Notes

• Results are approximate and should be verified for critical applications
• Input values should be within reasonable engineering ranges

Bending Stress Definition

The fundamental formula for bending stress in a beam is:

$\sigma = \frac{y \cdot M}{I}$

Where:

• $\sigma$ = bending stress (Pa, N/mm², psi)
• $y$ = perpendicular distance from the point of interest to the neutral axis (m, mm, in)
• $M$ = bending moment at the cross-section (Nm, lb·in)
• $I$ = second moment of area (moment of inertia) of the cross-section (m⁴, mm⁴, in⁴)

Beam Loading Formulas

Beam Supported at Both Ends with Uniform Distributed Load

Moment at position x: $M_x = \frac{q \cdot x \cdot (L - x)}{2}$

Maximum moment (at center, $x = L/2$): $M_{max} = \frac{q \cdot L^2}{8}$

Maximum stress: $\sigma_{max} = \frac{y_{max} \cdot q \cdot L^2}{8 \cdot I}$

Maximum deflection: $\delta_{max} = \frac{5 \cdot q \cdot L^4}{384 \cdot E \cdot I}$

Deflection at position x: $\delta_x = \frac{q \cdot x \cdot (L^3 - 2Lx^2 + x^3)}{24 \cdot E \cdot I}$

Support reactions: $R_1 = R_2 = \frac{q \cdot L}{2}$

Beam Supported at Both Ends with Single Center Load

Maximum moment (at center): $M_{max} = \frac{F \cdot L}{4}$

Maximum stress: $\sigma_{max} = \frac{y_{max} \cdot F \cdot L}{4 \cdot I}$

Maximum deflection: $\delta_{max} = \frac{F \cdot L^3}{48 \cdot E \cdot I}$

Support reactions: $R_1 = R_2 = \frac{F}{2}$

Unit Conversions

References

• Original Source