Understanding the Bending Moment Reaction Formulas for Simply Supported and Cantilever Beams

Introduction

The bending moment reaction formulas for simply supported beams and cantilever beams are fundamental concepts in structural engineering. These formulas help in determining the maximum bending moment a beam can withstand, which is crucial for designing beams that can safely support structural loads. This article delves into the derivation and application of these formulas for both types of beams.

Simply Supported Beam

To understand the bending moment reaction formulas for a simply supported beam, it's essential to consider the equilibrium conditions and the distribution of loads along the beam. A simply supported beam is supported at both ends, allowing it to rotate but not to translate vertically.

Configuration

A simply supported beam with a uniform load w (force per unit length) applied across the entire length L.

Diagram

A simply supported beam with a uniform load w applied over its entire length L.

Total Load

The total load on the beam is given by:

W w cdot L

Reactions at Supports

Let R_A and R_B be the reactions at supports A and B, respectively.

Sum of Vertical Forces

R_A R_B W w cdot L

Sum of Moments about A

Negative moments are taken as clockwise, while positive moments are taken as counterclockwise.

- The moment due to the load w at the center of the beam L/2 is:

M w cdot L cdot frac{L}{2} frac{wL^2}{2}

- Taking moments about point A:

R_B cdot L - frac{wL^2}{2} 0impliesR_B frac{wL}{2}

Substituting R_B into the first equation:

R_A frac{wL}{2} wLimpliesR_A frac{wL}{2}

Maximum Bending Moment

The maximum bending moment occurs at the midpoint L/2 of the beam:

- The bending moment at the midpoint due to the reaction at A is:

M R_A cdot frac{L}{2} frac{wL}{2} cdot frac{L}{2} frac{wL^2}{4}

However, we also consider the moment due to the load:

M frac{wL^2}{2}(moment due to the load)

Thus, the maximum bending moment in a simply supported beam under a uniform load is:

M_{text{max}} frac{wL^2}{8}

Cantilever Beam

A cantilever beam is fixed at one end and has no support at the other end, subjecting it to a uniform load w over its entire length L.

Configuration

A cantilever beam fixed at one end A and subjected to a uniform load w over the entire length L.

Diagram

A cantilever beam with a uniform load w applied over its entire length L.

Reactions at Support

Let R_A be the reaction at support A. Due to the fixed support, both a vertical reaction and a moment are present.

Total Load

The total load on the beam is again:

W w cdot L

Sum of Vertical Forces

R_A W wL

Sum of Moments about A

The moment at the fixed end A must balance the moment due to the load:

- The moment due to the load w at the center of the beam L/2 is:

M w cdot L cdot frac{L}{2} frac{wL^2}{2}

- The moment at the fixed end A:

M_A - frac{wL^2}{2} 0impliesM_A frac{wL^2}{2}

Maximum Bending Moment

The maximum bending moment in a cantilever beam occurs at the fixed end:

M_{text{max}} frac{wL^2}{2}

Summary

For a simply supported beam under a uniform load w:

Maximum Bending Moment: frac{wL^2}{8}

For a cantilever beam under a uniform load w:

Maximum Bending Moment: frac{wL^2}{2}

These formulas are essential in structural engineering for designing beams to withstand bending stresses.