Rectangular Beam Bending Stress Analysis

A rectangular section beam is designed to withstand a bending moment M that changes along its length. The crucial requirement for this beam is to maintain the same maximum bending stress at every cross-section. Additionally, the depth of the section is kept constant. Our goal is to determine how the width of the beam should vary to meet these conditions.

Bending Stress Formula Fundamentals

To analyze the stress in a beam, we use the fundamental formula for bending stress. This formula relates the bending moment, the distance from the neutral axis, and the moment of inertia. For the maximum bending stress, the formula simplifies to:

$$\sigma_b = \frac{M}{Z}$$

• \(\sigma_b\): This represents the maximum bending stress developed in the beam's cross-section.
• \(M\): This is the bending moment acting at the specific cross-section.
• \(Z\): This is the section modulus of the beam's cross-section. It's a geometric property that indicates the resistance of the section to bending.

Section Modulus for Rectangular Section

For a rectangular beam section, the section modulus \(Z\) is calculated based on its width \(b\) and depth \(d\). The formula for the section modulus of a rectangular section is:

$$Z = \frac{bd^2}{6}$$

• \(b\): This is the width of the rectangular beam section.
• \(d\): This is the depth of the rectangular beam section.

Deriving Beam Width Variation

Now, let's combine these formulas and apply the given conditions to find out how the beam width \(b\) must vary. We are given two key conditions:

1. The maximum bending stress \(\sigma_b\) is constant across all cross-sections. Let's denote this constant value as \(\sigma_{\text{constant}}\).
2. The depth of the section \(d\) is constant. Let's denote this constant value as \(d_{\text{constant}}\).

Substitute the expression for \(Z\) into the bending stress formula:

$$\sigma_b = \frac{M}{\left(\frac{bd^2}{6}\right)}$$

This can be rearranged as:

$$\sigma_b = \frac{6M}{bd^2}$$

Since \(\sigma_b\) and \(d\) are constants, we can write:

$$\sigma_{\text{constant}} = \frac{6M}{b(d_{\text{constant}})^2}$$

Our goal is to find how \(b\) varies. Let's rearrange the equation to solve for \(b\):

$$b = \frac{6M}{\sigma_{\text{constant}}(d_{\text{constant}})^2}$$

In this equation, \(6\), \(\sigma_{\text{constant}}\), and \((d_{\text{constant}})^2\) are all constant values. Therefore, we can express their combined constant value as \(K\):

$$K = \frac{6}{\sigma_{\text{constant}}(d_{\text{constant}})^2}$$

Substituting \(K\) back into the equation for \(b\), we get:

$$b = K \cdot M$$

This equation clearly shows that the width \(b\) is directly proportional to the bending moment \(M\). This means if the bending moment doubles, the width must also double to maintain the same maximum bending stress with a constant depth.

Summary of Beam Design Relationship

Based on the derivation, to achieve a constant maximum bending stress in a rectangular section beam with a constant depth, the beam's width must vary directly with the bending moment M acting on that section. This relationship is crucial for efficient and safe structural design, especially when dealing with varying load conditions.

Thus, the width will vary as \(M\).