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Worked Example: Beam Bending Stresses

This worked example demonstrates how to apply the elastic flexure formula to find the maximum bending stress in a beam.

1. Problem Statement

Consider a solid rectangular wooden beam with a width $b = 0.1 \text{ m}$ and height $h = 0.2 \text{ m}$ , subjected to an internal bending moment $M = 5000 \text{ N·m}$ . Calculate the maximum bending stress.

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3. Formula and Calculation

Substituting these values into the flexure formula:

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The maximum bending stress at the outer fibers is 7,500,000 Pa (or 7.5 MPa).

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1. Problem Statement

2. Given Data

Note: Calculate the Second Moment of Area $I = \frac{bh^3}{12} = \frac{(0.1)(0.2)^3}{12} = 6.667 \times 10^{-5} \text{ m}^4$ The distance $y$ to the extreme fiber from the neutral axis at mid-height is $y = \frac{h}{2} = \frac{0.2}{2} = 0.1 \text{ m}$

Bending Moment (M)

N·m

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3. Formula and Calculation

Substituting these values into the flexure formula:

\sigma_{max} = \frac{M y}{I}

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The maximum bending stress at the outer fibers is 7,500,000 Pa (or 7.5 MPa).

2. Given Data

Bending Moment (M)

N·m

Distance from Neutral Axis (y)

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Area Moment of Inertia (I)

m⁴

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\sigma_{max} = \frac{M y}{I}

Worked Example: Beam Bending Stresses

1. Problem Statement

3. Formula and Calculation

1. Problem Statement

2. Given Data

3. Formula and Calculation

Bending Stress Formula

Where:

Calculation:

2. Given Data

Bending Stress Formula

Where:

Calculation: