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Q2. If the Beam A shown here had Section 1, what is the highest stress at the top and bottom of the cross section? Are they tensions or compressions?
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Answer from Sia

Posted 26 days ago

Solution

a

To find the highest stress at the top and bottom of the cross-section for Beam A with Section 1, we first need to calculate the total load on the beam. The uniformly distributed load (w) is given as 4 kilopounds per foot, and the length of Beam A is 30 feet. Thus, the total load (W) can be calculated as: $W = w \times L = 4 \, \text{kip/ft} \times 30 \, \text{ft} = 120 \, \text{kip}$

b

The maximum bending moment (M) occurs at the center of the beam and can be calculated using the formula for a simply supported beam with a uniformly distributed load: $M = \frac{wL^2}{8} = \frac{4 \, \text{kip/ft} \times (30 \, \text{ft})^2}{8} = 450 \, \text{kip-ft}$

c

The moment of inertia (I) for Section 1 (a square cross-section) is given by the formula: $I = \frac{1}{12} b h^3$, where both the width (b) and height (h) are 9 inches. Converting inches to feet, we have: $b = h = \frac{9}{12} \, \text{ft} = 0.75 \, \text{ft}$. Thus, $I = \frac{1}{12} (0.75)(0.75)^3 = 0.0352 \, \text{ft}^4$

d

The stress (σ) at the top and bottom of the cross-section can be calculated using the bending stress formula: $\sigma = \frac{M c}{I}$, where c is the distance from the neutral axis to the outermost fiber. For a square section, $c = \frac{h}{2} = \frac{0.75}{2} = 0.375 \, \text{ft}$. Therefore, $\sigma = \frac{450 \, \text{kip-ft} \times 0.375 \, \text{ft}}{0.0352 \, \text{ft}^4} = 4785.23 \, \text{psi}$. The top of the beam experiences compression, while the bottom experiences tension

D

Answer

The highest stress at the top is 4785.23 psi (compression), and at the bottom is 4785.23 psi (tension).

Key Concept

Bending Stress in Beams: The bending stress in a beam is calculated using the formula $\sigma = \frac{M c}{I}$, where $M$ is the bending moment, $c$ is the distance from the neutral axis to the outermost fiber, and $I$ is the moment of inertia of the cross-section.

Explanation

The calculation of bending stress allows us to determine the maximum stress experienced by the beam under a given load, indicating whether the material is in tension or compression.

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