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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∻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: $$W = w imes L = 4 \, ext{kip/ft} imes 30 \, ext{ft} = 120 \, ext{kip}$$. ‖ ‖ 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: $$M = \frac{wL^2}{8} = \frac{4 \, ext{kip/ft} imes (30 \, ext{ft})^2}{8} = 450 \, ext{kip-ft}$$. ‖ ‖ 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: $$b = h = \frac{9}{12} \, ext{ft} = 0.75 \, ext{ft}$$. Thus, $$I = \frac{1}{12} (0.75)(0.75)^3 = 0.0352 \, ext{ft}^4$$. ‖ ‖ 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, $$c = \frac{h}{2} = \frac{0.75}{2} = 0.375 \, ext{ft}$$. Therefore, $$\sigma = \frac{450 \, ext{kip-ft} imes 0.375 \, ext{ft}}{0.0352 \, ext{ft}^4} = 4785.23 \, ext{psi}$$. 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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