Given a simply supported beam with length L[math] L =8.4m, and load W[math] W =7 kN, find the bending stress σ[math] \sigma at the top of the section in the midspan of the beam, remember to show if this stress is compressive or tensile. The elastic modulus for the top is ze[math] z_e =571733.2 mm3[math] mm^3 . Provide the answer in MPa (N/mm^2) rounded to two decimal places.

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BlackTom AI · Accepted Answer

Let me restate the given problem and then walk through the calculation step by step.
- Problem: A simply supported beam of length L = 8.4 m carries a central load W = 7 kN. Find the bending stress σ at the top fiber in the midspan, and indicate whether it's compressive or tensile. The elastic modulus for the top fiber is Ze = 571733.2 mm^3. Provide the answer in MPa, rounded to two decimals.
- Known values: L = 8.4 m = 8400 mm, W = 7 kN = 7000 N, Ze = 571733.2 mm^3.

Step 1: Determine the bending moment at midspan for a simply supported beam with a central point load.
- For a simply supported beam with a central load W, the maximum bending moment occurs at midspan and is Mmax = W * L / 4.
- Compute: Mmax = 7000 N * 8400 mm / 4 = 7000 * 2100 = 14,700,000 N·mm = 1.47 × 10^7 N·mm.

Step 2: Relate bending moment to bending stress using the section modulus.
- Bending stress at the extreme fiber is σ = M / Z, where Z is the section modulus (sometimes denoted Ze in some texts).
- Using the given Ze = 571,733.2 mm^3, compute: σ = (1.47 × 10^7 N·mm) / (571,733.2 mm^3) ≈ 25.73 N/mm^2 = 25.73 MPa.

Step 3: Determine the sign (compression vs tension) for the top fiber at midspan.
- In a simply supported beam under a central downward load, the top fibers at midspan experience compression while the bottom fibers experience tension (the beam sags downward in the center).
- Therefore, the bending stress at the top fiber is negative when using a convention that tensile stresses are positive.
- Thus σ_top ≈ -25.73 MPa (compressive at the top fiber, tensile at the bottom).

Step 4: Round and present the result.
- The calculated magnitude is 25.73 MPa, and with the top fiber being in compression, the final sign is negative: −25.73 MPa.

Note on options: In the provided data set, there aren’t multiple answer choices listed to compare against. The result derived here is consistent with the standard formulas for a simply supported beam with a central point load and with the given geometric and material parameters.

Dashboard ENG2081: Mechanics of Structures 2A (2025-26)

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A square section of side length [math: b] is subjected to a bending stress of [math: σ] \sigma = 5.4 MPa under a bending moment of [math: M]=6.8 kNm. Find the side length [math: b] of the square section. Provide your answer in mm, rounded to two decimal places.

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