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.

Question

BlackTom AI · Accepted Answer

Question restatement: A square cross-section of side length b is subjected to a bending stress of σ = 5.4 MPa under a bending moment M = 6.8 kNm. Find b in mm, rounded to two decimals.

Key formulas and relationships:
- For a square section about its centroidal axis, the moment of inertia is I = b^4 / 12.
- The maximum bending stress is σ = M * c / I, where c is the distance from the neutral axis to the outer fiber. For a square, c = b/2.
- Substituting c and I gives σ = M * (b/2) / (b^4 / 12) = M * (b/2) * (12 / b^4) = 6M / b^3.
- Solve for b: b^3 = 6M / σ, hence b = [6M / σ]^(1/3).

Unit consistency:
- Convert M to N·mm: M = 6.8 kNm = 6.8 × 10^3 N·m = 6.8 × 10^6 N·mm.
- σ is given in MPa, i.e., N/mm^2, so use σ = 5.4 N/mm^2.

Compute step by step:
- 6M = 6 × 6.8 × 10^6 = 40.8 × 10^6 N·mm.
- 6M / σ = (40.8 × 10^6) / 5.4 ≈ 7.555555 × 10^6.
- Take the cube root: b ≈ (7.555555 × 10^6)^(1/3).
- The cube root of 7.555555 × 10^6 is approximately 195.0 mm (since 195^3 ≈ 7.41 × 10^6 and 196^3 ≈ 7.54 × 10^6, the value is in that vicinity).

Result interpretation:
- The side length b is about 195.00 mm when rounded to two decimals.

Note:
- If a solution or given answer elsewhere shows 92.67 mm, that would correspond to a different configuration or a misapplication of the bending- stress formula (e.g., using I = b^3/12 or applying a different orientation). Based on the standard formula for a square cross-section bending about the centroidal axis, the correct derivation yields b ≈ 195.00 mm.

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.数值题

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