A simply supported beam of length, L, has a rectangular hollow cross-section of breadth b = 138 mm, d = depth 196 mm and thickness t = 8 mm. The bending stress must not exceed 111 N/mm2. If the beam is to carry a central point load of W = 15 kN, calculate the maximum span allowed, L (in m, two decimal places).

Question

BlackTom AI · Accepted Answer

To begin solving, I will set up the bending stress for a simply supported beam with a central point load.

- For a simply supported beam with a central load W, the maximum bending moment is Mmax = W·L/4.
- The bending stress at the outer fiber is σ = Mmax / S, where S is the section modulus of the cross-section.

Next, determine the section modulus S for the rectangular hollow cross-section.
- The outer dimensions are: breadth b_o = 138 mm and depth d_o = 196 mm.
- The inner hollow dimensions (subtracting the wall thickness t on both sides) are: breadth b_i = b_o − 2t = 138 − 16 = 122 mm, depth d_i = d_o − 2t = 196 − 16 = 180 mm.
- The second moment of area for a hollow rectangle about the horizontal centroidal axis is I = (b_o d_o^3 − b_i d_i^3)/12.
- Compute the two parts: I_o = (138)(196^3)/12 ≈ 86,589,664 mm^4 and I_i = (122)(180^3)/12 ≈ 59,292,000 mm^4.
- Therefore I = I_o − I_i ≈ 27,297,664 mm^4.
- The section modulus S for bending about the strong axis is S = I/(d/2) = I/98 (since the outer depth is 196 mm, the extreme fiber is 98 mm from the neutral axis).
- This yields S ≈ 27,297,664 / 98 ≈ 278,548 mm^3.

Apply the allowable stress condition to relate L to the given load W and allowable stress σ_allow = 111 N/mm^2.
- Set σ_allow = Mmax / S = (W·L/4) / S.
- Solve for L: L = 4·σ_allow·S / W.
- Substitute W = 15 kN = 15,000 N and the computed S: L = (4 × 111 × 278,548) / 15,000 mm.
- Numerically, 4 × 111 × 278,548 ≈ 123,675,312, and dividing by 15,000 gives L ≈ 8,245 mm.

Convert to meters: L ≈ 8,245 mm ≈ 8.25 m (two decimal places).

Summary of key steps:
- Compute inner and outer moments to get I and then S for a hollow rectangle.
- Use Mmax = W·L/4 for a central point load on a simply supported beam to relate L to allowable stress.
- Solve L = 4·σ_allow·S / W and convert units to obtain the maximum span in meters.

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