The simply supported beam in Fig. a has the cross-sectional area shown in Fig. b. Determine the absolute maximum bending stress (MPa) in the beam.

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BlackTom AI · Accepted Answer

To determine the absolute maximum bending stress in the simply supported beam, we must follow these steps and clearly identify the data required.

Step 1 — Identify the load and reaction framework
- The beam is simply supported and subjected to a uniformly distributed load (UDL) of 5 kN/m over a span of 6 m, as shown.
- For a simply supported beam with a uniform load, the maximum bending moment occurs at mid-span and is given by Mmax = w L^2 / 8.
- Here w = 5 kN/m and L = 6 m, so Mmax = (5 kN/m) × (6 m)^2 / 8 = 5 × 36 / 8 kN·m = 180 / 8 kN·m = 22.5 kN·m (which is 22,500 N·m).

Step 2 — Understand the cross‑section to obtain I and c
- The cross-section is shown in Fig. b as an I‑section (a T‑top and T‑bottom flange with a vertical web). To compute the bending stress, we need the section modulus Z or the second moment of area I and the distance c from the neutral axis to the outermost fiber: σmax = Mmax c / I.
- Specifically, we need:
  • The overall depth of the section (distance from extreme fiber top to extreme fiber bottom).
  • The flange dimensions: width and thickness (for both top and bottom flanges).
  • The web dimensions: thickness and the clear height between flanges (or the overall web height between flange inner faces).
  • The position of the neutral axis (centroid) to compute I accurately for this symmetric I/T shape.
- From the figure, we can read some numbers: flange width 250 mm, flange thickness 20 mm, web thickness 20 mm, and vertical spacing between flanges seems to show 150 mm distances on each side of the midline A (the neutral axis). However, the exact overall depth (to determine c for the top/bottom extreme fibers) and the web height need to be interpreted correctly from the drawing or confirmed with explicit dimensions.

Step 3 — Compute I and c (needs data)
- If the cross-section were perfectly symmetric about the mid-height, the neutral axis would lie at the mid-height, and the top and bottom extreme fibers would be equidistant from the NA. In that case c would be half the overall depth, and I would be computed by summing the moment of inertia of the two flanges and the web about the NA (using parallel-axis theorem).
- Without precise overall depth and the exact web height, I and c cannot be computed unambiguously.

What you should provide or confirm to proceed:
- The exact overall depth of the cross-section (top flange outer surface to bottom flange outer surface).
- The exact depth/height of the web and how much of that height is between the inner faces of the flanges (i.e., the clear web height).
- Confirmation that the top and bottom flanges are identical (width = 250 mm, thickness = 20 mm each).
- If there is any additional thickness at the web or any fillets, please specify.

Once these dimensions are confirmed, we can:
- compute I by summing the I of each rectangle (flanges and web) about the neutral axis using the parallel-axis theorem,
- determine c as the distance from the neutral axis to the outermost fiber (top or bottom),
- then calculate σmax = Mmax × c / I, converting units to MPa (1 MPa = 1 N/mm^2) as needed.

Summary:
- Step 1 yields Mmax = 22.5 kN·m.
- Step 2 requires exact cross‑sectional geometry (overall depth, web height, flange thickness, and widths) to compute I and c.
- Step 3 after obtaining I and c, apply σmax = Mmax c / I and convert to MPa.

If you provide the missing cross‑section dimensions or confirm the assumed ones, I can complete the numerical value for the absolute maximum bending stress in MPa.

ENGR20004_2025_SM2 Topic quiz - Bending stress

The simply supported beam in Fig. a has the cross-sectional area shown in Fig. b. Determine the absolute maximum bending stress (MPa) in the beam.

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