Flexural strengths measured from three-point transverse bending tests may be determined for which cross-sectional shapes?

• A. Rectangular cross-sections only
• B. Circular cross-sections only
• C. Rectangular and circular cross-sections
• D. Any cross-section shape

Answer: (C)

Three-point bending produces pure bending in the span, so the outermost fiber stress is set by the bending moment and the section’s geometry. The maximum stress follows sigma = M c / I, or equivalently sigma = M / S with S = I/c (the section modulus). For a simply supported beam with a center load, the bending moment at midspan is M = P L / 4. So, the cross-section you use determines how that moment translates into stress via its I and c (or S).

Rectangular cross-section: I = b h^3 / 12 and c = h/2, so S = I/c = (b h^3 / 12) / (h/2) = b h^2 / 6. The flexural strength becomes sigma = M / S = (P L / 4) / (b h^2 / 6) = 3 P L / (2 b h^2). This gives a simple, direct way to compute the failure stress from the test data for common rectangular bars.

Circular cross-section: I = π d^4 / 64 and c = d/2, so S = I/c = (π d^4 / 64) / (d/2) = π d^3 / 32. Then sigma = M / S = (P L / 4) / (π d^3 / 32) = 8 P L / (π d^3). This provides the straightforward relation for circular specimens.

These two shapes have simple, well-known expressions for I and c, making the flexural strength from three-point bending easily determined. For other cross-sections, you could in principle compute I and c and apply sigma = M c / I, but the standard, straightforward results are most commonly developed and used for rectangular and circular shapes.