What is the standard expression for the second moment of area of a rectangular cross-section with width b and height h, and which bending-stress equation uses it?

What is the standard expression for the second moment of area of a rectangular cross-section with width b and height h, and which bending-stress equation uses it?

• A. I = b*h^3/12; sigma = M*c/I
• B. I = b^3*h/12; sigma = M*c/I
• C. I = b*h^2/12; sigma = M*c/I
• D. I = h*b^3/6; sigma = M*c/I

Answer: (A)

The moment of inertia for a rectangle about its centroidal horizontal axis (through the mid-height) is I = b h^3 / 12. This comes from integrating y^2 over the cross-section, with width b constant and y measured from the neutral axis: I = ∫ dA y^2 = b ∫_{-h/2}^{h/2} y^2 dy = b(h^3/12). The bending-stress relation that uses this I is sigma = M c / I, where c is the distance from the neutral axis to the outer fiber. For the outer surface of the rectangle, c = h/2, so the maximum bending stress is sigma_max = M(h/2) / I.

The moment of inertia for a rectangle about its centroidal horizontal axis (through the mid-height) is I = b h^3 / 12. This comes from integrating y^2 over the cross-section, with width b constant and y measured from the neutral axis: I = ∫ dA y^2 = b ∫_{-h/2}^{h/2} y^2 dy = b(h^3/12).

The bending-stress relation that uses this I is sigma = M c / I, where c is the distance from the neutral axis to the outer fiber. For the outer surface of the rectangle, c = h/2, so the maximum bending stress is sigma_max = M(h/2) / I.