What is the maximum bending stress in a simply supported beam with a central load, and what is the related formula?

What is the maximum bending stress in a simply supported beam with a central load, and what is the related formula?

• A. Mmax = P*L/2; sigma_max = Mmax*c/I
• B. Mmax = P*L/4; sigma_max = Mmax*c/I with c = h/2 and I = b*h^3/12
• C. Mmax = P*L; sigma_max = Mmax*c/I
• D. Mmax = P*L^2/8; sigma_max = Mmax*c/I

Answer: (B)

For a simply supported beam with a central point load, the bending moment is greatest at mid-span and equals P times L divided by 4. The bending stress at the outer fiber is given by sigma = M times c over I, where c is the distance from the neutral axis to the outer fiber and I is the second moment of area. For a rectangular cross-section, c = h/2 and I = b h^3 / 12. Substituting these and Mmax = P L / 4 into sigma = M c / I gives: sigma_max = (P L / 4) * (h / 2) / (b h^3 / 12) = 3 P L / (2 b h^2). This matches the choice that uses Mmax = P L / 4 and the standard stress formula with c = h/2 and I = b h^3/12. The other moment values (P L/2, P L, P L^2/8) do not correspond to a simply supported beam with a central load.

For a simply supported beam with a central point load, the bending moment is greatest at mid-span and equals P times L divided by 4. The bending stress at the outer fiber is given by sigma = M times c over I, where c is the distance from the neutral axis to the outer fiber and I is the second moment of area.

For a rectangular cross-section, c = h/2 and I = b h^3 / 12. Substituting these and Mmax = P L / 4 into sigma = M c / I gives:

sigma_max = (P L / 4) * (h / 2) / (b h^3 / 12) = 3 P L / (2 b h^2).

This matches the choice that uses Mmax = P L / 4 and the standard stress formula with c = h/2 and I = b h^3/12. The other moment values (P L/2, P L, P L^2/8) do not correspond to a simply supported beam with a central load.