For a simply supported beam of span L with a central point load P, which expression gives the maximum deflection δ_max?

• A. δ_max = P L^4 /(8 E I)
• B. δ_max = P L^3 /(48 E I)
• C. δ_max = P L /(3 E I)
• D. δ_max = P L^2 /(16 E I)

Answer: (B)

When a simply supported beam carries a central point load, the end reactions are equal (each is P/2), and the bending moment is greatest at the center. This peak moment drives the maximum deflection there. Using the beam bending relation EI d^2y/dx^2 = M(x) and the symmetry about midspan, you set up M(x) = (P/2) x for the left half (0 ≤ x ≤ L/2). Integrating twice gives the deflection, with boundary conditions y(0) = 0 and the slope at midspan zero due to symmetry (y'(L/2) = 0). Solving yields the center deflection magnitude δ_max = P L^3 /(48 E I). The negative sign simply indicates downward deflection, so the magnitude is as stated.

This result matches the standard deflection for a simply supported beam under a mid-span point load. Other expressions either do not have the correct L dependence or fail dimensional consistency, so they do not represent the center deflection correctly.