In a simply supported beam with a central point load P, what is the maximum bending moment and where does it occur?

In a simply supported beam with a central point load P, what is the maximum bending moment and where does it occur?

• A. The maximum bending moment is P L / 4, occurring at midspan.
• B. The maximum bending moment is P L / 2, occurring at midspan.
• C. The maximum bending moment is P L / 4, occurring at the support.
• D. The maximum bending moment is P L / 8, occurring at quarter span.

Answer: (A)

For a simply supported beam with a central point load, the supports take equal reactions, so each support carries P/2. The shear is constant on each side of the load: +P/2 to the left of the load and −P/2 to the right. The bending moment is the integral of shear. On the left side, M(x) = (P/2) x, which grows from zero at the left support to (P/2)(L/2) = PL/4 at midspan. By symmetry, the right side gives the same value at midspan. The supports have zero moment in a simply supported beam, so this interior value PL/4 at midspan is the maximum moment. Why not PL/2? That would be larger than the moment produced by this loading configuration. The moment at any interior point cannot exceed PL/4 for a central point load on a simply supported span. The PL/8 option corresponds to a different loading, such as a uniform distributed load, not a single central point load.

For a simply supported beam with a central point load, the supports take equal reactions, so each support carries P/2. The shear is constant on each side of the load: +P/2 to the left of the load and −P/2 to the right. The bending moment is the integral of shear. On the left side, M(x) = (P/2) x, which grows from zero at the left support to (P/2)(L/2) = PL/4 at midspan. By symmetry, the right side gives the same value at midspan. The supports have zero moment in a simply supported beam, so this interior value PL/4 at midspan is the maximum moment.

Why not PL/2? That would be larger than the moment produced by this loading configuration. The moment at any interior point cannot exceed PL/4 for a central point load on a simply supported span. The PL/8 option corresponds to a different loading, such as a uniform distributed load, not a single central point load.