Simply supported beam with a central point load: Does the maximum bending moment occur at the point of loading (i.e., at midspan)?
-
ATrue
-
BFalse
-
COnly when the beam is fixed-ended
-
DOnly when the load is off-center
-
EOnly when shear is zero at ends
Answer
Correct Answer: True
Explanation
Introduction / Context:For a simply supported beam under a single central point load, identifying where the maximum bending moment occurs is fundamental. Designers use this to select section size, reinforcement, and to estimate deflection and stresses safely.
Given Data / Assumptions:
- Beam is simply supported at both ends.
- A single point load P acts at the midspan (center).
- Material is linearly elastic; self-weight ignored for clarity.
Concept / Approach:The shear force diagram changes sign at the location of maximum bending moment. For a symmetric beam and centrally placed load, the symmetry dictates that the maximum moment occurs at the midspan under the load.
Step-by-Step Solution:
Support reactions: R_A = R_B = P/2Shear just left of midspan: V = +P/2; just right: V = −P/2Bending moment at a distance x from left: M(x) = R_A * x for x < L/2At midspan: M_max = R_A * (L/2) = (P/2) * (L/2) = P * L / 4This is the maximum because the shear changes sign there.Verification / Alternative check:From calculus, dM/dx = V. Setting V = 0 gives the extremum at midspan; checking neighbouring values confirms a maximum.
Why Other Options Are Wrong:False or conditional statements contradict the shear sign-change rule and symmetry for a central point load on a simply supported beam.
Common Pitfalls:Confusing with uniformly distributed loads (UDL) or off-center point loads; mixing simply supported with fixed-ended boundary conditions.
Final Answer:
True