Beam diagrams – UDL case:\nFor a simply supported beam carrying a uniformly distributed load of intensity w per unit length over the whole span, the bending moment diagram has which shape?

Introduction / Context:
Shear force and bending moment diagrams are essential tools for beam design. Recognizing the characteristic shapes for common loading scenarios allows quick checks and aids in sizing members without re-deriving equations each time.

Given Data / Assumptions:

• Simply supported beam.
• Uniformly distributed load (UDL) of intensity w acting over the full span.
• Static, linear elastic behavior.

Concept / Approach:
Relationships between load w(x), shear force V(x), and bending moment M(x) are: dV/dx = −w(x) and dM/dx = V(x). For constant w, shear varies linearly with x, and bending moment varies quadratically, resulting in a parabolic bending moment diagram.

Step-by-Step Solution:

Verification / Alternative check:
Symmetry dictates V is zero at midspan for symmetric UDL; substituting x = l/2 into M(x) yields M_max = w * l^2 / 8, consistent with standard formulas and confirming the parabolic shape.

Why Other Options Are Wrong:

• Horizontal/vertical line: would imply zero or undefined change; inconsistent with quadratic bending moment under constant load.
• Inclined straight line: corresponds to constant shear (e.g., point load case), not UDL.

Common Pitfalls:
Confusing the shapes: for a point load at midspan, the bending moment diagram is triangular (piecewise linear), while for UDL it is parabolic.

Final Answer:
A parabolic curve