For a simply supported beam carrying a single concentrated load P at the mid-span (centre), sketch and identify the bending moment diagram (BMD). Explain which standard geometric shape best represents the BMD for this loading case, given that bending moment is zero at both supports and maximum at the mid-span.

Introduction:
In strength of materials and structural analysis, the bending moment diagram (BMD) shows how bending moment varies along the beam length for a given loading. For a simply supported beam with a central point load, understanding the qualitative BMD shape is essential for design and checks.

Given Data / Assumptions:

• Simply supported beam of span L.
• Single concentrated load P applied at mid-span.
• Supports provide no moment; hence M = 0 at both ends.
• Material is linearly elastic; small deflection theory applies.

Concept / Approach:
The shear force is constant between loads and changes value only at a point load. The bending moment is the area under the shear force diagram. With symmetric loading, the BMD must also be symmetric and linear between load application points.

Step-by-Step Solution:
1) Reactions at supports are equal: RA = RB = P/2 (by symmetry).2) Shear on left half is +P/2 up to the load; on right half it is -P/2 after the load.3) Bending moment varies linearly with the integral of shear: M(x) increases linearly from 0 at A to a maximum at mid-span, then decreases linearly to 0 at B.4) The straight-line rise and fall on two symmetric halves form a V-shaped diagram with equal slopes on each side — an isosceles triangle.

Verification / Alternative check:
M_max at mid-span = (P * L) / 4, occurring where shear changes sign (at the point load). Linear segments confirm a triangular BMD.

Why Other Options Are Wrong:
Right-angled triangle: would imply M ≠ 0 at one support; not true for simply supported ends.
Equilateral triangle: shape is triangular but 'equilateral' is a geometric side-length property, not applicable to BMD axes.
Rectangle: requires constant bending moment, which occurs only under pure couple loading, not a single mid-span point load.

Common Pitfalls:
Confusing the shear force diagram (which is rectangular here) with the BMD; also assuming curvature implies a parabolic BMD — parabolas arise under uniformly distributed loads, not point loads.

Final Answer:
Isosceles triangle