Question

A cantilever of length \(L\) carries a point load \(W\) at the free end. The bending moment diagram will be

• A. Triangle with maximum ordinate at the fixed end
• B. Parabola with maximum ordinate at the fixed end
• C. Triangle with maximum ordinate at the free end
• D. Parabola with maximum ordinate at the center of the beam

Hint:

For a cantilever beam carrying a point load at the free end: - Shear Force Diagram (SFD): Rectangle - Bending Moment Diagram (BMD): Triangle - Maximum bending moment: \[ \boxed{M_{\max}=WL} \] at the fixed end.

Answer 1

The Correct Option is A

Concept: For a cantilever beam carrying a concentrated load at its free end, - The shear force remains constant throughout the beam. - The bending moment varies linearly from zero at the free end to a maximum value at the fixed end. Hence, the bending moment diagram (BMD) is a triangle.

Step 1: Write the bending moment equation. Let \(x\) be the distance measured from the free end. The bending moment at any section is \[ M=Wx. \] Thus, the bending moment varies linearly with \(x\).

Step 2: Evaluate the bending moment at important points. At the free end, \[ x=0, \] so \[ M=0. \] At the fixed end, \[ x=L, \] therefore, \[ M=WL. \] This is the maximum bending moment.

Step 3: Identify the bending moment diagram. Since the bending moment increases uniformly from \[ 0 \quad \text{to} \quad WL, \] the BMD is a triangle having its maximum ordinate at the fixed end. Hence, \[ \boxed{\text{The bending moment diagram is a triangle with maximum ordinate at the fixed end.}} \] Therefore, the correct option is \[ \boxed{(A)\;\text{Triangle with maximum ordinate at the fixed end}.} \]