For a cantilever beam of length l carrying a load that varies linearly from zero at the free end to w per unit length at the fixed end, the shear force diagram (SFD) has what shape?

Introduction / Context:
Being able to sketch the qualitative shapes of shear-force and bending-moment diagrams for different loadings is a core skill in structural analysis and machine design.

Given Data / Assumptions:

• Cantilever beam, length l.
• Linearly varying distributed load: w(x) increases from 0 at free end to w at fixed end.
• Small-deflection beam theory; sign conventions standard.

Concept / Approach:
The differential relationships are:
dV/dx = −w(x)dM/dx = V(x)With w(x) linear in x, integrating once gives V(x) quadratic in x (a parabola), and integrating again gives M(x) cubic.

Step-by-Step Solution:

Verification / Alternative check:
Total shear at the fixed end equals the resultant of the triangular load: W_total = (1/2) * w * l. Evaluating V(l) = −(w/(2l)) * l^2 = −(w l)/2 matches the resultant, confirming the parabolic SFD.

Why Other Options Are Wrong:
Horizontal/vertical/inclined lines correspond to zero/impulse/constant load distributions, not a linearly varying load.Cubic curve describes the bending moment diagram for this case, not the SFD.

Common Pitfalls:
Using the wrong origin for x; forgetting that V is the integral of −w(x), hence one degree higher in polynomial order than w(x).

Final Answer:

Parabolic curve