Cantilever Beam SFD and BMD Analysis

This explanation covers the Shear Force Diagram (SFD) and Bending Moment Diagram (BMD) for a common structural mechanics scenario: a cantilever beam subjected to a concentrated load at its free end.

Understanding the Cantilever Beam Setup

A cantilever beam is fixed at one end and free at the other. In this case, the beam has a length denoted by l, and a concentrated load, W, is applied exactly at the free end.

Calculating Shear Force (SFD)

To find the shear force at any point along the beam, imagine cutting the beam at a distance x from the free end and considering the forces acting on one side of the cut. For a cantilever beam with a load W at the free end, the shear force is constant throughout the beam's length.

• Consider a section at distance x from the free end.
• The only vertical force acting is the load W at the free end.
• Therefore, the shear force $V(x)$ at any section is constant and equal to W.
• Mathematically, $V(x) = W$ for $0 \le x \le l$.

A constant shear force value across the length is represented graphically as a Rectangular shape in the Shear Force Diagram (SFD).

Calculating Bending Moment (BMD)

The bending moment at any point is calculated by summing the moments of the forces about that point. For this cantilever beam:

• Consider a section at distance x from the free end.
• The load W is acting at the free end (distance 0 from the free end).
• The moment caused by this load W at the section x is the load multiplied by the distance from the section to the point where the load is applied (which is x).
• So, the bending moment $M(x)$ at distance x from the free end is $M(x) = W \times x$.
• This is a linear equation. At the free end ($x=0$), the moment is $M(0) = W \times 0 = 0$. At the fixed end ($x=l$), the moment is $M(l) = W \times l$.

A bending moment that varies linearly with distance is represented graphically as a Triangular shape in the Bending Moment Diagram (BMD). The moment is maximum at the fixed end.

Conclusion

Based on the analysis:

• The Shear Force Diagram (SFD) is Rectangular (constant shear force).
• The Bending Moment Diagram (BMD) is Triangular (linearly varying bending moment).

Therefore, the correct description is Rectangular and Triangular respectively.