Question

A cantilever beam AB of span 3 m is fixed at A and free at B is subjected to a concentrated load of 50 kN acting at a distance of 2 m from the fixed end. The maximum bending moment in the beam is

• A. 50 kNm at A
• B. 100 kNm at A
• C. 150 kNm at A
• D. 100 kNm under the load.

Hint:

For any cantilever beam, the maximum bending moment and maximum shear force will always occur at the fixed support.
Simply calculate the total moment caused by all loads about the fixed support to find the maximum bending moment.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the maximum bending moment in a cantilever beam with a point load applied at a specific location.

Step 2: Key Formula or Approach:
For a cantilever beam, the bending moment at any section is the algebraic sum of the moments of the forces acting to one side of the section. The maximum bending moment for a cantilever beam always occurs at the fixed support.
The bending moment ($M$) caused by a point load ($P$) at a distance ($a$) from the point of interest is $M = P \times a$.

Step 3: Detailed Explanation:
The beam is a cantilever, fixed at A and free at B. The span is 3 m.
A point load of $P = 50$ kN is applied at a distance of 2 m from the fixed end A.
The bending moment diagram for this beam will be as follows:
- From the free end B to the point of the load (from $x=3$ m to $x=2$ m), there are no forces, so the bending moment is zero.
- From the point of the load to the fixed end A (from $x=2$ m to $x=0$ m), the bending moment increases linearly.
The maximum bending moment will occur at the fixed support A ($x=0$).
To calculate the moment at A, we consider the load of 50 kN acting at a distance of 2 m from A.
\[ M_{max} = M_A = P \times (\text{distance from A}) \] \[ M_{max} = 50 \text{ kN} \times 2 \text{ m} \] \[ M_{max} = 100 \text{ kNm} \] This is a hogging (negative) bending moment. The magnitude is 100 kNm, and it occurs at the fixed support A. The bending moment under the load itself is also 100 kNm because the moment is constant from the load to the support in this specific problem setup (this is a mistake in reasoning, moment increases linearly from load to support). The bending moment at the point of the load (looking from the free end) is zero, the moment at the support is the maximum. The bending moment at a distance 'x' from the free end is $M(x) = -50(x-1)$ for $x\gt 1$. The moment at the support A (x=3) is not well-defined. Let's assume the distance is from the fixed end. Let the fixed end be at x=0. The load is at x=2. The free end is at x=3. The bending moment at any section x is the moment of forces to its right. For $x \gt 2$, there are no forces, so $M(x)=0$. For $x \lt 2$, the moment is $M(x) = -50 \times (2-x)$. The maximum magnitude of this moment occurs at the fixed end, x=0. $M_{max} = |M(0)| = |-50 \times (2-0)| = 100$ kNm. This maximum moment is at the support A.

Step 4: Final Answer:
The maximum bending moment in the beam is 100 kNm at the fixed end A.