Question

If the shear force along a section of a beam is zero, the bending moment at the section is

• A. zero
• B. maximum
• C. minimum
• D. average of maximum-minimum

Hint:

To locate the section of maximum bending moment in a beam under complex loading:
1. Draw the Shear Force Diagram (SFD).
2. Identify the points where the shear force changes sign (crosses zero).
3. The bending moment will reach its peak value at these exact points.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks about the state of the bending moment at a cross-section of a beam where the shear force is zero.

Step 2: Key Formula or Approach:

• In beam theory, there is a fundamental differential relationship between the bending moment ($M$) and the shear force ($V$) along the length of the beam ($x$): \[ V = \frac{dM}{dx} \]

Step 3: Detailed Explanation:

• According to differential calculus, for any continuous function $M(x)$ to reach a local extremum (either a local maximum or a local minimum), its first derivative with respect to $x$ must equal zero: \[ \frac{dM}{dx} = 0 \]
• Substituting the relation $V = \frac{dM}{dx}$ into this mathematical condition: \[ V = 0 \]
• This implies that at any location along the span of the beam where the shear force curve passes through zero, the bending moment diagram will reach a local extremum.

• For most common loading scenarios (like uniformly distributed loads or concentrated loads on simply supported beams), this extremum corresponds to the maximum bending moment in the span.

• Therefore, the bending moment is maximum (or minimum/extremum) at the point of zero shear.

Step 4: Final Answer:
The bending moment at the section where the shear force is zero is maximum.