Question

The bending moment on a section is maximum where shearing force is ________

• A. One
• B. Maximum
• C. Minimum
• D. Changing sign

Hint:

To locate the maximum bending moment on a beam, draw the shear force diagram first.
Find the points where the shear force line crosses the zero line (changes sign), and compute the bending moment at those specific locations.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the condition of the shear force at a section along a beam where the bending moment reaches its maximum value.
This is a standard relational concept in beam theory and structural analysis.

Step 2: Key Formula or Approach: The relationship between load, shear force, and bending moment along a beam is represented by differential equations.
The rate of change of the bending moment \(M\) with respect to the position \(x\) along the beam is equal to the shear force \(V\):
\[ \frac{dM}{dx} = V \]

Step 3: Detailed Explanation:

• According to calculus, a continuous function \(M(x)\) reaches a local maximum or minimum at points where its first derivative with respect to \(x\) is zero.

• Since \(\frac{dM}{dx} = V\), the bending moment can reach a local maximum or minimum at points along the beam where the shear force \(V = 0\).

• In shear force diagrams, points where the shear force is zero often correspond to locations where the shear force curve crosses the reference axis, thereby changing its sign.

• Therefore, a sign change in the shear force diagram identifies the exact cross-section of the beam where the bending moment reaches a local extremum.

Step 4: Final Answer:
The bending moment on a section is maximum where the shearing force is changing sign.