Question

Consider the beam shown in the figure (not to scale), on a hinge support at end A and a roller support at end B. The beam has a constant flexural rigidity and is subjected to the external moments of magnitude \( M \) at one-third spans, as shown in the figure. Which of the following statements is/are TRUE? \includegraphics[width=0.5\linewidth]{image8.png}

• A. Support reactions are zero
• B. Shear force is zero everywhere
• C. Bending moment is zero everywhere
• D. Deflection is zero everywhere

Hint:

For beams with only external moments and no vertical loads, the support reactions and shear forces will be zero. The bending moment and deflection will still exist due to the applied moments.

Answer 1

The Correct Option is A, B

Let's analyze the given beam and its loading conditions: - The beam has hinge support at end \( A \) and roller support at end \( B \), meaning it is statically determinate. - There are external moments \( M \) applied at \( \frac{L}{3} \) from both ends of the beam.

Step 1: Support reactions
For the beam to be in equilibrium, the sum of all forces and moments must be zero. In this case, since there are no vertical loads applied to the beam, the support reactions at \( A \) and \( B \) must be zero. Thus, there are no reactions at the supports. This means statement (A) is true.

Step 2: Shear force
Since the beam has no external loads, only moments are applied. The shear force \( V \) is the change in internal force along the length of the beam. Since there are no vertical loads, the shear force is zero everywhere along the beam. Thus, statement (B) is true.

Step 3: Bending moment
The bending moment in the beam will be affected by the applied external moments \( M \). However, since the beam is subjected to moments at the one-third points, the internal bending moment will not be zero everywhere. The bending moment is only zero at certain points depending on the beam's length and external moments, so statement (C) is false.

Step 4: Deflection
The deflection of the beam depends on the bending moments and the beam's flexural rigidity. As there are no vertical loads and the beam is simply supported, there will be deflection at various points along the beam. Therefore, statement (D) is false. Thus, the correct answer is (A) and (B): Support reactions are zero and shear force is zero everywhere. \[ \boxed{\text{The correct answers are (A) and (B).}} \]