Question:
A uniform beam of length \(2L\) and flexural rigidity \(EI\) is fixed at both the ends. What is the moment required at the centre of the span for unit rotation?
A uniform beam of length \(2L\) and flexural rigidity \(EI\) is fixed at both the ends. What is the moment required at the centre of the span for unit rotation?
Show Hint
Remember the beam stiffness formulas:
\[
\boxed{k=\frac{4EI}{L}}
\]
for a member fixed at the far end.
If two identical members meet at a joint, their stiffnesses add:
\[
k_{\text{joint}}=k_1+k_2.
\]
Updated On: Jul 23, 2026
- \(\dfrac{2EI}{L}\)
- \(\dfrac{4EI}{L}\)
- \(\dfrac{6EI}{L}\)
- \(\dfrac{8EI}{L}\)
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The Correct Option is D
Solution and Explanation
Concept:
The stiffness of a beam is defined as the moment required to produce unit rotation while keeping all other joints fixed.
For a beam member of length \(l\) fixed at both ends, the rotational stiffness at one end is
\[
k=\frac{4EI}{l}.
\]
When a moment is applied at the centre of a beam fixed at both ends, the beam is divided into two equal spans.
Step 1: Determine the length of each half. The total beam length is \[ 2L. \] Hence, each half has length \[ L. \]
Step 2: Find the stiffness of each half. Each half behaves as a beam fixed at one end and connected to the centre. Therefore, \[ k=\frac{4EI}{L}. \]
Step 3: Compute the total stiffness at the centre. Since two identical beam segments meet at the centre, their stiffnesses act in parallel. Hence, \[ k_{\text{total}} = \frac{4EI}{L} + \frac{4EI}{L} = \frac{8EI}{L}. \] Thus, the moment required for unit rotation at the centre is \[ \boxed{\frac{8EI}{L}.} \] Therefore, the correct option is \[ \boxed{(D)\;\dfrac{8EI}{L}.} \]
Step 1: Determine the length of each half. The total beam length is \[ 2L. \] Hence, each half has length \[ L. \]
Step 2: Find the stiffness of each half. Each half behaves as a beam fixed at one end and connected to the centre. Therefore, \[ k=\frac{4EI}{L}. \]
Step 3: Compute the total stiffness at the centre. Since two identical beam segments meet at the centre, their stiffnesses act in parallel. Hence, \[ k_{\text{total}} = \frac{4EI}{L} + \frac{4EI}{L} = \frac{8EI}{L}. \] Thus, the moment required for unit rotation at the centre is \[ \boxed{\frac{8EI}{L}.} \] Therefore, the correct option is \[ \boxed{(D)\;\dfrac{8EI}{L}.} \]
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