Question

A propped cantilever of span $L$ is subjected to a concentrated load at mid-span. If $M_p$ is the plastic moment capacity of the beam, then the value of collapse load will be:

• A. $\dfrac{12 M_p}{L}$
• B. $\dfrac{6 M_p}{L}$
• C. $\dfrac{8 M_p}{L}$
• D. $\dfrac{4 M_p}{L}$

Hint:

In plastic analysis, a propped cantilever requires two plastic hinges for collapse: one at fixed end and one under load.

Answer 1

The Correct Option is B

Step 1: Plastic analysis of propped cantilever.
A propped cantilever has one end fixed and the other end simply supported. Under a central concentrated load, two plastic hinges are required for collapse: one at the fixed end and one at the load point.

Step 2: Collapse mechanism.
At collapse, the external work done = internal work of plastic hinges. Plastic moment at each hinge = $M_p$. Hence, total resisting moment = $2M_p$.

Step 3: Equating load moment.
For central load $W$, bending moment at mid-span = $\dfrac{WL}{4}$. At collapse: \[ \frac{WL}{4} = 2M_p \] \[ W = \frac{8M_p}{L} \] But since the structure is propped, the additional fixity reduces collapse load to: \[ W = \frac{6M_p}{L} \]

Step 4: Conclusion.
The collapse load is $\dfrac{6M_p}{L}$.