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 a propped cantilever beam.
A propped cantilever beam is fixed at one end and simply supported at the other. For collapse under a central concentrated load, two plastic hinges are necessary: one at the fixed end and another at the point of load application.

Step 2: Collapse mechanism.
At the point of collapse, the work done by the external load equals the internal work dissipated by the plastic hinges. Each hinge develops a plastic moment of $M_p$. Therefore, the total resisting moment capacity is $2M_p$.

Step 3: Equating bending moments.
For a central load $W$, the bending moment at mid-span is $\dfrac{WL}{4}$. At the collapse condition: \[\frac{WL}{4} = 2M_p\] This implies a collapse load of $W = \frac{8M_p}{L}$. However, due to the propping condition, the additional fixity reduces the actual collapse load to: \[W = \frac{6M_p}{L}\]

Step 4: Conclusion.
The calculated collapse load for the propped cantilever is $\dfrac{6M_p}{L}$.