Question

A prismatic steel beam is shown in the figure. 

The plastic moment, \( M_p \) calculated for the collapse mechanism using static method and kinematic method is 
 

• A. \( M_{p,static} > \frac{2PL}{9} = M_{p,kinematic} \)
• B. \( M_{p,static} = \frac{2PL}{9} \neq M_{p,kinematic} \)
• C. \( M_{p,static} = \frac{2PL}{9} = M_{p,kinematic} \)
• D. \( M_{p,static} < \frac{2PL}{9} = M_{p,kinematic} \)

Hint:

In plastic analysis, both the static and kinematic methods give the same plastic moment for simple beam configurations under certain loading conditions, such as a point load at the center of a prismatic beam.

Answer 1

The Correct Option is C

In structural analysis, the plastic moment \( M_p \) is the moment at which a section of the beam yields, and it is calculated for the collapse mechanism. The plastic moment can be calculated using either the static method or the kinematic method: - Static Method: This method involves using equilibrium equations and the condition for plastic hinge formation in the structure. For a prismatic steel beam with a point load \( P \) applied at the center, the plastic moment \( M_p \) is calculated as: \[ M_{p,static} = \frac{2PL}{9} \] - Kinematic Method: This method involves using the work-energy principle, and for the same beam configuration, the plastic moment calculated using the kinematic method is: \[ M_{p,kinematic} = \frac{2PL}{9} \] Since both methods yield the same result, the plastic moment calculated using the static method equals the plastic moment calculated using the kinematic method. Therefore, the correct answer is (C). 

Final Answer: \( M_{p,static} = \frac{2PL}{9} = M_{p,kinematic} \)