1. Solution to Provided MCQ
Question: In case of a simply supported rectangular beam of span $L$ and loaded with a central load $W$, the length of elasto-plastic zone of the plastic hinge is...
Justification: For a rectangular section, the Shape Factor ($S$) is 1.5. The length of the plastic zone ($L_p$) for a simply supported beam with a central point load is given by the formula:
$$L_p = L \left( 1 - \frac{1}{S} \right)$$
Substituting $S = 1.5$:
$$L_p = L \left( 1 - \frac{1}{1.5} \right) = L \left( 1 - \frac{2}{3} \right) = \frac{L}{3}$$
2. & 3. High-Yield MCQs (GATE, ESE, SSC JE)
Q1. What is the shape factor for a circular cross-section?
A) 1.50
B) 1.12
C) 1.70
D) 2.00
Correct Answer: C) 1.70
Q2. For a simply supported beam of span $L$ carrying a Uniformly Distributed Load (UDL) over the entire span, the length of the plastic zone is:
A) $L / \sqrt{3}$
B) $L / 3$
C) $L(1 - 1/\sqrt{S})$
D) $L \sqrt{1 - 1/S}$
Correct Answer: D) $L \sqrt{1 - 1/S}$ (For a rectangular section where $S=1.5$, this becomes approximately $0.577L$).
Q3. The ratio of plastic section modulus ($Z_p$) to the elastic section modulus ($Z_e$) is known as:
A) Aspect Ratio
B) Load Factor
C) Shape Factor
D) Factor of Safety
Correct Answer: C) Shape Factor
Q4. A fixed beam of span $L$ is subjected to a central concentrated load. The number of plastic hinges required to form a total collapse mechanism is:
A) 1
B) 2
C) 3
D) 4
Correct Answer: C) 3 (One at each support and one under the load).
Q5. In the plastic analysis of structures, the "Kinematic Theorem" (Upper Bound Theorem) states that the load obtained is:
A) Always less than or equal to the true collapse load
B) Always greater than or equal to the true collapse load
C) Exactly equal to the true collapse load
D) Independent of the collapse mechanism
Correct Answer: B) Always greater than or equal to the true collapse load
4. Core Theoretical Concepts: Plastic Hinge and Plastic Analysis
Plastic Hinge Concept: A plastic hinge is a yielded zone in a structural member that allowed large rotations to occur at a constant plastic moment ($M_p$). It occurs when every fiber of the cross-section reaches the yield stress ($\sigma_y$).
Shape Factor ($S$): It represents the reserve strength of a section beyond the elastic limit.
Plastic Moment ($M_p$): The maximum moment a section can resist, calculated as $M_p = \sigma_y \cdot Z_p$.
Length of Plastic Zone ($L_p$): This is the portion of the beam where the bending moment exceeds the yield moment ($M_y$). It depends on the loading type and the cross-sectional shape.
Collapse Mechanism: For a structure to collapse, it must become unstable by forming enough plastic hinges. For a propped cantilever, 2 hinges are needed; for a fixed beam, 3 hinges; and for a simply supported beam, only 1 hinge is required.
Load Factor: The ratio of the collapse load to the working load. It incorporates both the factor of safety and the shape factor.
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