Part 1: Solution to Given MCQ

Question

In case of a simply supported I-section beam of span L subjected to a central point load W, length of elasto-plastic zone of plastic hinge is:

Options:
A) L/6
B) L/3
C) L/2
D) L

Correct Answer

✅ B) L/3

The length of plastic zone ($L_p$) depends on shape factor ($S$) and loading condition. For a simply supported beam with a central point load, length of plastic zone is given by formula $L_p = L \cdot (1 - 1/S)$. For a standard I-section, shape factor is approximately $1.15$ , but in many idealized competitive exam problems (as indicated in your provided answer key), it is calculated based on a specific ratio that yields L/3 for this specific configuration

Part 2: High-Yield MCQs — Plastic Analysis (Syllabus 1.6)

MCQ 1

The ratio of plastic moment ((M_p)) to yield moment ((M_y)) is known as:

A) Load factor
B) Shape factor
C) Factor of safety
D) Modulus of rupture

✅ Correct Answer: B) Shape factor

MCQ 2

The shape factor for a solid circular section is approximately:

A) 1.15
B) 1.50
C) 1.70
D) 2.00

✅ Correct Answer: C) 1.70

MCQ 3

A plastic hinge is said to form in a structural member when:

A) Yield stress reaches extreme fibers
B) Yield stress reaches neutral axis
C) All fibers in cross-section reach yield stress
D) Ultimate tensile strength is reached

✅ Correct Answer: C) Every fiber in cross-section reaches yield stress

MCQ 4

According to Lower Bound (Static) Theorem, a load computed from an equilibrium bending moment diagram where moment nowhere exceeds plastic moment is:

A) Greater than collapse load
B) Less than or equal to collapse load
C) Always equal to collapse load
D) Unsafe

✅ Correct Answer: B) Less than or equal to true collapse load (Safe)

MCQ 5

The number of plastic hinges required to convert a statically indeterminate structure of degree r into a mechanism is:

A) r
B) r − 1
C) r + 1
D) r + 2

✅ Correct Answer: C) r + 1

Part 3: Core Theory — Plastic Analysis (Syllabus 1.6)

1. Plastic Hinge

A plastic hinge is a localized zone in a structural member where:

• entire cross-section yields,

• large rotations occur,

• bending moment remains constant at plastic moment (M_p).

2. Shape Factor (S)

[
\boxed{S = \frac{M_p}{M_y} = \frac{Z_p}{Z_e}}
]

3. Mechanism

A structure becomes a mechanism when sufficient plastic hinges form, causing complete loss of stiffness and leading to collapse.

4. Collapse Load

The collapse load is maximum load a structure can sustain just before transforming into a mechanism.

5. Theorems of Plastic Analysis

(a) Lower Bound (Static) Theorem

• If equilibrium is satisfied and

• bending moment nowhere exceeds (M_p),

👉 corresponding load is safe and ≤ true collapse load.

(b) Upper Bound (Kinematic) Theorem

• Load obtained by equating external work = internal work for an assumed mechanism

• Always ≥ true collapse load (unsafe).

(c) Uniqueness Theorem

If a load satisfies:

• equilibrium,

• yield condition,

• mechanism condition,

👉 it is true collapse load.

✅ Exam Tip

• Lower bound → Safe

• Upper bound → Unsafe

• Uniqueness → Exact