If all dimensions of prismatic bar are doubled, n maximum stress produced in it under its own weight will ............

Part 1: Solution to the Provided MCQ

Question: If all the dimensions of a prismatic bar are doubled, then the maximum stress produced in it under its own weight will ____________.

• Correct Option: C) increase to two times

• Justification: The maximum stress $\sigma$ in a vertical prismatic bar due to its own weight occurs at the support and is given by the formula $\sigma = \frac{\text{Total Weight}}{\text{Area of Cross-section}} = \frac{\gamma \cdot A \cdot L}{A} = \gamma \cdot L$, where $\gamma$ is the unit weight of the material and $L$ is the length. Since stress is directly proportional only to the length ($\sigma \propto L$), doubling all dimensions (including length) results in $2 \times L$, thus doubling the stress.

Part 2 & 3: High-Yield MCQs (Syllabus 1.2: Stresses and Strains)

These questions are frequently encountered in competitive exams like GATE and SSC JE regarding the behavior of materials under self-weight.

1. The total elongation produced in a prismatic bar of length $L$ hanging vertically under its own weight is:

• A) $\frac{WL}{AE}$

• B) $\frac{WL}{2AE}$

• C) $\frac{2WL}{AE}$

• D) $\frac{WL}{4AE}$

• Correct Answer: B (Note: This is equivalent to $\frac{\gamma L^2}{2E}$)

2. The ratio of elongation of a conical bar due to its own weight to that of a prismatic bar of the same length and material is:

• A) $1/2$

• B) $1/3$

• C) $1/4$

• D) $1$

• Correct Answer: B

3. If a material obeys Hooke's Law up to the breaking point, it is classified as:

• A) Ductile

• B) Brittle

• C) Elastic

• D) Plastic

• Correct Answer: B

4. The stress at which a material transforms from elastic behavior to plastic behavior is known as:

• A) Ultimate stress

• B) Breaking stress

• C) Yield stress

• D) Proof stress

• Correct Answer: C

5. For a bar of uniform strength, the cross-sectional area varies such that:

• A) Bending moment is constant

• B) Shear stress is constant

• C) Bending stress is constant at every section

• D) Deflection is zero

• Correct Answer: C

Part 4: Core Theoretical Concepts: Stresses Under Self-Weight

• Linear Stress Distribution: In a vertical prismatic bar, the stress is zero at the free bottom end and increases linearly to a maximum at the top support.

• Independence of Area: The maximum stress ($\sigma = \gamma L$) caused by self-weight is independent of the cross-sectional area; it depends only on the material's density and the length of the member.

• Elongation Comparison: The elongation of a prismatic bar under its own weight is exactly half of the elongation it would experience if an external load equal to its weight were applied at the free end.

• Conical Bar Behavior: A conical bar experiences significantly less elongation ($1/3$ of a prismatic bar) because its volume (weight) is concentrated toward the support where the area is larger.

• Prismatic Bar Doubling: If all dimensions are doubled, the volume increases $2^3 = 8$ times and the area increases $2^2 = 4$ times. Therefore, the stress ($\text{Weight} / \text{Area}$) increases by $8/4 = 2$ times.