Question

A plate of 100 mm wide and 10 mm thick is bent into an arc of a circle of radius 10 m. If the modulus of elasticity of the material is 200 GPa, then the maximum bending stress induced in the plate is

• A. 100 kN/m$^2$
• B. 100 N/mm$^2$
• C. 200 kN/m$^2$
• D. 200 N/mm$^2$

Hint:

The bending equation $\frac{\sigma}{y} = \frac{E}{R}$ is very useful when the radius of curvature is known.
It directly relates stress to the geometry of the bent shape and the material's elasticity, without needing to know the applied moment ($M$) or moment of inertia ($I$).
Again, ensure all units are consistent before calculating.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the maximum bending stress in a plate that is bent into a circular arc.

Step 2: Key Formula or Approach:
This problem uses the theory of simple bending. The fundamental bending equation is:
\[ \frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R} \] where:
$\sigma$ = Bending stress at a distance $y$ from the neutral axis
$E$ = Modulus of Elasticity
$R$ = Radius of curvature of the bent beam
We need to find the maximum bending stress, $\sigma_{max}$. From the equation, we can write:
\[ \sigma = \frac{E y}{R} \] The maximum stress occurs at the extreme fiber, where $y$ is maximum. For a rectangular plate of thickness $t$, the neutral axis is at the center, so $y_{max} = t/2$.
\[ \sigma_{max} = \frac{E (t/2)}{R} \]

Step 3: Detailed Explanation:
First, identify and convert all values to consistent units (N and mm).

• Plate thickness ($t$) = 10 mm.

• Maximum distance from neutral axis ($y_{max}$) = $t/2 = 10 \text{ mm} / 2 = 5$ mm.

• Radius of curvature ($R$) = 10 m = $10 \times 1000 = 10,000$ mm.

• Modulus of Elasticity ($E$) = 200 GPa = $200 \times 10^3$ N/mm$^2$.

• The width of the plate (100 mm) is not needed to calculate the stress.


Now, substitute these values into the formula:
\[ \sigma_{max} = \frac{E \cdot y_{max}}{R} \] \[ \sigma_{max} = \frac{(200 \times 10^3 \text{ N/mm}^2) \times (5 \text{ mm})}{10,000 \text{ mm}} \] \[ \sigma_{max} = \frac{1,000,000}{10,000} \text{ N/mm}^2 \] \[ \sigma_{max} = 100 \text{ N/mm}^2 \] Note that 1 N/mm$^2$ is equal to 1 MPa. So the stress is 100 MPa.

Step 4: Final Answer:
The maximum bending stress induced in the plate is 100 N/mm$^2$.