Understanding Stress and Strain in Steel

The question asks us to find the stress induced in a Steel plate when subjected to a tensile load, given its Young's modulus and the resulting strain. This problem involves fundamental concepts of material mechanics, specifically the relationship between stress and strain within the elastic limit of a material.

When a material like Steel is subjected to a force, it deforms. If the force is tensile (pulling), the material elongates, and this elongation relative to the original length is called strain. The force applied per unit area is called stress. For many materials, including Steel within its elastic limit, stress is directly proportional to strain. This relationship is described by Hooke's Law.

The constant of proportionality in Hooke's Law is known as Young's Modulus (E), also called the modulus of elasticity. It is a measure of the stiffness of the material. The formula relating stress ($\sigma$), strain ($\epsilon$), and Young's Modulus (E) is:

$$\sigma = E \times \epsilon$$

In this problem, we are given:

• Young's Modulus for Steel plate, $E = 200 \, \text{GPa}$
• Strain developed, $\epsilon = 2$

We need to find the stress ($\sigma$) induced in the Steel plate.

Stress Calculation using Young's Modulus

First, let's ensure our units are consistent. Young's Modulus is given in GPa (Gigapascals). Stress is typically measured in Pascals (Pa) or multiples like MPa (Megapascals) or GPa. Strain is a dimensionless quantity.

We know that $1 \, \text{GPa} = 10^9 \, \text{Pa}$.

So, $E = 200 \, \text{GPa} = 200 \times 10^9 \, \text{Pa}$.

Now, we can use the formula $\sigma = E \times \epsilon$ to calculate the stress:

$$\sigma = (200 \times 10^9 \, \text{Pa}) \times 2$$

$$\sigma = 400 \times 10^9 \, \text{Pa}$$

The options for stress are given in MPa. We need to convert our calculated stress from Pa to MPa. We know that $1 \, \text{MPa} = 10^6 \, \text{Pa}$. Therefore, $1 \, \text{Pa} = 10^{-6} \, \text{MPa}$.

So, $\sigma = 400 \times 10^9 \, \text{Pa} = 400 \times 10^9 \times 10^{-6} \, \text{MPa}$.

When multiplying powers of 10, we add the exponents: $10^9 \times 10^{-6} = 10^{(9-6)} = 10^3$.

$$\sigma = 400 \times 10^3 \, \text{MPa}$$

This calculated stress value matches one of the given options.

The stress induced in the Steel plate is $400 \times 10^3 \, \text{MPa}$.