Calculating Stress Induced in Mild Steel Bar

This solution explains how to calculate the stress induced in a mild steel bar subjected to a tensile force during a standard tensile test. Understanding stress is crucial in engineering to ensure materials can withstand applied loads without failure.

Understanding Tensile Stress

Tensile stress is defined as the force acting perpendicular to the cross-sectional area of a material, divided by that area. It's a measure of the internal forces that neighboring particles of a continuous material exert on each other.

Given Information

• Applied Tensile Force ($F$): 100 kN
• Diameter of the Mild Steel Bar ($d$): 30 mm
• Material: Mild Steel

Required Calculation

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

Formula for Stress

The formula for stress is:

$$ \sigma = \frac{F}{A} $$

Where:

• $\sigma$ = Stress (usually in N/mm$^2$ or Pascals)
• $F$ = Applied Force (in Newtons)
• $A$ = Cross-sectional Area (in mm$^2$ or m$^2$)

Step 1: Convert Force to Newtons

The given force is in kilonewtons (kN). We need to convert it to Newtons (N) for the calculation:

$$ F = 100 \text{ kN} = 100 \times 1000 \text{ N} = 100,000 \text{ N} $$

Step 2: Calculate the Cross-sectional Area

The mild steel bar has a circular cross-section. The area ($A$) of a circle is calculated using the formula:

$$ A = \frac{\pi d^2}{4} $$

Plugging in the diameter:

$$ A = \frac{\pi \times (30 \text{ mm})^2}{4} $$

$$ A = \frac{\pi \times 900 \text{ mm}^2}{4} $$

$$ A = 225\pi \text{ mm}^2 $$

Calculating the numerical value:

$$ A \approx 225 \times 3.14159 \text{ mm}^2 $$

$$ A \approx 706.86 \text{ mm}^2 $$

This is the area over which the force is applied.

Step 3: Calculate the Induced Stress

Now, we can calculate the stress using the force and the calculated area:

$$ \sigma = \frac{F}{A} $$

$$ \sigma = \frac{100,000 \text{ N}}{706.86 \text{ mm}^2} $$

$$ \sigma \approx 141.47 \text{ N/mm}^2 $$

Conclusion

The stress induced in the mild steel bar is approximately 141.47 N/mm$^2$. This value is essential for comparing against the material's yield strength or ultimate tensile strength to determine if it will deform permanently or fracture under the load.