Calculating Longitudinal Force in a Bar Given Poisson’s Ratio, Diameter, and Strain

Understanding the relationship between Poisson's Ratio, diameter, change in diameter, and modulus of elasticity is essential when calculating the longitudinal force along the axis of a bar. This article will guide you through the process with clear steps and key formulas.

Getting the Right Values

Before we jump into the calculations, let's clarify the given values:

The Poisson’s Ratio is given as 0.33. The diameter of the bar is 19 mm. The change in diameter is 2.5 x 10^-3 mm. The modulus of elasticity (Young's Modulus) is 69 GPa.

It's important to ensure that the values are consistent. If the change in diameter is meant to be 2.5 x 10^3 mm, this would be greater than the diameter, leading to an unrealistic lateral strain. Therefore, we will assume the change in diameter is 2.5 x 10^-3 mm or 2.5 microns.

Step 1: Determine the Longitudinal Strain

To find the longitudinal strain, we use the Poisson’s Ratio. Poisson’s Ratio (ν) is defined as the negative ratio of the transverse strain (change in diameter per unit diameter) to the axial strain (change in length per unit length):

Formula: Poisson's Ratio

ν εlat / εlon

Here, εlat is the lateral strain, and εlon is the longitudinal strain.

Calculating Lateral Strain

The lateral strain can be calculated as:

εlat Δd / d

εlat 2.5 × 10-3 mm / 19 mm ≈ 1.32 x 10-4

Calculating Longitudinal Strain

Now, using the Poisson’s Ratio, we can find the longitudinal strain:

εlon -ν × εlat

εlon -0.33 × 1.32 x 10-4 ≈ -4.36 x 10-5

Step 2: Determine the Longitudinal Stress

Once we have the longitudinal strain, we can calculate the longitudinal stress using the modulus of elasticity (Young's Modulus).

Formula: Modulus of Elasticity

σ E × εlon

σ 69 GPa × -4.36 x 10-5 ≈ -3.00 GPa

The negative sign indicates that the force is compressive.

Step 3: Determine the Longitudinal Force

Finally, we can calculate the longitudinal force by multiplying the longitudinal stress by the cross-sectional area of the bar.

Area of the Bar

The area (A) is given by the formula for the area of a circle:

A π × (d/2)2

A π × (19 mm / 2)2 ≈ 453.38 mm2

Formula: Force

F σ × A

F -3.00 GPa × 453.38 mm2 ≈ -1,360,140 N or -1.36 MN

The force is compressive, as expected given the positive Poisson’s Ratio.

Conclusion

By following these steps, you can accurately calculate the longitudinal force along the axis of a bar given the Poisson’s Ratio, diameter, and strain. This process is crucial for understanding the mechanical behavior of materials under stress.

For further reading and to improve your knowledge on similar topics, consider exploring:

Understanding Poisson's Ratio and its Applications Tensile and Compressive Stress in Materials Modulus of Elasticity and Its Measurement Techniques