Shear Modulus Experiment Report

Objectives

The objective of this experiment is to determine the shear modulus for a hollow brass and the shear modulus of a solid round brass bar.

Introduction and Theory

Torsion occurs when a shaft is subjected to torque, whether the shaft is rotating or not. Torque induces shear stress on the cross-section of the shaft, causing it to twist. The relationship between torque (T), force (F), and radius (R) is given by:

T = F × R

The angle of twist (Φ) and shear modulus (G) are essential parameters for understanding torsion.

The polar moment of inertia (J) of the cross-section of the bar is also crucial for calculating shear modulus, and it is determined by the difference in the outer and inner radii of the cross-section:

J = π/2 * (outer radius^4 - inner radius^4)

Apparatus

1. Hollow brass bar
2. Solid brass bar
3. Torsion device
4. Load hanger
5. Measuring tape

Methodology and Procedure

1. Measure the dimensions of the brass bars.
2. Place the brass bar in the torsion device.
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3. Attach the load to the load hanger.
4. Record the angle of twist as the load is applied.
5. Gradually increase the load and repeat the experiment to obtain multiple data points.

Data and Calculations

The following formulas are used for calculations:

T = F × R

Φ = (θ * L) / (G * J)

G = (T * L) / (Φ * J)

Where:

• T is the torque (N.mm)
• Φ is the angle of twist (radians)
• G is the shear modulus (MPa)
• J is the polar moment of inertia (m^4)
• L is the length of the bar (m)

The calculated polar moment of inertia for the hollow brass bar is J = 0.

08 m^4, and for the solid brass bar, J = 0.034 m^4.

Sample of Calculations

Here is a sample of the calculations performed:

For Hollow Brass Bar:

Using the formula: Φ = (θ * L) / (G * J)

For Load = 5 N, Torque = 550 N.mm, Polar Moment of Inertia (J) = 0.08 m^4

Angle of Twist (Φ) = (θ * 0.08 m^4) / (20220.58 MPa * 0.08 m^4) = θ / 20220.58 radians

Discussion

Torsion induces shear stress on the cross-section of a shaft, causing it to twist. The angle of twist, Φ, is directly related to shear strain, and as the angle of twist increases, so does the shear strain. The polar moment of inertia, J, is determined by the cross-sectional dimensions of the shaft and affects the calculation of shear modulus.

It is important to note that the length of the shaft does not affect the polar moment of inertia; rather, it depends on the diameter of the shaft. Additionally, predicting the exact point of failure due to torsion is challenging as it typically occurs at points with pre-existing cracks or weaknesses in the material.

Conclusions and Recommendations

In this experiment, we determined the shear modulus for both hollow and solid brass bars. However, it is essential to acknowledge that experimental errors may occur, as the apparatus may have experienced wear and tear over time, potentially affecting the results. To improve the accuracy of future experiments, it is recommended to regularly inspect and maintain the equipment used for torsion testing.