**Principle Stresses and Factor of Safety**

Principle stresses are the maximum and minimum normal stresses acting on a point in a material. These stresses determine the strength of the material and can be used to calculate the factor of safety against failure. The factor of safety is a measure of how much stronger a structure is compared to the maximum load it is expected to bear.

**Given Information**

In this case, the principle stresses are as follows:
- Maximum (tensile) stress = 100 MPa
- Minimum (tensile) stress = 50 MPa
- Minimum (compressive) stress = -25 MPa

The elastic limit in simple tension is given as 220 MPa and the Poisson's ratio is 0.03.

**Factor of Safety Calculation**

To calculate the factor of safety against failure, we will use three different theories: maximum normal stress theory, maximum shear stress theory, and maximum distortion energy theory.

1. **Maximum Normal Stress Theory**

According to this theory, failure occurs when the maximum stress reaches the yield strength of the material. In this case, the yield strength is given as 220 MPa.

The factor of safety based on maximum normal stress theory can be calculated as:
Factor of Safety = Yield Strength / Maximum Stress

For tensile stresses, the maximum stress is 100 MPa. Therefore,
Factor of Safety = 220 MPa / 100 MPa = 2.2

For compressive stresses, the maximum stress is 25 MPa. Therefore,
Factor of Safety = 220 MPa / 25 MPa = 8.8

2. **Maximum Shear Stress Theory**

According to this theory, failure occurs when the maximum shear stress reaches the yield strength divided by the factor of safety. The maximum shear stress can be calculated using the formula:

Maximum Shear Stress = (Maximum Stress - Minimum Stress) / 2

For the given principle stresses, the maximum shear stress can be calculated as:
Maximum Shear Stress = (100 MPa - (-25 MPa)) / 2 = 62.5 MPa

The factor of safety based on maximum shear stress theory can be calculated as:
Factor of Safety = Yield Strength / Maximum Shear Stress

Factor of Safety = 220 MPa / 62.5 MPa = 3.52

3. **Maximum Distortion Energy Theory**

According to this theory, failure occurs when the distortion energy per unit volume reaches the yield strength divided by the factor of safety. The distortion energy per unit volume can be calculated using the formula:

Distortion Energy per Unit Volume = (1/2) * (Maximum Stress^2 + Minimum Stress^2 - 2 * Poisson's Ratio * Maximum Stress * Minimum Stress)

For the given principle stresses and Poisson's ratio, the distortion energy per unit volume can be calculated as:
Distortion Energy per Unit Volume = (1/2) * (100^2 + 50^2 - 2 * 0.03 * 100 * 50) = 7125 MPa

The factor of safety based on maximum distortion energy theory can be calculated as:
Factor of Safety = Yield Strength / Distortion Energy per Unit Volume

Factor of Safety = 220 MPa / 7125 MPa = 0.0309

**Conclusion**

Based on the three theories, the factor of safety against failure is found to be:
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