Principal Stresses in a Circular Bar

The principal stresses in a circular bar occur when the material is subjected to an axial force and shear force. The principal stresses are the maximum and minimum normal stresses experienced by the material at a specific point. The difference between the two principal stresses is known as the stress range.

1. Definition of Principal Stresses:
Principal stresses are the normal stresses acting on planes that are perpendicular to each other at a specific point in a material. These stresses represent the maximum and minimum values of stress experienced by the material.

2. Axial Force:
Axial force is the force acting along the axis of the circular bar. It causes a uniform stress distribution across the cross-section of the bar, resulting in a normal stress. The axial stress can be calculated using the formula: σ_axial = P/A, where P is the axial force and A is the cross-sectional area of the bar.

3. Shear Force:
Shear force is the force acting perpendicular to the axis of the circular bar. It causes a non-uniform stress distribution across the cross-section of the bar, resulting in a shear stress. The shear stress can be calculated using the formula: τ_shear = VQ/Ib, where V is the shear force, Q is the first moment of area, I is the moment of inertia, and b is the width of the bar.

4. Principal Stresses:
The principal stresses in a circular bar can be determined by considering the combined effect of axial force and shear force. The principal stresses occur at angles of ±45 degrees to the axis of the bar. The formulas for the principal stresses are given as: σ_1 = (σ_axial + τ_shear) / 2 and σ_2 = (σ_axial - τ_shear) / 2.

5. Difference between Principal Stresses:
The difference between the two principal stresses is known as the stress range. In this case, the stress range is given as 120 MPa. Mathematically, it can be expressed as: σ_1 - σ_2 = 120 MPa. By substituting the formulas for σ_1 and σ_2, we can solve for the axial stress and shear stress.

6. Calculation of Axial Stress and Shear Stress:
Using the equations σ_1 = (σ_axial + τ_shear) / 2 and σ_2 = (σ_axial - τ_shear) / 2, we can rearrange the equations to solve for σ_axial and τ_shear. By substituting these values into the equation σ_1 - σ_2 = 120 MPa, we can solve for the unknowns.

7. Conclusion:
In conclusion, the difference between two principal stresses in a circular bar can be determined by considering the combined effect of axial force and shear force. By solving the equations for the principal stresses and using the given stress range, the axial stress and shear stress can be calculated. These values provide insight into the stress distribution and behavior of the circular bar under the applied loads.