The shear stress in a plane normal of which makes an angle of 30 degree with the direction of maximum principal stress will be ?
To determine the shear stress in a plane that is at an angle of 30 degrees with the direction of the maximum principal stress, you can use Mohr's circle or the stress transformation equations.
1. Using Mohr's Circle:
- Draw a Mohr's circle and label the center as the average stress (σavg) and the diameter as the difference between the maximum (σmax) and minimum (σmin) principal stresses.
- Determine the principal stresses (σ1 and σ2) by considering the given maximum principal stress direction.
- Locate the plane of interest on the circle at an angle of 30 degrees.
- The shear stress (τ) is given by the horizontal distance between the circle and the plane.
2. Using Stress Transformation Equations:
- The stress transformation equations relate the normal stress (σ) and the shear stress (τ) on a plane inclined at an angle (θ) to the direction of the maximum principal stress.
- Assuming the components of the state of stress are σx, σy, and τxy, the transformation equations for the shear stress (τ') and normal stress (σ') on a plane inclined at an angle (θ') are:
τ' = (σx - σy) / 2 * sin(2θ') + τxy * cos(2θ')
σ' = (σx + σy) / 2 + (σx - σy) / 2 * cos(2θ') - τxy * sin(2θ')
- Substitute the known values of σx, σy, and τxy according to the given problem, and θ' is the angle of the plane (30 degrees in this case).
- Solve for the shear stress (τ') to get the answer.
These methods should help you determine the shear stress in the plane at an angle of 30 degrees with the direction of the maximum principal stress.