For the given stale of stress, determine (a) the orientation of the planes of maximum in-plane shearing stress, (b) the maximum in-plane shearing stress, (c) the corresponding normal stress.

Step: 1

(a)

Calculate the orientation of the planes of the maximum in-plane shearing stress using the following formula:

Here, normal stress in the x-direction is , normal stress in the y-direction is , and the shear stress is .

Substitute,, and in the above equation.

Calculate the second shear plane angle by using the following relation:

Therefore, the orientation of the planes of the maximum in-plane shearing stress are .

Step: 2

(b)

Calculate the maximum in plane shearing stress using the following formula:

Therefore, the maximum in-plane shearing stress is .

Step: 3

(c)

Calculate the normal stress on the maximum shear stress plane:

Note: the normal stress on the maximum shear stress plane is equal to the average of the two normal stresses.

Therefore, the normal stress on the maximum shear stress plane is .