Page 228 - Introduction to Continuum Mechanics

Stress 213

(i) On the plane of maximum shearing stress, the normal stress is zero.
(ii) A plane with its normal in the direction of ej + 2e 2 - 2e$ has a stress vector
t =50ex + 100e2 - 100e3 MPa. It is a principal plane.
4.19. Why can the following two matrices not represent the same stress tensor?

4.20. Given a

(a) Find the magnitude of shearing stress on the plane whose normal is in the direction of
«i + e 2-
(b) Find the maximum and minimum normal stresses and the planes on which they act.
(c) Find the maximum shearing stress and the plane on which it acts.
4.21. The stress components at a point are given by
T u = lOOMPa, T 22 = 300 MPa, T 33 = 400 MPa, T 12 = T 13 = T 23 = 0
(a) Find the maximum shearing stress and the planes on which it acts.

(b) Find the normal stress on these planes.
(c) Are there any plane/planes on which the normal stress is 500 MPa?
4.22. The principal values of a stress tensor T are: Tj = 10 MPa , T 2 = -10 MPa and
?3 = 30 MPa. If the matrix of the stress is given by

find the value of TU and 733.

4.23. If the state of stress at a point is

find (a) the magnitude of the shearing stress on the plane whose normal is in the direction of
2&i + 2e 2 + e 3 , and (b)the maximum shearing stress.

4.24. Given