2.24 Exercises
53
14. Prove the following relationships
(i) (σ n ) oct =
1
3
(σ 1 + σ 2 + σ 3 )
(ii) τ
2
oct =
1
9
(σ 1 − σ 2 )
2
+ (σ 2 − σ 3 )
2
+ (σ 3 − σ 1 )
2
(iii) 9τ
2
oct = 2I
2
1 − 6I 2
15. The state of stress at a point in a body is given by
σ x = x
2 y + 20 τ xy = 3x
2 y
σ y = x
3 z + y
2
τ yz = yz
σ z = yz
2
+ 10 τ xz = xz
Determine the body forces distribution at the point (1, 2, 3) so that the stresses
are in equilibrium.
16. The state of stress at a point in a body is given with reference axes as
σ x = 200 N/m
2
τ xy = 100 N/m
2
σ y = 0
τ yz = 0
σ z = 500 N/m
2
τ xz = 0
if a new set of axis x
y
z
is formed by rotating x y z axes through 60° about
the z-axis in the anticlockwise direction, determine the components of stress for
the new axes. Also prove that the invariants remain unchanged.
17. The components of stress at a point are:
σ x = 10 kPa τ xy = 20 kPa
σ y = −20 kPa τ yz = 30 kPa
σ z = −20 kPa τ xz = 30 kPa
Determine
(a) The principal stresses at the point
(b) Deviatoric and spherical stress tensors.
18. The stress components at a point in cylindrical co-ordinates are:
σ r = r
3
θ + r τ r θ = r
2
θ
σ θ = r
2 z + θ
2
τ θ z = θ z + θ
2
σ z = r
2 z
2
+ θ z τ r z = r z
2