2.22 General State of Stress in Three Dimensions in Cylindrical …
33
Fig. 2.17 Stresses acting on
the three-dimensional
element
∂σ r
∂r
+
1
r
∂τ r θ
∂θ
+
∂τ zr
∂z
σ r − σ θ
r
= 0
(2.48)
∂τ r θ
∂r
+
1
r
∂σ θ
∂θ
+
∂τ θ z
∂z
2τ r θ
r
= 0
(2.49)
∂τ zr
∂r
+
1
r
∂τ θ z
∂θ
+
∂σ z
∂z
+
τ zr
r
= 0
(2.50)
2.23 Numerical Examples
Example 2.1 When the stress tensor at a point with reference to axes (x, y, z) is given
by the array,
⎡
⎣
4 1 2
1 6 0
2 0 8
⎤
⎦ MPa
show that by transformation of the axes by 45° about the z-axis, the stress invariants
remain unchanged.
33
Fig. 2.17 Stresses acting on
the three-dimensional
element
∂σ r
∂r
+
1
r
∂τ r θ
∂θ
+
∂τ zr
∂z
σ r − σ θ
r
= 0
(2.48)
∂τ r θ
∂r
+
1
r
∂σ θ
∂θ
+
∂τ θ z
∂z
2τ r θ
r
= 0
(2.49)
∂τ zr
∂r
+
1
r
∂τ θ z
∂θ
+
∂σ z
∂z
+
τ zr
r
= 0
(2.50)
2.23 Numerical Examples
Example 2.1 When the stress tensor at a point with reference to axes (x, y, z) is given
by the array,
⎡
⎣
4 1 2
1 6 0
2 0 8
⎤
⎦ MPa
show that by transformation of the axes by 45° about the z-axis, the stress invariants
remain unchanged.