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5 Two-Dimensional Problems in Cartesian Co-ordinate System
Fig. 5.13 Variation of τ xy
At y = + h, τ xy =
6Fh
h 2
−
6F
h 3
h
2
= 0
At y = y, τ xy = −
6F
h 2
h −
6F
h 3
(−h
2
) = −
12F
h
The variation of τ xy is shown in Fig. 5.13.
Example 5.4
Investigate what problem of plane stress is satisfied by the stress function
φ =
3F
4h
x y −
x y
3
3h 2
+
p
2
y
2
applied to the region included in y = 0, y = h, x = 0 on the side x positive.
Solution The given stress function may be written as
φ =
3F
4h
x y −
1
4
F x y
3
h 3
+
p
2
y
2
Now
∂
2
φ
∂ x 2 = 0
∂
2
φ
∂ y 2 = −
3 × 2
4
.
F x y
h 3
+
2 p
2
= p −
1.5
F
h 3
x y
and
∂
2
φ
∂ x∂ y
=
3F
4h
−
3
4
F y
2
h 3
5 Two-Dimensional Problems in Cartesian Co-ordinate System
Fig. 5.13 Variation of τ xy
At y = + h, τ xy =
6Fh
h 2
−
6F
h 3
h
2
= 0
At y = y, τ xy = −
6F
h 2
h −
6F
h 3
(−h
2
) = −
12F
h
The variation of τ xy is shown in Fig. 5.13.
Example 5.4
Investigate what problem of plane stress is satisfied by the stress function
φ =
3F
4h
x y −
x y
3
3h 2
+
p
2
y
2
applied to the region included in y = 0, y = h, x = 0 on the side x positive.
Solution The given stress function may be written as
φ =
3F
4h
x y −
1
4
F x y
3
h 3
+
p
2
y
2
Now
∂
2
φ
∂ x 2 = 0
∂
2
φ
∂ y 2 = −
3 × 2
4
.
F x y
h 3
+
2 p
2
= p −
1.5
F
h 3
x y
and
∂
2
φ
∂ x∂ y
=
3F
4h
−
3
4
F y
2
h 3