154
5 Two-Dimensional Problems in Cartesian Co-ordinate System
Condition (iii)
W
2
= −
+h
−h
τ xy dy
W
2
= −
+h
−h
−
b
2
y
2
− c
dy
W
2
=
+h
−h
b
2
y
2
+ c
dy
=
b
3
h
3
+ 2ch
Substituting for the value of c, we get
W
2
= −
2
3
bh
3
Hence,
b = −
3W
4h 3
and c = −
3W
8h
Condition (iv)
Now,
σ x = −
3W
4h 3
x y
Hence,
M =
+h
−h
−
3W
4h 3
x y
2 dy = 0
Therefore,
M = −
W x
2
5 Two-Dimensional Problems in Cartesian Co-ordinate System
Condition (iii)
W
2
= −
+h
−h
τ xy dy
W
2
= −
+h
−h
−
b
2
y
2
− c
dy
W
2
=
+h
−h
b
2
y
2
+ c
dy
=
b
3
h
3
+ 2ch
Substituting for the value of c, we get
W
2
= −
2
3
bh
3
Hence,
b = −
3W
4h 3
and c = −
3W
8h
Condition (iv)
Now,
σ x = −
3W
4h 3
x y
Hence,
M =
+h
−h
−
3W
4h 3
x y
2 dy = 0
Therefore,
M = −
W x
2