214
6 Two-Dimensional Problems in Elasticity …
W
A
B
W
2
W
2
R
M
N
M AB
M AB
The bending moment at any section MN can be determined by
M M N = −M AB +
W R
2
(1 − cos θ)
∴ At θ = 0,
M mn = −M AB
But
M AB =
W R
2
1 −
2
π
∴ M AB =
W × 100
2
1 −
2
π
= 18.17 W
Now,
σ A =
P
A
+
M A
AR
1 +
y A
m(R + y A )
=
W
2 A
+
M A
AR
1 +
y A
m(R + y A )
=
W
2 × 1500
+
(−18.17 W )
1500 × 100
1 +
(−25)
0.0217(100 − 25)
∴ σ A = 0.002073 W (Tensile)
and
σ B =
P
A
+
M A
AR
1 +
y B
m(R + y B )
=
w
2 × 1500
−
18.17W
1500 × 100
1 +
25
0.0217(100 + 25)
∴ σ B = −0.00090423W (Compression)
6 Two-Dimensional Problems in Elasticity …
W
A
B
W
2
W
2
R
M
N
M AB
M AB
The bending moment at any section MN can be determined by
M M N = −M AB +
W R
2
(1 − cos θ)
∴ At θ = 0,
M mn = −M AB
But
M AB =
W R
2
1 −
2
π
∴ M AB =
W × 100
2
1 −
2
π
= 18.17 W
Now,
σ A =
P
A
+
M A
AR
1 +
y A
m(R + y A )
=
W
2 A
+
M A
AR
1 +
y A
m(R + y A )
=
W
2 × 1500
+
(−18.17 W )
1500 × 100
1 +
(−25)
0.0217(100 − 25)
∴ σ A = 0.002073 W (Tensile)
and
σ B =
P
A
+
M A
AR
1 +
y B
m(R + y B )
=
w
2 × 1500
−
18.17W
1500 × 100
1 +
25
0.0217(100 + 25)
∴ σ B = −0.00090423W (Compression)