6.15 Numerical Examples
207
m = −1 +
R
Ah
[b 1 h + (R + c 1 )(b − b 1 )] ln
R + c 1
R − c
− (b − b 1 )h
But for rectangular section, c = c 1 , b = b 1 ,
Therefore
m = −1 +
R
Ah
[bh + (R + c)(0)] ln
R + c
R − c
− (0)
m = −1 +
6
8 × 4
[2 × 4 + (6 + 2)(0)] ln
6 + 2
6 − 2
Therefore m = 0.0397.
Now, stress at
A = σ A =
P
A
+
M
AR
1 +
y A
m(R + y A )
= −
2000
8
+
2000 × 6
8 × 6
1 +
(−2)
0.0397(6 − 2)
∴ σ A = −3148.6 kg/cm
2 (Compression)
Stress at
B = σ B =
P
A
+
M
AR
1 +
y B
m(R + y B )
=
−2000
8
+
2000 × 6
8 × 6
1 +
2
0.0397(6 + 2)
Therefore, σ B = +1574.31 kg/cm
2 (Tension).
To compute the stresses at C and D.
At section CD, the bending moment,
207
m = −1 +
R
Ah
[b 1 h + (R + c 1 )(b − b 1 )] ln
R + c 1
R − c
− (b − b 1 )h
But for rectangular section, c = c 1 , b = b 1 ,
Therefore
m = −1 +
R
Ah
[bh + (R + c)(0)] ln
R + c
R − c
− (0)
m = −1 +
6
8 × 4
[2 × 4 + (6 + 2)(0)] ln
6 + 2
6 − 2
Therefore m = 0.0397.
Now, stress at
A = σ A =
P
A
+
M
AR
1 +
y A
m(R + y A )
= −
2000
8
+
2000 × 6
8 × 6
1 +
(−2)
0.0397(6 − 2)
∴ σ A = −3148.6 kg/cm
2 (Compression)
Stress at
B = σ B =
P
A
+
M
AR
1 +
y B
m(R + y B )
=
−2000
8
+
2000 × 6
8 × 6
1 +
2
0.0397(6 + 2)
Therefore, σ B = +1574.31 kg/cm
2 (Tension).
To compute the stresses at C and D.
At section CD, the bending moment,