Torsion

146 CHAPTER 5. TORSION
The predictions at point A are
a ′ b ′2 |τA1|
M1
= 9 b
2
′ 9 b
=
a ′ 2 a ;
a ′ b ′2 |τA2|
M1
= 3
5
2 2
tanh γ
, (5.87)
1 − (tanh γ)/γ
where τA1 and τA2 are the stress components predicted by the crude and refined solutions, respectively.
Fig. 5.20 shows the predictions of the crude and refined solutions, together with the exact solution. Here
again, good agreement is found between the refined and exact solutions. Comparing the results at points
A and B, it is clear that the maximum shear stress component occurs in the middle of the long side of the
rectangular section, i.e. at point B if a > b.
NONDIMENSIONAL SHEAR STRESS AT A
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
1 2 3 4 5 6 7 8 9 10 11 12
a/b
Figure 5.20: Nondimensional shear stress at point A versus aspect ratio a/b for the crude (◦), refined (△)
and exact (▽) solutions.
5.3.5 Problems
Problem 5.2
b
i 3
B
b
r
Figure 5.21: Circular shaft with a keyway.
Consider a circular shaft of radius a with a semi-circular keyway of radius b, as depicted in fig. 5.21. The shaft
is subjected to torsion.
(1) Find the stress function for this problem starting from
a
a
Φ = A x 2 2 + x 2 3 − 2ax2 1 − b2
x 2 2 + x2 3
A
r
i 2
.