106
ʹ͸ͶǤͲ͵ ൌ ߪଵ
͸ͶǤͲ͵
ͳ͵ͷǤͻ͹ ൌ ߪଶ
ߪଷpsi
psi
psi
28. ȂȂ͵
107
psi
psi
psi
psi
ߪଷൌ െͺ͸͹ǤͲ͸
ߪଷ=ߪଵ
െ͵Ͳ͹Ǥͻ ൌ ߪଶ
279.56kPa
Ͷ͵͵Ǥͷ͵ ൌ ߬௠௔௫
29. Both principal stresses are compressive – same sign – 3D Solution.
108
150.02kPa
168.76kPa
psi
24.48psi
ߪଶൌെ͵͹ǤͶͺ
XͲAxis
30. Both principal stresses are compressive – same sign – 3D Solution
ߪ௬ൌͲ
ߪ௫ൌ ʹͲͷ͵ͺ 
ߪଶ20538
ܺ
-Axis
ʹ߶ఛ
ߪଶ
ߪଵ
͸ͻǤͻͺι
ߪ௔௩௚
31. ܦ ൌ ͲǤͷͲͲǤ Stress element on bottom.
From Figure 3-20: ܯ஻ൌ ʹͷʹήǢܶ஻ൌ ͵ͲͲή
ଷȀ͵ʹ ൌ ߨሺͲǤͷͲ݅݊ሻଷȀ͵ʹ ൌ ͲǤͲͳʹ͹͹ଷ
ߪଵൌ ͳͲʹ͸ͻ൅ͳͷͻ͸͸ ൌ ʹ͸ʹ͵ͷ
߶ఛൌ ͸ͻǤͻͺι
V1 = 26 235 psi
110
ɒൌͳͷͻ͸͸
ɒ
ɐʹʹͲͷ͵ͺ
ͳͲʹ͸ͻ ɐͳ
ɐ
ɒൌͳʹʹʹͷ
ɐ
ʹ
ɒ
ʹ
Ԅ
ɐ
Ͳ
V
1
V
2
V
3
31. 3D Mohr’s Circle Solution
3D Principal Stresses:
V
1
26235
p
s
i
3
111
߬஻
ߪଶ
߬௠௔௫ ൌ ͳͳͺ͹Ͳ
ߪ௔௩ ൌ ͸ͻͲͲ
ߪ௔௩.
ߪଶߪଵ
߬
32. Use ܦ ൌ ͳǤͷͲͲ. Stress element on bottom.
From solution for Problem P3-48, ܯ஻ൌ Ͷͷ͹ʹήǢܶ஻ൌ ͸ͶͲͲή
ܼൌߨܦ
ଷȀ͵ʹ ൌ ͲǤ͵͵ͳ͵ଷǢߪ஻ൌܯ
஻Ȁܼ ൌ Ͷͷ͹ʹήȀͲǤ͵͵ͳ͵ଷൌ ͳ͵ͺͲͲ ൌ ߪ௫
ଷȀͳ͸ ൌ ʹܼ ൌ ͲǤ͸͸ʹ͹ଷǢ߬஻ൌܶ
ʹ߶ఙൌݐܽ݊
଺ଽ଴଴ቁ ൌ ͷͶǤͶ͸ιǢ߶ఙൌ ʹ͹Ǥʹι
ʹ߶ఛൌʹ߶
ఙ൅ͻͲι ൌ ͳͶͶǤͶ͸ιǢ߶ఛൌ ͹ʹǤʹι
112
32. 3D Mohr’s Circle Solution
3D Principal Stresses:
V
1
18770
p
s
i
V
2
0 psi
V
3
4970 psi
18770 psi
-4970 psi 0 psi V
W
113
ߪଶ
ʹ߶ఛ
ߪଶߪଵߪ
3867
ߪ௔௩.
߬
33. ܦ ൌ ʹǤʹͷǤǣǤ 
 ͵ǦͶͻǡܯ஻ൌ ͺ͸ͶͲήǢܶ஻ൌ ͶͲͲή
ܼൌߨܦ
ଷȀ͵ʹ ൌ ͳǤͳͳͺଷǢߪ஻ൌ ܯȀܼ ൌ ሺͺ͸ͶͲήሻȀͳǤͳͳͺଷൌ ͹͹ʹ͸
ଷȀͳ͸ ൌ ʹܼ ൌ ʹǤʹ͵͹ଷǢ߬஻ൌܶ
ʹ߶ఙൌݐܽ݊
߶ఙൌ ͳǤ͵͵ι
ʹ߶ఛൌ ͻͲιെʹ߶ఙൌ ͺ͹Ǥ͵ͷι
X-axis
33. 3D Mohr’s Circle Solution
3D Principal Stresses:
V
p
si
115
ߪ
26.48
ʹ߶
ఙ
ߪ
஺௏
ߙ
15.28
ݔ
߬
௠௔௫
34. Use ܦ ൌ ͷͲǤͲ for shaft A-B: Stress element on bottom.
From solution for Problem 3-50, ܯ
஺
ൌ െͲǤ͸ͷͲήǢܶ
஺
ൌ ͲǤ͵͹ͷή
ܼൌߨܦ
ଷ
Ȁ͵ʹ ൌ ͳʹʹ͹ʹ
ଷ
଺ହ଴୒ή୫
ଵ଴
య
୫୫
ʹ߶
ఙ
ൌ ͳͺͲιെߙ ൌ ͳͷͲι
ʹ߶
ఛ
ൌ ͻͲι൅ߙ ൌ ͳʹͲι
116
34. 3D Mohr’s Circle Solution
3D Principal Stresses:
V
1
4.09 MP
a
ܶ ൌ ʹͲͲͲͲή
ܨ ൌ ͹ͷͲͲͲ
ݕ
߬௠௔௫
ܶ
ߪଶ
ߪଵ߶ఙߪ௔௩௚
߬௠௔௫ ߶ఛݔ
35. ܦ ൌ ͶǤͲͲǢܣ ൌ ߨܦଶȀͶ ൌ ͳʹǤͷ͹ଶǢܼ௣ൌߨܦ
ଷȀͳ͸ ൌ ͳʹǤͷ͹ଷ
ߪൌିி
஺ൌି଻ହ଴଴଴୪ୠ
ଵଶǤହ଻୧୬మൌ െͷͻ͸ͺ
߶ఛൌ ͵ͲǤͻ͸ι
ߪ௔௩
௚
118
35. 3D Mohr’s Circle Solution
3D Principal Stresses:
V
1
398
p
s
i
V
2
0 psi
V
3
6366 psi
398 psi
-6366 psi 0 psi
V
W
ܶ ൌ ͶͷͲή
36. ܦ ൌ ʹͲǢܣ ൌ
గ஽
మ
ସ
ൌ ͵ͳͶ
ଶ
120
36. 3D Mohr’s Circle Solution
3D Principal Stresses:
V
1
349.5 MP
a