8 Mohar Circle Torsion and Axial PDF
8 Mohar Circle Torsion and Axial PDF
8 Mohar Circle Torsion and Axial PDF
Mohr’s Circle
max
Design Criteria, allow , allow (0, )
2 1
2 1
1
2 s
2
max
1 2 1 max
Stress Trajectories
Tc
max
J
2
2 1
1
Tc
1
J
Torsional Failure Modes
Tc Tc
max 1
J J
Tc Tc
max 1
J J
My VQ
I Ib
max
(0, )
2 1
Mohr’s Circle
( , )
Axial + Torsional Loading
Problem 971: The solid shaft in small hydraulic turbine is 100mm in diameter and supports an
axial compressive load of 140π kN. Determine the maximum power that can be developed at
4 Hz without exceeding a maximum shearing stress of 70 MPa or Maximum Normal Stress of
90 MPa
D2 (0.1) 2
A 2.5 103 m 2
4 4
D4 (0.1)4
J 3.125 106 m4
32 32
D 100 mm
70 MPa
90 MPa P 140 kN
P (140000 )
5.6 10 7
N/m 2
56 MPa
A 2.5 10
3
Tc T (0.05) 1.6 106 1.6T
T( ) N/m 2
MPa
J 3.125 106
Mohr’s Circle
56 MPa max
(0, 1.6 T )
1.6T
MPa
C 28 MPa 2 (28,0) 1
C
1.6T
max R (28) 2 ( ) 2 70 MPa
(56, 1.6 T )
1.6T
2 C R 28 (28) 2 ( ) 2 90 MPa
P 2 f T
1.6T 2
(28) 2 ( ) 62 MPa
P 2 (4)(1086)
D2 (4) 2
A 12.57 in 2
4 4
D4 (4) 4
J 25.13 in 4
300 32 32
P (36) P 36 kips
2.86 Ksi
A 12.57
T 2.5 kip-ft
Tc T (2) (2.4 12)2
2.29 Ksi
J 25.13 25.13
2.86 0 2.86 0
30 cos 2(30) 2.29sin 2(30) 0.16 Ksi 161 Psi
2 2
2.86 (0)
xy sin 2(30) 2.29cos 2(30) 2.38 Ksi = 2383 Psi
2
Bending + Torsional Loading
D4 (0.1)4
I 1.5625 106 m4
64 64
D4 (0.1)4
J 3.125 106 m4
32 32
D 100 mm
80 MPa
100 MPa
M 2500 N.m
Mc (2500 )(0.05)
8 10 7
N/m 2
80 MPa
I 1.5625 10
6
P 2 f T
1.6T 2
402 ( ) 60 MPa
P 2 (30)(87.81)
Mc 4M
3
I r
Tc 2T
3
I r
4M
r3
Mohr’s Circle
max
(0, 2T
r3
)
2T
r3
2 (40,0) 1
C
( 4 Mr3 , 2rT3 )
C 2M /( r 3 )
2M 2 2T 2 2
max R ( ) ( ) M 2
T 2
r3 r3 r3
1 C R
2
r3
M M 2
T 2
900 12 max 10 ksi
If T 900 lb-ft 10.8 kips-in
1000
600 12 max 16 ksi
M 600 lb-ft 7.2 kips-in
1000
2
max M 2
T 2
r3
2 8.263
3 7.22 10.82 3 ksi 10 ksi r 0.938 in.
r r
1
2
r3
M M 2
T 2
2
r
3 7.2 7.2 10.8
2 2
12.847
r 3
ksi 16 ksi r 0.929 in.
750 N.m
2500 N
2500 N
3750 N
4000 N
750 N.m
750 N.m 2875 N 2500 N
2500 N
1250 N 3750 N
1500 N.m
4000 N
1250 N
1500 N.m
3625 N
750 N.m
2500 N 4000 N 2500 N
2500 N 750 N.m
3750 N 750 N.m
4000 N 1500 N.m
2875 N.m
3625 N
BMzD
2500 N
750 N.m 750 N.m 1500 N.m
1500 N.m 750 N.m
1250 N 4m 3750 N 2m TMD
BMyD
1250 N.m
3750 N.m
5000 N.m
3625 N.m
750 N.m
2500 N E 2875 N.m
2500 N D
3750 N
BMzD
4000 N C
1250 N A BMyD
1500 N.m
1250 N.m
| M | M z2 M y2
4725.2 N.m 5000 N.m
My
3834.5 N.m
Mz
|M|
A B C D E
TMD
From Prob. 951 and this problem. 4M
3625 N.m Mohr’s Circle
2 2 r3
max 3 M T 2 70 MPa
r (0, )
max
2875 N.m
2
r 2
1 3 M M T 2 120 MPa 2T
r3 2 1
BMzD
At section C
BMyD
2
max 4725.2 2
1500 2
1000 mm 70 MPa 1250 N.m
r 3
( , )
r 35.6 mm 3750 N.m
5000 N.m
1
2
r 3
4725.2 4725.22 15002 1000 120 MPa
4725.2 N.m 5000 N.m
r 37.2 mm
3834.5 N.m
At section D
|M|
2 A B C D E
max 50002 7502 1000 mm 70 MPa
r 3
r 35.8 mm
1500 N.m
750 N.m
2
1 3 5000 50002 7502 1000 120 MPa
r TMD
r 37.7 mm r ≥ 37.7 mm
state of stress on
the element on the
surface of vessel
1 67.5 R
2 67.5 R
x y
2
R2 xy
2
2
V 30 kN 250 mm
P My 40 7500 20
A I 20 120 2.88 106
P 40 kN
68.75 MPa
C (C , 0) (
x y
, 0) Mohr’s Circle at point A
2
68.75 0
(
2
, 0) (34.375, 0) max (68.75,16.67)
x y
R ( ) 2 xy2
2
(
68.75 2
) 16.67 2 38.20 MPa
2 C 2 1
2 (34.375,0)
1 , 2 C R 34.375 38.20
72.578, 3.825 MPa (0,16.67)
72.58 3.83
xy
16.67
sin 2 12.94
R 38.20
12.94O 3.83 72.58
20 mm 20 mm
120 mm N.A.
B
40 mm
20 1203 Q (20 40) 40
I =2.88 106 mm4
12 =3.2 104 mm3
V 30 kN 300 mm
P My 40 9000 (20)
A I 20 120 2.88 106
P 40 kN
45.83 MPa
45.83 MPa
P 40 kN
16.67 MPa
M 9000 kN.mm
45.83 0 45.83 0
30 cos 2(30) 16.67sin 2(30) 48.8 MPa
2 2
45.83 (0)
xy sin 2(30) 16.67 cos 2(30) 11.5 MPa
2 Mohr’s Circle at point B
11.51 MPa
48.81 MPa
(45.83,16.67)
L1
L2
1.2D L3
L4
1.2D
D
Also find the maximum shearing stress at point A. Show your results on a
complete sketch of a differential element.
P
L