PROBLEM 7.75
For the state of stress shown, determine the value of Wxy for which the
maximum shearing stress is 80 MPa.
SOLUTION
Vx
1
(V x V y )
2
V ave
Vx Vy
120 70
2
2
Assume
V min
0
V max
2W max
Va
R
V max
Vy
120 MPa
70 MPa
95 MPa
25 MPa
160 MPa
V ave R
V max V ave
160 95
65 MPa
2
R2
2
W xy
§Vx Vy ·
2
¨
¸ W xy
2
©
¹
§Vx Vy ·
R ¨
¸
2
©
¹
2
2
652 252
602
W xy
Vb
V a 2R
160 130
30 MPa t 0
r60.0 MPa W
(o.k.)
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1110
PROBLEM 7.81
The state of plane stress shown occurs in a machine component made of a steel with
V Y 325 MPa. Using the maximum-distortion-energy criterion, determine whether
yield will occur when (a) V 0 200 MPa, (b) V 0 240 MPa, (c) V 0 280 MPa. If
yield does not occur, determine the corresponding factor of safety.
SOLUTION
§ Vx V y ·
2
¨
¸ W xy
2
©
¹
2
V ave V 0
(a)
V0
200 MPa
R
V ave 200 MPa
Va
V ave R
100 MPa,
V a2 V b2 V aV b
F . S.
(b)
V0
240 MPa
F . S.
280 MPa
V ave R
300 MPa
264.56 MPa < 325 MPa
(No yielding)
325
264.56
F . S . 1.228 W
V ave R
140 MPa,
V a2 V b2 V aV b
V0
Vb
V ave 240 MPa
Va
(c)
100 MPa
Vb
V ave R
340 MPa
295.97 MPa < 325 MPa
(No yielding)
325
295.97
F . S . 1.098 W
V ave 280 MPa
Va
V a2 V b2 V aV b
V ave R
180 MPa,
329.24 MPa > 325 MPa
Vb
V ave R
380 MPa
(Yielding occurs) W
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1119
PRO
OBLEM 7.1
144
45
3
2
Deterrmine the strain H x , knowingg that the folloowing strains have been dettermined by
use of
o the rosette shown:
s
30
x
15
1
H1
4800P
T1
15q
T2
30q
T3
75q
H2
1220P
H3
800P
SOLUTION
H x cos
c 2 T1 H y sinn 2 T1 J xy sinn T1 cos T1
0.9330H x 0.06699H y 0.25J xy
H x coos2 T 2 H y sinn 2 T 2 J xy sin T 2 cos T 2
0.75H x 0.25H y 0..4330J xy
H x coos2 T3 H y sinn 2 T3 J xy sin T3 cos T3
0.06699H x 0.9330H y 0.25J xy
H1
4880P
(1)
H2
120
1 P
(2)
H3
800P
(3)
Solvving (1), (2), and
a (3) simultaaneously,
Hx
253P ,
Hy
307 P ,
J xy
8893P
Hx
253P W
PRO
OPRIETARY MAT
TERIAL. Copyrigght © 2015 McGrraw-Hill Educatioon. This is proprietary material soleely for authorizedd instructor use.
Not authorized
a
for salee or distribution inn any manner. Thiis document may not
n be copied, scannned, duplicated, forwarded, distribbuted, or posted
on a website,
w
in whole or part.
1187
PROBLEM 7.146
The rosette shown has been used to determine the following strains at a point
on the surface of a crane hook:
3
45
45
4
2
420 u 106 in./in. H 2
H1
45
1
45 u 106 in./in. H 4
165 u 106 in./in.
x
(a) What should be the reading of gage 3? (b) Determine the principal strains
and the maximum in-plane shearing strain.
SOLUTION
(a) Gages 2 and 4 are 90q apart.
H ave
H ave
1
(H 2 H 4 )
2
1
(45 u 106 165 u 106 )
2
60 u 106 in./in.
Gages 1 and 3 are also 90q apart.
H ave
H3
1
(H1 H 3 )
2
2H ave H1 (2)(60 u 106 ) 420 u 106
H3
(b) H x
J xy
H1
420 u 10 6 in./in. H y
2H 2 H1 H 3
H3
300 u 106 in./in. W
300 u 10 6 in./in.
(2)(45 u 106 ) 420 u 106 (300 u 106 )
210 u 106 in./in.
§ H x H y · § J xy ·
¨
¸ ¨
¸
© 2 ¹ © 2 ¹
2
R
2
2
§ 420 u 106 (300 u 106 ) · § 210 u 106
¨¨
¸¸ ¨¨
2
2
©
¹ ©
·
¸¸
¹
2
375 u 106 in./in.
Ha
H ave R 60 u 106 375 u 106
Hb
H ave R 60 u 106 375 u 106
J max (in-plane)
Ha
Hb
J max (in-plane)
2R
435 u 106 in./in. W
315 u 106 in./in. W
750 u 10 6 in./in. W
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
1189
PROBLEM 7.149
The strains determined by the use of the rosette attached as shown during the test of a
machine element are
3
75
2
x
75
1
H1
93.1 u 106 in./in.
H2
385 u 106 in./in.
H3
210 u 106 in./in.
Determine (a) the orientation and magnitude of the principal strains in the plane of the
rosette, (b) the maximum in-plane shearing strain.
SOLUTION
Use H xc
J xy
1
1
(H x H y ) (H x H y ) cos 2T sin 2T
2
2
2
where
and
From Eq. (2),
T
75q
for gage 1,
T
0
for gage 2,
T
75q
for gage 3.
H1
J xy
1
1
(H x H y ) (H x H y ) cos (150q) sin (150q)
2
2
2
(1)
H2
J xy
1
1
(H x H y ) (H x H y ) cos 0 sin 0
2
2
2
(2)
H3
J xy
1
1
(H x H y ) (H x H y ) cos (150q) sin (150q)
2
2
2
(3)
Hx
Hz
385 u 106 in./in.
Adding Eqs. (1) and (3),
H1 H 3
Hy
(H x H y ) (H x H y ) cos 150q
H x (1 cos 150q) H y (1 cos 150q)
H1 H 3 H x (1 cos 150q)
(1 cos 150q)
93.1 u 106 210 u 106 385 u 106 (1 cos 150q)
1 cos 150q
35.0 u 106 in./in.
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
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1192
PROBLEM 7.149 (Continued)
Subtracting Eq. (1) from Eq. (3),
H 3 H1 J xy sin 150q
H 3 H1
J xy
sin 150q
210 u 106 (93.1 u 106 )
sin 150q
606.2 u 106 in./in.
J xy
Hx H y
tan 2T p
606.2 u 106
385 u 106 35.0 u 106
§ H x H y · § J xy ·
¨
¸ ¨
¸
© 2 ¹ © 2 ¹
2
R
2
30.0q, Tb
120.0q W
2
§ 385 u 106 35.0 u 106
¨¨
2
©
J max (in-plane)
(a) T a
1
1
(H x H y )
(385 u 106 35.0 u 106 )
2
2
210 u 106 in./in.
H ave
(b)
1.732
2
· § 606.2 ·2
¸¸ ¨
¸
¹ © 2 ¹
Ha
H ave R 210 u 106 350.0 u 106
Hb
H ave R 210 u 106 350.0 u 106
R
350.0 u 106 in./in.
350.0 u 106
Ha
Hb
560 u 106 in./in. W
140.0 u 106 in./in. W
J max (in-plane)
700 u 10 6 in./in. W
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
1193