sm8 128

_______________________________________________________________________
PROBLEM (*8.128) The strain gage readings from a 60o rosette mounted at a point A on a loaded
clamp (Fig. P8.128) are as follows:
a
ε
= 920
μ
, b
ε
= 360
μ
, c
ε
= -80
μ
Determine the principal strains. Show results on a properly oriented deformed element.
*SOLUTION
920 ( 0 ), 360 ( 60 ), 80 ( 120 )
oo
aabb cc
εμθ εμθ εμθ ====−=−=−
o
a
Therefore
22
cos sin sin cos
ax ay axy a
ε
εθεθγθ =++ θ
10 ) cos 0 sin 0 sin0 cos0
ooo
xyxy
εεγ
−=++ 920
x
920( ,
62 2 o
ε
μ =
b
Similarly,
22
cos sin sin cos
bx by bxy b
ε
εθεθγθ =++ θ
o 62 2
360(10 ) cos ( 60 ) sin ( 60 ) sin( 60 )cos( 60 )
ooo
xyxy
εεγ
−=−+−+− −
6 360(10 ) 0.25 0.75 0.443
x
yxy
ε
ε
−=+−
γ
c
(1)
and,
22
cos sin sin cos
cx cy cxy c
ε
εθεθγθ =++ θ
o
−+−+− − −=
62 2
80(10 ) cos ( 120 ) sin ( 120 ) sin( 120 )cos( 120 )
ooo
xyxy
εεγ
−
6 80(10 ) 0.25 0.75 0.443
x
yxy
ε
ε
−
γ ++−= (2)
Subtract Eq. (2) from Eq. (1):
440 0.833
x
y
μ
γ =−
So
528.2 105
xy y
γ
με =− =− μ
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or
instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is unlawful.
1(
2
avg x y )
ε
εε =+
1(920 105) 407.5
2
μ
=− =
1
2
22
528.2
[(920 407.5) ( ) ]
2
R=− +−
576.6
μ
=
1,2 avg
R
ε
ε =±
1407.5 576.6 984
ε
μ =+=
2169
ε
μ =−
γ
(μ)
2
ε
(
μ
)
5407. 2'
O
C
p
θ
920
264.1
A
x
R
y
Continued on next slide

Continued on next slide
264.1
tan2 ' , ' 13.63
920 407.5 o
pp
θθ ==
−
A
984
μ
13.63o
x’
x
y
’
y
169
μ
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or
instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is unlawful.