sm12 27

________________________________________________________________________
PROBLEM (12.27) At a point in a structure subjected to plane stress, the stresses have the
magnitudes and direction shown acting on an element in Fig. P12.27. Determine the strain-energy density
and its components at the point.
Given: E = 200 GPa,
ν
= 1/3
SOLUTION
Refer to Fig. P12.27.
We have
9
200 10 75
2(1 1 3)
GG
×
==
+Pa
and
22
1,2 72 72
()(5)
22
σ
++
=± +
;
or
12
10.09 1.09
M
Pa MPa
σ
σ ==−
Equations(12.17), (12.19), and (12.20) become:
22
012120 1
11
(2) (
26
v
UU
EE
2
2
2
)
ν
σ
σνσσ σσ
−
=+− = +
222 22
0121212
11
[( ) ] ( )
12 6
d
UGG
12
σ
σσσ σσσσ =−++=+−
Substitute the given data:
12 22
09
10 1
[10.09 1.09 2( )(10.9 1.09)]
2(200 10 ) 3
U=++×
×
3
276 Jm=
12 2
09
12(13)10(10.09 1.09)
6(200 10 )
v
U−
=−
×
3
23 Jm=
12 22
09
10 [10.09 ( 1.09) 10.9 1.09]
6(75 10 )
d
U=+−+×
×
3
253 Jm=
Check:
3
00 0
276
vd
UU U Jm=+=
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