Civil Engineering Reference
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into surface integrals. If we assume that the solid is subjected to a non-uniform
volumetric strain (caused for example by a temperature increase) given by
H
H
§
·
x
0
İ
¨
¸
(13.40)
¨
¸
0
©
¹
y
0
additional work will be done.
S
¦
(, )
PQ
xy
y
uQ
()
x
TPQ
(, )
Q
H
dy
x
uQ
()
dS
Q
y
0
dy
Q
¦
xy
PQ
(, )
x
TPQ
(, )
dx
Q
H
dx
P
P1
x
x
0
P
Load case 1
Load case 2
Figure 13.4
Application of the Betti theorem including the effect of initial strains
Referring to figure 13.4, the work done by the displacements/strains of load case 1 times
the forces/stresses of load case 2 is given by:
³
Wu
(
QT
P Qu
,
QT
P QdS Q
,
)
(
)
12
x
xx
y
xy
(13.41)
S
³³
H
dx
6
P Q dy
,
H
dy
6
P Q dx
,
x
0
xx
y
0
yx
where
are the stresses at
Q
due to a unit force in
x
direction at
P
. Here we assume that only volumetric initial strains are present, even though it is
obvious that shear strains could easily be included. The work done by the displacements
of load case 2 times the forces/stresses of load case 1 is the same as for the case where
no initial strains are applied.
6
P,
Q
and
6
P,
Q
xx
yx
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