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No credit card required ELASTOSTATICS 123
5.2 WEIGHTED RESIDUAL STATEMENTS
Following
the
same ideas
as
developed
in the
previous chapters
we
can write
an
extended weighted residual equation taking into
consideration
the
equilibrium equations (5.2)
and the two
types
of
boundary conditions (5.5) and (5.6). This statement
is,
\(j£
+b
*)
u
*
άΩ
=
\
*-
)uidr+
\
(
"*-
u
"
)p
*
dr
°. '
Γΐ Γχ
(5.13)
where wjf, p? are the displacements and surface forces corresponding
to
the
weighting field:
Pt
=
nja% (5.14)
The strain-displacement relationship
(5.7) and the
constitutive
equations (5.8) are assumed to apply for both, the approximating and
the weighting fields.
We
can now
integrate (5.13)
by
parts which gives,
- I
a
jk
sf
k
dQ
+ I
M?dß=
- ί
Ρκ^άΓ-
\
ρ
4
ιι?<1Γ +
+ ί
k
-u
k
)pidr
(5.15)
Let
us
consider
for
generality that,
<^
=
<7;*
+ <
(5.16)
where
σ
ι
are the
total stresses and
σ° the
initial stresses.
Taking this into consideration
we can
write (5.15)
as
- I
σ),ε%άΩ- \σ%ε%άΩ+
\
b
k
utaQ
JQ JQ JQ
=
- ί
p
k
u*
k
ar-
\
p
k
u*
k
ar+
f
k
-u
k
)ptdr (5.17)
Jr
2
Jr
{
Jr
{
Integrating
by
parts again
the
first term
in
(5.17) and taking into

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