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

=

\

(ρ

*-p«

)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 (ü

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

(ü

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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