Civil Engineering Reference
In-Depth Information
Assuming that the cross-section of the
b
olt is small and that the initial stress does not
vary across it, the initial stress at point
Q
is only a function of [ i.e.
2(3)
¦
c
V[
(( )
Q
N
()
[ V
(18.24)
0
x
i
0
xn
n
1
Substitution into (18.23) gives
2(3)
C
¦¦
E
³
c
(18.25)
R
PQ
,
N
V
dV
i
i
n
0
xn
c
c
1
n
1
V
c
or
CN
¦¦
E
cc
R
'
V
(18.26)
i
ni
0
xn
cn
11
where
'
EE
c
ni
(,)
PQNdV
(18.27)
i
n
c
V
c
If
n
is not node
i
then (18.27) may be evaluated using Gauss Quadrature
K
¦
E
c
ni
(18.28)
'
E
A
PQ
,
(
[
)
N
[
JW
i
j
n
j
j
k
1
where A is the cross-sectional area of the inclusion, which is assumed constant along the
cell here and
K
is the number of Gauss points required.
If
n
is node
i
then the Kernel becomes singular. In this case an analytical integration can
be carried out
2
ª
º
1
³³
«
»
'
E
E
(,)
PQ dAN
()
J
[ [
(18.29)
ni
i
n
«
»
¬
¼
1
A
The increment in displacement due to the body force
^
R
is computed by
> @
^` ^`
7
uR
(18.30)
The increment in displacement
u
at a cell node
P
due to an increment in displacement
(computed by equation 18.30) is computed for the subsequent iteration steps as
E
N
C
N
¦¦
¦¦
E
ee
c c
(18.31)
u
P
'
T
u
'
V
a
n
n
n
0
xn
en
11
cn
11
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