Civil Engineering Reference
In-Depth Information
(3.53)
V
V
[
V
u
3
K
that is
-½-½-
w
x
w
x
ww ww
y
z
y
z
½
°°°°°
°
w
[
w
K
w w
[ K K [
w w
°°°°°
°
°°°°°
°
w
y
w
y
ww ww
zx
zx
V
u
®¾®¾®
¾
3
(3.54)
w
[
w
K
w w
[ K K [
w w
°°°°°
°
°°°°°
°
w
z
w
z
w w
x
y
w w
x
y
°°°°°
°
w
[
w
K
w w
[ K K [
w w
¯¿¯¿¯
¿
As indicated previously the unit normal vector
v
3
is obtained by first computing the
length of the vector:
2
2
2
VVVVJ
(3.55)
3
3
x
3
y
3
z
This is also the real area of a segment of size 1x1 in the local coordinate system, or the
Jacobian
of the transformation. The normalised vector in the direction perpendicular to
the surface of the element is given by
1
V
v
V
(3.56)
3
3
3
vv
are not orthogonal to each other. An orthogonal
system of axes is required for the definition of strains and stresses needed later. Here we
assume that the first axis defined by vector
v
1
is in the direction of
v
[
. The second axis is
defined by:
It should be noted here that
[ K
vvv
u
(3.57)
2
1
3
The computation of the normal vector requires the derivatives of the shape functions.
These are computed by SUBROUTINE Serendip_deriv shown below.
SUBROUTINE
Serendip_deriv(DNi,xsi,eta,ldim,nodes,inci)
!---------------------------------
!
Computes Derivatives of Serendipity shape functions
!
for one and two-dimensional (linear/parabolic)
!
finite boundary elements
!---------------------------------
REAL,INTENT(OUT) ::
DNi(:,:) !
Derivatives of Ni
REAL, INTENT(IN) ::
xsi,eta !
intrinsic coordinates
INTEGER,INTENT(IN)::
ldim !
element dimension
INTEGER,INTENT(IN)::
nodes !
number of nodes
INTEGER,INTENT(IN)::
inci(:) !
element incidences
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