Civil Engineering Reference
In-Depth Information
becomes,
∂
∂r
0
0
∂
∂z
0
0
1
r
1
r
∂
∂θ
0
[
A
]
==
(5.3)
∂
∂z
∂
∂r
0
∂
∂θ
∂
∂z
1
r
0
1
r
∂
∂θ
∂
1
r
∂r
−
0
For each harmonic
i
, the strain-displacement relationship provided by the library subroutine
bmat nonaxi
is of the form,
B
i
1
B
i
2
B
i
3
B
i
4
···
B
i
nod
B
i
B
j
···
=
where
nod
is the number of nodes in an element.
A typical submatrix from the above expression for symmetric loading is given by
∂N
j
∂r
cos
iθ
0
0
∂N
j
∂z
0
cos
iθ
0
iN
j
r
N
j
r
cos
iθ
0
cos
iθ
[
B
]
j
=
(5.4)
∂N
j
∂z
∂N
j
∂r
cos
iθ
cos
iθ
0
iN
j
r
∂N
j
∂z
−
0
sin
iθ
sin
iθ
∂N
j
∂r
sin
iθ
iN
j
r
N
j
r
−
−
sin
iθ
0
The equivalent expression for antisymmetry is similar to equation (5.6) but with the
sine and cosine terms interchanged and the signs of elements (3,3), (5,2) and (6,1) reversed.
Additional
INTEGER
variables required by this subroutine are
iflag
and
lth
. The vari-
able
iflag
is set to 1 or -1 for symmetry or antisymmetry respectively, and the variable
lth
gives the harmonic on which loads are to be applied. An additional variable input
to this program is the angle
chi
(in degrees in the range 0
◦
to 360
◦
). This is the angle
at which stresses are evaluated and printed. Naturally, stresses could be printed at other
locations if required. It should be noted that if
lth=0
and
iflag=1
, the analysis reduces
to ordinary axisymmetry as demonstrated by the fifth example with Program 5.1.
The program uses 8-node quadrilateral elements and can be considered a variant of
Program 5.1 with nodes and elements numbered in the
dir='r'
direction. Each element has
24 degrees of freedom, as shown in Figure 5.19 and reduced integration (
nip=4
) is assumed.
