5.7
PIPE UNDER INTERNAL PRESSURE, TORUS NO.8
Copy the example files B7_*
to Z88 entry files Z88*:
B7_X.DXF --> Z88X.DXF
input file for the CAD converter Z88X
B7_2.TXT --> Z88I2.TXT boundary conditions
B7_3.TXT --> Z88I3.TXT heading parameter for tension processor
CAD:
Import Z88X.DXF into your CAD program and view the superstructure. You
usually
would have designed this example in a CAD system and then exported it
as
Z88X.DXF.
Z88:
Z88X,
conversion "from Z88X.DXF to Z88NI.TXT"
Z88O, structure file Z88NI, look at the
super structure
Z88N, mesh generator, produces Z88I1.TXT
Z88O, structure file Z88I1.TXT, undeflected
FE structure
Z88X, conversion, "from Z88I* . TXT
to Z88X.DXF"
CAD:
Import Z88X.DXF
into your CAD program and look at it. You usually would have now added
the
boundary conditions and control information Z88I3.TXT into CAD and then
exported it as Z88X.DXF.
Z88:
Z88X,
conversion, "from Z88X.DXF to Z88I* . TXT"
Z88F calculates deflections
Z88D calculates stresses
Z88O, plots FE structure, now also
deflected and stresses display
Z88E, nodal force calculation
We look at a pipe under
internal pressure. Pipe inside diameter 80 mm, pipe outside diameter
160 mm,
length 40 mm. For Torus elements the cross-section of the pipe is
important.
The inside radius shall be
expanded by 0.1 mm = rd (press fit). Attach this displacements to the
nodes
from 1 to 11. To fix the structure in space, e.g. fix node 6 in Z
direction.
One calculates
analytically:
p = rd*E/ri*(1/((1+qa)/(1-qa)
+nue ) ) = 262 N/mm2 = 2.620 bar
with qa= ri2/ra2
= 0.25 and E = Young's modulus and nue = Poisson's ratio
Radial stresses:
SIGRR i = -p = -262 N/mm2
SIGRR a = 0 = 0
Tangential stresses:
SIGTE i = p*((1+qa)/(1-qa) ) = 437 N/mm2
SIGTE a = 2p*qa/(1-qa) =
175 N/mm2
Because stresses are
calculated in the Gauss points, use linear extrapolations to get the
stresses
directly in the inside diameter and the outside diameter.
The force: F = p*A =
p*2*Pi*ri*l = 2,633,911 N.
This confirms the sum of
the forces of the elements 1-5 for the nodes 1-11 in Z88O4.TXT.
5.7.1
Input
General: The entries for
the mesh generator contain merely a single Torus No.8
as super element. It is subdivided into
40 finite elements. A Torus No.12 also could, of course, be used as
super
element. Yet this is quite useless for this simple super structure,
being
designed of straight lines. Torus elements No.12
are more powerful than Torus
elements No.8 if the super structure has many curvilinear edges because
they
feature cubic shape functions, but Torus No.8 uses only square
parables. Thus,
many curvilinear structures allow a better approach with few Torus
elements
No.12 due to the higher curve function.
Make sure that cylindrical
coordinates are always expected for Torus No.6, No.8 and No.12, i.e.
radius R
(replaces X) and height coordinate Z (replaces Y). R and Z must feature
always
positive values ! KFLAG must be zero!
With
CAD program:
Proceed after the
description chapter
2.7.2.
Do not forget to write on the layer Z88EIO the super element
descriptions by
TEXT function:
SE 1 8 8
L 5 e (subdivide
8x into X geometrical ascending and 5x equidistant into Y)
Write the general
information and material information on the layer Z88GEN,
Z88NI.TXT 2
8 1 16
1 0 0 0
0 0 (2D,
8 nodes, 1 SE, 16 DOF, 1 mat info,
all flags 0)
MAT 1
1
1
206000 0.3
3
0 (SE1
to SE1:Young's,Poisson's,INTORD for
FE, QPARA=0)
Export the drawing as DXF
file with the name Z88X.DXF and start the CAD converter Z88X with the
option
"from Z88X.DXF to Z88NI.TXT". Z88X will produce the mesh generator
input file Z88NI.TXT. You
should have a look at it with Z88O:
Super structure Z88NI.TXT
With
editor:
Write the mesh
generator input file Z88NI.TXT (cf.
chapter 3.3) with an editor:
2 8 1 16 1 0 0 0 0 0 (2D, 8
nodes, 1 SE, 16 DOF, 1 mat info,
all flags 0)
1 2 40
0 (1st
node, 2 DOF, R and Z coordinate)
2 2 80
0 (2nd
node, 2 DOF, R and Z coordinate)
3 2 80
40
4 2 40
40
5 2 60
0
6 2 80
20
7 2 60
40
8 2 40
20
1 8 (superelement
1, type Torus No.8)
1 2 3
4 5 6
7 8 (coincidence
1st SE)
1 1 206000
0.3 3
0 (SE1
to SE1: Young's,Poisson's,INTORD for
FE,QPARA=0)
1 8 (subdivide
SE1 into Torus elements No.8
and subdivide)
8 L 5
E (8
times geometrical ascending into X and
5 times equidistant into Y)
CAD
and editor:
Start the mesh generator
Z88N to produce the
final Z88 structure file Z88I1.TXT. Look at it either
* in the CAD program (from Z88I1.TXT to Z88X.DXF) after conversion with
Z88X or
* with the Z88 plot program Z88O for
defining the boundary conditions:
We force displacements of
0.1 mm upon the inside margin. Every node receives the same value as
the load
division in accordance with section 2.4 applies to forces only. Take
care that
the structure is fixed in space again. Therefore fix the degree of
freedom 2
for the node 6. Any other nodes are possible, too.
With
CAD program:
Switch to the layer Z88RBD and write with the TEXT function into any
free
place:
Z88I2.TXT 12
(12
boundary conditions)
RBD 1 1 1 2 0.1 (RB 1: node 1, at DOF 1, i.e into R, a displacement
of 0.1
mm)
RBD 2 2 1 2 0.1
RBD 3 3 1 2 0.1
RBD 4 4 1 2 0.1
RBD 5 5 1 2 0.1
RBD 5 6 1 2 0.1
RBD 7 6 2 2 0 (BC 7:
for fixing
structure in space)
RBD 8 7 1 2 0.1
RBD 9 8 1 2 0.1
RBD 10 9 1 2 0.1
RBD 11 10 1 2 0.1
RBD 12 11 1 2 0.1
With
editor:
Create the file of the boundary conditions Z88I2.TXT
by editing:
12 (12 boundary conditions)
1 1 2
0.1 (RB
1: node 1, at DOF 1, i.e into R, a displacement of 0.1 mm)
2 1
2 0.1
3 1 2
0.1
4 1
2 0.1
5 1 2
0.1
6 1 2
0.1
6 2 2
0 (BC
7: for fixing structure in space)
7 1
2 0.1
8 1 2
0.1
9 1 2
0.1
10 1 2
0.1
11 1 2 0.1
Input for stress
calculation:
In
the CAD program:
Switch to the layer Z88GEN and write with the TEXT function into any
free
place:
Z88I3.TXT 3 0
1 (3
x 3 Gauss points per FE, KFLAG 0, von Mises stresses)
KFLAG always 0, because
additional output of radial and tangential stresses is useless for
torus elements.
SIGRR (radial stresses) and SIGTE (tangential stresses) are calculated
for
torus elements anyway, cf. section 4.12.
Export the drawing as DXF
file with the name Z88X.DXF, then start the CAD converter Z88X with the
option "from
Z88X.DXF to Z88I*.TXT". The CAD converter produces the three Z88 input
files Z88I1.TXT, Z88I2.TXT, Z88I3.TXT.
With
editor
Enter in the
parameter file for the stress processor Z88I3.TXT
(cf. Chapter 3.5):
3 0 1 (3x3 Gauss points for stresses, KFLAG 0, von Mises stresses)
FE mesh Z88I1.TXT
CAD
and editor:
Now launch the Cholesky
solver Z88F and
then the stress
processor Z88D. Compute nodal forces with
Z88E.
5.7.2
Results
The Cholesky solver Z88F
provides the following output files:
Z88O0.TXT stores the processed structure data. For documentation
purposes.
Z88O1.TXT stores the processed boundary conditions: For
documentation
purposes.
Z88O2.TXT, the displacements, the main task and solution of the
FEA problem.
The stress processor Z88D uses internally the calculated
displacements
from Z88F and stores Z88O3.TXT, the calculated stresses. The
results in
Z88O3.TXT depend on the header parameters in Z88I3.TXT.
The nodal force processor Z88E uses internally the calculated
deflections of Z88F and stores Z88O4.TXT, the computed nodal
forces.
Stresses display of the
torus structure