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