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<div class="Section1">
<p><b><span lang="EN-GB" style="font-size: 18pt;">5.6
PIPE UNDER INTERNAL PRESSURE, PLAIN STRESS ELEMENT NO.7</span></b><span
 lang="EN-GB" style=""><o:p></o:p></span></p>
<p><span lang="EN-GB" style="">Copy the example file
B6_X.DXF to Z88X.DXF. <br>
B6_X.DXF ---&gt; Z88X.DXF input file for CAD converter Z88X<o:p></o:p></span></p>
<p><b><i><span lang="EN-GB" style="font-size: 13.5pt;">CAD:</span></i></b><span
 lang="EN-GB" style=""><o:p></o:p></span></p>
<p><span lang="EN-GB" style="">Import Z88X.DXF into your
CAD program and look at it. Usually you would have designed this
example in a
CAD system and then exported it as Z88X.DXF.<o:p></o:p></span></p>
<p><b><i><span lang="EN-GB" style="font-size: 13.5pt;">Z88:</span></i></b><span
 lang="EN-GB" style=""> (in reduced form, more detailed
instructions cf. examples </span><a href="e88s1.htm"><span lang="EN-GB"
 style="">5.1</span></a><span lang="EN-GB" style="">, </span><a
 href="e88s2.htm"><span lang="EN-GB" style="">5.2</span></a><span
 lang="EN-GB" style=""> and </span><a href="e88s3.htm"><span
 lang="EN-GB" style="">5.3</span></a><span lang="EN-GB" style="">) <br>
</span><a href="e88x.htm"><span lang="EN-GB" style="">Z88X</span></a><span
 lang="EN-GB" style="">, conversion, "from Z88X.DXF to
Z88I*.TXT" <br>
</span><a href="e88o.htm"><span lang="EN-GB" style="">Z88O</span></a><span
 lang="EN-GB" style="">, looking at structure, structure
file Z88I1.TXT <br>
</span><a href="e88f.htm"><span lang="EN-GB" style="">Z88F</span></a><span
 lang="EN-GB" style=""> calculates deflections <br>
</span><a href="e88d.htm"><span lang="EN-GB" style="">Z88D</span></a><span
 lang="EN-GB" style=""> calculates stresses <br>
</span><a href="e88e.htm"><span lang="EN-GB" style="">Z88E</span></a><span
 lang="EN-GB" style=""> calculates nodal forces <br>
</span><a href="e88o.htm"><span lang="EN-GB" style="">Z88O</span></a><span
 lang="EN-GB" style="">, plot FE structure, now also
deflected (FUX, FUY, FUZ per 100.)<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">We deal with a pipe under
internal pressure of 1,000 bar (=100 N/mm<sup>2</sup>). Inside diameter
of the
pipe is 80 mm, outside diameter of the pipe is 160 mm. The length is 40
mm. If
one chooses the supports cleverly, a quarter of the pipe is enough to
reflect
the problem.<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">Such structures are best
suited for polar coordinates. The internal pressure of 1,000 bar
corresponds to
a force of 251,327 N which is loaded onto the inside quadrant. The
251,327 N
have to be distributed onto the nodes 1,6,9,14,17,22,25,30 and 33 in
accordance
with the rules for </span><a href="e88i2.htm"><span lang="EN-GB"
 style="">boundary conditions</span></a><span lang="EN-GB" style="">
(cf. chapter 3.4):<o:p></o:p></span></p>
<p><span lang="FR" style="">"1/6 points": 10,472 N <br>
"2/3 points": 41,888 N <br>
"2/6 points": 20,944 N<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">Control:
2*10,472+4*41,888+3*20,944 = 251,328 okay<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">These forces have an
outwardly directed radial effect. Thus, they must be subdivided into X
and Y
components for boundary conditions. E.g. the node 6 as "2/3 point" is
subdivided into X = 41,083 N and into Y = 8,172 N, because node 6 has
an angle
Phi = 11.25 degrees.<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">When dealing with a
rotationally symmetrical structure, the additional calculation of
radial
stresses and tangential stresses can be interesting: Set KFLAG to 1 in
Z88I3.TXT. As stresses are calculated in the Gauss points, use linear
extrapolations to get the stresses directly in the inside diameter and
the
outside diameter.<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">This problem is simple to
check analytically. Consult appropriate machine element books for
proper
calculation formulas or see </span><a href="e88s7.htm"><span
 lang="EN-GB" style="">chapter 5.7</span></a><span lang="EN-GB" style="">.
<o:p></o:p></span></p>
<p><img border="0" width="531" height="383" id="_x0000_i1025"
 src="g88s601.gif"></p>
<p><b><span lang="EN-GB" style="">Plot of the undeflected
structure</span></b><span lang="EN-GB" style=""><o:p></o:p></span></p>
<p><b><span lang="EN-GB" style="font-size: 13.5pt;">5.6.1
Input</span></b><span lang="EN-GB" style=""><o:p></o:p></span></p>
<p><b><i><span lang="EN-GB" style="font-size: 13.5pt;">With
CAD program:</span></i></b><span lang="EN-GB" style=""> <br>
Proceed after the description </span><a href="e88x.htm"><span
 lang="EN-GB" style="">chapter 2.7.2</span></a><span lang="EN-GB"
 style="">. Do
not forget to write on the layer Z88EIO
the element descriptions by TEXT function: <o:p></o:p></span></p>
<p><span lang="EN-GB" style="">FE<span style="">&nbsp;&nbsp; </span>1<span
 style="">&nbsp;&nbsp;
</span>7<span style="">&nbsp;&nbsp; </span><i>(1st finite element type
7)</i>
<br>
FE<span style="">&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>7<span style="">&nbsp;&nbsp; </span><i>(2nd finite element type
7)</i>
<br>
<i>.........<span style="">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>(element
3 to 7
dropped here) <br>
</i>FE<span style="">&nbsp;&nbsp; </span>8<span style="">&nbsp;&nbsp; </span>7<span
 style="">&nbsp; </span><i>(
8th finite element type 7)</i><o:p></o:p></span></p>
<p><span lang="EN-GB" style="">Write the general
information and material information on the layer Z88GEN, like<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">Z88I1.TXT<span style="">&nbsp;&nbsp; </span>2<span
 style="">&nbsp;&nbsp;
</span>37<span style="">&nbsp;&nbsp; </span>8<span style="">&nbsp;&nbsp;
</span>74<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>0<span style="">&nbsp;&nbsp;
</span>0<span style="">&nbsp;&nbsp; </span>0<span style="">&nbsp; </span><i>(2D,
37 nodes, 8 ele, 74 DOF, 1 mat info,
polar coor., IBFLAG 0, IPFLAG 0, IQFLAG 0)</i> <br>
MAT <span style="">&nbsp;&nbsp;</span>1<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>8<span style="">&nbsp;&nbsp;
</span>206000<span style="">&nbsp;&nbsp;
</span>0.3<span style="">&nbsp;&nbsp; </span>3<span style="">&nbsp;&nbsp;
</span>40<span style="">&nbsp;&nbsp; </span><i>(1st
mat info: Ele 1 to 8: Young's, Poisson's,INTORD= 3, QPARA = thickness =
40) </i><o:p></o:p></span></p>
<p><span lang="EN-GB" style="">Write the boundary
conditions with the TEXT function onto the layer Z88RBD. Here we have
the </span><a href="e88i2.htm"><span lang="EN-GB" style="">case of
edge
loads</span></a><span lang="EN-GB" style=""> for the
boundary conditions. You should consult chapter 3.4. and take into
account the
explanation and sketches for load distributions. <o:p></o:p></span></p>
<p><span lang="EN-GB" style="">Z88I2.TXT 26 <i>(26
boundary conditions)</i> <br>
RBD 1 1 1 1 10472 <i>(1st BC: Node 1, DOF 1(= X), a load of 10,472 N)</i>
<br>
RBD 2 1 2 2 0 <i>(2nd BC: Node 1, DOF 2 (=Y), a displacement of 0
(=fixed))</i>
<br>
RBD 3 2 2 2 0 <br>
RBD 4 3 2 2 0 <br>
RBD 5 4 2 2 0 <br>
RBD 6 5 2 2 0 <br>
RBD 7 6 1 1 41083 <br>
RBD 8 6 2 1 8172 <br>
RBD 9 9 1 1 19350 <br>
RBD 10 9 2 1 8015 <br>
RBD 11 14 1 1 34829 <br>
RBD 12 14 2 1 23272 <br>
RBD 13 17 1 1 14810 <br>
RBD 14 17 2 1 14810 <br>
RBD 15 22 1 1 23272 <br>
RBD 16 22 2 1 34829 <br>
RBD 17 25 1 1 8015 <br>
RBD 18 25 2 1 19350 <br>
RBD 19 30 1 1 8172 <br>
RBD 20 30 2 1 41083 <br>
RBD 21 33 1 2 0 <br>
RBD 22 33 2 1 10472 <br>
RBD 23 34 1 2 0 <br>
RBD 24 35 1 2 0 <br>
RBD 25 36 1 2 0 <br>
RBD 26 37 1 2 0<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">Switch to the layer Z88GEN
and write into any free place:<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">Z88I3.TXT<span style="">&nbsp;&nbsp; </span>3<span
 style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span><i>(3x3 Gauss points for stresses, KFLAG=1
i.e. additional calculation of radial and tangential stresses, von
Mises
stresses)</i><o:p></o:p></span></p>
<p><span lang="EN-GB" style="">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" (DXF -&gt; I*). The CAD converter
produces the three Z88 input files Z88I1.TXT, Z88I2.TXT, Z88I3.TXT.<o:p></o:p></span></p>
<p><b><i><span lang="EN-GB" style="font-size: 13.5pt;">With
editor:<br>
</span></i></b><span lang="EN-GB" style="">Write the
structure data file </span><a href="e88i1.htm"><span lang="EN-GB"
 style="">Z88I1.TXT</span></a><span lang="EN-GB" style=""> (cf.
chapter 3.2) with an editor:<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">2<span style="">&nbsp;&nbsp; </span>37<span
 style="">&nbsp;&nbsp;
</span>8<span style="">&nbsp;&nbsp; </span>74<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>0<span style="">&nbsp;&nbsp;
</span>0<span style="">&nbsp;&nbsp;
</span>0<span style="">&nbsp;&nbsp; </span><i>( 2D, 37 nodes, 8
elements,
74 DOF, 1 mat info, Polar coor., beam &amp; plate &amp; surface loads<span
 style="">&nbsp; </span>flag 0, each)</i> <br>
1<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;&nbsp;
</span>40<span style="">&nbsp;&nbsp; </span>0<span style="">&nbsp;&nbsp;
</span><i>(1st node, 2 DOF, R and Phi coordinate)</i>
<br>
2<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;&nbsp;
</span>48<span style="">&nbsp;&nbsp; </span>0<span style="">&nbsp;&nbsp;
</span><i>(2nd node, 2 DOF, R and Phi coordinate)</i>
<br>
3<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;&nbsp;
</span>56<span style="">&nbsp;&nbsp; </span>0 <br>
4<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;&nbsp;
</span>68<span style="">&nbsp;&nbsp; </span>0 <br>
5<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;&nbsp;
</span>80<span style="">&nbsp;&nbsp; </span>0 <br>
6<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;&nbsp;
</span>40<span style="">&nbsp;&nbsp; </span>11.25 <br>
7<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;&nbsp;
</span>56<span style="">&nbsp;&nbsp; </span>11.25 <br>
8<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;&nbsp;
</span>80<span style="">&nbsp;&nbsp; </span>11.25 <br>
9<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;&nbsp;
</span>40<span style="">&nbsp;&nbsp; </span>22.5 <br>
<i>.....<span style="">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>(Nodes
10.. 35 dropped here
)</i> <br>
36<span style="">&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>68<span style="">&nbsp;&nbsp; </span>90 <br>
37<span style="">&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>80<span style="">&nbsp;&nbsp; </span>90 <br>
1<span style="">&nbsp;&nbsp; </span>7<span style="">&nbsp;&nbsp;
</span><i>(element 1, Plain Stress Element No.7)</i> <br>
1<span style="">&nbsp;&nbsp; </span>3<span style="">&nbsp;&nbsp;
</span>11<span style="">&nbsp;&nbsp; </span>9<span style="">&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;
</span>7<span style="">&nbsp;&nbsp; </span>10<span style="">&nbsp;&nbsp;
</span>6<span style="">&nbsp;&nbsp; </span><i>(coincidence
1st element)</i> <br>
2<span style="">&nbsp;&nbsp; </span>7 <br>
3<span style="">&nbsp;&nbsp; </span>5<span style="">&nbsp;&nbsp;
</span>13<span style="">&nbsp;&nbsp; </span>11<span style="">&nbsp;&nbsp;
</span>4<span style="">&nbsp;&nbsp;
</span>8<span style="">&nbsp;&nbsp; </span>12<span style="">&nbsp;&nbsp;
</span>7 <br>
<i>..... (elements 3 .. 7 dropped here)</i> <br>
8<span style="">&nbsp;&nbsp; </span>7<span style="">&nbsp;&nbsp;
</span><i>(element 8, Plain Stress Element No.7)</i> <br>
27<span style="">&nbsp;&nbsp; </span>29<span style="">&nbsp;&nbsp;
</span>37<span style="">&nbsp;&nbsp; </span>35<span style="">&nbsp;&nbsp;
</span>28<span style="">&nbsp;&nbsp;
</span>32<span style="">&nbsp;&nbsp; </span>36<span style="">&nbsp;&nbsp;
</span>31<span style="">&nbsp;&nbsp; </span><i>(coincidence
8th element)</i> <br>
1<span style="">&nbsp;&nbsp; </span>8<span style="">&nbsp;&nbsp;
</span>206000<span style="">&nbsp;&nbsp; </span>0.3<span style="">&nbsp;&nbsp;
</span>3<span style="">&nbsp;&nbsp;
</span>40<span style="">&nbsp;&nbsp; </span><i>(Ele 1 to 8: Young's,
Poisson's, INTORD = 3, thickness = 40)</i><o:p></o:p></span></p>
<p><span lang="EN-GB" style="">Here we have the </span><a
 href="e88i2.htm"><span lang="EN-GB" style="">case of edge
loads</span></a><span lang="EN-GB" style=""> for the
boundary conditions. Consult chapter 3.4. and take into account the
explanation
and sketches for load distributions. Here is </span><a href="e88i2.htm"><span
 lang="EN-GB" style="">Z88I2.TXT</span></a><span lang="EN-GB" style="">:
<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">26<span style="">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><i>(26 boundary
conditions)</i> <br>
1<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>10472<span style="">&nbsp;&nbsp;
</span><i>(Node 1, DOF 1(= X), a load of 10,472 N)</i>
<br>
1<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0<span style="">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><i>(Node 1, DOF 2 (=Y), a
displacement of 0 (=fixed))</i> <br>
2<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0<br>
3<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0 <br>
4<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0 <br>
5<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0 <br>
6<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp; </span>41083 <br>
6<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp; </span>8172 <br>
9<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp; </span>19350 <br>
9<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp; </span>8015 <br>
14<span style="">&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp; </span>34829 <br>
14<span style="">&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp; </span>23272 <br>
17<span style="">&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>14810 <br>
17<span style="">&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>14810 <br>
22<span style="">&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>23272 <br>
22<span style="">&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>34829 <br>
25<span style="">&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>8015 <br>
25<span style="">&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>19350 <br>
30<span style="">&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>8172 <br>
30<span style="">&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>41083 <br>
33<span style="">&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp; </span>0 <br>
33<span style="">&nbsp;&nbsp; </span>2<span style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span>10472 <br>
34<span style="">&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp; </span>0 <br>
35<span style="">&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp; </span>0 <br>
36<span style="">&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp; </span>0 <br>
37<span style="">&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp; </span>0<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">This example is very nice
for experiments with the boundary conditions: Enter deflections rather
than
forces into X and Y, e.g. 0.01 mm in radial direction to the outside.
At node 1
you can enter the 0.01 mm directly as X displacement and at node 33 you
can
enter directly the Y displacement of 0.01 mm, but for the other nodes
the
radial displacements of 0.01 mm must be subdivided into X and Y
components
respectively (via sine and cosine). Or enter mixed BC: A couple of
nodes with
displacements, the others with forces.. In practice nobody would do so
for such
a task, however, but Z88 can handle it.<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">A broad experimenting field
also opens up with </span><a href="e88i3.htm"><span lang="EN-GB"
 style="">Z88I3.TXT</span></a><span lang="EN-GB" style="">: You
have 5 possibilities for the first value
and two possibilities each for the second and third value, cf. Chapters
</span><a href="e88i3.htm"><span lang="EN-GB" style="">3.5</span></a><span
 lang="EN-GB" style=""> and </span><a href="e88e7.htm"><span
 lang="EN-GB" style="">4.7</span></a><span lang="EN-GB" style="">. Now
we can
produce plenty of results: <o:p></o:p></span></p>
<p><span lang="EN-GB" style="">Here is </span><a href="e88i3.htm"><span
 lang="EN-GB" style="">Z88I3.TXT</span></a><span lang="EN-GB" style="">:<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">3<span style="">&nbsp;&nbsp; </span>1<span
 style="">&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp; </span><i>(3x3 Gauss points for
stresses, KFLAG=1 i.e. additional calculation of radial and tangential
stresses, von Mises stresses) </i><o:p></o:p></span></p>
<p><b><i><span lang="EN-GB" style="font-size: 13.5pt;">CAD
and editor:</span></i></b><span lang="EN-GB" style=""> <br>
The structure data Z88I1.TXT, the boundary conditions Z88I2.TXT and the
header
file for the stress processor Z88I3.TXT (with any contents) are ready
to go.
Now launch <o:p></o:p></span></p>
<p><span lang="EN-GB" style="">&gt; </span><a href="e88f.htm"><span
 lang="EN-GB" style="">Z88F the
Cholesky solver</span></a><span lang="EN-GB" style=""> <br>
&gt; </span><a href="e88d.htm"><span lang="EN-GB" style="">Z88D
the stress processor</span></a><span lang="EN-GB" style="">
<br>
&gt; </span><a href="e88e.htm"><span lang="EN-GB" style="">Z88E
the nodal force processor</span></a><span lang="EN-GB" style=""><o:p></o:p></span></p>
<p class="UU"><i style=""><span lang="EN-GB"
 style="font-size: 14pt; font-family: &quot;Times New Roman&quot;;">Boundary
conditions by edge loads:<o:p></o:p></span></i></p>
<p class="TA"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">The
data entry by single forces was somewhat
cumbersome because of dividing the force of 251.327 N to several nodal
points
with respect to the actual angle position. It is much more easier to
enter edge
loads by the surface and pressure loads file Z88I5.TXT. The edge load
is:<o:p></o:p></span></p>
<p><span lang="IT" style="">q= F/l = F/(r*Phi) =
251327/(40*Pi/2) = 4000 N/mm<o:p></o:p></span></p>
<p class="TA"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">This
edge load acts onto the elements 1, 3, 5 and 7.
The edge of <a href="e88e7.htm">element 1</a> is the edge defined by
the corner
nodes 9 and 1 and the middle node 6 etc. The edge load points normally
to the
edge, there are no tangential loads. Thus,<span style="">&nbsp;
</span>the surface and pressure loads file <a href="e88i5.htm">Z88I5.TXT</a>
is:<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">4<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">1<span
 style="">&nbsp;&nbsp; </span>4000.<span style="">&nbsp;&nbsp; </span>0.<span
 style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>9<span style="">&nbsp;&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;&nbsp;
</span>6<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">3<span
 style="">&nbsp;&nbsp; </span>4000.<span style="">&nbsp;&nbsp; </span>0.<span
 style="">&nbsp;&nbsp;
</span>17<span style="">&nbsp;&nbsp;&nbsp; </span>9<span style="">&nbsp;
</span>14<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">5<span
 style="">&nbsp;&nbsp; </span>4000.<span style="">&nbsp;&nbsp; </span>0.<span
 style="">&nbsp;&nbsp;
</span>25<span style="">&nbsp; </span>17<span style="">&nbsp; </span>22<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">7<span
 style="">&nbsp;&nbsp; </span>4000.<span style="">&nbsp;&nbsp; </span>0.<span
 style="">&nbsp;&nbsp;
</span>33<span style="">&nbsp; </span>25<span style="">&nbsp; </span>30<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><o:p>&nbsp;</o:p></span></p>
<p class="TA"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">To
make the solvers reading in the surface and
pressure loads file you are to <u>set the surface and pressure loads
flag to 1</u>
in the first line of the general data file <a href="e88i1.htm">Z88I1.TXT</a>:<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp;
</span>37<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>8<span style="">&nbsp;&nbsp;&nbsp; </span>74<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>0<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>0<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>0<span style="">&nbsp; </span><span style="">&nbsp;&nbsp;</span><b
 style="">1</b><o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><span style="">&nbsp;</span><span style="">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span></span><span
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">|</span><span
 lang="EN-GB" style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span
 style="">&nbsp; </span>surface and pressure loads flag IQFLAG<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><o:p>&nbsp;</o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">Now
edit the boundary conditions file <a href="e88i2.htm">Z88I2.TXT</a>:
Skip all forces:<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><o:p>&nbsp;</o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="">&nbsp;&nbsp;&nbsp;
</span>10<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0.<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp; </span><span style="">&nbsp;&nbsp;</span>0.<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>3<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0.<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>4<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0.<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>5<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0.<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="">&nbsp;&nbsp;&nbsp;
</span>33<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0.<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="">&nbsp;&nbsp;&nbsp;
</span>34<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0.<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="">&nbsp;&nbsp;&nbsp;
</span>35<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0.<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="">&nbsp;&nbsp;&nbsp;
</span>36<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0.<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="">&nbsp;&nbsp;&nbsp;
</span>37<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp; </span>0.<o:p></o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><o:p>&nbsp;</o:p></span></p>
<p class="T0Char"><span lang="EN-GB"
 style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">Please
see the sample files B6_Q_1.TXT, B6_Q_2.TXT and
B6_Q_5.TXT on the CD-ROM or the Internet packages. Now compute
deflections,
stresses and nodal forces as usual:<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 36pt; text-indent: -18pt;"><!--[if !supportLists]--><span
 lang="EN-GB" style="font-family: Symbol;"><span style="">&middot;<span
 style="font-family: &quot;Times New Roman&quot;; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;">&nbsp;&nbsp;
</span></span></span><!--[endif]--><span lang="EN-GB" style="">Z88F<span
 style=""> </span>the
Cholesky solver<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 36pt; text-indent: -18pt;"><!--[if !supportLists]--><span
 lang="EN-GB" style="font-family: Symbol;"><span style="">&middot;<span
 style="font-family: &quot;Times New Roman&quot;; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;">&nbsp;&nbsp;
</span></span></span><!--[endif]--><span lang="EN-GB" style="">Z88D<span
 style=""> </span>the
stress processor<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 36pt; text-indent: -18pt;"><!--[if !supportLists]--><span
 lang="EN-GB" style="font-family: Symbol;"><span style="">&middot;<span
 style="font-family: &quot;Times New Roman&quot;; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;">&nbsp;&nbsp;
</span></span></span><!--[endif]--><span lang="EN-GB" style="">Z88E<span
 style=""> </span>the
nodal force processor<o:p></o:p></span></p>
<p><b><span lang="EN-GB" style="font-size: 13.5pt;">5.6.2 Results:</span></b><span
 lang="EN-GB" style=""><o:p></o:p></span></p>
<p><span lang="EN-GB" style="">The Cholesky solver Z88F
provides the following output files: <br>
<b>Z88O0.TXT</b> stores the processed structure data. For documentation
purposes. <br>
<b>Z88O1.TXT</b> stores the processed boundary conditions: For
documentation
purposes. <br>
<b>Z88O2.TXT</b>, the displacements, the main task and solution of the
FEA
problem.<br>
The stress processor <b>Z88D</b> internally uses the calculated
displacements
from Z88F and stores <b>Z88O3.TXT</b>, the calculated stresses.<br>
The nodal force processor <b>Z88E</b> internally uses the calculated
deflections of Z88F and stores <b>Z88O4.TXT</b>, the computed nodal
forces.<o:p></o:p></span></p>
<p><span lang="EN-GB" style="">This example is very
suitable to demonstrate all the possibilities of the stress calculation
with </span><a href="e88d.htm"><span lang="EN-GB" style="">Z88D</span></a><span
 lang="EN-GB" style=""> and </span><a href="e88e7.htm"><span
 lang="EN-GB" style="">Plain Stress Elements No.7</span></a><span
 lang="EN-GB" style=""> (or </span><a href="e88e11.htm"><span
 lang="EN-GB" style="">Plain Stress Elements No.11</span></a><span
 lang="EN-GB" style="">). We recall: </span><a href="e88i3.htm"><span
 lang="EN-GB" style="">Z88I3.TXT</span></a><span lang="EN-GB" style="">
was: 3 1
1, i.e. 3 x 3 Gauss points,
additional calculation of radial and tangential stresses (which is very
meaningful for this example) and von Mises stresses calculation. Enter
in
Z88I3.TXT: 3 0 1, so that you will get von Mises stresses but no radial
and
tangential stresses. The results with 2 0 0 become even shorter (only
still 2 x
2 Gauss points, no radial/tangential stresses and no von Mises
stresses). You
will get the stresses with 0 0 0 into the corner nodes instead of into
the
Gauss points. </span><span lang="EN-GB" style="">Now
experiment.. you have 5 x 2 x 2 = 20 possibilities. <o:p></o:p></span></p>
<p><img alt="" src="e88s602.jpg"><br>
</p>
<span style="font-weight: bold;"><span lang="EN-GB" style="">Plot of
the <i>von Mises</i>
stresses</span></span><span lang="EN-GB" style=""><span
 style="font-weight: bold;"> </span><br>
<br>
</span><span style=""></span>
<p class="MsoNormal" style="text-align: justify;"><span lang="EN-GB"
 style="font-size: 12pt;">Now you should plot the
boundary conditions. The problem here is that some nodes have more than
one BC
e.g. node 1 features a force in X direction and is fixed in Y
direction,
too:<span style="">&nbsp; </span><o:p></o:p></span></p>
<p class="MsoNormal" style="text-align: justify;"><span lang="EN-GB"
 style="font-size: 12pt;"><span style="">&nbsp; </span><span style="">&nbsp;&nbsp;&nbsp;</span>1<span
 style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>1<span style="">&nbsp;&nbsp;
</span>+1.04720E+004<o:p></o:p></span></p>
<p class="MsoNormal" style="text-align: justify;"><span lang="EN-GB"
 style="font-size: 12pt;"><span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><span style="font-size: 12pt;">1<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style="">&nbsp;&nbsp; </span>+0.00000E+000<o:p></o:p></span></p>
<p class="MsoNormal" style="text-align: justify;"><span lang="EN-GB"
 style="font-size: 12pt;"><o:p>&nbsp;</o:p></span></p>
<p class="MsoNormal" style="text-align: justify;"><span lang="EN-GB"
 style="font-size: 12pt;">Thus, you select the BC you
want to see. <o:p></o:p></span></p>
<span style=""></span><span lang="EN-GB" style=""><br>
<img alt="" src="e88s603.jpg"><br>
<br>
Selective plot of the BCs with Z88O.<br>
<br>
</span><span style=""></span>
<p class="MsoNormal" style="text-align: justify;"><span lang="EN-GB"
 style="font-size: 12pt;">Suppose you would now choose a
hardened steel. This means to compute the <i style="">principal
stresses</i> not the <i style="">von Mises</i>
stresses. Thus, the third entry in Z88I3.TXT would now read 2:<o:p></o:p></span></p>
<p class="MsoNormal" style="text-align: justify;"><span
 style="font-size: 12pt;"><span style="">&nbsp;&nbsp;&nbsp; </span>3<span
 style="">&nbsp;&nbsp;&nbsp;&nbsp;
</span>1<span style="">&nbsp;&nbsp;&nbsp;&nbsp; </span>2<o:p></o:p></span></p>
<p class="MsoNormal" style="text-align: justify;"><span lang="EN-GB"
 style="font-size: 12pt;">Then do a stress calculation
again with Z88D.<o:p></o:p></span></p>
<span style=""><br>
</span><span lang="EN-GB" style=""><br>
<br>
<o:p></o:p></span></div>
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