@UEBER = 5.8 BEISPIEL 7: ROHR UNTER INNENDRUCK, TORI NR.7

5.9 RECTANGULAR PLATE, PLATE NO. 19

We want to compute a thick rectangular plate of steel.

Data:

- Dimensions: 1.000 x 1.000 x 100 mm
- Area load 46,42 N/mm**2
- Young's Modulus 206.000 N/mm**2
- Poisson's Ratio 0.3

We will draw the plate super structure in AutoCAD. Draw one single super element plate type No.20, which will be subdivided by the mesher Z88N into 8 x 8 = 64 plates of type No.19, i.e. with 16 nodes each. Of course, for this example you could use an editor and generate the mesh generator input file by hand at the same pace:

(Windows: AutoCAD LT 97, drawing the rectangular plate)

You'll find the exact procedure plottet in chapter 2.7 - however, try it by yourself and export the drawing as Z88X.DXF into the Z88 directory. If it doesn't work at all (but it really does):

Copy B18_X.DXF into Z88X.DXF

(Windows: CAD converter Z88X. Looks very similar on UNIX machines)

Choose from Z88X to Z88NI.TXT. Then, launch the mesher Z88N:

(Windows: mesh generator Z88N. Looks very similar on UNIX machines)

Now you may look at the structure with Z88P:

(Windows: Plot program Z88P, undeflected structure. Looks similar on UNIX computers)

Now you've got some work: you must read off the node numbers for the boundary conditions in Z88P. We have to decide how to support the plate. We'll choose "cutting edges", i.e. the boundaries are supported by a "bezel" above and below. This allows angular movement crosswise to the bezels, but fixture in direction of the bezels.

If you want to support the boundary in front, i.e. running in X direction, with cutting edges, then you must fix the degree of freedom 1 (the Z direction) and the degree of freedom 3 (the rotation around the Y axis).

We've got 625 nodes in total. Which to support ? Good question ! In order to save some work (seldom a good idea) we'll try to fix only the corner nodes of the elements, which lay on the boundaries. This nodes are

- left boundary: 1, 4, 7, 10, 13, 16, 19, 22, 25
- lower boundary: 1, 76, 151, 226, 301, 376, 451, 526, 601
- upper boundary: 25, 100, 175, 250, 325, 400, 475, 550, 625
- right boundary: 601, 604, 607, 610, 613, 616, 619, 622, 625

(Windows: read off the the nodes with Z88P. Looks similar on UNIX machines)

See the beginning and the end of the boundary conditions file Z88I2.TXT (if you are too lazy to do the work of entering the boundary conditions: B18_2ROU.TXT) :

`68
1 1 2 0.
1 2 2 0.
1 3 2 0.
4 1 2 0.
4 2 2 0.
....
622 1 2 0.
622 2 2 0.
625 1 2 0.
625 2 2 0.
625 3 2 0.`

We may now launch one of the solvers. Because the structure is really tiny, the Cholesky solver is the right choice. The displacement file Z88O2.TXT gives us the information for node 313, which lies exactly in the middle of the plate:

`313 +1.1236511E+001 -2.1751298E-008 +2.1751298E-008
`

The deflection U2 (i.e. the rotation around the X axis) and U3 (i.e. the rotation around the Y axis) are zero, looks good. The deflection U1, i.e. w, is 11,24 mm. "Analytically" (this is also only an approximation for thin plates, ref. to the classical mechanics literature) one computes:

f= (0.71 * p * b**4) / (E * h**3)= (0,71 * 46,42 * 500**4) / (206.000 * 100**3) = 10 mm

This results in a variety of (10 - 11,24) / 10 * 100 = 12%.

Here's why. Firstly, the analytical formulae in the literature are thin plates of the Kirchhoff type neglecting the shear forces, secondly, this formulae were won with series expansion and thirdly, we could truely put some more work into a better formulation of the boundary conditions. Here's how our plot looks with a magnification factor of 50:

See how the boundaries raise between the corner nodes? Guess we must swallow the bitter pill and support all the nodes laying on boundaries (copy file B18_2.TXT to Z88I2.TXT). This results in:

w at node 313: 10,5 mm, variety to the analytical calculation about 5 % (the analytical calculation supplies thin plates and is not very exact here. This thin plates should feature a thickness of about 1/50, 1/100 or fewer of the main dimensions!)

We may calculate the stresses "analytically":

x = y = (1,15 * p * b**2) / h**2 = (1,15 * 46,42 * 500**2) / 100**2 = 1.335 N/mm**2

The stress parameter file Z88I3.TXT needs the following entries for computing the stresses in the corner nodes:

`0 0 0
`

After running Z88D you may read off the stresses of node 313 from the elemente 28, 29, 36 or 37; it is the node with XX= 600 and YY= 600: x = y = 1.334 N/mm**2.

(Now the boundaries are supported properly)

Finally, we'll compute the stresses in the Gauss points and, thus, adjust Z88I3.TXT as follows:

`4 0 1
`

After a Z88D run we may look at the von Mises stresses:

(Windows: Plot of the von Mises stresses in the 4 x 4 Gauss points. Z88P.Looks similar on UNIX machines)

(Windows: Plot of the Z displacements. Z88O. Looks similar on UNIX machines)
Now you've got a small impression of plate calculation. Consult the devil (and Daniel Webster) when computing deflections and stresses for plates! I recommend parabolic tetrahedrons or hexahedrons in contrary for (thick) plate calculations, that means more input expense but the results are always save and free of suspicious interpretation constraints.