Beam Problem 5 first created 08/11/09 - last modified 08/11/09 Page Author: Ty Harness
A silicon beam 1[mm] long and a square cross section 10um x 10um with clamped end conditions is shown in figure 1 is subjected to a end force,P of 1[um]. Find the transverse deflection using an analytic method and compare with any available finite element analysis program. Work together in groups of 2 or 3 and present your solutions at the next tutorial meet.

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Figure 1 - Encastre Beam Problem 5


Use the following material properties: A typical value of 1.08e11 can be used for the Young's Modulus for silicon[1].

The second moment of area:

$I = (b*d^3)/12 = (10*10^-6 (10*10^-6)^3) / 12 = 8.333*10^-22 [m]^4 $

The analytical solution to the above beam can be in the majority mechanics text books. For instance the end deflection

$y = (P* L^3) / (12*E*I) $

taken from Stephens[2].

$y = (1*10^-6(1*10^-3)^3) / (12*1.08*10^11*8.33*10^-22) = 92.6[um] $
I've chosen to use Prof. Rieg's Z88 Finite Elements Program because it's free and unlimited and ideal for student use. The theory behind FEM is beyond the scope of this tutorial but you can learn a great deal from the Z88 manual. The input files Z88I1.txt and Z88I2.txt are below if you wish to repeat the analysis using the Z88 software.

Z88I1.txt

2 10 9 30 1 0 1 0 Z88I1.TXT,typed in by Ty
1 3 +0.00000E+000 +0.00000E+000 +0.00000E+000 node #1
2 3 +1.11111E-004 +0.00000E+000 +0.00000E+000 node #2
3 3 +2.22222E-004 +0.00000E+000 +0.00000E+000 node #3
4 3 +3.33333E-004 +0.00000E+000 +0.00000E+000 node #4
5 3 +4.44444E-004 +0.00000E+000 +0.00000E+000 node #5
6 3 +5.55556E-004 +0.00000E+000 +0.00000E+000 node #6
7 3 +6.66667E-004 +0.00000E+000 +0.00000E+000 node #7
8 3 +7.77778E-004 +0.00000E+000 +0.00000E+000 node #8
9 3 +8.88889E-004 +0.00000E+000 +0.00000E+000 node #9
10 3 +1.00000E-003 +0.00000E+000 +0.00000E+000 node #10
1 13 element #1
1 2
2 13 element #2
2 3
3 13 element #3
3 4
4 13 element #4
4 5
5 13 element #5
5 6
6 13 element #6
6 7
7 13 element #7
7 8
8 13 element #8
8 9
9 13 element #9
9 10
1 9 +1.08000E+011 +3.00000E-001 1 +1.00000E-010 0 0 +8.33333E-022 +5.00000E-006 0 0

Z88I2.txt

5 , typed in from ty
1 1 2 0.00000E+000
1 2 2 0.00000E+000
1 3 2 0.00000E+000
10 3 2 0.00000E+000
10 2 1 -1.00000E-006


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Figure 2 - Z88PV10 screen shot


Z88o2.txt

output file Z88O2.TXT : displacements, computed by Z88F V10
*************

Knoten U(1) U(2) U(3) U(4) U(5) U(6)

1 +0.0000000E+000 +0.0000000E+000 +0.0000000E+000
2 +0.0000000E+000 -3.1753241E-008 -5.4869658E-004
3 +0.0000000E+000 -1.1685194E-007 -9.6021918E-004
4 +0.0000000E+000 -2.4005455E-007 -1.2345678E-003
5 +0.0000000E+000 -3.8611955E-007 -1.3717424E-003
6 +0.0000000E+000 -5.3980675E-007 -1.3717424E-003
7 +0.0000000E+000 -6.8587174E-007 -1.2345678E-003
8 +0.0000000E+000 -8.0907436E-007 -9.6021918E-004
9 +0.0000000E+000 -8.9417305E-007 -5.4869658E-004
10 +0.0000000E+000 -9.2592630E-007 +0.0000000E+000
References
[1] Zhang L M, Uttamcham i D., "Measurement of the Mechanical Properties of Silicon Microresonators", Sensors and Actuators, A 29 79-84. [2] Stephens, R.C., Strength of Materials, Edward Arnold Ltd., 1st, London: (1982), 122.

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