Beam Problem 6 first created 09/11/09 - last modified 09/11/09 Page Author: Ty Harness
A silicon beam 1[mm] long and a square cross section 10um x 10um fixed one end and free the other is shown in figure 1. The free end is subjected to moment,M of 1[uNm]. Find the transverse deflection and slope at the free end 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 - A silicon cantilever beam subjected to an end moment at the free end


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 = (M* L^2) / (2*E*I) $ and the end slope is $theta = (M*L)/(E*I)

taken from Den Hartog[2].

$y = (1*10^-6 * (1*10^-3)^2)/ (2*1.08*10^11*8.333*10^-22) = 5.6*10^-3 [m] $

$theta = (1*10^-6 * 1*10^-3) / (1.08*10^11*8.333*10^-22) = 11.12$
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

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


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Figure 2 - Z88P v10 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 -6.8586996E-005 -1.2345672E+000
3 +0.0000000E+000 -2.7434798E-004 -2.4691343E+000
4 +0.0000000E+000 -6.1728296E-004 -3.7037015E+000
5 +0.0000000E+000 -1.0973919E-003 -4.9382686E+000
6 +0.0000000E+000 -1.7146811E-003 -6.1728469E+000
7 +0.0000000E+000 -2.4691393E-003 -7.4074141E+000
8 +0.0000000E+000 -3.3607714E-003 -8.6419812E+000
9 +0.0000000E+000 -4.3895776E-003 -9.8765484E+000
10 +0.0000000E+000 -5.5555578E-003 -1.1111116E+001
References
[1] Zhang L M, Uttamcham i D., "Measurement of the Mechanical Properties of Silicon Microresonators", Sensors and Actuators, A 29 79-84.

[2] Hartog, J.,P.,, Strength of Materials, Dover Publications Inc., 1st, New York: (1961), 317.

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