Lab 03
Problem Statement: Using the Lab03 matlab script as a starting point, fill in the required code to solve the simply supported beam with
a uniform distributed load of w(x) = 100 lb/ft in the downward direction. Use the two variable formulation to obtain nodal values of moment
and deflection and then post-process to obtain the nodal values of shear and slope from their definitions. Perform a uniform mesh refinement
study for the linear basis only in Enorm of convergence study, identify the mesh for your most accurate solution and present a
shear/moment/slope/deflection graph for this mesh.
Analysis: The recipe was applied to the differential equations for Moment ( L(M) = -d2M(x)/dx2+ w(x) = 0 ) and also for Deflection
( L(y) = -EI(x)d2y(x)/dx2 + M(x) = 0 ). The Moment is solved for first and then used to find the deflection.
For the Moment Analysis ------ {DIFFM}*{M} = -{LOAD} + {BSHR}
For the Deflection Analysis ------ {DIFFY}*{Y} = {MOM} + {BSHR}
After the values of Moment and Deflection are found we can use these values to post-process and find the shear and slope. Applying the
recipe to the Shear differential equation ( L(V) = V(x) – dM(x)/dx = 0 ) and the Slope differential equation ( L(Q) = Q(x) – dy(x)/dx = 0 )
For the Shear Analysis ------ {MASS}*{V}={DMOM}
For the Slope Analysis ------ {MASS}*{Q}={DDEF}
This data was entered into matlab code along with the modification to the dirichlet data. MatLab Code for Lab-03
Results:
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Table 1. Bending Moment Convergence Table in Euler Bernoulli Beam |
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|
# Elements |
Mleft |
Mcenter |
Mright |
Enorm |
DEnorm |
eh/n |
Slope |
|
3 |
0 |
1111.1 |
0 |
370370.37 |
------- |
------- |
------- |
|
6 |
0 |
1250 |
0 |
405092.59 |
34722.2222 |
11574.0741 |
------- |
|
12 |
0 |
1250 |
0 |
413773.15 |
8680.5556 |
2893.5185 |
2.00000000000 |
|
24 |
0 |
1250 |
0 |
415943.29 |
2170.1389 |
723.3796 |
2.00000000000 |
|
48 |
0 |
1250 |
0 |
416485.82 |
542.5347 |
180.8449 |
2.00000000002 |
|
96 |
0 |
1250 |
0 |
416621.46 |
135.6337 |
45.2112 |
2.00000000013 |
|
192 |
0 |
1250 |
0 |
416655.36 |
33.9084 |
11.3028 |
2.00000000104 |
|
384 |
0 |
1250 |
0 |
416663.84 |
8.4771 |
2.8257 |
2.00000001749 |
|
768 |
0 |
1250 |
0 |
416665.96 |
2.1193 |
0.7064 |
2.00000125411 |
|
1536 |
0 |
1250 |
0 |
416666.49 |
0.5298 |
0.1766 |
2.00000381556 |
|
3072 |
0 |
1250 |
0 |
416666.62 |
0.1324 |
0.0441 |
2.00025027684 |
|
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Table 2. Deflection Convergence Table in Euler Bernoulli Beam |
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|
# Elements |
Yleft |
Ycenter |
Yright |
Enorm |
DEnorm |
eh/n |
Slope |
|
3 |
0 |
0.000140018 |
0 |
0.4321554 |
------- |
------- |
------- |
|
6 |
0 |
0.000173273 |
0 |
0.5354675 |
0.10331213854 |
0.034437379 |
------- |
|
12 |
0 |
0.000176226 |
0 |
0.5640445 |
0.02857701112 |
0.00952567037 |
1.854082835 |
|
24 |
0 |
0.000176965 |
0 |
0.5713679 |
0.00732340806 |
0.00244113602 |
1.964267941 |
|
48 |
0 |
0.000177149 |
0 |
0.5732101 |
0.00184216397 |
0.00061405466 |
1.991113701 |
|
96 |
0 |
0.000177195 |
0 |
0.5736713 |
0.00046124978 |
0.00015374993 |
1.997781343 |
|
192 |
0 |
0.000177207 |
0 |
0.5737867 |
0.00011535677 |
0.00003845226 |
1.999445519 |
|
384 |
0 |
0.000177210 |
0 |
0.5738155 |
0.00002884196 |
0.00000961399 |
1.99986142 |
|
768 |
0 |
0.000177210 |
0 |
0.5738227 |
0.00000721066 |
0.00000240355 |
1.99996598 |
|
1536 |
0 |
0.000177211 |
0 |
0.5738245 |
0.00000180267 |
0.00000060089 |
1.999993641 |
|
3072 |
0 |
0.000177211 |
0 |
0.5738250 |
0.00000045060 |
0.00000015020 |
2.000220219 |
|
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Table 3. Post Processing Values for Shear and Slope |
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|
# Elements |
Vleft |
Vcenter |
Vright |
Qleft |
Qcenter |
Qright |
|
|
3 |
400.000 |
0 |
-400.000 |
0.0000504 |
0 |
-0.0000504 |
|
|
6 |
451.923 |
0 |
-451.923 |
0.0000551 |
0 |
-0.0000551 |
|
|
12 |
475.944 |
0 |
-475.944 |
0.0000563 |
0 |
-0.0000563 |
|
|
24 |
487.972 |
0 |
-487.972 |
0.0000566 |
0 |
-0.0000566 |
|
|
48 |
493.986 |
0 |
-493.986 |
0.0000567 |
0 |
-0.0000567 |
|
|
96 |
496.993 |
0 |
-496.993 |
0.0000567 |
0 |
-0.0000567 |
|
|
192 |
498.496 |
0 |
-498.496 |
0.0000567 |
0 |
-0.0000567 |
|
|
384 |
499.248 |
0 |
-499.248 |
0.0000567 |
0 |
-0.0000567 |
|
|
768 |
499.624 |
0 |
-499.624 |
0.0000567 |
0 |
-0.0000567 |
|
|
1536 |
499.812 |
0 |
-499.812 |
0.0000567 |
0 |
-0.0000567 |
|
|
3072 |
499.906 |
0 |
-499.906 |
0.0000567 |
0 |
-0.0000567 |
|
Conclusion: For a simply supported beam with a distributed load all the way across the beam (which is what we have) the equation for the Max moment is
Mmax = wx/2(l-x). The length of our beam is 10 foot and the distributed weight is 100 lbs/ft. So if you solve this equation you get the following
Mmax = [(100)(5)/2](10-5) = 1250lb*ft. Also for the slope the equation is: Ymax = (wx/24EI)*(2lx2-x3-l3) If you plug in the appropriate numbers
you get –1.772E-4 in. These values are almost exactly the same as what we got with our FEA Results for Moment/Deflection/Shear/Slope.
From the Convergence Graph of Moment and Deflection you can see that the mesh was in the adequate mesh zone since the slope of the lines stay at 2. I couldn’t
refine the mesh enough to get round off error, my computer would crap out before I could there.
