User:EML4500.f08.RAMROD.E/HW6

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Lecture Notes 11/5/08 Meeting 30[edit]

To solve continuous Principle of Virtual Work problem, we must use integration by parts. Recall from calculus integration by parts shown below.

Principle of Virtual Work

For:

and

  • Note that the left-hand side of this equation,, equals .

Finally, simplifying we get the formula for integration by parts:

Applying this principle to the equation for continuous PVW equation:

Denoting the first term, , as and as , we can plug these terms into the above formula and simplify the continuous PVW equation using integration by parts.

Where the first term is denoted as and the second term is denoted by .


<\br>Now, return to the model problem of a beam shown below.

Beam model.jpeg

The boundary conditions for this model are:

and

At x=0, select the weighting function, , such that . (i.e. kinematically admissible)


Principle of Virtual Work
Motivation of this technique

The motivation for this method is shown by the discrete PVW applied to Equation form 9/17 shown below:

for all
with
where and are the known applied forces while ,,, and are the unknown reactions.

Since can be selected arbitrarily, select such that so to eliminate equations involving the unknowns, (i.e. eliminate rows 1,2,5 & 6 of the k matrix).

Such that:

  • Note: for all .

Where,


Back to Continuous PVW

By selecting , unknown reaction, term cancels.

So, continuous PVW simplifies down to:

for all such that .


Matlab Portion[edit]

In order to construct the lumped mass matrix, a matrix that stores the mass associated with each degree of freedom at each node, the density, 7.8 g/cm^3, is multiplied by the cross sectional area, 4 cm^2, and by half of the length of the associated element. To do this, a length matrix was also created using the x and y coordinates of each node and the Pythagorean theorem. After the lumped mass matrix, M, was created, a simpler mass matrix, m, was made from the diagonals of M. Lastly, assuming K_bar and M_bar are the simplified stiffness and mass matrices using PVW, we can solve for the eigen values and eigen vectors of the generalized eigenvector problem:

In this code, matrix D are the eigenvectors and the diagonal of matrix V are the eigenvalues.

Finally, we isolate only the three smallest eigenvalue/eigenvector pairs and use them to plot the associated three deformations of the structure with respect to the undeformed structure. This is shown in the three figures following the matlab code below. This code is intended to be executed following the above techniques for plotting the deformed shape.

L=zeros(n_elem,1);
for i=1:n_elem;
      node_1 = node_connect(1,i);
      node_2 = node_connect(2,i);
L(i,1)=abs(sqrt(((x(node_1)-x(node_2))^2)+((y(node_1)-y(node_2))^2)));
end

m=zeros(n_node,1);

for i=1:((n_elem)-1);
   m(2*i,1)=0.5*7.8*4*L(i,1);
   m((2*i)-1,1)=m(2*i,1);
end

M=zeros(n_node,1);

for i=1:n_node;
    M(2*i,2*i)=m(i,1);
    M((2*i)-1,(2*i)-1)=m(i,1);
    
end

K_bar=K;
M_bar=M;

[V,D]=eig(K_bar,M_bar)

A=sort(D);
B=sort(A(1,:));
min_eval=[B(1,1);B(1,2);B(1,3)];

min_evec1=V(:,1)
min_evec2=V(:,2)
min_evec3=V(:,3)

y=zeros(92,1);
y(1,1)=0;

for i=2:92;
    y(i,1)=1+y(i-1);
end

The following are the results and plots from the above Matlab Code:

V =

  Columns 1 through 7

   -0.0014944     -0.05031     0.038896     0.014198    -0.032696    -0.001106    0.0071974
   -0.0094129     0.021545    -0.011552    0.0065082     0.020884    0.0011283   -0.0094644
   -0.0026749    0.0029728    -0.063282     0.017282    -0.032582  -0.00085494    0.0066026
   -0.0020615     0.009503     -0.01669     0.018202    -0.020147  -0.00055093    0.0079385
   -0.0049418    -0.019243   -0.0096861     0.011171    -0.015833  -0.00026065  -0.00036456
   -0.0083028      0.01154    0.0040931    0.0074832     0.015165   0.00083623   -0.0067624
   -0.0049418    -0.019243   -0.0096861     0.011171    -0.016113  -0.00029973  -0.00025865
   -0.0049431    -0.018737      0.05144     0.010434   0.00010155   0.00012578  -0.00032828
   -0.0049418    -0.019243   -0.0096861     0.011171    -0.015761  -0.00031107  -0.00042716
   -0.0027531    0.0027253  -0.00033855     0.016338    -0.014736  -0.00037307    0.0055664
   -0.0073676     -0.01539   -0.0077489    0.0073001  -0.00091268   0.00031983   -0.0065232
   -0.0075273     0.010308    0.0034739    0.0087205    0.0092429   0.00055298   -0.0040796
   -0.0073676     -0.01539   -0.0077489    0.0073001  -0.00087416   0.00031888   -0.0064171
   -0.0055158    0.0071134    0.0018676      0.01193  9.7725e-005   0.00035942  -0.00046686
   -0.0073676     -0.01539   -0.0077489    0.0073001  -0.00091143   0.00032318   -0.0065745
   -0.0035286     0.003957   0.00028073       0.0151   -0.0088887  -0.00015875    0.0029767
   -0.0097911    -0.011541   -0.0058137    0.0034335     0.011117   0.00045368    -0.009729
   -0.0067275    0.0090379    0.0028352    0.0099965    0.0040127   0.00040091   -0.0021783
   -0.0097911    -0.011541   -0.0058137    0.0034335     0.011098    0.0010523   -0.0096099
   -0.0043041    0.0051888       0.0009     0.013863   -0.0038229  8.9657e-005    0.0012635
    -0.011003   -0.0096163   -0.0048461    0.0015002     0.014537   0.00035216    -0.011152
    -0.007188    0.0097693    0.0032029    0.0092618    0.0040206    0.0005884   -0.0025633
    -0.011003   -0.0096163   -0.0048461    0.0015002     0.014342     0.000869    -0.010619
   -0.0055158    0.0071134    0.0018676      0.01193  8.9107e-005  -0.00070175  -0.00060388
    -0.011003   -0.0096163   -0.0048461    0.0015002     0.014591    0.0014347    -0.011022
   -0.0038679    0.0044959   0.00055166     0.014559    -0.003822   0.00025881    0.0017894
    -0.012215   -0.0076917   -0.0038785  -0.00043309     0.016733    0.0014591    -0.011545
   -0.0076242     0.010462    0.0035513    0.0085658    0.0033657    0.0017022   -0.0021194
    -0.012215   -0.0076917   -0.0038785  -0.00043309     0.016631    0.0014314    -0.011354
   -0.0065094    0.0086915    0.0026611     0.010344    0.0030594  -0.00063799   -0.0029699
    -0.012215   -0.0076917   -0.0038785  -0.00043309     0.016621   0.00044098    -0.011493
   -0.0045222    0.0055352    0.0010742     0.013515   -0.0028964   -0.0009389    0.0017509
    -0.012215   -0.0076917   -0.0038785  -0.00043309     0.016676   0.00043757    -0.011595
   -0.0034316    0.0038031   0.00020332     0.015255   -0.0031524    0.0014629    0.0018065
    -0.012988   -0.0064638   -0.0032611   -0.0016665       0.0163    0.0036404   -0.0098054
   -0.0071395    0.0096923    0.0031642    0.0093392    0.0036554    0.0003512   -0.0032446
    -0.012988   -0.0064638   -0.0032611   -0.0016665     0.016242   -0.0019621    -0.010752
   -0.0038921    0.0045344   0.00057101      0.01452   -0.0034357   3.967e-005    0.0023552
    -0.014616   -0.0038772   -0.0019607   -0.0042649    0.0050086      0.01472    0.0038994
   -0.0084967     0.011848     0.004248    0.0071739   -0.0038492     0.007872    0.0048045
    -0.014616   -0.0038772   -0.0019607   -0.0042649    0.0048501     -0.01516   -0.0019165
   -0.0025592    0.0024174  -0.00049337     0.016647    0.0040403     0.008784   -0.0022205
    -0.016623  -0.00069005  -0.00035828   -0.0074664 -1.7925e-005    0.0063344     0.015577
    -0.012859     0.018776    0.0077314   0.00021403    -0.026786    -0.046864    -0.055085
    -0.016623  -0.00069005  -0.00035828   -0.0074664  -0.00066765    0.0044429     0.011469
     -0.01029     0.014696      0.00568    0.0043126    -0.010806   -0.0066796   -0.0028508
    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0010076    0.0035273    0.0095149
   -0.0092237     0.013003    0.0048285    0.0060139   -0.0064944    0.0032482    0.0044614
    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0012259    0.0028835    0.0082753
   -0.0084482     0.011771    0.0042093    0.0072512   -0.0039643    0.0092514    0.0047445
    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0015742    0.0018451    0.0073598
   -0.0074788     0.010231    0.0034352    0.0087978   -0.0013382     0.014987    0.0044037
    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0019923   0.00059961    0.0069186
   -0.0062913    0.0083451    0.0024869     0.010692  -0.00020806     0.019347    0.0042055
    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0019997   -0.0011384    0.0065969
   -0.0047645    0.0059201    0.0012677     0.013128   0.00055104     0.019693    0.0025555
    -0.016623  -0.00069005  -0.00035828   -0.0074664    -0.001584    -0.002481    0.0065811
   -0.0035528    0.0039955   0.00030008     0.015062    0.0016607      0.01567   0.00083891
    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0012435   -0.0036022    0.0071142
   -0.0026077    0.0024944  -0.00045466      0.01657    0.0041473      0.01023   -0.0015388
    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0010142   -0.0043633     0.008214
   -0.0018321    0.0012626   -0.0010739     0.017807     0.006589    0.0042216   -0.0034572
    -0.016623  -0.00069005  -0.00035828   -0.0074664  -0.00064793    -0.005506     0.010106
  -0.00074159   -0.0004695   -0.0019448     0.019547     0.010882   -0.0063904     0.001122
    -0.016623  -0.00069005  -0.00035828   -0.0074664   0.00015962   -0.0080805     0.014385
     0.001803   -0.0045111   -0.0039768     0.023607     0.027895    -0.053301     0.047945
     -0.01718   0.00019526  8.6827e-005   -0.0083557   -0.0032142    -0.001658    0.0057069
     -0.01092     0.015697    0.0061832    0.0033073    -0.013746     -0.01368   -0.0095212
     -0.01718   0.00019526  8.6827e-005   -0.0083557   -0.0032288      0.00119    0.0056899
  -0.00013572   -0.0014318   -0.0024286     0.020514     0.013702    -0.013691     0.005942
    -0.017483   0.00067641   0.00032873    -0.008839   -0.0042235   -0.0039517    0.0046073
   -0.0098538     0.014003    0.0053317    0.0050086   -0.0089283   -0.0023228   0.00075412
    -0.017483   0.00067641   0.00032873    -0.008839   -0.0041914    0.0034874    0.0054104
   -0.0011778   0.00022335   -0.0015965     0.018851    0.0090606   -0.0018008   -0.0012768
    -0.017665   0.00096509   0.00047387    -0.009129   -0.0046243   -0.0048842    0.0049671
   -0.0092237     0.013003    0.0048285    0.0060139   -0.0066272    0.0030477    0.0034716
    -0.017665   0.00096509   0.00047387    -0.009129   -0.0041038   -0.0036227    0.0063496
    -0.008012     0.011078    0.0038609    0.0079472   -0.0026624     0.012219    0.0047374
    -0.017665   0.00096509   0.00047387    -0.009129   -0.0035153    -0.002223    0.0069672
   -0.0068487    0.0092304     0.002932    0.0098031  -0.00061368     0.017688    0.0044436
    -0.017665   0.00096509   0.00047387    -0.009129   -0.0032428  -0.00042462    0.0074362
   -0.0055158    0.0071134    0.0018676      0.01193   0.00018008     0.020205    0.0034991
    -0.017665   0.00096509   0.00047387    -0.009129   -0.0034892    0.0014233    0.0075404
   -0.0041829    0.0049963   0.00080324     0.014056   0.00095992     0.018184    0.0017389
    -0.017665   0.00096509   0.00047387    -0.009129   -0.0040526    0.0029032    0.0073444
   -0.0030439    0.0031872  -0.00010632     0.015874    0.0028869     0.013143  -0.00040913
    -0.017665   0.00096509   0.00047387    -0.009129   -0.0045661    0.0043338     0.006219
   -0.0018321    0.0012626   -0.0010739     0.017807    0.0067302    0.0039641   -0.0024242
    -0.020088    0.0048143    0.0024091    -0.012996    -0.012773    -0.023915    0.0046152
   -0.0092237     0.013003    0.0048285    0.0060139    -0.006803    0.0026545    0.0036709
    -0.020088    0.0048143    0.0024091    -0.012996    -0.012464     0.023128     0.013457
   -0.0018321    0.0012626   -0.0010739     0.017807    0.0069032    0.0035928   -0.0028846

  Columns 8 through 14

    0.0066546     0.018818   -0.0072996     0.021862     0.015029    -0.015567    -0.013693
    -0.014606     -0.01366     0.014578    -0.018684     -0.01909     0.026004      0.02282
    -0.007662     0.017069    0.0086158     0.024413     0.014404     0.016827      0.02152
    -0.015646    0.0099807     0.015714     0.023035     0.017577     0.022624     0.023672
   0.00057936    0.0052153   -0.0013058  -0.00028167   -0.0042208   -0.0033513   -0.0055633
    -0.012112   -0.0078425     0.011319    -0.008877     -0.01045     0.016385     0.013797
  -5.195e-006    0.0062641  5.8781e-005    0.0018738   -0.0032159    0.0019347    0.0019017
   -0.0068736  -0.00084155    0.0073392    0.0011035  -0.00035545     0.013768     0.014389
  -0.00056303    0.0047573    0.0013334 -4.0125e-006   -0.0043484    0.0065872    0.0073062
    -0.012905    0.0050606     0.012051     0.012486    0.0096137     0.014271     0.011743
   0.00045534   -0.0076039  -0.00051373     -0.01946    -0.017665    0.0013262   -0.0011838
    -0.011249   0.00072992    0.0090261    0.0041759   -0.0011735    0.0074295    0.0048084
   0.00031187   -0.0071523  -0.00022058    -0.017611    -0.016037    0.0015853  -0.00048388
    -0.010646  -0.00080776    0.0077078    0.0014474  1.6662e-006    0.0018068   -0.0037741
   0.00015335   -0.0078603   0.00012241    -0.020033    -0.018305    0.0027887  -0.00053652
    -0.012009    -0.002967    0.0099424   -0.0011645   0.00070286    0.0072079    0.0011141
   0.00016652   -0.0049453  -0.00037902   -0.0042007   -0.0036596    -0.002754   0.00075953
   -0.0098021    0.0042999    0.0059142    0.0055826  -0.00061762  -0.00063858   -0.0050555
    0.0012193   -0.0047806  -0.00090763   -0.0042104   -0.0035975   -0.0026283  -0.00019841
    -0.010273   -0.0056355    0.0064372   -0.0031484   0.00083897   -0.0027709   -0.0073692
   0.00020412    0.0040957  -0.00099802    0.0073957  -0.00031887   -0.0037235  -0.00036719
   -0.0083677     0.010547    0.0035844    0.0099322   -0.0013981     -0.00368   -0.0084516
   0.00090541    0.0024708   -0.0008352    0.0048302  -0.00022073   -0.0018631   0.00089764
    -0.010716  -0.00061777    0.0049711    0.0010323   0.00016098   -0.0068412   -0.0094633
    0.0018121    0.0041723   -0.0010859    0.0074942  -0.00032555   0.00064445    0.0041615
    -0.009112    -0.011122    0.0044496   -0.0078457    0.0019855   -0.0086869    -0.012305
    0.0015331     0.012465   -0.0007433     0.015077    0.0015499    0.0003494  -0.00090943
   -0.0057456       0.0148    0.0016457     0.010274   -0.0026705   -0.0032403    -0.010062
    0.0015559      0.01156  -0.00076085     0.013719    0.0014912    0.0004219   0.00010342
   -0.0099528    0.0079996     0.003304    0.0089868   -0.0024377   -0.0077263   -0.0097806
   0.00056878     0.011285   -0.0011587     0.013286    0.0014097   -0.0009701    0.0021191
    -0.010692   -0.0087847    0.0044835   -0.0069132    0.0029417     -0.01184    -0.015201
   0.00061577     0.011764   -0.0012315     0.013934    0.0014093  -0.00070067    0.0028758
   -0.0063287    -0.014351    0.0021133   -0.0082928    0.0035201   -0.0098438    -0.011506
    0.0046923     0.015608  -0.00088502     0.013999    0.0030114    0.0050119   -0.0016117
   -0.0077337     0.013114    0.0017569     0.011323   -0.0035634   -0.0061465     -0.01006
   -0.0029975     0.014531 -3.2679e-005     0.013237    0.0026879   -0.0043136   -0.0011128
   -0.0085031     -0.01298    0.0028626   -0.0090424    0.0042917    -0.012174    -0.014322
     0.022278     0.013709   -0.0036124    -0.014494    0.0087426     0.020364   -0.0045793
    0.0063637     0.015648    -0.003688   -0.0075924    -0.003578    0.0028635   -0.0095753
    -0.024356     0.011616    0.0099995    -0.010778    0.0070027    -0.020287    -0.023433
    0.0076003    -0.014085   -0.0069026    0.0056614    0.0055495    -0.004477    0.0014002
   -0.0045508 -5.7031e-005   -0.0096163   -0.0060887    0.0067607    0.0038396     0.016364
     0.038374    -0.010291     0.016216   -0.0023307     0.017948     0.016688     -0.01344
  -0.00035847   -0.0020919   -0.0047725   -0.0062583     0.011705     0.012888      0.00285
    0.0023535     0.013259    -0.014923   -0.0039882    -0.016594    -0.024673     0.027007
    0.0014884   -0.0023001   -0.0034625   -0.0062617     0.011893     0.011306    0.0031097
    0.0024398     0.014651    -0.010587   -0.0061214   -0.0097526   -0.0033283   -0.0021663
    0.0024289   -0.0027026   -0.0019896   -0.0060187     0.011988    0.0083103    0.0046988
    0.0094551     0.021195   -0.0062669    -0.012177   -0.0069277    0.0023437    -0.010074
    0.0020794   -0.0050287   -0.0019391   -0.0036606     0.011002    0.0058826    0.0043496
     0.016292     0.022583   -0.0029114    -0.014072   -0.0031642    0.0091915    -0.014098
   0.00083095   -0.0075368  -0.00055325   -0.0018046     0.010501   0.00060967     0.004508
     0.021099     0.011645   -0.0038619   -0.0082723  -0.00021821     0.016225    -0.014995
   -0.0012132   -0.0076824    0.0018541   -0.0013014    0.0098783    -0.006317    0.0025992
     0.021881   -0.0078466   -0.0063384    0.0034517    0.0029722       0.0111   -0.0034253
   -0.0027469    -0.005289    0.0039517   -0.0025361    0.0095021    -0.010008   -0.0016801
     0.017834    -0.019778   -0.0080751     0.010463    0.0053227   -0.0010602    0.0067537
   -0.0034107    -0.003034    0.0047744   -0.0044908    0.0098783    -0.010375   -0.0046068
     0.010829     -0.01892    -0.010311    0.0088316      0.00942   -0.0089837    0.0054424
   -0.0026886   -0.0025977    0.0062134   -0.0043238    0.0088284    -0.010947   -0.0058995
    0.0030455    -0.012722    -0.011821    0.0030047     0.012374    -0.010863   0.00050371
   -0.0011435   -0.0023285     0.007443    -0.004095    0.0078891     -0.01078   -0.0064964
    0.0011195    -0.010779    -0.013715   0.00069984     0.016164    -0.018132  -0.00037073
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  Columns 15 through 21

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  Columns 22 through 28

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  Columns 29 through 35

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  Columns 36 through 42

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  Columns 43 through 49

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  Columns 50 through 56

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  Columns 57 through 63

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  Columns 64 through 70

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  Columns 71 through 77

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  Columns 78 through 84

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  Columns 85 through 91

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  Column 92

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  Columns 36 through 42

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  Columns 71 through 77

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min_evec1 =

   -0.0014944
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   -0.0083028
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   -0.0065094
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   -0.0045222
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   -0.0034316
    -0.012988
   -0.0071395
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   -0.0038921
    -0.014616
   -0.0084967
    -0.014616
   -0.0025592
    -0.016623
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     -0.01029
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   -0.0092237
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   -0.0026077
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    -0.016623
  -0.00074159
    -0.016623
     0.001803
     -0.01718
     -0.01092
     -0.01718
  -0.00013572
    -0.017483
   -0.0098538
    -0.017483
   -0.0011778
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   -0.0030439
    -0.017665
   -0.0018321
    -0.020088
   -0.0092237
    -0.020088
   -0.0018321


min_evec2 =

     -0.05031
     0.021545
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     0.009503
    -0.019243
      0.01154
    -0.019243
    -0.018737
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    0.0027253
     -0.01539
     0.010308
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    0.0071134
     -0.01539
     0.003957
    -0.011541
    0.0090379
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    0.0051888
   -0.0096163
    0.0097693
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    0.0071134
   -0.0096163
    0.0044959
   -0.0076917
     0.010462
   -0.0076917
    0.0086915
   -0.0076917
    0.0055352
   -0.0076917
    0.0038031
   -0.0064638
    0.0096923
   -0.0064638
    0.0045344
   -0.0038772
     0.011848
   -0.0038772
    0.0024174
  -0.00069005
     0.018776
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     0.014696
  -0.00069005
     0.013003
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     0.011771
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     0.010231
  -0.00069005
    0.0083451
  -0.00069005
    0.0059201
  -0.00069005
    0.0039955
  -0.00069005
    0.0024944
  -0.00069005
    0.0012626
  -0.00069005
   -0.0004695
  -0.00069005
   -0.0045111
   0.00019526
     0.015697
   0.00019526
   -0.0014318
   0.00067641
     0.014003
   0.00067641
   0.00022335
   0.00096509
     0.013003
   0.00096509
     0.011078
   0.00096509
    0.0092304
   0.00096509
    0.0071134
   0.00096509
    0.0049963
   0.00096509
    0.0031872
   0.00096509
    0.0012626
    0.0048143
     0.013003
    0.0048143
    0.0012626


min_evec3 =

     0.038896
    -0.011552
    -0.063282
     -0.01669
   -0.0096861
    0.0040931
   -0.0096861
      0.05144
   -0.0096861
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   -0.0077489
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   -0.0077489
   0.00028073
   -0.0058137
    0.0028352
   -0.0058137
       0.0009
   -0.0048461
    0.0032029
   -0.0048461
    0.0018676
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   0.00055166
   -0.0038785
    0.0035513
   -0.0038785
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    0.0010742
   -0.0038785
   0.00020332
   -0.0032611
    0.0031642
   -0.0032611
   0.00057101
   -0.0019607
     0.004248
   -0.0019607
  -0.00049337
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    0.0077314
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      0.00568
  -0.00035828
    0.0048285
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    0.0042093
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    0.0034352
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    0.0024869
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    0.0012677
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   0.00030008
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  -0.00035828
   -0.0010739
  -0.00035828
   -0.0019448
  -0.00035828
   -0.0039768
  8.6827e-005
    0.0061832
  8.6827e-005
   -0.0024286
   0.00032873
    0.0053317
   0.00032873
   -0.0015965
   0.00047387
    0.0048285
   0.00047387
    0.0038609
   0.00047387
     0.002932
   0.00047387
    0.0018676
   0.00047387
   0.00080324
   0.00047387
  -0.00010632
   0.00047387
   -0.0010739
    0.0024091
    0.0048285
    0.0024091
   -0.0010739


Min evec1.jpg
Min evec3.jpg