Calfem - A finite element toolbox to MATLAB, version 3.3

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$Institut fÃ¼r Mathematik PD Dr. Matthias Ehrhardt Dipl.-Math.$

5.2 Spring elementThe spring element, shown below, can be used for the analysis of one-dimensional springsystems and for a variety of analogous physical problems.k●u 1 u 2●Quantities corresponding to the variables of the spring are listed in Table 1.Problem type Spring Nodal dis- Element Springstiffness placement force forceSpring k u P NBarEALu P NThermal conductionλALT ¯Q QElectrical circuit1RU Ī IGroundwater flowkALφ ¯Q QPipe networkπD 4128µLp ¯Q QTable 1: Analogous quantities5.2 – 1 ELEMENT
Interpretations of the spring elementProblem type Quantities DesignationsSpringu 1 , P 1N●●k●●u 2 , P 2NkuPNspring stiffnessdisplacementelement forcespring forceBaru 1 , P 1NLE, Au 2 , P 2NLEAuPNlengthmodulus of elasticityarea of cross sectiondisplacementelement forcenormal forceThermalconductionQ 1T 1●λQLT 2●Q 2LλT¯QQlengththermal conductivitytemperatureelement heat flowinternal heat flowElectricalcircuitI 1U 1●IRU 2●I 2RŪIIresistancepotentialelement currentinternal currentGroundwaterflowLQ 1 φ 1Q φ 2 Q 2 k●kL●φ¯QQlengthpermeabilitypiezometric headelement water flowinternal water flowPipenetwork(laminarflow)Q 1p 1 D, µp 2QLQ 2LDµp¯QQlengthpipe diameterviscositypressureelement fluid flowinternal fluid flowTable 2: Quantities used in different types of problemsELEMENT 5.2 – 2
Page 6: Contents1 Introduction 1 - 12 Gener
Page 9 and 10: The available functions are organiz
Page 11 and 12: clearPurpose:Syntax:Remove variable
Page 13 and 14: dispPurpose:Syntax:Display a variab
Page 15 and 16: formatPurpose:Syntax:Control the ou
Page 17 and 18: loadPurpose:Syntax:Retrieve variabl
Page 19 and 20: savePurpose:Syntax:Save workspace v
Page 21 and 22: whatPurpose:Syntax:Directory listin
Page 23 and 24: ...Purpose:Continuation.Syntax:...D
Page 26 and 27: 3 Matrix functionsThe group of matr
Page 28 and 29: :Purpose:Create vectors and do matr
Page 30 and 31: + − ∗ /Purpose:Syntax:Matrix ar
Page 32 and 33: detPurpose:Syntax:Matrix determinan
Page 34 and 35: fullPurpose:Syntax:Convert sparse m
Page 36 and 37: lengthPurpose:Syntax:Vector length.
Page 38 and 39: minPurpose:Syntax:Minimum element(s
Page 40 and 41: edPurpose:Syntax:Reduce the size of
Page 42 and 43: sparsePurpose:Syntax:Create sparse
Page 44 and 45: sqrtPurpose:Syntax:Square root.B=sq
Page 46: zerosPurpose:Syntax:Generate a zero
Page 49 and 50: hookePurpose:Syntax:Compute materia
Page 51 and 52: dmisesPurpose:Syntax:Form the elast
Page 56 and 57: The following functions are availab
Page 58: Spring elementspring1sPurpose:Compu
Page 61 and 62: ar2eTwo dimensional bar elementPurp
Page 63 and 64: ar2gTwo dimensional bar elementThe
Page 65 and 66: ar3eThree dimensional bar elementPu
Page 68 and 69: 5.4 Heat flow elementsHeat flow ele
Page 70 and 71: Two dimensional heat flow elementsf
Page 82 and 83: Three dimensional heat flow element
Page 84 and 85: 5.5 Solid elementsSolid elements ar
Page 86 and 87: planteplantsplantfplanqeplanqsplanr
Page 88 and 89: Two dimensional solid elementsplant
Page 92 and 93: Two dimensional solid elementsplanq
Page 94 and 95: Two dimensional solid elementsplanq
Page 96 and 97: Two dimensional solid elementsplanr
Page 98 and 99: Two dimensional solid elementsplanr
Page 104 and 105:
Two dimensional solid elementsplani
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Three dimensional solid elementssol
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Page 120:
Page 123 and 124:
eam2eTwo dimensional beam elementPu
Page 125 and 126:
eam2eTwo dimensional beam elementwh
Page 127 and 128:
eam2sTwo dimensional beam elementTh
Page 129 and 130:
eam2tTwo dimensional Timoshenko bea
Page 131 and 132:
eam2tsTwo dimensional Timoshenko be
Page 133 and 134:
eam2wTwo dimensional beam elementTh
Page 135 and 136:
eam2wsTwo dimensional beam elementT
Page 137 and 138:
eam2gTwo dimensional beam elementTh
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eam2gsTwo dimensional beam elementP
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eam2dTwo dimensional beam elementPu
Page 143 and 144:
eam2dsTwo dimensional beam elementP
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Three dimensional beam elementbeam3
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Page 150:
Page 153 and 154:
platrePlate elementPurpose:Compute
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platrePlate elementwhere∗∇ =⎡
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platrsPlate elementTheory:The curva
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6.2 Static system functionsThe grou
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Static system functionscoordxtrPurp
Page 164 and 165:
Static system functionseigenPurpose
Page 166 and 167:
Static system functionsextractExamp
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Static system functionssolveqPurpos
Page 170 and 171:
6.3 Dynamic system functionsThe gro
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Dynamic system functionsdyna2fPurpo
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Dynamic system functionsfreqrespPur
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Dynamic system functionsifftPurpose
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Dynamic system functionsspectraPurp
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Dynamic system functionsstep1⎡f =
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Dynamic system functionsstep2⎡f =
Page 184 and 185:
7 Statements and macrosStatements d
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forPurpose:Syntax:Initiate a loop.f
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functionPurpose:Syntax:Define a new
Page 190 and 191:
8 Graphics functionsThe group of gr
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clfPurpose:Syntax:Clear current fig
Page 194 and 195:
eldisp2Purpose:Syntax:Draw the defo
Page 196 and 197:
eldia2The magnification factor magn
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eliso2Purpose:Syntax:Display elemen
Page 200 and 201:
figurePurpose:Syntax:Create figures
Page 202 and 203:
gridPurpose:Syntax:Grid lines for 2
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plotPurpose:Syntax:Linear two dimen
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textPurpose:Syntax:Add text to curr
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xlabel, ylabel, zlabelPurpose:Synta
Page 212 and 213:
9.2 MATLAB introductionThe examples
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MATLAB introductionexi1>> whosName
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MATLAB introductionexi2Purpose:Show
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MATLAB introductionexi2The determin
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MATLAB introductionexi3B =C =3 95 7
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MATLAB introductionexi5Purpose:How
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9.3 Static analysisThis section ill
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Static analysisexs1and the load vec
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Static analysisexs1The spring force
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Static analysisexs2>> Edof=[1 1 22
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Static analysisexs3Purpose:Analysis
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Static analysisexs3Resultsa = Q =0
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Static analysisexs4A global stiffne
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Static analysisexs4>> N8=bar2s(ex8,
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Static analysisexs5>> I1=2510e-8; I
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Static analysisexs5Section force di
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Static analysisexs5543210300000−1
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Static analysisexs6Purpose:Set up a
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Static analysisexs6% ----- Create a
Page 250 and 251:
Static analysisexs7Purpose:Analysis
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Static analysisexs7Resultsa= Q=1.0e
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Static analysisexs7[a,Q]=solveq(K,f
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9.4 Dynamic analysisThis section co
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Dynamic analysisexd1% --- topology
Page 260 and 261:
Dynamic analysisexd1An attractive w
Page 262 and 263:
Dynamic analysis2exd2[Dsnap,D,V,A]=
Page 264 and 265:
Dynamic analysisexd3Purpose:This ex
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Dynamic analysisexd4Purpose:This ex
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Dynamic analysisexd4Snapshots (sec)
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exn1Nonlinear analysisPurpose:Analy
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exn1Nonlinear analysisbc=[1 0;2 0;3
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exn2Nonlinear analysisPurpose:Buckl
Page 277 and 278:
exn2Nonlinear analysisThe following
Page 279 and 280:
Indexabs . . . . . . . . . . . . 3
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