Solution and Answer Guide
m m m
to Accompany
m
A First Course in the Fini
m m m m m
te Element Method
m m
Enhanced 6 m
th
m EDITION
DARYL L. LOGAN
M M
, Contents
Chapter m1.............................................................................................................................. 1
Chapter m2.............................................................................................................................. 3
Chapter m3............................................................................................................................ 25
Chapter m4.......................................................................................................................... 137
Chapter m5.......................................................................................................................... 203
Chapter m6.......................................................................................................................... 315
Chapter m7.......................................................................................................................... 363
Chapter m8.......................................................................................................................... 383
Chapter m9.......................................................................................................................... 397
Chapter m10........................................................................................................................ 423
Chapter m11........................................................................................................................ 449
Chapter m12........................................................................................................................ 477
Chapter m13........................................................................................................................ 499
Chapter m14........................................................................................................................ 539
Chapter m15........................................................................................................................ 561
Chapter m16........................................................................................................................ 591
AppendixmA ....................................................................................................................... 629
AppendixmB: ...................................................................................................................... 635
AppendixmD....................................................................................................................... 641
, Chapter 1 m
1.1. Am finitem elementm ism am smallm bodym orm unitm interconnectedm to m otherm units m tom model m a m l
argermstructuremormsystem.
1.2. Discretizationmmeansmdividingmthembodym(system) mintomanmequivalentmsystemmofmfinitemelem
entsmwithmassociatedmnodesmandmelements.
1.3. Them modernm developmentm ofm them finitem elementm methodm beganm in m 1941m with m the m wor
km ofmHrennikoff minmthemfieldmofmstructuralmengineering.
1.4. Them directm stiffnessm methodm wasm introducedm inm 1941 m bym Hrennikoff. m However, m itm was
m notmcommonlymknownmasmthemdirectmstiffnessmmethodmuntilm1956.
1.5. Ammatrixmismamrectangular marraymofmquantitiesmarrangedminmrowsmandmcolumnsmthatmismoftenm
usedmtomaidminmexpressingmandmsolvingmamsystemmofmalgebraicmequations.
1.6. Asm computerm developedm itm madem possiblem tom solve m thousands m ofm equationsm in m am matte
rm ofmminutes.
1.7. Themfollowingmaremthemgeneralmstepsmofmthemfinitemelementmmeth
od. mStepm1
Dividem them bodym intom anm equivalentm systemm ofm finitem elementsm with m associated
nodesmandmchoosemthemmostmappropriatemelementmtype. mCho
Stepm2
osemamdisplacementmfunctionmwithinmeachmelement.
Stepm3
Relatemthemstressesmtomthemstrainsmthroughmthemstress/strainmlaw—
generallymcalledmthemconstitutivemlaw.
Stepm4
Derivemthemelementm stiffnessmmatrixmandmequations. mUsemthemdirectmequilibriumm
method, mamworkmormenergymmethod, mormammethodmofmweightedmresidualsmtomrelate
mthemnodalmforcesmtomnodalmdisplacements.
Stepm5
Assemblemthemelementmequationsmtomobtainmthemglobalmormtotalmequationsmandmint
roducemboundarymconditions.
Stepm6
Solvemformthemunknownmdegreesmofmfreedomm(ormgeneralizedmdisplacements). mS
Stepm7
olvemformthemelementmstrainsmandmstresses.
Stepm8
Interpretmandmanalyzemthemresultsmformuseminmthemdesign/analysismprocess.
1.8. Themdisplacementmmethodmassumesmdisplacements mofmthemnodes mas mthemunknowns mofmthempr
oblem. mThemproblemmismformulatedmsuch mthatmam setmofmsimultaneousmequationsmism solvedmfor
mnodalmdisplacements.
1.9. Fourmcommonmtypesmofmelementsmare:msimplemlinemelements, msimplemtwo-
dimensionalmelements, msimplemthree-
dimensionalmelements, mandmsimplemaxisymmetricmelements.
1.10m Threemcommonmmethodsmusedmtomderivemthemelementmstiffnessmmatrixmandmequationsmare
(1) directmequilibriummmethod
(2) workmormenergymmethods
1
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3
, (3) methodsmofmweightedmresiduals
1.11. Themtermm‘degreesmofmfreedom’ mrefersmtomrotationsmandmdisplacementsmthatmaremassociatedmw
ithmeachmnode.
1.12. Fivemtypicalmareasmwheremthemfinitemelementmismappliedmaremasmfollows.
(1) Structural/stressm analysis
(2) Heatmtransfermanalysis
(3) Fluidmflowmanalysis
(4) Electricmormmagneticmpotentialmdistributionmanalysis
(5) Biomechanicalm engineering
1.13. Fivemadvantagesmofmthemfinitemelementmmethodmaremthemabilitymto
(1) Modelmirregularlymshapedmbodiesmquitemeasily
(2) Handlemgeneralmloadmconditionsmwithoutmdifficulty
(3) Modelm bodiesm composedmofmseveralm differentm materialsm becausem elementm equationsm a
remevaluatedmindividually
(4) Handlemunlimitedmnumbersmandmkindsmofmboundarymconditions
(5) Varymthemsizemofmthemelementsmtommakemitmpossiblemtomusemsmallmelementsmwheremnecessary
2
©m202
3
m m m
to Accompany
m
A First Course in the Fini
m m m m m
te Element Method
m m
Enhanced 6 m
th
m EDITION
DARYL L. LOGAN
M M
, Contents
Chapter m1.............................................................................................................................. 1
Chapter m2.............................................................................................................................. 3
Chapter m3............................................................................................................................ 25
Chapter m4.......................................................................................................................... 137
Chapter m5.......................................................................................................................... 203
Chapter m6.......................................................................................................................... 315
Chapter m7.......................................................................................................................... 363
Chapter m8.......................................................................................................................... 383
Chapter m9.......................................................................................................................... 397
Chapter m10........................................................................................................................ 423
Chapter m11........................................................................................................................ 449
Chapter m12........................................................................................................................ 477
Chapter m13........................................................................................................................ 499
Chapter m14........................................................................................................................ 539
Chapter m15........................................................................................................................ 561
Chapter m16........................................................................................................................ 591
AppendixmA ....................................................................................................................... 629
AppendixmB: ...................................................................................................................... 635
AppendixmD....................................................................................................................... 641
, Chapter 1 m
1.1. Am finitem elementm ism am smallm bodym orm unitm interconnectedm to m otherm units m tom model m a m l
argermstructuremormsystem.
1.2. Discretizationmmeansmdividingmthembodym(system) mintomanmequivalentmsystemmofmfinitemelem
entsmwithmassociatedmnodesmandmelements.
1.3. Them modernm developmentm ofm them finitem elementm methodm beganm in m 1941m with m the m wor
km ofmHrennikoff minmthemfieldmofmstructuralmengineering.
1.4. Them directm stiffnessm methodm wasm introducedm inm 1941 m bym Hrennikoff. m However, m itm was
m notmcommonlymknownmasmthemdirectmstiffnessmmethodmuntilm1956.
1.5. Ammatrixmismamrectangular marraymofmquantitiesmarrangedminmrowsmandmcolumnsmthatmismoftenm
usedmtomaidminmexpressingmandmsolvingmamsystemmofmalgebraicmequations.
1.6. Asm computerm developedm itm madem possiblem tom solve m thousands m ofm equationsm in m am matte
rm ofmminutes.
1.7. Themfollowingmaremthemgeneralmstepsmofmthemfinitemelementmmeth
od. mStepm1
Dividem them bodym intom anm equivalentm systemm ofm finitem elementsm with m associated
nodesmandmchoosemthemmostmappropriatemelementmtype. mCho
Stepm2
osemamdisplacementmfunctionmwithinmeachmelement.
Stepm3
Relatemthemstressesmtomthemstrainsmthroughmthemstress/strainmlaw—
generallymcalledmthemconstitutivemlaw.
Stepm4
Derivemthemelementm stiffnessmmatrixmandmequations. mUsemthemdirectmequilibriumm
method, mamworkmormenergymmethod, mormammethodmofmweightedmresidualsmtomrelate
mthemnodalmforcesmtomnodalmdisplacements.
Stepm5
Assemblemthemelementmequationsmtomobtainmthemglobalmormtotalmequationsmandmint
roducemboundarymconditions.
Stepm6
Solvemformthemunknownmdegreesmofmfreedomm(ormgeneralizedmdisplacements). mS
Stepm7
olvemformthemelementmstrainsmandmstresses.
Stepm8
Interpretmandmanalyzemthemresultsmformuseminmthemdesign/analysismprocess.
1.8. Themdisplacementmmethodmassumesmdisplacements mofmthemnodes mas mthemunknowns mofmthempr
oblem. mThemproblemmismformulatedmsuch mthatmam setmofmsimultaneousmequationsmism solvedmfor
mnodalmdisplacements.
1.9. Fourmcommonmtypesmofmelementsmare:msimplemlinemelements, msimplemtwo-
dimensionalmelements, msimplemthree-
dimensionalmelements, mandmsimplemaxisymmetricmelements.
1.10m Threemcommonmmethodsmusedmtomderivemthemelementmstiffnessmmatrixmandmequationsmare
(1) directmequilibriummmethod
(2) workmormenergymmethods
1
©m202
3
, (3) methodsmofmweightedmresiduals
1.11. Themtermm‘degreesmofmfreedom’ mrefersmtomrotationsmandmdisplacementsmthatmaremassociatedmw
ithmeachmnode.
1.12. Fivemtypicalmareasmwheremthemfinitemelementmismappliedmaremasmfollows.
(1) Structural/stressm analysis
(2) Heatmtransfermanalysis
(3) Fluidmflowmanalysis
(4) Electricmormmagneticmpotentialmdistributionmanalysis
(5) Biomechanicalm engineering
1.13. Fivemadvantagesmofmthemfinitemelementmmethodmaremthemabilitymto
(1) Modelmirregularlymshapedmbodiesmquitemeasily
(2) Handlemgeneralmloadmconditionsmwithoutmdifficulty
(3) Modelm bodiesm composedmofmseveralm differentm materialsm becausem elementm equationsm a
remevaluatedmindividually
(4) Handlemunlimitedmnumbersmandmkindsmofmboundarymconditions
(5) Varymthemsizemofmthemelementsmtommakemitmpossiblemtomusemsmallmelementsmwheremnecessary
2
©m202
3
