beam element
TRANSCRIPT
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BeamElement
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ElementaryBeamTheory:
Elementstiffnessequation(localnode:i,jor1,2):
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FormalApproach
Toderivethisapproach,theshapefunctionsareintroduced,
whichisacubicfunction.
whichimplies
that
the
rigid
body
Thedeflectioncanberepresentedas,
mo on srepresen e y e
assumeddeformedshapeofthe
beam.
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Curvatureofthebeamis,
wherethestraindisplacementmatrixBisgivenby,
Strainenergystoredinthebeamelementis
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Thestiffnessmatrixforthesimplebeamelementis
Combiningtheaxialstiffness(barelement),thestiffnessmatrixofageneral2Dbeam
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Example1The
beam
shown
above
is
clamped
at
the
two
endsandacteduponbytheforcePandmomentMinthemidspan.Determinethedeflectionandrotationatthecenternodeandthereaction
.
Elementstiffness
matrices
are,
Element1:
Element2:
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Loadsandconstraints(BCs)are,
,
FromglobalFEequation,weobtainthereactionforcesandmoments,
Stressesinthebeamatthetwoendscanbecalculatedusingtheformula,
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Distributedloads
Ifq(x)=0,thenexactsolutionforthedeflectionvisacubicfunctionofx,whichis.
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EquivalentNodalLoadsofDistributedTransverseLoad
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Example2Acantileverbeamissubjectedtodistributed
lateralloadp.Determinethedeflectionandrotationattherightend,thereactionforceand
.
Theworkequivalentnodalloadsare
ApplyingtheFEequation:
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Loadandconstraints(BCs)are,
Reducede uationis,
Thesenodalvaluesarethesameastheexactsolution.
Notethat
the
deflection
v(x)(for0
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GlobalFEequationis,
inwhich