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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