Euler-Bernoulli BeamEdit
DisplacementsEdit
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Strain-Displacement RelationsEditEquilibrium EquationsEdit
Balance of forcesEdit
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Stress ResultantsEdit
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Constitutive RelationsEditWeak FormsEditFinite Element ModelEdit
Finite element model for Euler Bernoulli beam
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where .
Hermite Cubic Shape FunctionsEdit
Hermite shape functions for beam
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Finite Element EquationsEdit
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where
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Symmetric Stiffness MatrixEdit
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Load VectorEdit
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Newton-Raphson SolutionEdit
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where
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The residual is
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For Newton iterations, we use the algorithm
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where the tangent stiffness matrix is given by
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Tangent Stiffness MatrixEdit
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Load StepsEdit
Recall
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- Divide load into small increments.
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- Compute and for first load step,
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Stiffness of Euler-Bernoulli beam.
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- Compute and for second load step,
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- Continue until F is reached.
Membrane LockingEdit
Recall
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where
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Mebrane locking in Euler-Bernoulli beam
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For Hinged-HingedEdit
Membrane strain:
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or
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Hence, shape functions should be such that
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linear, cubic Element Locks!
Too stiff.
Selective Reduced IntegrationEdit