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7c-TimoshenkoBeamExample

# 7c-TimoshenkoBeamExample - MAE M168/CEE M135C Introduction...

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© William Klug MAE M168/CEE M135C Introduction to Finite Element Methods Lecture 7c Galerkin for Bernoulli-Euler and and Timoshenko Beam problems

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© William Klug Bernoulli-Euler Theory: Kinematics Plane sections remain plane and normal to the neutral axis (NA) Cross-section Rotation: Axial Displacement: Axial Strain: Curvature of the NA:
© William Klug Bernoulli-Euler Theory: Resultants Bending moment Note: If we assume Hooke’s law and neglect stresses and strains in the plane of the cross-section: (Moment of Inertia)

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© William Klug Bernoulli-Euler Theory: Virtual Work Admissibility Conditions: 1) Real and virtual curvatures must be bounded, i.e., continuous slope with no kinks 2) Must satisfy essential boundary conditions (EBC): Displacement or slope may be specified on boundary Hjelmstad
© William Klug FEM approximation Galerkin/Rayleigh-Ritz idea: – Approximate unknown field (displacement) by piecewise polynomial interpolation over each element

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7c-TimoshenkoBeamExample - MAE M168/CEE M135C Introduction...

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