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Finite Element Methods · Weak formulations, Galerkin projection, and variational structure

A two-node Timoshenko beam element of length L=1.2 m has local degrees of freedom…

Problem

A two-node Timoshenko beam element of length \(L=1.2\,\text{m}\) has local degrees of freedom \([w_1,\phi_1,w_2,\phi_2]\), where \(w\) is transverse displacement and \(\phi\) is the independent cross-section rotation. Interpolate both fields linearly with \(N_1=1-x/L\) and \(N_2=x/L\). Use curvature \(\kappa=d\phi/dx\), engineering shear strain \(\gamma=dw/dx-\phi\), and strain energy \(U=\tfrac12\int_0^L(EI\kappa^2+kGA\gamma^2)\,dx\). Derive the complete \(4\times4\) element stiffness matrix using exact integration and evaluate it for \(EI=24{,}000\,\text{kN m}^2\) and \(kGA=150{,}000\,\text{kN}\). Verify from the evaluated matrix that the two rigid-body modes produce zero internal nodal force.

Hint

Form separate strain-displacement rows for curvature and engineering shear strain.

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