A two-node Timoshenko beam element of length L=2 uses linear interpolation for…

A two-node Timoshenko beam element of length \(L=2\) uses linear interpolation for transverse displacement \(w\) and cross-section rotation \(\theta\): \(w=N_1w_1+N_2w_2\), \(\theta=N_1\theta_1+N_2\theta_2\), where \(N_1=(1-\xi)/2\), \(N_2=(1+\xi)/2\), and \(x=L(1+\xi)/2\). Its bending strain is \(\kappa=d\theta/dx\), its shear strain is \(\gamma=\theta-dw/dx\), and its energy is \(\tfrac12\int_0^L[EI\kappa^2+kGA\gamma^2],dx\), with \(EI=1\) and \(kGA=1000\). Derive the \(4\times4\) element stiffness matrix in the degree-of-freedom order \([w_1,\theta_1,w_2,\theta_2]\) for (a) full two-point integration of both terms and (b) selective reduced integration using two points for bending and one point for shear. The one-point rule is \((\xi,w)=(0,2)\), and the two-point rule uses \(\xi=\pm1/\sqrt{3}\) with unit weights. With \(w_1=\theta_1=0\) and a unit transverse nodal force applied to \(w_2\), solve for \(w_2\) and \(\theta_2\) in both cases and compare the tip displacement with \(PL^3/(3EI)+PL/(kGA)\) for \(P=1\).

Write separate row vectors for curvature and shear strain before integrating their outer products.