Skip to main content
ProofAnvil

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

A prismatic Euler-Bernoulli beam element has length L=2.4 m and local degree-of-freedom…

Problem

A prismatic Euler-Bernoulli beam element has length \(L=2.4\,\text{m}\) and local degree-of-freedom vector \(\mathbf d=[w_1,\theta_1,w_2,\theta_2]^T\), where \(w\) is positive upward and \(\theta=dw/dx\). With \(\xi=x/L\), its Hermite interpolation is \(w(x)=N_1w_1+N_2\theta_1+N_3w_2+N_4\theta_2\), where \(N_1=1-3\xi^2+2\xi^3\), \(N_2=L(\xi-2\xi^2+\xi^3)\), \(N_3=3\xi^2-2\xi^3\), and \(N_4=L(-\xi^2+\xi^3)\). For \(\mathbf d=[4.0\,\text{mm},-3.0\,\text{mrad},-2.0\,\text{mm},5.0\,\text{mrad}]^T\), calculate \(w\), \(dw/dx\), and the Euler-Bernoulli curvature \(d^2w/dx^2\) at \(x=0.90\,\text{m}\). Use metres and radians in the calculation.

Hint

Convert millimetres to metres and milliradians to radians before multiplying by the shape functions.

Check your work

Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.

Check your answerSee the full solution

Create a free account to check your answer and see the solution. Create a free account.

More Finite Element Methods practice problems