Intro

A CPP code developed to solve 1D Bernoulli Euler beam problem in FEM using continuous Hermite-shape function.

Phsical problem

$u_{,xxxx} = l(x)$ on $\mathbf{\Omega} = (0,L)$ (transverse equilibrium)

$u(L) = 0$ (zero transverse displacement)

$u_{,x}(L) = 0$ (zero slope)

$EI u_{,xx}(0) = M$ (prescribed moment)

$EI u_{,xxx}(0) = Q$ (prescribed shear)

Notes: $E$ is Young's modulus and $I$ is the moment of inertia, both of which are assumed to be constant.

Exact solution

We can solve exact solution of this 4th order ODE with four boundary conditions

Code details

void Gauss(int N, double a, double * qp, double * wq) assigns values to qp and wq.

double Hermitebasis(const double &x1, const double &x2, int i, int der, double &x) returns specific value of Hermite-shape function.

Hermite-shape function has $C^1$ continuity on the element nodes.

uu = solver.solve(F) is filled with $[u_{1},u_{1,x},u_{2},u_{2,x},...]^T$.

Run the code

1. Run mkdir build under FEM-Bernoulli_Euler_beam/ to create a new directory.
2. Run cd build to step in the new directory.
3. Run cmake .. to create makefile using CMakeLists.txt from upper directory.
4. Run make to create an executable file named Bernoulli_Euler_beam.
5. Run ./Bernoulli_Euler_beam.