Buckling of a compressive beam

In this case study, we examine the classical Euler buckling phenomenon of a slender beam under axial compression. The beam is fixed at both ends, with a gradually increasing compressive load applied at the right end. As the load increases, the beam transitions from a straight configuration to a buckled state, illustrating the onset of structural instability. The case highlights the ability of the DDG model to accurately capture buckling behavior in slender structures.

Simulation Initialization

To initialize the simulation, the following inputs are used:

1. Geometry and connection:
   • (i) Nodal positions: the position of the nodes, $\mathbf{q}(t=0)$, with a total of $N=40$, with the beam length $L=1.0\mathrm{m}$.
   • (ii) Stretching elements: connection of every two consecutive nodes, with a total of $N_s=39$.
   • (iii) Bending elements: connection of every two consecutive edges, with a total of $N_b=38$.
2. Physical parameters:
   • (i) Young’s modulus, $E=10\mathrm{MPa}$.
   • (ii) Material density, $\rho =1000\mathrm{kg}/{\mathrm{m}}^3$.
   • (iii) Cross-sectional radius, $r_0=0.01\mathrm{m}$.
   • (iv) Damping viscosity, $\mu =0.1$.
   • (v) Gravitational field, $\mathbf{g}=[0.0,0.1{]}^T\mathrm{m}/{\mathrm{s}}^2$.
   • (vi) The overall simulation is static, i.e., $\mathrm{ifStatic}=1$.
3. Numerical parameters:
   • (i) Total simulation time, $T=10.0\mathrm{s}$.
   • (ii) Time step size, $\mathrm{dt}=0.01\mathrm{s}$.
   • (iii) Numerical force tolerance, $\mathrm{tol}=1\times {10}^{-4}$.
   • (iv) Maximum iterations, $N_{\mathrm{iter}}=10$.
4. Boundary conditions:
   • The first two nodes, $\{{\mathbf{x}}_1,{\mathbf{x}}_2\}$, and the last two nodes, $\{{\mathbf{x}}_{39},{\mathbf{x}}_{40}\}$, are fixed to achieve clamped-clamped boundary conditions, thus $\mathcal{FIX}=[1,2,3,4,77,78,79,80{]}^T$.
5. Initial conditions:
   • (i) Initial position is input from the nodal positions.
   • (ii) Initial velocity is set to zeros.
6. Loading steps:
   • (i) Perturbation step: a small perturbation to the initial horizontal configuration is created by applying a small gravitational force (with $\mathbf{g}=[0.0,0.1{]}^T\mathrm{m}/{\mathrm{s}}^2$) when $t\le 1.0\mathrm{s}$.
   • (ii) Compression step: when $t>1.0\mathrm{s}$, a displacement is applied to the last two nodes along the negative $X$-axis with speed $v_0=0.1\mathrm{m}/\mathrm{s}$, until the compression distance is larger than the target value, $\mathrm{\Delta }x\ge 0.6\mathrm{m}$.

Dynamic Rendering