Continuum Cantilever

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Dynamic analysis of a cantilever beam, modeled with 8-node brick elements.

In this example a simple problem in solid dynamics is considered. The structure is a cantilever beam modelled with three dimensional solid elements.

For three dimensional analysis, a typical solid element is defined as a volume in three dimensional space. Each node of the analysis has three displacement degrees of freedom. Thus the model is defined with ndm = 3 and ndf = 3.

For this model, a mesh is generated using the block3D command. The number of nodes in the local $x$ -direction of the block is nx, the number of nodes in the local $y$ -direction of the block is ny and the number of nodes in the local $z$ -direction of the block is nz. The block3D generation nodes {1,2,3,4,5,6,7,8} are prescribed to define the three dimensional domain of the beam, which is of size $2 \times 2 \times 10$ .

# mesh generation
block3D $nx $ny $nz   1 1  $element  $eleArgs {
    1   -1     -1      0
    2    1     -1      0
    3    1      1      0
    4   -1      1      0 
    5   -1     -1     10
    6    1     -1     10
    7    1      1     10
    8   -1      1     10
}

model.block3D(nx, ny, nz, 1, 1, Brick, 1, {
              1: [-1.0, -1.0,  0.0],
              2: [ 1.0, -1.0,  0.0],
              3: [ 1.0,  1.0,  0.0],
              4: [-1.0,  1.0,  0.0],
              5: [-1.0, -1.0, 10.0],
              6: [ 1.0, -1.0, 10.0],
              7: [ 1.0,  1.0, 10.0],
              8: [-1.0,  1.0, 10.0]})

Two possible brick elements can be used for the analysis. These may be created using the terms StdBrick or BbarBrick. An elastic isotropic material is used.

For initial gravity load analysis, a single load pattern with a linear time series and a single nodal loads is used.

Boundary conditions are applied using the fixZ command. In this case, all the nodes whose $z$ -coordiate is $0.0$ have the boundary condition {1,1,1}, fully fixed.

A solution algorithm of type Newton is used for the problem. The solution algorithm uses a ConvergenceTest which tests convergence on the norm of the energy increment vector. Five static load steps are performed.

Subsequent to the static analysis, the wipeAnalysis and remove loadPatern commands are used to remove the nodal loads and create a new analysis. The nodal displacements have not changed. However, with the external loads removed the structure is no longer in static equilibrium.

The integrator for the dynamic analysis if of type GeneralizedMidpoint with $\alpha = 0.5$ . This choice is uconditionally stable and energy conserving for linear problems. Additionally, this integrator conserves linear and angular momentum for both linear and non-linear problems. The dynamic analysis is performed using $100$ time increments with a time step $\Delta t = 2.0$ .

The deformed shape at the end of the analysis is rendered below:

The results consist of the file cantilever.out, which contains a line for every time step. Each line contains the time and the horizontal displacement at the upper right corner the beam. This is plotted in the figure below:

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