2.1.3 Cantilever beam analyzed with CAXA and SAXA elements

Product: ABAQUS/Standard  

ABAQUS provides a family of elements that are intended for the nonlinear analysis of structures that are initially axisymmetric but undergo nonlinear, nonaxisymmetric deformation. These elements, continuum elements named CAXA and shell elements named SAXA, are often used to model cylindrical or pipe structures in which the deformation is assumed to be symmetric with respect to  0° and the bending of the structure occurs about the  90°-axis. The elements are written for arbitrarily large deformation in geometrically nonlinear analysis. The nonlinear capability is particularly useful for slender structures. The elements use standard isoparametric interpolation within the rz plane, combined with Fourier interpolation with respect to . Up to four Fourier modes are allowed. As a simple large-deformation demonstration problem, the cantilever problem in Geometrically nonlinear analysis of a cantilever beam, Section 2.1.2, is solved with both CAXA and SAXA elements. The cantilever is loaded at its tip by a load of constant direction. This example evaluates the accuracy of the second-order (8-node for CAXA and 3-node for SAXA) and the first-order (4-node for CAXA and 2-node for SAXA) elements in a single large-displacement case and compares the results to those obtained with beam theory.

This example is also used to analyze the frequency response of the tip-loaded cantilever beam modeled with CAXA and SAXA elements. The results are compared to those obtained with beam theory.

Problem description

Loading and boundary conditions

The load on the tip of the cantilever is increased to a value of 20000, which causes the tip to deflect more than 75 units. CAXA elements have rigid body modes in both the global x- and z-directions. The rigid body mode in the z-direction is removed by fixing the z-displacement of node set BASE at the fixed end of the pipe. The rigid body mode in the x-direction is eliminated by fixing the r-displacements at the midside nodes located at the fixed end of the pipe. The ovalization of the fixed end is also restricted by these boundary conditions. All other cross-sectional planes can ovalize. The concentrated load is split in two, with half applied to midside nodes in each of the  0° and  180° planes on the loaded end of the pipe. To avoid any deformation through the wall thickness in the CAXA model due to the application of concentrated loads on the loaded end, the radial displacements at the midside nodes are constrained to be equal to the average radial motion of the nodes at the inside and outside radii. This is accomplished with the *EQUATION option (Linear constraint equations, Section 28.2.1 of the ABAQUS Analysis User's Manual).

The general loading step forms the base state for the frequency analysis step that follows. In the frequency analysis step the load and boundary conditions are maintained as defined in the previous step.

Results and discussion

Input files

Figures

Figure 2.1.3–1 Progressive deformation of pipe.

Figure 2.1.3–2 Comparison of load-displacement curves for CAXA elements.

Figure 2.1.3–3 Comparison of load-displacement curves for SAXA elements. (Only SAXA22 is shown since all elements SAXA1n and SAXA2n for n=2, 3, or 4 give identical results.)