Cantilever beam analyzed with CAXA and SAXA elements

This example verifies the accuracy of the second-order (8-node for CAXA and 3-node for SAXA) elements and the first-order (4-node for CAXA and 2-node for SAXA) elements in a single large-displacement case. It also analyzes the frequency response of the tip-loaded cantilever beam modeled with CAXA and SAXA elements. In both cases the results are compared to those obtained with beam theory.

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ProductsAbaqus/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 θ=​ 0° and the bending of the structure occurs about the θ=​ 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 r–z plane, combined with Fourier interpolation with respect to θ. 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 is solved with both CAXA and SAXA elements. The cantilever is loaded at its tip by a load of constant direction.

Problem description

The cantilever, a pipe 100 units long, has a cross-section with outer radius 1.2675 and wall thickness 0.2. This pipe is moderately slender (L/R=​ 78.9). This type of problem becomes considerably more difficult numerically as the slenderness ratio increases. Young's modulus is chosen as 30 × 106, and Poisson's ratio is 0.3. The motion of the pipe axis is entirely in a plane, so any of the CAXA or SAXA elements would be suitable except for those elements using only one Fourier mode in the θ-direction. (Due to the finite rotation of the pipe, the projection of the cross-section on the r–z plane becomes an ellipse.) Since the Fourier modes are defined in a fixed r–z system, the use of second-order Fourier expansion (including ovalization) is the minimum required. The finite element model uses the second-order elements with 10 elements along the length (z-direction) of the pipe and one element in the r-direction for the CAXA model. The first-order SAXA model uses 20 elements. A finite element model using the first-order CAXA4n (n=2, 3, or 4) elements is expected to give a stiffer response as a result of shear locking. However, a model using the first-order elements with reduced integration and hourglass control, CAXA4Rn (n= 2, 3, or 4), is capable of giving a much more accurate response. Without any mesh convergence study, we solve the problem by using a 2 × 20 mesh of fully integrated first-order elements and a 4 × 40 mesh with the corresponding reduced-integration elements with hourglass control.

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 θ=​ 0° and θ=​ 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 an equation constraint (Linear Constraint Equations).

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

Figure 1 shows the progressive deformation of the pipe modeled with element type CAXA82. The results for the CAXA elements, in terms of the motion of the tip of the cantilever, are shown in Figure 2, where they are compared to the beam solution obtained with the B22 beam elements. It is apparent that the displacement solutions with the CAXA8n, CAXA8Rn, and CAXA4Rn (n=2, 3, or 4) elements predict almost precisely the results obtained by the model using the B22 beam elements. As expected, the fully integrated, first-order CAXA models have a much stiffer response, while the counterpart elements with reduced integration and hourglass control give more accurate results. The results for the SAXA elements are shown in Figure 3, where once again the results are compared to the B22 solution. For clarity, only the SAXA22 results are plotted since all elements SAXA1n and SAXA2n (n=2, 3, or 4) produce nearly identical results.

The frequency response of the tip-loaded cantilever beam modeled with CAXA8n, CAXA8Rn, and CAXA4Rn (n=2, 3, or 4) elements is very close to that obtained from the model using B22 beam elements. The natural frequencies for the fully integrated, first-order CAXA elements are higher because of the stiffer response of these elements.

Input files

cantilevercaxasaxa_caxa42.inp

CAXA42 element model.

cantilevercaxasaxa_caxa43.inp

CAXA43 element model.

cantilevercaxasaxa_caxa44.inp

CAXA44 element model.

cantilevercaxasaxa_caxa4r2.inp

CAXA4R2 element model.

cantilevercaxasaxa_caxa4r3.inp

CAXA4R3 element model.

cantilevercaxasaxa_caxa4r4.inp

CAXA4R4 element model.

cantilevercaxasaxa_caxa82.inp

CAXA82 element model.

cantilevercaxasaxa_caxa83.inp

CAXA83 element model.

cantilevercaxasaxa_caxa84.inp

CAXA84 element model.

cantilevercaxasaxa_caxa8r2.inp

CAXA8R2 element model.

cantilevercaxasaxa_caxa8r3.inp

CAXA8R3 element model.

cantilevercaxasaxa_caxa8r4.inp

CAXA8R4 element model.

cantilevercaxasaxa_saxa12.inp

SAXA12 element model.

cantilevercaxasaxa_saxa13.inp

SAXA13 element model.

cantilevercaxasaxa_saxa14.inp

SAXA14 element model.

cantilevercaxasaxa_saxa22.inp

SAXA22 element model.

cantilevercaxasaxa_saxa23.inp

SAXA23 element model.

cantilevercaxasaxa_saxa24.inp

SAXA24 element model.

Figures

Figure 1. Progressive deformation of pipe.

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

Figure 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.)