Modal dynamic and transient dynamic analysis with CAXA and SAXA elements

A cantilever pipe 100 units long with an outer radius of 1.2675 units and a wall thickness of 0.2 units subjected to tip loading is analyzed. The pipe is modeled with all the elements listed above. The first-order, fully integrated CAXA model consists of 2 × 20 elements in the mesh, while the CAXA4Rn and the CAXA4RHn models consist of 4 × 40 elements in the mesh. The second-order CAXA models use 20 elements along the length of the pipe. The first-order SAXAn model uses 20 elements along the length of the pipe, while 10 elements are used in the SAXA2n model. The material behavior is assumed to be isotropic elastic with a Young's modulus of 30.E6 and Poisson's ratio of 0.3.

The modal dynamic and steady-state dynamic procedures, the direct-solution steady-state procedure, the subspace-based steady-state procedure, and the transient dynamic procedure are used in the verification tests. A sinusoidal load with a maximum amplitude of 1.0E4 units is applied to the tip of the cantilever pipe. The concentrated load is split in two, with half applied to the midside nodes in each of the $\theta =$ 0° and $\theta =$ 180° planes on the loaded end of the pipe. All the nodes on one end of the pipe are fixed. To avoid any deformation through the wall thickness in the CAXA model caused by 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.

The cantilever pipe described in the previous section is used in these verification tests. A white noise power spectral density is used to describe the applied ground accelerations. The material definition is assumed to be isotropic elastic. The values are not important.

Since random response analysis is a modal-based procedure, an eigenfrequency extraction step is required to obtain the mode shapes and natural frequencies of the system. The first ten modes are used in the random response steps with a damping ratio of 0.01 for each mode. The base motion is applied only to degree of freedom 1.

The model consists of a cylinder 300 units in length with an outer radius of 2 units. The finite element mesh consists of a single element that has nodes lying on the axis from each of the planes forming the element. The nodes on the axis are tied such that the element can simulate a solid cylinder. The material properties are assumed to be isotropic elastic. The values are not important.

The spectrum of peak displacement values as a function of frequency and damping ratio is specified in the spectrum step, and the base motion is applied in directions 1 (r-direction) and 2 (z-direction) using the response spectrum.

This problem is similar to the verification problem pmodbas3.inp using CAX4H elements described in Modal dynamic analysis with baseline correction. CAXA4H2 elements are used in the present verification test. The test illustrates the use of baseline correction and base motion for CAXA elements.

The structure analyzed is a cylinder made of rubberlike material. An 8 × 8 mesh of CAXA4H2 elements is employed. The nodes on the axis of the cylinder are constrained such that they do not move away from the axis after deformation.

The structure is preloaded statically in compression in the axial direction by a rigid platen. The response to applied axial excitation at the rigid surface is sought. The acceleration records are the same as those used in the problem pmodbase.inp (see Modal dynamic analysis with baseline correction).

This problem is similar to the problem described in FV41: Free cylinder: axisymmetric vibration, where axisymmetric elements are used.

The axisymmetric behavior is simulated by imposing the condition that the radial and axial displacements of the nodes on planes other than the $\theta =$ 0° plane be the same as the nodes on the $\theta =$ 0° plane.

This problem is similar to the problem described in FV42: Thick hollow sphere: uniform radial vibration, where axisymmetric elements are used.