Ansys Eigenvalue Buckling Prestress

[Maximina]

The following image shows a thin part subject to a pressure on the thin top face, and simply supported at the ends. The support is from Remote Displacement objects that prevent UX, UY, ROTY and ROTZ. The far end also prevents UZ. The part will bend like a simply supported beam. When sufficiently loaded, the part will buckle out of plane, in the UX direction.

Note that the applied loading is a pressure on the thin upper face. In the large-displacement analysis, this pressure will act as a follower load, changing the direction in which it pushes as the face rotates. Users may prefer to apply a force to the face, so that the direction is not modified during large-displacement analysis.

The buckled shape can be seen further below. In a nonlinear large-displacement analysis, this example fails to buckle because of the perfect symmetry of the model and load. To trigger nonlinear buckling, some sort of UX offset loading or geometric imperfection needs to be applied. An often-preferred technique is to perform a linear eigenvalue buckling analysis based on the applied loads, and use a buckling mode deformation to apply a slight distortion to the unloaded mesh employed in the nonlinear large-displacement buckling analysis. This can be facilitated via the UPGEOM command of Ansys. In Workbench, the deformation can be applied by APDL Commands Objects.

Application of deformation of the unloaded mesh in a shape based on the result of a linear eigenvalue buckling analysis can be applied with UPGEOM, which adds displacements from a previous analysis (in this case a linear eigenvalue buckling analysis) and updates the geometry (node positions) of the finite element model mesh to the deformed configuration. The command includes selection of a load step/substep from a previous analysis, and a node displacement amplitude scaling factor. It is typical in pre-deformed nonlinear large displacement buckling analysis to apply a maximum displacement magnitude on the order of maximum manufacturing variation. The user must be aware of the units employed in solving to get the scaling factor to be appropriate.

This article illustrates nonlinear buckling with a pre-buckled linear buckling shape applied, using the Workbench Mechanical interface. Two small APDL Commands Object snippets are included in the model Outline that convey the result of a linear eigenvalue buckling analysis to the shape of the unloaded mesh in the nonlinear large-displacement buckling analysis.

There is a second Nonlinear Buckling Analysis schematic in the lower right corner of the Project Schematic image. This last analysis shares the same model. This final schematic was used to examine the result of applying a larger amplitude deformation by a linear eigenvalue buckling shape to the same model and loading.

The next image shows the Workbench Mechanical Outline resulting from the above Project Schematic. The static structural linear analysis environment automatically sends its loads into the linked linear buckling analysis environment. Below that are the two large-displacement nonlinear buckling analyses.

Note the presence of a Commands Object in the linear buckling postprocessing area. It makes a copy of the RST results file from the Linear Buckling analysis. There is another Commands Object in each nonlinear buckling environment, which applies a distortion taken from a chosen linear buckling mode shape in the RST file copy, using it to slightly distort the unloaded mesh in a nonlinear buckling run.

This shape can be chosen to drive a mesh deformation for the non-loaded mesh in the nonlinear buckling analysis that follows. The deformation is intended to help initiate this buckling shape in the nonlinear analysis.

The following APDL Commands Object is inserted at the environment level of the nonlinear buckling analysis. It will use the deformed buckling shape calculated for a chosen buckling mode to distort the mesh prior to large-displacement nonlinear buckling analysis. The user should set the factor in the UPGEOM command, and choose the load step/substep for which the distorted shape should be read. In this example, Load Step is 1 (implying the loading and boundary conditions in the Linear Eigenvalue Buckling analysis), and the Substep is also 1, implying the first buckling mode shape.

The factor used in UPGEOM below is 0.50. Since the units in the analysis are set to millimeters, this factor multiplied by the highest deformation of 1.0 in the above buckling mode implies a maximum non-loaded mesh deformation from the original geometry of 0.50 mm.

A second nonlinear buckling analysis was inserted into the Workbench Mechanical outline. It uses a larger factor on the UPGEOM command. This causes the buckling deformation to build up more gradually. With only very slight deformations of the original mesh, the onset of buckling is typically sudden, when the load gets close to the eigenvalue buckling load. This image shows the buildup of extreme deformations as the load is ramped up through substeps in a nonlinear buckling analysis with the mesh pre-deformed by the first linear eigenvalue buckling mode:

Finally, we hope that you can make productive use of our Nonlinear Buckling with Pre-Buckled Shape Distortion resource article for Ansys Mechanical Workbench. For more information, or to request engineering expertise for a particular use-case scenario surrounding nonlinear or ambiguous buckling, pre-buckled shape distortion or other related projects, please contact us here.

The stress stiffness matrix is computed based on the stressstate of the previous equilibrium iteration. Thus, to generate avalid stress-stiffened problem, at least two iterations are normallyrequired, with the first iteration being used to determine the stressstate that will be used to generate the stress stiffness matrix ofthe second iteration. If this additional stiffness affects the stresses,more iterations need to be done to obtain a converged solution.

In some linear analyses, the static (or initial) stress statemay be large enough that the additional stiffness effects must beincluded for accuracy. Modal (ANTYPE,MODAL) andsubstructure (ANTYPE,SUBSTR) analyses are linearanalyses for which the prestressing effects can be requested to beincluded (PSTRES,ON command). Note that in thesecases the stress stiffness matrix is constant, so that the stressescomputed in the analysis are assumed small compared to the prestressstress.

The linear buckling load can be calculated directly by addingan unknown multiplier of the stress stiffness matrix to the regularstiffness matrix and performing an eigenvalue buckling problem (ANTYPE,BUCKLE) to calculate the value of the unknown multiplier. This is discussed in more detail in Buckling Analysis.

The strain-displacement equations for the general motion ofa differential length fiber are derived below. Two different resultshave been obtained and these are both discussed below. Consider themotion of a differential fiber, originally at dS, and then at ds afterdeformation.

where u is the displacement parallel to the original orientationof the fiber. This is shown in Figure 3.6: Motion of a Fiber with Rigid Body Motion Removed. Notethat X, Y, and Z represent global Cartesian axes, and x, y, and zrepresent axes based on the original orientation of the fiber. Bythe Pythagorean theorem,

One further case requires some explanation: axisymmetric structureswith nonaxisymmetric deformations. As any stiffening effects mayonly be axisymmetric, only axisymmetric cases are used for the prestresscase. Axisymmetric cases are defined as (input as MODE on MODE command) = 0. Then, any subsequent load steps with any value of (including 0 itself) uses that same stressstate, until another, more recent, = 0 case is available. Also, torsionalstresses are not incorporated into any stress stiffening effects.

Specializing this to SHELL61 (Axisymmetric-HarmonicStructural Shell), only two stresses are used for prestressing: σs, σθ, the meridionaland hoop stresses, respectively. The element stress stiffness matrixis:

The effect of change of direction and/or area of an appliedpressure is responsible for the pressure load stiffness matrix ([Spr]) (see section 6.5.2 of Bonet and Wood([237])). It is used either for a large-deflectionanalysis (NLGEOM,ON), for an eigenvalue bucklinganalysis, or for a modal, linear transient, or harmonic analysis thathas prestressing flagged (PSTRES,ON command).

[Spr] is derived as an unsymmetricmatrix. Symmetricizing is done, unless the command NROPT,UNSYM is used. Processing unsymmetric matrices takes more runningtime and storage, but may be more convergent.

In a nonlinear analysis (ANTYPE,STATIC or ANTYPE,TRANS), the stress stiffness contribution is activatedand then added to the stiffness matrix. When not using large deformations(NLGEOM,OFF), the rotations are presumed to besmall and the additional stiffness induced by the stress state isincluded. When using large deformations (NLGEOM,ON), the stress stiffness augments the tangent matrix, affectingthe rate of convergence but not the final converged solution.

The document provides an overview of buckling analysis in ANSYS. It discusses buckling of columns with well-defined end conditions, buckling of a special column, and second order analysis of a simple beam. The preprocessing, solution, and postprocessing phases of ANSYS are outlined. Step-by-step instructions are given for modeling each example and obtaining the buckling load using eigenvalue buckling analysis. Manual calculations are also shown for comparison.Read less

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