Nonlinear Buckling Analysis with Initial Imperfection

CUSTOMER SUPPORT NOTE
Nonlinear Buckling Analysis with Initial
Imperfection
Note Number:
CSN/LUSAS/1014
This support note is issued as a guideline only.
© Finite Element Analysis Ltd.
© Finite Element Analysis Ltd 2011
CSN/LUSAS/1014
Table of Contents
1.
INTRODUCTION
1
2.
DESCRIPTION
2
CSN/LUSAS/1014
1. Introduction
In a nonlinear buckling analysis, a small perturbation load could be applied to encourage
convergence past buckling and onto a post-buckling path. This load should be small enough
such that it does not affect the results, but just disturbs the symmetry of the mesh enough to
encourage a particular post-buckling path.
An alternative method for a post-buckling solution is to use slightly deformed initial mesh or
geometry. This may be achieved when defining the geometry, or by using a deformed mesh
from a separate analysis. For example the deformed mesh of an eigen value buckling
analysis may be used as a starting point for the mesh in a subsequent and separate nonlinear
buckling analysis. This latter option may well be convenient if an eigenvalue buckling or a
frequency analysis has already been performed on the same model.
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2. Description
To achieve a post-buckling solution, an initial deformed shape can be either created by
actually building the model with an imperfection in the geometry or by using the deformed
shape from another analysis (say from eigenvalue, static or linear eigenvalue buckling
analysis).
A linear (eigenvalue) buckling analysis will produce critical buckling loads and buckled
shapes. The results of this analysis are normalised displacements due to eigenvector unity
normalisation. A factored deformed shape from this linear buckling analysis can then be used
as the initial mesh for a further nonlinear buckling analysis to develop post-buckling
behaviour.
Load
Initial
imperfect
geometry
Critical
buckling
load
Development
of post-buckled
shape under NL
incremental
loading
Incremental nonlinear buckling
analysis
Linear Buckling Analysis
Non-linear Analysis
Formulate model as linear problem with
perfect geometry
Solve eigenproblem for critical loads and
buckled configurations
If required, ‘tabulate’ reduced deformed
shape as imperfect initial geometry for
non-linear analysis
Formulate GNL analysis based on
initially imperfect geometry
Increment loading and progressively
develop true post-buckled
configuration
For example, an eigenvalue buckling analysis of the plate girder has been run and the
buckling load has been calculated for the girder. In figure 1 the deformed shape of the girder
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is shown. When an eigenvalue buckling analysis is performed the displacements are unity
normalised, in this case as the model has been built in metres unity being one metre.
Figure 1
To use the deformed mesh of such an analysis follow the steps below.
1. Create a new model for a subsequent nonlinear buckling analysis. It is suggested to
save a copy of the model used for the eigenvalue buckling analysis (or any other
acceptable model) with a different name. This will ensure that the mesh used is that
same in both models (an essential condition of using an initial deformed mesh). This
model can then be set up to perform a nonlinear analysis with geometric nonlinearity.
2. In this new model, to specify a nonlinear solution, you must add a Nonlinear and
Transient Control to the Loadcase (right-click on the loadcase>Controls>Nonlinear
and Transient) in place of the Eigenvalue control.
3. The inclusion of Geometric Nonlinearity (GNL) is required in a nonlinear buckling
analysis. For a Nonlinear Buckling Analysis, elements that have a geometric
nonlinearity formulation must therefore be used. The geometric nonlinearity
formulations supported by the elements being used can be checked in the Element
Reference Manual. The chosen suitable formulation to be used is then set in the
Nonlinear Options in the Loadcase TreeView (or via File>Model
Properties>Solution tab>Nonlinear Options). If different elements are used with
different formulations for GNL, more than one option can be selected in the
Nonlinear Options.
4. Once the model is ready to analyse, the results file of the first model (eigenvalue
buckling analysis in this example) can be opened on top of the current, nonlinear
model. To now use the deformed shape from the initial analysis the required results
from the required loadcase must first be set active (see Figure 2).
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Figure 2
5. Then go to Utilities>Mesh>Use Deformations… menu option, as shown in Figure 3.
Enter a scaling factor that is required. In this case by entering the value of 0.015 it
will scale the one metre deflection down to 15mm. ( 1 x 0.015 = 0.015m = 15 mm).
Note that for a nonlinear buckling analysis where you are using an initial
imperfection, that the minimal imperfection required should be used; just enough to
encourage convergence past buckling and onto a post-buckling path.
Figure 3
6. Once the scaling factor has been entered click OK and then solve the new nonlinear
buckling analysis.
This method adjusts the coordinates of all nodes in the mesh to include the imperfection using
the supplied deformed mesh factor. It is important to note that the deformed mesh will be
saved in the model until the mesh is reset (Utilities>Mesh>Mesh Reset), even if the model is
closed and reopened. Please see the Modeller Reference Manual>Chapter 7 : Running
an Analysis>LUSAS Analyses Types>Nonlinear Analysis and the guidance in the user
area of our website for further information.
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