Linear and Nonlinear Buckling Analysis - Predictive Engineering

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Linear and Nonlinear Buckling
Analysis
with Femap V10.3.1, Nastran 8.1 and
LS‐DYNA v971 r6.0
www.PredictiveEngineering.com
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Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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Introduction
Calculating the buckling force for a ideal column is quite simple. As long as you know the length, second moment of inertia and
elastic modulus of the beam, you can calculate the force by hand. What if you don’t have an ideal column. What if you don’t have
column or beam geometry at all but buckling is still a concern for your design?
This white paper will walk you through the NX Nastran Linear Buckling Analysis. Additionally, the paper will show you how to
validate your linear buckling analysis with a non‐liner static analysis.
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Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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Shown above are the cross sectional properties for the beam we will be analyzing. Given a length of
300”, the critical load is 57,562 lbf.

Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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Boundary Conditions
The beam is pinned at both ends. The vertical degree of
freedom is released at the upper constraint. Rotation
about the vertical axis is prevented at the lower
constraint. An arbitrary load is applied to the upper most
node.
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Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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The title of the Output Set will include a load factor
value. This factor multiplied by the applied load is equal
to the critical load. 5.751*10,000lbf = 57,510 lbf,
correlating to the hand calculation of 57,562 within
0.1%.

Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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Additional Results
The linear buckling analysis returns two output sets.
The first output set (titled “NX NASTRAN Case 1” by
default) is simply a linear elastic analysis of the
boundary conditions specified in the analysis.
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The second output set (in this case, titled “Eigenvalue
5.750754”) shows the first mode shape of the structure
with the given constraints. The peak deflection of the
structure is set to 1. Stresses are reaction forces
correspond to the deflection of the mode shape. The
only valuable piece of quantitative information that
comes from this output set is the load factor value.

Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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Additional Resources
A similar example of Classic Euler Buckling
can be found in the NX Nastran User’s
Guide.
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Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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Verification
More common than not, the solution cannot be verified
with a hand calculation (this is why we are doing FEA,
right?). The solution can be verified with the slightly trickier
Nonlinear Static analysis. For this validation, the load has
been increased to 100,000 lbf such that the critical load will
be evaluated in the intermediate steps of the nonlinear
analysis. Additionally, a small bending moment (Mx = 100
in‐lbf) has been added to direct the model to the first
buckling mode.

Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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2012 –All Rights Reserved
The non‐linear solution diverges at time = 0.5755. With the
load increasing from 0 to 100,000 lbf as time increases
from 0 to 1, this correlates to a load of 57,550 lbf.

Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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0.1
Euler Beam Buckling: Defelction vs Load
0.09
0.085
0.08
0.09
0.08
Deflection (in)
0.075
0.07
0.065
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Deflection (in)
0.07
0.06
0.05
0.04
0.06
0.055
0.05
57400 57450 57500 57550 57600
Load (lbf)
0.03
0.02
0.01
10000 20000 30000 40000 50000 60000
Load (lbf)
To further illustrate the buckling load, the vertical deflection
is plotted as a function of load. The point at which small
increase in load results in a large increase in deflection is the
buckling load.
Additional data at the buckling load was acquired using the
Arc‐Length Method option within the Analysis Manager,
Nonlinear Control Options.
“The arc‐length method should be used if the problem
involves snap‐through or postbuckling deformation”
–NX Nastran 7 Basic Nonlinear Analysis User’s Guide

Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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Once you are confident analyzing simple members for
buckling with both the linear and nonlinear approach,
move on to more complicated geometry. Above is an
example from the NX Nastran Handbook of Nonlinear
Analysis for an Imperfect Spherical Shell.

Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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A more advanced example of linear and nonlinear buckling is provided here. A simple
thin walled aluminum cylinder (i.e. beer can) is modeled with plate elements. The
load case is an equally distributed axial force. The beer can is carefully constrained to
avoid end effects.

Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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With a perfect cylinder, the Eigenvalue buckling solution indicates that the cylinder should buckle at a load of 73 lbf (0.073*1,000 lbf). The
nonlinear solution just compresses the cylinder. In the perfect numerical world, this makes perfect sense.
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Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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A common trick to initiate buckling in perfect geometry / mesh, is to take the buckled mode shape and map it onto the mesh. This can be
done using a Femap API within the Custom Tools > PostProcessing > Nodes Move By Deform with Options. We have scaled the mode
shape by 0.001. The concept is that you just want to perturb the perfect geometry such that it will start to buckle. It should be noted that
the two solutions correlate within 2%.
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Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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If we adjust the constraints to restrict horizontal movement on the top edge and to pin the bottom edge, the buckling force increases to
around 200 lbf. These constraints simulate a real beer can more closely. This seems reasonable for the support weight of an idealized beer
can.
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Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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A nonlinear analysis provides the buckling stress at the point of collapse. The utility of the geometrically nonlinear approach is that one
can gain insight into the structure prior to its buckled condition. The image on the left shows the can with a load of 200 lbf (0.2*1,000 lbf)
while the image on the right shows the buckled condition is at 216 lbf. It is also useful to note that if the material had a yield strength in
excess of 42,000 psi, the buckling behavior would be completely independent of material nonlinear effects.
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Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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Another confirmation using LS‐DYNA gave similar results for the beer can with buckling occurring around 190 lbf. However, the question
that arises whether this buckled shape is really stable or an artifact of our boundary conditions. In modeling, a model is just a model and
can’t really be trusted unless it has been validated with a test.
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Linear and Nonlinear Buckling Analysis with NX Nastran and LS‐DYNA
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In this example, the nodes of the can were tweaked, or perturbed, to create a completely random surface. This operation was done within
LS‐DYNA and allows one to specify a completely random displacement of the nodes. Results indicate that the can would buckle right
around 190 lbf.
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