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© 2014 CAE Associates
Nonlinear Buckling Analysis Using Workbench v15
Michael Bak November 2014
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CAE Associates Inc.
Engineering Consulting Firm in Middlebury, CT specializing in FEA and CFD analysis.
ANSYS Channel Partner since 1985.
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Background on Structural Stability
Many structures require an evaluation of their structural stability. — Thin columns, compression members, and vacuum tanks are all examples of
structures where stability considerations are important. — At the onset of instability (buckling) a structure will have a very large change in
displacement {∆u} under essentially no change in the load (beyond a small load perturbation).
F F
Stable Unstable
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Background on Structural Stability
In real structures the critical load can rarely be achieved. — A structure generally becomes unstable at a load lower than the critical load
because of imperfections and nonlinear behavior. Analysis techniques for pre-buckling and collapse load analysis include:
— Linear eigenvalue buckling. — Nonlinear buckling analysis.
F
u
Idealized Load Path
Imperfect Structure’s Load Path
Pre-buckling
Linear Eigenvalue Buckling
Nonlinear Buckling
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Linear Eigenvalue Buckling
Linear eigenvalue buckling analysis predicts the theoretical buckling strength (the bifurcation point) of an ideal linear elastic structure.
— This method corresponds to the textbook approach of linear elastic buckling analysis.
— The eigenvalue buckling solution of a Euler column will match the classical Euler solution.
However, imperfections and nonlinear behavior prevent most real world structures from achieving their theoretical elastic buckling strength.
Eigenvalue buckling generally yields non-conservative results and should be used with caution. However, there are two advantages to performing an eigenvalue buckling analysis:
— Relatively inexpensive (fast) analysis. — The buckled mode shapes can be used as an initial geometric imperfection
for a nonlinear buckling analysis in order to provide more realistic results.
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Linear Eigenvalue Buckling
To develop the eigenvalue problem, first solve the load-displacement relationship for a linear elastic pre-buckling load state {P0}:
— Given {P0} solve for
to obtain
— {u0} = the displacements resulting from the applied load {P0} — {σ} = the stresses resulting from {u0}
{ } [ ]{ }00 uKP e=
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Linear Eigenvalue Buckling
Assuming the pre-buckling displacements are small, the incremental equilibrium equations at an arbitrary state ({P}, {u}, {σ}) are given by
where
[Ke] = elastic stiffness matrix
[Kσ(σ)] = initial stress matrix evaluated at the stress state {σ}
{ } [ ] ( )[ ][ ]{ }uKKP e ∆+=∆ σσ
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Linear Eigenvalue Buckling
Assuming pre-buckling behavior is a linear function of the applied load {P0},
then we can show that
Thus, the incremental equilibrium equations expressed for the entire pre-buckling range become
{ } [ ] ( )[ ][ ]{ }uKKP e ∆+=∆ 0σλ σ
{ } { } { } { } { } { }000 σλσλλ === uuPP
( )[ ] ( )[ ]0σλσ σσ KK =
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Linear Eigenvalue Buckling
At the onset of instability (the buckling load {Pcr}), the structure can exhibit a change in deformation {∆u} in the case of no additional loading:
By substituting the above expression into the previous incremental equilibrium equations for the pre-buckling range we have:
In order to satisfy the previous relationship, we must have:
[ ] ( )[ ][ ]{ } { }00 =∆+ uKKe σλ σ
{ } { }0≈∆P
[ ] ( )[ ][ ] { }0det 0 =+ σλ σKKe
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Linear Eigenvalue Buckling
In a finite element model with n degrees of freedom, the above equation yields an nth order polynomial in λ (the eigenvalues).
— The eigenvectors {∆u}n in this case represent the deformation superimposed on the system during buckling.
— The elastic critical load {Pcr} is given by the lowest value of λ calculated.
Demonstration of linear eigenvalue buckling problem:
— Buckling of a cylinder.
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Nonlinear Buckling
A nonlinear buckling analysis employs a nonlinear static analysis with gradually increasing loads to seek the load level at which a structure becomes unstable.
Using a nonlinear buckling analysis, you can include features such as initial imperfections, plastic behavior, contact, large-deformation response, and other nonlinear behavior.
Bifurcation Point, Eigenvalue Buckling Nonlinear Buckling
u
F
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Nonlinear Buckling
In a nonlinear buckling analysis, the goal is to find the first limit point (the maximum load before the solution becomes unstable).
Nonlinear buckling is more accurate than eigenvalue buckling and is therefore recommended for the design or evaluation of structures.
Post-buckling can also be modeled in nonlinear buckling.
u
F Post-buckling
Nonlinear Buckling
First Limit Point
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Nonlinear Buckling Procedure
Performing a nonlinear buckling analysis is similar to most other nonlinear analyses with the following additional points:
— A small perturbation (such as a small force) or geometric imperfection is often required to initiate buckling.
• The buckled mode shape from an eigenvalue buckling analysis can be used to generate an initial geometric imperfection.
— The applied load should be set to a value slightly higher (10 to 20%) than the critical load predicted by the eigenvalue buckling analysis.
— The analysis must be run with geometric nonlinearities activated.
— Write out results for a sufficient number of results steps so that you can examine the load deflection curve.
— Modeling a symmetric sector will only predict symmetric buckled shapes.
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Nonlinear Buckling Procedure
Some notes on applying an initial perturbation/imperfection:
— The magnitude of the initial perturbation/imperfection will influence the results of the nonlinear buckling analysis.
— The initial perturbation/imperfection will remove the sharp discontinuity in the load-deflection response.
— The value of the imperfection (or imperfection generated by the perturbation) should be small relative to the overall dimensions of the structure.
• This value should match the size of the imperfection (real or postulated) in the real structure.
— Manufacturing tolerances can be used to estimate the magnitude of the perturbation/imperfection.
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Nonlinear Buckling Procedure
The buckling load can be found by reviewing the load deflection curve. — A flat (or near-flat) curve indicates buckling. The tangent stiffness will approach
zero as the structure nears its buckling load.
Recognize that an un-converged solution does not necessarily mean that the structure has reached its maximum load.
Numerical Instability Physical Instability (buckling)
u
Fapp Unconverged Solution
Last Converged Solution
KT > 0
u
Fapp Unconverged Solution
Last Converged Solution
KT → 0
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Nonlinear Buckling Procedure
ANSYS can predict post-buckling behavior. — It is possible to predict buckling but continue to converge if the structure can
still carry load.
— Therefore, it is imperative to review the entire results history to determine when buckling has occurred – cannot blindly assume buckling has occurred at non-convergence.
The STABILIZE option can help achieve post-buckling behavior. — Adds artificial damping to provide a resistive force when buckling occurs.
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Nonlinear Buckling Demonstration
Buckling of cylinder with imperfection. Post-buckling of roof structure.
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Thank you!
Thank you for attending CAE Associates webinar on nonlinear buckling. You will receive via email a survey to fill out and return. We welcome any
comments or additional questions on the content.
A transcript of this presentation can be downloaded from our website:
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