Chapter Nine
Geometric Instability
Geometric Instability
Chapter Overview
Training Manual
– Eigenvalue Buckling
– Load Control
– Displacement Control
– Arc-Length Method
Advanced Structural Nonlinearities 6.0
• This chapter will deal with geometric instability problems,
namely those concerning buckling. The following techniques
will be addressed:
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... Chapter Overview
A. Background on Structural Stability
B. Linear (Eigenvalue) Buckling Procedure
C. Background on Nonlinear Buckling Techniques
D. Nonlinear Pre-Buckling Procedure
E. Nonlinear Post-Buckling Procedure
Advanced Structural Nonlinearities 6.0
• In this chapter, we will cover the following topics:
Training Manual
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A. Background on Structural Stability
Training Manual
• 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
Advanced Structural Nonlinearities 6.0
• 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.
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... Background on Structural Stability
Training Manual
F
F
F
Bifurcation Point
Unstable Equilibrium
Neutral Equilibrium
Fcr
u
Stable Equilibrium
u
Advanced Structural Nonlinearities 6.0
• An idealized fixed end column will exhibit the following
behavior under increasing axial loads (F).
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... Background on Structural Stability
Training Manual
• A bifurcation point is a point in load history where two
branches of the solution are possible.
• In the case of the idealized fixed end column,
F
at the critical load (Fcr), the column can buckle
to the left or to the right. Thus two load paths
are possible. In the case of real structures the
P
existence of geometric imperfections or force
perturbations (P  0) will determine the direction
of the load path.
F
u
Advanced Structural Nonlinearities 6.0
Bifurcation Point
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... Background on Structural Stability
Training Manual
• Consider the equilibrium of the ball shown below. If the
surface is concave upward the equilibrium is stable, the ball
will return to its original position if perturbed. If the surface
is concave downward the equilibrium is unstable, if perturbed
the ball will roll away. If the surface is flat the ball is in neutral
equilibrium, if perturbed the ball will remain in its new
position.
Stable
Unstable
Neutral
Advanced Structural Nonlinearities 6.0
Stable, Unstable, and Neutral Equilibrium
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... Background on Structural Stability
Training Manual
• At a value of F < Fcr the column is in stable equilibrium. If a
small perturbing force (P  0) is introduced and then
removed, the column will return to its original position. At
values F > Fcr the column is in unstable equilibrium, any
perturbing force will cause collapse. At F = Fcr the column is
in neutral equilibrium, and this is defined as the critical load.
Advanced Structural Nonlinearities 6.0
Critical Load
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... Background on Structural Stability
Training Manual
• In real structures the critical load can rarely be achieved. A
structure generally will become unstable at a load lower than
the critical load because of imperfections and nonlinear
behavior.
F
Bifurcation Point
Fcr
Actual structural response,
instability occurs below
critical load.
u
Advanced Structural Nonlinearities 6.0
Limit Load
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B. Linear Eigenvalue Buckling
Training Manual
– Linear Eigenvalue Buckling
– Nonlinear Buckling Analysis
• In this section, we will focus on the first method, linear
eigenvalue buckling.
F
Linear
Eigenvalue
Buckling
Pre-buckling
Nonlinear
Buckling
Idealized Load Path
Imperfect Structure’s
Load Path
u
Advanced Structural Nonlinearities 6.0
• Analysis techniques for pre-buckling and collapse load
analysis include:
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... Linear Eigenvalue Buckling
Training Manual
• The eigenvalue formulation determines the bifurcation points
of a 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.
Advanced Structural Nonlinearities 6.0
• Eigenvalue buckling analysis predicts the theoretical
buckling strength (the bifurcation point) of an ideal linear
elastic structure.
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... Linear Eigenvalue Buckling
Training Manual
F
Limit Load
Idealized Load Path
Bifurcation
Point
Pre-buckling
Imperfect Structure’s
Load Path
u
Advanced Structural Nonlinearities 6.0
• However, imperfections and nonlinear behavior prevent most
real world structures from achieving their theoretical elastic
buckling strength. Eigenvalue buckling generally yields
unconservative results, and should be used with caution.
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... Linear Eigenvalue Buckling
Training Manual
– 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.
Advanced Structural Nonlinearities 6.0
• Although eigenvalue buckling typically yields unconservative
results there are two advantages to performing a linear
buckling analysis:
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... Linear Eigenvalue Buckling
Training Manual
{P0} = [Ke]{u0}
to obtain
{u0} = the displacements resulting from the
applied load {P0}
{s} = the stresses resulting from {u0}
Advanced Structural Nonlinearities 6.0
• A linear buckling analysis is based on a classic eigenvalue
problem. To develop the eigenvalue problem, first solve the
load-displacement relationship for a linear elastic prebuckling load state {P0}; i.e. given {P0} solve for
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... Linear Eigenvalue Buckling
Training Manual
{P} = [[Ke] + [Ks(s)]]{u}
where
[Ke]
= elastic stiffness matrix
[Ks(s)] = initial stress matrix evaluated at the stress state {s}
Advanced Structural Nonlinearities 6.0
• Assuming the pre-buckling displacements are small, the
incremental equilibrium equations at an arbitrary state ({P},
{u}, {s}) are given by
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... Linear Eigenvalue Buckling
Training Manual
{P} = l{P0}
{u} = l{u0}
{s} = l{s0}
then we can show that
[Ks(s)] = l[Ks(s0)]
• Thus, the incremental equilibrium equations expressed for
the entire pre-buckling range become
{P} = [[Ke] + l[Ks(s0)]]{u}
Advanced Structural Nonlinearities 6.0
• Assuming pre-buckling behavior is a linear function of the
applied load {P0},
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... Linear Eigenvalue Buckling
Training Manual
{P}  0
• By substituting the above expression into the previous
incremental equilibrium equations for the pre-buckling range
we have
[[Ke] + l[Ks(s0)]]{u} = {0}
The above relation represents a classic eigenvalue problem.
Advanced Structural Nonlinearities 6.0
• At the onset of instability (the buckling load {Pcr}), the
structure can exhibit a change in deformation {u} in the case
of
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... Linear Eigenvalue Buckling
Training Manual
det[[Ke] + l[Ks(s0)]] = 0
• In a finite element model with n degrees of freedom, the
above equation yields an nth order polynomial in l (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 l calculated.
Advanced Structural Nonlinearities 6.0
• In order to satisfy the previous relationship, we must have
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... Eigenvalue Buckling Procedure
Training Manual
1. Build the Model
2. Obtain the Static Solution with Prestress
3. Obtain the Eigenvalue Buckling Solution
4. Review the Results
Advanced Structural Nonlinearities 6.0
• An eigenvalue buckling analysis includes the following four
main steps:
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... Eigenvalue Buckling Procedure
Training Manual
• This task is similar to most other analyses with the following
two additional points:
– Only linear behavior is valid. Nonlinear elements are treated as
linear. Their stiffnesses are based on their initial status and are
not changed.
– Young’s Modulus must be defined. Material properties may be
linear, isotropic or orthotropic, nonlinear properties are ignored.
Advanced Structural Nonlinearities 6.0
Build the Model
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... Eigenvalue Buckling Procedure
Training Manual
• When obtaining the static solution the prestress flag must be
set to perform a subsequent eigenvalue buckling analysis.
– Main Menu > Preprocessor > Loads > Analysis Options …
– or issue the command: PSTRES,ON
Advanced Structural Nonlinearities 6.0
Obtain the Static Solution with Prestress
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... Eigenvalue Buckling Procedure
Training Manual
• Unit loads are usually sufficient. The eigenvalues calculated
represent the buckling load factors on the applied load.
• Note that the eigenvalues represent scale factors for all
loads. If certain loads are constant while other loads are
variable you will need to ensure that the stress stiffness
matrix from the constant loads is not factored (discussed
later).
• Solve the model
– Main Menu > Solution > -Solve- Current LS …
– or issue the command: SOLVE
Advanced Structural Nonlinearities 6.0
Obtain the Static Solution with Prestress
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... Eigenvalue Buckling Procedure
Training Manual
• After completing the static solution, exit and re-enter
solution, and specify eigenvalue buckling as the analysis
type:
– Main Menu > Finish
– Main Menu > Solution > -Analysis Type- New Analysis …
– or issue the commands:
FINISH
/SOLU
ANTYPE,BUCKLE
Advanced Structural Nonlinearities 6.0
Obtain the Eigenvalue Buckling Solution
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... Eigenvalue Buckling Procedure
Training Manual
• Specify the method of eigenvalue extraction and number of
buckled modes to extract:
– Main Menu > Solution > Analysis Options …
– or issue the command: BUCOPT,LANB,3,0
Block Lanczos is the
recommended eigenvalue
extraction method. In this
example, 3 modes have
been requested.
Advanced Structural Nonlinearities 6.0
Obtain the Eigenvalue Buckling Solution
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... Eigenvalue Buckling Procedure
Training Manual
• Specify the number of modes to write to the results file.
– Main Menu > Solution > -Load Step Opts - Expansion Pass >
Expand Modes ...
– or issue the command: BUCOPT,LANB,3,0
The relative stress
distribution can also
be calculated.
Advanced Structural Nonlinearities 6.0
Obtain the Eigenvalue Buckling Solution
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... Eigenvalue Buckling Procedure
Training Manual
• You can iterate on the eigensolution, adjusting the variable
loads until the eigenvalue becomes 1.0 or nearly 1.0.
Consider the example of a pole with self weight WO and an
externally applied load A. You can iterate, adjusting the value
of A until l = 1.0.
Advanced Structural Nonlinearities 6.0
Note on Constant and Variable Loads
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... Eigenvalue Buckling Procedure
Training Manual
• Results from an eigenvalue buckling analysis can be
reviewed in the General Postprocessor. The results consist of
load factors, buckled mode shapes, and the relative stress
distributions.
– Main Menu > General Postproc > Results Summary ...
– or issue the command: SET,LIST
The “Set” column
indicates the buckling
mode number, and the
value for “Time”
represents the corresponding load factor.
Advanced Structural Nonlinearities 6.0
Review the Results
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... Eigenvalue Buckling Procedure
Training Manual
• The maximum displacement of a buckled mode shape is
normalized to 1.0. Therefore, the displacements do not
represent actual deformations and the stresses are relative to
the buckled mode shape.
• It is usually beneficial to review the first few buckling mode
shapes. In a subsequent nonlinear buckling analysis the
higher buckling modes of the structure may be important.
• If there are closely spaced eigenvalues, this indicates that the
structure is imperfection sensitive. A nonlinear buckling
analysis should be executed with the appropriate
imperfection or perturbation.
Advanced Structural Nonlinearities 6.0
Review the Results
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... Eigenvalue Buckling Procedure
Training Manual
• In some cases negative eigenvalues are computed in an
eigenvalue buckling analysis. This occurs when the
eigenvalue extraction procedure encounters numerical
difficulties. In this case, the shift point for eigenvalue
extraction can be specified (BUCOPT). Eigenvalue extraction
is most accurate near the shift point. This will require that
you have some knowledge of the value of the critical load.
Advanced Structural Nonlinearities 6.0
Additional Considerations:
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... Eigenvalue Buckling Procedure
Training Manual
• The pressure-load stiffness matrix is often important in
buckling analyses to accurately calculate the load factor.
ANSYS automatically includes the pressure-load stiffness
matrix by default for eigenvalue buckling analyses. Although
not recommended, the user can manually activate or
deactivate the inclusion of the pressure-load stiffness via:
– Main Menu > Solution > Unabridged Menu
– Main Menu > Solution > -Load Step Opts- Solution Ctrl …
– or issue the command: SOLCONTROL,,,INCP
Advanced Structural Nonlinearities 6.0
Additional Considerations:
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C. Background on Nonlinear Buckling
Training Manual
F
Idealized Load Path
Limit Point
Bifurcation
Point
Imperfect Structure’s
Load Path
Idealized
Static
Behavior
Pre-buckling
Post-buckling
Actual Dynamic
Response
u
Advanced Structural Nonlinearities 6.0
• Shown below is a generalized nonlinear load deflection
curve. This figure illustrates the idealized load path, an
imperfect structure’s load path, and the actual dynamic
response of the structure.
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... Background on Nonlinear Buckling
Training Manual
• There are various analysis techniques available for
calculating the nonlinear static force deflection response of a
structure. These techniques include:
– Load Control
– Displacement Control
– Arc-Length Method
Advanced Structural Nonlinearities 6.0
• Earlier, in Section B, the procedure for linear eigenvalue
buckling was discussed (idealized load path on previous
slide).
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... Background on Nonlinear Buckling
Training Manual
• Consider the snap through analysis of the shallow arch
shown below. When the solution to a problem is performed
with incrementally applied forces (F) the solution is
performed using load control.
F
F
F
Fapp
Can Fapp be
achieved with
load control?
u
Advanced Structural Nonlinearities 6.0
Load Control:
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... Background on Nonlinear Buckling
Training Manual
• The difficulty of using load control with the Newton-Raphson
is that the solution can not progress past a point of
instability. At the point of instability (Fcr) the tangent stiffness
matrix KT is singular. Using load control, the NewtonRaphson method will not converge. However, this type of
analysis can be useful to characterize the pre-buckling
behavior of a structure.
Fapp
Fcr
Only Fcr can be
achieved using
load control.
KT = 0
KT < 0
u
Advanced Structural Nonlinearities 6.0
Load Control:
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... Background on Nonlinear Buckling
Training Manual
• When the arch is loaded with an incrementally applied
displacement, as opposed to a force, the solution is
performed using displacement control. The advantage of
displacement control is that it produces a stable solution
beyond Fcr. (The imposed displacement provides an
additional constraint at the point of instability.)
UY
Fapp
UY
UY
u
Fapp can be achieved
with displacement
control. (Fapp is now
the reaction force at
the imposed
displacement UY.)
Advanced Structural Nonlinearities 6.0
Displacement Control:
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... Background on Nonlinear Buckling
Training Manual
• The disadvantage of displacement control is that it only
works when you know what displacements to impose! If the
arch is loaded with a pressure load as opposed to a
concentrated force, displacement control is not possible.
P
With a more complicated
loading it is generally not
clear which displacements
to impose.
Advanced Structural Nonlinearities 6.0
Displacement Control:
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... Background on Nonlinear Buckling
Training Manual
• The arc-length is a solution method used to obtain
numerically stable solutions for problems with instabilities
(KT  0), or negative tangent stiffnesses (KT < 0).
• The arc-length method can be used for static problems with
proportional loading.
• Although the arc-length method
can solve problems with a
complicated force-deflection
response, it is best suited to
solve problems where the
response is smooth without
sudden bifurcation points.
F
u
Advanced Structural Nonlinearities 6.0
Arc-Length Method:
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... Background on Nonlinear Buckling
Training Manual
• The arc-length method solves for load and displacement
simultaneously in a manner similar to the Newton-Raphson
method. However, an additional unknown in introduced in
the solution, the load factor l (-1 < l < 1). The force
equilibrium equations can then be rewritten as,
[KT]{u} = l {Fa} - {Fnr}
• In order to accommodate the additional unknown a constraint
equation must be introduced, the arc-length . The arc-length
relates the load factor l and displacement increments {u} in
the arc-length iterations.
Advanced Structural Nonlinearities 6.0
Arc-Length Method:
– Note that the Arc-Length Method reduces to the full NewtonRaphson method if this constraint  is removed.
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... Background on Nonlinear Buckling
Training Manual
• Another way to view the difference between Arc-Length and
(full) Newton-Raphson method is that Newton-Raphson
methods use a fixed applied load vector {Fa} per substep.
Arc-Length methods, on the other hand, use a variable load
vector l{Fa} per substep.
F
F
3
4
3
2
4
2
1
1
u
Newton-Raphson Method

u
Arc-Length Method
Advanced Structural Nonlinearities 6.0
Arc-Length Method:
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... Background on Nonlinear Buckling
The arc-length relates the
incremental load factor
l to the incremental
displacement u via a
spherical arc. Shown
here are the incremental
load factor l and the
incremental
displacement u for the
arc-length method with
the full Newton-Raphson.
Arc Length Radius  un2  l2
Advanced Structural Nonlinearities 6.0
Arc-Length Method:
Training Manual
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... Background on Nonlinear Buckling
Training Manual
• By forcing the arc-length iterations to converge along a
spherical arc which intersects the equilibrium path, solutions
can be obtained for structures undergoing zero or negative
stiffness behaviors.
F
ri Arc Length Radii
ri
Converged Substeps
ri
ri
ri
Equilibrium Path
u
Advanced Structural Nonlinearities 6.0
Arc-Length Method:
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... Background on Nonlinear Buckling
Training Manual
• Load control, displacement control, and arc-length method
are summarized below. These are three techniques used in
the solution of nonlinear static buckling problems.
• There is an additional method, which will be discussed later,
which can solve buckling problems via dynamics.
Loading
Load Control
Solution Method Pre-Buckling Post-Buckling Restriction
Newton-Raphson
Yes
No
Response must have a one-to-one
Method
relationship with respect to force.
Displacement Newton-Raphson
Yes
Yes
Response must have a one-to-one
Control
Method
relationship with respect to
displacement. Sometimes,
imposed displacements are not
possible (they may not characterize
loading conditions well)
Either
Arc-Length
Yes
Yes
The arc-length constraint must be
Method
satisfied. May not handle
responses which are not smooth.
For proportional loading only.
Advanced Structural Nonlinearities 6.0
Summary of Three Nonlinear Buckling Techniques:
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D. Nonlinear Buckling
Training Manual
• Using a nonlinear buckling analysis, you can include features
such as initial imperfections, plastic behavior, contact, largedeformation response, and other nonlinear behavior.
F
Bifurcation Point,
Eigenvalue Buckling
Nonlinear Buckling
u
Advanced Structural Nonlinearities 6.0
• A nonlinear buckling analysis employs a nonlinear static
analysis with gradually increasing loads to seek the load level
at which a structure becomes unstable.
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... Nonlinear Buckling
Training Manual
• Nonlinear buckling is more accurate than eigenvalue
buckling, and is therefore recommended for the design or
evaluation of structures.
Nonlinear
Buckling
Arc-Length
Method
F
First Limit Point
u
Advanced Structural Nonlinearities 6.0
• In a nonlinear buckling analysis, the goal is to find the first
limit point (the largest value of the load before the solution
becomes unstable). The arc-length method can be used to
follow the post-buckling behavior.
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... Nonlinear Pre-Buckling Procedure
Training Manual
1. Build the Model
(Including an initial imperfection or perturbation)
2. Obtain the Solution
3. Review the Results
Advanced Structural Nonlinearities 6.0
• A nonlinear buckling analysis includes the following three
main steps:
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... Nonlinear Pre-Buckling Procedure
Training Manual
• This task is similar to most other analyses with the following
additional points:
– A small perturbation (such as a small force) or geometric
imperfection is required to initiate buckling.
– The buckled mode shape from an eigenvalue buckling analysis
can be used to generate an initial imperfection.
– The value of the applied load should be set to a value slightly
higher (10 to 20%) than critical load predicted by the eigenvalue
buckling analysis.
Advanced Structural Nonlinearities 6.0
Build the Model
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... Nonlinear Pre-Buckling Procedure
Training Manual
• Creating an initial imperfection from the buckled mode shape.
– Main Menu > Preprocessor > -Modeling- Update Geom …
– or issue the command: UPGEOM
Multiplier for the
displacements to be
added to the original
geometry.
Mode Number
Results file from the
eigenvalue buckling
analysis.
Advanced Structural Nonlinearities 6.0
Build the Model - Initial Imperfection
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... Nonlinear Pre-Buckling Procedure
Training Manual
• The magnitude of the initial imperfection will influence the
results of the nonlinear buckling analysis. The initial
imperfection will remove the sharp discontinuity in the loaddeflection response.
• The value of the imperfection should be small relative to the
overall dimensions of the structure. The 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 imperfection.
Advanced Structural Nonlinearities 6.0
Build the Model - Initial Imperfection
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... Nonlinear Pre-Buckling Procedure
Training Manual
• A nonlinear buckling analysis is a static analysis with
nonlinear geometric effects extended to the point where the
structure reaches its limit load. Be sure that nonlinear
geometry is activated (NLGEOM,ON).
• The use of solution control is recommended (default).
• Use the full Newton-Raphson option without adaptive
descent (default with solution control).
Advanced Structural Nonlinearities 6.0
Obtaining the Solution
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... Nonlinear Pre-Buckling Procedure
Training Manual
• Invoke automatic time stepping (default with solution
control). With automatic time stepping on, the program
automatically seeks out the buckling load. If the solution
does not converge at a given load, the program bisects and
attempts a new solution at a smaller load. As such the
minimum time step will affect the precision of your results.
Fapp
Fapp
Flimit
1,3,5
6
4
7
Substep
Number
2
u
“Time”
Advanced Structural Nonlinearities 6.0
Obtain the Solution
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... Nonlinear Pre-Buckling Procedure
Training Manual
• It is possible (although unlikely) that if the time steps are
large you might “jump over” the instability and obtain a
solution for a “snapped through” configuration. Be sure to
plot the load deflection curve in the time history
postprocessor.
F
Stable
F
Fapp
Unstable
F
Stable
u
Advanced Structural Nonlinearities 6.0
Obtaining the Solution
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... Nonlinear Pre-Buckling Procedure
Training Manual
• Be sure to set a small minimum time step to allow bisection.
• A value 10 to 20 percent higher than the eigenvalue buckling
load will often be a good choice for the value of the applied
load. Recall that you can set “time” equal to the value of the
applied load for easier postprocessing.
• Be sure to write out results for a sufficient number of
substeps (OUTRES), so that you can examine the load
deflection curve in the general postprocessor.
Advanced Structural Nonlinearities 6.0
Obtain the Solution
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... Nonlinear Pre-Buckling Procedure
Training Manual
• Review the load deflection curve in the time history
postprocessor. In the case of an in-plane loading you will
need to plot the out-of-plane (lateral) deflection versus the
load.
F
F
Plot force (F) versus
displacement in the
x-direction (UX).
y
UX
x
Advanced Structural Nonlinearities 6.0
Review the Results
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... Nonlinear Pre-Buckling Procedure
Training Manual
• The load deflection curve can help determine whether or not
a physical instability or a numerical instability is the cause of
the solution divergence. Recognize that an unconverged
solution does not necessarily mean that the structure has
reached its maximum load!
Advanced Structural Nonlinearities 6.0
Additional Considerations:
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... Nonlinear Pre-Buckling Procedure
Training Manual
• The tangent stiffness will approach zero as a structure nears
its buckling load. A numerical or physical instability can be
determined from the slope of the load-deflection curve
Fapp
Unconverged
Solution
Fapp
KT > 0
KT  0
Last Converged
Solution
Last Converged
Solution
u
Numerical Instability
Unconverged
Solution
u
Physical Instability (buckling)
Advanced Structural Nonlinearities 6.0
Additional Considerations:
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... Workshop Exercise
• Workshop 12: Buckling of Arch
Advanced Structural Nonlinearities 6.0
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Training Manual
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Geometric Instability
E. Nonlinear Post-Buckling Procedure
Training Manual
– Displacement Control
– Dynamics
– Arc-Length Method
F
Limit Point
Imperfect Structure’s
Static Load Path
Idealized
Static
Behavior
Post-buckling
Actual Dynamic
Response
u
Advanced Structural Nonlinearities 6.0
• Analysis techniques for post-buckling include:
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... Nonlinear Post-Buckling Procedure
Training Manual
• As discussed earlier, displacement control can be used to
predict a post-buckled response for simple loading
conditions. The main disadvantage to displacement control
is that for complicated loading it is generally not clear what
displacements to impose.
UY
Fapp can be
achieved with
displacement
control.
Fapp
UY
UY
u
Advanced Structural Nonlinearities 6.0
Displacement Control:
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... Nonlinear Post-Buckling Procedure
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• A static stability problem using load control, can be solved by
using a nonlinear transient dynamic analysis. With a
dynamic analysis the softening response will not be
calculated when the structure “snaps-through” (it will be a
dynamic snap). The main disadvantage of dynamics is that it
is not always easy to damp out unwanted dynamic effects
(ringing).
F
F
F
Fapp
Dynamic
Response
u
Advanced Structural Nonlinearities 6.0
Dynamics:
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... Nonlinear Post-Buckling Procedure
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• The arc-length is a solution method used to obtain
numerically stable solutions to static problems with
proportional loading experiencing instabilities (KT  0), or
negative tangent stiffnesses (KT < 0).
F
ri Arc Length Radii
ri
Converged Substeps
ri
ri
ri
Equilibrium Path
u
Advanced Structural Nonlinearities 6.0
Arc-Length Method:
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... Nonlinear Post-Buckling Procedure
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1. Build the Model
2. Obtain the Solution
3. Review the Results
Advanced Structural Nonlinearities 6.0
• As with any other ANSYS analysis there are three main steps
to perform an analysis using the arc-length:
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... Nonlinear Post-Buckling Procedure
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• This task is similar to most other analyses with the following
additional considerations:
– The arc-length is only valid for a force controlled proportional
loading analysis. No tabular loads are permitted.
– The load factor is applied to all loads. Therefore, including a
geometric imperfection to initiate an instability may be
preferable to using a perturbation.
– The value of the applied load should be set to a value slightly
higher (10 to 20%) than critical load predicted by the eigenvalue
buckling analysis.
Advanced Structural Nonlinearities 6.0
Build the Model
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... Nonlinear Post-Buckling Procedure
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• The arc-length is only valid for a static analysis
(ANTYPE,STATIC). Be sure that nonlinear geometry is
activated (NLGEOM,ON).
• The use of solution control is recommended (default).
• Do not use the line search (LNSRCH), the predictor (PRED),
adaptive descent, automatic time stepping (AUTOTS,
DELTIM), time (TIME), or time integration (TIMINT) with the
arc-length method!
Advanced Structural Nonlinearities 6.0
Obtaining the Solution
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• A value 10 to 20 percent higher than the eigenvalue buckling
load will often be a good choice for the value of the applied
load. Do not set a value for “time”, when using the arclength. “Time” in an arc-length analysis is related to the load
factor.
• Be sure to write out results for a sufficient number of
substeps (OUTRES), so that you can examine the load
deflection curve in the general postprocessor.
Advanced Structural Nonlinearities 6.0
Obtain the Solution
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... Nonlinear Post-Buckling Procedure
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• The reference arc-length is calculated from the value of the
applied load and the number of substeps using the following
equation:
Reference Arc-Length Radius = Total Load/NSBSTP
• where NSBSTP is the number of substeps specified. Using
more substeps will allow the program to more closely follow
the load deflection response but will also result in a longer
solution time.
Advanced Structural Nonlinearities 6.0
Obtaining the Solution
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... Nonlinear Post-Buckling Procedure
• To activate the arc-length use:
– Main Menu > Solution > -Analysis Type- Sol’n Control …
– Solution Control > Advanced NL tab
– or issue the command: ARCLEN
Advanced Structural Nonlinearities 6.0
Obtaining the Solution
Training Manual
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... Nonlinear Post-Buckling Procedure
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• The factors MAXARC and MINARC are arc-length radius
multipliers which are used to define the limits of the arclength radius. (Similar to setting the number of substeps for
automatic time stepping.)
MAXARC defaults to 10
MINARC defaults to 0.001
Advanced Structural Nonlinearities 6.0
Obtaining the Solution - Arc-Length Parameters
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• The arc-length solution can be terminated by either achieving
a load factor (l) of 1.0 or by setting limits on the solution.
• The arc-length solution can be terminated by reaching the
first limit point or achieving a maximum displacement criteria
at a specific node.
Advanced Structural Nonlinearities 6.0
Obtaining the Solution - Arc- Length Termination Criteria
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... Nonlinear Post-Buckling Procedure
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• Time in an arc-length analysis is related to the load factor.
Therefore do not reference results by a “time” value when
postprocessing an arc-length analysis. You should always
reference the results by load step and substep number.
• “Time” in an arc-length analysis is not always increasing, and
in some analyses it can be negative. In the time history
postprocessor the total arc-length load factor (ALLF) and the
arc-length load factor increment (ALDL) can be stored by
using a solution summary (SOLU) item.
Advanced Structural Nonlinearities 6.0
Review the Results
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• Be sure to review the load deflection curve in the time history
postprocessor as in the case of a nonlinear buckling
analysis.
• Note that the load factor (TIME) can increase or decrease, and
can even become negative.
Advanced Structural Nonlinearities 6.0
Review the Results
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... Nonlinear Post-Buckling Procedure
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• “Drifting back” in which the analysis retraces its steps back
along the load deflection curve is one difficulty that is caused
by using too large or too small of an arc-length radius. You
can use the number of substeps (NSUBST) and arc-length
radius multipliers (MAXARC and MINARC) to adjust the arclength radius.
Advanced Structural Nonlinearities 6.0
Additional Considerations:
• If the arc-length fails to converge, reducing the initial arclength radius (NSUBST) can enhance convergence.
Decreasing the lower arc-length radius multiplier (MINARC)
can also improve convergence.
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• Be sure to plot the load-deflection curve. Establishing when
the structure becomes unstable in the load history can be
very useful when debugging an analysis.
Advanced Structural Nonlinearities 6.0
Additional Considerations:
• Some trial and error may be required to determine the optimal
settings for the arc-length radius. If the arc-length radius is
too small the solution is very inefficient. If it is too large the
solution may miss the buckling point or the “snap through”
regime. The arc-length radius multipliers need to be set
carefully.
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Advanced Structural Nonlinearities 6.0
Additional Considerations:
• In some cases, the arc-length procedure requires an initial
geometric imperfection to trigger the nonlinear buckling
mode. In such cases, use an eigenvalue buckling analysis to
determine the mode, and add a geometric imperfection. For
example, lateral torsional buckling of a cantilever beam would
require a geometric imperfection, whereas a snap through
analysis of a shallow arch would not require a geometric
imperfection.
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... Workshop Exercise
• Workshop 13: Post-Buckling Response of Frame
Advanced Structural Nonlinearities 6.0
Please refer to your Workshop Supplement:
Training Manual
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