nonlinear buckling web

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NonlinearBuckling

Prediction inANSYS

August 2009



Buckling Overview

Prediction of buckling of engineering structures is a challenging problem

for several reasons:

— A real structure contains imperfections that can greatly effect the buckling

factor.

— Many engineering structures will retain load-carrying capability after initial

buckling.

— Determining when an analysis has reached instability is often difficult to verify.



Buckling Overview

ANSYS provides two techniques for simulating buckling:

— Eigenvalue, or linear, buckling.

• This approach performs an eigenvalue solution to obtain the theoretical buckling

load factor for an ideal elastic structure.

• Considered non-conservative since imperfections and nonlinearities prevent most

structures from achieving their theoretical buckling strength.

• Typically used as a first pass on the buckling load and shape.

— Nonlinear buckling.

• Consists of running a nonlinear, large deflection solution until the analysis stops

converging, indicating an instability.

• Detailed review of the nonlinear behavior must be used to determine if analysis has

reached a true structural instability.• Factors such as the presence of imperfections, element formulation, step size,

element mesh size, and nonlinear convergence settings will play a role in the

prediction of the instability.

• Post-buckling, i.e. the ability of a structure to carry load after buckling has occurred,

can be predicted.



Recommended Buckling Analysis Procedure

A typical buckling analysis will include the following steps:

— Perform eigenvalue buckling solution to determine estimates for buckling factors andexpected buckling modes.

• If modeling a portion or sector of a full structure, the predicted modes will only be those that can

be represented with that sector.

— Perform nonlinear analysis to determine more accurate buckling solution.

• Most physical structures will contain imperfections that will cause buckling well below theoreticalbuckling limits.

• Imperfections can be added into the analysis via:

— Small applied displacements.

— Small applied loading.

— Small changes in geometry – can use an eigenvalue mode to apply a small perturbation in the geometry

using the ANSYS UPCOORD command.

• Run nonlinear solution until it stops converging, indicating an instability.

• Review force-displacement behavior at key locations to determine if non-convergence is due to

structure instability (buckling) or numerical instability.

• Consider that many engineering structures can buckle locally but continue to carry additional

loading (post-buckling).

• Use of newer element technology and updating default nonlinear settings may help in obtaining

accurate solutions.



Buckling Study

Consider the sector model of the

cylindrical shell with stiffener.

— The geometry modeled is acylinder with stiffeners, loaded

with radial point forces as

shown in the one-sector model

below.

— Model is fixed at one end in all

DOF, and symmetry conditions

are used at the cyclic

boundaries.

— Various ANSYS shell element

types and solution settings were

used to predict the largedeflection nonlinear analysis

which in some cases includes

both a pre- and post-buckling

response.



Eigenvalue Solution

Eigenvalue buckling analysis was performed to provide a starting point for

this analysis.

— Eigenvalue analyses were run with the SHELL63, SHELL181 and SHELL281.

For the one-sector model:

— All models predicted the same results, with the first two modes shown below:



Eigenvalue Solution

The first mode occurs at a load factor of 0.204.

— This mode predicts adjacent cylindrical segments moving in opposite radial

directions.— This mode is characterized as having a nodal diameter of 4, since the

deflected shape repeats 4 times around the circumference.

— In this mode, the stiffener bends in the tangential direction.

The second mode occurs at a load factor of 0.275.— The mode is a nodal diameter of 8.

— This mode is more likely to occur when the stiffener bends in the radial

direction but remains planar.



Eigenvalue Solution

If modeling the full 360-degree structure, the lowest eigenvalue buckling

mode is shown below with the following characteristics:

— Nodal diameter of 3.— Load factor = 0.088 is lower than lowest sector load factor of 0.204.

— Sector model cannot predict this mode.



Nonlinear Buckling Solution

Nonlinear buckling analyses were run for several cases:

— Element types:

• SHELL63• SHELL181

• SHELL281

— Nonlinear settings:

• Default settings.

• PRED,OFF and a tighter convergence tolerance of 1x10

-6

.— 200 fixed substeps over applied loading of 750000 N.

— Large deflections turned on.

— Ideal geometry and loading (no imperfections modeled).




A review of the final displaced shapes indicates that these analyses

deformed into the second eigenvalue (nodal diameter = 8) mode shape.

— An example deformed shape from the default SHELL181 case:




A plot of the force versus radial deflection at the location of the point load

at all substeps for all models is presented:




Observations from the force-deflection behavior:

— All analyses predict the same general behavior. Differences occur only in

nonlinear convergence.— There are several inflection points in the force-deflection curve, but there is no

indication that the structure has become completely unstable.

— The ―correct‖ buckling load cannot be inferred by just looking at the final non-

converged value. Nonlinear buckling requires a review of the force-deflection

behavior.

— Since the newer ANSYS elements contain the latest features to aid in

obtaining converged solutions, it is not surprising that they can predict more

―post-buckling‖.

— The default nonlinear settings in ANSYS are intended for basic nonlinear

analyses. Advanced nonlinearities such as buckling prediction usually require

modification of the default settings.— Experience and careful review of results can determine when it is appropriate

to change these settings.




A close inspection of the deflected shapes and animations indicates the

following buckling stages:

— At approximately 0.30: buckling of cylindrical panels inward.— At approximately 0.55: buckling of center of stiffener inward.

— At approximately 0.80: buckling of free end of cylinder inward.




Based on the nonlinear large deflection analyses, the following summary

can be made:

— Assuming a geometry without imperfections, the structure will initially buckle inmode 2, at a load factor of approximately 0.31.

— Analyses that continue to converge past this load are predicting a post-

buckling behavior. For the real problem this behavior may not be physical due

to imperfections, plasticity, etc, where this post-buckling strength may be

numerically over-estimated.

— The newer shell elements have better convergence capabilities and thus are

able to predict more of the post-buckling behavior.

A real structure would most likely not buckle in this manner.

— Slight imperfections would most likely cause the model to become unstable

sooner.

— If the true ultimate capacity of the structure is of interest, it is recommended to

perform a series of simulations including geometric imperfections, material

plasticity, variations in loading to be included in the analysis model to provide a

relative comparison of limit loads.




Since the lowest buckling mode for this structure is not excited by the

previous models, it is recommended that a nonlinear buckling analysis be

performed using an ―imperfect‖ geometry.— A standard procedure in this case is to provide a slight imperfection by using

the ANSYS UPCOORD feature.

— The mode 1 mode shape from the eigenvalue solution, with a very small scale

factor applied, was used to update the nodal locations used in a subsequent

nonlinear analysis.

— The imperfection applied to the model is on the order of manufacturing

tolerances and is not visibly noticeable.




The subsequent analysis using SHELL181s stopped converging at load

factor = 0.375 with the following buckled shape:

li kli l i




The characteristics of

this buckled shape:

— Due to the slightimperfection, bending

of the stiffener in the

tangential direction

occurs, enabling this

buckling mode.

— This mode is moresevere and less likely

to continue to post-

buckle, as indicated by

the flat force-deflection

behavior.

N li B kli S l i




It is important to note that the sector model is limited to predictions of

buckled shapes that can be represented by the sector geometry and

boundary conditions.— As an illustration of this concept, if a full model is run with no imperfections

(using the SHELL181 model), the full model will buckle sooner than the

sector model.

— The force-deflection curves from the two analyses are exact up until the full

model buckles. The final buckled shape shows the stiffeners deflecting

tangentially.

N li B kli S l ti




By adding an imperfection to the full model, an even lower buckling

mode can be obtained.

— The eigenvalue buckling analysis of the full model indicated that the firstbuckling mode (nodal diameter = 3) occurs at a much lower load.

— By running a nonlinear analysis of the full model, including a geometric

imperfection by using UPCOORD based on the first mode, a buckling factor

of 0.085 is found with the buckled shape shown below.

B kli St d S



Buckling Study Summary

This buckling study illustrates the following points:

— Analyzing the nonlinear behavior of the sector model, the final load predicted

at non-convergence can be different based on element formulation andnonlinear settings.

— Review of the results indicates that all the analyses predicted similar results,

but the newer element formulations were able to model more of the post-

buckling behavior.

— In this case, lower buckling factors and more physically realistic and consistent

results can be found by including initial imperfections into the geometry.

— In addition, even lower buckling factors can be determined when modeling the

full structure, which can buckle into lower modes than the sector model.

Therefore, for this structure and based on all of the analyses performed

and assumptions made, a load factor of 8.5% of the full loading would beconsidered an accurate assessment of the buckling load of this structure.

B kli A l i S



Buckling Analysis Summary

Accurate calculation of physically realistic buckling loads using ANSYS

requires the following considerations:

— Many buckling problems require adding small imperfections to allow thestructure to deflect into more physically realistic buckling modes.

— Results between full and sector models will typically not provide the same

buckling factors, due to the full model not having the constraints of sector-

symmetry and thus having the ability to buckle into lower energy modes.

— Default nonlinear settings are valid for many cases, but they often require

adjustment for advanced nonlinear behavior such as post-buckling.

— Nonlinear analyses require detailed review of the overall force-deflection

behavior throughout the entire analysis, not just a review of the final point.

— It is recommended that eigenvalue buckling always be performed prior to

nonlinear buckling to provide understanding in how the nonlinear model will

behave.

nonlinear buckling web

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