Nonlinear Buckling Prediction in ANSYS. August 2009

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1 Nonlinear Buckling Prediction in ANSYS August 2009

2 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.

3 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.

4 Recommended Buckling Analysis Procedure A typical buckling analysis will include the following steps: Perform eigenvalue buckling solution to determine estimates for buckling factors and expected 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 theoretical buckling 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.

below. Model is fixed at one end in all DOF, and symmetry conditions are used at the cyclic boundaries.

5 Buckling Study Consider the sector model of the cylindrical shell with stiffener. The geometry modeled is a cylinder 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 large deflection nonlinear analysis which in some cases includes both a pre- and post-buckling response.

6 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:

7 Eigenvalue Solution The first mode occurs at a load factor of 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 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.

8 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 = is lower than lowest sector load factor of Sector model cannot predict this mode.

9 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 1x fixed substeps over applied loading of N. Large deflections turned on. Ideal geometry and loading (no imperfections modeled).

10 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:

11 A plot of the force versus radial deflection at the location of the point load at all substeps for all models is presented:

12 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 nonconverged 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.

13 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.

14 Based on the nonlinear large deflection analyses, the following summary can be made: Assuming a geometry without imperfections, the structure will initially buckle in mode 2, at a load factor of approximately Analyses that continue to converge past this load are predicting a postbuckling 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.

15 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.

16 The subsequent analysis using SHELL181s stopped converging at load factor = with the following buckled shape:

17 The characteristics of this buckled shape: Due to the slight imperfection, bending of the stiffener in the tangential direction occurs, enabling this buckling mode. This mode is more severe and less likely to continue to postbuckle, as indicated by the flat force-deflection behavior.

18 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.

19 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 first buckling 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 is found with the buckled shape shown below.

20 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 and nonlinear 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 postbuckling 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 be considered an accurate assessment of the buckling load of this structure.

21 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 the structure 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 sectorsymmetry 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.

Using Energy History Data to Obtain Load vs. Deflection Curves from Quasi-Static Abaqus/Explicit Analyses

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