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# An Introduction to Modelling Buckling and Collapse

**The term ‘collapse’ is often used to refer to the sudden loss of structural integrity and does not distinguish between the two major categories leading to structural failure: (a) material failure or (b) structural instability due to loss of structural stiffness within the elastic limit of the material. The term ‘buckling’ is often reserved specifically for the latter category. At an elementary level, predicting material failure may be accomplished using linear finite element analysis. The strains and corresponding stresses obtained from this analysis may be compared to design stress (or strain) allowables anywhere within the structure. If the finite element solution indicates regions where these allowables are exceeded, it is assumed that material failure has occurred. Design allowables are based on experimentally-derived material strengths and usually include a safety margin. This type of analysis will give an adequate prediction for statically determinate metallic structures undergoing small deformations.**

If the structure is statically indeterminate, as indeed the majority of practical structures are, the analyst may want to assess the evolution of material failure. The load at which material failure initiates may be significantly lower than that which leads to eventual collapse. The analyst may also want to know the nature of this failure progression, that is, is it gradual or does it occur rapidly? Material failure may be the result of plasticity in metallic structures or fracture, which is a more prevalent form of failure in brittle materials such as ceramics or carbon-fibre reinforced plastics.

The structure may also undergo large deformations before or during material failure and it is therefore apparent that the presence of geometric and material non-linearities requires more sophisticated finite element solution schemes.

Buckling refers to the loss of stability of a structure and in its simplest form, is independent of material strength where it is assumed that this loss of stability occurs within the elastic range of the material. It is primarily characterised by a loss of structural stiffness and cannot be modelled using basic linear finite element analysis.

Slender or thin-walled structures under compressive loading are susceptible to buckling. Buckling may also be stable or unstable and different geometries lead to different forms of buckling stability. A distinction is also made between classical buckling and other forms of structural instability. Without resorting to formal definitions and distinctions at this stage, one physical manifestation of these differences is that classical buckling results in the structure deforming primarily in a direction orthogonal to that of the applied loading causing this instability.

Another form of structural instability, termed snap buckling, is where the deformation is primarily in the direction of the applied loading.

Material failure and buckling may both occur in sequence leading to structural collapse. A thin-walled or slender structure may initially buckle elastically but the resulting high deformations may cause localised material failure. A thicker-walled or less slender structure under high compressive loading may exhibit localised inelastic behaviour, such as plasticity, which lowers the stiffness leading to buckling.

## Scope

This book provides an introduction to the various mechanisms which may lead to structural buckling or collapse and the strategies employed in modelling this behaviour using finite element analysis. While the use of some jargon is unavoidable, every effort has been made to ensure that definitions are given when required, either within the text or as an entry in the Glossary at the end of this book. It is also the authors’ view that the introduction of mathematical equations is central to giving the analyst further insight into the suitability of the various numerical strategies available in most finite element systems. If the reader is particularly averse to the mathematical treatment given herein, he or she may skip over these and still obtain useful and practical information on modelling buckling and collapsing structures.

The development of new finite element methodologies and formulations, to model highly non-linear behaviour such as buckling and collapse, continues to advance at a rapid pace. The majority of the methodologies and modelling strategies presented in this book may be found in current commercial finite element packages. To give this book relevance beyond the immediate time, recent developments which are still at the research stage are also discussed. While these may not be available commercially at the time of writing, there is enough information given for the interested reader to implement these schemes in non-commercial (research) codes or, if possible, as user subroutines in commercial finite element packages.

## Contents

**1. Introduction**

1.1. Scope1.2. Readership1.3. Layout

### **2. A Brief Overview of Elastic Stability**

2.1. Introduction2.2. Stiffening Behaviour2.3. ‘Snap-Through’ Behaviour2.4. Classical Buckling2.5. Stable Symmetric Buckling2.6. Unstable Symmetric Buckling2.7. Asymmetric Buckling

###

**3. Linear Buckling Analysis**

3.1. Introduction3.2. Linear Buckling Analysis3.3. Steps for a Finite Element Eigenvalue Analysis3.4. Methods for Finding Eigenvalues and Eigenvectors3.5. Example 3A – Linearised Plate Buckling3.6. Buckling Analysis for Pre-Loaded Structures3.7. Symmetry3.8. Example 3B – Using Symmetry for Isotropic Plate Buckling

###

**4. Geometric Non-Linear Analysis**

4.1. Introduction4.2. Non-Linear Analysis Formulation4.3. Newton-Raphson Methods4.4. Limitation of Newton-Raphson Schemes4.5. Direct Load Incrementation Scheme4.6. Energy dissipation scheme4.7. Arc-Length Methods4.8. Follower Force Loading4.9. Example 4A – Clamped Curved Beam with Central Point Load4.10. Example 4B – Postbuckling Stiffened Carbon-Fibre CompositePanels

**5. Dynamic Analysis for Solving Non-Linear Problems**

5.1. Introduction5.2. Dynamic Solution Methods for Non-Linear Problems5.3. Implicit Solution5.4. Explicit Solution5.5. Other Forms of Explicit/Implicit Solution Methods5.6. Choice of Solution Method5.7. Mass-Scaling5.8. Load Rate5.9. Extended Explicit Solution Method5.10. Combined Quasi-Static/Pseudo-Transient Method5.11. Example 5A – Clamped Curved Beam with Central Point LoadUsing Explicit Dynamic Analysis5.12. Example 5B – Postbuckling Response of a Cantilever BeamUsing Explicit Dynamic Analysis5.13. Example 5C – Clamped Curved Beam with Central Point LoadUsing a Combined Quasi-Static/Pseudo-Transient Method5.14. Example 5D – Postbuckling of a Blade-Stiffened Panel UsingExplicit Analysis5.15. Example 5E – Postbuckling of an I-Stiffened Panel UsingCombined Quasi-Static/Pseudo-Transient Method5.16. Remarks

**6. Modelling Collapse Through Material Plasticity**

6.1. Introduction6.2. Basic Plasticity6.3. Limit Load Analysis (Plastic Collapse)6.4. Application to Structural Collapse – Limit Load Analysis

6.4.1. The First Theorem of Limit Analysis (Static Principle)6.4.2. The Second Theorem of Limit Analysis

(Kinematic Principle)6.5. General Elastic-Plastic Analysis6.6. Finite Element Modelling for a Plastic Collapse Analysis6.7. Mesh and Element Considerations6.8. Other Considerations6.9. Example 6A - Finite Element Plastic Collapse Solution

**7. Collapse due to Fracture**

7.1. Introduction7.2. Fracture Mechanics Basics7.3. Example 7A - Stress Intensity Factor Calculation7.4. Quarter-Point Isoparametric Elements7.5. Virtual Crack Closure Technique7.6. Interface Elements7.7. Example 7B – VCCT Applied to Skin-Stiffener DebondingProblem7.8. Example 7C – Mixed-Mode Multiple Delamination UsingInterface Elements

**8. Practical Guide to Selecting a Solution Scheme**

**9. Concluding Remarks**

**10. References**

**11. Appendix: Solution Procedures for Eigenproblems**

11.1. Subspace Iteration11.2. Lanczos Method

**12. Glossary**

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# About

Falzon & Hitchings

First Published - September 2006

Softback, 136 Pages