Initial Post-buckling Analysis Research Articles

A reduced model for nonlinear analysis and design of thin-walled structures prone to multi-modal buckling

A reduced model for nonlinear analysis and design of thin-walled structures prone to multi-modal buckling

Closed-Form Solution for Secondary Perturbation Displacement in Finite-Element-Implemented Koiter’s Theory

The finite-element-implemented Koiter’s initial postbuckling analysis has not managed to convince commercial finite element code developers to incorporate it into their codes to benefit structural designers and analysts. Three key obstacles have been identified. With two of them being resolved recently, the present paper addressed the remaining one, namely, the efficient solution of the second-order perturbation equation. A closed-form solution is obtained, which is more computationally efficient than any approach adopted in the past. The closed-form solution is obtained mathematically rigorously, and in the meantime it is computational efficient. The approach applies also to problems where buckling is of multiple modes. The solution procedure established in this paper has been verified and demonstrated through a structural application.

Closed solution for initial post-buckling behavior analysis of a composite beam with shear deformation effect

Closed solution for initial post-buckling behavior analysis of a composite beam with shear deformation effect

An efficient finite strip procedure for initial post-buckling analysis of thin-walled members

An efficient procedure based on the semi-analytical finite strip method with invariant matrices is developed and applied to analyze the initial post-buckling of thin-walled members. Nonlinear strain–displacement equations are introduced in the manner of the von Karman assumption for the classical thin plate theory, and the formulations of the finite strip methods are deduced from the principle of the minimum potential energy. In order to improve the computational efficiency, an analytical integral of the stiffness matrix is transformed into matrix multiple calculation with introducing invariant matrices which can be integrated in advance only once. Three commonly employed benchmark problems are tested with proposed method and other state-of-the-art methods. The corresponding comparison results show that: (1) this finite strip method is proved to be a feasible and accurate tool; (2) compared with the calculation process of the conventional finite strip methods, the proposed procedure is much more efficient since it requires the integration of the stiffness matrix only once no matter how many iterations are needed; and (3) the advantage of time-saving is greatly remarkable as the number of iterations increases.

An efficient finite strip procedure for initial post-buckling analysis of composite laminated members

An efficient procedure based on the semi-analytical finite strip method with newly introduced invariant matrices is developed to analyze the initial post-buckling of composite laminated members. The nonlinear strain-displacement equations obtained from the Von-Karman assumption and three plate theories, which are classical thin plate theory, first-order shear deformation plate theory, and high-order shear deformation plate theory can be used to evaluate the initial post-buckling performance of the composite laminated members. According to the principle of the minimum potential energy, the formulations of the finite strip method can be deduced. In order to improve the computational efficiency, the pre-integrated invariant matrices are introduced, which can convert the complicated analytical integral calculation of the stiffness matrix into a relatively simple matrix multiplication calculation. Several benchmark problems are tested based on the proposed method and other conventional methods. The corresponding comparison results show that: (1) the proposed method is proved to be feasible and accurate for those three different theories; (2) compared with the other conventional finite strip methods, the proposed method is much more efficient since it requires the integration of the stiffness matrix only once no matter how many iterations are needed, (3) and the advantage of time-saving is increasingly remarkable as the number of iterations increases.

Initial post-buckling analysis of a fixed–fixed strut

In this paper, the initial post-buckling of a fixed–fixed strut in compression at the first bifurcation point is analyzed. Using the Fourier series of the lateral deflection, the second variation of the potential energy is proved, analytically, to be semi-positive definite when the compression is equal to the Euler critical load. The fourth variation of the potential energy is positive when the disturbance of the lateral deflection matches the buckling mode. Based on Koiter’s initial post-buckling theory, the equilibrium of the straight state of the strut is stable at the stability limit; when the compression slightly exceeds the Euler critical load, the curved shape at initial post-buckling is stable.

An isogeometric formulation of the Koiter’s theory for buckling and initial post-buckling analysis of composite shells

Numerical formulations of the Koiter theory allow the efficient prediction, through a reduced model, of the behavior of shell structures when failure is dominated by buckling. In this work, we propose an isogeometric version of the method based on a solid-shell model. A NURBS interpolation is employed on the middle surface of the shell to accurately describe the geometry and the high continuity typical of the displacement field in buckling problems and to directly link the CAD model to the structural one. A linear interpolation is then adopted through the thickness together with a modified generalized constitutive matrix, which allows us to easily eliminate thickness locking and model multi-layered composites. Reduced integration schemes, which take into account the continuity of the shape functions, are used to avoid interpolation locking and make the integration faster. A Mixed Integration Point strategy makes it possible to transform the displacement model into a mixed (stress–displacement) one, required by the Koiter method to obtain accurate predictions, without introducing stress interpolation functions. The result is an efficient numerical tool for buckling and initial post-buckling analysis of composite shells, characterized by a low number of DOFs and integration points and by a simple and quick construction of the reduced model.

Modeling for Initial Post-Buckling Analysis of Laminated Plates and Shells by FEM

Nonlinear analysis of plate and shell structures can explain the phenomenon which cannot been explained by classical stability theory, and can obtain better validation of experimental results. Stability problems are essentially nonlinear and their nonlinear finite element solutions ultimately result in solving nonlinear algebraic equations and nonlinear eigenvalue problems. The solutions can define the shape of basic path and determine critical load by using the incremental method, The perturbation methods are used near the critical point, and the basic formulas are given for initial post-buckling analysis by FEM.

An Investigation on Relative Post-Buckling Stiffness Variations of Symmetrically Cross-Ply Laminates

In this paper, the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of symmetrically cross-ply laminates are presented. The so-called exact finite strip is developed based on the concept that it is effectively a plate. In the development process, the Von-Karman’s equilibrium equation is solved exactly to obtain the buckling loads and the corresponding form of out-of-plane buckling deflection modes. The investigation of thin flat plate buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. The post-buckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. In this method, the Von-Karman’s compatibility equation governing the behavior of symmetrically laminated composite plates is used together with a consideration of the total strain energy of the plate. Through the solution of the compatibility equation, the in-plane displacement functions are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. These in-plane and out-of-plane deflected functions are then substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficient. The developed method is subsequently applied to investigate the relative post-buckling stiffness variations of some representative thin symmetric cross-ply laminates for which the results are also obtained through the application of a semi-analytical finite strip method.

Correlation of constraint conditions in Koiter's initial postbuckling analysis and in the consistently linearized eigenvalue problem

AbstractKoiter's postbuckling analysis and the consistently linearized eigenvalue problem are used in sensitivity analysis of the postbuckling behavior of elastic structures. A classification of structures can be made on basis of the eigenvector of the eigenvalue problem. This classification distinguishes between structures with different quality of conversion from imperfections sensitivity into imperfection insensitivity, which is expressed in terms of the coefficients of Koiter's initial postbuckling analysis. Mathematically, this connection is expressed in the correlation of constraint equations. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Necessary and Sufficient Conditions for Zero‐stiffness Postbuckling

AbstractIn the course of sensitivity analysis of the initial postbuckling behavior of elastic structures, a special case may occur which is referred to as zero‐stiffness postbuckling. The purpose of this paper is to present sufficient and necessary conditions for this special case which is associated with a favorable form of transition from imperfection sensitivity to imperfection insensitivity in the course of sensitivity analysis of the initial postbuckling path. Koiter's initial postbuckling analysis is chosen as the vehicle for quantification of the initial postbuckling behavior. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An exact finite strip for the initial postbuckling analysis of channel section struts

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of channel section struts. The presented method provides an efficient and extremely accurate buckling solution. The Von-Karman’s equilibrium equation is solved exactly to obtain the buckling loads and mode shapes for the channel section struts. The investigation of buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. Through the solution of the Von-Karman’s compatibility equation, the in-plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. All the displacement functions are then substituted in the total strain energy expressions. The theorem of minimum total potential energy is subsequently applied to solve for the unknown coefficient. The developed method is subsequently applied to analyze the initial post-buckling behavior of some representative channel sections for which the results were also obtained through the application of a semi-energy finite strip method. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.

On the predictability of zero‐stiffness postbuckling

AbstractThe purpose of this paper is to derive sufficient and necessary conditions for zero‐stiffness postbuckling which is bound up with a favorable form of transition from imperfection sensitivity to imperfection insensitivity in the course of sensitivity analysis of the initial postbuckling path. Zero‐stiffness postbuckling as such is shown to be imperfection insensitive. Choosing Koiter's initial postbuckling analysis as the vehicle for answering the question of predictability of zero‐stiffness postbuckling seems, at first sight, to be a contradiction in terms, since zero‐stiffness postbuckling includes the entire postbuckling path whereas Koiter's initial postbuckling analysis is practically restricted to the neighborhood of the bifurcation point. The theoretical findings presented in this work are verified numerically.

Relative Post-Buckling Stiffness Calculation of Symmetrically Laminated Composite Plates Using an Exact Finite Strip

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of symmetrically laminated composite plates. The so-called exact finite strip is developed based on the concept that it is effectively a plate. The present method, which is designated by the name Full-analytical Finite Strip Method in this paper, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman’s equilibrium equation is solved exactly to obtain the buckling loads and the corresponding form of out-of-plane buckling deflection modes. The investigation of thin flat plate buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. The post-buckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. In this method, the Von-Karman’s compatibility equation governing the behavior of symmetrically laminated composite plates is used together with a consideration of the total strain energy of the plate. Through the solution of the compatibility equation, the in-plane displacement functions are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. These in-plane and out-of-plane deflected functions are then substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficient. The developed method is subsequently applied to analyze the initial post-buckling behavior of some representative thin flat plates for which the results are also obtained through the application of a semi-analytical finite strip method. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.

Initial thermo-mechanical post-buckling of beams with temperature-dependent physical properties

The objective of this work is to analyze the elastic buckling and initial post-buckling behavior of slender beams subjected to uniform heating. The beams are assumed to be double-hinged with fixed ends, preventing thermal expansion. Consequently, destabilizing compressive forces arise that may lead to beam buckling. When the temperature is further increased, the beam experiences finite displacements, with the result that the analysis is geometrically non-linear. The modulus of elasticity and the thermal induced strain, key material properties for this problem, are temperature-dependent. Thus, the coefficients of the governing equations are not constant. This suggests the physical non-linearity of the mathematical model. Hence, the analysis is geometrically and physically non-linear. The analysis is sensitive to the beam initial temperature, as the thermal strain is a function of the initial and final temperatures. The material is considered to be linear elastic, and consequently viscoelastic and plastic effects are not taken into account. Furthermore, the beam cross-section properties are assumed to be constant, which is consistent with the small strain formulation. A perturbation method is applied to the governing non-linear differential equations so that the initial post-buckling behavior may be analytically determined when temperatures above the critical temperature are applied to the beam. To illustrate the application of the formulation we present a case study for the aluminum 7075-T6 alloy, a material commonly used in aerospace and naval industries. Nonetheless, it is expected similar behavior for other metallic materials. The curves that define the variation of the modulus of elasticity, the thermal strain and the yield stress with temperature are considered in our analysis. The change in length, reaction forces at the supports and geometric configurations are obtained as a function of temperature and the beam slenderness ratio. The critical buckling loads and temperatures and the initial post-buckling analysis are also calculated in the context of the temperature-independent physical properties. Our results emphasize the importance of modeling the material's non-linearity if accuracy is required. However, from a practical application point of view results are acceptable if temperature-independent physical properties are employed, especially for large slenderness ratios.

Imperfection Sensitivity or Insensitivity of Zero‐stiffness Postbuckling … that is the Question

AbstractZero‐stiffness postbuckling of a structure is characterized by a secondary load‐displacement path along which the load remains constant. In sensitivity analysis of the (initial) postbuckling path it is usually considered as a borderline case between imperfection sensitivity and imperfection insensitivity. However, it is unclear whether zero‐stiffness postbuckling as such is imperfection sensitive or insensitive. In this paper, Koiter's initial postbuckling analysis is used as a tool for sensitivity analysis. Distinction between two kinds of imperfections is made on the basis of the behavior of the equilibrium path of the imperfect structure. New definitions of imperfection insensitivity of the postbuckling behavior are provided according to the classification of imperfections. A structure with two degrees of freedom with a zero‐stiffness postbuckling path is studied, considering four different imperfections. The results from this example show that zero‐stiffness postbuckling is a case of transition from imperfection sensitivity to imperfection insensitivity for imperfections of the first kind and that it is imperfection insensitive for imperfections of the second kind. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Finite element based coupled mode initial post-buckling analysis of a composite cylindrical shell

This paper presents a finite element formulation of Koiter's initial post-buckling theory using a multi-mode approach. Initial post-buckling theory provides direct information about the imperfection sensitivity of a structure under compressive loading, and is also the basis of a nonlinear reduced order model. The objective of the present work is to illustrate the capability of the implementation for buckling analysis of shell structures including modal interaction. A coupled mode initial post-buckling analysis for a composite cylindrical shell under axial compression, including the effect of a nonlinear pre-buckling state, has been carried out using a small number of representative modes. For small imperfection amplitudes the limit-point buckling loads obtained with the reduced order model compare reasonably well with full model nonlinear analysis, illustrating that a fast prediction of the coupled mode response of imperfect shells is possible using the approach proposed.

An exact finite strip for the calculation of relative post-buckling stiffness of isotropic plates

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of isotropic flat plates. The so-called exact finite strip is assumed to be simply supported out-of-plane at the loaded ends. The strip is developed based on the concept that it is effectively a plate. The present method, which is designated by the name Full-analytical Finite Strip Method in this paper, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and the corresponding form of out-of-plane buckling deflection modes. The investigation of thin flat plate buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. It is noted that in the present method, only one of the calculated out-of-plane buckling deflection modes, corresponding to the lowest buckling load, i.e., the first mode is used for the initial post-buckling study. Thus, the postbuckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. In this method, the Von-Karman's compatibility equation governing the behavior of isotropic flat plates is used together with a consideration of the total strain energy of the plate. Through the solution of the compatibility equation, the in-plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. These in-plane and out-of-plane deflected functions are then substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficient. The developed method is subsequently applied to analyze the initial postbuckling behavior of some representative thin flat plates for which the results are also obtained through the application of a semi-analytical finite strip method. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.

An exact finite strip for the calculation of relative post-buckling stiffness of I-section struts

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of I-section struts. The so-called exact strip is developed based on the concept that it is effectively a plate. The presented method, which is designated by the name Full-analytical Finite Strip Method, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and mode shapes for the I-section struts. The investigation of buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. The current post-buckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. Through the solution of the Von-Karman's compatibility equation, the in-plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. All the displacement functions are then substituted in the total strain energy expressions. The theorem of minimum total potential energy is subsequently applied to solve for the unknown coefficient. Finally, the developed method is subsequently applied to analyze the initial post-buckling behavior of some representative I-sections for which the results were also obtained through the application of a Semi-energy Finite Strip Method [Ovesy HR. The development and application of a semi-energy post-local buckling finite strip. PhD dissertation, Cranfield University, UK, 1998]. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.

Conditions for symmetric, antisymmetric, and zero-stiffness bifurcation in view of imperfection sensitivity and insensitivity

In general, it is not clear how to design structures such that they are ab initio imperfection insensitive, i.e., without modifications of the original design after the diagnosis of imperfection sensitivity. Symmetry and antisymmetry of bifurcation paths, representing qualitative properties of a structure, not only simplify the postbuckling analysis, but also have an influence on the behavior of real, i.e., imperfect structures. The special case of a zero-stiffness postbuckling path incorporates both, symmetry and antisymmetry. Mathematical definitions of the three categories symmetric, antisymmetric, and zero-stiffness equilibrium paths are given. It is shown that symmetry as well as antisymmetry causes the vanishing of specific coefficients in an asymptotic series expansion, following from Koiter’s initial postbuckling analysis. Thereupon, methods of checking for the three categories of bifurcation behavior are discussed. Finally, the three categories are investigated in terms of necessary and/or sufficient conditions for imperfection insensitivity. For instance, a horizontal tangent of the postbuckling path at the bifurcation point is required for imperfection insensitivity. Four examples illustrate non-symmetric, symmetric, and antisymmetric bifurcation as well as zero-stiffness postbuckling behavior. For the first two examples, the approach of increasing the stiffness results in the conversion of the system from imperfection sensitivity into insensitivity. Remarkably, this happens without changing the prebuckling behavior and the buckling load of the structure. The third example demonstrates that imperfection sensitivity is an inevitable implication of antisymmetric bifurcation.