Initial Postbuckling Research Articles

An efficient method for postbuckling optimization of composite laminated plates with holes under compression and shear loading

Laminated composite plates with hole are commonly used in aeronautic and aerospace structure design, subjecting to the in-plane compression and shear loading. The existing of hole reduces both the in-plane and out-of-plane stiffness and influences the postbuckling behavior of the plate. The structural lightweight design can be further explored when the external load is allowed entering into the postbuckling regime. However, the lamination optimizations to maximize the postbuckling stiffness have high computational burden when the standard nonlinear finite element method is applied in each function call for postbuckling analysis. Thus, we propose a reduced-order modeling method based on the improved Koiter perturbation theory for buckling and postbuckling analysis. The low-scale reduced order model, instead of the full-scale FE model, is constructed and solved to obtain the initial postbuckling response. The postbuckling stiffness is defined to be the slope of the load-end shortening relation curve at the bifurcation point. A multi-start searching strategy based on the gradient-based algorithm is developed to maximize the postbuckling stiffness, efficiently and accurately. Laminated composite plates with different hole sizes, loading conditions and ply numbers, are considered to validate the excellent performance of proposed method.

A corotational mixed flat shell finite element for the efficient geometrically nonlinear analysis of laminated composite structures

AbstractA corotational flat shell element for the geometrically nonlinear analysis of laminated composite structures is presented. The element is obtained from the Hellinger–Reissner variational principle with assumed stress and displacement fields. The stress interpolation is derived from the linear elastic solution for symmetric composite materials. The element is isostatic, namely the stress interpolation is ruled by the minimum number of parameters. Displacement and rotation fields are only assumed along the contour of the element. As such, all the operators are efficiently obtained through analytical contour integration. The geometrical nonlinearity is introduced by means of a corotational formulation. The proposed finite element, named MISS‐4c, proves to be locking free and shows no rank defectiveness. A multimodal Koiter's algorithm is used to obtain the initial postbuckling response. Results show good accuracy and high convergence rate in the geometrically nonlinear analysis of composite shell structures.

Postbuckling analysis and optimization of composite laminated panels using a novel perturbation-based approximation FE method

When composite laminated panels under compression are allowed to continuously loaded into the moderately deep postbuckling regime, further weight savings can be achieved. For this purpose, this work presents an optimization scheme for the design of postbuckling behavior of composite laminates. An efficient tool, termed the Koiter perturbation-based approximation FE method, is developed to obtain the postbuckling behavior of composite laminated panel. The initial postbuckling response is represented by the nonlinear predictor solved by the reduced order model, while the deformations in the moderately deep postbuckling regime are obtained by applying the correction phase under the design load. Five different indices, including the critical buckling load, postbuckling stiffness, stiffness residual, and in-plane and out-of-plane deformations under the design load, are proposed to assess various postbuckling performances of the compressed panel. The postbuckling indices can be flexibly selected to be either the objective or the nonlinear constraints in the lamination optimization. A genetic algorithm is subsequently used to determine the optimal lamination configuration for the most favorable postbuckling performance. The optimizations of composite laminates under uniaxial and biaxial compression for different boundary conditions are carefully studied and discussed with the consideration of initial geometric imperfections.

Influence of chirality on buckling and initial postbuckling of inextensible rings subject to central loadings

A variational model is used to study buckling and initial postbuckling of flexible but inextensible rings made from filaments with or without chirality and subject to two cases of uniform central loading and inverse-square central loading. In contrast to previous works in which only in-plane bifurcation are included, we allow for both in-plane and out-of-plane bifurcation. For a ring made from a filament with no chirality, circular rings subject to uniform central loading or inverse-square central loading undergo, respectively, out-of-plane bifurcation or in-plane bifurcation at critical values of loading lower than or equal to those previously published. Additionally, the relative load difference to attain the post-buckled shapes for the former or the latter is, respectively, greater than or equal to that previously published. For a ring made from a filament with chirality, on the other hand, circular rings undergo coupling between in-plane and out-of-plane bifurcation at critical values of loading lower than those at which buckling would occur with no chirality, regardless of loading scenario considered. Moreover, dependent on the twist-to-bend ratio (torsional rigidity to flexural rigidity of the filaments), the chirality has a softening or stiffening effect on postbuckling behavior of elastic rings in the sense that the relative load difference to attain the post-buckled shapes for both types of central loading is, respectively, lower or greater than that which would attain with no chirality.

Optimal Postbuckling Design of Variable Angle Tow Composite Plates

Perturbation-based approximation methods are widely used in preliminary design studies of thin-walled structures. In this paper, postbuckling analysis of a variable-angle-tow composite plate is performed using the perturbation-based asymptotic numerical method, which transforms the nonlinear problem into a set of well-posed recursive linear problems. These linear problems are solved using a novel generalized differential-integral quadrature method, and the postbuckling solutions are sought over a finite load step size around the critical buckling point using asymptotic expansions. The accuracy of the asymptotic numerical method in evaluating the initial postbuckling of variable-angle-tow plates under compression is investigated. Subsequently, a novel postbuckling optimization approach based on asymptotic numerical method results is proposed for the design of variable-angle-tow laminates. The postbuckling features obtained from asymptotic numerical method are used in an efficient two-level optimization framework for the design of variable-angle-tow plates. At the first level, a globally convergent method of moving asymptotes is adopted to determine the optimal lamination parameter distributions that maximize the postbuckling performance of the variable-angle-tow plate. At the second level, a genetic algorithm is used to convert the optimal lamination parameter distributions into realistic variable-angle-tow layups. The optimization studies are performed for square variable-angle-tow plates for axial/biaxial compression under different in-plane boundary conditions.

Analytical Solution for Initial Postbuckling Deformation of the Sandwich Beams Including Transverse Shear

This paper presents analytical approximate solutions for the initial postbuckling deformation of sandwich beams including transverse shear. The approximate procedure is based on the nonlinear beam equation (with transverse shear included) by coupling the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials; the governing differential equation with sinusoidal nonlinearity can be reduced to form a cubic-nonlinear equation. Analytical approximations to the resulting boundary condition problem are established by combining the methods of Newton linearization and harmonic balance. Unlike the classical method of harmonic balance, the linearization is performed prior to proceeding with harmonic balancing thus resulting in a set of linear algebraic equations instead of one of nonlinear algebraic equations, thereby establishing analytical approximate solutions. Illustrative examples are presented for a few typical sandwich construction configurations, and it is shown that the proposed approximate solutions are in excellent agreement with the numerical solution obtained via the shooting method for both small and large angles of rotation.

The effects of warping on the postbuckling behaviour of thin-walled structures

A direct 1D model able to describe the mechanical behaviour of thin-walled beams is used to analyse the initial postbuckling of some sample frames. The model allows to take into account the boundary conditions on warping in a very simple way and proves to be a very efficient tool for the analysis of thin-walled frames. Making use of the FEM based COMSOL multiphysics software, numerical results for a number of L frames are obtained and shown.

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)

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.

Modes of transition from imperfection sensitivity to imperfection insensitivity

AbstractIn order to a priori find the mode of conversion of a structure from imperfection sensitivity to insensitivity in the course of sensitivity analysis of the initial postbuckling path (cf. [2]), terms appearing in the consistently linearized eigenvalue problem (cf. [1]) are studied. Two main classes can be identified (cf. [4]). The first one is characterized by the restriction of the prebuckling deformations to axial deformations. For this class, the quality of the aforementioned conversion is better than the one for the other class, for which the above restriction does not hold. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Initial post buckling of toroidal shell segments pressurized by external load

Initial post buckling of toroidal shell segments pressurized by external load

Postbuckling of truss-lattice shear panels using exact theory

A new solution is developed to model the stable postbuckling behaviour of a truss-lattice shear panel. The mode shapes are derived through load equilibrium conditions and are based on the load in the structural members. The load in each member of a single-cell panel can be calculated exactly, without the need for an iterative postbuckling path, and the method produces excellent results in initial and advanced postbuckling. Comparisons are drawn with an alternative analytical method and the commonplace finite element approach. A Rayleigh‐Ritz method based on a Fourier approximation to the mode shape provides the useful progression from an unbuckled to a buckled structure giving excellent results in initial postbuckling, although is limited for advanced postbuckling. The standard finite element method for this problem produces accurate results but with limited detail around the buckling load owing to the presence of an imperfection in the shape of the initial mode, which is required to initiate the postbuckling analysis.

Initial post dynamic buckling of a quadratic-cubic column pressurized by a sinusoidally slowly varying dynamic load

In this investigation, we determine the dynamic buckling load of an imperfect finite column resting on a mixed quadratic-cubic nonlinear elastic foundation trapped by an explicitly time dependent sinusoidally slowly varying dynamic load .The resultant coefficients are dynamically slowly varying and the formulation contains two small but Mathematically unrelated parameters upon which regular perturbation procedure are used in asymptotic expansions of the variables. The imperfection is assumed in the shape of the mth term in the Fourier sine series expansion with small (in absolute value) Fourier coefficients. A generalization of Lindsted-Poincare procedure is used and the structure under investigation is, on the main, a nonlinear oscillatory system with small perturbations. The results obtained are strictly asymptotic and are valid for small absolute values of the perturbation parameters. Journal of the Nigerian Association of Mathematical Physics Vol. 9 2005: pp. 199-214

Initial Postbuckling of Elastic Rods Subjected to Thermal Loads and Resting on an Elastic Foundation

In practical applications encountered in some mechanical systems initially straight slender elements laterally supported by elastic foundations may be prevented from thermal expansion. In such conditions a compressive force is created and a serious equilibrium instability phenomenon may take place when a critical temperature gradient is imparted to the structure. If the temperature is further increased the rod may experience relatively large deflections, and a geometrically non-linear two-point boundary value problem describes this behaviour. However, if finite but moderate deflections are admitted, an approximate solution may be obtained if a classical perturbation expansion method is applied to the non-linear ordinary differential governing equations. This approach renders a set of linear equations which can be sequentially solved. This work proposes an analytical solution for the assessment of the initial post-buckling configuration of slender elastic rods supported on linear elastic foundations and subjected to a uniform temperature gradient. The rod ends are assumed hinged and immovable and the thermal strain-temperature relationship is linear. The governing equations are derived and made non-dimensional so it is seen that two parameters rule the problem: the foundation stiffness modulus and the rod slenderness ratio. Results are presented and discussed for one slenderness ratio and a range of foundation stiffness that corresponds to the fourth buckling mode. Generally an increase in the applied temperature is associated with a decrease in the compressive force, but it is shown particular conditions where an opposite behavior is observed. The initial post-buckling results for unsupported rods are compared with existing full post-buckling solution and show excellent agreement, given that in practical situations the deflections are moderate.

A simplified nonlinear stability analysis of an imperfect rectangular two-bar frame

A simplified nonlinear stability analysis for moderately large rotations and small strains is performed on a rectangular two-bar frame subjected to a concentrated eccentrically applied joint load. Such a simplification consisting of adopting linear kinematic relations leads to very reliable results for the initial postbuckling path in the vicinity of the critical point of the above imperfect frame. The existence of an asymmetric bifurcation point is thoroughly discussed and a direct evaluation of the bifurcational load is readily obtained. Using this technique the effect of imperfection sensitivity is also addressed. A qualitative analysis associated with the physical phenomenon yields a substantial reduction of the computational work. The efficiency and reliability of this approximate nonlinear stability analysis proposed herein is illustrated by means of several examples for which a lot of numerical results based on a more accurate nonlinear analysis are available.

Shell Theory Accuracy with Regard to Initial Postbuckling Behavior of Cylindrical Shell

are the normal strains in the prebuckling state. The buckling anal- ysis for the nondebonded imperfect beams is then similar to that presented earlier for the free-vibration problem. In the case of a generally debonded beam, an exact buckling solution cannot be obtained and approximate treatments should be employed. How- ever, if the beam is with through-length delamination at the kth interface, for example, an exact solution can be obtained, just as it can for the nondebonded imperfect beam, but with K (k)

Buckling and Initial Postbuckling Behavior of Sandwich Beams Including Transverse Shear

An asymptotic solution is presented for the buckling and initial postbuckling behavior of sandwich beams. The effect of transverse shear is included, and the shear correction is calculated from energy equivalency. The asymptotic procedure is based on the nonlinear beam equation (with transverse shear included), and closed-form solutions are derived for the critical load and for the load and midpoint deflection and axial shortening vs applied compressive load during the initial postbuckling phase. Illustrative results are presented for a few typical sandwich construction configurations, in particular, with regard to the effect of face sheet and core material system.

On Initial Postbuckling of Composite Plate Assemblies by Semianalytical Procedure

This article deals with the buckling and postbuckling analysis of thin-walled composite structures made of assemblies of plates by means of a semianalytical approach. Both open and closed sections are analyzed. An orthotropic constitutive model is introduced in order to evaluate composite material behavior. The proposed method is shown to be quite efficient from a computational point of view and to offer significative advantages in optimization design procedures over the finite-element method. Some examples are discussed in detail.

On Initial Postbuckling of Composite Plate Assemblies by Semianalytical Procedure

This article deals with the buckling and postbuckling analysis of thin-walled composite structures made of assemblies of plates by means of a semianalytical approach. Both open and closed sections are analyzed. An orthotropic constitutive model is introduced in order to evaluate composite material behavior. The proposed method is shown to be quite efficient from a computational point of view and to offer significative advantages in optimization design procedures over the finite-element method. Some examples are discussed in detail.

An asymptotic analysis of delamination buckling and growth in layered plates

Delamination buckling and growth in compressively loaded elastic-layered plates is analyzed. The delamination growth process is assumed to start from an initial interlaminar adhesion defect and propagate as induced by buckling. The relevant governing equations for buckling and initial postbuckling are developed by employing asymptotic analysis results. The energy release rate concept is then applied to analyze delamination growth. Two techniques are analyzed, namely the global energy approach and the local J-integral approach. For the two cases, the effects of the asymptotic approach accuracy on the postbuckling and delamination growth are investigated. A general model of the plate is proposed, in which a global instability of the whole plate can occur together with a local instability of the layers. In addition, simplified models are examined which may be useful in many engineering applications, although based on hypotheses about the plate geometry. Finally, the numerical results show the effectiveness of the developed models. Comparisons between the general model and the thin film approximation show the convergence of the former to the latter.