A new finite element for buckling analysis of laminated stiffened plates

A new stiffened plate element for stability analysis of laminated stiffened plates has been presented. The basic plate element is a combination of Allman's plane stress triangular element and a Discrete Kirchhoff–Mindlin plate bending element. The element includes transverse shear effects. The model accommodates any number of arbitrarily oriented stiffeners within the plate element and eliminates constraints on the mesh division of the plate. The element has no problem associated with shear locking – a phenomenon usually encountered in isoparametric elements. The stability analysis of laminated stiffened plates has been carried out under different loading conditions with the present element.     

Progressive failure analysis for advanced grid stiffened composite plates/shells

A progressive failure methodology is developed to simulate the initiation and propagation of multi-failure modes for advanced grid stiffened (AGS) composite plates/shells on the basis of a stiffened element model. Failures of both skin and ribs are taken into consideration, which are matrix cracking, fiber failure, fiber–matrix shear failure, delamination in skin and fiber failure in rib. All these failures are defined using a set of 2-D stress-based polynomial failure criteria wherein the transverse shear stresses at centroid of the stiffened element are calculated by employing an integrated approach of finite element and finite difference method. Corresponding material and stiffness degradation behavior is introduced after the initiation of individual failure mechanisms. The progressive failure behavior of a composite orthotropic-grid curved panel with a centrally located cutout under compressive load is evaluated using the method.     

Efficient modelling of delamination buckling in composite cylindrical shells under axial compression

Composite cylindrical shells and panels are widely used in aerospace structures. These are often subjected to defects and damage from both in-service and manufacturing events. Delamination is the most important of these defects. This paper deals with the computational modelling of delamination in isotropic and laminated composite cylindrical shells. The use of three-dimensional finite elements for predicting the delamination buckling of these structures is computationally expensive. Here combined double-layer and single-layer of shell elements are employed to study the effect of delamination on the global load-carrying capacity of such systems under axial compressive load. It is shown that through-the-thickness delamination can be modelled and analysed effectively without requiring a great deal of computing time and memory. A parametric study is carried out to study the influence of the delamination size, orientation and through-the-width position of a series of laminated cylinders. The effect of material properties is also investigated. Some of the results are compared with the corresponding analytical results. It is shown that ignoring the contact between the delaminated layers can result in wrong estimations of the critical buckling loads in cylindrical shells under compressive load.     

Measurement techniques for buckling sensitive composite shells

This paper describes a measurement system that was developed for an experimental study on buckling of glass reinforced composite cylinders. An automated non-contact laser device operating inside the test specimen provided a three dimensional scanning system for measuring the deformation of the shell wall. The measurement system was used to obtain the initial geometric imperfections, as well as the deformation under varying control axial displacement. All loading and data acquisition operations were carried out using advanced computer controlled techniques. Due to the large number of measurements undertaken in each test, data collection was followed by processing and reduction techniques, thus delivering the data in a form suitable for finite element analysis and comparative studies. Typical results are presented in order to demonstrate the reliability, accuracy and versatility of the system.     

Delamination buckling and postbuckling in composite cylindrical shells under combined axial compression and external pressure

A series of finite element analyses on the delaminated composite cylindrical shells subject to combined axial compression and pressure are carried out varying the delamination thickness and length, material properties and stacking sequence. Based on the FE results, the characteristics of the buckling and postbuckling behaviour of delaminated composite cylindrical shells are investigated. The combined double-layer and single-layer of shell elements are employed which in comparison with the three-dimensional finite elements requires less computing time and space for the same level of accuracy. The effect of contact in the buckling mode has been considered, by employing contact elements between the delaminated layers. The interactive buckling curves and postbuckling response of delaminated cylindrical shells have been obtained. In the analysis of post-buckled delaminations, a study using the virtual crack closure technique has been performed to find the distribution of the local strain energy release rate along the delamination front. The results are compared with the previous results obtained by the author on the buckling and postbuckling of delaminated composite cylindrical shells under the axial compression and external pressure, applied individually.     

A semi-analytical finite element model for the analysis of laminated 3D axisymmetric shells: Bending, free vibration and buckling

A general semi-analytical finite element model is developed for bending, free vibration and buckling analysis of shells of revolution made of laminated orthotropic elastic material. The 3D elasticity theory is used and the equations of motion are obtained by expanding the displacement field and load in the Fourier series in terms of the circumferential coordinate, θ. The coefficients of the expansion are functions of (r, z), and they are approximated using the finite element method. This leads to a semi-analytical finite element in the (r, z) plane. The element is validated by comparing the present results with the analytical and numerical solutions available in the literature.     

Finite-element modelling and buckling analysis of anisogrid composite lattice cylindrical shells

The paper investigates the buckling behaviour of anisogrid composite lattice cylindrical shells under axial compression, transverse bending, pure bending, and torsion. The lattice shells are modelled as three-dimensional frame structures composed of curvilinear ribs subjected to the tension/compression, bending in two planes and torsion. The specialised finite-element model generation procedure (model generator/design modeller) is developed to control the orientation of the beam elements allowing the original twisted geometry of the curvilinear ribs to be closely approximated. The effects of varying the length of the shells, the number of helical ribs and the angles of their orientation on the buckling behaviour of lattice structures are examined using parametric analyses. Buckling of the lattice shells with cutouts is also analysed. The results of these studies indicate that the modelling approach presented in this work can be successfully applied to the solution of design problems.     

Closed-form buckling analysis of stiffened composite plates and identification of minimum stiffener requirements

Longitudinal stiffeners attached to composite plates may significantly increase the overall buckling loads of the resultant stiffened structure. As long as the bending stiffness EI of the stiffener remains beneath some a priori unknown threshold value EI_min (with E being Young’s modulus and I being the bending moment of inertia), the buckling pattern will usually be of a more or less global nature, meaning that both the plate and the stiffener will exhibit some certain buckling modes. Once a threshold value EI_min for the bending stiffness, also called minimum stiffness, is exceeded, the stiffener more or less remains in its original position in the state of the onset of buckling while the buckling pattern of the stiffened plate is dominated by local buckling modes of the plate itself. The knowledge of this minimum bending stiffness EI_min of longitudinal stiffeners of composite plates is of high practical importance and a predominant design criterion and will be considered in this paper. For the basic load cases uniform compression and pure shear and their combination, simple closed-form analytical approaches will be presented which enable a straightforward and quick yet accurate estimation of the buckling loads (compression) and (shear) of stiffened composite plates on the one hand, and the minimum bending stiffness EI_min of the attached stiffeners on the other hand.     

Simultaneous buckling design of stiffened shells with multiple cutouts

Buckling is usually initiated from a local region near the cutout for cylindrical stiffened shells under axial compression, and then the evolution of buckling waves is governed by the combined effects of local and global stiffness, which limit the load-carrying capacity. Therefore, a simultaneous buckling pattern is crucial for improving the structural efficiency. In this study, a multi-step optimization strategy for the integrated design of near and far fields away from cutouts is proposed, and the convergence criterion of buckling optimization is improved as a deformation-based index. The numerical implementation of the asymptotic homogenization method is utilized to construct an efficient finite element model for post-buckling analysis. A 5?m diameter stiffened shell in a launch vehicle demonstrates that the proposed framework can provide a simultaneous buckling design with high structural efficiency in an efficient manner. Both the buckling deformations and stress of the optimum design are more uniform compared to other optimum designs.     

Genetic optimization of stiffened plates and shells

This work gives two examples of application of stochastic techniques for the optimization of stiffened plates or shells. The research strategy consists in substituting, for finite‐element calculations in the optimization process, an approximate response of a neural network, or an approximate response from the Ritz method. More precisely, the paper describes the use of a backpropagation neural network or the Ritz method in creating function approximations for use in computationally intensive design optimization based on genetic algorithms. Two examples of applications are presented; the first one deals with the optimization of stiffeners on a plate by varying their positions, while having well‐defined dimensions; the second example deals with the optimization of a thin shell subject to buckling. Copyright © 2001 John Wiley & Sons, Ltd.     

Semi-analytic probabilistic analysis of axially compressed stiffened composite panels

The influence of scattering input parameters on the response of axially compressed stiffened composite panels is investigated. In order to estimate the stochastic distributions and the correlations between the first buckling load (or local buckling load), the global buckling load and the collapse load, a semi-analytic probabilistic analysis is performed. A procedure is given for evaluating the probability of failure of stiffened panels from the determined stochastic distributions, and probabilistically justified safety factors are derived.     

Buckling of conical composite shells

A semi-analytical approach is proposed to obtain the linear buckling response of conical composite shells under axial compression load. A first order shear deformation shell theory along with linear strain-displacement relations is assumed. Using the principle of minimum total potential energy, the governing equilibrium equations are found and Ritz method is applied to solve them. Parametric study is performed by finding the effect of cone angle and fiber orientation on the critical buckling load of the conical composite shells.     

Determination of robustness for a stiffened composite structure using stochastic analysis

It is a common practice to only consider the nominal means as input variables for both classical solid mechanics and finite element (FE) analysis problems. A single solution based on the mean values is then used in design. In reality all input variables are stochastic, existing within a range of possible values. Different combinations of these stochastic input variables will lead to differing output responses and the introduction of variability will cause each structure to have a response that deviates from the original specification, sometimes with catastrophic consequences. In this paper two variables, influence and sensitivity, have been identified as parameters affecting structural robustness. Variability and uncertainty in loads, geometry and lamina stiffness are introduced via a stochastic finite element analysis (SFEA) procedure. The procedure is applied to the design of composite yacht hulls comparing the robustness of designs aimed at satisfying a range of performance and cost requirements. It is shown that influence and sensitivity are useful in identifying designs that lead to imperfection tolerant structures.     

Effects of nonlinear strain components on the buckling response of stiffened shear-deformable composite plates

A detailed investigation of the weight of each non linear term of the Green–Lagrange strain displacement equation is presented, with reference to the buckling of orthotropic, both flat and prismatic, Mindlin plates. Usually in the literature, in buckling analysis only the second order terms related to the out-of-plane displacement are considered. Such heuristic simplification, known as von Kármán hypothesis, starts by the consideration that the buckling mode of a flat plate is described by dominant out-of-plane displacement and disregards the non-linear terms of the Green–Lagrange strain tensor depending on the in plane displacement components, whose role is confined to first order, say pre-critical, deformation. The present paper shows that disregarding the non linear terms related to the in-plane strain–displacement is equivalent to neglect shear induced rotation. In the work, the governing equations are derived using the principle of strain energy minimum and the differential equations solution is gained by using the general Levy-type method. The obtained results show that the von Kármán model overestimates the critical load when, in buckling mode, magnitudes of shear rotation, in-plane and out-of-plane displacements are comparable.     

The use of initial imperfection approach in design process and buckling failure evaluation of axially compressed composite cylindrical shells

Thin-walled cylindrical shells are susceptible to buckling failures caused by the axial compressive loading. During the design process or the buckling failure evaluation of axially-compressed cylindrical shells, initial geometric and loading imperfections are of important parameters for the analyses. Therefore, the engineers/designers are expected to well understand the physical behaviours of shell buckling to prevent unexpected serious failure in structures. In particular, it is widely reported that no efficient guidelines for modelling imperfections in composite structures are available. Knowledge obtained from the relevant works is open for updates and highly sought. In this work, we study the influence of imperfections on the critical buckling of axially compressed cylindrical shells for different geometries and composite materials (Glass Fibre Reinforced Polymer (GFRP), Carbon Fibre Reinforced Polymer (CFRP)) and aluminium using the finite element (FE) analysis. Two different imperfection techniques called eigenmode-affine method and single perturbation load approach (SPLA) were adopted. Validations of the present results with the published experimental data were presented. The use of the SPLA for introducing an imperfection in axially compressed composite cylindrical shells seemed to be desirable in a preliminary design process and an investigation of a buckling failure. The knockdown factors produced by the SPLA were becoming attractive to account for uncertainties in the structure.     

Vibrations and buckling of composite orthotropic cylindrical shells with nonuniform axial loads

The vibrational response of orthotropic composite cylindrical shells, subjected to circumferentially nonuniform axial loads, is investigated based on Flügge-type field equations. The use of a complex finite Fourier transform provides a simple method for handling any arbitrary nonuniform load but introduces modal coupling between the transformed equations. For simply supported boundaries (conditions SS3) the determination of the critical buckling load reduces to finding the eigenvalues of a finite matrix. Two different nonuniform loads are considered, having forms proportional to (1+2cos θ) and (θ*−θ), where is the Heaviside function, θ is the circumferential coordinate and aθ* is the width of an axial strip of the shell of radius a. Computed results indicate the sensitivity of the critical buckling loads and free vibrational frequency to the type of nonuniform load and the material lay-ups of the cylinders.     

Buckling response of laminated composite stiffened plates subjected to partial in-plane edge loading

This article presents the buckling analysis of laminated composite stiffened plates subjected to partial in-plane edge loading. The finite element method is used to carry out the analysis. The eight-noded isoparametric degenerated shell element with C⁰ continuity and first-order shear deformation and a compatible three-noded curved beam element are used to model the plate skin and the stiffeners, respectively. The eigen value analysis is carried out to track the buckling load. The convergence study is performed for some specific problems and the results are compared with the available results in the literature. It is observed that the convergence of results is very fast for this finite element model. Effect of different parameters like orientation of fibers, number of layers, and loading types are considered in the present investigation. It is also observed that all these parameters have significant effect on the buckling response of the composite stiffened plate.     

Three-dimensional thermomechanical buckling analysis for functionally graded composite plates

Three-dimensional thermomechanical buckling analysis is investigated for functionally graded composite structures that composed of ceramic, functionally graded material (FGM), and metal layers. Material properties are assumed to be temperature dependent, and in FGM layer, they are varied continuously in the thickness direction according to a simple power law distribution in terms of the ceramic and metal volume fractions. The finite element model is adopted by using an 18-node solid element to analyze more accurately the variation of material properties and temperature field in the thickness direction. Temperature at each node is obtained by solving the thermomechanical equations. For a time discretization, Crank–Nicolson method is used. In numerical results, the thermal buckling behavior of FGM composite structures due to FGM thickness ratios, volume fraction distributions, and system geometric parameters are analyzed.     

A computationally efficient method for the buckling analysis of shells with stochastic imperfections

A computationally efficient method is presented for the buckling analysis of shells with random imperfections, based on a linearized buckling approximation of the limit load of the shell. A Stochastic Finite Element Method approach is used for the analysis of the “imperfect” shell structure involving random geometric deviations from its perfect geometry, as well as spatial variability of the modulus of elasticity and thickness of the shell, modeled as random fields. A corresponding eigenproblem for the prediction of the buckling load is solved at each MCS using a Rayleigh quotient-based formulation of the Preconditioned Conjugate Gradient method. It is shown that the use of the proposed method reduces drastically the computational effort involved in each MCS, making the implementation of such stochastic analyses in real-world structures affordable.