Finite Strip Method Research Articles

Large displacement stability analysis of thin plate structures: Scope of MPI/OpenMP parallelization in harmonic coupled finite strip analysis

The paper presents large displacement stability analysis of orthotropic thin plate structures with different boundary conditions along the diaphragm-supported edges. A semi-analytical harmonic coupled finite strip method (HCFSM) is used to solve the large deflection and the post-buckling problems or may be applied to both problems simultaneously. The stability of equilibrium states is assessed by looking at the eigenvalues of tangent stiffness matrix of structure. In the HCFSM formulation the coupling of all series terms dramatically increases computation time when a large number of series terms are used. Therefore it is natural to use parallel programming standards, such as MPI and OpenMP to speed up computation. The examples provided justify the proposed improvements in the conventional FSM and are in accordance with the experimental data.

Dynamical analysis of stiffened plates using the compound strip method

The harmonic compound finite strip method has been applied to linear transient vibration analysis of stiffened plates. In this method, eigenfunctions of Bernoulli-Euler beam have been used as the displacement interpolation functions in longitudinal direction, while finite element shape functions have been used for it in transverse direction. The Kirchhoff–Love thin plate theory has been used and the equation of motion of structure is derived from Lagrange’s equation of motion. The governing equations have been solved by the mode superposition where step-by-step procedure has been used for the solution of modal equation. The stiffener has been modeled so that it may lie anywhere within the plate strip which helps to increase the flexibility in mesh generation. The formulation is applicable for rectangular plates stiffened with longitudinal and transverse beams and supported on columns. The proposed method is validated through several examples. The strips with free end give erroneous results for non-zero Poisson’s ratio.

Geometric Non-Linear Analysis of Composite Laminated Plates Using Higher Order Finite Strip Element

A finite strip method for geometric non-linear static analysis based on the tangential stiffness matrix has been developed using the new concept of polynomial finite strip elements, with Reissner (higher order shear deformable element) plate-bending theory for composite plates. A finite strip analysis programming package, which is capable of performing non-linear analysis for composite flat panels, has also been developed with Reissner plate bending element. Good agreement with the finite element results has been observed through various test cases, confirming the accuracy and reliability of the new developed finite strip method.

Thin plate bending analysis and treatment of material discontinuities using the generalised RKP-FSM

A finite strip method (FSM) utilising the generalised reproducing kernel particle method (RKPM) [Behzadan, Shodja, and Khezri (2011)] is developed for the bending analysis of thin plates. In this innovative approach, the spline functions in the conventional spline finite strip method (SFSM) are replaced with generalised RKPM 1-D shape functions in the longitudinal direction, while the transverse cubic functions which are used in the conventional formulations are retained. Since the generalised RKPM is one of the class of meshfree methods which deal efficiently with derivative-type essential boundary conditions, its introduction in the FSM is beneficial for solving boundary value problems such as the bending of thin plates in which a number of essential boundary conditions can include first derivatives of the displacement function. In this paper, the formulation for the generalised RKP-FSM is derived for the analysis of thin plates, and its accuracy and convergence are examined through a series of numerical studies. Moreover, by modifying the concept of the augmented corrected collocation method [Shodja, Khezri, Hashemian, and Behzadan (2010)] a new feature is added to the conventional SFSM and the generalised RKP-FSM which allows for the exact treatment of material discontinuities in bending analysis. Oscillatory behaviour of solutions near the interface of material discontinuities due to Gibb's phenomenon is successfully eliminated in both methods.

Analysis of functionally graded sandwich plate structures with piezoelectric skins, using B-spline finite strip method

Functionally graded materials are composite materials wherein the composition of the constituent phases can vary in a smooth continuous way with a gradation which is function of its spatial coordinates. This characteristic proves to be an important issue as it can minimize abrupt variations of the material properties which are usually responsible for localized high values of stresses, and simultaneously providing an effective thermal barrier in specific applications. In the present work, it is studied the static and free vibration behaviour of functionally graded sandwich plate type structures, using B-spline finite strip element models based on different shear deformation theories. The effective properties of functionally graded materials are estimated according to Mori–Tanaka homogenization scheme.These sandwich structures can also consider the existence of outer skins of piezoelectric materials, thus achieving them adaptive characteristics. The performance of the models, are illustrated through a set of test cases.

A constrained finite strip method for prismatic members with branches and/or closed parts

The decomposition of buckling modes of thin-walled members subjected to axial stresses is a topic of great practical interest which can be achieved using the generalised beam theory (GBT) or the constrained finite strip method (cFSM). However, the latter is not general enough to study prismatic members with arbitrary cross-sections and the objective of this paper is to extend the cFSM to allow the buckling modes decomposition for prismatic members with branches and/or closed parts. To define the combined GD buckling mode, two assumptions are used: (i) cylindrical plate bending and (ii) negligible in-plane transverse/shear strains. The corresponding constraint matrix, RGD, is derived in a simple and general way. The methodology used to separate the global and distortional modes is similar to that used in the original cFSM while the derivation of the constraint matrix for local modes remains identical. Some examples are considered and the pure buckling curves are compared to the conventional FSM results. The conclusion is that the new cFSM has successfully computed the GD and the L modes of these sections.

Free and transient vibration analysis of an un-symmetric cross-ply laminated plate by spectral finite elements

In this study, a spectral finite element formulation for free and transient vibration analysis of a rectangular un-symmetric cross-ply laminated composite plate of Levy-type is developed. Plate kinematics obeys classical lamination plate theory. The formulation is based on dynamic shape functions in frequency-domain derived from the exact solution of the governing wave equations. Based on Levy-type solution and finite strip method, 2/D space-time partial differential equations (PDEs) of motion are transformed to 1/D space-time PDEs. Spectral elements are used to transform 1/D space-time PDEs to temporal ordinary differential equations. For investigating validity and accuracy of the model presented, numerical solutions are established for both free and transient vibrations. Comparison of spectral finite element method (SFEM) time responses with an exact analytical method demonstrates the superiority of SFEM in both reducing computational cost and increasing numerical accuracy. It is correct especially for excitations of high-frequency contents. A comparison of SFEM and FEM shows the same result.

Inelastic local buckling of curved plates with or without thickness-tapered sections using finite strip method

This paper addresses the inelastic local buckling of the curved plates using finite strip method in which buckling modes and displacements of the curved plate are calculated using sinusoidal shape functions in the longitudinal direction and polynomial functions in the transverse direction. A virtual work formulation is employed to establish the stiffness and stability matrices of the curved plate whilst the governing equations are then solved using a matrix eigenvalue problem. The accuracy and efficiency of the proposed finite strip model is verified with finite element model using ABAQUS as well as the results reported elsewhere while a good agreement is achieved. In order to illustrate the proposed model, a comprehensive parametric study is performed on the steel and aluminium curved plates in which the effects of curvature, the length of the curved plate as well as circumferential boundary conditions on the critical buckling stress are investigated. The developed finite strip method is also used to determine the buckling loads of the curved plates with thickness-tapered sections as well as critical stresses of the aluminium cylindrical sectors that are subjected to uniform longitudinal stresses.

Closed-form solution to thin-walled box girders considering effects of shear deformation and shear lag

Considering three longitudinal displacement functions and uniform axial displacement functions for shear lag effect and uniform axial deformation of thin-walled box girder with varying depths, a simple and efficient method with high precision to analyze the shear lag effect of thin-walled box girders was proposed. The governing differential equations and boundary conditions of the box girder under lateral loading were derived based on the energy-variational method, and closed-form solutions to stress and deflection corresponding to lateral loading were obtained. Analysis and calculations were carried out with respect to a trapezoidal box girder under concentrated loading or uniform loading and a rectangular box girder under concentrated loading. The analytical results were compared with numerical solutions derived according to the high order finite strip element method and the experimental results. The investigation shows that the closed-form solution is in good agreement with the numerical solutions derived according to the high order finite strip method and the experimental results, and has good stability. Because of the shear lag effect, the stress in cross-section centroid is no longer zero, thus it is not reasonable enough to assume that the strain in cross-section centroid is zero without considering uniform axial deformation.

Parametric instability of laminated longitudinally stiffened curved panels with cutout using higher order FSM

The dynamic instability of longitudinally stiffened panels having rectangular internal cutouts under parametric in-plane loading is studied by using a developed finite strip method (FSM). The loading is considered as uniform stresses throughout the whole area. The effects of perforations on the instability load frequency regions are investigated using the Bolotin’s first order method and a negative stiffness modeling approach. In order to demonstrate the capabilities of the developed methods in predicting the structural dynamic behavior, some representing results are obtained and compared with those in the literature wherever available.

Linear buckling analysis of perforated steel storage rack columns with the Finite Strip Method

An investigation on the use of the Finite Strip Method (FSM) to calculate elastic buckling loads of perforated cold formed storage rack columns is presented. Nowadays, this calculation can be accurately performed by means of the Finite Element Method (FEM), because the effect of perforations can be explicitly considered in the analysis. However, the FSM is preferred in cold-formed steel design since it is implemented in much convenient and easy to use software. The problem with FSM is that holes cannot be easily modeled. In this paper, the concept of the reduced thickness of the perforated strip is applied to take into account their effect. A formulation is presented for the reduced thickness that has been calibrated with loads obtained in eigen-buckling FEM analyses. Its accuracy has been verified carrying out analyses on real rack columns with different end conditions.

A semi-energy finite strip non-linear analysis of imperfect composite laminates subjected to end-shortening

The geometric non-linear analysis of imperfect composite laminates is presented based on the concept of the Semi-Energy finite strip method (FSM). The initial out-of-plane imperfection is assumed to be of the sinusoidal shape in the longitudinal direction, and of different shapes in the transverse direction. The validity and numerical efficiency of the developed semi-energy FSM is investigated by comparing the results with those obtained from a fairly conventional and well exploited finite strip technique, namely the Full-Energy FSM as well as the evaluated results from ANSYS.

Self-shape optimisation application: Optimisation of cold-formed steel columns

This paper presents the optimisation of cold-formed steel open columns using the recently developed self-shape optimisation method that aims to discover new profile shapes. The strength of the cold-formed steel sections is calculated using the Direct Strength Method, and the rules developed in the present work to automatically determine the local and distortional elastic buckling stresses from the Finite Strip and constrained Finite Strip Methods are discussed. The rules are verified against conventional and optimum sections yielded in this research, and found to accurately predict the elastic buckling stresses. The optimisation method is applied to singly-symmetric (mono-symmetric) cold-formed steel columns, and the operators behind the method for the special case of singly-symmetric open profiles are introduced in this paper. “Optimum” cross-sections for simply supported columns, 1.2mm thick, free to warp and subjected to a compressive axial load of 75kN are presented for column lengths ranging from 1000 to 2500mm. Results show that the optimum cross-sections are found in a relatively low number of generations, and typically shape to non-conventional “bean”, “oval” or rounded “Σ” sections. The algorithm optimises for distortional and global buckling, therefore likely subjecting the cross-sections to buckling interaction. A manual attempt to redraw the “optimum” cross-sections to include limitations of current manufacturing processes is made. Future developments of the method for practical applications are also discussed.

Inelastic local buckling behaviour of perforated plates and sections under compression

A recently developed inelastic large displacement isoparametric spline finite strip method was used to investigate the inelastic stress distributions, load transfers and failure modes for a range of hole shapes, sizes and spacings. While such study is available for elastic material behaviour, the influence of yielding on the load transfers in and failure modes of perforated plates and sections is not well understood. This study considers perforated simply supported plates under compression and perforated C-section columns in compression failing by local buckling. Several inelastic material models were included in the study, including elastic perfectly-plastic and linear hardening models, and a model with nonlinear yielding and isotropic hardening. Comparisons are made with elastic analyses and conclusions are drawn about the influence of yielding and strain hardening on the stress distributions, load transfers and failure modes of typical thin-walled metallic plates and structural sections.

A constrained finite strip method for prismatic members with branches and/or closed parts

A constrained finite strip method for prismatic members with branches and/or closed parts

Elastic buckling of cold-formed channel sections in shear

The paper studies and provides solutions to the elastic buckling of whole channel sections including flanges and lips where the sections are loaded in pure shear parallel with the web. The elastic buckling analyses of thin-walled channel sections in pure shear have been implemented by means of the Spline Finite Strip Method (SFSM) to determine the elastic buckling loads (Vcr) of the sections. The shear buckling coefficients (kv) of the web for design of a section are back-calculated from the shear buckling loads. The main variables are the flange widths, member lengths and lip sizes. The boundary conditions at two end sections are simply supported. The results of the analyses plotted in the format of interaction charts are given as design guidelines for designers to predict the elastic buckling shear coefficient (kv) without using the Spline Finite Strip Method (SFSM) software. Typical buckling modes at different flange widths, member lengths and lip sizes are also included in this paper. The modes include local web and flange buckling, section distortional buckling and section twisting.

High accuracy postbuckling analysis of box section struts

AbstractThis paper presents the theoretical developments of two finite strip methods (i.e. semi‐analytical and full‐analytical) for the post‐buckling analysis of some box section struts. In the semi‐analytical finite strip approach, all the displacements are postulated by the appropriate shape functions while in the development process of the full‐analytical approach, the von‐Kármán'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 struts buckling behaviour is then extended to the post‐buckling study with the assumption that the deflected form after the buckling is the combination of first, second and higher (if required) modes of buckling. Thus, the full‐analytical post‐buckling study is effectively a multi term analysis, which is attempted by utilizing the so‐called semi‐energy method. In this method the von‐Kármán compatibility equation is used together with a consideration of the total strain energy of the strut. 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 coefficients in the assumed out‐of‐plane deflection function. The in‐plane and out‐of‐plane deflection functions are substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficients. It is noted that the Classical Plate Theory (CPT) is applied throughout the theoretical developments. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of these methods is significantly promoted.

Parametric instability analysis of moderately thick FGM cylindrical panels using FSM

In the current paper, it is the first time that a dynamic stability analysis of moderately thick cylindrical panels made from FG materials is conducted by employing finite strip formulations based on a Reddy-type third order shear deformation theory (HSDT). Two versions of finite strip methods (FSM), namely semi-analytical and B-spline methods are developed. The mechanical properties of FGM are assumed to change in thickness direction according to a power-law function. The temperature effects are ignored. The strain terms are expressed in terms of the Koiter–Sanders theory of shallow shells. In order to demonstrate the capabilities of the developed methods in predicting parametric behavior of the subject structures, some representative results are obtained and compared with those in the literature wherever available.

Vehicle strip element in the analysis of stiffened plate under realistic moving loading

In this article, a new vehicle strip element is proposed for a vehicle–bridge interaction modelling. Finite strip method principle is applied on the vehicle dynamic system in order to model and couple vehicle with the structure, which is discretised with flat shell strip elements. Vehicle degrees of freedom are transformed and embedded into the structural ones, such that any new conditions or model changes can be easily integrated. Combined with the shell finite strip it enabled observing all degrees of freedom of the structure and the vehicle, reducing modelling and calculation time and effort in the same time. A full vertical vehicle model with nine degrees of freedom is adopted here. The bridge is modelled as a simply supported stiffened plate with use of a flat shell strip elements. The main benefit of the vehicle strip element is possibility to analyse vehicle–structure interaction in a greater detail. This resulted in more realistic and comprehensive analysis of a bridge under the moving loading. Such realistic examples of bridge analysis are rare in literature due to the large effort they require using standard finite elements. A parametric study is made to compare the vehicle models with different level of complexity and to enquire the relationship between model simplification and the reliability of results. In contrast to beam or plate models, application of the flat shell strip element enabled the analysis with the realistic bridge cross section.

Buckling analysis of functionally graded plates under thermal loadings using the finite strip method

In the present article, a finite strip method is applied for analyzing the buckling behavior of rectangular functionally graded plates (FGPs) under thermal loadings. The material properties of FGPs are assumed to vary continuously through the thickness of the plate, according to the simple power law distribution. Derivations of equations are based on the classical plate theory (CPT). The fundamental equations for rectangular plates of functionally graded material (FGM) are obtained by discretizing the plate into some finite strips. The solution is obtained by the minimization of the total potential energy and solving the corresponding eigenvalue problem. In addition, numerical results for a variety of functionally graded plates with different boundary conditions are presented and compared with those available in the literature. Moreover, the effects of geometrical parameters and material properties on the FGPs’ buckling temperature difference are determined and discussed.