Finite Strip Method Research Articles

Effective width method for distortional buckling design of cold-formed lipped channel sections

The aim of this paper is to improve the effective width method for distortional buckling strength of cold-formed steel lipped channel members using the energy method and the large deflection theory based on nonlinear post-buckling strength analysis of the partially stiffened element. The accuracy of the proposed method was verified using the finite element method. Comparison on post-distortional buckling strength is conducted between the energy method and the effective width method in the Chinese code Technical code of cold-formed thin-walled steel structures (GB50018-2002). The compared results show that the energy method is very efficient and the effective width method formula in Chinese code can be used to calculate the post-distortional buckling strength. Then, The half-wave length, elastic buckling stress of distortional buckling and the corresponding stability coefficient of partially stiffened elements are developed based on the energy method. With comparison on the calculated results of elastic buckling stress using the proposed method and the Finite Strip Method, suitability and good precision of the developed method are illuminated. Finally, a uniform formula for the stability coefficient of partially stiffened elements considering both local and distortional buckling effect is established. Comparison with experimental results and calculated results predicted with North America specification indicates that the proposed method is reasonable and has a high accuracy and reliability for the distortional buckling load carrying capacities of cold-formed lipped channel members.

Shear buckling of plates and thin-walled channel sections with holes

Thin-walled structural channel members are commonly manufactured with cut-outs to allow access for building services such as plumbing, electrical and heating systems in the walls and ceilings. The presence of holes in the members will cause changes in the stress distribution and there will be consequently changes in the buckling characteristic and ultimate strength capacity. Recent work by Pham and Hancock has provided solutions to determine the shear buckling load using the Spline Finite Strip Method (SFSM) for whole thin-walled channel sections without holes. In this paper, the same methodology is utilised to study and provide solutions to the elastic shear buckling firstly for the perforated square plates and subsequently for the whole thin-walled lipped channel sections with centrally located holes. Both circular and square holes with the same sizes and diameters were chosen for investigation. The main variables are the diameters of the circular holes and the sizes of the square holes. While there is only uniform pure shear applied throughout in square plates, three different cases for shear loading in the channel are considered to maintain longitudinal equilibrium. The method is also benchmarked against the Finite Element Method (FEM) using software package ABAQUS/Standard in all cases. Comparisons between hole shapes, loading cases and buckling modes of both square plates and channels are included. For design purposes, approximate equations for shear buckling coefficients of square plate and channel section containing central circular and square holes are also proposed in this paper. Design example is also provided for design purposes.

Buckling analysis of laminated composite cylindrical shell subjected to lateral displacement-dependent pressure using semi-analytical finite strip method

The objective of this paper is to investigate buckling behavior of composite laminated cylinders by using semi-analytical finite strip method. The shell is subjected to deformation-dependent loads which remain normal to the shell middle surface throughout the deformation process. The load stiffness matrix, which is responsible for variation of load direction, is also throughout the deformation process. The shell is divided into several closed strips with alignment of their nodal lines in the circumferential direction. The governing equations are derived based on the first-order shear deformation theory with Sanders-type of kinematic nonlinearity. Displacements and rotations of the shell middle surface are approximated by combining polynomial functions in the meridional direction and truncated Fourier series along with an appropriate number of harmonic terms in the circumferential direction. The load stiffness matrix, which is responsible for variation of load direction, is also derived for each strip and after assembling, global load stiffness matrix of the shell is formed. The numerical illustrations concern the pressure stiffness effect on buckling pressure under various conditions. The results indicate that considering pressure stiffness causes buckling pressure reduction which in turn depends on various parameters such as geometry and lay-ups of the shell.

Semi-analytical model based on generalized plane strain assumption for unidirectional composites with matrix micro cracks

ABSTRACTGeneralized plane strain state is assumed and stress-based finite strip method is formulated for analysis of unidirectional laminates with matrix microcracks. Total complementary potential energy is minimized and fourth-order Euler Lagrange governing equations are presented. This stress-based generalized plane strain approach analyzes general layup and loading conditions. It provides flexibility to control the number of finite strip nodal lines within each lamina; hence, stress behavior can be predicted across each lamina at the desired location of the structure. Along with all of the capabilities which are common with finite strip methodology based on plane strain assumption, this current work has extended the analysis of the cracked laminate. For example, by incorporating behavior of the out of the plane shear stress, boundary conditions including natural boundary conditions are imposed appropriately to solve governing Euler's equations. Results are compared with previously developed displacement-based formulation in the literature for cracked laminates. It has already been shown that a stress-based plane strain approach enhances variation-based cracked laminate analysis where only the case of cross-ply laminate under tension is considered. This current work applies generalized plane strain-based finite strip methodology to carry out analysis under different loading conditions.

Buckling and free vibration of the FGM thin micro-plate based on the modified strain gradient theory and the spline finite strip method

Abstract In the present article the mechanical instability and free vibration of FGM micro-plate based on the modified strain gradient theory were studied using the spline finite strip method. By daily increase in the application of micro-scale structures, developing theories were become essential to account in a way for the size-reduction effect. The modified strain gradient theory based on three length scale parameters, has the capability of evaluating structures at the micro size level. Considering the obtained results, it was clear that increasing the length-scale parameter would increase the critical buckling load and the vibration frequency, similar to the macroscopic case. In addition, increasing the power of volume fraction module decreases the critical load and the natural frequency of micro plate. Finally, the effect of length-scale parameter, boundary conditions, volume fraction module and dimensions of the micro-plate on critical loading and natural frequency of micro-plate were studied.

Simplified semi-analytical model for elastic distortional buckling prediction of cold-formed steel flexural members

The expressions for elastic distortional buckling stress predictions available in literature are used with reasonable accuracy, but are either iterative or involve increased calculation effort for design. It is an accepted fact among researchers that these expressions are not evaluated enough for the distortional stress prediction of sections with complex lip stiffeners. In this paper, a candidate model for distortional buckling stress predictions is presented which is semi analytical in nature and is simple to incorporate in the direct strength method (DSM) of cold-formed steel design. The proposed expression incorporates the effect of complex lip stiffeners on the elastic distortional buckling capacities of cold formed steel flexural members. In the proposed model, the (i) translational stiffness at lip-flange junction and (ii) rotational stiffness at the flange-web junction, are derived from regression analysis of wide range of cross sectional dimensions. The proposed method is calibrated with semi-analytical finite strip method presented in the literature and this formulation has been demonstrated to be good in comparison with recently published numerical based distortional buckling predictions.

Buckling analysis of laminated plate structures with elastic edges using a novel semi-analytical finite strip method

A novel semi-analytical finite strip method is presented for buckling analysis of composite plate structures with boundary edges elastically supported. A set of unique Fourier series functions is introduced to represent the longitudinal variation of deflection along a strip, and they are capable of handling elastic edges with translational and rotational spring supports. The proposed hybrid method overcomes limitation of classical finite strip method only capable of handling simple end boundary conditions of structures, and it avoids the ill-conditioning when a set of standard Fourier series functions is used for buckling analysis. Accuracy and validity of the proposed method are demonstrated by the convergence and comparative studies in comparison with the numerical finite element method. As an example, the present method is applied to buckling analysis of a composite Z-stiffened panel under pure shear, and its capability and efficiency of treating different edge conditions in the panel skin and stiffeners are illustrated.

Buckling analysis of thin-walled members via semi-analytical finite strip transfer matrix method

Slender thin-walled members are main components of modern engineering structures, whose buckling behavior has been studied widely. In this article, thin-walled members with simply supported loaded edges can be discretized in the cross-section by semi-analytical finite strip technology. Then, the control equations of the strip elements will be rewritten as the transfer equations by transfer matrix method. This new method, named as semi-analytical finite strip transfer matrix method, expands the advantages of semi-analytical finite strip method and transfer matrix method. This method requires no global stiffness matrix, reduces the size of matrix, and improves the computational efficiency. Compared with finite element method’s results of three different cross-sections under axial force, the method is proved to be reliable and effective.

Shell element for constrained finite element analysis of thin-walled structural members

In this paper a novel shell finite element is introduced, specifically proposed for constrained shell finite element analysis. The proposed element is derived from the finite strips used in the semi-analytical finite strip method. The new finite element shares the most fundamental feature of the finite strips, namely: transverse and longitudinal directions are distinguished. Moreover, the new element keeps the transverse interpolation functions of finite strips, however, the longitudinal interpolation functions are changed from trigonometric functions (or function series) to classic polynomials. It is found that the proper selection of the polynomial longitudinal interpolation functions makes it possible to perform modal decomposition similarly as in the constrained finite strip method (cFSM). This requires an unusual combination of otherwise well-known shape functions. If the so-constructed shell finite elements are used to model a thin-walled member, (hence, with using discretization in both the transverse and the longitudinal directions,) modal decomposition can be done essentially identically as in cFSM, whilst the practical applicability of the method is significantly extended (e.g., various restraints, holes, certain cross-section changes can easily be handled). In this paper the focus is on the derivation of the novel shell finite element. Constraining capability is illustrated by some basic examples. Practical application of the novel element will be presented in subsequent papers.

Buckling of thick deep laminated composite shell of revolution under follower forces

Laminated composite shells are commonly used in various engineering applications including aerospace and marine structures. In this paper, using semi-analytical finite strip method, the buckling behavior of laminated composite deep as well as thick shells of revolution under follower forces which remain normal to the shell is investigated. The stiffness caused by pressure is calculated for the follower forces subjected to external fibers in thick shells. The shell is divided into several closed strips with alignment of their nodal lines in the circumferential direction. The governing equations are derived based on first-order shear deformation theory which accounts for through thickness-shear flexibility. Displacements and rotations in the middle surface of shell are approximated by combining polynomial functions in the meridional direction as well as truncated Fourier series with an appropriate number of harmonic terms in the circumferential direction. The load stiffness matrix which accounts for variation of loads direction will be derived for each strip of the shell. Assembling of these matrices results in global load stiffness matrix which may be un-symmetric. Upon forming linear elastic stiffness matrix called constitutive stiffness matrix, geometric stiffness matrix and load stiffness matrix, the required elements for the second step analysis which is an eigenvalue problem are provided. In this study, different parameter effects are investigated including shell geometry, material properties, and different boundary conditions. Afterwards, the outcomes are compared with other researches. By considering the results of this article, it can be concluded that the deformation-dependent pressure assumption can entail to decrease the calculated buckling load in shells. This characteristic is studied for different examples.

Cold-formed steel structures: Research review 2013–2014

This article reviews research on cold-formed steel structures published in 2013 and 2014 in three leading journals: the Journal of Structural Engineering, ASCE, Thin-Walled Structures and the Journal of Constructional Steel Research. It also reviews papers published in the three main conferences in the area over the same period. These are Eurosteel 2014 (Naples, Italy), the 7th International Conference on Thin-Walled Structures (Busan, Korea) and the 22nd International Specialty Conference on Cold-Formed Steel Structures (St Louis, MO, USA). Three research areas which have recently been incorporated in the North American Specification NAS S100:2012 or are being incorporated in the Australian/New Zealand Standard AS/NZS 4600 have been highlighted. These are the works on the semi-analytical finite strip method for sections in shear by the author and his colleagues at the University of Sydney, net section rupture by Associate Professor Lip Teh at the University of Wollongong and fire design by Professor Mahendran at Queensland University of Technology.

FSM에 의한 다층 원통쉘의 동적 특성에 관한 연구

This study was presented how to interpret a laminated cylindrical shell having both ends supported condition are simple, and by identifying the three-dimensional stress characteristics. The governing equations were using the concept of finite element assuming the conversion into ordinary differential equations and partial differential equations by numerical analysis using the finite strip method. In particular, a function performed for the three-dimensional laminated cylindrical shell having a simple support condition were analyzed for the case composed of a specific function of the beam consisting of a trigonometric function. Layer material and layer thickness of a steel or concrete, the cylindrical length and the like by varying the parameters variously examine the effects of multi-layer cylindrical shell.

Seismic Analysis of Long-Span Cable-Stayed Bridges by an Integrated Finite Strip Method

AbstractThis paper provides a very efficient, integrated framework for seismic analyses of long-span cable-stayed bridges. The efficiency comes from the dramatic reduction in formation time and the degrees of freedom (DOF) associated with the structure, using the integrated finite strip method (IFSM) along with the application of a very robust and efficient time history method (THM) using the Newmark scheme for dynamic analysis of the bridge structure. The previous versions of the finite strip method are limited to modeling the bridge deck only, whereas other structural components are replaced by assumed boundary conditions. Using the IFSM, all components of the long-span cable-stayed bridge can be modeled in a unified system, and consequently, the real dynamic behavior including the interactions between deck, piers, and cables can be perfectly considered. To verify the solution, the geometric and dynamic properties of the Kap Shui Mun (KSM) Bridge, as a real example of a long-span cable-stayed bridge, ar...

Optimization Of Curved Plated Structures With The Finite Strip And Finite Element Methods

Abstract The aim of this study is to compare two available numerical tools for solving of partial differential equations for the optimal design of structures. In the past years numerous methods were developed for topology optimization, from these we have adopted the optimality criteria (OC) approach. The main idea is that we state the optimal conditions, that the minimizer has to fulfil at the end of an iterative proves. This method however is not a general one, only advantageous in the case of separable problems, but comes with fast speed, easy programming, and a relative insensitivity of computational time to the number of variables. In the paper we suggest a new method for the elimination of a numerical error, the so called ‘checkerboard pattern’. In the presented examples we applied one loading case and an elastic material behaviour. The cost function is the net weight of the structure and upper bound of the compliance is set as the optimality constraint.

Nonlocal buckling and vibration analysis of thick rectangular nanoplates using finite strip method based on refined plate theory

The buckling and vibration of thick rectangular nanoplates is analyzed in this article. A graphene sheet is theoretically assumed and modeled as a nanoplate in this study. The two-variable refined plate theory (RPT) is applied to obtain the differential equations of the nanoplate. The theory accounts for parabolic variation of transverse shear stress through the thickness of the plate without using a shear correction factor. Besides, the analysis is based on the nonlocal theory of elasticity to take the small-scale effects into account. For the first time, the finite strip method (FSM) based on RPT is employed to study the vibration and buckling behavior of nanoplates and graphene sheets. Hamilton’s principle is employed to obtain the differential equations of the nanoplate. The stiffness, stability and mass matrices of the nanoplate are formed using the FSM. The displacement functions of the strips are evaluated using continuous harmonic function series which satisfy the boundary conditions in one direction and a piecewise interpolation polynomial in the other direction. A matrix eigenvalue problem is solved to find the free vibration frequency and buckling load of the nanoplates subjected to different types of in-plane loadings including the uniform and nonuniform uni-axial and biaxial compression. Comparison studies are presented to verify the validity and accuracy of the proposed nonlocal refined finite strip method. Furthermore, a number of examples are presented to investigate the effects of various parameters (e.g., boundary conditions, nonlocal parameter, aspect ratio, type of loading) on the results.

High accuracy post-buckling analysis of moderately thick composite plates using an exact finite strip

A high accuracy finite strip for the buckling and post-buckling analysis of moderately thick symmetric cross-ply composite plates is presented in this paper by using first order shear deformation theory (FSDT). The presented method, which is designated by the name Full-analytical Finite Strip Method (F-a FSM), provides an efficient and extremely accurate buckling solution in which the Von-Karman’s equilibrium set of equations is solved exactly, and the out-of-plane mode shapes and critical loads are obtained. In the post-buckling stage, the Von-Karman’s compatibility equation is solved exactly with the assumption that the deflected form after buckling is a combination of obtained buckling modes (single or multiple mode shapes). The principle of minimum potential energy is invoked to solve for the unknown coefficients in the assumed out-of-plane deflection and rotations functions.

Stability of thin FGM microplate subjected to mechanical and thermal loading based on the modified couple stress theory and spline finite strip method

The buckling of a thin FGM microplate subjected to mechanical and thermal loading is evaluated using the spline finite strip method, based on modified couple stress theory. Regarding thickness, the features of the FGM plate are assumed as being variable according to the law's proposed model. The spline finite strip method are used for calculating the buckling load and critical temperature by solving an eigenvalue problem. The results showed that reduction in size did not affect interaction of forces. Also, the length scale parameter, volume fraction modulus, different boundary conditions and the dimensions of the plate are considered.

Forced vibration analysis of composite cylindrical shells using spline finite strip method

In this study, forced vibration behavior of thin-walled composite circular cylindrical shells is investigated using the spline finite strip method (spline FSM). Spline FSM is one of the versions of finite element method (FEM) employing a special element called finite strip. The shells considered in this study are assumed to be thin; therefore, the classical bending theory of shells and Sanders–Koiter's strain–displacement relation are used in the theoretical formulation of developed method. Time-history response of shells concerning arbitrary type of forces, boundary conditions and damping effects are obtained using developed spline FSM. To check the validity of results a comparison has been performed with the results obtained by conventional finite element method using ANSYS software. As far as the time history response of cylindrical shells is available the natural frequencies are extracted using Fast Fourier Transform (FFT) approach. The evaluated natural frequencies of composite shells are then compared with those obtained from an eigenvalue analysis. The comparison of results revealed the fact that developed spline FSM is an accurate tool that can be used in transient and harmonic analyses of composite laminated cylindrical shells.

Finite strip modeling of cementitious laminates reinforced with sisal fibers

The use of natural fibers to reinforce of cementitious matrices permits the development of sustainable materials with many potential applications. However, the improvement of such materials strongly depends on experimental studies, making the process slow and expensive. The numerical methods currently available for modeling composites can adequately describe the behavior of polymeric laminates, but have limited application in cementitious laminates that can have multiple cracking. This paper proposed a numerical model that combines the Finite Strip Method, Classical Theory of Laminates and Damage Mechanics to model the flexural behavior of laminated composites. The method can monitor the stresses developed in each layer and verify the ruptures resulting from traction, compression or shear. To validate the numerical method, flexural tests were carried out on cementitious laminates reinforced with 2 or 3 layers of long sisal fiber, with fiber content of 3% or 6%. Comparisons between the numerical and experimental results demonstrate good agreement, especially for composites reinforced with high fiber content. The adequate prediction of internal stress allows for the use of this numerical method to design cementitious laminates reinforced with sisal fibers that possess properties suitable for various applications.

Multiobjective optimization of cold-formed steel columns

The optimal design of cold-formed steel columns is addressed in this paper, with two objectives: maximize the local-global buckling strength and maximize the distortional buckling strength. The design variables of the problem are the angles of orientation of cross-section wall elements—the thickness and width of the steel sheet that forms the cross-section are fixed. The elastic local, distortional and global buckling loads are determined using Finite Strip Method (CUFSM) and the strength of cold-formed steel columns (with given length) is calculated using the Direct Strength Method (DSM). The bi-objective optimization problem is solved using the Direct MultiSearch (DMS) method, which does not use any derivatives of the objective functions. Trade-off Pareto optimal fronts are obtained separately for symmetric and anti-symmetric cross-section shapes. The results are analyzed and further discussed, and some interesting conclusions about the individual strengths (local-global and distortional) are found.