Finite Strip Method Research Articles

A modified Direct Strength Method for nonlinear twisting model of simply-supported C-section purlins

This paper proposes a framework to predict the buckling limit states of C-section purlins under wind uplift load with top flange restrained by the sheeting. The prediction framework includes a nonlinear twisting model to account for the flexural-torsional behavior and the application on the Direct Strength Method. The torsional effect induced by load eccentricity and the diaphragm effects provided by sheeting are incorporated into the flexural-torsional differential equations of the nonlinear twisting model. The biaxial bending moments and bi-moment, the flexural-torsional buckling strength and the stress distribution of the purlin can be obtained from the proposed model. On the basis of the Finite Strip Method (FSM), the elastic buckling load can be calculated through reducing the applied bending moments by the load factor. Thereafter, the Direct Strength Method is modified to account for the buckling interaction behavior influenced by the warping-torsion and diaphragm effects. A database of 17 simple-span purlin tests is used to validate the proposed model.

Constrained spline Finite Strip Method for thin-walled members with open and closed cross-sections

The present paper aims at introducing the constrained spline Finite Strip Method (csFSM). The proposed approach is basically a spline Finite Strip Method (spline FSM) that allows the modal decomposition. Similarly to the constrained Finite Strip Method (cFSM), some mechanical assumptions are made in order to constrain the general spline FSM model to buckle in specific modes, for example to enforce the member to buckle in the local-plate mode, or distortional mode. Derivation of matrices that define the distortional (D) and global (G) modes for thin-walled members with unbranched open and closed cross-sections is the main objective of this paper. To define these subspaces, a standard practice is followed which consists in forming RGD, the constraint matrix of the combined GD space, then, RG and RD the constraint matrices of pure global and distortional buckling modes, respectively. Mechanical criteria are used to derive RGD and RG matrices, while orthogonality conditions are used to derive RD matrix. The implementation of the mechanical criteria is done by using FEM procedure rather than the cFSM one. Moreover, some practical aspects on how to constrain a spline FSM model are also discussed, including how to force the torsional mode of closed cross-sections. Numerical examples of modal decomposition are provided for a column - beam problem, with standard boundary conditions. The distortional and global buckling loads obtained are found to be in good agreement with those calculated via the cFSM and the Generalized Beam Theory (GBT). The paper concludes with a discussion on the applicability of csFSM in cold-formed steel member design.

Buckling of Thick Laminated Composite Plates Based on Zigzag and Third order Shear Deformation Theories using the Finite Strip Method

Buckling of Thick Laminated Composite Plates Based on Zigzag and Third order Shear Deformation Theories using the Finite Strip Method

Free vibration analysis of functionally graded cylindrical shells with different shell theories using semi-analytical method

In this study, the semi-analytical finite strip method is adopted to examine the free vibration of cylindrical shells made up of functionally graded material. The properties of functionally graded shells are assumed to be temperature-dependent and vary continuously in the thickness direction according to a simple power law distribution in terms of the volume fraction of ceramic and metal. The material properties of the shells and stiffeners are assumed to be continuously graded in the thickness direction. Theoretical formulations based on the smeared stiffeners technique and the classical shell theory with first-order shear deformation theory which accounts for through thickness shear flexibility are employed. The finite strip method is applied to five different shell theories, namely, Donnell, Reissner, Sanders, Novozhilov, and Teng. The approximate procedure is compared favorably with three-dimensional finite elements. Finally, a detailed numerical study is carried out to bring out the effects of power-law index of the functional graded material, stiffeners, and geometry of the shells on the difference between various shell theories. Finally, the importance of choosing the shell theory in simulating the functionally graded cylindrical shells is addressed.

Buckling analysis of steel jacking pipes embedded in elastic tensionless foundation based on spline finite strip method

Buckling of steel jacking pipes subjected to axial compression and surrounded by soil is investigated. The jacking pipes are simplified as simply-supported circular cylinders embedded in tensionless Winkler-Pasternak foundation that reacts to soil compression only. The governing equation for tensionless buckling analysis is established based on the spline finite strip method (SFSM), and the critical buckling load is obtained by solving the governing equation through an iterative procedure. Numerical examples are analyzed, and the comparison of buckling loads with available solutions and finite element results shows good accuracy of the SFSM. Based on a parametric study, the variation of tensionless buckling loads with slenderness ratio of jacking pipes is first studied. Moreover, the effects of diameter-thickness ratio and foundation stiffness are also investigated. The present SFSM-based tensionless buckling analysis is applicable to cylinders of any length and on tensionless foundation, and it is capable of efficiently and effectively predicting buckling of steel jacking pipes subjected to axial compression and surrounded by soil.

Dynamic and buckling analysis of composite laminated plates with and without strip delamination under hygrothermal effects using finite strip method

The effect of hygrothermal conditions such as temperature and moisture on free vibration frequency and buckling load of composite laminated plates was investigated in present article. For this purpose, the effect of changing in material characteristics with changing in temperature and moisture on buckling capacity and natural frequency of plates with different end conditions and biaxial loading was evaluated. In addition, the effect of delamination of layers on buckling load and natural frequency of plate was studied in different situations. The finite strip method was used in present paper to calculate the critical load of plate considering first-order shear deformation theory. In finite strip formulation for evaluating the displacement field of each strip, the trigonometric shape functions were used in longitudinal direction and the Hermitian and linear shape functions were used for out–of–plane and in–plane transverse direction, respectively. The place and dimension of delaminating layers was modelled by separating the adjacent elements and reconstructing the standard, geometric, force and mass matrices, so, the critical load and natural frequency of laminated plates was calculated in different configurations.

Finite Strip Modeling of the Varying Dynamics of Shell-Like Structures During Machining Processes

The efficiency of the finite strip method (FSM) in modeling the varying dynamics of shell-like structures during machining operations is investigated. The workpiece is modeled as a shallow, helicoidal, cantilevered shell, and the natural modes are computed using FSM. In the FSM solution, the workpiece is discretized only in the chordwise direction, and the membrane and bending displacement fields of the shell in the spanwise direction are approximated by a set of basis functions that satisfy clamped-free boundary conditions. The displacement fields in the chordwise direction are approximated using polynomial functions. The efficiency of the presented FSM is investigated by comparing the computed natural vibration modes against the ones obtained using the finite element method (FEM). The FSM model was found to yield results of greater or comparable accuracy, even with up to 40% fewer degrees-of-freedom (DOFs). Also, the accuracy of the presented model is verified by comparing the predicted frequency response functions (FRFs) against the FRFs that were measured by conducting impulse hammer tests in various stages of machining a generic curved blade.

Panel flutter of curvilinear composite laminated plates in the presence of delamination

The effect of variable fiber placement angle on the supersonic linear flutter of rectangular composite panels containing square delamination zone is investigated using an enhanced spline version of finite strip method (FSM). The location dependent stiffness characteristics and mass matrices due to variable fiber orientation angles within every ply are extracted. The structural formulation is based on the higher-order shear deformation theory while the first-order piston theory is utilized to predict the loading effects of the supersonic airflow. Laminated composite material with varying fiber orientation angles along the axial direction is considered. The effect of aerodynamic damping is overlooked. The flutter coalescence of vibration modes is then traced using a standard eigenvalue procedure. Some representative results are provided to show the accuracy and capability of the developed formulation. The effects of material layup as well as geometry on the flutter behavior of laminated panels are then studied and the variation of critical aerodynamic pressure considering different delamination size, delamination depth, and boundary conditions are examined.

Elastic and inelastic buckling of square and skew FGM plates with cutout resting on elastic foundation using isoparametric spline finite strip method

The elastic buckling of square and skew thin functionally graded material (FGM) plates with cutout resting on an elastic foundation is investigated using the isoparametric spline finite strip method. It is assumed that the material properties of FGM plates vary smoothly through the thickness according to the power law model. The elastic foundation is simulated by the Winkler and two-parameter Pasternak models. The isoparametric spline finite strip method is also applied to investigate initial inelastic local buckling loads of skew homogenous plates with cutout based on the Ramberg–Osgood representation of the stress–strain curve using the deformation theory of plasticity. The critical buckling load of the plate is achieved by minimizing its total potential energy and solving the corresponding eigenvalue problem. In addition, the effects of elastic foundation coefficients and hole size on the local buckling of plates with different boundary conditions are determined and discussed.

Free vibration and buckling analysis of thin plates subjected to high gradients stresses using the combination of finite strip and boundary element methods

Free vibration and buckling analysis of thin plates under arbitrary loading is investigated taking the advantages of both boundary element and finite strip methods. In the first step, the governing differential equation is converted to equivalent integral equation using static fundamental solutions of bi harmonic equation. Moreover, domain integrals due to inertia and in-plane forces are evaluated applying the finite strip interpolation functions to express the displacement field. This leads to a reduction of the size of the problem so that the mesh generation becomes easier and the computational efforts are reduced. In the buckling analysis, the near boundary stresses are calculated using a transformation method to cope with singularities. The proposed method can be easily applied to arbitrarily distributed and concentrated edge loading for different boundary conditions. The accuracy and efficiency of the presented method has been investigated through the solution of some sample problems for rectangular and skew plates.

Development of a viscoelastic spline finite strip formulation for transient analysis of plates

In this study, the forced vibration behavior of viscoelastic plates is investigated using the spline finite strip method (spline FSM). It is assumed that the viscoelastic behavior of materials considered in this research is linear, which means the material constants are independent of load level and are a function of time only. The shear modulus is thus represented as a function of the so-called Prony series. Moreover, it is assumed that the bulk modulus remains constant. For spline FSM, the equilibrium equations are derived using the principle of virtual work and the classical bending plate theory in matrix form. A numerical solution is developed for forced vibration analyses using the Newmark's method. The damping effects corresponding to the viscoelastic nature of materials are incorporated in the analysis by a novel approach that defines a fictitious damping force vector. The convergence of the spline finite strip method for a viscoelastic plate under different loading conditions is studied. Additionally, the fast fourier transform (FFT) of forced vibration responses is applied to evaluate the natural frequencies of plates. The accuracy and efficiency of the proposed spline finite strip model are verified by comparing the simulation outcomes presented in this work with the results obtained from the finite element method using ANSYS. Through a comparison of the results, we found the spline finite strip results to be in good agreement with the finite element results.

Stability of tow-steered curved panels with geometrical defects using higher order FSM

For the first time, the parametric instability characteristics of tow-steered variable stiffness composite laminated (VSCL) cylindrical panels is investigated using B-spline finite strip method (FSM). The panel is considered containing geometrical defects including cutout and delamination. The material properties are assumed to vary along the panel axial length of any lamina according to a linear fiber-orientation variation. A uniformly distributed inplane longitudinal loading varies harmoni-cally with time is considered. The instability load frequency regions corresponding to the assumed in-plane parametric load-ing is derived using the Bolotin's first order approximation through an energy approach. In order to demonstrate the capabili-ties of the developed formulation in predicting stability behavior of the thin-walled VSCL structures, some representative results are obtained and compared with those in the literature wherever available. It is shown that the B-spline FSM is a proper tool for extracting the stability boundaries of perforated delaminated VSCL panels.

A simplified flange–lip model for distortional buckling of cold-formed steel channel-sections with stiffened web

This paper presents a simplified analytical model for determining the critical stress of distortional buckling of lipped channel-sections with stiffened web made from cold-form steel (CFS). Lipped channel-section with stiffened web have been shown to have a distinct advantage in resisting local buckling and are associated with the higher distortional buckling stress, when compared to the channel-section without stiffeners. It is widely used as a substitution for standard channel-section in cold form steel construction applications. In the current work, CFS channel-sections with stiffened web are investigated based on the flange–lip model. In order to determine the stiffness of rotational springs representing the restraining effect of the web to the flange–lip system, the web with different type of stiffeners is modeled as an orthopedic plate. Using the total potential energy principle, the formula for calculating the local buckling stress of the stiffened web considering loading scenarios including a pure compression and a pure bending moment are derived. The stiffness of rotational spring can be obtained. Finally, the prediction of distortional buckling critical stress of lipped channel-section with different type of stiffened webs is carried out, which is shown to be in good agreement with those calculated by the finite strip method (FSM).

Elastic buckling analysis of cold-formed steel built-up sections with discrete fasteners using the compound strip method

In this paper, the compound strip method is applied to the stability analysis of cold-formed steel built-up sections. A beam element with adjustable stiffness properties is adopted to represent the utilised fastener and the associated stiffnesses of the connection elements are incorporated in the global stiffness matrix of the built-up sections. The presented method allows for modelling arbitrarily-located discrete fasteners in the context of the semi-analytical finite strip method. The proposed numerical technique is verified against finite element solutions through various numerical examples and shown to be both accurate and versatile. Some typical and also complex built-up sections with various fastener configuration and end boundary conditions are analysed to evaluate the influence of fastener spacing. The extent of composite behaviour in built-up sections is determined by investigating the enhancement of buckling capacity and changes in the corresponding buckling modes. The simplicity of the proposed technique expedites extensive parametric studies of cold-formed built-up sections and facilitates the search for optimal placement of fasteners and choice of section geometry.

Buckling analysis of thin-walled members with open-branched cross section via semi-analytical finite strip transfer matrix method

Thin-walled members with open-branched cross section have been used in many modern engineering structures, and their buckling performance has been widely studied. In this paper, two transfer methods are developed to tackle the transfer problems at the junction of the open-branched cross-section of thin-walled members by using semi-analytical finite strip transfer matrix method (FS-TMM) which is a combined use of semi-analytical finite strip method (SA-FSM) and transfer matrix method (TMM) for the buckling analysis. Compared to traditional SA-FSM, this method has a smaller matrix and higher computational efficiency due to no global stiffness matrix generated. An asymmetric E-section member, a symmetric I-section member and a X-section member with loaded edges simply supported are analyzed by the derived formulation. All the results are compared with SA-FSM's or finite element method's results to prove the reliability and efficiency of this method.

ULTIMATE STRENGTH OF COMPRESSED SLABS AND BOX GIRDERS

<jats:p>A theoretical investigation into the effectiveness of a plate thickness against the ultimate strength of a compressed slabs and box girders is carried out. Series of the buckling analyses, the elastic, the viscoplastic and the ultimate strength are performed by the rheological-dynamical inelastic theory and the finite strip method on a slabs and box girders under thrust. In the analytical method, rheological-dynamical analogy (RDA) is introduced to express the critical stresses of slabs and box girders in the range of viscoplastic strains and strain hardening. Applying the finite strip method (FSM) as a semi-analytical method, the fundamental equilibrium equations are derived based on the principle of minimum total potential energy. Apart from the quantitative research the qualitative research is presented to demonstrate the capabilities of the present theory.</jats:p>

Stability analysis of variable stiffness composite laminated cylindrical panels containing delamination

A finite strip method is formulated in order to extract the stability characteristics of laminated composite cylindrical panels containing embedded square delamination region. The material properties are assumed to vary along the panel axial length of any lamina according to a linear fiber orientation variation. The effects of existence of a square delamination region are studied on the stability characteristics of the curved panel. The friction effects, contact conditions, and delamination growth phenomenon at delaminated interfaces and edges are overlooked. The natural frequencies of the panel have been investigated based on two composite micromechanical schemes. The dynamic behavior of the delaminated panel has been extracted through using an enhanced B-spline finite strip method formulated based on the principle of virtual work. A third-order Reddy-type shear deformation theory is utilized in order to bring the effects of moderately thick laminate in to account besides a Koiter–Sanders strain–displacement theory of shallow shells. The effects of material volume fractions, layup, micromechanical model, delamination size, and location on dynamic stability characteristics of the structure are studied. In order to demonstrate the capabilities of the developed method in predicting the structural dynamic behavior, some representing results are obtained and compared with those available in the literature.

Time depended deformation and buckling of viscoelastic thick plates by a fully discretized finite strip method using Third order shear deformation theory

Abstract Time depended deformation and critical buckling load of viscoelastic thick plates were studied using finite strip method with the trigonometric functions in longitudinal direction and the polynomial functions in transverse direction. The plates were considered to be thick and the third order shear deformation theory was used to consider the effect of shear stresses in thickness. The mechanical properties of the material were considered to be linear viscoelastic by expressing the relaxation modulus in terms of Prony series. Time history of maximum deflection of viscoelastic plates subjected to transverse loading and unloading on plates was calculated using a fully discretized formulation. In addition, the critical in–plane load of plates was calculated by a nonlinear procedure in different times of loading. Moreover, the effect of thickness and the interaction of biaxial in–plane loading on critical load of plate were studied.

Buckling of functionally graded plates under thermal, axial, and shear in-plane loading using complex finite strip formulation

ABSTRACTA complex finite strip method was used to study the buckling of functionally graded plates (FGPs) under thermal and mechanical (longitudinal, transverse, and shear in-plane) loading. The mechanical characteristics of FGPs were assumed to vary through the thickness, according to power law distribution. The nonlinear temperature distribution in the direction of the plate thickness was assumed according to thermal conduction steady state conditions. In complex finite strip method, the polynomial Hermitian functions were assumed in the transverse direction and the complex exponential functions were used in the longitudinal direction to evaluate the standard and geometric stiffness matrices that have the ability of calculating the critical shear stress in contrast to trigonometric shape functions. The solution was obtained by the minimization of the total potential energy and solving the corresponding eigenvalue problem. In addition, numerical results for FGPs with different boundary conditions were presented and compared with those available in the literature and the interaction curves of mechanical and thermal buckling capacity of FGPs were obtained.

Constrained shell finite element method for thin-walled members with holes

In this paper the constrained finite element method is applied for the buckling analysis of thin-walled members with holes. The method is basically a shell finite element method, but constraints are applied which enforce the member to deform in accordance with specific mechanical criteria, e.g., to force the member to buckle in flexural, or flexural-torsional or distortional mode. The method is essentially similar to the constrained finite strip method, but the trigonometric longitudinal shape functions of the finite strip method are replaced by polynomial longitudinal shape functions, and longitudinal discretization is used, which transform the finite strip into multiple finite elements. As the method has already been presented in earlier papers, here it is just briefly summarized. More detailed discussion of the secondary cross-section modes is provided, since the presence of holes might have important consequences on these secondary modes. Two basic approaches for handling the holes are discussed. Numerical examples are also presented. Comparison of linear buckling results to available results from the literature fully validates the proposed constrained finite element method. Further numerical examples illustrate the difficulties and uncertainties involved in regular unconstrained buckling analysis of members with holes, and also show that the proposed constrained finite element method eliminates the uncertainties in assessing the effect of holes on the critical loads, as well as, in general, helps to understand the effect of holes on the buckling behavior.