Complex Finite Strip Method Research Articles

Inelastic buckling analysis of cold-formed steel members with residual stresses based on CFSM

Residual stresses in cold-formed steel members are generally composed of a membrane component and a bending component, which may result in the inelastic buckling of cold-formed members. The complex finite strip method (CFSM) has been widely used to analyze the buckling of thin-walled members and it can handle shear effects easily. As for inelastic buckling, the available CFSM mainly focuses on the local buckling of plates and cannot be applied to the buckling of cold-formed members that can buckle in a distortional or global mode. Besides, the bending component may be hard to be incorporated into the finite strip method. In the present study, a modified CFSM is proposed to analyze the inelastic buckling of cold-formed steel members with residual stresses. The effects of the membrane and bending components on inelastic buckling are considered through the applied stresses and modifying material properties, respectively. The results calculated by the proposed model are compared to the available solutions of several examples in previous studies, and good agreements are observed. Based on the proposed model, the effects of residual stresses on the inelastic buckling of plates and cold-formed members are investigated. The results show that the inelastic buckling stress is significantly lower than the elastic buckling stress when the elastic buckling stress exceeds the yield stress. The membrane component has a significant influence on the inelastic buckling stress of plates. The bending component mainly affects the inelastic local buckling of members under compression and the inelastic global buckling of members under compression as well as bending and shear.

Bubble Complex Finite Strip Method in the Stability and Vibration Analysis of Orthotropic Laminated Composite Plates

In this study, a bubble complex finite strip method (BCFSM) with the higher-order zigzag theory is formulated for mechanical buckling and free vibration analysis of laminated composite plates, including cross-ply and angle-ply laminates. Few studies have been done to obtain the analytical solutions for clamped and free boundary conditions in the longitudinal and transverse edges. Therefore, this study, for the first time, investigates the effects of various boundary conditions on the stability and vibration results of laminated composite plates subjected to axial or pure shear forces with the use of higher-order zigzag theory and BCFSM. Following this, both the interlaminar continuity conditions of transverse shear stresses and the shear-free surface conditions are satisfied by applying a cubic displacement and a zigzag linear varying displacement with the same number of unknowns as the first-order shear deformation theories. Moreover, the effects of width-to-thickness ratio, fiber orientation, number of modes, different dimensional ratios of the plate, and finally, the number of layers are investigated through numerical examples. The bubble shape functions are exploited in the transverse direction to improve the convergence of the method. Finally, the shearing and axial interaction diagrams of composite laminated plates are presented for various types of boundary conditions.

Stability of stiffened cruciform steel columns under shear and compression by the complex finite strip method

This paper focuses on the stability of steel columns with cruciform section under uniform compressive and shear stresses. The complex finite strip method (CFSM) is employed to solve the buckling of open thin-walled steel columns. CFSM results are validated against other numerical techniques. CFSM has high convergence, accuracy and superior computational efficiency. Due to the use of complex functions, it can also capture shear buckling. The influence of longitudinal stiffeners and section geometry on the buckling behavior of stiffened cruciform sections is investigated. The buckling behavior of cruciform sections is compared against U, I, and T section steel columns.

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.

Buckling analysis of thin-walled cold-formed steel structural members using complex finite strip method

In this paper, a generalised complex finite strip method is proposed for buckling analysis of thin-walled cold-formed steel structures. The main advantage of this method over the ordinary finite strip method is that it can handle the shear effects due to the use of complex functions. In addition, distortional buckling as well as all other buckling modes of cold-formed steel sections like local and global modes can be investigated by the suggested complex finite strip method. A combination of general loading including bending, compression, shear and transverse compression forces is considered in the analytical model. For validation purposes, the results are compared with those obtained by the Generalized Beam Theory analysis. In order to illustrate the capabilities of complex finite strip method in modelling the buckling behavior of cold-formed steel structures, a number of case studies with different applications are presented. The studies are on both stiffened and unstiffened cold-formed steel members.

Elastic distortional buckling of doubly symmetric steel I-section beams with slender webs

Doubly symmetric steel I-section members with thin webs and stocky flanges have a tendency to buckle in a so-called distortional buckling mode, involving distortion of the web of the I-section in the plane of its cross-section. As this mode is more complicated than the local or global buckling, analytical expressions take empirical forms and vary in between different proposals. This paper has two main objectives. The first objective is to propose a complex finite strip method for distortional buckling analysis of I-section beams with slender webs and check the suitability for such analysis by comparing its accuracy with other methods and the second objective is to propose a simple method for predicting the distortional buckling loads of I-beams. The latter objective is superior to current methods with respect to the weight of the sections.

On the use of bubble complex finite strip method in the nonlocal buckling and vibration analysis of single-layered graphene sheets

In the present work, the buckling and vibration of rectangular single-layered graphene sheets is analyzed based on the nonlocal theory of elasticity which takes the small scale effects into account. The graphene sheet is assumed as a thin plate, and the classical plate theory is applied to obtain the differential equation of the sheet. For the first time, the complex finite strip method is employed to study the vibration and buckling behavior of graphene sheets. The weighted residual method is employed to obtain the stiffness, stability and the mass matrices of the graphene sheet which is assumed to be an isotropic nanoplate. A sinusoidal displacement function is used for the longitudinal direction , which satisfies the simply supported boundary condition , while piecewise interpolation polynomials including the Hermitian and bubble functions are assumed for the other direction. A matrix eigenvalue problem is solved to find the vibration frequency and buckling load of graphene sheets subjected to different types of in-plane loadings including the uniform and non-uniform uniaxial and biaxial compressions as well as shear loading. The accuracy of the proposed model is validated by comparing the results with those reported by the available references. Furthermore, a number of examples are presented to investigate the effects of various parameters (e.g., boundary conditions, nonlocal parameter, aspect ratio, and type of loading) on the results. The bubble complex finite strip method is proposed to study the buckling and vibration characteristics of rectangular single-layered graphene sheets, based on the nonlocal classical plate theory. • Buckling and free vibration and of single layered graphene sheets are considered. • The bubble complex finite strip method is applied to study the GS׳s problems for the first time. • Nonlocal continuum mechanics is employed to include the small scale-effects. • Contrary to the most of the studies, different boundary conditions are considered.

Inelastic local buckling of flat, thin-walled structures containing thickness-tapered plates

This paper is concerned with the inelastic local buckling of flat plate structures that contain plates with variable thicknesses. Use is made of the semi-analytical complex finite strip method, which is augmented with transverse bubble functions. Stiffness and stability matrices are derived for inclusion in the finite strip method, which is based on the deformation theory of plasticity. The numerical scheme is programmed, and several numerical examples are presented to illustrate the prowess and scope of the procedure. The inelastic local buckling of tapered plates subjected to compression and shear with different boundary conditions is studied first, and the method is then applied to the inelastic local buckling of channel sections with tapered flanges and stiffened plates with variable thickness and different geometries. Slenderness limits for channels that delineate between local buckling and yield (and that define the transition from non-compact to slender cross-sections) are discussed.

Inelastic local buckling of plates and plate assemblies using bubble functions

A complex finite strip method of analysis that includes inelasticity is described. The finite strip displacement functions are augmented with so-called bubble functions, which are extra modes associated with internal or nodeless degrees-of-freedom. It is shown that the use of bubble functions significantly improves the convergence of the inelastic finite strip method. The analytical procedure is then used to study the inelastic local buckling of plates in compression and shear, stiffened plates in compression and I-section beams in shear.

Local Buckling by Complex Finite Strip Method Using Bubble Functions

Bubble functions are finite element shapes that are zero on the boundary of the element, but are nonzero at the other points. They have been hitherto used to augment finite element formulations to obtain rapid convergence. The paper augments bubble functions to the ordinary semi‐analytical complex finite strip treatment in order to calculate the elastic local buckling stress of plate assemblies. The results show that the use of bubble functions improves significantly the convergence of finite strip method in terms of strip subdivision, and leads to much smaller storage required for the structure stiffness and stability matrices. Numerical studies are included, including long plates in compression and shear, a channel section in pure compression and shear, a stiffened plate, and an I‐section with a longitudinal stiffener. These studies illustrate the power of the bubble function based method for studying the elastic stability of plates and plate assemblies.

Local buckling of I-sections bent about the minor axis

Abstract The paper presents a modification to an existing semi-analytical complex finite strip method of analysis to incorporate inelastic local buckling based on the deformation theory of plasticity. The method is programmed to investigate the local buckling of doubly-symmetric I-section beams bent about the minor axis. Buckling curves are derived for a variety of geometries. The limiting width to thickness ratios given in the Australian AS4100 for beams bent about the minor axis are shown to agree well with the finite strip predictions.

Elastic local buckling of composite tee-beams with longitudinal stiffeners

The semi-analytical complex finite strip method is used to study the elastic local buckling of composite tee-section beams in negative moment and shear which contain a longitudinal stiffener attached to the web. The optimum position of this stiffener for different positions of the neutral axis is calculated in order to maximize the local buckling stress. Design graphs for longitudinal stiffeners are presented. Interaction curves between shear and bending for different positions of the neutral axis in a stiffened girder are given. Key words: composite beams, elasticity, finite strips, local buckling, longitudinal stiffeners, webs.

Design of transversely stiffened webs of I-beams

Abstract The design rules for webs in shear given in the limit states British and Australian steel standards are reviewed. The semi-analytical complex finite strip method of buckling analysis is then used to obtain the elastic local buckling coefficients of the webs of I-beams subjected to shear when the restraint provided by the flanges is included. It is shown quantitatively how the flange restraint increases the local buckling coefficient above the value for a web with simply supported edges, the latter forming the basis of the strength rules in the present standards. The use of the rationally-derived local buckling coefficients to enhance the strength of webs in shear implicit in the design standards is then described. Use of these local buckling coefficients should lead to economies in steel.

Local buckling of I-section beams with longitudinal web stiffeners

The semi-analytical complex finite strip method is used to study the elastic local buckling of I-sections in bending containing a longitudinal stiffener attached to the web. The optimum position of this stiffener is calculated to maximise the web local buckling stress. The local buckling of the stiffener is then considered, and recommendations are made regarding the stiffener proportions that would cause yield of the stiffener before it buckled locally.

Stability of monosymmetric beam-columns with thin webs

Abstract The principle of virtual work is used to derive stiffness and stability matrices which govern the elastic distortional buckling of beam-columns of monosymmetric I-section. The matrices are of order four, so that extraction of the buckling eigenvalue can be achieved rapidly. The solutions are compared with those of the more complex finite strip method, and the agreement is shown to be good. Distortional buckling solutions are obtained for tee-section beams, columns and beam-columns. Based on the computer-generated solutions, approximate formulae for the distortional buckling loads and moments are given.

Buckling and vibration of shear deformable prismatic plate structures by a complex finite strip method

A finite strip method is presented for the determination of buckling stresses and natural frequencies of vibration of prismatic plate structures assembled from plate flats, which generally are laminates of fibre-reinforced composite material. The finite strip method is of the single-term type, corresponding to the assumption of sinusoidal longitudinal spatial variation of displacement and force quantities. Anisotropic material behaviour and applied in-plane shear stress are accommodated by expressing the strip displacement field in terms of complex quantities. The out-of-plane properties of plate flats are based upon the use of first-order shear deformation plate theory. A family of finite strip models is described and a sub-structuring procedure is utilised to reduce the size of the eigenvalue problem. Presented numerical results reveal the high accuracy and good convergence characteristics of the method, as well as indicating the influence of through-thickness shear effects in a range of circumstances.

Buckling under combined loading of thin, flat‐walled structures by a complex finite strip method

AbstractA finite strip method is presented for determining the initial buckling stresses of any structure consisting of a series of thin flat isotropic plates rigidly connected together at their longitudinal edges. Each plate may be subjected to a combination of longitudinal and transverse compression, longitudinal in‐plane bending, and shear, and it is assumed that the buckling mode, of whatever type, is sinusoidal in the longitudinal direction. Due to the presence of shear, the perturbation forces and displacements which occur at the edges of component plates during buckling are out of phase, and this is accounted for by defining their magnitudes in terms of complex quantities. Stiffness matrices relating the amplitudes of these forces and displacements are derived using an approximate method based upon assumed displacement functions across the width of the plate. It is shown how the method can be used to calculate natural frequencies of prismatic structures, and finally an indication of the accuracy of the method is given together with some illustrative results.