Finite Strip Method Research Articles

Effect of Bracing on Linear Free Vibration Characteristics of Thin-Walled Beams with Open Cross Section

The paper presents an analysis of the coupled vibration of beams with arbitrary thin-walled open cross section, braced with identical transversal header beams uniformly distributed along their length. The explicit form of analytic solution is derived by directly solving the governing differential equations of motion. The development is based on Vlasov theory which includes the effect of flexural-torsion coupling, the constrained torsion warping, and rotary inertia. The governing differential equations for coupled bending-torsional vibrations are performed using the principle of virtual displacements. In the case of simply supported beam, exact explicit expressions are derived to predict the natural frequencies and the corresponding mode shapes. The frequency equation, given in determinantal form, is expanded in an explicit analytical form, and then solved using the symbolic computing package Mathcad. The expressions are concise and very simple and as such convenient to be used by a practicing engineer who does not need to go into detail of thin-walled beam theory. Also, the use of explicit expressions gives significant savings in computing time compared with the alternative numerical methods [finite-element method (FEM), finite strip method, differential transform method, etc.]. To demonstrate the validity of this method the natural frequencies of braced thin-walled beams, having coupled deformation modes, are evaluated and compared with FEM.

Stub column tests of thin-walled complex section with intermediate stiffeners

A series of stub column tests on complex sections with intermediate stiffeners is presented in this paper. Initial geometric imperfections and material properties of the test specimens were measured. It is shown that the intermediate stiffeners could effectively enhance the local buckling stress of thin-walled sections. The test strengths are compared with the design strengths calculated using the direct strength method in the North American Specification and Australian/New Zealand Standard for cold-formed steel structures. It is shown that the direct strength method using finite strip method to obtain the buckling stresses is very conservative. Therefore, finite element method was used to predict the elastic buckling stresses. It is shown that the design strengths calculated using direct strength method based on the buckling stresses obtained from finite element analysis results generally agree with the test results well.

A new model for simulation of cold roll-forming of tubes by using spline finite strip method

A new model, called object model, for the simulation of cold roll-forming of tubes is presented. The model inherits the advantages of old models and is the embodiment of forming process that the strip is rolled step by step from feed rollers to last rolling pass. The elastic-plastic large deformation spline finite strip method based on updated Lagrangian method has been developed by improving the stiffness and transition matrix. Combined theory formulas and new analytical model, the forming process of a tube has been simulated successfully as an example. The analytical results are submitted and indicate that the proposed simulation method and new model are applicable.

Dynamic instability analysis of composite laminated thin-walled structures using two versions of FSM

Abstract Two versions of finite strip method (FSM) namely semi-analytical and spline methods are developed to calculate the stability and instability regions in the case of flat and curved thin-walled composite laminated structures under harmonic axial in-plane loads in the context of so-called parametric loading. 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 representing results are obtained and compared with those in the literature. Good accuracy in the results is achieved.

Exact post‐buckling stiffness calculation of box section struts

PurposeThe purpose of this paper is to develop and apply an exact finite strip (F‐a FSM) for the buckling and initial post‐buckling analyses of box section struts.Design/methodology/approachThe Von‐Karman's equilibrium equation is solved exactly to obtain the buckling loads and deflection modes for the struts. The investigation is then extended to an initial post‐buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. Through the solution of the Von‐Karman's compatibility equation, the in‐plane displacement functions are developed in terms of the unknown coefficient. These in‐plane and out‐of‐plane deflected functions are then substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficient.FindingsThe F‐a FSM is applied to analyze the buckling and initial post‐buckling behavior of some representative box sections for which the results were also obtained through the application of a semi‐energy finite strip method (S‐e FSM). For a given degree of accuracy in the results, however, the F‐a FSM analysis requires less computational effort.Research limitations/implicationsIn the present F‐a FSM, only one of the calculated deflection modes is used for the initial post‐buckling study.Practical implicationsA very useful and computationally economical methodology is developed for the initial design of struts which encounter post‐buckling.Originality/valueThe originality of the paper is the fact that by incorporating a rigorous buckling solution into the Von‐Karman's compatibility equation, and solving it, a fairly efficient method for post‐buckling stiffness calculation is achieved.

Spline finite strip method incorporating different plate theories for thick piezoelectric composite plates

In the present analysis, the spline finite strip with higher-order shear deformation is formulated for the static analysis of piezoelectric composite plates. The proposed method incorporates Reddy's third-order shear deformation theory, Touratier's "Sine" model, Afaq's exponential model, Cho's higher-order zigzag laminate theory, as well as the classic plate theory and the first-order plate theory. Thus, the analysis can be conducted based on any of the above-mentioned theories. The selection of a specific method is done by simply changing a few terms in a 2 by 2 square matrix and the results, obtained according to different plate theories, can be compared to each other. Numerical examples are presented for piezoelectric composite plates subjected to mechanical loading. The results based on different shear deformation theories are compared with the three-dimensional solutions. The behaviours of piezoelectric composite plates with different length-to-thickness ratios, fibre orientations, and boundary conditions are also investigated in these examples.

Optimum buckling design of composite stiffened panels using ant colony algorithm

Optimal design of laminated composite stiffened panels of symmetric and balanced layup with different number of T-shape stiffeners is investigated and presented. The stiffened panels are simply supported and subjected to uniform biaxial compressive load. In the optimization for the maximum buckling load without weight penalty, the panel skin and the stiffened laminate stacking sequence, thickness and the height of the stiffeners are chosen as design variables. The optimization is carried out by applying an ant colony algorithm (ACA) with the ply contiguous constraint taken into account. The finite strip method is employed in the buckling analysis of the stiffened panels. The results shows that the buckling load increases dramatically with the number of stiffeners at first, and then has only a small improvement after the number of stiffeners reaches a certain value. An optimal layup of the skin and stiffener laminate has also been obtained by using the ACA. The methods presented in this paper should be applicable to the design of stiffened composite panels in similar loading conditions.

Buckling Analysis of Laminated Composite Plates Using Higher Order Semi—Analytical Finite Strip Method

Rectangular plates made of laminated composite material because of the advantageously high strength and stiffness to weight ratio are used frequently as structural component in various branches of engineering, chief of which are aerospace and marine engineering. Design concepts of these plates that lead to the increase in the buckling load can directly lower the structural cost and/or weight. The finite strip method is one of a number of procedures which can be used to solve the buckling problem of plate structures. In the present work the main concern is with the buckling behavior of plates with simply supported ends subjected to uni-axial pure compression loads. The solution is sought by implementing the higher order semi-analytical finite strip method which incorporates additional degrees of freedom for each nodal line by using Reddy’s higher order plate theory. Therefore the current method is more universal in dealing with different plate thicknesses. In addition, in this semi-analytical finite strip method, all the displacements are postulated by the appropriate harmonic shape functions in the longitudinal direction and polynomial interpolation functions in the transverse direction. The solution is based on the concept of principle of minimum potential energy and an eigen-value analysis is subsequently carried out. From the presented results it can be concluded that the higher order semi-analytical finite strip method is very reliable for the preliminary design of composite plates especially in the case of buckling analysis of relatively thick plates.

The influence of post-local buckling mechanics on the stress variations, axial stiffness and ultimate failure strength of uniformly compressed thin-walled i-section struts

It is well known that thin-walled compression members are subject to the effects of local buckling and that due to these local effects the compressive carrying capability of short strut members can be significantly reduced. Finite element simulation is employed in this paper to examine the post-buckled response of thin-walled sections giving due consideration to the influence of geometric imperfections and to elasto-plastic material behaviour. The findings from this work highlight the complete loading history of the compression struts from the onset of elastic local buckling through the nonlinear elastic and elasto-plastic post-buckling phases of behaviour to final collapse and unloading. A detailed account of the growth and redistribution of stresses as well as the influence of yielding and yield propagation throughout loading is given in the paper. The results from the finite element simulations are shown to compare well with independent simulations using the finite strip method of analysis.

Elastic buckling of cold-formed steel columns and beams with holes

Simplified methods for approximating the global, distortional, and local critical elastic buckling loads of cold-formed steel columns and beams with holes are developed and summarized. These methods provide engineering approximations appropriate for design, but are intended to be general enough to accommodate the range of hole shapes, locations, and spacings common in industry. The simplified methods are developed as a convenient alternative to shell finite element eigen-buckling analysis, which require laborious and subjective visual identification methods as well as commercial software not generally accessible to the engineering community. Global buckling of cold-formed steel beams and columns, including the influence of holes, is predicted with approximate “weighted average” cross-section properties formulated from classical energy-based stability solutions. Distortional and local buckling of a cold-formed steel member with holes are determined with the semi-analytical finite strip method, considering appropriate modifications to the element thickness and choice of buckling half-wavelength. The proposed methods are verified with shell finite element eigen-buckling studies. Unambiguous, simple methods for elastic buckling prediction of members with holes is central to the extension of the Direct Strength Method (DSM) for cold-formed steel member ultimate strength determination.

Assessing the Susceptibility to Local Buckling at the Femoral Neck Cortex to Age-Related Bone Loss

Although it is known that long cortical bone structurally alter their area moment of inertia with age related bone loss maintaining their bending strength, the incidence of fragility fractures associated with cortical thinning still prevails. We hypothesize that cortical thinning with aging increases the local buckling susceptibility under abnormal or eccentric loads, and initiates fracture. The paper presents a series of 3D geometrical model derived from CT scans of a human femoral neck used to simulate age-related bone loss. The purpose of the model is to predict the susceptibility of local buckling at the femoral neck in falls by elderly folks. Geometric three-dimensional models of femoral neck cortices were developed from 7 human cadaver femurs (4 female, 3 male, 52-68 years). Three age related femoral neck models were simulated by either reducing (young age-related model) or increasing (old age-related model) the outer cortical surfaces in the radial-direction, by 1-mm. The control model was the middle-age related model. The inner cortex diameter was also adjusted to equilibrate the compressive stresses, based on the load-profile of a single-legged stance. Based on the old age related model, two additional "fragile" models were simulated by reducing the compressive load profile by 10 and 20% changing the inner cortex diameter, respectively. Using these models for each specimen, the consequence of a fall on the greater trochanter was evaluated. The Finite Strip Method (FSM) was used to investigate the association between local buckling at the femoral neck and the load to failure. Under constant loading, buckling progressively reduced the load to failure with aging, as seen in 2/7 of the middle age (by 9-15%) and 5/7 of the old age (by 7-32%) related models. In the fragile models, a 51% reduction in the load to failure was noted. Structural adaptation to age-related bone loss might preserves the bending strength under physiologic loads, but cortical thinning effects the buckling ratio reaching a critical threshold that would make the bone susceptible to local buckling at the femoral neck increasing the risk of fracture in a fall.

Lateral stiffness of steel plate shear wall systems

The accuracy of the finite element method and strip method of analysis for calculating the lateral stiffness of steel plate shear wall (SPSW) systems is assessed by making comparisons with experimental findings. Comparisons revealed that while both methods provide acceptable accuracy, they also require the generation of sophisticated computer models. In this paper, two alternative methods are developed. The first one is an approximate hand method based on the deep beam theory. The classical deep beam theory is modified in the light of parametric studies performed on restrained thin plates under pure shear and pure bending. The second one is a computer method based on the truss analogy. Stiffness predictions using the two alternative methods are found to compare well with the experimental findings. In addition, lateral stiffness predictions of the alternate methods are compared against the solutions provided using finite element and strip methods of analysis for a class of test structures. These comparisons reveal that the developed methods provide estimates with acceptable accuracy and are simpler than the traditional analysis techniques.

Buckling of composite FRP structural plates using the complex finite strip method

Abstract A finite strip method is presented to predict local, distortional, and lateral buckling of composite fiber reinforced plastic (FRP) structural plates. Each plate may be subjected to a combination of longitudinal compression, longitudinal in-plane bending, or shear stress. A sinusoidal function is assumed in the longitudinal direction for a buckling mode of whatever type and a polynomial function is used in the transversal direction. The critical stress and critical moment of I-shape and box and channel sections under bending and uniform loading are obtained by solving an eigenvalue problem. Using this solution technique, a simple expression is developed for prediction of I-shape section beams’ buckling stresses in three design curves. The critical stresses for different E x / E y values are calculated using the design curves.

Dynamic behaviour of multi-stiffened plates

The paper investigates the dynamic behaviour of stiffened panels. The coupled differential equations for eccentric stiffening configuration are first derived. Then a semi-analytical procedure for dynamic analysis of stiffened panels is presented. Unlike finite element or finite strip methods, where the plate is discretized into a set of elements or strips, the plate in this procedure is treated as a single element. The potential energy of the structure is first expressed in terms generalized functions that describe the longitudinal and transverse displacement profiles. The resulting non-linear strain energy functions are then transformed into unconstrained optimization problem in which mathematical programming techniques are employed to determine the magnitude of the lowest natural frequency and the associated mode shape for pre-selected plate/stiffener geometric parameters. The described procedure is verified with other numerical methods for several stiffened panels. Results are then presented showing the variation of the natural frequency with plate/stiffener geometric parameters for various stiffening configurations.

Calculation of pure distortional elastic buckling loads of members subjected to compression via the finite element method

An analysis procedure is presented which allows to calculate pure distortional elastic buckling loads by means of the finite element method (FEM). The calculation is carried out using finite element models constrained according to uncoupled buckling deformation modes. The procedure consists of two steps: the first one is a generalised beam theory (GBT) analysis of the member cross-section, from which the constraints to apply to the finite element model are deduced; in the second step, a linear buckling analysis of the constrained FEM model is performed to determine the pure distortional loads. The proposed procedure is applied to thin-walled members with open cross-section, similar to those produced by cold-forming. The distortional loads obtained are rather accurate. They are in agreement with the loads given by GBT and the constrained finite strip method (cFSM).

Buckling analysis of thin-walled sections under general loading conditions

This paper presents an investigation of the buckling behaviour of thin-walled sections subjected to general loading conditions. The semi-analytical finite strip method is used. The existing results are only for sections subjected to a uniform loading, namely: uniform compression, uniform bending and uniform distributed loads, which are applied at the shear centre. For a general loading condition, we proposed the realizing linear analysis first to give longitudinal stresses. The stiffness matrix is provided in the standard manner. Each strip is divided into cells and longitudinal stresses are recorded in these cells. The integrations are performed on each cell domain and the sum of them provides the geometric matrix of the strip.

Finite-Strip Method for the Analysis of Cracked Plates with Application to Plate-Girder Bridges

The finite-strip method (FSM) is one of the most efficient methods for structural analysis of bridges, reducing the time required for analysis without affecting the degree of accuracy. FSM is therefore an ideal platform for traditional time-consuming fracture analysis. This paper presents a new bending crack strip (BCS) that combines the shape functions of the spline finite strip with the eigenfunction solutions of the differential equations that govern the deflection around a crack. When this new BCS is implemented in FSM, complex three-dimensional plate structures with embedded cracks that are perpendicular to the longitudinal axis of the strips can be analyzed efficiently. The stress intensity factors of such structures can be directly computed with BCS. Extra nodal points are used around the cracked area to increase the precision of the numerical method. A substructuring technique is employed to determine the corresponding degrees of freedom of the boundary nodes, resulting in efficient analyses of cracked plates and plate-girder structures. This paper presents two examples to illustrate the accuracy and convergence rate of BCS as well as to demonstrate applications of the proposed model in the analysis of plates and plate girder bridges.

An exact finite strip for the calculation of relative post-buckling stiffness of isotropic plates

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of isotropic flat plates. The so-called exact finite strip is assumed to be simply supported out-of-plane at the loaded ends. The strip is developed based on the concept that it is effectively a plate. The present method, which is designated by the name Full-analytical Finite Strip Method in this paper, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman'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 thin flat plate buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. It is noted that in the present method, only one of the calculated out-of-plane buckling deflection modes, corresponding to the lowest buckling load, i.e., the first mode is used for the initial post-buckling study. Thus, the postbuckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. In this method, the Von-Karman's compatibility equation governing the behavior of isotropic flat plates is used together with a consideration of the total strain energy of the plate. 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 coefficient in the assumed out-of-plane deflection function. These in-plane and out-of-plane deflected functions are then substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficient. The developed method is subsequently applied to analyze the initial postbuckling behavior of some representative thin flat plates for which the results are also obtained through the application of a semi-analytical finite strip method. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.

Strength Evaluation of Cold-Formed Steel Columns Using The Results of Finite Strip And Finite Element Linear Stability Analysis

Most cold-formed steel columns display open and rather thin-walled cross-sections which mean that their structural behaviour is strongly affected by local and global buckling. Th e local mode, that occurs for shorter profi les, is characterized by (i) the local plate mode (LPM) characterized by the simultaneous flexural buckling of the web and fl anges and (ii) by the distortional mode (DM) characterized by the displacements of flange-stiff ener edges (that remain plane). The global mode occurring for long profi les is characterized by (i) the fl exural mode (FM) characterized by the translation of the whole section in the direction of the major principal axis and (ii) by the fl exural-torsional mode (FTM) characterized by the simultaneous translation and rotation of the whole section. Th e possibility of using the results of linear stability analysis in the national codes of thin-walled cold-formed steel structural elements (for instance, European and Brazilian Codes) arises, i.e. local and global buckling instability modes and corresponding bifurcation stresses determining the ultimate strength of members. Two powerful numerical methods are chosen to perform a linear stability analysis of a cold-formed steel structural member: (i) the Finite Strip Method, (i1) the Semi-Analytical Finite Strip Method (trigonometric functions are used in the approximation of displacement) used for simply supported boundary conditions, (i2) the Spline Finite Strip Method (‘spline’ functions are used in the approximation of displacement) used other boundary conditions and (ii) the Finite Element Method. The linear local and global stability results of for Z, C and rack cold-formed columns are used to obtain ultimate strength through the procedures adopted in the Eurocode 3, Part 1.3 and in the Brazilian Code (NBR 14.762/2001). The obtained numerical estimates by specifi cations are compared with experimental results available in literature.

有限帯板法による航空機用複合材補強パネルの簡易的座屈解析法

有限帯板法による航空機用複合材補強パネルの簡易的座屈解析法