Compound Strip Method Research Articles

A non-uniform compound strip method for effective buckling analysis of stiffened structures

In this paper, a versatile compound strip method (CSM) with non-uniform spline functions is developed for buckling analysis of stiffened structures. The proposed method further extends the existing CSM by permitting the flexible modification of local knots to place the stiffeners along the strip arbitrarily. In addition, a reasonable knot arrangement can achieve an accurate solving procedure with improved convergence. The displacements of stiffeners are expressed compatibly by the fundamental parameters of skin according to the beam-plate model. Consequently, the reinforcement of stiffeners is naturally incorporated by adding the beam stiffness matrix to the corresponding strips. Based on the first-order shear deformation plate theory and non-uniform spline functions, the semi-analytical formulations of stiffened structures are established. The present method provides the possibility to achieve the local mesh refinement in the finite strip methods. The convergence and validation study is demonstrated through the comparisons with the results of the existing literature solution and finite element method. In conclusion, a more efficient and applicable CSM is proposed to analyze the buckling behaviors of stiffened structures.

A compound strip method based on the Mindlin plate theory for static and buckling analyses of thin-walled members with transverse stiffeners

The compound strip method origins from the finite strip method (FSM) aimed to analyze thin-walled members with additional parts such as bolts and stiffeners. Traditionally, FSM mainly solves the buckling problem of prismatic members with longitudinally uniform sections. However, existing FSM could not take into account the stiffeners’ effect in the present form. The paper proposes a compound strip method based on the Mindlin plate theory (MCSM), in which members are modeled by finite strips while stiffeners are by MITC4 shell elements. This combination can effectively connect the stiffeners with the flange/web of the members. Meanwhile, the contribution from the thickness of stiffeners to buckling strengths could be considered accurately. The validity of MCSM is demonstrated through the comparison of the deformation and capacity from static and buckling analyses with solutions of the finite element method (FEM). The developed MCSM is then used to explore the impact of the number and thickness of stiffeners on the members. Besides, the discussion on how sectional characteristics affect stiffeners in improving the buckling loads is carried out. To ensure the calculation in MCSM is accurate and efficient, a study on meshing schemes is also presented.

A compound strip method for static and buckling analyses of thin-walled members with transverse stiffeners

The objective of this study is to propose and develop a Compound Strip Method (CSM) for static and buckling analyses of thin-walled members with transverse stiffeners. Traditionally thin-walled members are analyzed as prismatic members using computationally efficient methods such as the popular Finite Strip Method (FSM) — spline or semi-analytical. However, with structural systems getting more complicated and also for seeking better performance, transverse components that render the members no longer prismatic are possible, such as spaced transverse stiffeners along the member length. Existing FSM cannot take into account the stiffeners’ effect in the present form. Hence, this study presents a new compound strip method (CSM) based on the spline FSM for static and buckling analyses of thin-walled members with transverse stiffeners. The transverse stiffeners are modeled with shell element formulation by considering in-plane deformation. Transverse stiffeners can be placed flexibly along the member length in the proposed CSM. The developed CSM is then used to analyze plates and thin-walled members with transverse stiffeners. The accuracy of the method is validated with shell finite element solutions through static and buckling analyses of those members. Meanwhile, the impact of transverse stiffeners on the members’ buckling modes is also revealed by the analyses.

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.

07.20: Application of the compound strip method in buckling analysis of cold‐formed steel built‐up sections

ABSTRACTIn this paper, the compound strip method is utilised and extended for the stability analysis of cold‐formed steel built‐up sections. The discrete fasteners are incorporated in the analysis such that they can be placed anywhere within the length of the member. A beam element with geometrical and material properties of the fastener is adopted to model the connection between the individual sections. The stiffness matrix terms for the connection element are derived and added directly to the global stiffness matrix of the considered built‐up sections. The proposed numerical technique is verified through various examples and the results are compared with the finite element solutions. The results obtained demonstrate that the developed technique can accurately and efficiently model the behaviour of cold‐formed steel built‐up sections. Furthermore, the influence of the fasteners properties, their longitudinal spacing and also the boundary conditions on the overall buckling behaviour of built‐up sections is examined through numerical examples. The simplicity of the proposed technique facilitates an extensive parametric study of cold‐formed built‐up sections including the search for optimal placement of fasteners and combinations of component sections in built‐up profiles.

07.13: On extending the direct strength method to the design of cold‐formed steel built‐up columns

ABSTRACTThis paper presents a numerical investigation of the effects of discrete fasteners on the buckling behaviour of cold‐formed steel (CFS) built‐up columns and concludes with the assessment of the applicability of the current direct strength method (DSM) to the design of these members. To this end, the compound strip method is first utilised to accurately model the buckling behaviour of CFS built‐up columns under axial compression via introducing connecting elements to represent the contribution of discrete fasteners. A series of elastic buckling analyses with different levels of composite action are carried out to obtain the “signature curves” corresponding to the considered CFS built‐up sections. The performed studies comprise of variations in cross‐sections, end boundary conditions and fastener spacing. Utilising the developed numerical techniques, the ultimate strength of the sections are estimated and an adjustment procedure is suggested to extend the current DSM equations to the design of CFS built‐up columns. The proposed adjustment is verified against the existing experimental data and shown to be capable and accurate in capturing the expected behaviour of CFS built‐up members with or without stiffeners.

Dynamical analysis of stiffened plates using the compound strip method

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

Vibration localization in plates rib-stiffened in two orthogonal directions

This study deals with the vibration localization in plates rib-stiffened in two orthogonal directions. The effect of small misplacements of stiffeners, with various combinations of flexural and torsional rigidities, on both the free vibration modes and forced vibration responses of a plate clamped at four edges is presented. An extended finite strip approach, or the compound strip method, is employed to obtain the natural frequencies and the corresponding vibration modes. Galerkin's method is applied to determine the forced vibration response of the rib-stiffened plate subjected to a harmonic concentrated load at the center of one of its four panels. It is observed that a small disorder of the rib-stiffeners may significantly affect the vibration localization of the whole plate. By deliberately introducing irregularity into the stiffeners, the vibration magnitudes of some of the panels of the plate may be reduced.

Vibration analysis of stiffened plates

A spline compound strip method has been presented for the free vibration analysis of stiffened plates. The plate was discretized and modelled as strip elements. The displacement function of a strip element has been expressed as the product of the conventional transverse shape functions and longitudinal cubic B-splines. The flexural, torsional, and axial effect of the stiffeners towards the strip were derived and incorporated into a direct methodology. The theory has been illustrated with several examples including one-directional stiffened plates and cross stiffened plate. The results of the present analysis were adequately put into comparison with those from other previous methods.

B‐Spline Compound Strip Formulation for Braced Thin‐Walled Structures

A B‐spline bracing element of arbitrary orientation is developed and combined with the B‐spline compound strip method (BCSM) to analyze structures having bracing members. In the BCSM, a problem is first partially discretized to an ordinary differential equation by the finite element method. Then, the equation is solved using the Ritz‐Galerkin method with unequally spaced cubic B‐spline functions. The BCSM uses a special element “strip.” The strip is described as a product of the longitudinal cubic B‐spline series and transverse finite element polynomial shape functions. The bracing element is treated as a space frame element. For including the stiffness contribution of the bracing element, transformation and interpolation matrices are derived. A direct stiffness formulation is used. This technique eliminates the repetitive solutions required for each redundant in a flexibility approach based on the traditional finite strip method (FSM). The simplicity, accuracy, and convenience of this presented method ar...

Compound Strip Method for Folded Plates with Connecting Elements

The compound strip method is extended to include folded plate structures with connecting elements. The potential energy formulation common to the finite strip method is retained, and a direct stiffness technique for incorporating connecting elements is developed. The stiffnesses of connecting elements attached to the plate system at discrete points are written with respect to local element degrees of freedom, which are then transformed into forms consistent with the nodal parameters of the finite strips. The additional stiffnesses of grounded columns and connecting elements are derived in this manner and added directly to the finite strip stiffness. The combined use of rotation transformations and displacement parameter interpolation to model connecting elements with arbitrary orientations has not previously been applied to the finite strip method.

Applications of Compound Strip Method for Folded Plates with Connecting Elements

The compound strip method (CSM) is extended to include folded plate structures with frame elements in this paper, the second of two. The mathematical development of the CSM is described in the first paper and the application of the CSM is illustrated herein. Example analyses include a folded plate roof supported by rigid diaphragms, a folded plate roof supported by a flexible frame, a contrived folded plate system consisting of two plates connected by two skewed strut elements, and a box‐girder bridge with various cross‐bracing configurations. Experimental data in conjunction with conventional finite elements are employed for comparison of accuracy and computational effort.

Compound strip method for box girders and folded plates

The compound strip method (CSM) for plates is an expansion of the finite strip method (FSM) and was developed to incorporate the effects of the support elements in the analysis of linear elastic plate systems. In this paper the CSM is further expanded to analyze structures such as folded plates and box girders, whose behavior under load consists of bending and membrane actions. The stiffness of the support elements is added to the stiffness of the strip and summed over the entire structure. The displacement approach is used in the formulation and the resulting simultaneous equations are solved numerically. Several intermediately supported folded plate examples are solved using this approach and the results compare favorably with available analytical and experimental data.

Spline compound strip analysis of folded plate structures with intermediate supports

A B-spline column element in 3D space is derived and combined with the B- spline compound strip method for the analysis of plate-type structures (e.g. folded plates, box-girders, etc.) with intermediate supports. A direct stiffness method is used. Numerical examples demonstrate the advantages of this method: accuracy, efficiency and simplicity.

B-spline compound strip analysis of stiffened plates under transverse loading

An improved compound strip method based on the unequally spaced cubic B-spline is presented. The in-plane displacements are also incorporated in this approach to analyze the concentrically and eccentrically stiffened plates. Several of the available theoretical, experimental, and numerical results have been compared with those obtained from the proposed method. The accuracy and efficiency of this method is verified.

Compound strip method for the frequency analysis of continuous sector plates

A finite-strip method is developed for the frequency analysis of linear elastic flat curved plate systems continuous over flexible supports. The method presented incorporates the effects of support elements using a direct stiffness methodology. Mass matrices for the support elements are derived and presented in the form of strip matrices. These strip matrices add directly to the plate strip matrices at the element level. This summation creates a substructure called a compound strip. Curved multipanel plate systems are analyzed and the results are presented for illustration. Natural frequencies and mode shapes obtained from the compound strip analyses compare closely to those obtained from the finite-element method.

Discussion of “ Compound Strip Method for Analysis of Plate Systems ” by Jay A. Puckett and Richard M. Gutkowski (January 1986, Vol. 112, No. 1)

Discussion of “ <i>Compound Strip Method for Analysis of Plate Systems</i> ” by Jay A. Puckett and Richard M. Gutkowski (January 1986, Vol. 112, No. 1)

Compound strip method for the buckling analysis of continuous plates

A special finite strip method is developed for the analysis of linear buckling of flat plate systems that are continuous over non-rigid supports. This approach incorporates the effect of support elements in a direct stiffness methodology. The stiffness contribution of the support elements adds directly to the plate strip stiffness matrices at the element level prior to assembly. This summation of plate and support stiffness contributions forms a substructure, which is termed a compound strip. The compound strip methodology may be readily employed for the enhancement of computer programs based on traditional finite strip procedures. The validity of the compound strip method for elastic buckling analysis is demonstrated in two examples. The critical loads based on compound strip methodology compare favorably with those obtained with the finite element method.

Compound Strip Method for Free Vibration Analysis of Continuous Plates

A finite strip method is developed for the analysis of continuous elastic plate systems. This approach incorporates the effect of support element stiffness and mass in a direct manner. The stiffness and mass contributions of the support elements are derived and given as strip matrices. These matrices add directly with the plate strip matrices at the element level. This summation constitutes a substructure termed a compound strip. This methodology may be readily incorporated into existing finite strip programs based on lower order elements. The theory is illustrated with an example analysis. The compound strip methodology compares favorably with the finite element method for frequency and mode shape.

Compound Strip Method for Continuous Sector Plates

A finite strip method is developed for the analysis of linear elastic curved plate systems that are continuous over non‐rigid supports. This approach incorporates the effect of the support elements in a direct stiffness methodology. The stiffness contribution of the support elements are derived and given as strip stiffness matrices which are combined with the plate strip matrices at the element level prior to assembly. This summation of plate and support stiffness contributions forms a substructure which is termed a compound strip. The compound strip methodology may be used readily to enhance computer programs based on traditional finite strip procedures. The validity of the compound strip method is demonstrated in several illustrative analyses. The compound strip methodology compares favorably to the finite element method for displacement and moment.