A Review of the Finite Strip Method

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A review of the finite strip method Y K Cheung and L G Tham University of Hong Kong, Hong Kong

Summary The first paper on finite strip analysis of structures  was published by Cheung in the late 1960s. Since then, many more papers have been published and they they have have demonst demonstrat rated ed that that the finite finite strip strip method method has higher higher effici efficienc ency y than than the finite finite elemen elementt method method as a smaller number of degrees of freedom are

involved in the solution. As a recently published book by Cheung and Tham has reported a compre comprehen hensiv sive e review review on papers papers publish published ed befor before e 996, the present article will focus mainly on the publications in the past three years but references are also also made made to some some key papers papers publis published hed earlie earlier. r.

Prog. Struct. Engng Mater. 2000; 2: 369 d 375

The finite strip method was devised for structural analysis in the late 1960s. It was applied to various types of structures such as plates, shells, box girder  bridges and tall buildings, etc. The displacements of  a conventional strip are described by functions which are given as products of trigonometrical/hyperbolic series and polynomials. The series have to satisfy a priori boundary conditions at the end of the strips. For a simply supported plate, it can be shown readily that a sine series should be chosen. In the early 1980s, the spline finite strips, which used B-3 spline series for interpolation, were also proposed. The spline series satisfy the continuity requirements and they can be easily modified to satisfy various boundary conditions. Other developments allowed other series, such as exponential series and computed shape functi function ons, s, to be emplo employe yed d as well. Hanke Hankell and Laplac Laplacee transforms were also used to analyse geotechnical problems. As a comprehensive review on the contributions  before 1996 can be found in Ref. [1], the present article will focus on the post-1995 publications grouped under the following headings: 1. Developm Developments ents of various types of strips. strips. 2. Impleme Implementati ntation on of finite strip in parallel programming programming environment. environment. 3. Applications in vibration vibration and stability analyses. analyses.

Developments of various types of strips Cocchi[2] had employed the series of  trigonometrical/hyperbolic functions for the descr descript iption ion of the the displa displacem cemen ents ts to analys analysee plates plates with with various support conditions. He demonstrated that the Copyright

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orthogonality conditions were satisfied not only for simply supported case but also for other support conditions, and therefore, the characteristic matrices would be uncoupled. As a result, the computation efficiency was greatly improved. Gagnon et al.[3] used spline finite strip to analyse rectangular plates. They employed various order spline function (linear, quadratic and cubic) for the interpolation in the longitudinal direction and comparison of their efficiency was carried out. Kong et al [4] tried to improve the convergence by replacing the polynomial components of the shape functions by natural shape functions which were obtained by solving the fourth-order differential equation describing the bending action in the transverse direction. Taking a simply supported strip as an example, it was shown readily that the interpolation functions in the transverse direction (x-direction) can be written as Xm (x)"a1m cosh m x#a2m sinh m x a x cosh m x#a4m x sinh m x

# 3m

(1)

where m"m/L (L"length of the strip). a1m , a2m , a3m and a4m are undetermined coefficients. Having the shape functions determined, the solution process was conducted in the usual manner. Such strips can overcome a number of difficulties encountered in the standard strips. These difficulties include rigid body modes, constant strains, spurious zero-energy modes, etc. Applying the U-transformation, Li et al [5] were able to unco uncoup uple le the the equa equati tion onss of the the fini finite te stri strips ps so that that the the problem of the whole structure can be simplified into an equivalent problem of single strip. Prog. Struct. Engng Mater. 2000; 2:369}375

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STRUCTURAL ANALYSIS AND CAD

Zhong et al[6] subdivided each strip into a number of  intervals and carried out precise integration to generate the shape functions. The approach is based on the method of matrix exponentiation and theory of  structural mechanics optimal control. The method was proven to improve the accuracy in modelling the effects of the supports and point loads. Another approach to improve the convergence and accuracy is to include augmented functions in addition to the usual shape functions. Such augmented functions can be applied to the shape functions in either the longitudinal or transverse directions. In treating shear walls/plates the thickness of which changed in stepwise manner, Cheung et al [13] augmented the trigonometrical/hyperbolic series by a piecewise continuous functions so that the required continuity at locations of abrupt changes of thickness could be taken care. The augmenting functions were piecewise linear polynomials for plane stress strip whereas it was necessary to use piecewise cubic polynomials to achieve the C1 continuity as required in the bending strip. The continuity and equilibrium requirements at the junctions where there were abrupt changes in thickness are taken into account in determining the coefficients for the polynomials. For example, the conditions to be applied in the case of 

a bending strip were the continuities of the bending moment and shear force at each junction level. Applications of the augment strips in static, vibration as well as stability analyses of plates and shear walls were documented in Refs [7,8]. The examples quoted fully demonstrated the improvements in accuracy as well as rate of convergence that one could achieve by incorporating the augmenting functions. On the other hand, Azhari & Bradford [9] used  bubble functions as the augmented functions in their study on the local buckling of plate structures. In addition to the cubic polynomial shape functions for interpolation of the deflection in the transverse direction (x-axis), the following bubble functions were introduced.  A Zs" 2n (1#)n (1!)n, n"2, 32 2

(2)

where "2x/b. b is the width of the strip and A is a multiplier. It was shown that the fourth-order bubble function had a major effect on the accuracy of the method, and therefore, it was recommended that the deflection for a bubble strip should be expressed as



w"

1 f b ()2+ d fn , Yn

(3)

1,2,2, r 

n

Fig. 1 Typical isoparametric spline strips [11] Copyright

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FINITE STRIP METHOD where



1 f b ()2"

1 b (1! )2 (1# ); (1! )2 (2# ); 8 4 b (1! )(1# )2 8

!

;

1 1 (1# )2 (2! ); 4 (1# )2 (1! )2 4 2



Yn is the interpolation functions in the y-direction. +d fn , denotes the deflection vector. For structures having non-rectangular geometry, it is possible to carry out geometric mapping by isoparametric transformation and spline finite strips can then be used in the transformed coordinate system. The transformation can be carried out for the whole structure before it is discretised into a number of strips[10]. The approach can be made more flexible if  the structure was discretised into a number of  substructures[11,12] (Fig. 1) and the geometric transformation was carried out for each substructure. Each substructure was then modelled as an assemblage of spline strips. Compatibility between adjacent substructures was ensured by forcing displacements and rotations at the connecting knots to  be equal.

Implementation of finite strip in parallel programming environment It is noted that the ‘structure’ of the finite strip programme is very suitable for the parallel processing, and therefore, attempts to parallelize the finite strip

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programme in shared-memory machines were made  by Puckett & Schmidt[13] as well as Chen & He [14] , Chen[15] , and Chen & Byreddy [16] implemented parallelisation for the finite strip programme on a network of SUN work stations interconnected by the ethernet. Two parallisation strategies were proposed.

MASTER`SLAVE SCHEME Such scheme consists of a master processor and a number of slave processors (Fig. 2a). The master processor is responsible for the creation of the slave processors; reading of the input data; broadcasting of  the data to all slaves using the broadcast commands and computation of the final results from the computed results of each slave processor. The slave processors are responsible for processing of the assigned tasks and sending back the computed results to the master.

MULTI MASTER`SLAVE SCHEME This scheme has a tree-like structure (Fig. 2b). The processor at the root and the leaves are referred to as the root and leaf processors, respectively. Processors in between the root and leaves are called the intermediate processors. Each intermediate processor serve as the master of the slave processors that it created and the slave of the master that created it. Information is communicated between master and slave processors at each level. Such scheme is very flexible and can optimise the usage of the processors. Cheung and Tham [1] also implemented a finite strip programme for the analysis of box girder in the

Fig. 2 Parallel programming schemes: (a) masterdslave scheme; (b) multi masterdslave scheme Copyright

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STRUCTURAL ANALYSIS AND CAD

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well as the load vector + F , for the assigned number of  terms and solves the equilibrium equation (EQUILEQ). The results thus obtained are transmitted  back to the master processor for the calculation of the stresses and displacements.

Applications

Fig. 3 Flow chart for the finite strip programme for box girder analysis

parallel programming environment of the IBM Scalable POWER parallel SP2 system. The programme is based on the master-slave scheme. The flow chart of  the programme is as given in Fig. 3 and the listing of the programmes can be found in Ref. [1]. After reading the input data, the master processor  broadcasts the information to the slave processors. Each slave processor forms the stiffness matrix [ K] as

Hu[17] studied the convergence of the finite strip method for higher modes of vibration of cylindrical honeycomb panels with respect to the strip division and support conditions. He reported that the finite strip results included one extra mode than those predicted by the analytical solutions. It is very likely that these spurious modes are due to the integration schemes adopted and they may be eliminated by employing selective integration in forming the characteristic matrices. The isoparametric spline strips were applied to study the free vibration behaviour of various types of  shell by Au et al [12]. Fig. 4 shows the mode shapes for a spherical shell of square planform (width of the planform is ‘a’). The shell is supported at four corners. Other pertinent dimensions are as follows: ‘a/R’"0.2; ‘a/t’"100, where R and t are the radius and thickness of the shell, respectively. The Poisson’s ratio ( ) is 0.3. The fundamental frequencies for a simply supported bi-directionally stepped square plates were computed by using the augmented shape functions. Table 1 tabulates the frequency coefficients 1 (1"( 1/4a2 ) ( D1/h1 , where 1 is the fundamental angular frequency,  is the density and D1 is the flexural rigidity for two different discretisation schemes: 1. Scheme 1: three strips, that is one central strip through the thickened section and two side strips of  constant thickness. One term is used for the trigonometrical series.

Fig. 4 Modes shapes for a corner point supported spherical shell [12] Copyright

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FINITE STRIP METHOD Table 1 Frequency coefficient for stepped square plate

h 2/h 1

Scheme 1 Scheme 2 .........................................................................................................................

0.70 15.560 15.477 0.75 16.309 16.189 0.80 17.024 16.917 0.85 17.715 17.635 0.90 18.410 18.361 0.95 19.091 19.051 .........................................................................................................................

2. Scheme 2: eight strips of three equal width and three terms for the trigonometrical series. The results tabulated in Table 1 indicate that the convergence is very fast. In the late 1970s, the conventional finite strip method was also applied to analyse buckling problems. The applications of the finite strip method to buckling analysis were attempted by Delcourt [18] as well as Plank & Wittrick[19] . Examples of buckling of  plates and folded plate structures were reported and the success of the work had laid the foundation for the development of the finite strip method in non-linear analyses. A sequel of papers were later published by Bradford and his collaborators[20–25] to illustrate the improvement in convergence that one could achieve in employing bubble functions. They covered applications in the buckling analyses of plates with various support conditions, plates under different loading conditions, composite steel-concrete members and inelastic buckling of plates and plate assemblies. In Table 2, the buckling coefficients of flat plates under compression were given. The coefficient ( kc ) is defined as

c"kc

2E



t 2 12(1!  ) b

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clamped. They also reported that by adopting the  bubble functions, the convergence could be significantly improved. Conventional finite strip would require four times more strips to achieve the same accuracy. Hancock carried out extensive studies on the applications of the spline finite strip method in the stability analyses of thin-walled structural sections. Lau & Hancock[27] applied the spline finite strip method to the inelastic buckling analysis which could take into account the non-linear nature of the stressstrain curves and strain hardening of the material and residual stress distribution. Kwon & Hancock [28] further developed the method to handle local, distortional and overall buckling mode in the post buckling behaviour range, and the interaction between the various modes. A computer package (THINWALL)[29] is now available for such analysis. Zhu & Cheung [30] also carried out the study of the post-buckling behaviour of shells using the spline finite strip method. In a recent publication [31] , they extended the method to study the behaviour of  cylindrical shells under combined loading (axial compression and external pressure). They successfully traced the post-buckling equilibrium path and the post buckling radial deflections thus obtained were in good agreement with experimental results. Fig. 5 shows the relation between the axial load (P) and edge shortening () for a cylinder under confining pressure ( p). Other pertinent parameters of the shell are as follows: Radius (R)"100 mm, thickness ( t )"0.247, length (L)"50.9 mm, Young’s modulus"5.56 GPa, Poisson’s ratio"0.3.

2

(4)

where c is the critical stress, t is the plate thickness, b is the plate width, E is the elastic modulus and   is the Poisson’s ratio. The loaded edges were simply supported whereas the longitudinal edges were simply supportedTable 2 Buckling coefficient of flat plates [22] Length/thickness 0.728 0.79 0.889 ......................................................................................................................... Buckling coefficient (two bubble strips) 5.47 5.41 5.50 ......................................................................................................................... Buckling coefficient [26] 5.47 5.41 5.51 ......................................................................................................................... Copyright

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Fig. 5 Effect of hydrostatic pressure on the post-buckling relations between P  and  [31] Prog. Struct. Engng Mater. 2000; 2:369}375

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It was shown that the theoretical critical hydrostatic pressure (pc ) for this cylinder under completely clamped conditions was 4.33 kPa. In the figure, Z"( (1! 2 L/Rt, n is number of buckling wave number and Rh"p/pc . Dawe and his collaborators[33–36] studied the  buckling of laminated and composite plates using both conventional and spline strips. Comparisons of the  buckling load factors and mid-point deflection(wc ) obtained by the finite strip using trigonometrical series[36] and spline series are made in Table 3. The results for Ref.[36] were obtained by using six and three series terms for inplane displacements and out-ofplane displacement respectively. Kong & Cheung [37] proposed that beam vibration functions can be used to analyse composite plates with intermediate line support. Vibration of plates with point supports was studied by Cheung & Zhou [38] using the static beam functions. Other investigators also carried out studies of   buckling of laminated plates or composite structures using the finite strip approach and some of the recent publications are given in the reference list [39–41] . Wang & Rammerstorfer[42] determined the effective  breadth and effective width of stiffened plates not only in the pre-buckling but also in the post-buckling range. Though the stiffness and geometric matrices are uncoupled in the prebuckling range, they will be coupled in the post-buckling range. The solution process will  become inefficient as the sizes of the matrices increase with number of terms used for the interpolation and therefore, an interaction scheme was proposed. Ignoring the coupling terms of the stiffness matrices, the standard iteration process was used to obtain the displacements. The additional error introduced by neglecting the coupling terms of the stiffness matrices was eliminated by further equilibrium iterations. Kong[43] studied the response of plates under various forms of loadings by the spline strips. In the analysis, the non-linearity due to the membrane action of the plates undergoing large deformation was taken

into account. The Ritz vector was adopted to improve the efficiency of the solution scheme. On the other hand, Zhu[44] solved the non-linear vibration problems using spline strips by the incremental harmonic  balance method[45] .

Table 3 Results for four-layer cross-ply square laminate (03/903/03/903 ) [35] End-shortening strain"0.02%

End-shortening strain"0.10%

........................................ .......................................... Load w c /thickness Load w c /thickness factor factor .........................................................................................................................

Spline 1 section 27.69 0.5002 82.11 1.811 2 sections 27.23 0.5315 78.15 1.886 3 sections 27.17 0.5311 77.23 1.915 4 sections 27.14 0.5302 76.87 1.909 ......................................................................................................................... Ref [36] 27.28 0.5351 77.22 1.909 ......................................................................................................................... Copyright

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References and recommended reading Papers of particular interest have been marked: * Special interest ** Exceptional interest [1] CheungYK& ThamLG. Finite strip method . Boca Raton FL: CRC Press. 1997: 392. [2] CocchiG M. Finite stripmethod in the analysis of thin platestructures with various edge restraints. Computers and Structures  1996: 61 (2): 303d313. [3] GagnonP, GosselinC & Cloutier L. Finite strip element for the analysis of variable thickness. Computers and Structures  1997: 63 (2): 349 d362. [4] Kong J, Cleghorn WL & Cheung YK. Analysis of plates using enhanced finite strips. Proceedings of the 3rd Asian Pacific Conference on Computational  Mechanics, Seoul, Korea, 16 d18 September 1996. 197 d202. [5] Li Y, Chan HC & Cheung YK. A novel finite strip method for vibration analysis of plates. Proceedings of the 3rd Asian Pacific Conference on Computational  Mechanics, Seoul, Korea, 16 d18 September 1996. 209 d214. [6] Zhong WX, Cheung YK & Li Y. The precise finite strip method. Computers and Structures  1998: 69 (6): 773d783. [7] Cheung YK, Au FTK & Zheng DY. Finite strip analysis of shear walls and thin plates with abrupt changes of thickness by using modified beam vibration modes. Proceedingsof 2nd InternationalConferenceon Thin walled Structures, 1998. 53}64. [8] Au FTK, Zheng DY & Cheung YK. Vibration and stability of nonuniform beams with abrupt changes of cross-section by using C 1 modified beam vibration functions. Applied Mathematical Modeling  1999: 29: 19}34. [9] Azhari M & Bradford MA. Local buckling by complex finite strip method using bubble functions. Journal of Engineering Mechanics  (ASCE) 1994: 120 (1): 43d57. [10] LiWY. Spline finite stripanalysis of arbitrarily shaped plates and shells. PhD Thesis, Department of Civil Engineering, University of Hong Kong. 1998. [11] Au FTK & CheungYK. Static and free vibration analysis of variable-depth bridges of arbitrary alignments using the isoparametric spline finite strip method. Thin-Walled Structures  1996: 24: 19 d15. [12] Au FTK & Cheung YK. Free vibration and stability analysis of shells by the isoparametric spline finite strip method. Thin-Walled Structures 1996: 24: 53d 82. [13] Puckett JA and Schmidt RJ. Finite strip method for groundwater modeling in a parallel computing environment. Engineering Computations  1990: 7: 167d172. [14] Chen HC and He AF. Implementation of the finite strip method for structural analysis on a parallel computer. Proceedings of the 1990 International  Conference on Parallel Processing (vol III: Algorithms and Applications). 1990: 372d373. [15] ChenHC. Increasing parllelism in the finite strip formulation static analysis. International Journal of Nueral, Parallel Scientific Computing  1994: 2: 273d298. [16] Chen HC & Byreddy V. Solving plate bending problems using finite strips on networked workstations. Computers and Structures  1997: 62 (2): 227d236. [17] Hu X. Free vibration analysis of symmetrical cylindrical honeycomb panels by using he finite strip method. Journal of Vibration and Control  1997: 3 (1): 19d32. [18] Delcourt CRC. Linear and geometrical non-linear analysis of flat-walled structures by the finite strip method. PhD Thesis, Department of Civil Engineering, University of Adelaide. 1978. [19] Plank RJ & Wittrick WH. Buckling under combined loading of thin flat walled structures by a complex finite strip method. International Journal for  Numerical Methods in Engineering  1974: 8 (2): 323d339. [20] Bradford MA & Azhari M. Buckling of plates with different end conditions using the finite strip method. Computers and Structures  1995: 56 (1): 75d 83. [21] Bradford MA & Azhari M. Inelastic local buckling of plates and plate assemblies using bubble functions. Engineering Structures  1995: 17 (2): 95d103. [22] Bradford MA & Azhari M. Use of bubble functions for the stability of  plates with different end conditions. Engineering Structures  1997: 19 (2): 151d161. [23] Uy B & Bradford MA. Elastic local buckling of steel plates in composite steeldconcrete members. Engineering Structures  1996: 18 (3): 193d200.

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FINITE STRIP METHOD [24] Uy B & Bradford MA. Inelastic local buckling behaviour of thin steel plates in profiled composite beams. Structural Engineering  1994: 72 (16): 259 d 267. [25] Uy B & Bradford MA. Local buckling of cold formed steel sheeting in profiled composite beams at service loads. Structural Engineering Review  1995: 7 (4): 289 d 300. [26] Handbook of structural stability  (Eds Column Research Committee of   Japan). Tokyo: Corona Publishing Company. 1971. [27] Lau SCW & Hancock GJ. Inelastic buckling analyses of beams, columns and plates using the spline finite strip method. Thin-Walled Structures  1989: 7: 213d238. [28] Kwon YB & Hancock GJ. A nonlinear elastic spline finite strip analyses for thin-walled section. Thin-Walled Structures  1991: 12: 295d319. [29] Papangelis JP & Hancock GJ. Computer analysis of thin-walled structural members. Computers and Structures  1995: 56: 157d176. [30] Zhu DS & CheungYK. Post buckling analysis of shells by spline finite strip method. Computers and Structures  1989: 31: 357d364. [31] Zhu DS & Cheung YK. Post-buckling analysis of circular cylindrical shell under combined load. Computers and Structures  1996: 58 (1): 21d26. [32] Yamaki N. Elastic stability of circular cylindrical shells. Amsterdam: Elsevier. 1984. [33] Dawe DJ & Peshkam V. Note on finite strip buckling analysis of  composite plate structures. Composite Structures 1996: 34 (2): 163d168. [34] Wang S & Dawe DJ. Finite strip large deflection and post-overall-buckling analysis of diaphragm-supported plate structures. Computers and Structures  1996: 61 (1): 155d170. [35] Dawe DJ & Wang S. Post-buckling analysis of thin rectangular laminated plates by spline FSM. Thin-Walled Structures  1998: 30 (14): 159 d179.

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[36] Dwe DJ, Lam SSE & Azizian ZG. Non-linear finite strip analysis of  rectangular laminates under end shortening, using classical plate theory. International   Journal for Numerical Methods in Engineering  1992: 35: 1087 d1110. [37] Kong J & Cheung YK. Vibration of shear-deformable plates with intermediate line supports: A finite layer approach. Journal of Sound and Vibration  1995: 184 (3): 639 d 649. [38] Cheung YK& ZhouD. The free vibration of rectangular composite plates with point-supports using static beam function. Composite Structures  1999: 44: 145}154. [39] Wang WJ, Tseng YP & Lin KJ. Stability of laminated plates using finite strip method based on a higher order plate theory. Composite Structures  1996: 34 (1): 65d76. [40] Rhodes J. Semi-analytical approach to buckling analysis for composite structure. Composite Structures  1996: 35 (1): 93d99. [41] Loughlan J. Buckling of composite stiffened box section subjected to compression and bending. Composite Structures  1996: 35 (1): 101d116. [42] Wang X & Rammerstorfer F G. Determination of effective breadth and effective widthof stiffened plates by finite strip analyses. Thin-WalledStructures 1996: 26 (4): 261 d286. [43] Kong J. Analysis of plate-type structures by finite strip, finite prism and finite layer method. PhD Thesis, Department of Civil and Structural Engineering. University of Hong Kong. 1994. [44] Zhu DS. Nonlinear static and dynamic analysis of plates and shells by spline finite strip method PhD Thesis, Department of Civil and Structural Engineering. University of Hong Kong. 1988. [45] Lau SL. Incremental harmonic balance methods for nonlinear structural. vibration. PhD Thesis, Department of Civil and Structural Engineering. University of  Hong Kong. 1982.

Y K Cheung

Department of Civil Engineering, University of Hong Kong, Pokfulam Road, Hong Kong L G Tham

Department of Civil Engineering, University of Hong Kong, Pokfulam Road, Hong Kong

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