Finite Element Method - Semi - analytical finite element processes - use of orthogonal functions and finite strip methods _09 This monograph presents in detail the novel "wave" approach to finite element modeling of transient processes in solids. Strong discontinuities of stress, deformation, and velocity wave fronts as well as a finite magnitude of wave propagation speed over elements are considered. These phenomena, such as explosions, shocks, and seismic waves, involve problems with a time scale near the wave propagation time. Software packages for 1D and 2D problems yield significantly better results than classical FEA, so some FORTRAN programs with the necessary comments are given in the appendix. The book is written for researchers, lecturers, and advanced students interested in problems of numerical modeling of non-stationary dynamic processes in deformable bodies and continua, and also for engineers and researchers involved designing machines and structures, in which shock, vibro-impact, and other unsteady dynamics and waves processes play a significant role.
Semi-analytical finite element processes - use of orthogonal functions and ’finite strip’ methods 9.1 Introduction Standard finite element methods have been shown to be capable, in principle, of dealing with any two- or three- (or even four-)* dimensional situations Nevertheless, the cost of solutions increases greatly with each dimension added and indeed, on occasion, overtaxes the available computer capability It is therefore always desirable to search for alternatives that may reduce computational effort One such class of processes of quite wide applicability will be illustrated in this chapter In many physical problems the situation is such that the geometry and material properties not vary along one coordinate direction However, the ‘load’ terms may still exhibit a variation in that direction, preventing the use of such simplifying assumptions as those that, for instance, permitted a two-dimensional plane strain or axisymmetric analysis to be substituted for a full three-dimensional treatment In such cases it is possible still to consider a ‘substitute’ problem, not involving the particular coordinate (along which the geometry and properties not vary), and to synthesize the true answer from a series of such simplified solutions The method to be described is of quite general use and, obviously, is not limited to structural situations It will be convenient, however, to use the nomenclature of structural mechanics and to use potential energy minimization as an example We shall confine our attention to problems of minimizing a quadratic functional such as described in Chapters 2-6 of Volume The interpretation of the process involved as the application of partial discretization in Chapter of Volume followed (or preceded) by the use of a Fourier series expansion should be noted Let (x,y , z ) be the coordinates describing the domain (in this context these not necessarily have to be the Cartesian coordinates) The last one of these, z , is the coordinate along which the geometry and material properties not change and which is limited to lie between two values O h3 + 3h2(x- X j - ) 3h(x - x;-*)2- 3(x - x ; ) , xi-1 d x d xi =6h3 h3 - h ( ~ - ~ i + l ) + h ( ~ - ~ ; + r ) + ( ~ - ~ i +xi1 d) 3x ,d x j + l + ( ~ i + 2-x13, 0, Xj+I d x d xi+2 x > x;+2 (9.40) The use of B3 splines offers certain distinct advantages when compared with the conventional finite element method and the semi-analytical finite strip method It is computationally efficient When using B3 splines as displacement functions, continuity is ensured up to the second order (C2 continuity) However, to achieve 308 Semi-analyticalfinite element processes the same continuity conditions for conventional finite elements, it is necessary to have three times as many unknowns at the nodes (e.g quintic Hermitian functions) It is more flexible than the semi-analytical finite strip method in the boundary condition treatment Only the local splines around the boundary point need to be amended to fit any specified boundary condition It has wider applications than the semi-analytical finite strip method The spline finite strip method can be used to analyse plates with arbitrary shapes.28 In this case any domain bounded by four curved (or straight) sides can be mapped into a rectangular one (see Chapter of Volume 1) and all operations for one system (x,y) can be transformed to corresponding ones for the other system (c, q) The finite strip methods have proved effective in a large number of engineering applications, many listed in the text by C h e ~ n g ’References ~ 29-39 list some of the typical linear problems solved in statics, vibrations, and buckling analysis of structures Indeed, non-linear problems of the type described in Sec 4.20 have also been successfully tackled.40i41 Of considerable interest also is the extension of the procedures to the analysis of stratified (layered) media such as may be encountered in laminar structures or foundation^.^^-^ 9.8 Concluding remarks A fairly general process combining some of the advantages of finite element analysis with the economy of expansion in terms of generally orthogonal functions has been illustrated in several applications Certainly, these only touch on the possibilities offered, but it should be borne in mind that the economy is achieved only in certain geometrically constrained situations and those to which the number of terms requiring solution is limited Similarly, other ‘prismatic’ situations can be dealt with in which only a segment of a body of revolution is developed (Fig 9.12) Clearly, the expansion must now be taken in terms of the angle lnO/a, but otherwise the approach is identical to that described previously.2 In the methods of this chapter it was assumed that material properties remain constant with one coordinate direction This restriction can on occasion be lifted with the same general process maintained An early example of this type was presented by Stricklin and DeA~ ~ d rad e Inclusion ~ ’ of inelastic behaviour has also been successfully treated.46-49 In Chapter 17 of Volume dealing with semi-discretization we considered general classes of problems for time All the problems we have described in this chapter could be derived in terms of similar semi-discretization We would thusJirst semi-discretize, describing the problem in terms of an ordinary differential equation in z of the form d2a da Kl - - + K - + K a + f = dz dz Second, the above equation system would be solved in the domain < z < a by means of orthogonal functions that naturally enter the problem as solutions of ordinary References 309 Fig 9.1 Other segmental, prismatic situations differential equations with constant coeficients This second solution step is most easily found by using a diagonalization process described in dynamic applications (see Chapter 17, Volume 1) Clearly, the final result of such computations would turn out to be identical with the procedures here described, but on occasion the above formulation is more self-evident References P.M Morse and H Feshbach Methods of Theoretical Physics, McGraw-Hill, New York, 1953 O.C Zienkiewicz and J.J.M Too The finite prism in analysis of thick simply supported bridge boxes Proc Inst Civ Eng., 53, 147-72, 1972 Y.K Cheung The finite strip method in the analysis of elastic plates with two opposite simply supported ends Proc Inst Civ Eng., 40, 1-7, 1968 Y.K Cheung Finite strip method of analysis of elastic slabs Proc Am SOC.Civ Eng., 94(EM6), 1365-78, 1968 Y.K Cheung Folded plates by the finite strip method Proc Am SOC.Civ Eng., 95(ST2), 963-79, 1969 Y.K Cheung The analysis of cylindrical orthotropic curved bridge decks Publ Int Ass Struct Eng., 29-11, 41-52, 1969 Y.K Cheung, M.S Cheung and A Ghali Analysis of slab and girder bridges by the finite strip method Building Sci., 5, 95-104, 1970 Y.C Loo and A.R Cusens Development of the finite strip method in the analysis of cellular bridge decks In K Rockey et al (eds), Con$ on Developments in Bridge Design and Construction, Crosby Lockwood, London, 1971 Y.K Cheung and M.S Cheung Static and dynamic behaviour of rectangular plates using higher order finite strips Building Sci., , 151-8, 1972 3 10 Semi-analyticalfinite element processes 10 G.S Tadros and A Ghali Convergence of semi-analytical solution of plates, Proc Am SOC.Civ Eng., 99(EM5), 1023-35, 1973 11 A.R Cusens and Y.C Loo Application of the finite strip method in the analysis of concrete box bridges Proc Inst Civ Eng., 57-11, 251-73, 1974 12 T.G Brown and A Ghali Semi-analytic solution of skew plates in bending Proc Inst Civ Eng., 57-11, 165-75, 1974 13 A.S Mawenya and J.D Davies Finite strip analysis of plate bending including transverse shear effects Building Sci., 9, 175-80, 1974 14 P.R Benson and E Hinton A thick finite strip solution for static, free vibration and stability problems Int J Num Meth Eng., 10, 665-78, 1976 15 E Hinton and O.C Zienkiewicz A note on a simple thick finite strip Int J Num Meth Eng., 11, 905-9, 1977 16 H.C Chan and Foo Buckling of multilayer plates by the finite strip method Int J Mech Sci., 19, 47-56, 1977 17 Y.K Cheung Finite Strip Method in Structural Analysis, Pergamon Press, Oxford, 1976 18 I.S Sokolnikoff The Mathematical Theory of Elasticity, 2nd edition, McGraw-Hill, New York, 1956 19 S.P Timoshenko and J.N Goodier Theory of Elasticity, 3rd edition, McGraw-Hill, New York, 1969 20 O.C Zienkiewicz, P.L Arlett and A.K Bahrani Solution of three-dimensional field problems by the finite element method The Engineer, October 1967 21 E.L Wilson Structural analysis of axi-symmetric solids Journal of AIAA, 3, 2269-74, 1965 22 V.V Novozhilov Theory of Thin Shells, Noordhoff, Dordrecht, 1959 [English translation] 23 P.E Grafton and D.R Strome Analysis of axi-symmetric shells by the direct stiffness method Journal of AIAA, 1, 2342-7, 1963 24 Foo Application of Finite Strip Method in Structural Analysis with Particular Reference to Sandwich Plate Structure, PhD thesis, The Queen’s University of Belfast, Belfast, 1977 25 Y.K Cheung Computer analysis of tall buildings In Proc 3rd Int Conf on Tall Buildings, pp 8-15, Hong Kong and Guangzhou, 1984 26 I.J Schoenberg Contributions to the problem of approximation of equidistant data by analytic functions Q Appl Math., 4, 45-99 and 112-1 14, 1946 27 W.Y Li, Y.K Cheung and L.G Tham Spline finite strip analysis of general plates J Eng Mech., ASCE, 112(EM1), 43-54, 1986 28 Y.K Cheung, L.G Tham and W.Y Li Free vibration and static analysis of general plates by spline finite strip Comp Mech., 3, 187-97, 1988 29 Y.K Cheung Orthotropic right bridges by the finite strip method In Concrete Bridge Design, pp, 812-905, Report SP-26, American Concrete Institute, Farmington, MI, 1971 30 H.C Chan and Y.K Cheung Static and dynamic analysis of multilayered sandwich plates Int J Mech Sci., 14, 399-406, 1972 31 D.J Dawe Finite strip buckling of curved plate assemblies under biaxial loading Int J Solids Struct., 13, 1141-55, 1977 32 D.J Dawe Finite strip models for vibration of Mindlin plates J Sound Vib., 59,44-52, 1978 33 Y.K Cheung and C Delcourt Buckling and vibration of thin, flat-walled structures continuous over several spans Proc Inst Civ Eng., 64-11, 93-103, 1977 34 Y.K Cheung and S Swaddiwudhipong Analysis of frame shear wall structures using finite strip elements Proc Inst Civ Eng., 65-11, 517-35, 1978 35 D Bucco, J Mazumdax and G Sved Application of the finite strip method combined with the deflection contour method to plate bending problems J Comp Struct., 10, 827-30, 1979 References 1 36 C Meyer and A.C Scordelis Analysis of curved folded plate structures Proc Am SOC Civ Eng., 97(STlO), 2459-80, 1979 37 Y.K Cheung, L.G Tham and W.Y Li Application of spline-finite strip method in the analysis of curved slab bridge Proc Inst Civ Eng., 81-11, 111-24, 1986 38 W.Y Li, L.G Tham and Y.K Cheung Curved box-girder bridges Proc Am SOC.Civ Eng., 114(ST6), 1324-38, 1988 39 Y.K Cheung, W.Y Li and L.G Tham Free vibration of singly curved shell by spline finite strip method J Sound Vibr., 128, 411-22, 1989 40 Y.K Cheung and D.S Zhu Large deflection analysis of arbitrary shaped thin plates J Comp Struct., 26, 811-14, 1987 41 D.S Zhu and Y.K Cheung Postbuckling analysis of shells by spline finite strip method J Comp Struct., 31, 357-64, 1989 42 S.B Dong and R.B Nelson On natural vibrations and waves in laminated orthotropic plates J Appl Mech., 30, 739, 1972 43 D.J Guo, L.G Tham and Y.K Cheung Infinite layer for the analysis of a single pile J Comp Geotechnics, 3, 229-49, 1987 44 Y.K Cheung, L.G Tham and D.J Guo Analysis of pile group by infinite layer method Geotehnique, 38,415-31, 1988 45 J.A Stricklin, J.C DeAndrade, F.J Stebbins and A.J Cwertny Jr Linear and non-linear analysis of shells of revolution with asymmetrical stiffness properties In L Berke, R.M Bader, W.J Mykytow, J.S Przemienicki and M.H Shirk (eds), Proc 2nd Conf: Matrix Methods in Structural Mechanics, Volume AFFDL-TR-68-150, pp 1231-5 1, Air Force Flight Dynamics Laboratory, Wright Patterson Air Force Base, OH, October 1968 46 L.A Winnicki and O.C Zienkiewicz Plastic or visco-plastic behaviour of axisymmetric bodies subject to non-symmetric loading; semi-analytical finite element solution Znt J Num Meth Eng., 14, 1399-412, 1979 47 W Wunderlich, H Cramer and H Obrecht Application of ring elements in the nonlinear analysis of shells of revolution under nonaxisymmetric loading Comp Meth Appl Mech Eng., 51, 259-75, 1985 48 H Obrecht, F Schnabel and W Wunderlich Elastic-plastic creep buckling of circular cylindrical shells under axial compression Z Angew Math Mech., 67, T118-Tl20, 1987 49 W Wunderlich and C Seiler Nonlinear treatment of liquid-filled storage tanks under earthquake excitation by a quasistatic approach In B.H.V Topping (ed.), Advances in Computational Structural Mechanics, Proceedings 4th International Conference on Computational Structures, pp 283-91, August 1998 ... centre and is defined by I x < xi-2 0, Q; xi-2 d x d X j - (x-xi-2) > h3 + 3h2(x- X j - ) 3h(x - x ;-* ) 2- 3(x - x ; ) , xi-1 d x d xi =6h3 h3 - h ( ~ - ~ i + l ) + h ( ~ - ~ ; + r ) + ( ~ - ~ i... functions N ( x ) of standard type and the Y,,(y) series or spline function part 306 Semi- analyticalfinite element processes The most commonly used series are the basic functions" (or eigenfunctions)... - -1 N j / r sin I0 r - 0 N,,=cos 10 I N j / r cos 16 0 N,,, cos ie N,,z sin 19 -I N , / r sin I0 ( N j , r- N i / r ) sin 16 - (9.27) 300 Semi- analytical finite element processes component and