Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 Developing an Innovative Finite Element Analysis for Free-form Shells RAJI, S.A., Civil Engineering Department, University of Ilorin, P.M.B. 1515 Ilorin, Nigeria. e-mail address : saraji@unilorin.edu.ng Abstract Different finite elements have been developed for the analysis and optimization of free-form shells. Ahmad et.al.[3] first developed a degenerated , Mindlin-Reissner type, curved shell element which is quite efficient and simple. In the degenerated shell element formulation of the isoparametric element concept is extended to shells. While the original degenerated shell elements are capable of dealing satisfactorily with thick plates and shells, their performance deteriorate as the plate or shell become thin. This phenomenon has been attributed to shear and membrane locking. Several researchers modified and extended the original degenerated shell element to improve its behavior for both thick and thin shell situations. In this paper, historical review of shell elements that have been developed for accurate analysis of curved shell structures is made and the formulation of an improved nine-node Lagrangian shell element with special emphasis on selective and reduced integration is presented and it is hoped that this would assist in overcoming some of the inherent problems of shear and membrane locking in thin shells. Keywords: degenerated shell element, Mindlin-Reissner element, Lagrangian 1. Introduction 2 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 Although shell structures have been featuring in building construction over a long period of time, like the mosque of Santa Sophia in Istanbul built in 538 A.D. [ 1 ] , it was not until the 1920s that the thin shell roof emerged as a practical means for spanning large distance. The first general theory of thin shell structures was established by Love [2] in 1888. However, analytical solutions to thin shell structural problems are limited in scope and are not applicable to arbitrary shapes, load conditions, irregular support conditions, cut –outs other aspect of practical designs. These limitations are however overcome by the development of finite element method in the twentieth century [ 3]. This method is widely used in various engineering and structural mechanics problems including the equilibrium in which the parameters within the system do not change with time; Eigen-value problems in which critical values of certain parameters must be determined; propagation problems in which the time dimension has to be determined. During the last two decades there has been extensive research on developing efficient, accurate and reliable shell elements for analysing structures with complex geometries and loadings. Consequently, the various types of elements described in the next section are now available for the analysis of thin shells. 2. Finite Elements for Thin Shells Several types of element exist for the analysis of thin shells. These include : Flat plate (facet) elements with quadrilateral and triangular shapes. Curved elements of quadrilateral and triangular shapes derived from classical shell theory. Elements degenerated from three-dimensional solid elements. 2.1 Flat Plate (Facet) Elements 3 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 Various authors [4 to 9] have studied the advantages and disadvantages of the various types of these elements. Greene et al. [6] first suggested the concept of the use of 'faceted' form flat element in shell analysis. However, the success of such analysis was delayed until formulations for plate elements in bending became well grounded. The historical account of the formulations and developments for the flat elements are obtainable in textbooks [7 to10] and in survey papers[l1,12]. Meek and Tan [13] developed a triangular flat element with 'Loof nodes' with good performance and gave an interesting survey of developments in flat elements for shell analysis. Some of the major attractive features of this modeling are [8]: (a) it is simple to formulate; (b) it is simple to define the geometry; (c) it is easy to mix with other types of element; (d) it is capable of modeling rigid body motion without including strains; and (e) the requirement of using a relatively large number of elements provides the advantages of convenience in incorporating complex loading and boundary conditions. In general, adequate performance can be obtained when the sought response is predominantly either membrane or bending [14]. However, when the membrane and bending stiffness are strongly coupled, the performance is extremely poor. For example, in an attempt to model the torsional behavior of a slit cylinder (one of Morley's [15] 'sensitive' solution) the flat elements failed completely. The use of flat modeling of shell has been a major feature in almost all the popular finite element code, such as ABAQUS [16], AD1NA [17], ANSYS [18], MARC [19], NASTRAN series [20] and SAP series[21]. When only approximate solution is sufficient, flat plate elements have been used to solve shell problems with static [22, 23], dynamic [24,25], buckling [26] and other physical phenomena especially for practical problems in the industry. Some of the problems encountered in the application of the flat elements in the analysis of shells include [7]: 4 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 (a) The actual coupling of stretching and bending in the element is neglected in this formulation. Such coupling is a major contributor towards load carrying mechanism in shells and other curved members; (b) The junctions where elements are coplanar presents problems of null stiffness corresponding to rotation about the axis normal to the plane ( z ) (c) The restriction to triangular shapes when general shells are to be treated. (d) The influence of the geometric approximation upon the solution for imperfection-sensitive structures. (e) When the stiffness equations from the 'faceted' form are used to analyse curved shell structure, 'discontinuity' bending moments which are not present in the continuously curved structure at the element juncture lines were observed. 2.2 Curved Elements Derived From Classical Shell Theories The limitations of flat elements in shell analysis arise from the flatness of the elements and the discontinuity in geometry caused by these elements in representing a curved surface. The singly curved elements have been developed with a view to overcoming such limitations. They are divided into singly and doubly curved shell elements. The singly curved elements have gone through various improvements by various researchers [27 to 49], each improving on the previous researcher's work. The singly curved elements were first developed in the axisymmetric form for the analysis of shells of revolution. The conical segment element for the analysis of shell of revolution was introduced by Grafton and Strome[27]. Improvements in the derivation of element stiffness were presented by Popov et. al [28]. Percy et al [29] extended this formulation for orthotropic and laminated materials. Dong [30] presented results for laminated 5 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 spherical caps which were in good agreement with analytical results. However, Navaratna [31] showed a significant discontinuity in stress at the transition point when using two conical segments of different sizes for a spherical cap. He also showed that the stress discontinuity disappeared when curved rather than conical segments, which are a special case of flat elements, were used. Fulton et al [32] and Beitch [33] recognized the need for curved elements for modeling shell structures. Gradually, researchers have increased the sophistication of these elements by improving the geometric approximations [34] and the displacement interpolation functions [35] and by extending the applicability to include anisotropic materials [36], dynamic response [37, 38], elastic stability [39] and nonlinear effects [40]. In practical applications, shell structures have been most commonly designed in axisymmetric shape [41, 42] because elements of axisymmetric shape are more efficient than elements of general shape. Gallagher [9] presented a 24 degree of freedom conforming element using the complete bicubic Hermite interpolation functions. Connor and Brebbia[43] developed a 20 degree of freedom, non-conforming rectangular cylindrical shell element with a complete bilinear expansion for middle surface displacement u and v, and an incomplete bicubic expansion for transverse deflection w. Cantin and Clough [44] modified the Gallagher's element[5] to a cylindrical shell element and added the six rigid body modes explicitly by introducing the appropriate trigonometrical terms into the displacement functions. This 24 degree of freedom element was conforming and well behaved, with monotonic convergence of the results towards correct solutions with mesh refinement. Their displacement functions did not contain the constant circumferential strain mode since the constant term was missing from the w-function and the inter-element continuity was violated. Sabir and Lock [45] and Ashwell and Sabir [46] showed that if certain terms were omitted from the displacement function of [44], a nonconforming element with smaller stiffness resulted but without any apparent loss in accuracy. Fonder and Clough[47] explicitly added four missing rigid body modes in a 6 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 cylindrical shell element with 24 degree of freedom and found significant improvement over the perfomance of the same element without this addition. Jones and Strome [48] and Stricklin et al .[49] were among the first to apply doubly curved elements to the shells of revolution. They used curved meridional elements rather than conical segments. Attempts was made by Utku [50] to develop a general curved shallow shell element by accounting for the shear deformation with linear variation of strain across the thickness in the 15 degree of freedom triangular element with nonzero Gaussian curvature. Thus, the displacement fields could be represented with simple linear functions. As the distribution of the transverse displacement component was not compatible with the assumed linear distribution of rotations, the element experienced considerable strain under rigid body rotation[57]. Brebbia and Deb Nath [51] reviewed the shallow shell elements and compared their relative convergence rate with an established finite difference solution for a clamped hypar shell under uniformly distributed normal load. Yang [52] also developed a high order 48 degree of freedom shallow shell finite element with two principal radii of curvature and a twist radius. 2.3 Elements Degenerated from Three-Dimensional Solid Elements Although, the hypothesis underlying the degenerated shell element and the classical shell theory are essentially the same, the reduction to resultant form is typically carried out numerically in the former, and analytically in the latter. The isoparametric concept for three dimensional elements has strong appeal for shell application since relatively few isoparametric elements can be used to model curved shell structures. Also, complex mathematical expressions and assumptions of shell theories could be avoided [53]. 7 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 In the use of a 20-node brick element, the basic assumption made is that no strain occurs across the thickness and that the direct strain vary linearly in that direction. However, the displacement pattern for pure bending reveal a curved upper and lower surfaces, rather than straight as derived from the above assumptions. Wilson et. al[46] adds displacement modes called 'bubble modes' to describe the curvatures of the upper and lower surfaces but difficulties have been encountered with non-rectangular forms of this element. A number of curved shell elements have been developed using the three-dimensional formulation. The SHEBA family developed by Argyris and Scharpf[6] are series of complex elements based on the displacement method. The inadequate membrane representation in 'faceted’ flat elements were overcome, but the simplicity of the elements was lost. This is because analytical functions describing the shell geometry and additional degrees of freedom (first- and second-order derivatives) were necessary. Ahmad [60] developed a shell element for moderately thick shells by applying the so-called 'degeneration' process to the three dimensional element. The 3-dimensional formulation was degenerated by introducing the assumption that originals normal to the middle surface are inextensible and remain straight (which could be linked with the Winkler's theory of thick curved beams [6l]), and that the elastic modulus in normal direction is zero. This procedure permitted a measure of the transverse shear deformation since the mid-surface normal are not necessarily to be kept normal during deformation, it may make angles of other than right angle with the deformed surface, thus, accommodating an important feature in thick shell situations. Although the Ahmad's shell element is attractive due to its simplicity and ability of reproducing shear deformation, unfortunately the results obtain worsened and finally completely failed as the thickness of the element was reduced [53]. In addition, the element was too stiff and showed a very slow convergence rate. Too [54] observed that 2x2 Gaussian integration gives remarkably improved results in the 8 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 quadratic stack, provided the final stresses are calculated at the same 2 x 2 points. This prompted the improvement of the Ahmad's element by means of the totally reduced integration method by Zienkiewicz el. al. [55] which greatly improve the results in shell applications. Another developed family of isoparametric shell elements is the SemiLoof shell elements by Irons[58].These elements are general curved triangles and curved quadrilaterals with three degrees of freedom u,v, and w at each of four corners and four midsides and normal rotations at the two Gauss points (the so-called 'Loof’ nodes) along each side. The discrete Kirchhoff hypothesis is imposed on the 32-degrees of freedom element at the 'Loof’ nodes and a reduced integration technique is applied. The resulting shell element obtained is capable of representing shells with junctions and discontinuities to a great extent [56]. MacNeal [59] developed a simple 4-node quadrilateral element, QUAD4, which is basically a thick shell element in which the shear stiffness is improved by adjusting the shear modulus of rigidity according to the actual plate thickness and by using a special form of the reduced integration technique for the shear terms. The membrane component of the element stiffness is computed by applying selective integration since the shear component is integrated using only one integration point at center of the element. The resulting shell element can be used for both thick and thin shell problems with high accuracy. Structured and unstructured finite-element meshes based on bilinear quadrilateral and linear triangular elements were used in discretizing assumed gradient algorithm in solving the level set equation [64,65]. Several other researchers had modified and extended the original degenerated shell element to improve its behaviour for both thick and thin shell situations. Among them is the element formulated by Huang and Hinton [62]. This element has exhibited excellent general behaviour and eliminated several drawbacks associated with earlier degenerated shell elements. The basic finite element formulation for 9 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 the new proposed nine-node degenerated shell element is described in the next section. 3. FORMULATION OF THE LAGRANGIAN SHELL ELEMENT IMPROVED 9-NODE The Huang-Hinton element has been used successfully in series of applications and the general performance has been found to be better compare to Lagrangian and Heterosis elements, especially for shear force distribution. However, the element performed poorly for skewed plate when very high span/thickness ratios were considered with parallelogramic methods.[62]. The proposed new 9-Node Lagrangian element seeks to use a recent and more efficient method to rectify the locking and mechanism problems. The method uses the sampling points at which no locking effect exists and the new shear strains constructed from these sampling points are used instead of the original displacement fields. 3.1 Element Displacement Field As far as transverse shear effect is concerned, the Mindlin plate(Fig.1) and shell elements(Fig.2) are quit similar. Hence, the Mindlin plate bending theory is used to determine the sampling points for transverse shear strain of the shell element. For a typical point (x, y, z) on the Mindlin plate, the displacement components due to bending could be expressed as 10 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 =-1 =+1 11 =+1 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 u = z x (x, y) v = z y (x, y) w = w(x, y) (1) where u and v are the in-plane displacement components and w is the transverse deflection. Ѳx and Ѳy are the normal rotation in x z and y z planes respectively. For conventional n-node Mindlin plate element, the displacement and rotations are expressed as w wi n ( 2) x N i xi i 1 y yi where, wi, xi and yi are unknown displacement values at node i Ni is the shape function associated with node i As mentioned earlier, in order to eliminate shear locking, the assumed strain method which required appropriate sampling points for transverse shear strain will be used. Hence, only transverse shear strain component will be discussed. The moment curvature and shear force-shear strain relationship for plate are as for degenerated shell element in which the in-plane stresses are p' ( x' , y' , xy' )T (3) and the transverse shear stresses are s' ( xz' , yz' ) T (4) The shear strain components in the global coordinate system are s (u,z ' w, x ' ), (v,z ' w, y ' ) T (5) The above equation can be rewritten in the view of Equation (1) as 12 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 s ( x w, x ' ), ( y w, y ' ) T (6) When we use the same shape function for normal rotations and transverse deflection w, the order of the two right-hand side terms of Equation (6) are not equal. The desirable displacement field will be introduced to establish the consistency of the two terms. The natural transverse shear strain can be written as, x , xz ( 7) y , yz Substituting into Equation (6) into Equation (7) yields x , x w (8) y, w y giving w w ( 9) The real strain components in the global coordinates could be obtained by inversion of equation (7) as, , x xz (10) yz , y As shown in equation (7), natural strain tensor can be expressed as polynomial terms in the natural coordinates. Through this relation, the natural strain components will be used instead of the real strain components. Transverse shear will vanish if the span/thickness ratio is extremely large. 13 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 w w 0 (11) w (12) w For a nine-node element, the polynomial term for , w, and , or w, are [2] 1, , , , 2 , 2 , 2, 2 , 2 2 w, w, 1, , , , 2 , 2 1, , , , 2 , 2 , 2, 2 , 2 2 w, w, 1, , , , 2 , 2 (13) It is clear that the polynomial of term and of equation (13) are generally one order higher than that of the expression of w, and w, . Two ways of resolving this inconsistency are: 1. The coefficients of higher order term in and have to be enforced to be zero. 2. All terms presented in the expression and have their counterpart in the expression of w, and w, . In adopting the second approach, a desirable deflection field w and w .are chosen to express the transverse shear strain in Equation (13) such that that the polynomial terms of w, and w, should include all terms of the and . 14 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 Therefore, inconsistency of both terms of equation 13, hence, the shear locking problem, can be solved by the above definition and the desirable displacement field will express the realistic relationship between the deflection and the rotational fields. 3.2 Sampling Points for Transverse Shear Strain For nine-node Lagrange element, Fig. 3, the rotation and deflection fields are w x y wi N i xi i 1 yi 9 (14) 15 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 where N i , is the quadratic Lagrange shape function associated with node i. The desirable deflection field for expression of may be expressed as w 3 c pq pq (15) p ,q 0 Let the biquadratic approximation of deflection be 9 N w w i i 1 (16) i where wi is nodal desirable displacement value at node i Enforcing the equality of derivatives. w, w , (17) yields the six sampling points of : (ε,η)i , i=1,6 1 1 1 3, 1 1: 3, 0 : 1 3 ,1 2 : 1 3, 0 4 (18) 3 3 , 1, : 1 3 , 1 5 6 (19) The above six points are suitable for expression of the shear strain in an element of arbitrary element geometry. Another set of six points for the shear strain can be found by same procedure. Location of sampling points of : (ε,η)i , i=1,6 are 1, 1 1, 1 3 1: 3 2 : 0, 1 3 0, 1 3 : 1, 1 3 :1, 1 3 4 (20) 3 5 6 3.3 Evaluation of Strain Energy Components 16 (21) Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 The membrane, bending and shear strain energy components for a Mindlin-Reissner degenerated shell element can be evaluated from the membrane, bending and shear stress resultants M , M , M Q , Q 'm N x ' , N y ' , N x ' y ' 'b 's T T x' y' x'y' T x' (22) y' where x’,y’,z’ is a locally directed coordinate system in which the x’,y’ plane is tangential to the shell mid-surface. These stress resultants for a degenerated Mindlin-Reissner shell element can be obtained by appropriately integrating the corresponding stress components with respect to the thickness coordinate[63]. For a typical element the bending, membrane and shear strain energies can be evaluated by the expressions, ^ 2 wb 1 Db ' b T ' b dA A ^ 2 wm 1 Dm ' m T ' m dA A ^ 2 ws 1 Ds dA ' s T ' s (23) A where for surface integration x / x / dA y / x y / dd z / z / (24) 17 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 ^ Note that Db ^ ^ ,D m , D are the matrix of flexural, membrane and s shear rigidities respectively where, ^ ^ t /2 Dm Dp z 'dz ' , t /2 ^ t /2 Db Dp ( z ' ) 2 dz ' , t /2 Dz t /2 D dz s ' (25) t /2 The accumulated contributions to the bending, membrane and shear energies are obtained by summing the contributions from each element. The total strain energy of the finite element solution is then computed using the expression 2 w 2 2 2 wb wm ws (26) 4. Conclusions An historical review of different finite element has been presented with a view to develop an improved nine-node Lagrangian shell element that would behave well in the analysis and optimization of plates and shells. Benchmark tests including plates and shells of different geometries and loading would be carried out in order to study the behaviour of this element. The bending, membrane and shear strain energies would be evaluated to observe the contribution of each form of energy. It is hoped that this element would solve the spurious mechanism problem which produces singular element stiffness matrix and causes shear and membrane locking in thin shells. Acknowledgements The author wishes to thank University of Ilorin for the Senate Research Grant used for this research work. 18 Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16 REFERENCES 1. Ashwell, G.H. and Galagher, R.H.(eds), Finite element for thin shells and curved members, Wiley, New York. 2. Love , A.E.H. 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