A comparison between different finite elements for elastic and aero-elastic analyses

In the present paper, a comparison between five different shell finite elements, including the Linear Triangular Element, Linear Quadrilateral Element, Linear Quadrilateral Element based on deformation modes, 8-node Quadrilateral Element, and 9-Node Quadrilateral Element was presented. The shape functions and the element equations related to each element were presented through a detailed mathematical formulation. Additionally, the Jacobian matrix for the second order derivatives was simplified and used to derive each element’s strain-displacement matrix in bending. The elements were compared using carefully selected elastic and aero-elastic bench mark problems, regarding the number of elements needed to reach convergence, the resulting accuracy, and the needed computation time. The best suitable element for elastic free vibration analysis was found to be the Linear Quadrilateral Element with deformation-based shape functions, whereas the most suitable element for stress analysis was the 8- Node Quadrilateral Element, and the most suitable element for aero-elastic analysis was the 9-Node Quadrilateral Element. Although the linear triangular element was the last choice for modal and stress analyses, it establishes more accurate results in aero-elastic analyses, however, with much longer computation time. Additionally, the nine-node quadrilateral element was found to be the best choice for laminated composite plates analysis.

Journal of Advanced Research (2017) 635–648 Contents lists available at ScienceDirect Journal of Advanced Research journal homepage: www.elsevier.com/locate/jare Original Article A comparison between different finite elements for elastic and aero-elastic analyses Mohamed Mahran a,⇑, Adel ELsabbagh b, Hani Negm a a b Aerospace Engineering Department, Cairo University, Giza 12613, Egypt Asu Sound and Vibration Lab, Design and Production Engineering Department, Ain Shams University, Abbaseya, Cairo 11517, Egypt g r a p h i c a l a b s t r a c t a r t i c l e i n f o Article history: Received April 2017 Revised 23 June 2017 Accepted 28 June 2017 Available online July 2017 Keywords: Finite element method Triangular element Quadrilateral element Free vibration analysis Stress analysis Aero-elastic analysis a b s t r a c t In the present paper, a comparison between five different shell finite elements, including the Linear Triangular Element, Linear Quadrilateral Element, Linear Quadrilateral Element based on deformation modes, 8-node Quadrilateral Element, and 9-Node Quadrilateral Element was presented The shape functions and the element equations related to each element were presented through a detailed mathematical formulation Additionally, the Jacobian matrix for the second order derivatives was simplified and used to derive each element’s strain-displacement matrix in bending The elements were compared using carefully selected elastic and aero-elastic bench mark problems, regarding the number of elements needed to reach convergence, the resulting accuracy, and the needed computation time The best suitable element for elastic free vibration analysis was found to be the Linear Quadrilateral Element with deformation-based shape functions, whereas the most suitable element for stress analysis was the 8Node Quadrilateral Element, and the most suitable element for aero-elastic analysis was the 9-Node Quadrilateral Element Although the linear triangular element was the last choice for modal and stress analyses, it establishes more accurate results in aero-elastic analyses, however, with much longer computation time Additionally, the nine-node quadrilateral element was found to be the best choice for laminated composite plates analysis Ó 2017 Production and hosting by Elsevier B.V on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer review under responsibility of Cairo University ⇑ Corresponding author E-mail address: abdu_aerospace@eng.cu.edu.eg (M Mahran) http://dx.doi.org/10.1016/j.jare.2017.06.009 2090-1232/Ó 2017 Production and hosting by Elsevier B.V on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) 636 M Mahran et al / Journal of Advanced Research (2017) 635–648 Nomenclature Symbol A As Asd Avlm B d D J K M N w W coefficient matrix for in-plane action steady aerodynamic coefficient matrix in structural coordinates unsteady aerodynamic coefficient matrix in structural coordinates steady aerodynamic coefficient matrix strain-displacement matrix displacement in global coordinates stress-strain matrix (isotropic material properties matrix) Jacobian Matrix for first order derivatives stiffness matrix mass matrix shape function matrix structural bending displacement field structural bending nodal displacements Introduction Numerical methods are usually the first choice for many researchers and engineers to analyze complicated systems because of their accessibility, flexibility and ability to solve complex systems The Finite Element Method (FEM) as one of the powerful numerical methods for structural analysis comes at the top of the list of all numerical methods As introduced in many Refs [1–6], the method is mainly based on dividing the whole structure into a finite number of elements connected at nodes The properties of the whole structure such as mass and stiffness, which are continuous in nature, are discretized over the elements and approximate solutions are obtained for the governing equations The elements equations are assembled together to reach a global system of algebraic equations, which can be solved for the unknown solution variables of the structure The accuracy of the FEM solution depends on many factors, such as the interpolation polynomials and subsequently the element shape functions, the number of degrees of freedoms selected for each element, the mesh size, and the type of element used The model accuracy is a result of the deep understanding of the effect of each factor on the final results The selection of the element interpolation functions is a key factor in the accuracy of the FEM solution For this reason, intensive researches have been made to develop new finite elements having different shapes and interpolation functions There are numerous types of elements for different structural problems In this paper, the main focus is on two-dimensional shell elements Finite shell elements such as triangular elements [7–9], quadrilateral elements [10,11], higher order elements [12–17], and improved elements [18] are all tested and approved to achieve an acceptable level of accuracy Although a vast number of elements are available in literature, researchers cannot easily figure out which element is the most suitable to select for their particular problem The selection problem is even more difficult for engineers who are mainly interested in the application rather than the theoretical background Additionally, the detailed mathematical formulation of some thin shell bending elements, especially the higher order ones, cannot be easily found in the literature Considering aero-elasticity in which the structural model is coupled to an aerodynamic model adds more complications to the problem, and makes the choice of the suitable element more challenging Aero-elasticity is crucial for structures such as aircraft, wind turbines, and several other applications in which divergence and flutter phenomena may occur leading to catastrophic failures x, y, z d X, Y, Z V E V inf q1 r br JJ k t x e n, g q element local coordinates displacement vector in local coordinates structural global coordinates volume elasticity modulus of the wing material flow speed dynamic pressure stress reference length (half the wing root chord) Jacobian matrix for second order derivatives reduced frequency plate wing thickness wing damping ratio flutter frequency strain vector reference element coordinates air density of the structure Therefore, designers of these structures are constrained by the design limits and definitely need accurate FEM without being computationally expensive Therefore, the aim of the present work is to present a detailed mathematical formulation for different thin shell finite elements along with a complete comparison between them for specific problems in structures and aero-elasticity The results of the selected elements are compared based on (1) solution accuracy of each element, (2) number of elements needed to achieve convergence, and (3) computational time The comparison is for free vibration analysis, stress analysis, aero-elastic analysis, and laminated composite analysis Five different elements are selected for the present comparison with different nature These finite elements are (1) Three-node linear triangular element [1] denoted as LINTRI (2) Four-node linear quadrilateral element [1] denoted as LINQUAD (3) Four-node linear quadrilateral element based on deformation modes (MKQ12 [18]) (4) Eight-node quadrilateral element denoted as QUAD8NOD (5) Nine-node quadrilateral element denoted as QUAD9NOD These elements are selected with different nature ranging from linear to higher order, triangular to quadrilateral, and improved to regular elements to provide wide range of variety to the present comparison All these elements are tested using bench mark problems from the literature [19,20] for elastic and aero-elastic analyses with analytical results and/or experimental measurements The element shape functions are derived using MATHEMATICA [21] software and then implemented into MATLAB [22] codes to solve the selected problems The finite elements’ formulation The present finite element model is based on either the classical plate theory for metallic materials or laminated plate theory for composite materials Both are based on the Kirchhoff assumptions which neglect the transverse shear and transverse normal effects [2] To formulate a finite shell element there is a standard procedure that is usually followed (1) Start from the weak (integral) form of the governing equation 637 M Mahran et al / Journal of Advanced Research (2017) 635–648 (2) Assume suitable interpolation polynomials for both the inplane Pp and bending Pb displacement fields (3) Calculate the coefficients of these polynomials by applying the nodal movement conditions (4) Determine the shape functions for both the in-plane Np and bending Nb actions (5) Derive the strain-displacement matrix B from the shape functions’ derivatives (6) Integrate to obtain the element stiffness matrix K knowing the material elasticity matrix and the strain displacement relationships (7) Calculate the element mass matrix Me from the element shape functions and the material density q, and finally, (8) The structural matrices K and M can be obtained by assembling the element matrices obtained in steps and All these steps were followed for each element considered in the current study to derive the element shape functions, straindisplacement relationships, stiffness, and mass matrices for both the in-plane and bending actions The element shape functions, presented in this section, are derived using MATHEMATICA software All the strain-displacement matrices, stiffness, and mass matrices are numerically integrated using MATLAB software General formulation In the present section, general formulation of the element shape functions and strain displacement matrices is developed Based on this formulation all the shell elements’ shape functions and subsequently the elements’ equations are derived In-plane action u ¼ Pp a ð1Þ where u is the in-plane displacement field at any point through the element and a is a vector of constants to be determined from the nodal in-plane displacements U U ẳ Aa 2ị u1 > > > > > > > > > > v > > > > < = U¼ > > > > > > > > unnod > > > > > > : ; ð3Þ v nnod nnod represents the total number of nodes in an element The size of U equals the total element in-plane degrees of freedoms Finally, the in-plane shape functions can be obtained, N p ẳ Pp A 4ị The in-plane strain displacement matrix can be obtained from the shape functions’ derivatives by using the Jacobian matrix definition 9 ex > Np;x > = > < u;x > = > < = ey ¼ u;y Np;y ¼ Bp ¼ > > > :c > ; > : ; > : ; u;y ỵ v ;x Np;y ỵ Np;x xy ¼ J1 J3 J2 J4 ! J4 ÀJ J det J4 À J3 ÀJ J1 J1 À J2 & ' Np;n U Np;g ð5Þ where J ; J2 ; J3 ; J are the elements of the Jacobian matrix and J det is the Jacobian determinant ¼ x;n y;n x; g y ;g # ; J det ¼ J J À J J ð6Þ Bending action An interpolation function is chosen either from Pascal’s Triangle or based on the displacements modes, w ¼ Pb a ð7Þ where w is the bending displacement at any point on the element, from which we can obtain the rotation around the x-axis (hx) and the y-axis (hy) using the Jacobian matrix hx ¼ dw dw ; and hy ¼ À dy dx ð8Þ Note that the Jacobian matrix elements are rearranged in Jà so that the displacement rotations are defined as, & hx hy ' ¼ ÀJ J1 J det ÀJ J2 !& w;n ' w;g ẳ J1 & w;n ' 9ị w;g Then, the three bending displacements can be calculated from the equation, 9 > ! > Pb > ! = = < Pb;n a ¼ hx ¼ À1 À1 c a > > Jà > Jà : > : ; ; Pb;g hy ð10Þ a is the coefficients vector to be determined from the out of plane nodal displacements W The bending shape functions will have the form Nb ¼ Pb CÀ1 An interpolation function is chosen from Pascal’s Triangle [1,2], > < Jẳ " ! 11ị J where the C matrix is calculated from the c matrix after applying the nodal boundary conditions, and w1 > > > > > > > > > hx1 > > > > > > > > > > h > > y1 > w > > > > < = < = hx ¼ Nb ¼ Nb W > > > : > ; > > > > hy > > > > wnnod > > > > > > > > > h > xnnod > > > > : ; hynnod ð12Þ Then the strain-displacement matrix can be derived and simplified from the Jacobian definition for second order derivatives They were derived and simplified by the authors to have the form 8 9 > > < wxx > < wnn > = = wyy Bb ¼ ¼ JJÀ1 wgg > > > > : : ; ; 2wng 2wxy ð13Þ where JJ is the Jacobian Matrix for the second order derivatives, which can be calculated using the elements of the regular Jacobian matrix to have the form, 2 J1 > < wnn > = wgg ¼ J 23 > > : ; 2wng 2J J J 22 J 24 2J J 9 38 > > < wxx > < wxx > = = ¼ JJ: wyy wyy J3 J4 > > > > : : ; ; 2wxy 2wxy J2 J3 ỵ J1 J4 J1 J2 14ị The inverse of the Jacobian Matrix for the second order derivatives is 638 M Mahran et al / Journal of Advanced Research (2017) 635–648 À1 JJ ¼ J det J 24 J 22 J 23 J 21 ÀJ J À2J J 2J J J2 J3 ỵ J1 J4 J J Pp ¼ f 1; n; g; gn g and subsequently the in-plane shape functions have the form Based on this simple and detailed mathematical implementation, the considered elements’ equations can be derived All the shape functions for those elements are presented in the following sections Notice that Pb and Pp are represented as row vectors all over the present paper The LINTRI thin-shell element has three nodes The element has six degrees of freedom per node with a total of 18 degrees of freedom Fig 1a shows a schematic of the element with the element global, local, and reference coordinates The element interpolation and shape functions are derived in the following For in-plane action The interpolation polynomial for in-plane action has the form ð16Þ and subsequently the in-plane shape functions have the form Np ¼ f À g g À n n g ð17Þ For bending action The interpolation polynomial for bending action based on the element area coordinates has the form ( Pb ¼ n; g; À g À n; gn; g1 ỵ g ỵ nị; n1 ỵ g ỵ nị; gn2 ; ) g2 ỵ g ỵ nị; n1 þ g þ nÞ2 ð18Þ and subsequently the bending shape functions are N b1 ẳ ỵ g ỵ nị2g2 þ gðÀ1 þ 2nÞ þ ðÀ1 þ nÞð1 þ 2nÞÞ N b2 ẳ ỵ g ỵ nịJ þ gÞg þ J nðÀ1 þ g þ nÞÞ N b3 ẳ ỵ g ỵ nịJ þ gÞg þ J nðÀ1 þ g þ nÞÞ N b4 ẳ g2g2 ỵ 21 ỵ nịn ỵ g3 ỵ 2nịị N b5 ẳ gJ ỵ nịn þ J ðg2 þ gðÀ1 þ nÞ þ ðÀ1 ỵ nịnịị Np ẳ f gị1 nị gị1 ỵ nị ỵ gị1 ỵ nị ỵ gị1 nị g 21ị For bending action The interpolation basis functions for bending action selected from Pascal’s Triangle has the form The linear triangular element (LINTRI) Pp ẳ f 1; n; g g 20ị 15ị 19ị Pb ẳ f1; n; g; n2 ; gn; g2 ; n3 ; gn2 ; g2 n; g3 ; gn3 ; g3 ng 22ị Nb1 ẳ ỵ gị1 ỵ nị2 ỵ g ỵ g2 ỵ n ỵ n2 ị Nb2 ẳ ỵ gị1 þ nÞðJ ðÀ1 þ g2 Þ þ J ỵ n2 ịị Nb3 ẳ ỵ gị1 ỵ nịJ ỵ g2 ị ỵ J 1 ỵ n2 ịị Nb4 ẳ þ gÞð1 þ nÞðg þ g2 þ ðÀ2 þ nÞð1 ỵ nịị Nb5 ẳ ỵ gị1 ỵ nịJ ỵ J ỵ g2 ị J n2 ị Nb6 ẳ ỵ gị1 ỵ nịJ J g2 ỵ J 1 ỵ n2 ịị 23ị and subsequently the bending shape functions are Nb7 ẳ ỵ gị1 ỵ nị1 ỵ gịg ỵ ỵ nị1 ỵ nịị Nb8 ẳ ỵ gị1 ỵ nịJ ỵ g2 ị ỵ J ỵ n2 ịị Nb9 ẳ ỵ gị1 þ nÞðJ ðÀ1 þ g2 Þ þ J 1 ỵ n2 ịị Nb10 ẳ ỵ gị1 ỵ nị2 ỵ ỵ gịg ỵ n ỵ n2 ị Nb11 ẳ ỵ gị1 ỵ nịJ J g2 ỵ J ỵ n2 ịị Nb12 ẳ ỵ gị1 þ nÞðJ þ J ðÀ1 þ g2 Þ J n2 ị 24ị N b6 ẳ gJ 1 ỵ nịn ỵ J g2 ỵ g1 ỵ nị ỵ ỵ nịnịị N b7 ẳ n3n ỵ 2g2 ỵ g1 ỵ nị ỵ n2 ịị N b8 ẳ nJ gn ỵ J g2 þ gðÀ1 þ nÞ þ ðÀ1 þ nÞnÞÞ N b9 ẳ nJ gn ỵ J g2 ỵ g1 þ nÞ þ ðÀ1 þ nÞnÞÞ The linear quadrilateral element (LINQUAD) The LINQUAD element consists of four nodes It has six degrees of freedom per node with a total of 24 degrees of freedom Fig 1b shows a schematic of the element with the global, local, and reference coordinates The element interpolation and shape functions are derived to be as follows For in-plane action The interpolation polynomial for in-plane action selected from Pascal’s Triangle has the form The linear quadrilateral element based on deformation modes (MKQ12) The MKQ12 element has four nodes It has six degrees of freedom per node with a total of 24 degrees of freedom It has the global, local, and reference coordinates shown in Fig 1b This element was introduced by Karkon and Rezaiee-Pajand [18] It has the same in-plane shape functions of the LINQUAD element but with improved bending shape functions based on the deformation modes 2 2 > < 1;n; g; gn;0:51 ỵ n ị; 0:51 ỵ g ị; 0:5n1 þ n Þ; 0:5gðÀ1 þ g Þ; > = 2 Pb ẳ 0:251 ỵ g2 ịn3 n ị; 0:25g3 g2 ị1 ỵ n ị; > > : ; 0:25g1 ỵ g2 ịn3 n2 ị;0:25g3 g2 ịn1 ỵ n2 ị 25ị The shape functions are then 639 M Mahran et al / Journal of Advanced Research (2017) 635–648 Fig Finite elements local and reference coordinates N b1 ẳ ỵ gị1 ỵ nị2 g g2 n ỵ gn ỵ g2 n n2 ỵ gn2 ỵ g2 n2 ị N b2 ẳ ỵ gị2 ỵ nị2 2J1 ỵ 2J ỵ J g ỵ 2J g ỵ 2J1 n ỵ J3 n ỵ J gn ỵ J3 gnị 16 N b3 ẳ ỵ gị2 ỵ nị2 2J2 ỵ 2J þ J g þ 2J g þ 2J2 n ỵ J4 n ỵ J gn ỵ J4 gnị 16 N b4 ẳ ỵ gị1 ỵ nị2 ỵ g ỵ g2 n ỵ gn ỵ g2 n ỵ n2 gn2 g2 n2 ị N b5 ẳ ỵ gị2 ỵ nị2 2J1 ỵ 2J J g ỵ 2J g ỵ 2J1 n J3 n ỵ J gn À J3 gnÞ 16 N b6 ẳ ỵ gị2 ỵ nị2 2J2 ỵ 2J J g ỵ 2J g ỵ 2J2 n J4 n ỵ J gn J4 gnị 16 N b7 ẳ ỵ gị1 þ nÞð2 þ g À g2 þ n þ gn g2 n n2 gn2 ỵ g2 n2 ị N b8 ẳ ỵ gị2 þ nÞ2 ðÀ2J1 À 2J3 þ J1 g þ 2J g ỵ 2J n ỵ J3 n J1 gn À J gnÞ 16 N b9 ẳ ỵ gị2 ỵ nị2 2J2 2J4 þ J2 g þ 2J g þ 2J n ỵ J4 n J2 gn J gnị 16 N b10 ẳ ỵ gị1 ỵ nị2 g ỵ g2 ỵ n ỵ gn g2 n ỵ n2 ỵ gn2 g2 n2 ị N b11 ẳ ỵ gị2 ỵ nị2 2J 2J J1 g ỵ 2J3 g ỵ 2J1 n J n J1 gn ỵ J3 gnị 16 N b12 ẳ ỵ gị2 ỵ nị2 2J 2J J2 g ỵ 2J4 g ỵ 2J2 n J n J2 gn ỵ J4 gnị 16 ð26Þ The eight-node quadrilateral element (QUAD8NOD) The QUAD8NOD element has eight nodes It has six degrees of freedom per node with a total of 48 degrees of freedom Fig 1c shows a schematic of the element with the global, local, and reference coordinates The element interpolation and shape functions were derived in the following Pp ¼ 1; n; g; gn; n2 ; g2 ; n2 g; g2 n É ð27Þ and subsequently the in-plane shape functions have the form 1 Np1 ẳ gị1 nị1 g nị; Np2 ẳ gị1 ỵ nị1 g ỵ nị 4 1 Np3 ẳ ỵ gị1 ỵ nị1 ỵ g ỵ nị; Np4 ẳ ỵ gị1 nị1 ỵ g nị 4 1 Np5 ẳ gị1 nị1 ỵ nị; Np6 ẳ gị1 ỵ gị1 ỵ nị 2 1 Np7 ẳ ỵ gị1 nị1 ỵ nị; Np8 ẳ gị1 þ gÞð1 À nÞ 2 ð28Þ For bending action The interpolation polynomial for bending action was selected carefully from the well-known Pascal Triangle Initially, the following basis functions were selected; ( Pb ¼ 1; n; g; n2 ; gn; g2 ; n3 ; gn2 ; g2 n; g3 ; n4 ; gn3 ; g3 n; g2 n2 ) ð29Þ g4 ; n5 ; gn4 ; g2 n3 ; g3 n2 ; g4 n; g5 ; g2 n4 ; g3 n3 ; g4 n2 Using the above basis functions yields a singular C matrix The rank of the matrix turns out to be 22 instead of 24, which indicates that two terms result in repeated equations Different terms have been replaced with higher order terms to detect the reason for the singularity The analysis revealed that the bilinear term gn, the biquadratic term g2 n2 , and the bicubic term g3 n3 all yield similar equations Therefore, the biquadratic and bicubic terms were replaced with gn5 and g5 n Finally, the bending interpolation function for the QUAD8NOD element is; For in-plane action The interpolation polynomial for in-plane action selected from Pascal’s Triangle has the form È ( Pb ¼ 1; n; g; n2 ; gn; g2 ; n3 ; gn2 ; g2 n; g3 ; n4 ; gn3 ; g3 n; g4 ; n5 ; gn4 ; g2 n3 ; g3 n2 ; g4 n; g5 ; gn5 ; g2 n4 ; g4 n2 ; g5 n ) ð30Þ 640 M Mahran et al / Journal of Advanced Research (2017) 635–648 This choice eliminates all the singularities, and subsequently the bending shape functions are bal, local, and reference coordinates shown in Fig 1d The element interpolation and shape functions were derived in the following N b1 ¼ À ỵ gị1 ỵ nịg3 ỵ 3g4 g1 ỵ nị2 g2 ỵ nị ỵ ỵ nị1 þ nÞ2 ðÀ2 þ 3nÞÞ ! J ỵ g2 ị1 ỵ nị3g3 ỵ g2 2nị ỵ 3g1 ỵ nị ỵ ỵ nị1 ỵ nị2 ị N b2 ẳ 24 ỵJ ỵ gị1 ỵ n2 ị1 ỵ g2 ỵ g3 ỵ g1 þ ð3 À 2nÞnÞ þ nð3 þ n À 3n2 ịị N b3 ẳ J ỵ gị1 þ 3g3 þ n À 3gð1 þ nÞ À n2 ỵ nị ỵ g2 ỵ 2nịị ỵ gị1 ỵ nị 24 J 1 ỵ nị1 ỵ g2 ỵ g3 ỵ g1 ỵ 2nịnị ỵ n3 ỵ n 3n2 ịị ! 1 ỵ gị1 ỵ nịg3 ỵ 3g4 ỵ g2 ỵ nị g1 ỵ nị2 ỵ ỵ nị2 ỵ nị2 ỵ 3nịị ! J ỵ gị3g3 ỵ 3g1 ỵ nị ỵ ỵ nị2 þ nÞ À g2 ð1 þ 2nÞÞ ðÀ1 þ gÞð1 ỵ nị ẳ 24 J ỵ nịg2 ỵ g3 g1 ỵ nị1 ỵ 2nị ỵ ỵ nị1 ỵ nị1 ỵ 3nịị N b4 ẳ N b5 N b6 J ỵ gị3g3 ỵ 3g1 þ nÞ þ ðÀ1 þ nÞ2 ð1 þ nÞ À g2 ỵ 2nịị ẳ ỵ gị1 ỵ nị 24 J 1 ỵ nịg2 ỵ g3 g1 ỵ nị1 ỵ 2nị ỵ ỵ nị1 ỵ nị1 ỵ 3nịị N b7 ẳ ỵ gị1 ỵ nịg3 ỵ 3g4 ỵ g2 ỵ nị þ gðÀ1 þ nÞ2 þ ðÀ1 þ nÞ2 ð1 þ nị2 ỵ 3nịị ! J ỵ gị1 ỵ g3 ỵ g ỵ 3g2 ị ỵ n þ gð3 þ 2gÞn þ n2 À n3 Þ ð1 ỵ gị1 ỵ nị N b8 ẳ 24 ỵJ ỵ nịg2 g3 ỵ g1 ỵ nị1 ỵ 2nị ỵ ỵ nị1 ỵ nị1 ỵ 3nịị N b9 ẳ J ỵ gị1 ỵ g3 ỵ g ỵ 3g2 ị ỵ n ỵ g3 þ 2gÞn þ n2 À n3 Þ ð1 þ gÞð1 þ nÞ 24 þJ ðÀ1 þ nÞðg2 À g3 þ gð1 þ nÞð1 þ 2nÞ þ ðÀ1 þ nÞð1 þ nÞð1 þ 3nÞÞ ! ð31Þ ! ð1 þ gị1 ỵ nị1 ỵ gị2 ỵ gị2 ỵ 3gị ỵ gị2 n ỵ ỵ gịn2 ỵ n3 ỵ 3n4 ị ! J ỵ gị1 ỵ n2 ị1 ỵ gị2 ỵ gị ỵ 31 ỵ gịn ỵ 2gịn2 ỵ 3n3 ị ẳ 24 J ỵ g2 ị1 ỵ nị3g3 ỵ g2 2nị 3g1 ỵ nị þ ðÀ1 þ nÞð1 þ nÞ2 Þ ! J 1 ỵ gị1 ỵ n2 ị1 ỵ gị ỵ gị ỵ 31 ỵ gịn ỵ 2gịn2 ỵ 3n3 ị ẳ 24 ỵJ ỵ g2 ị1 ỵ nị3g3 ỵ g2 2nị 3g1 ỵ nị ỵ ỵ nị1 ỵ nị2 ị N b10 ¼ N b11 N b12 N b13 ẳ ỵ gị1 ỵ n2 ị2 ỵ g þ g2 þ 2n2 Þ ðÀ1 þ gÞðÀ1 þ n2 ÞðJ ð1 þ gÞð1 þ gðÀ3 þ 2gị n2 ị 6J n1 ỵ n2 ÞÞ N b14 ¼ 12 N b15 ¼ À ỵ gị1 ỵ n2 ịJ ỵ gị1 þ gðÀ3 þ 2gÞ À n2 Þ À 6J n1 ỵ n2 ịị 12 Nb16 ẳ ỵ g2 ị1 ỵ nị2g2 ỵ ỵ nị1 ỵ nịị Nb17 ẳ ỵ g2 ị1 ỵ nị6J g1 ỵ g2 ị ỵ J ỵ nịg2 ỵ nị1 ỵ 2nịịị 12 Nb18 ẳ ỵ g2 ị1 ỵ nị6J g1 þ g2 Þ þ J ðÀ1 þ nÞðg2 À ỵ nị1 ỵ 2nịịị 12 Nb19 ẳ þ gÞðÀ1 þ n2 ÞðÀ2 À g þ g2 þ 2n2 ị Nb20 ẳ ỵ gị1 þ n2 ÞðJ4 ðÀ1 þ gÞð1 þ gð3 þ 2gÞ n2 ị 6J n1 ỵ n2 ịị 12 N b21 ẳ ỵ gị1 ỵ n2 ịJ3 ỵ gị1 ỵ g3 ỵ 2gị n2 ị 6J1 n1 ỵ n2 ịị 12 Nb22 ẳ ỵ g2 ị1 ỵ nị2 ỵ 2g2 ỵ n ỵ n2 ị Nb23 ẳ ỵ g2 ị1 ỵ nị6J4 g1 ỵ g2 ị þ J ð1 þ nÞðÀ1 þ g2 þ ð3 2nịnịị 12 N b24 ẳ ỵ g2 ị1 ỵ nị6J3 g1 ỵ g2 ị ỵ J11 ỵ nị1 ỵ g2 ỵ 2nịnịị 12 32ị For in-plane action The interpolation polynomial for in-plane action selected from Pascal’s Triangle has the form Pp ¼ È 1; n; g; gn; n2 ; g2 ; n2 g; g2 n; n2 g2 É ð33Þ and subsequently the in-plane shape functions have the form 1 g ỵ g2 ịn ỵ n2 ị; Np2 ẳ g ỵ g2 ịn ỵ n2 ị 4 1 2 ẳ g ỵ g ịn ỵ n ị; N p4 ẳ g ỵ g2 ịn ỵ n2 ị 4 1 2 ẳ g ỵ g ị1 n ị; Np6 ẳ g2 ịn ỵ n2 ị 2 1 2 ẳ g ỵ g ị1 n ị; Np8 ẳ g2 ịn ỵ n2 Þ 2 ¼ ð1 À g2 Þð1 À n2 Þ Np1 ¼ Np3 Np5 Np7 Np9 ð34Þ For bending action The nine-node quadrilateral element (QUAD9NOD) The QUAD9NOD element has nine nodes It has six degrees of freedom per node with a total of 54 degrees of freedom It has the glo- The interpolation polynomial for bending action was selected carefully from the well-known Pascal Triangle Initially, the following basis functions were selected; M Mahran et al / Journal of Advanced Research (2017) 635–648 ( Pb ¼ 1; n; g; n2 ; gn; g2 ; n3 ; gn2 ; g2 n; g3 ; n4 ; gn3 ; g3 n; g4 ; g2 n2 ; n5 3 4 ) ; gn ; g n ; g n ; g n; g ; gn ; g n ; g n ; g n; g n ; g n ; 5 ð35Þ Using the above basis functions yielded singular C matrix The rank of the matrix turned out to be 23 instead of 24, which indicated that two terms result in repeated equations Different terms have been replaced with higher order terms to detect the origin of the singularity Finally, the bending interpolation function for the QUAD9NOD element is; ( Pb ¼ 1; n; g; gn; n2 ; g2 ; n3 ; gn2 ; g2 n; g3 ; n4 ; gn3 ; g3 n; g4 ; n5 ; gn4 ; 641 Nb23 ẳ ỵ g2 ị1 ỵ nịn2J g1 ỵ g2 ị ỵ J g ỵ ỵ gịn ỵ n3 ịị Nb24 ẳ ỵ g2 ị1 ỵ nịn2J3 g1 þ g2 Þ þ J ðg þ ðÀ1 þ gịn ỵ n3 ịị Nb25 ẳ ỵ g2 ị1 ỵ n2 ị1 ỵ g2 3gn ỵ n2 ị Nb26 ẳ ỵ gị1 ỵ gị1 ỵ nị1 þ nÞðJ gðÀ1 þ g2 Þ þ J2 nðÀ1 ỵ n2 ịị Nb27 ẳ ỵ gị1 ỵ gị1 þ nÞð1 þ nÞðJ3 gðÀ1 þ g2 Þ þ J n1 ỵ n2 ịị 38ị ) The test problems g4 n; g5 ; gn5 ; g2 n4 ; g4 n2 ; g5 n; g2 n5 ; g5 n2 ; 12 g2 n2 ỵ g3 n3 ị 36ị and subsequently the bending shape functions are It has been mentioned earlier that the aim of the present work was to compare between different shell finite elements with different behavior for elastic and aero-elastic analyses This will enable 1 ỵ gịg1 þ nÞnð4 þ g2 þ 3g3 þ gð2 þ 6nÞ þ nð2 þ n þ 3n2 ÞÞ Nb2 ẳ ỵ gịg1 ỵ nịnJ ỵ g3 þ n þ gnÞ þ J ð1 þ g ỵ gn ỵ n3 ịị Nb3 ẳ ỵ gịg1 ỵ nịnJ ỵ g3 ỵ n ỵ gnị ỵ J 1 ỵ g ỵ gn ỵ n3 ịị Nb4 ẳ ỵ gịgn1 ỵ nị8 ỵ g2 ỵ gị5 ỵ 3gị ỵ 10n ỵ 6gn ỵ n2 3n3 ị Nb5 ẳ ỵ gịgn1 ỵ nịJ ỵ g3 ỵ g2 ỵ nị ỵ nị ỵ J ỵ g ỵ gịn ỵ n3 ịị Nb6 ẳ ỵ gịgn1 ỵ nịJ ỵ g3 ỵ g2 ỵ nị ỵ nị þ J ð1 þ g À ð2 þ gÞn ỵ n3 ịị Nb7 ẳ g1 ỵ gịn1 ỵ nịg2 ỵ 3g3 ỵ g2 6nị ỵ ỵ nị4 ỵ n2 ỵ 3nịịị Nb8 ẳ g1 ỵ gịn1 ỵ nịJ ỵ g3 þ n À gnÞ þ J ðÀ1 þ g gn ỵ n3 ịị Nb9 ẳ g1 ỵ gịn1 ỵ nịJ ỵ g3 ỵ n gnị ỵ J 1 ỵ g gn ỵ n3 ịị Nb10 ẳ g1 þ gÞðÀ1 þ nÞnð8 þ ðÀ2 þ gÞgð5 þ 3gÞ þ 10n À 6gn À n2 À 3n3 Þ Nb11 ẳ g1 ỵ gị1 ỵ nịnJ þ g þ ðÀ2 þ gÞn þ n3 Þ þ J ỵ g3 ỵ n g2 ỵ nịịị Nb12 ẳ g1 ỵ gị1 ỵ nịnJ 1 ỵ g ỵ ỵ gịn ỵ n3 ị ỵ J ỵ g3 ỵ n g2 ỵ nịịị Nb13 ẳ þ gÞgðgðÀ4 þ g þ 3g2 Þ þ 6ð1 þ gịn 2n2 ị1 ỵ n2 ị Nb14 ẳ ỵ gịg1 ỵ n2 ịJ g3 þ gðÀ1 þ nÞ þ nÞ À 2J nðÀ1 ỵ n2 ịị Nb15 ẳ ỵ gịg1 þ n2 ÞðJ ðg3 þ gðÀ1 þ nÞ þ nị 2J n1 ỵ n2 ịị Nb16 ẳ ỵ g2 ịn1 ỵ nị2g2 6g1 þ nÞ þ nð1 þ nÞðÀ4 þ 3nÞÞ Nb1 ẳ Nb17 ẳ ỵ g2 ịn1 ỵ nị2J g1 ỵ g2 ị ỵ J g1 ỵ nị ỵ n n3 ịị Nb18 ẳ ỵ g2 ịn1 ỵ nị2J g1 ỵ g2 ị ỵ J1 g1 ỵ nị ỵ n n3 ịị Nb19 ẳ g1 ỵ gị1 ỵ n2 ịg1 ỵ gị4 ỵ 3gị 61 ỵ gịn ỵ 2n2 ị Nb20 ẳ g1 ỵ gị1 ỵ n2 ịJ ỵ gịg ỵ g2 nị 2J n1 ỵ n2 ịị Nb21 ẳ g1 ỵ gị1 ỵ n2 ịJ ỵ gịg ỵ g2 nị 2J n1 ỵ n2 ịị Nb22 ẳ ỵ g2 ị1 ỵ nịn2g2 6g1 ỵ nị n4 þ n þ 3n2 ÞÞ ð37Þ any researcher to select the shell element which best suits his/ her specific application Different test benchmark problems are considered for elastic and aero-elastic analyses These problems are described in detail in this section in addition to their mathematical models These models are implemented in the next section into MATLAB codes, which are carefully constructed and validated For each problem, a suitable number of elements was selected based on convergence analysis The number of elements was increased till the response converged to a certain value Then the results were compared with published experimental or analytical solutions 642 M Mahran et al / Journal of Advanced Research (2017) 635–648 Study of the natural frequencies of free vibration of an elastic square plate Problem formulation This problem was presented by Safizadeh et al [20] in which an analytical solution was provided It deals with a square plate (1 m  m) with thickness t equal 0.003 m and the material properties, elastic modulus, mass density, and Poisson’s ratio E ¼ 71 Gpa; q ¼ 2700 kg=m3 ; and The mathematical model m ¼ 0:3 The plate is fixed along all sides The mathematical model The plate natural frequencies are calculated by solving the eigenvalue problem Kb x Mb ịW ẳ 39ị where the subscript b refers to the bending action and the mass and stiffness matrices are calculated from the strain displacement matrix and the shape functions Z Keb ¼ Meb ¼ Z V V t BTb Ds Bb dV ¼ NTb Nb dV 0 t 12 ZZ First the stiffness matrix for both the in-plane and bending actions are derived The stiffness matrix of the bending action Keb is given in Eq (40) and the in-plane stiffness matrix Kep has the form Z Kep ¼ BTb Ds Bb J det dndg NTb INb J det dnd g V ZZ BTp Ds Bp dV ¼ t BTp Ds Bp J det dndg > > < rx = ry ¼ Ds ðBp Á dp À zBb Á db Þ > sxy ; ð40Þ t3 12 07 Wing aero-elastic analysis 0 t3 12 Problem formulation The superscript e means that these matrices are calculated over each element and then assembled in the global coordinates Ds is the isotropic material stiffness matrix Ds ¼ m E 4m 1 À m2 0 0 ð43Þ where dp is the local in-plane displacements vector and db is the local bending displacements vector I ẳ 40 42ị The elastic problem was solved to find the displacements, then the stresses are calculated using the equation > : ZZ ¼q formed to select the suitable number of elements for both the aerodynamic and finite element analyses The material properties are Young’s Modulus = 98E9 Pa, Poison’s ratio = 0.28, plate thickness = 0.001 m The flow properties are Speed = 30 m/s, density = 1.225 kg/m3, AOA = 3° AOA is the flow angle of attack ð41Þ 1Àm Stress and deformation analysis of metallic plate wing The problem formulation The elastic stress and displacement analysis performance of the considered elements was tested by analyzing a plate-like straightrectangular wing under aerodynamic load The wing geometry was shown in Fig The aerodynamic analysis was performed by using the Doublet lattice method [23] A convergence analysis was per- The finite element selection in aero-elastic analysis is always a problem The point is to select the suitable element for accurate aero-elastic calculations and load transformation between the aerodynamic model and the structural model and vice versa For this purpose, a comparison was made between different finite elements for aero-elastic analysis The results are compared with published experimental results For all the finite elements, the shape functions are used in the aero-elastic coupling, rather than the conventional spline interpolation, to make the model more accurate and consistent [24] A straight-rectangular plate wing model made of laminated composite materials was analyzed with different laminate configurations The wing has the same plane form shown in Fig The suitable number of elements was selected for both the aerodynamic and finite element analyses throughout convergence analyses The lamina material properties were EL = 98 Gpa, ET = 7.9 Gpa, GLT = 5.6 Gpa, tL = 0.28, q = 1520 kg/m3, t = 0.134eÀ3m Mathematical model Fig Plate wing plane form geometry The need to decrease the aircraft structural weight for economic purposes leads to an increase in the aircraft flexibility, and subsequently the tendency for aero-elastic instability The wing aeroelastic instability was categorized into divergence and flutter analyses [25–29] In the former, analysts were interested in determining the minimum speed at which wing static torsional instability takes place In the latter, analysts were interested in determining the minimum speed at which wing dynamic instability flutter takes place For both analyses, the doublet lattice method was used for the steady and unsteady aerodynamic analyses [23] for all elements An exception was the linear triangular element where the vortex lattice method [30] was used in the steady aerodynamic analysis because it produces more accurate results, although it M Mahran et al / Journal of Advanced Research (2017) 635–648 needs more elements and subsequently longer computation time For this reason, it was not considered for the rest of elements, as the doublet lattice method was enough and smaller computational time The problem was solved by developing two models; one for the structural analysis and the other for the aerodynamic analysis Then, the aerodynamic coefficient or stiffness matrix was transformed into the structural nodes by means of either spline interpolation or by the same shape functions of the finite element The use of the shape functions of the finite element in the connection between the finite element model and the aerodynamic model was found to be more accurate and consistent than the spline method [24] The mathematical models for both the divergence and flutter analyses are presented in this section The divergence analysis Divergence can be regarded as a static torsional instability that occurs for aircraft wings at a certain flight speed The divergence speed can be calculated by solving the eigenvalue problem Kb q1 As ịW ẳ ð44Þ where As is the aerodynamic stiffness transformed from the aerodynamic control points into the finite element nodes by using the element shape functions [31], and qdiv represents the dynamic pressure at which divergence takes place qdiv ¼ qV 2div ð45Þ where V div is the velocity at which the divergence occurs The flutter analysis Flutter can be regarded as a dynamic instability that occurs to aircraft wings at a certain flight speed The flutter speed was determined by solving the eigenvalue problem K1 Mb ỵ b qb2r 2k A sd ! ỵ i1 x2 ! I Wẳ0 46ị where br is a reference length (chosen to be half the wing root chord), Asd is the unsteady aerodynamic stiffness matrix transformed from the aerodynamic control points to the finite elements nodes by using the element shape functions [31], while k is known as the reduced frequency which is defined as k¼ br x Vf ð47Þ Eq (46) comprises two unknowns; the speed V and the frequency x, and both can be obtained by iteration where the flutter occurs at zero damping coefficient It is worth noting that Eq (46) is nested and solved using the kmethod The aerodynamic coefficient matrix As for steady aerodynamic analysis was calculated using either the Vortex Lattice Method (VLM) [30] or the Doublet Lattice Method (DLM) [23] Then, they were transformed to the structural coordinates as following ð48Þ where Asteady is the steady aerodynamic coefficient matrix at the aerodynamic control points GN1 and GN2d are transformation matrices calculated from the element bending shape functions ZZ GN1 ¼ TT NTb dxdyK T and K are geometric transformation matrices that connect between the global structural coordinates and the element local coordinates GN2d ẳ TT n X NT K 50ị iẳ1 n represents the number of aerodynamic control points in each element In flutter analysis, the unsteady aerodynamic coefficient matrix Asd was calculated using the DLM Asd ¼ GN1 Ầ1 unsteady BcGN2f ð51Þ Aunsteady is the unsteady aerodynamic coefficient matrix at the aerodynamic control points GN2f is calculated as GN2d, by considering the lateral displacement w Bc is a boundary conditions matrix calculated as Bc ¼ iÃk ; À1 br ! ð52Þ Laminated plate elastic analysis Problem formulation To present a complete picture regarding the differences between the considered finite elements, a comparison between them for the analysis of a composite laminated plate was established A square plate was considered with 25 cm side length and cm total thickness [32] The lamina material properties were EL = 52.5 MPa, ET = 2.1 MPa, GLT = 1.05 MPa, tL = 0.25 Reddy [32] presents an analytical solution for the maximum displacement of the square plate subject to a distributed pressure of N/cm2 Two laminate configurations were considered as well as two different boundary conditions The analytical results are represented in the following section The mathematical model for this problem is exactly the same as that of the elastic deformation problem, but with imposing the effect of composite material on the stress-strain relation More details can be found in Reddy [32] Results and discussion The selected finite elements are tested by the four problems described in the previous section, and the results are demonstrated in this section All the analyses have been performed on a personal computer with an i7 processor CPU @ 3.6 GHz, intel core and 16 GB RAM The results for each analysis are shown in the next subsection The dynamic elastic analysis Aerodynamic analysis and aero-elastic coupling As ẳ GN1 A1 steady GN2d 643 49ị The problem of the square plate presented in the previous section was implemented on a computer code using MATLAB software Table shows the predicted natural frequencies The first column contains the analytical solution [20] for the first five natural frequencies, while the finite element results are listed in the rest of the columns Results have shown that the natural frequencies predicted by the different types of finite elements are in general less than those predicted by the analytical model The frequency values obtained by using the Linear Triangular Element are farthest from the analytical ones and the number of elements (N_elem) needed to reach convergence is a maximum On the other hand, the frequencies obtained by using the Linear Quadrilateral Element based on deformation modes are closest to the analytical solution The 644 M Mahran et al / Journal of Advanced Research (2017) 635–648 Table The natural frequencies of clamped square plate [Hz] Mode Analytical (20) LINTRI LINQUAD MKQ12 QUAD8NO QUAD9NOD Nelem 28.47 58.06 58.06 85.66 104.09 – 26.2 53.35 53.35 78.35 95.54 800 26.54 54.06 54.06 79.09 96.91 225 26.81 54.78 54.78 81.49 98.29 100 26.68 54.39 54.39 80.3 97.42 100 26.61 54.27 54.25 80.01 97.24 100 Fig The average error and execution time of each finite element in the frequency analysis Table The product of the average error percentage and processing time for various finite elements Mode LINTRI LINQUAD MKQ12 QUAD8NO QUAD9NOD avg  Time 125.5 23.8 7.7 25.8 101 Table Max displacement and stress over the plate wing under aerodynamic load Element LINTRI LINQUAD MKQ12 QUAD8NO QUAD9NOD dmax [mm] rmax [MPa] Nelem Time [s] 19.1 28.9 192 20.4 20.3 32.9 120 5.2 20.3 33.2 120 8.1 20.4 34.4 60 19.9 34.1 60 7.9 MKQ12 performed even better than the higher order elements, which were expected to produce accurate results in bending analysis The number of elements needed to reach convergence in the linear quadrilateral element based on deformation modes is also the same as those of higher order elements The average error percentage and processing time of each element are demonstrated in Fig The product of the average error percent and processing time can be used as a measure of excellence of the finite element, where the best element has the minimum value for the product This product is given in Table 2, which shows that the linear quadrilateral element based on deformation modes is the best element to use in this type of problems Plate wing stress and displacement analysis The elastic performance of a plate wing under steady aerodynamic load was studied using the five finite elements under consideration The values of the maximum Von Mises stresses and maximum displacements are tabulated in Table and plotted in Fig together with the execution time Fig shows the distribution of the Von Mises stress in the wing for each element type The stresses are calculated in the case of triangular elements over the mid-side points and then averaged over the element In case of the quadrilateral elements, the stresses are determined at the element integration points, and then averaged over the element Since there was neither analytical nor experimental data available for this model, the error percentage cannot be computed for this particular problem However, it is clear from Fig that the stress distribution resulting from using higher order elements (QUAD8NOD and QUAD9NOD) is the smoothest and most realistic This can be attributed to the higher order interpolation functions for displacements, which render the stress distribution (derived from the displacement derivatives) continuous Hence, if all the results are considered together, the best performing element can be considered to be the QUAD8NOD element, which results in accurate displacement and stress distributions, together with a reasonable computational time Following this element comes the QUAD9NOD in the second place On the other hand, the LINTRI element comes as the worst element for wing stress analysis from the point of view of computation time and stress distribution as seen in Fig 645 M Mahran et al / Journal of Advanced Research (2017) 635–648 Tme 32.9 LINTRI LINQUAD 7.9 19.9 20.4 8.1 20.3 5.2 20.3 19.1 20.4 28.9 34.1 34.4 σ 33.2 d MKQ12 QUAD8NOD QUAD9NOD Fig Max displacement and stress of the plate wing in addition to the executing time Fig The Von Mises stresses for each element model 646 M Mahran et al / Journal of Advanced Research (2017) 635–648 Table Divergence speed of the plate wing related to different laminate configurations Laminate configuration [0 90 90 0] [45 À45 0 À45 45] [45 45 0 45 45] [À45 À45 0 À45 À45] [30 30 0 30 30] [À30 À30 0 À30 À30] Nelem FE mesh DLM mesh Divergence/Flutter Speed [m/s] Exp [19] LINTRI [31] LINQUAD MKQ12 QUAD8NOD QUAD9NOD 25F >32 28F 12.5D 27F 11.7D – – – 25.4/26.4 47.5F 27.8F 12.7/29.1 27.4F 12.8/48.1 48 3Â8  12 25.4/24.47 43.8F 26.1F 11.4/26.9 26.1F 11.58/33.7 96  16  12 25.5/25.9 46.8F 27.6F 11.5/29 27.16F 11.67/35.09 96  16  12 25.4F 45.6F 29.2F 12.86/23.5 28.1F 13/31.4 12 2Â6  12 52.6/25.3 46.6F 29F 12.88/32.4 27.8F 13/30.6 12 2Â6  12 Table The error % in each analysis and the computation time Laminate configuration [0 90 90 0] [45 À45 0 À45 45] [45 45 0 45 45] [À45 À45 0 À45 À45] [30 30 0 30 30] [À30 À30 0 À30 À30] avg % Time [s] avg  Time Error % LINTRI LINQUAD MKQ12 QUAD8NOD QUAD9NOD 5.6 – 0.7 1.6 1.5 5.6 2.3 120 276 2.1 – 6.8 8.8 3.3 2.1 5.2 20.3 106 3.7 – 1.4 7.7 0.6 3.7 3.4 22.8 77.5 1.6 – 4.4 2.9 4.2 1.6 3.3 17.6 58.1 1.3 – 3.6 3.1 3.2 1.3 2.8 18.6 52.1 choice for wing aero-elastic analysis, followed by the QUAD8NOD element Composite plate wing aero-elastic analysis The static and dynamic aero-elastic analysis of a composite plate wing was carried out using the five elements The results are listed in Table for the smaller of the divergence and Flutter speeds The wing aero-elastic analysis was performed for different laminate configurations The subscript D refers to the Divergence speed while subscript F refers to the Flutter speed The error percent, the average error, and the computation time are listed in Table and plotted in Fig Considering the minimum value of the product of the average error and execution time to be the sign of excellence, we find that the QUAD9NOD element was the best Laminated plate elastic analysis The square laminated plate was analyzed for [0, 90]o and [À45, 45]° laminate configurations Two boundary conditions were considered; in the first all the plate sides were simply supported, and in the second all the plate sides were clamped [32] The analytical results are listed in Table 6, and the average error and computation time for each element are depicted in Fig The analyses are obtained for the maximum normalized bending displacement, Fig The average error and computation time for each element 647 M Mahran et al / Journal of Advanced Research (2017) 635–648 Table Laminated elastic plate maximum normalized displacement BC’s Laminate config Analytical LINTRI LINQUAD MKQ12 QUAD8NOD QUAD9NOD Simply supported [0, 90] [À45, 45] [0, 90] [À45, 45] Nelem 1.6955 0.6773 0.3814 0.3891 – 1.719 0.6902 0.3952 0.3901 200 1.61 0.8554 0.3806 0.294 144 1.606 0.8721 0.371 0.2865 144 1.6958 0.7198 0.4096 0.4229 36 1.6996 0.6925 0.3968 0.4078 25 Fixed Fig The average error and computational time of laminated plate elastic analysis  ¼ w 100wmax ET t3tot L4 P ð53Þ where wmax is the plate maximum thickness, t tot is the total laminate thickness, L is the side length, and P is the applied pressure load The results have shown that the linear triangular and the 9-node quadrilateral elements are the best elements for laminated composite analysis from the accuracy point of view However, the linear triangular element (LINTRI) needs higher number of elements, and subsequently, longer computational time On the other hand, the worst elements for laminated composite analysis were the LINQUAD and the MKQ12 elements as they have the maximum relative average error Conclusions In the present paper, five different thin shell finite elements were considered The five elements were the Linear Triangular Element (LINTRI), the Linear Quadrilateral Element (LINQUAD), the Linear Quadrilateral Element Based on Deformation Modes (MKQ12), the 8-Node Quadrilateral Element (QUAD8NOD), and the 9-Node Quadrilateral Element (QUAD9NOD) A simple and detailed mathematical model to derive the interpolation functions and the stiffness matrix of each element was presented The basis functions were selected from the well-known Pascal Triangle to minimize the order of the interpolation functions However, singularities existed and specific terms had to be removed and replaced with other terms to eliminate the source of singularities The five elements were tested using several elastic and aero-elastic analyses through three carefully selected bench mark problems with analytical or experimental results available in the literature In order to have a fair comparison, a convergence analysis was conducted for each element and the minimum number of elements needed for convergence was used in the comparison From the present investigation, it was found that the MKQ12 element was the best choice for elastic free vibration analysis of a plate, since it yields the most accurate results with the minimum execution time The second choice is the QUAD8NOD element, and the worst results are produced by the LINTRI element In case of elastic thin shells, and if the stress analysis was sought, the most accurate elements are naturally the higher order elements From the point of view of time and accuracy, the best element was found to be the QUAD8NOD element, and the QUAD9NOD element comes out second The worst element for this kind of analysis was the LINTRI element, which requires longer computational times and produces discontinuous stress distributions For aero-elastic analysis, the most accurate results are obtained by using the LINTRI element, however, it requires the longest computational time The best element for this type of analysis was found to be the QUAD9NOD element considering its accuracy and computational time The second choice was the QUAD8NOD element It is worth noting that in spite of its bad performance in elastic analysis, the LINTRI element was found to be more consistent to use with the Vortex Lattice Method in the aero-elastic analysis, but it requires long computation time In laminated composite plate analysis, the best recommended elements are either the LINTRI or QUAD9NOD However, the LINTRI element requires dense mesh, and subsequently longer computational time The present results can serve many researchers and engineers interested in elastic and aero-elastic analyses, especially those who find difficulties in finding the detailed formulation of the finite elements, and those who are confused in selecting specific elements for specific application 648 M Mahran et al / Journal of Advanced Research 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wing was carried... The finite element selection in aero -elastic analysis is always a problem The point is to select the suitable element for accurate aero -elastic calculations and load transformation between the aerodynamic... literature [19,20] for elastic and aero -elastic analyses with analytical results and/ or experimental measurements The element shape functions are derived using MATHEMATICA [21] software and then implemented