Nonlinear dynamic analysis of shells with the triangular element TRIC

Transcription

1 Comput. Methods Appl. Mech. Engrg. 19 (00) Nonlinear dynamic analysis of shells with the triangular element TRIC John Argyris a, Manolis Papadrakakis b, *, acharias S. Mouroutis b a Institute for Computer Applications, University of Stuttgart, D-059 Stuttgart 80, Germany b Institute of Structural Analysis &Seismic Research, National Technical University Athens, ografou Campus, Athens 1580, Greece Received 9 December 00; received in revised form April 00 Abstract TRIC is a facet triangular shell element, which is based on the natural mode method. It has been shown that the TRIC shell element satisfies the individual element test and in the framework of the nonconsistent formulation the convergence requirements are fulfilled, while it has been proved to be very efficient in linear and nonlinear static problems. Moreover, another major advantage in the formulation of this element is the incorporation of the transverse shear deformations in a way that defies the shear-locking phenomenon. In this work the derivation of the consistent and lumped mass matrices of the TRIC element is presented so that it can be used in linear and nonlinear dynamic problems. Both translational and rotational inertia are included in the consistent mass matrix, which is conceived, using kinematical and geometrical arguments consistent with the assumed natural rigid body and straining modes of the element. All the kinematical and geometrical arguments that are invoked for the derivation of the consistent mass matrix are briefly presented. Moreover, two formulations of the lumped mass matrix of TRIC are derived. The first formulation is based entirely on geometrical considerations whereas the second is based on lumping the consistent mass matrix of TRIC. Finally, the elementõs robustness and accuracy will be shown by applying it to properly selected benchmark examples of nonlinear shell dynamics, while its computational efficiency will be demonstrated by comparing the CPU performance of the element with the other available shell elements. Ó 00 Elsevier B.V. All rights reserved. 1. Introduction When faced with the challenge of investigating time-dependent nonlinear phenomena with the finite element method engineers and scientists are usually confronted with a major constraint that is the high computational cost involved in the simulations. This problem is further aggravated with the use of higher order elements with many degrees of freedom. Furthermore, the extension of these types of elements to the nonlinear range, and especially, to time-dependent problems, is not straightforward. For this reason, simple * Corresponding author. Tel.: ; fax: addresses: mpapadra@central.ntua.gr (M. Papadrakakis), zachmour@central.ntua.gr (.S. Mouroutis) /0/$ - see front matter Ó 00 Elsevier B.V. All rights reserved. doi:10.101/s (0)0015-

2 00 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) plate and shell finite elements, incorporating a number of features that make them competent in the study of intricate nonlinear phenomena, both static and dynamic, are highly desirable. To this extent Argyris and co-workers [] conceived a three node multilayered triangular element based on the natural mode finite element method, a method introduced by Argyris in the 190s, that separates the pure deformational modes also called natural modes from the rigid body movements of the element. Furthermore, this element has the following characteristics that make it very attractive: Simplicity, since no complex transformations or numerical integration are required. Material generality achieved through a multilayered construction. The inclusion of transverse shear deformation is performed in a way that eliminates the shear-locking effect in a physical manner, based on a first order shear deformable beam theory. Finally, possessing only displacement and rotational degrees of freedom the connection with other element types is facilitated, while with its triangular shape geometric generality is easily realized. The element has been applied with great success to a number of linear and geometrically nonlinear bending and elastic stability problems for isotropic and composite plates and shells. The main goal of this paper is to extend the use of the TRIC (TRIangular Composite) shell element to linear and nonlinear dynamic analysis of plates and shells. It is the next step in the continuing attempt to develop an accurate and efficient, yet robust, shell finite element for the analysis of thin and moderately thick isotropic and composite shell structures. An open question in dynamic finite element analysis is whether a consistent mass matrix or a lumped mass matrix should be used for the implicit time integration algorithms. A mass matrix consistent with the underlying variational formulation, in the sense that the assumed displacement fields are the same as those used for calculating the stiffness matrix, has an undeniably logical justification. On the other hand, there are arguments of simplicity and computational efficiency, which support the use of a mass matrix which is calculated merely by the device of an equal lumping of the distributed mass of a finite element at its nodes. In this paper a consistent mass matrix is formulated in the natural coordinate system of the TRIC shell element by constructing geometrically the modal matrix that relates the nodal displacements with the natural modes of the element. This formulation is based on the same principles followed for the derivation of the consistent mass matrix of a predecessor shell element of TRIC []. Furthermore, lumped masses are also produced either by using geometrical arguments or by applying the HR lumping technique [18] on the consistent mass matrix. From the standpoint of rigor, the argument in favor of the consistent mass matrix is obviously preferred, but the popularity of the lumped mass matrix approach in applications is conceded and the efficiency of both types of mass matrices is investigated. Nevertheless, the use of the consistent mass matrix in the present work allows the rotatory inertia associated with the drilling rotation degree of freedom to be properly included in the formulation, in addition to the usual bending rotatory inertias, at minimal extra computational cost and without requiring any prejudgement of its significance relative to other inertial properties. An additional problem when dealing with nonlinear dynamics is the implementation of suitable time integration algorithms that exhibit robustness and efficiency, since algorithms which are unconditionally stable for linear dynamics often lose this stability in the nonlinear case. Consequently, several modifications of the classical integration schemes have been carried out since the beginning of the 1990s in order to avoid nonconvergence. Simo and Tarnow [9] have been the first to design an algorithm denoted as Energy Momentum Method, which guarantees unconditional stability in nonlinear dynamics of three-dimensional elastic bodies. As this algorithm requires a modified calculation of the stress tensor on the element level, a lot of research effort was directed since then to the implementation of this concept to different finite element concepts allowing for finite deformations and rotations. In order to control the numerical dissipation of high frequencies Kuhl and Ramm [] extended the Constraint Energy Method by Hughes et al. [19] and the Generalized Energy Momentum Method by Kuhl and Crisfield [] to nonlinear dynamics of shells. The incentive behind these modifications of the classical time integration schemes were the numerical damping characteristics of those schemes allowing for larger time steps (see also [10,14,1,,]).

3 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) Therefore, another goal in this paper is to investigate up to what extent the TRIC shell element can produce stable solutions in nonlinear shell dynamics by applying standard time integration methods, such as the Newmark method, with acceptable time step sizes, or modified time integration schemes such as the aforementioned would be necessary in order to produce accurate and computationally efficient results. To this extent a number of benchmark tests are performed to investigate the numerical performance of the TRIC element with consistent and lumped mass matrices in nonlinear shell dynamics.. The stiffness matrix of TRIC A brief description In this section the basic features of the TRIC shell element are briefly discussed, while all necessary relations for the derivation of the consistent mass matrix are described in greater detail. For a complete presentation of the TRIC shell element the reader is referred to a series of papers published by Argyris and co-workers [1 ] where theoretical details of the TRIC formulation as well as extensions to elastoplasticity and stochastic shell analysis are presented, while the basic features of the element are given in Appendix A..1. Kinematics of the element For the multilayered composite triangular shell element the following coordinate systems shown in Fig. 1 are adopted. The natural coordinate system, which has the three axes parallel to the sides of the triangle. The local Cartesian coordinate system, placed at the triangleõs centroid, and the global Cartesian coordinate system where global equilibrium refers to. Finally, for each ply of the triangle, a material coordinate system 1,, is defined with axis 1 being parallel to the direction of the fibers. The use of these different coordinate systems makes TRIC a suitable element in modeling multilayer anisotropic shell structures that can degenerate, as special case, to a sandwich or single-layer configuration. Fig. depicts the three total natural axial strains, which are measured parallel to the edges of the triangle and replace the Cartesian strains in the natural mode formulation. These strains c t are measured directly parallel to the triangleõs sides, while by definition straining of one side leaves all other triangular sides unstrained. Fig. 1. The multilayer triangular TRIC element; coordinate systems.

4 008 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) Fig.. Natural axial strains at the middle surface. 8 < c t ¼ : c ta c tb c tc 9 = ; : ð1þ Similarly, the total natural transverse shear strains c s are defined for each one of the triangleõs edges: 8 9 < c a = c s ¼ c : b ; : ðþ c c Fig. depicts the total natural transverse shear strain for side a of the triangle. As shown, transverse shearing of one side leaves all other side angles along the height of the triangle orthogonal. The total natural axial strains c t are related to the three in-plane local Cartesian strains c 0 according to the expression 8 9 pffiffi 8 9 < c ta = c ax 0 s ax 0 sax 0c ax 0 c t ¼ B t c 0 () c : tb ; ¼ pffiffi >< c x 0 x 0 >= c bx 0 s bx 0 sbx 0c bx 0 c 4 pffiffi 5 y 0 y c >: pffiffiffi 0 >; ; ðþ tc c cx 0 s cx 0 scx 0c cx 0 cx 0 y 0 where c ix 0 ¼ coshi; x 0 i; i ¼ a; b; c; ð4þ s ix 0 ¼ sinhi; x 0 i; i ¼ a; b; c and ha; x 0 i, hb; x 0 i, hc; x 0 i are the angles that the triangleõs edges a, b and c form with the local x 0 axis, respectively. The total transverse shear strains c s are related to the two out-of-plane transverse shear strains c 0 s via 8 9 < c a = c ax0 s ax0 c s ¼ T s c 0 s () c : b ; ¼ 4 c bx 0 s bx 0 5 c x 0 z 0 : ð5þ c c cx 0 s y 0 z 0 cx 0 c c Fig.. Total natural transverse shear strain for side a.

5 The corresponding natural stresses r c to the total natural axial strains c t are grouped in the vector 8 9 < r ca = r c ¼ r cb ðþ : ; r cc while the corresponding natural transverse shear stresses are 8 9 < r sa = r s ¼ r sb : ; : ðþ r sc The constitutive relations between the natural stresses and the total natural strains are established by initiating the following sequence of coordinate system transformations Material system! Local system! Natural system With simple geometric transformations and by contemplating the invariance of the strain energy density in the different coordinate systems, one can easily reach to an expression for the constitutive matrix in the natural coordinate system for both axial and transverse deformations r c r s r ¼ j ct v s r c t c s r or frg ¼½jŠ½eŠ ð8bþ valid for each layer r. Matrix j ct defines the constitutive matrices of axial deformation and symmetrical bending, while matrix v s corresponds to antisymmetrical bending and transverse shear modes. The stresses strains are decomposed to updated engineering stresses strains that correspond to the natural modes and those produced by the rigid body modes in the framework of a co-rotational formulation. Additional information for the derivation of the natural constitutive matrix can be found in []... Natural modes and generalized forces and moments The multilayered triangular shell element TRIC has Cartesian degrees of freedom per node. Its natural stiffness is only based on deformations and not on associated rigid-body motions. The element has 18 degrees of freedom but the actual number of straining modes is 1: 18 Cartesian d:o:f: rigid body d:o:f: ¼ 1 straining modes: The rigid body and 1 straining modes, which are illustrated in Figs. 4 and 5, are grouped in the vector q 0 e ¼ ð181þ 4 q 0 ð1þ q N ð11þ J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) ð9þ in which q 0, q N represent the rigid body and the straining modes, respectively, with corresponding entries q 0 ¼ ½q 01 q 0 q 0 q 04 q 05 q 0 Š t ; t: ð10þ q N ¼ c 0 ta c 0 tb c 0 tc w Sa w Aa w Sb w Ab w Sc w Ac w a w b w c The following subvectors are contained in (10): c 0 t ¼ c 0 ta c 0 tb c 0 t tc axial straining mode; ð11þ t w S ¼ w Sa w Sb w Sc symmetric bending mode; ð1þ ð8aþ

6 010 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) Fig. 4. The natural rigid-body modes q 0. o 1 / γ tβ o 1 / γ tγ 1 / o γ tα o 1 / γ tα o 1 / γ tϕ 1 / ψ s β o 1 / γ tγ 1 / ψ sγ 1 / ψ sα 1 / ψ sα 1 / ψ sβ 1 / ψ sγ b 1 / ψ Aβ b 1 / ψ Aγ b 1 / ψ Aα b 1 / ψ Aα s 1 / ψ Aβ 1 / b Aβ b 1 / ψ Aγ s 1/ ψ Aγ s 1 / ψ Aα s 1 / ψ Aα ψ α s 1 / ψ Aβ s 1 / ψ Aγ ψ β ψ γ Fig. 5. The 1 natural straining modes q N.

7 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) t w A ¼ w Aa w Ab w Ac antisymmetric bending þ shearing mode; ð1þ t w z ¼ w a w b w c azimuth rotational mode: ð14þ The antisymmetrical mode is the sum of the antisymmetric bending mode plus the antisymmetric shearing mode w Ai ¼ w b Ai þ ws Ai ; i ¼ a; b; c ð15þ while the axial straining modes c 0 t along the middle surface of the element, the total strains c t and the symmetric bending modes, shown in Fig. 5, are connected via c ta ¼ c 0 ta þ z0 w Sa l a ; c tb ¼ c 0 tb þ z0 w Sb l b ; ð1þ c tc ¼ c 0 tc þ z0 w Sc l c ; where l i ði ¼ a; b; cþ is the length of side i and z 0 is the distance from the middle surface along z 0 axis of the element. The elemental displacement vectors including the nodal degrees of freedom in the local and global coordinate systems are defined by * q ¼ u 0 v 0 w 0 h 0 u 0 w 0 t i ; i ¼ 1; ; ; q ¼ ½u v w h u wš t i ; i ¼ 1; ; : ð1þ The natural modes q N are related to the elemental Cartesian q via q N ¼ a N q ð18þ and the total axial strains are related to q N, following (1), via c t ¼ a N q N : ð19þ Matrices a N and a N are always related to the current geometry of the element only. The local Cartesian elemental vector q is connected to the global Cartesian elemental vector q via q ¼ T 0 q; ð0þ where T 0 is a matrix containing direction cosines. Using (0), Eq. (18) may be written as q N ¼ a N q ¼ a N T 0 q: ð1þ.. Axial and symmetric bending stiffness terms The natural stiffness matrix corresponding to the axial and symmetric bending modes can be produced from the statement of variation of the strain energy with respect to the natural coordinates: du ¼ r t c dc t dv : ðþ V Following (18) (0) dc t may be written as dc t ¼ a N dq N ¼ a N a N dq ¼ a N a N T 0 dq: ðþ

8 01 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) Substitution of () in () leads to du ¼ r t c a N dv dq N : V ð4þ The combination of Eqs. (8), which corresponds to the constitutive relation of r c and c t and (4) gives the following expression du ¼ q t N V a t N j cta N dv dq N from which the natural stiffness matrix containing contributions from the axial and symmetric bending modes can be reduced: k N ðc 0 t ; w sþ¼ a t N j cta N dv : ðþ V Transformation procedures can now be initiated to transform the natural matrix first to the local coordinate system and then to the global coordinate system K ax&s:b: global ¼ T t 0 a t N a t N j cta N dv 4 V 5 a N T 0 : ðþ fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} 4 natural coord: ð11þ 5 fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} local coord: ð1818þ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} global coord: ð1818þ ð5þ.4. Antisymmetric bending and shearing and azimuth stiffness terms Similar to the formulation for the axial and symmetric bending modes the Cartesian contribution to the elementõs stiffness matrix of the antisymmetric bending modes is expressed as follows: K antis:b: global ¼ T t 0 a t N a t Nb j cta Nb dv 4 V 5 a N T 0 ; ð8þ fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} 4 natural coord: ð11þ 5 fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} local coord: ð1818þ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} global coord: ð1818þ where a Nb is the matrix that relates the axial natural strains c b t via: c b t ¼ a Nb w b A : with the antisymmetric bending modes w b A ð9þ

9 The contribution to the elementõs stiffness matrix of the antisymmetric shearing modes takes the form: K antis:sh: global ¼ T t 0 a t N a t NS v Sa NS dv 4 V 5 a N T 0 ; ð0þ fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} 4 natural coord: ð11þ 5 fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} local coord: ð1818þ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} global coord: ð1818þ where a NS is the matrix that relates the natural shearing strains c S with the antisymmetric shearing modes w S A via: < c a = 1= >< w S Aa >= c S t ¼ a NS w S A ) c : b ; ¼ 4 1= 5 w S Ab 1= >: >; : ð1þ c c w S Ac Finally, the contribution to the elementõs stiffness matrix of the azimuth rotational modes is given by the matrix: 1 0:5 0:5 K azimuth natural ¼ k zz 4 0:5 1 0:5 5; ðþ 0:5 0:5 1 where k zz is an arbitrary constant. Details concerning their derivation of all the elementõs full natural and Cartesian stiffness matrices are given in []..5. The geometric stiffness J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) The geometric stiffness is based on large deflections but small strains and consists of two parts. A simplified geometric stiffness matrix that is generated by the rigid-body movements of the element and the natural geometric stiffness matrix due to the coupling between the axial forces and the symmetric bending modes (stiffening or softening effect). The simplified geometric stiffness includes only those natural forces which produce rigid-body moments when element undergoes rigid-body rotations. To the rigid body and straining modes q 0, q N correspond generalized forces and moments as follows: q 1 0 ¼ ½ q 01 q 0 q 0 Š t ; P 0 ¼ ½P 01 P 0 P 0 Š t ; ðþ q 0 ¼ ½ q 04 q 05 q 0 Š t ; M 0 ¼ ½M 01 M 0 M 0 Š t ; ð4þ c 0 t ¼ c 0 ta c 0 tb t; c 0 tc PN ¼ ½P a l a P b l b P c l c Š t ; ð5þ w S ¼ w Sa w Sb t w Sc ; M S ¼ ½M Sa M Sb M Sc Š t ; ðþ w A ¼ w Aa w Ab t w Ac ; M A ¼ ½M Aa M Ab M Ac Š t ; ðþ w z ¼ w a w b t w c ; M z ¼ ½M a M b M c Š t : ð8þ

10 014 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) To construct the geometric stiffness we will focus on small rigid-body rotation increments about x 0 y 0 z 0 combined in the vector dq 0 ¼ ½ dq 04 dq 05 dq 0 Š t : ð9þ These rigid-body rotation increments correspond to elemental Cartesian moments dm 0. By making use of the fact that the resultants of all forces produced by rigid-body motion must vanish, an expression for dm 0 is established: dm 0 ð1þ ¼ k GR dq 0 ðþð1þ ; ð40þ where k GR is the local rigid-body rotational geometric stiffness. As seen in [,4], k GR has the simple analytical form P a ya þ P byb þ P cyc P ax a y a þ P bx b yb þ P! cx c yc 0 l a l b l c l a l b l c kgr ¼ ðþ P ax a y a þ P bx b yb þ P! cx c yc P a x a þ P bx b þ P cx c ; ð41þ 0 4 l a l b l c l a l b l 5 c 0 0 P a þ P b þ P c where P a, P b, P c are the middle plane axial natural forces and x a ¼ l a c ax ¼ x 0 x0 ; x b ¼ l b c bx ¼ x 0 1 x0 ; x c ¼ l c c cx ¼ x 0 x0 1 ; ð4þ y a ¼ l a s ax ¼ y 0 y0 ; y b ¼ l b s bx ¼ y 0 1 y0 ; y c ¼ l c s cx ¼ y 0 y0 1 ; ð4þ are geometric expressions with x 0 1, y0 1, x0, y0, x0, y0 being the x0, y 0 coordinates of the three vertices of the facet triangle in the local Cartesian system. A transformation of k GR to the local coordinate system follows from kg ¼ a t 0R kgr a 0R ; ð44þ ð1818þ ð18þðþð18þ where a 0R is the transformation matrix relating the natural rigid-body rotations q 0 to the Cartesian nodal displacements and rotations q q 0 ¼ a 0R q: ð45þ kg is the so-called simplified geometric stiffness with respect to axes x 0 y 0 z 0. The term simplified refers to the fact that only the middle plane axial natural forces P a, P b, P c are included in k G which fully represent the prestress state within the material. Once the simplified geometric stiffness is formed it may transformed to the global coordinate system. As mentioned before, nearly all geometric stiffness arises from the rigid-body movements of the element. However, in buckling phenomena quite often the membrane forces are relatively large and in this case it may be worth considering an additional approximate natural geometric stiffness arising from the coupling between the axial forces and the symmetric bending mode (stiffening or softening effect). This natural geometric stiffness comprises the following diagonal matrix. P k NG ¼ a l a P a l a P b l b P b l b P c l c P c l c : ð4þ A derivation of this expression can be found in [1]. The natural geometric stiffness is then transformed first to local and ultimately to the global coordinates.

11 . Mass matrix of TRIC consistent and lumped formulations.1. The principal of virtual works in dynamics Argyris et al. [1] showed that for linear and geometrically nonlinear static analysis the principle of virtual work in terms of the elastic Cartesian stresses and strains takes the following form: r t dedv ¼ p t V du dv þ p t S dudv þ Rt dq; ð4þ V V S where r, e denote the stress and strain vectors, p V, p S, are the distributed body and surface forces, u, q are the displacement field and nodal displacement vector, respectively, while and R is the vector of concentrated forces and moments. The above relation can be applied to any coordinate system due to the invariance of the total energy. Employing the constitutive relation (8b): r ¼ je ð48þ Eq. (4) becomes: e t jdedv ¼ p t V dudv þ p t S dudv þ Rt dq: ð49þ V V S Eq. (49) mathematically expresses that in the course of a static deformation, a state of equilibrium is established between the external forces and the corresponding elastic restoring forces. Likewise, in a dynamic analysis, an equilibrium state may be established at any time t. The dynamic equilibrium state can be mathematically defined using dõalembertõs principle. In a static analysis, the application of external forces is slow, while in realizing a dynamic equilibrium, additional forces must be taken into account; inertia forces R I ðtþ related to the acceleration of the structure and damping (frictional) forces R D ðtþ related to velocity. The effect of the damping forces will be neglected in the subsequent formulation. The inertia forces for a infinitesimal volume dv of a continuous system of mass density q d and mass m ¼ q d dv are given by dr I ¼ q d dv u: ð50þ Assuming a modal distribution for the displacement function u of the form uðx; tþ ð1þ J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) uðx; y; z; tþ ¼ 4 vðx; y; z; tþ 5 ¼ xðxþ qðtþ ; ð51þ wðx; y; z; tþ ð18þð181þ where xðxþ is the modal matrix, Eq. (50) become dr I ¼ q d dv u ¼ q d xðxþdv qðtþ: Thus, the work produced by the inertia forces over the entire volume of the element is given by W I ¼ u t dr I ¼ dq t q d x t xdv q: V V ð5þ ð5þ The work W ext produced by the external body and surface forces p V and p S that appear in Eqs. (4) and (49) is transformed by introducing the modal distribution of Eq. (51) as follows: W ext ¼ dq t p V x t dv þ p S x t dv : ð54þ V S

12 01 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) Since, the work produced by the external forces must be equal to the work consumed by the elastic forces W el and the inertia forces W I : W el ðtþþw I ðtþ ¼W ext ðtþþr t dq; the principle of virtual works in dynamics takes the form: e t jðtþdedv þ dq t q d x t xdv qðtþ ¼dq t p V x t dv þ dq t V V V S p S x t dv þ R t dq: By substituting Eqs. () for all components of strain terms into the first integral of the above equation, the reference Eq. (5) takes the following form at the element level: q t ðtþ ½½a N T 0 Š t ½a t N ja NŠ½a N T 0 ŠŠ e dv dq þ q t ðtþ q d x t xdv dq V e V e ¼ p V ðtþx t dv dq þ p S ðtþx t dv dq þ R t e ðtþdq: ð5þ V e S e From Eq. (5) it is straightforward to distinguish that the first parenthesis is the stiffness matrix of the TRIC shell element, as shown in Eqs. () () and the second parenthesis represents the mass matrix. In the right hand side of Eq. (5) the first parenthesis represents the elemental body forces and the second the elemental surface forces applied externally on each element. Finally R t eðtþ is the part of the vector of externally applied nodal forces on the structure corresponding to the nodes of element e. Hence, Eq. (5) can take the following well-known form: M q þ Kq ¼ R ext : ð58þ Since the stiffness matrix of the element has already been presented it is the mass matrix of the element that has to be derived in order to perform to dynamic analysis of shell structures with the TRIC shell element... The consistent mass matrix ð55þ ð5þ The computation of the elemental mass matrix in the local coordinate system M e ¼ x t x dv ; ð1818þ ð18þð18þ V e q d ð59þ prerequisites the computation of the modal matrix x. The modal matrix x can be produced by invoking kinematical and geometrical arguments. It has to be pointed out, that similar to static analysis, the rotational inertia forces resulting from antisymmetric bending and shearing deformation are assumed uncoupled from the other forces, and as such they are treated independently. The modal matrix x can divided into five submatrices corresponding to rigid body, axial straining, symmetric bending, antisymmetric bending and shearing as well as azimuth rotational modes: x ð18þ h ¼ x 0 ðþ x t ðþ x S ðþ x A ðþ x z ðþ i : ð0þ Fig. 4 presents all the geometrical and kinematical arguments that lead to the computation of the modal submatrix x 0 : z 0 y 0 x 0 ¼ z 0 0 x 0 5: ð1þ ðþ y 0 x 0 0

13 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) Fig.. Axial straining mode c ta. The modal functions that relate the displacement vector u with the axial straining modes can be easily produced with the aid of Fig. where the axial straining mode (c ta ) along side a is shown with the corresponding displacements of nodes B, C. It is assumed that the elongation of side a occurs half at the side of node B and half at the side of node C. It can be deduced that the displacements of nodes B, C are: l a u 0 B ¼ sin _ cosðc _ p=þc 0 ta ; ðþ B l a v 0 B ¼ sin _ sinðc _ p=þc 0 ta ; ðþ B l a u 0 C ¼ sin _ cosðb _ p=þc 0 ta c ; ð4þ l a v 0 C ¼ sin _ sinðb _ p=þc 0 ta c ; ð5þ where _ B, C _ are the angles of the triangle at nodes B, C, respectively, _ b, _ c are the angles that form the sides b, c, with the x 0 axis of the local coordinate system, respectively, and u 0 B, u0 C, v0 B, v0 C are the displacements of nodes B, C along the local x 0, y 0 axes. Employing the following relations that hold for any triangle: sin _ A ¼ X ; sin _ B ¼ X ; sin C _ ¼ X ; l b l c l a l c l a l b where X is the triangle area. Since it also holds that: l a cos a _ ¼ x a ; l b cos b _ ¼ x b ; l c cos c _ ¼ x c ; ðþ l a sin a _ ¼ y a ; l b sin b _ ¼ y b ; l c sin c _ ¼ y c ; ð8þ Eqs. () (5) are transformed to: ðþ u 0 B ¼ l a y c 4X c0 ta ; u 0 C ¼ l a y b 4X c0 ta ; v0 B ¼ l a x c 4X c0 ta ; v0 B ¼ l a x b 4X c0 ta : ð9þ ð0þ

14 018 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) Similar expressions can be deduced for the other axial straining modes c tb and c tc. Hence, the displacements u, v produced by the axial straining modes c 0 ta, c0 tb and c0 tc have the following form: u 0 A ¼ l b y c 4X c0 tb þ l c y b 4X c0 tc ; u 0 B ¼ l a y c 4X c ta þ l c y a 4X c0 tc ; v0 A ¼ l b x c 4X c0 tb l c x b 4X c0 tc ; v0 B ¼ l a x c 4X c ta l c x a 4X c0 tc ; ð1þ ðþ u 0 C ¼ l a y b 4X c ta þ l b y a 4X c0 tb ; v0 C ¼ l a x b 4X c ta l b x a 4X c0 tb : Following the standard interpolation expressions for triangular elements: u 0 ¼ f a u 0 A þ f bu 0 B þ f cu 0 C ; ðþ ð4þ v 0 ¼ f a v 0 A þ f bv 0 B þ f cv 0 C ; where f a, f b and f c are the area coordinates of a point that belongs to the plane of the triangle, the relations between the displacements u 0, v 0 of any point within the TRIC shell element and the axial straining modes are derived by substituting Eqs. (1) () into Eqs. (4) and (5): ð5þ u 0 ¼ l a 4X ðy cf b þ y b f c Þc 0 ta þ l b 4X ðy cf a þ y a f c Þc 0 tb þ l c 4X ðy bf a þ y a f b Þc 0 tc ; ðþ v 0 ¼ l a 4X ðx cf b þ x b f c Þc 0 ta l b 4X ðx cf a þ x a f c Þc 0 tb l c 4X ðx bf a þ x a f b Þc 0 tc : From these equations the submatrix x t takes the form: l a 4X ðy l b cf b þ y b f c Þ 4X ðy l c cf a þ y a f c Þ 4X ðy af b þ y b f a Þ x t ¼ l ðþ a 4X ðx l a cf b þ x b f c Þ 4X ðx l c cf a þ x a f c Þ 4X ðx : ð8þ 4 af b þ x b f a Þ The third row of submatrix x t is equal to zero since the axial straining modes are confined into the plane of the element. The derivation of the submatrices x S and x A will be performed following the expression of displacement w as a function of the symmetric bending and antisymmetric bending and shearing modes, respectively. Thus, according to [5] the following relation for displacement w 0 holds: ðþ w 0 ¼ 1 l af b f c w Sa þ 1 l bf a f c w Sb þ 1 l cf a f b w Sc þ 1 l af b f c ðf b f c Þw Aa þ 1 l bf a f c ðf c f a Þw Ab þ 1 l cf a f b ðf a f b Þw Ac : ð9þ By applying the Kirchhoff Love assumption: u ¼ zow=ox 0 ; v ¼ zow=oy 0 ð80þ the in plane displacement components are expressed by ow 0 of u ¼ z a þ ow0 of b þ ow0 of c ; v ¼ z of a ox 0 of b ox 0 of c ox 0 ow 0 of a of a oy 0 þ ow0 of b of b oy þ ow0 of c 0 of c oy 0 ð81þ

15 and using f a ¼ x 0 B x 0 C x 0 X yb 0 yc 0 y 0 ¼ ðx0 B y0 C x0 C y0 B Þ X then of a ¼ y a ox 0 X and of a ¼ x a oy 0 X and of b ox 0 ¼ y b X ; of b oy 0 ¼ x b X ; þ ðy0 B y0 C Þ X x 0 þ ðx0 C x0 B Þ y 0 X of c ox 0 ¼ y c X and of c oy 0 ¼ x c X : Since the natural modes are independent to each other the derivation for the symmetric bending and antisymmetric bending and shearing modes can be carried out separately for each of the two types of natural modes. Starting with the first symmetric bending mode w Sa and making use of the expressions (8) and (84) (for w Sa ¼ 0, w Sb ¼ w Sc ¼ w Aa ¼ w Ab ¼ w Ac ¼ 0): u 0 ¼ z 0 l a 4X ðf cy b þ f b y c Þw Sa ; ð8þ ð8þ ð84þ ð85aþ v 0 ¼ z 0 l a 4X ðf cx b þ f b x c Þw Sa : ð85bþ In a similar manner for the analogous expressions are derived for the other symmetric bending modes: u 0 ¼ z 0 l b 4X ðf cy a þ f a y c Þw Sb ; J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) ð8aþ v 0 ¼ z 0 l b 4X ðf cx a þ f a x c Þw Sb ; u 0 ¼ z 0 l c 4X ðf by a þ f a y b Þw Sc ; ð8bþ ð8aþ v 0 ¼ z 0 l c 4X ðf bx a þ f a x b Þw Sc : ð8bþ From the above equations the submatrix x S of the modal matrix x can be formulated: z 0 l a 4X ðy z 0 l b cf b þ y b f c Þ 4X ðy z 0 l c cf a þ y a f c Þ 4X ðy af b þ y b f a Þ x S ¼ z0 l a ðþ 4X ðx cf b þ x b f c Þ z0 l b 4X ðx cf a þ x a f c Þ z0 l c 4X ðx af b þ x b f a Þ : ð88þ 4 l a f l b bf c f l 5 c af c f bf a For the antisymmetric bending and shearing mode w Aa Eq. (9) (for w Aa ¼ 0, w Ab ¼ w Ac ¼ w Sa ¼ w Sb ¼ w Sc ¼ 0) becomes: or u 0 ¼ z 0 l a f b f c f c X u 0 ¼ z 0 l a X y bf c f b y b f bf c f b y c!w X Aa ð89þ f c y c f b f c f b w Aa ð90aþ

16 00 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) and for displacement v 0 : v 0 ¼ z 0 l a X x f c bf c f b x c f b f b f c w Aa ð90bþ and the other antisymmetric bending and shearing modes in an analogous manner: u 0 ¼ z 0 l b X y cf a f c f a y a f c f a f c w Ab ; v 0 ¼ z 0 l b X x cf a f a u 0 ¼ z 0 l c X y af b f a v 0 ¼ z 0 l c X x af b f b f c x a f c f b y b f a f a x b f a f c f b f a ð91aþ f a w Ab ; ð91bþ f a w Ac ; ð9aþ f b w Ac : ð9bþ From the above equations the submatrix x A of the modal matrix x can be formulated and it is shown in Fig.. The last submatrix that remains to be derived is related to the azimuth straining modes. Its derivation is straightforward since by definition the in plane displacements of the element are functions of the azimuth straining modes []: u 0 ¼ x a l a ðf a f a Þw a; u 0 ¼ x b l b ðf b f b Þw b; u 0 ¼ x c l c ðf c f c Þw c; v 0 ¼ y a l a ðf a f a Þw a; v 0 ¼ y b l b ðf b f b Þw b; v 0 ¼ y c l c ðf c f c Þw c ð9þ ð94þ ð95þ and w 0 ¼ 0 since the azimuth straining modes produce only in plane displacements. Hence, the submatrix x z has the following form: x a ðf a l f a Þ x b ðf b a l f b Þ x c ðf c b l f c Þ c x z ¼ y a ðf ðþ a l f a Þ y b ðf b a l f b Þ y c ðf c b l f c Þ 4 c 5 : ð9þ The formulation of the modal matrix x is now completed and the mass matrix of the TRIC shell element is ready to be calculated by applying the relation (59). Symbolic computation is employed in order to carry out the lengthy but otherwise straightforward matrix multiplications that are implied by Eq. (59). It is one of the major advantages of the TRIC shell element that both the stiffness and the mass matrix are formulated analytically and that no numerical integration is required in the computation of any matrix. In order to calculate in an exact manner all the integrals the following formula will be used: 1 X X f p a fq a fr a dx ¼!p!q!r! ð þ p þ q þ rþ! : ð9þ

17 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) Fig.. Submatrix x A.

18 0 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) The lumped mass matrix An important advantage of using a lumped mass matrix is that the matrix is diagonal, which facilitates the solution of explicit time integration of the equations of motion and reduces significantly the computational cost of the finite element solution at each time step. On the other hand, the lumped matrix being an approximation of the consistent mass matrix often leads to reduced accuracy and instability, especially when coarse finite element meshes or higher order elements are used or when severe nonlinearities are present. There are two ways of producing the lumped mass matrix of a finite element. In the first and simplest one, the mass matrix is produced using geometrical arguments. Thus, instead of performing the integrations leading to the consistent mass matrix, an approximate mass matrix can be evaluated by lumping equal parts of the total element mass to the nodal points of the element. Each nodal mass essentially corresponds to the mass of the elementõs contributing volume around the node, leading to the conclusion that by using this procedure of lumping the mass, it is assumed, in essence, that the accelerations of the contributing volume to a node are constant and equal to the nodal values. In the case of a triangular element each node of the element is assigned to a third of the total mass of the element in each of the directions of the local or global coordinate systems. Hence, the lumped mass matrix produced by geometrical arguments is simply a diagonal matrix with the value q d V = for each of the translational degree of freedom, where q d and V are the density and the volume of the element, respectively. For the second lumping scheme the HR [18] approach was adopted. This is somewhat more complicated since it requires the calculation of the diagonal elements of the consistent mass matrix beforehand and can be applied for any arbitrary finite element. The idea behind this approach is to use the diagonal terms of the consistent mass matrix properly scaled so that the total mass of the element is preserved. Specifically, the procedural steps are as follows: 1. Compute only the diagonal coefficients of the consistent mass matrix.. Compute the total mass of the element, m. Compute a number s by adding the diagonal coefficients m ii associated with translational d.o.f. (but not rotational) that are mutually parallel and in the same direction. 4. Scale all (including the coefficients corresponding to rotational d.o.f.) the diagonal coefficients by multiplying them by the ratio m=s, in order to preserve the total mass of the element. This second lumping technique, although more complex than the first one since it requires the computation of the diagonal terms of the consistent matrix, has the advantage that the rotational d.o.f. are represented in the mass matrix and that all the diagonal terms in the mass matrix are nonzero, making it readily applicable to integration techniques that require the inverse of the mass matrix. 4. Time integration methods for the solution of nonlinear equations in structural dynamics The nonlinear dynamic response of a finite element system can be obtained by applying the time integration schemes used in linear dynamic analysis modified in a way that the effect of nonlinearities is incorporated together with numerical stability considerations that are necessary for reassuring the robustness of the time integration scheme. Explicit time integration schemes are easier to implement and as long as stability is achieved the resulting accuracy is usually satisfactory. However, they still suffer from excessive computational cost due to the restriction to the size of the time steps compared to implicit schemes, despite

19 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) the fact that stability requirements impose constraints to the size of these steps in implicit structural dynamic integration rules. The integration schemes used for implicit nonlinear structural dynamic equations are based on the Newmark family of algorithms [5]. The Newmark algorithms are unconditionally stable for linear problems but only conditionally stable in the nonlinear range. Energy conserving schemes such as the Energy Momentum Method were developed by Simo et al. [8] and Simo and Tarnow [0], especially for the case of geometrically nonlinear shell dynamics in order to guarantee stable solutions. The Energy Momentum Method is modification of the Newmark Method that ensures the preservation of energy as well as linear and angular momentum. Another point of interest in structural dynamics is the presence of higher modes in the dynamic response of a structure. Higher frequencies often lead to numerical collapse unless small time steps are used. This feature has led to modifications of the standard Newmark method such as Hilber-a method [0], the Bossak-a method [1] and the Generalized-a method [15] that introduce numerical dissipation, which damps out the high frequencies. One important remark, though, on numerical dissipation of high frequencies is that when combined with energy preservation, as in the case of the Constrained Energy Momentum Algorithm proposed by Kuhl and Ramm [], the energy transfer from higher to lower modes is inevitable, a transfer that has no physical basis. Hence, Armero and Pet ocz [8,9] introduced a modification of the Energy Momentum Method applied to frictionless and frictional dynamic contact problems, denoted as Modified Energy Momentum Method, Crisfield et al. [1] applied the aforementioned method to three-dimensional beams and Kuhl and Crisfield [] introduced the Generalized Energy Momentum Method for trusses, Kuhl and Ramm [4] applied the same method to shell structures, while Botasso et al. [1] proposed an Energy Decaying Scheme for nonlinear dynamics of shells. All these new modifications are simultaneously numerically dissipative, energy decaying and are aimed at producing stable and converged solutions with the largest possible time step for computational efficiency. Since this is the first implementation of the TRIC shell element to linear and nonlinear dynamic analysis of shell structures it is attempted to use as a first choice the standard Newmark scheme (trapezoidal rule). The computed response of the structures in the numerical tests performed in the next section will be discussed on the basis of the above-mentioned rationale Explicit integration The general form of the equilibrium equations (Eq. (58)) in dynamic analysis is M q þ C _q þ Kq ¼ R: ð98þ One of the most common explicit time integration operator used in nonlinear dynamic analysis is probably the central difference operator. The solution for the displacement vector tþdt q of Eq. (98) is obtained using the central difference approximations of the acceleration and velocity: t q ¼ 1 Dt ðt Dt q t q þ tþdt qþ; t _q ¼ 1 Dt ðtþdt q t Dt qþ ð99þ and substituting them into Eq. (98) for time t: M tþdt q ¼ t R t F þ M t q M t Dt q; ð100þ Dt Dt Dt where the damping matrix is neglected. It is clear from the form of Eq. (100) that the factorization of the effective stiffness matrix is performed only once, in the case of a consistent mass matrix, irrespective of the type of problem whether it is linear or nonlinear.

20 04 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) The shortcoming in the use of the central difference method lies in the severe time step restriction: for stability, the time step size Dt must be smaller than a critical time step Dt cr, which is equal to T n =p, where T n is the smallest period in finite element system. This time restriction was derived considering a linear system, but the result is also applicable to nonlinear analysis, since for each time step the nonlinear response calculation may be thought of in an approximate way as a linear analysis. However, whereas in a linear analysis the stiffness properties of a finite element system remain constant, in a nonlinear analysis these properties change during the response calculations. Hence, T n changes during the solution process and the time integration step size must remain always below the changing T n which means that either a very small time step size is required form the beginning of the solution process of that it adjusts in the process. 4.. Implicit integration All the implicit time integration schemes developed for linear dynamic analysis can also be applied to nonlinear dynamic response calculations. Using, for example the Newton Raphson iteration the governing dynamic equilibrium equations are (again neglecting the effects of a damping matrix): M tþdt q ðkþ1þ þ tþdt K ðkþ Dq ðkþ1þ ¼ tþdt R tþdt F ðkþ ; ð101þ where k, k þ 1 represent the iterations within the time step Dt. With the Newmark approximations: tþdt q ¼ 1 bdt ðtþdt q t qþ 1 bdt t _q 1 t q þ t q b ð10þ and introducing iterations k, k þ 1 within the time step tþdt q ðkþ1þ ¼ 1 bdt ðtþdt q ðkþ t q þ Dq ðkþ1þ Þ 1 bdt t _q 1 t q þ t q b ð10þ Eq. (101) becomes: tþdt K ðkþ þ M bdt Dq ðkþ1þ ¼ tþdt R tþdt F ðkþ M 1 bdt ðtþdt q ðkþ tþdt qþ 1 bdt t _q 1 b 1 t q : Eq. (104) constitutes the Newton Raphson Newmark solution (with b ¼ 0:5, trapezoidal rule) of Eq. (101), which will be used in our test cases with the TRIC shell element. ð104þ 5. Numerical examples In this section two benchmark test examples are examined to illustrate the efficiency of the TRIC shell element in dealing with nonlinear dynamic analysis of shell structures Spherical cap under uniform impulse loading In the first example the dynamic response of a spherical cap subjected to uniform impulse loading p ¼ 00 psi is examined.

21 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) Fig. 8. Spherical shell: geometry and material properties. The geometry and the material properties of the spherical shell are shown in Fig. 8. Due to the symmetry of the structure and the loading only one quarter of the spherical cap is analyzed. The different meshes used for the discretization of the structure are shown in Fig. 9. Both explicit and implicit time integration rules are applied. For the implicit integration, the Newmark method, with b ¼ 0:5 and c ¼ 0:5, was applied with Dt ¼ 10 5 s, for the dense mesh and 10 5 s, for the coarser meshes. On the other hand, for the explicit integration the time step used was Dt ¼ 0: s, when the coarse mesh ( d.o.f.) was analyzed, Dt ¼ 0: 10 5 s, for the middle mesh (541 d.o.f.) and Dt ¼ 0: 10 5 s for the denser mesh (81 d.o.f.). These time step increments were found to be maximum allowable for ensuring stability and accuracy. In the implicit time integration the consistent mass formulation was tested alongside the two lumped formulations (lumped from consistent and lumped from geometry). It was found that all three formulations produced almost identical results to those predicted by the explicit integration as can be seen in Fig. 10 where the time response of the central deflection of the spherical cap is plotted. Fig. 11 depicts the dynamic response predicted by the three different finite element meshes used. Finally, the solution obtained with the TRIC shell element is compared in Fig. 1 to the following results reported in the literature: The BST rotation-free triangle element by Onate et al. [], the DKT-15 shell element that combines the standard plate element by Batoz et al. [11] and the d.o.f. constant strain triangle by ienkiewicz and Taylor [] and the eight-node isoparametric shell element by Belytschko et al. [1]. A summary of the results for the central deflection at significant times is given in Table 1 where the results shown for the TRIC shell element correspond to the implicit solution with the dense mesh and the consistent mass matrix. These results demonstrate the efficiency and the accuracy of the TRIC shell element in nonlinear dynamic analysis both in explicit and implicit time integration schemes and verify the consistent mass matrix formulation presented in the previous sections. In this example it is not possible to compare the efficiency of the TRIC shell element as far as the size of the integration time step is concerned since the relevant data is not available. The CPU times required for the time integration of the nonlinear dynamic response of the spherical cap from time t ¼ 0 ms to time t ¼ 1 ms in a PIV processor.4 GHz with 1 GB DDRAM memory together with

22 0 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) Fig. 9. Spherical shell: finite element discretization. the computing time required by the ABAQUS finite element code are summarized in Table. The ABA- QUS element is the S -node triangular general purpose shell element. From Table it can be seen that the explicit integration is more expensive in this test example in all three finite element meshes and that the CPU time required by the TRIC shell element code is around 5% of the time required by ABAQUS.

23 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) Fig. 10. Central deflection response for different mass formulations. Fig. 11. Central deflection response for different finite element discretizations.

24 08 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) Fig. 1. Comparison with other shell formulations. Table 1 Spherical shell: computed central deflection Central deflection (in.) t ¼ 0: ms t ¼ 0:4 ms t ¼ 0: ms TRIC )0.048 ) BST [] ) ) DKT15 [11] )0.045 ) Belytschko et al. [1] )0.099 ) Table Comparison of CPU times required by TRIC and ABAQUS CPU time (s) for different meshes 81 d.o.f. 541 d.o.f. d.o.f. Implicit consist Implicit lumped geom Implicit lumped consist Explicit ABAQUS (Implicit) Cylindrical shell with dynamic snap through The second example is a cylindrical shell exhibiting dynamic snap through and more severe nonlinearities than the spherical cap. This is a typical benchmark example that has been used extensively as a test bed for

25 J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 19 (00) all nonlinear shell dynamics formulations presented so far. Snap through problems in shells produce higher dynamical modes and this is the reason why it is believed that the standard integration schemes such as the Newmark method are not adequate to produce a stable and accurate solution and that only algorithms with numerical dissipation and energy decaying schemes can be applied with an acceptable time step. Despite this widespread belief that is connected to other types of shell elements, it will be shown that using the TRIC shell element it is possible to obtain stable and accurate solutions with the Newmark integration method using a time step that is comparable to required maximum time step of more sophisticated time integration rules. Moreover, it will be shown that the solution obtained by the TRIC shell element for this highly nonlinear problem is computationally significantly less expensive in terms of CPU time compared to ABAQUS. The geometry and the elastic properties of the cylindrical shell are shown in Fig. 1. The two straight edges of the shell are simply supported, while the two curved edges are free. A concentrated load is applied at the central node of the shell. The value of this load increases linearly from 0 to kn in 0. s, then is held constant at that value. Due to the symmetry of the structure only a quarter of the shell is examined. The mesh used for the discretization consists of 81 vertices, 18 elements and 40 d.o.f. and is shown in Fig. 14. As before, both explicit and implicit time integration rules are applied and the dynamic response of all three mass formulations is investigated. For the implicit integration, the Newmark method, with b ¼ 0:5 and c ¼ 0:5, is applied with Dt ¼ 10 s. On the other hand for the explicit integration the maximum time step achieved is Dt ¼ : s. The results produced by the three different mass formulations for the Newmark implicit integration and the explicit integration are shown in Figs. 15 and 1. In all these plots of the shell apex deflection the pre-buckling, the buckling and the post-buckling behavior of structure are clearly illustrated. These figures demonstrate the stability and accuracy of the predicted nonlinear dynamic response of the cylindrical shell with regard to the mass formulation used in the analysis. As shown in Fig. 1 the lumped mass formulations do not converge after a while in the post-buckling range where the structure is vibrating around its new equilibrium position, while the consistent mass matrix formulation produces a stable solution until the end of the reported time. It is important to note that in order to produce Fig. 1. Cylindrical shell: geometry and material properties.