Corotational Nonlinear Dynamic Analysis of Thin-ShellStructures with Finite Rotations

Jinsong Yang∗ and Pinqi Xia†

Nanjing University of Aeronautics and Astronautics, 210016 Nanjing,

People’s Republic of China

DOI: 10.2514/1.J053147

A new formulation for corotational nonlinear dynamic analysis of thin-shell structures involving large

displacements and finite rotations has been presented in this paper. By introducing the kinematic description of the

geometrically exact quasi-static shell model to describe the motion of an arbitrary material point in a triangular shell

element, the new expressions for the inertial force vector and tangent inertiamatrix were derived systematically. The

elastic deformation was handled by using the consistent symmetrizable equilibrated corotational formulation, which

is independent with respect to the local finite element modeling. An energy-momentum-conserving algorithm and an

energy-decaying/momentum-conserving algorithm were developed for nonlinear dynamic analysis by applying the

generalized energy-momentummethod in the corotational formulation. Three classic numerical examples, including

the dynamic response, free motion, and dynamic buckling of thin-shell structures, were computed to verify the

accuracy and capability of the presented formulation for solving the nonlinear dynamic problems of thin-shell

structures with finite rotations. In the relevant numerical example, the conservations of linear momentum and

angular momentum of a shell structure due to the time-integration algorithms in the corotational formulation were

investigated.

Nomenclature

A = area of a triangular elementCiner = element gyroscopic matrix

d, _d, �d = global displacement, velocity, andacceleration of an material point

�dda = local pure deformational displacement of node a

E0, ER = element initial and corotated frames, respectivelyf iner,f int, f ext

= element global inertial, internal, and externalforce vectors, respectively

�f int = element local internal force vector�f int;P = element projected local internal force vectorG = global frameg = element residual dynamic force vectorH�θ� = tangent transformation matrix associated

with the rotation vector θh = thickness of an elementI3 = 3 × 3 identity matrix

I0ρ, Iρ = rotational inertia tensors in material formi = iteration index within the Newton–Raphson

iterationJ = angular momentum of an elementK = kinetic energy of an elementKDyn = element tangent inertia matrixKElem = total tangent matrix of an elementKGM = element moment-correction geometric

stiffness matrixKGP = element equilibrium projection geometric

stiffness matrixKGR = element rotational geometric stiffness matrixKiner = element dynamic stiffness matrix due to

centrifugal forces

�Km = element local material stiffness matrixKStat = element tangent stiffness matrixKStat;G = element geometric stiffness matrixKStat;M = element material stiffness matrixL = linear momentum of an elementM = element mass matrixMρ = shell mass per unit areaN1, N2, N3 = linear interpolation functions for a

triangular element�P = element projection matrix�pd = vector of element local pure deformational

displacements and rotations�pda = vector of local pure deformational displacements

and rotations of node aq = vector of element global displacements and

rotationsR = rotation matrixS = strain energy of an elementS0O, SO = initial and deformed local frames located at an

arbitrary material point O on the middle planeof a shell element, respectively

S0a, Sa = initial and deformed triads at node a

TGE0 , TGE = transformation matrices from global frame toelement initial and corotated frames, respectively

TGS0O, TGSO = transformation matrices from global frame to

local frames S0O and SO, respectivelyt = timeu = incremental displacement in a time stepx = position vector in global frame

�x = position vector in local frameαm, αf = integration parameters in the generalized-α time-

integration algorithmβ, γ = integration parameters in the Newmark formulaΔd = iterative displacement in a Newton–Raphson

iterationΔt = time-step sizeΔθ = additive iterative rotation vector related to θΔω = iterative spatial rotation variableδω = spatial spin variableΘ = incremental rotation vector in material form in a

time stepθ = incremental rotation vector in spatial form in a

time step

Received 2 November 2013; revision received 15 April 2014; accepted forpublication 12 June 2014; published online 30 September 2014. Copyright ©2014 by the American Institute of Aeronautics and Astronautics, Inc. Allrights reserved. Copies of this paper may be made for personal or internal use,on condition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 1533-385X/14 and $10.00 in correspondence with the CCC.

*Ph.D. Candidate, Laboratory of Rotorcraft Aeromechanics, College ofAerospace Engineering.

†Professor, Laboratory ofRotorcraftAeromechanics, College ofAerospaceEngineering; xiapq@nuaa.edu.cn (Corresponding Author).

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Λ = element corotational transformation matrixξ = additional numerical dissipation coefficientΩ, _Ω = angular velocity and acceleration in

material form03 = 3 × 3 zero matrix

I. Introduction

T HIN-SHELL structures with superior mechanical propertieshave been widely used in various structural engineering

projects, such as large deployable space structures, aircrafts, ships,and other civil structures. Under the action of dynamic loads, a thin-shell structure often exhibit a nonlinear dynamic behavior with largedisplacements and finite rotations, but the strains remain small. Thecorrect prediction for nonlinear dynamic response of the structure isof great importance for structural design and maintenance.For geometric nonlinear finite element analysis, the existing

formulations can be divided into three categories [1]: total La-grangian formulation, updated Lagrangian formulation, and co-rotational formulation. The main idea of corotational formulation [1]can be summarized as 1) introducing a local frame into each element,and the local frame continuously translates and rotates with theelement tracking rigid-body motion; 2) extracting the pure defor-mational motion from the total motion by eliminating the rigid-body motion; and 3) modeling the internal force in reference to thelocal frame with a linear finite element analysis, and then trans-forming the local internal force into the global frame. Compared withthe Lagrangian formulations, the corotational formulation has someadvantages. The corotational formulation can apply the existinglinear elements with high performance to the geometric nonlinearanalysis. The corotational formulation incorporates the geometricnonlinearity into a transformation matrix, which relates to the localand global quantities. This transformation matrix is independent ofthe shape functions of the local element. Hence, the corotationalformulation is called the element-independent corotational for-mulation. The corotational formulation can effectively deal with thelarge deformation problems, especially with problems involvingarbitrarily finite rotations.Currently, the corotational formulation has been extensively

applied to geometric nonlinear static analysis with different types ofelements such as beams, shells, and solids. Some representativestudies were conducted by Belytschko and Glaum [2], Rankin andBrogan [3], Nour-Omid and Rankin [4], Crisfield and Moita [5],Crisfield [6], Felippa and Haugen [1], Pacoste [7], Mohan andKapania [8], Battini and Pacoste [9,10], Hsiao [11], Li [12], and soon. There were a few of corotational formulations in the nonlinearstructural dynamic analysis and flexible multibody system dynamicsby Zhong and Crisfield [13], Meek and Wang [14], Relvas andSuleman [15],Almeida andAwruch [16], Chimakurthi et al. [17], andYang and Xia [18]. When extending and applying the corotationalformulation to nonlinear dynamic analysis, two basic issues need tobe handled carefully: 1) an appropriate formulation for the inertialpart; and 2) a stable time-integration algorithm.For shell dynamic analysis with the corotational formulation, it is

extremely difficult to obtain a consistent formulation for the inertialpart with the same local interpolation as adopted in the formulationfor the elastic part because of the decomposition of total motionof an element. To avoid this difficulty, the inertial force vector wasformulated directly by multiplying a constant mass matrix by theinertial acceleration vector in many studies [13–16,18]. Suchformulation was feasible with several numerical tests, but there weresome problems. First, such a formulation lacked a necessary shellkinematic description for the inertial part and limited the applicationof the corotational formulation to the dynamic analysis of a flexiblemultibody system. Second, such a formulation did not have the inter-polation information clear enough and could cause some uncertainmodeling errors. Crisfield et al. [19] proposed a suitable solution forthe corotational dynamic formulation with spatial beam elements.The kinematic description of the geometrically exact beam theorywas introduced to derive the inertial force vector of a corotationalbeam element. The formulations for the inertial part and elastic part

were conducted separately. The inertia part was interpolated by theglobal quantities, and the elastic part was interpolated by the localquantities. Recently, Le et al. [20] followed this idea and summarizedfour formulations for the inertial part in a corotational nonlineardynamic analysis with spatial beam elements.Regarding the time-integration algorithms, many studies [21–23]

have shown that the algorithms that are unconditionally stable inlinear dynamic analysis often encounter severe numerical instabilityproblems when applied to nonlinear dynamic analysis. This situationprompted researchers to develop new stable algorithms for nonlinearanalysis. In the development of new algorithms, most of them werebased on the idea of the conservation laws of a Hamilton system, theconservations of energy, linear momentum, and angular momentum,which guarantees the numerical stability in nonlinear cases. Theenergy-momentummethod, proposed by Simo and Tarrow [22], wasone of the most successful methods and has been applied to thenonlinear dynamic analysis of beam and shell structures in thecontext of the geometric exact beam or shell theory.Crisfield and Shi [24] proposed the approximately energy-

conserving algorithm for the corotational formulation of a shellelement in nonlinear dynamic analysis, whichwas adopted by Zhongand Crisfield [13] and Almeida and Awruch [16]. This algorithmcan be viewed as a special case of the energy-momentum method.Recently, the generalized energy-momentum method, a generalizedversion of the energy-momentum method given by Kuhl and Ramm[25], was introduced into the corotational formulation by Yang andXia [18], producing an energy-conserving algorithm and an energy-decaying algorithm. However, they applied this method directlyto the calculations of inertial, internal, and external force vectors atthe generalized midpoint, which was inconsistent with the originalmethod.In this paper, the kinematic description of the geometrically exact

quasi-static shell model [26] was introduced to develop a newformulation for the inertial part in corotational nonlinear dynamicanalysis with triangular shell elements. In this shell model, therotation of an arbitrary material point on the middle plane of a shellelement was represented by a rotation matrix that was completely thesame as in the corotational formulation of shell elements. In the newformulation, the derivation began with the kinematics of an arbitrarymaterial point in a shell element, was followed by the variation ofthe element kinetic energy, and then obtained the inertial force vectorin an integral form. A linear interpolation with the iterative andlinearized global quantities of nodal displacements and rotationparameters was adopted to derive the tangent inertia matrix.In this paper, the generalized energy-momentum method was

applied to the calculations of acceleration, velocity, displacement, andother kinematic variables at the generalized midpoint, which is morein accordance with the algorithm idea of the generalized energy-momentum method. Without and with introducing of numericaldissipation into the method, an energy-momentum-conserving algo-rithm and an energy-decaying/momentum-conserving algorithm withcontrollable high-frequency response dissipation can be obtained.In this paper, three classic numerical examples were considered to

demonstrate the capability of the corotational formulation presentedin this paper. These numerical examples dealt with the nonlineardynamic responses, large overall free motion, and dynamic bucklingof shell structures. In the second example, the conservations ofstructural linear and angular momentum due to time-integrationalgorithms in corotational nonlinear dynamic analysis were inves-tigated. The results obtained in this paper were compared with theresults in the literature.The outline of this paper is as follows. Section II presents the

parameterization and computational aspects of the finite rotationsassociating with the present formulation. Section III focuses on thenew formulation for the inertial part. The inertial force vector andthe tangent inertia matrix are derived systematically by introducingthe kinematic description of the geometrically exact quasi-static shellmodel. Section IV is devoted to the corotational formulation for theelastic part, in which the internal force vector and the tangentstiffness matrix are briefly presented. The solution procedure of thenonlinear motion equations by a time-integration algorithm and the

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Newton–Raphson iterative technique are addressed in Sec. V. InSec. VI, three classic numerical examples are considered to verify thecapability of the present formulation. Finally, the conclusions aredrawn in Sec. VII.

II. Finite Rotations

For nonlinear dynamics associated with three-dimensional (3-D)finite rotations, the parameterization of the finite rotations plays animportant role in the whole formulation. There are many ways toparameterize a finite rotation, such as rotation matrix, rotationalvector, spatial spin variable, and Euler quaternion. In this section,according to the solution procedure of a nonlinear dynamic equationby a time-integration algorithm and the Newton iteration methodinvolved in this paper, the relevant parameterizations of the finiterotations are briefly described. More detailed descriptions can bereferred to in the literature [27,28]. In the solution procedure, the timeduration is divided into a number of time steps: t0 < t1 < · · · tn <tn�1 < · · · < tend, and several iterations are carried out within each

time step. In this paper, the notations tin�1 and ti�1n�1 indicate the ith and

(i� 1)th iterations in a typical time step tn → tn�1, respectively.In a coordinate system, the coordinate of a vector at time t0 is x0.

After two successive finite rotations, the coordinate becomes xn attime tn and xin�1 at time tin�1. If the intrinsic parameterization

by a rotation matrix is used to represent a finite rotation, theaforementioned rotation process can be expressed by

xn � Rnx0; xin�1 � Rxn � RRnx0 � Rin�1x0 (1)

where the rotation matrix Rn represents an existing rotation in thetime interval t0 → tn,R represents an increment of rotation occurredin the time step tn → tin�1, andR

in�1 represents the total rotation. In

this paper, the existing and total rotations of each point of interest aredescribed with rotation matrices.Owing to the orthogonality of rotation matrix, a finite rotation can

be parameterized with only three independent parameters in manypossible ways. Among them, the rotation vector and spatial spinvariable are used to parameterize the incremental and iterativerotations of each point of interest in this paper. For the increment ofrotationR occurred in the time step tn → tin�1, as shown in Fig. 1, thecorresponding rotation vector is defined by

θ � nθ � f θx θy θz gT (2)

wheren is an unit vector that defines the rotation axis, and θ ���������θTθp

is the rotation angle.The rotation matrixR is a nonlinear function of the rotation vector

θ, and they are related by the Rodrigues formula:

R � I3 �sin θ

θ~θ� 1 − cos θ

θ2~θ ~θ � exp� ~θ� (3)

where I3 is the 3 × 3 identity matrix, and the bar (∼) over a vectordenotes the skew-symmetric matrix associated with this vector. Forthe rotation vector θ, the matrix ~θ is written as

~θ �

24 0 −θz θx

θz 0 −θy−θx θy 0

35 (4)

The rotationvector θ can be inversely extracted from a rotationmatrixR, and the extraction method is given by

θ � nθ � arcsin v

vv;

v � 1

2fR32 − R23 R13 − R31 R21 − R12 gT; v �

��������vTv

p(5)

where Rij�i; j � 1; 2; 3� is the element of the rotation matrix R. Toavoid numerical instabilities, when v is very small, arcsin v∕v ≈ 1can be taken for approximation.At the end of the (i� 1)th iteration in the time step tn → tn�1, an

iterative rotation is obtained and superposed on the incrementalrotation to form a compound rotation. Usually, the iterative rotation isparameterized with a spatial spin variable Δω, and the new rotationmatrix is updated as

Rnew � exp�gΔω�R (6)

where exp�gΔω� is the iterative rotation matrix. If this compoundrotation is updated by using rotation vector, then the new rotationvector is updated as

θnew � θ� Δθ (7)

where Δθ is the iterative rotation vector corresponding to the spatialspin variableΔω, and the relation connectingΔθwithΔω is given by

Δθ �H�θ�Δω (8)

where H�θ� is the tangent transformation operator [23,24], and itsexpression is

H�θ� � I3 −1

2~θ� 1

θ2

�1 −

θ∕2tan�θ∕2�

�~θ ~θ (9)

It is shown that the vector addition rule between the rotation vectorand the spatial spin variable does not meet, i.e., θnew ≠ θ�Δω; thereason is that θ and Δω belong to different vector spaces. Hence,before calculating θnew, one should transform Δω into the samevector space that θ belongs to. With the tangent transformationoperator H�θ�, the spatial spin variable Δω is transformed to be anadditive vector of the rotation vector θ, i.e., Δθ. In addition, therelation given in Eq. (8) becomes singular at θ � 2π, with the resultthat the rotation angle cannot exceed 2π when using the rotationvector for parameterization. However, in many nonlinear dynamicanalyses, the rotation angle often exceeds this limitation. To over-come the singularity problem, a usual remedy is to apply the tangenttransformation only within a time step [27].The variation of the rotation matrix is given by

δR � fδωR (10)

where fδω denotes an infinitesimal rotation superposed onto therotationR. Similar to Eq. (8), the corresponding variation of rotationvector is given by

δθ �H�θ�δω (11)

Mathematically, because the rotationmatrix is a two-point tensor, thecalculations concerning the finite rotations can be represented in twoFig. 1 Schematic diagram of finite rotation.

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forms: the material form and the spatial form. The time derivative ofthe rotation matrix in material form is expressed as

_R � R ~Ω (12)

where Ω is the angular velocity in material form. In addition, theangular acceleration vector in material form is denoted by _Ω.It should be noted that the time-integration algorithm applied in

this paper is the generalized energy-momentum method [25], whichstems from thewell-known generalized-α time-integration algorithm[29] in which the standard Newmark formula is employed forupdating the velocities and accelerations. For finite rotations,according to Simo andVu-Quoc [28] and Ibrahimbegovic [27], it hasno physicalmeaning to directly apply the standardNewmark formulato update the angular velocity and angular acceleration in spatialform, because these quantities at different time belong to differentvector spaces. However, the angular velocity and angular accel-eration can be updated correctly in material form, i.e.,

Ωi�1a;n�1 �

γ

βΔtΘi�1a;n�1 −

�γ

β− 1

�Ωa;n −

�γ

2β− 1

�Δt _Ωa;n (13)

_Ωi�1a;n�1 �

1

βΔt2Θi�1a;n�1 −

1

βΔtΩa;n −

�1

2β− 1

�_Ωa;n (14)

where Δt is the time-step size, and β and γ are the integration

parameters. Θi�1a;n�1 � RTa;nθ

i�1a;n�1, θ

i�1a;n�1 is the incremental rotation

vector in spatial form in the time step tn → ti�1n�1 and can be extracted

from the rotation matrix exp�eθi�1a;n�1� � Ri�1a;n�1R

Ta;n.

In the following derivations, the linearizations of the angularvelocity and acceleration in material form are also required. For the(i� 1)th iteration in the time step, tn → tn�1, i.e., they are ob-tained as

ΔΩi�1n�1 �

γ

βΔtRTnH�θin�1�Δωi�1n�1 (15)

Δ _Ωi�1n�1 �

1

βΔt2RTnH�θin�1�Δωi�1n�1 (16)

III. Inertial Force Vector and Tangent Inertia Matrix

In this section, a new formulation for the inertial part in thecorotational dynamic analysis with triangular shell elements waspresented. The kinematic description of the geometrically exactquasi-shell model [26] was introduced to define the motion of anarbitrary material point in a shell element, obtaining the position,velocity, and acceleration vectors. The new inertial force vector wasderived through the variation of the kinetic energy of the element, andthen an exact linearizationwas performed to obtain the tangent inertiamatrix.

A. Kinematics of Shell Element

In the geometrically exact quasi-static shell element model [26],the motion of an arbitrary material point in a shell element can bedefined by the translation of the corresponding middle plane pointand the rotation of a local frame fixed to this middle plane point. As

shown in Fig. 2, an orthogonal local frame S0O is defined at an

arbitrary material point O0 on the middle plane of the initial

configuration C0 of the shell element; and the base vectors are

denoted by �s0O;1; s0O;2; s0O;3�, in which s0O;1 and s0O;2 are placed on themiddle plane, s0O;1 is oriented parallel to the element edge 1–2, and

s0O;3 is perpendicular to the middle plane. In the shell element, an

arbitrary material point located on the axis s0O;3 is denoted by P0. In

the deformed configuration CD, the local frame S0O is rotated to SO,and the base vectors �sO;1; sO;2; sO;3� remain orthogonally. The

material points corresponding to the points O0 and P0 are O and P,respectively.The coordinate transformationmatrices from the global frameG to

the local frames S0O and SO can be written, respectively, as

TGS0O� � s0O;1 s0O;1 s0O;1 �T; TGSO � � sO;1 sO;2 sO;3 �T

(17)

In the global frame, the position vector of the point P is given by

xP � x0O0 � dO � xOP (18)

where

xOP � RxO0P0 � RTTGS0

O

�xO0P0 (19)

Fig. 2 Geometrically exact quasi-static shell element model.

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where �xO0P0 are the coordinates of the vectorO0P0, which is referredto in the local frame S0O; and �x0OP � f 0 0 ζ gT , ζ �H ∈ �−h∕2; h∕2�, in which h is the thickness of the shell element.The rotationmatrixR represents the rotation of the base vectors from

�s0O;1; s0O;2; s0O;3� to �sO;1; sO;2; sO;3�, andR � TTGSOTGS0O . It is shown

that Eq. (19) implies a basic assumption of shell motion: the local

frame S0O rotates as a rigid body, and the thickness of shell does notchange, i.e., �xO0P0 � �xOP, with first-order shear deformations beingaccounted for [26].Substituting Eq. (19) into Eq. (18) gives

xP � x0O0 � d�RTTGS0

O

�xO0P0 (20)

Considering the arbitrariness of point P and ignoring the relevantsuperscript and subscript, one can rewrite the position vector as

x � x0 � d�RTT0 �x0 � x0 � d�R �x (21)

where �x � TT0 �x0.Taking the time derivative of Eq. (21) yields the velocity vector of

an arbitrary point in the shell element as

_x � _d�R ~Ω �x (22)

B. Linear and Angular Momentum and Kinetic Energy of Shell

Element

In the global frame, the linear momentum L and angularmomentum J of a shell element can be formulated, respectively, asfollows:

L �ZVρ _x dV (23)

J �ZVρ�x × _x� dV (24)

where ρ is the material density of the shell element.Substituting Eq. (22) into Eqs. (23) and (24), one obtains the linear

momentum and angular momentum of the element expressed interms of the material angular velocity Ω, respectively, as

L �ZVρ� _d�R ~Ω �x� dV (25)

J �ZVρ�x × _x� dV �

ZVρfx × _d� �exO0 � d�R �x�R ~Ω �xg dV

(26)

Integrating Eqs. (25) and (26) over the shell thickness, H ∈�−h∕2; h∕2� leads to

L �ZAMρ

_d dA (27)

J �ZAfx × �Mρ

_d� � IρΩg dA (28)

where the following properties are considered:

~Ω �x � − ~�xΩ; �fR �x� � R ~�xRT (29)

Mρ �Zh∕2

−�h∕2�ρ dH � ρh (30)

Zh∕2

−�h∕2�ρ ~�x dH � 03 (31)

Iρ � −Zh∕2

−�h∕2�ρ ~�x ~�x dH � TT0 I

0ρT0;

I0ρ � −Zh∕2

−�h∕2�ρe�x0 e�x0 dH � diag

hρh3

12ρh3

120

i (32)

whereMρ is the mass per unit area of the shell element, and I0ρ and Iρare the rotational inertia tensors in material form.The kinetic energy of a shell element is written as

K � 1

2

ZVρ _xT _x dV (33)

Substituting Eq. (22) into Eq. (33), one obtains the kinetic energy ofthe element expressed by the angular velocity in material form as

K � 1

2

ZAf _dTMρ

_d�ΩTIρΩg dA (34)

C. Inertial Force Vector of Shell Element

The variation of kinetic energy of the shell element is obtained as

δK � −ZAfδdTMρ

�d� δωTR�Iρ _Ω� ~ΩIρΩ�g dA (35)

By the straightforward relationship between the variation of thekinetic energy and the virtual work done by the inertial force, theinertial force vector is derived as

δK � −δWiner � −δqTf iner (36)

where δq � f δdT1 δωT1 δdT2 δωT2 δdT3 δωT3 gT is the varia-tion of the vector of global displacements and rotations of theelement.Because δd and δω are infinitesimal quantities, their values at an

arbitrary material point on the middle plane of the shell element canbe linearly interpolated with the nodal values as follows:

δd � N1δd1 � N2δd2 � N3δd3 (37)

δω � N1δω1 � N2δω2 � N3δω3 (38)

N1 � ς1; N2 � ς2; N3 � 1 − ς1 − ς2 (39)

where ς1 and ς2 are the area coordinates of a triangular element.Substituting Eqs. (37) and (38) into Eq. (35), and in terms of

Eq. (36), one can obtain the inertial force vector as

f iner �

8>>>>>>>><>>>>>>>>:

RA N1Mρ

�ddARA N1R�Iρ _Ω� ~ΩIρΩ�dAR

A N2Mρ�ddAR

A N2R�Iρ _Ω� ~ΩIρΩ�dARA N3Mρ

�ddARA N3R�Iρ _Ω� ~ΩIρΩ�dA

9>>>>>>>>=>>>>>>>>;

(40)

It is observed that the new inertial force vector obtained in this paperis formulated in an integral formwhich is differentwith the traditionalformulation [13,16,18]. In the traditional formulation, the inertialforce vector is formulated by multiplying a constant mass matrix bythe acceleration vector. However, if the centrifugal term ~ΩIρΩ in

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Eq. (40) was neglected, the rotation matrixR was assigned with thevalue of its corresponding node, and the accelerations �d and _Ω werelinearly interpolated with the nodal values; then, the new formulationcould be reduced to the traditional one. Comparedwith the traditionalformulation, the new formulation is more accurate, especially fordynamic problems dominated by centrifugal forces. In addition, be-cause the new formulation was derived with a compatible kinematicdescription, it can be further applied to the dynamic analysis of aflexible multibody system of thin-shell structures.

D. Tangent Inertia Matrix of Shell Element

If aNewton iterativemethod is used to solve the nonlinear dynamicequilibrium equation, a consistent linearization of every term of thenonlinear equation is required to be performed to provide the tangentmatrix. In this subsection, the linearization of the inertial force vectoris addressed. For the (i� 1)th iteration in the time step tn → tn�1, theelement configuration at the ith iteration is the known latestconfiguration, and the linearization is performed referring to this

configuration. The notation �·�i�1n�1 � �·�in�1 � Δ�·�i�1n�1 is introducedin the following derivations:For the inertial force vector, we have

f i�1iner;n�1 � f iiner;n�1 � Δf i�1iner;n�1 (41)

The tangent inertia matrix of the shell element is defined as thechanging rate of the inertial force vector f iner with respect to thevector of global displacements and rotations q, i.e.,

Δf i�1iner;n�1 � KiDyn;n�1Δq

i�1n�1; Ki

Dyn;n�1 �∂f iner∂q

����in�1

(42)

The linearizations of the translational part and the rotational part ofthe inertial force vector at node a are, respectively, given by

Δf i�1iner;a;Trans;n�1 �ZANaMρΔ �di�1n�1 dA (43)

Δf i�1iner;a;Rot;n�1 �ZANaΔRi�1

n�1ain�1 dA�

ZANaR

in�1Δa

i�1n�1 dA

(44)

where a � Iρ _Ω� ~ΩIρΩ.The first integral on the right-hand side of Eq. (44) can be

straightforwardly reformulated as

�ΔRi�1n�1�ain�1 �eΔωi�1n�1R

in�1a

in�1 � −�gRi

n�1ain�1�Δωi�1n�1 (45)

where the properties b × c � ~bc � −c × b � − ~cb are used, inwhich b, c are arbitrary spatial vectors.The second integral on the right-hand side of Eq. (44) contains the

linearizations of the angular velocity and angular acceleration inmaterial form. The derived expressions are obtained as

ai�1n�1 � ain�1 � Δai�1n�1

� Iρ _Ωin�1 �eΩi

n�1IρΩin�1 � IρΔ _Ωi�1

n�1 �eΔΩi�1n�1IρΩ

in�1

�eΩin�1IρΔΩ

i�1n�1

(46)

Rin�1Δa

i�1n�1 � Ri

n�1fIρΔ _Ωi�1n�1 � �eΩi

n�1Iρ − �eIρΩi

n�1��ΔΩi�1n�1g(47)

During the iterations, the standard Newmark formula is used toupdate the velocities and accelerations. Hence, the linearization of thetranslational acceleration is given by

Δ �di�1n�1 �1

βΔt2Δdi�1n�1 (48)

The linearizations of the angular velocity and angular acceleration inmaterial form are given by Eqs. (15) and (16). Substituting Eqs. (15),(16), and (48) into the preceding relevant equations, one obtains

Δf i�1iner;a;Trans;n�1 �1

βΔt2

ZANaMρΔdi�1n�1 dA (49)

Δf i�1iner;a;Rot;n�1 � −ZANa�gRi

n�1ain�1�Δωi�1n�1 dA

� 1

βΔt2

ZANaR

in�1IρR

TnH�θin�1�Δωi�1n�1 dA

� γ

βΔt

ZANaR

in�1�eΩi

n�1Iρ − �eIρΩi

n�1��RTnH�θin�1�Δωi�1n�1 dA

(50)

Similarly, the same linear interpolations as for δd and δω are adopted

for the linearization quantitiesΔdi�1n�1 andΔωi�1n�1. Finally, the tangent

inertia matrix can be obtained, which is composed of 6 × 6 subblockmatrices. The subblock matrix associated with nodes a and b�a; b �1; 2; 3� is expressed by

KiDyn;n�1�a;b�

� 1

βΔt2

��RANaNbMρdA

�I3 03

03RANaNbR

in�1IρR

TnH�θin�1�dA

� γ

βΔt

�03 0303

RANaNbR

in�1feΩin�1Iρ− �eIρΩi

n�1�gRTnH�θin�1�dA

−�03 0303

RANaNb�eRin�1a

in�1�dA

(51)

The tangent inertia matrix of an element is grouped as

KiDyn;n�1�

24Ki

Dyn;n�1�1;1� KiDyn;n�1�1;2� Ki

Dyn;n�1�1;3�Ki

Dyn;n�1�2;1� KiDyn;n�1�2;2� Ki

Dyn;n�1�2;3�Ki

Dyn;n�1�3;1� KiDyn;n�1�3;2� Ki

Dyn;n�1�3;3�

35�18×18�

(52)

According to the preceding expression, the tangent inertia matrix canbe considered as a sum of mass matrix Mi

n�1, gyroscopic matrix

Ciiner;n�1, and centrifugal stiffness matrix Ki

iner;n�1, i.e.,

KiDyn;n�1 �

1

βΔt2Mi

n�1 �γ

βΔtCi

iner;n�1 �Kiiner;n�1 (53)

In the implementations of the present formulation, the linearmomentum, angular momentum, kinetic energy, inertial force vector,and tangent inertia matrix of the shell element are numericallyintegrated by using the seven-point Gauss quadrature. The position

vector x, translational velocity _d, and acceleration �d in integral termsare linearly interpolated with the nodal values. The variablesassociated with the finite rotations such as the rotation matrix R,

angular velocityΩ, and acceleration _Ω need to be stored and updatedat each Gauss integration point.

IV. Internal Force Vector and Tangent Stiffness Matrix

In this paper, the consistent symmetrizable equilibrated corota-tional formulation theory [1] was used to handle the formulation ofthe elastic part with triangular shell elements. A linear triangular flat

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shell elementwith three nodes and 18 deg of freedomwas extended tononlinear dynamic analysis. The internal force vector and the tangentstiffness matrix were briefly presented; more detailed derivations canbe found in [1].

A. Elastic Deformation of Shell Element

In the corotational formulation, three kinds of configurations for

an element are defined. They are the initial configuration C0, thecorotated configuration CR , and the deformed configuration CD,which correspond to the initial state, an intermediate state only takingplace in rigid translation and rotation, and the final state subsequentlyoccurring in elastic deformation, respectively, as shown in Fig. 3.Two kinds of coordinate systems are defined by the global frame andthe element frames, in which the element initial frame defined in the

initial configuration is denoted byE0 and the element corotated framedefined in the corotated configuration and the deformed config-uration is denoted by ER � E. There are many choices [9] for

establishing the element frame. In this paper, the origin of E0 is

selected to locate at the centroidC0 of the triangle. The local axis e01 isoriented parallel to the element edge 1–2, whereas e03 is perpendicularto the element plane. Denoting the coordinate transformation

matrices from the global frameG to the element initial frame E0 andto the element corotated frame ER � E as TGE0 and TGE,respectively, we have

TGE0 � � e01 e02 e03 �T; TGE � � e1 e2 e3 �T (54)

In addition, in order to describe the rotations of each node, a triadthat is rigidly tied to the node and rotates with the node is defined. Fornode a, the nodal triad in the initial configuration, the corotatedconfiguration, and the deformed configuration are denoted by S0a, S

Ra ,

and Sa, respectively. In this paper, S0a is selected to be parallel to the

element initial frame E0.In the global frame, for node a, the translation is expressed by a

vector da and the rotation is expressed by a rotation matrix Ra,(a � 1, 2, 3). In terms of the idea of corotational formulation, themotion process of an element from its initial configuration to thedeformed configuration needs two steps. The first step is a rigid-body

motion C0 → CR, in which the rigid translation is given by thecentroid displacement dC and the rigid rotation is represented by therotation matrix Re � TTGETGE0 . The second step is a pure elastic

deformation, i.e., CR → CD. This pure deformation is measured in theelement corotated frame, in which the local deformational translationis denoted by �dda and the local deformational rotation is expressed by

the rotation matrix �Rda. These local deformational variables areformulated as

�dda � �xCa − �x0C0a

(55)

�Rda � TGERaTTGE0 (56)

where

�xCa � TGExCa � TGE�xa − xC�;�x0C0a� TGE0x0

C0a� TGE0�x0a − x0C0� (57)

where xa, x0a and xC, x

0C0 are the position vectors of node a and the

element centroid in the global frame corresponding to the deformedand the initial configurations, respectively.With Eq. (5), the local deformational rotation vector �θda can be

extracted from the rotation matrix �Rda; writing it together with �ddagives the local deformational displacement and rotation of nodea in avector form:

�pda � f �dTda �θTda gT (58)

For the triangular shell element, the vector of local deformationaldisplacements and rotations is written as

�pd � f �dTd1 �θTd1 �dTd2 �θTd2 �dTd3 �θTd3 gT (59)

B. Internal Force Vector of Shell Element

In the element corotated frame, assuming that the local deforma-tion satisfies the small deformation condition, then the local internalforce of an element can be modeled with a linear finite elementanalysis as follows:

�f int � �Km �pd (60)

where �Km is the local material stiffness matrix of a linear triangularflat shell element, which is formulated with the nodal local coor-dinates in the initial configuration. In this paper, the linear triangularshell element adopted is constructed by a combination of the optimalmembrane element [30] with drilling degrees of freedom and thediscrete Kirchhoff triangular plate element [31], and the shell elementhas three nodes and 18 degrees of freedom.

Fig. 3 Corotational description of a shell element.

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The strain energy of element can be written as

S � 1

2�pTd

�Km �pd (61)

The virtual work of the element internal force can be expressed bylocal variables as

δWint � δ �pTd�f int (62)

Similarly, the virtual work of the element internal force can also beexpressed by global variables as

δWint � δqTf int (63)

where δq is the variation of the vector of global displacements androtations of an element, and it has been defined in Sec. III. In theconsistent symmetrizable equilibrated corotational formulation, aftersome algebraicmanipulations, the relationship between δq and δ �pd isconnected as

δ �pd � Λδq (64)

where Λ � �He�PTGE;e is the corotational transformation matrix

[1,16,18], and

�He � diag� I3 H��θd1� I3 H��θd2� I3 H��θd3� � (65)

and

TGE;e � diag�TGE TGE TGE TGE TGE TGE � (66)

�P is the equilibrium projection matrix [1], and its 6 × 6 subblockmatrix is

�P�a; b� ��Uab � �Sa �Gu;b 03

03 δabI3

(67)

Uab ��δab −

1

3

�I3; �Sa � ~�xa (68)

�Gu;1 �1

2A

26640 0 �x32

0 0 �y32

0 − 2Al12

0

3775; �Gu;2 �

1

2A

26640 0 �x13

0 0 �y13

0 2Al12

0

3775;

�Gu;3 �1

2A

26640 0 �x21

0 0 �y21

0 0 0

3775 (69)

where a; b � 1; 2; 3, �xij � �xi − �xj, �yij � �yi − �yj�i; j � 1; 2; 3�, �xand �y are the nodal local coordinate components, and l12 is the lengthof the element edge 1–2; these variables are associated with thedeformed configuration.In terms of the equality of virtual work by an internal force

δqTf int � δ �pTd�f int and Eq. (64), the global internal force vector can

be obtained as

f int � ΛT �f int � ΛT �Km �pd (70)

C. Tangent Stiffness Matrix of Shell Element

The tangent stiffness matrix of the shell element is defined as thechanging rate of the global internal force vector f int with respect tothe vector of global displacements and rotations q. For the (i� 1)thiteration in the time step tn → tn�1, we have

Δf i�1int;n�1 � KiStat;n�1Δq

i�1n�1; Ki

Stat;n�1 �∂f int∂q

����in�1

(71)

By Eq. (70), we have

Δf i�1int;n�1 � ΔΛi�1;Tn�1�f iint;n�1 � Λi;Tn�1Δ �f i�1int;n�1 (72)

In terms of the definition of corotational transformation matrix

Λ � �He�PTGE;e, the first term on the right-hand side of Eq. (72) can

be written as

ΔΛi�1;Tn�1�f iint;n�1 � ΔTi�1;TGE;e;n�1

�Pi;Tn�1 �Hi;Te;n�1

�f iint;n�1

� Ti;TGE;e;n�1Δ �Pi�1;Tn�1�Hi;Te;n�1

�f iint;n�1

� Ti;TGE;e;n�1 �Pi;Tn�1Δ �Hi�1;T

e;n�1�f iint;n�1 (73)

With the derivations by Felippa and Haugen [1], Eq. (73) can finallybe written as

ΔΛi�1;Tn�1�f iint;n�1 � Ki

Stat;G;n�1Δqi�1n�1

� �KiGR;n�1 �Ki

GP;n�1 �KiGM;n�1�Δqi�1n�1 (74)

where KiStat;G;n�1 is the geometric stiffness matrix composed of

three components, which are the rotational geometric stiffness

matrix KiGR;n�1, the equilibrium projection geometric stiffness

matrix KiGP;n�1, and the moment-correction geometric stiffness

matrixKiGM;n�1. In this paper, only their final expressions are given

and the detailed derivations can be found in [1].The rotational geometric stiffness matrixKi

GR;n�1 is produced bythe changing in the global internal force vector caused by rigidrotation of the element, and its expression is given by

KiGR;n�1 � −Ti;TGE;e;n�1 �F

inm;n�1 �G

in�1T

iGE;e;n�1 (75)

where

�Finm;n�1�−�e�niP;1;n�1e�miP;1;n�1e�niP;2;n�1e�mi

P;2;n�1e�niP;3;n�1e�mi

P;3;n�1�T

(76)

�Gin�1 � � �Giu;1;n�1 03 �Giu;2;n�1 03 �Giu;3;n�1 03 � (77)

where �niP;a;n�1 and �miP;a;n�1 are the projected local internal force and

moment for node a, respectively, and they are taken from the

projected local internal forcevector �f iint;P;n�1 � �Pi;Tn�1 �Hi;Te;n�1

�f iint;n�1.The equilibrium projection geometric stiffness matrix Ki

GP;n�1links the changing in the global internal force vector with thechanging in the projection matrix, and its expression is given by

KiGP;n�1 � −Ti;TGE;e;n�1 �G

i;Tn�1

�Fi;Tn;n�1�Pin�1T

iGE;e;n�1 (78)

where

�Fin;n�1 � −�e�niP;1;n�103e�niP;2;n�103e�niP;3;n�103�T (79)

The moment-correction geometric stiffness matrix KiGM;n�1 is

generated by changing in the tangent transformation operation �He,and its expression is given by

KiGM;n�1 � Ti;TGE;e;n�1 �P

i;Tn�1

�Min�1 �P

in�1T

iGE;e;n�1 (80)

where

�Min�1 � diag� 03 �Mi

1;n�1 03 �Mi2;n�1 03 �Mi

3;n�1 � (81)

The subblock matrix �Mia;n�1 for node a is written as

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�Mia;n�1�

η���θi;Tda;n�1 �mi

a;n�1�I3� �θida;n�1 �mi;Ta;n�1−2 �mi

a;n�1�θi;Tda;n�1�

�μe�θida;n�12 �mia;n�1

�θi;Tda;n�1−1

2e�mia;n�1

�H��θida;n�1�

(82)

where the coefficients η and μ are expressed by

η � 1

��θida;n�1�2�1 −

�θida;n�1∕2tan��θida;n�1∕2�

�

μ � ��θida;n�1�2 � 4 cos��θida;n�1� � �θida;n�1 sin��θida;n�1� − 4

4��θida;n�1�4 sin2 ��θida;n�1∕2�(83)

It should be noted that, when �θida;n�1 is very small, the approxi-mations η ≈ 1∕12 and μ ≈ 1∕360 can be taken to avoid numericalinstabilities.The second termon the right-hand side of Eq. (72) can bewritten as

Λi;Tn�1Δ �f i�1int;n�1 � Λi;Tn�1 �KmΔ �pi�1d;n�1 � Λi;Tn�1 �KmΛin�1Δqi�1n�1

� KiStat;M;n�1Δq

i�1n�1 (84)

whereKiStat;M;n�1 � Λi;Tn�1 �KmΛin�1 is the material stiffness matrix of

the element in the global frame.Finally, the tangent stiffness matrix of shell element is obtained as

KiStat;n�1 � Ki

Stat;M;n�1 �KiStat;G;n�1 (85)

V. Nonlinear Dynamic Equation

In this section, the generalized energy-momentum method [25]was applied in conjunction with the Newton–Raphson iterativetechnique to solve the nonlinear dynamic equilibrium equationobtained after spatial discretization by a corotational shell elementformulation. An energy-momentum-conserving algorithm and anenergy-decaying/momentum-conserving algorithm were obtained.The pertinent updating methods for kinematic variables at each nodeand integration point were addressed.

A. Generalized Energy-Momentum Method

The generalized energy-momentum method [25] was developedfrom the generalized-α time-integration algorithm [29] and based onthe idea of energy and momentum conservations of a Hamiltonsystem, and it possesses second-order accuracy, minimal numericaldissipation for low-frequency response, and controllable numericaldissipation for high-frequency response.In the global frame, for the (i� 1)th iteration in the time step

tn → tn�1, the dynamic equilibrium equation of shell element isformulated as

f i�1iner;n�1 � f i�1int;n�1 − f ext;n�1 � 0 (86)

Bymeans of the generalized energy-momentummethod, Eq. (86) canbe rewritten as

f i�1iner;n�1−αm � fi�1int;n�1−αf − f ext;n�1−αf � 0 (87)

whereαm,αf are the integration parameters in the generalized-α time-integration algorithm. It can be seen from Eq. (87) that, in thegeneralized energy-momentum method, the dynamic equilibriumcondition of an element is not established at tn�1 (the end of a timestep), but it is established at tn�1−α, which is a generalized midpointof the time step tn → tn�1.The inertial force f i�1iner;n�1−αm , internal force f i�1int;n�1−αf , and

external force f ext;n�1−αf at the generalized midpoint are defined,

respectively, by

f i�1iner;n�1−αm � f iner� �di�1n�1−αm ;

_Ωi�1n�1−αm ;Ω

i�1n�1−αf ;R

i�1n�1−αf � (88)

f i�1int;n�1−αf � Λi�1;Tn�1−αf�f i�1int;n�1−αf � Λi�1;Tn�1−αf

�Km �pi�1d;n�1−αf (89)

f ext;n�1−αf � �1 − αf�f ext;n�1 � αff ext;n (90)

where, �di�1n�1−αm is the translational acceleration at the generalized

midpoint, and _Ωi�1n�1−αm andΩ

i�1n�1−αf are the angular acceleration and

angular velocity in material form at the generalized midpoint,

respectively. Ri�1n�1−αf is the rotation matrix at the generalized

midpoint. Λi�1n�1−αf is the corotational transformation matrix at the

generalized midpoint. The local internal force at the generalized

midpoint is �f i�1int;n�1−αf . The vector of local deformational displace-

ments and rotations at the generalized midpoint is �pi�1d;n�1−αf . Their

expressions are given as follows:

�di�1n�1−αm � �1 − αm� �di�1n�1 � αm �dn

_Ωi�1n�1−αm � �1 − αm� _Ωi�1

n�1 � αm _Ωn

Ωi�1n�1−αf � �1 − αf�Ωi�1

n�1 � αfΩn

Ri�1n�1−αf � �1 − αf�Ri�1

n�1 � αfRn

Λi�1n�1−αf � �1 − αf�Λi�1n�1 � αfΛn�pi�1d;n�1−αf � �1 − αf� �pi�1d;n�1 � αf �pd;n (91)

It can be seen from Eq. (91) that, in this paper, the integrationparameters αm, αf in the generalized energy-momentum method are

used to form the kinematic variables at the generalizedmidpoint, suchas accelerations, velocities, rotation matrix, corotational trans-formation matrix, and local deformational variables. In the study byYang and Xia [18], these parameters were used to calculate theinertial, internal, and external forces at the generalized midpoint.From the formal point of view, Eq. (91) is more consistent with thealgorithm idea of the generalized energy-momentum method.The integration parameters αm, αf, β, and γ can be defined by using

a single parameter ρ∞ called the spectral radius (also called the high-frequency dissipation coefficientρ∞ ∈ �0; 1�). The expressions ofαm,αf, β, and γ are, respectively, given by

αm �2ρ∞ − 1

ρ∞ � 1; αf �

ρ∞ρ∞ � 1

;

β � 1

4�1 − αm � αf�2; γ � 1

2− αm � αf (92)

Many numerical studies show that the numerical dissipation intro-duced with the choice ρ∞ < 1 in the generalized-α time-integra-tion algorithm cannot always be extended to nonlinear dynamicanalysis [25]. As a remedy, an additional numerical dissipation pro-posed by Armero and Petocz [32] is introduced into the generalizedenergy-momentum method. In the corotational formulation, thisnumerical dissipation coefficient ξ only influences the calculation oflocal internal force at the generalized midpoint. At this point,�f i�1int;n�1−αf in Eq. (91) is rewritten as

�f i�1int;n�1−αf;ξ � �1 − αf � ξ� �f i�1int;n�1 � �αf − ξ� �f int;n� �Km��1 − αf � ξ� �pi�1d;n�1 � �αf − ξ� �pd;n�

(93)

So far, the algorithm expressed by Eqs. (88–93) is a special version ofthe generalized energy-momentum method developed for corota-tional dynamic analysis of thin shells. The choice ρ∞ � 1, ξ � 0, i.e.,

αm � αf � 2β � γ � 12, ξ � 0, corresponding to the case of no

numerical dissipation, yields the energy-momentum-conservingalgorithm, whereas the choice ρ∞ < 1, ξ > 0 results in an energy-decaying/momentum-conserving algorithm.

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B. Iterative Solution and Updating Process

At the ith iteration in the time step tn → tn�1, taking the first-ordertruncated Taylor series of f i�1iner;n�1−αm leads to

f i�1iner;n�1−αm � fiiner;n�1−αm � Δf i�1iner;n�1−αm

� f iiner;n�1−αm �KiDyn;n�1−αmΔq

i�1n�1 (94)

where

KiDyn;n�1−αm �

�1 − αm�βΔt2

Min�1−α �

γ�1 − αf�βΔt

Ciiner;n�1−α

� �1 − αf�Kiiner;n�1−α (95)

where f iiner;n�1−αm , Min�1−α, C

iiner;n�1−α, and Ki

iner;n�1−α are theinertial force and the components of the tangent inertia matrix at thegeneralizedmidpoint, and their expressions are the same as what waspresented in Sec. III, except that the values at the generalizedmidpoint defined in Eq. (92) should be used instead.Similarly, taking the first-order truncated Taylor series of

f i�1int;n�1−αf;ξ at the ith iteration in the time step tn → tn�1, we canobtain

f i�1int;n�1−αf;ξ � fiint;n�1−αf;ξ � Δf i�1int;n�1−αf;ξ

� f iint;n�1−αf;ξ �KiStat;n�1−αf ;ξΔq

i�1n�1 (96)

Combining the corotational formulation addressed in Sec. IV, wehave

KiStat;n�1−αf ;ξ � �1 − αf�Ki

Stat;G;n�1−αf;ξ �KiStat;M;n�1−αf (97)

where the material stiffness matrix at the generalized midpoint

is KiStat;M;n�1−αf � �1 − αf � ξ�Λi;Tn�1−αf �KmΛin�1. In addition, the

geometric stiffness matrix KiStat;G;n�1−αf;ξ can be calculated just by

replacing �f iint;n�1 with �f iint;n�1−αf;ξ.

Inserting Eqs. (94) and (96) into Eq. (87) gives the iterativeequation of an element as

KiElem;n�1−αΔq

i�1n�1 � −gin�1−α (98)

whereKiElem;n�1−α is the tangent matrix of the element, gin�1−α is the

residual force vector, and their expressions are given by

KiElem;n�1−α � Ki

Dyn;n�1−αm �KiStat;n�1−αf;ξ (99)

gin�1−α � f iiner;n�1−αm � fiint;n�1−αf;ξ − f ext;n�1−αf (100)

Assembling Eq. (98) can get the iterative equation of the wholestructure. The iterative convergence criterion adopted in this paper isbased on the magnitude of iterative displacement of the wholestructure, which is kΔQi�1n�1k ≤ 10−6. After the iterative equationwas solved, the iterative displacement and spatial spin variable

at node a extracted from ΔQi�1n�1 are denoted by Δqi�1a;n�1 �

fΔdi�1;Ta;n�1 Δωi�1;Ta;n�1 gT . The following formulas are used for up-

dating the nodal incremental displacement, global coordinate, andnodal rotation matrix, respectively:

ui�1a;n�1 � uia;n�1 � Δdi�1a;n�1 (101)

xi�1a;n�1 � xia;n�1 � Δdi�1a;n�1 (102)

Ri�1a;n�1 � exp�eΔωi�1a;n�1�Ri

a;n�1 (103)

The nodal translational velocity and acceleration are updated by thestandard Newmark formula as follows:

_di�1a;n�1 �γ

βΔtui�1a;n�1 −

�γ

β− 1

�_da;n −

�γ

2β− 1

�Δt �da;n (104)

�di�1a;n�1 �1

βΔt2ui�1a;n�1 −

1

βΔt_da;n −

�1

2β− 1

��da;n (105)

At each Gauss integration point of an element, the spatial spin

variable Δωi�1G;n�1 is linearly interpolated with the nodal values first,

and then the rotation matrix, angular velocity, and angularacceleration in material form are updated, respectively, by usingEqs. (103), (14), and (15). It can be seen that, in addition to the storageof the translational variables and rotation matrix at every node,the present formulation also needs extra storages of rotation variablesat every Gauss integration point, i.e., RG;n, Ri

G;n�1, ΩiG;n�1,

and _ΩiG;n�1.

VI. Numerical Examples

Three classic examples were considered to test the accuracy andcapability of the corotational formulation presented in this paper.These numerical examples dealwith dynamic responses, freemotion,and dynamic buckling of thin-shell structures. The performances ofthe energy-momentum-conserving (EMC) algorithm and the energy-decaying/momentum-conserving (EDMC) algorithms with differentchoices of integration parameters were presented. Comparisonsbetween the results obtained by the present formulation and thatreported by the literature were conducted.

A. Dynamic Responses of Clamped Spherical Cap Shell

This classic numerical example was proposed by Bathe et al. [33].The initial geometry and cross section of the spherical cap shell areshown in Fig. 4. The geometric sizes are radiusR � 0.1209 m, angleα � 10.9°, thickness h � 0.0004 m, and height H � 0.002182 m.The elastic modulus, Poisson’s ratio, and density are E �68.94 GPa, ν � 0.3, and ρ � 2618 kg∕m3, respectively. Thespherical cap shell is clamped, and it is subjected to a stepconcentrated force F at its top point A; F � 444.8 N, t ≥ 0 s.By using the symmetry, one-quarter of the spherical cap shell

structure was modeled. The number of triangular elements was 200,and the time-step size was Δt � 2 μs � 2 × 10−6 s. The used time-

Fig. 4 Clamped spherical cap shell subjected to a step concentrated force at top point.

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integration algorithms were the energy-momentum conservationalgorithm and the energy-dissipation/momentum-conservation(EDMC1 for ρ∞ � 0.80, ξ � 0; and EDMC2 for ρ∞ � 1.0, ξ �0.05) algorithm.The deformed configurations of the clamped spherical cap shell at

different times are shown in Fig. 5. The ratiow∕H of the deflection atpoint A in the z direction and the height of the spherical cap shellvarying with time are shown in Fig. 6. Under the action of the stepconcentrated force, the spherical cap shell produces a periodicdynamic response dominated by its fundamental frequency. Thenumerical results in this paper have a certain difference with theresults by Bathe et al. [33], but they are in very good agreement withthe results by Saigal and Yang [34] and Mondkar and Powell [35].The vertical velocity curves at pointA are shown in Fig. 7. The resultsgivenby theEMCalgorithm and by theEDMC1algorithm are almostcoincident, which further verifies that the numerical dissipationintroduced by ρ∞ < 1 cannot be definitely extended to the nonlineardynamics analysis [25]. Apart from the fundamental frequencycomponent, a large number of high-frequency components areobserved in the velocity responses obtained by the preceding twoalgorithms. However, in the results given by the EDMC2 algorithm,the curve is relatively smoother. The additional numerical dissipationintroduced by ξ > 0 rapidly suppresses the high-frequency com-ponents, whereas it has almost no effect on its fundamental frequencycomponent. The time histories of structural energies are shown inFig. 8, in which K, S, and K� S denote the kinetic, strain, and totalenergies of the shell structure, respectively.

B. Free Motion of Rectangular Thin Shell

This classic numerical example was proposed by Kuhl and Ramn[25]. The rectangular thin shell is shown in Fig. 9. The geometricsizes are length of L � 0.3 m, width of w � 0.06 m, and thicknessof h � 0.002 m. The elastic modulus, Poisson’s ratio, and densityare E � 206 GPa, ν � 0, and ρ � 7800 kg∕m3, respectively. Therectangular thin shell is free of constraint and subjected to the external

forces at the nodes of positions shown in Fig. 9. The expression offorce f�t� is

f�t� �

8<:2.0 × 107t; t ≤ 0.002

8.0 × 104 − 2.0 × 107t; 0.002 < t ≤ 0.004

0; t > 0.004

(106)

The number of elementswas 15 × 3 × 2, and the time-step sizewasΔt � 5.0 × 10−5 s. The used time-integration algorithms were theenergy–momentum-conserving (EMC for ρ∞ � 1, ξ � 0) algorithmand the energy-decaying/momentum-conserving (EDMC1 for ρ∞ �0.95, ξ � 0; and EDMC2 for ρ∞ � 1.0, ξ � 0.02) algorithm.Under the action of the external force, the rectangular thin shell

produces large three-dimensional bending and torsion deformations

Fig. 5 Deformed configurations of clamped spherical cap shell at different times.

Fig. 6 Ratio of vertical deflection at top point and height of clamped

spherical cap shell.

Fig. 7 Vertical velocity responses at top point of clamped spherical cap

shell.

Fig. 8 Kinetic energy K, strain energy S, and total energy K� S of

clamped spherical cap shell.

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while accompanying large rigid-body motions. The deformedconfigurations at different times in the time duration 0–0.04 s areshown in Fig. 10, in which the time interval is 0.004 s. The linearmomentum and angular momentum of the structure are shown inFigs. 11 and 12, respectively. It can be seen that, after the externalforce became zero, all the algorithms conserve the linear momentumand angular momentum of the structure, and the results are in verygood agreement with the results given by Kuhl and Ramn [25]. Itshould be noted that, according to the authors’ knowledge, this wasthe first time the conservations of linear momentum and angularmomentum of a structure due to a time-integration algorithm incorotational nonlinear dynamic analysis were investigated. The timeevolution of the energy of the structure is shown in Fig. 13. Becausethere is not any numerical dissipation in the EMC algorithm, the totalenergy of the structurewas also conserved by this algorithm.However,the total energy of the structurewas dissipated by about 11 and 20%bythe EDMC1 and EDMC2 algorithms, respectively, between 0.004 and0.1 s. Relatively, the effect of numerical dissipation introduced byusing ξ > 0 is larger than that by using ρ∞ < 1.

C. Dynamic Buckling of Top Opened Spherical Cap Shell

This classic numerical example was proposed by Brank et al. [36].The geometry and cross section of the top opened spherical cap shellare shown in Fig. 14. The geometric sizes are radius ofR � 12.16 m,angles of α1 � 18.594° and α2 � 55.668°, and thickness ofh � 0.4 m. The elastic modulus, Poisson’s ratio, and density areE � 1000 Pa, ν � 0.3, and ρ � 0.1 kg∕m3, respectively. The dis-placements in the z direction at the nodes of the bottom ring are set tobe zero.Anodal force is applied in the−zdirection at each node of thetop ring. The shell is meshed into 48 × 10 × 2 triangular elements.The expression of the nodal force f�t� is

f�t� �t; t ≤ 1.0

1.0; t > 1.0(107)

The time-step size was Δt � 0.01 s. The used time-integrationalgorithms were the energy–momentum-conserving (EMC forρ∞ � 1, ξ � 0) algorithm and the energy-decaying/momentum-conserving (EDMC1 for ρ∞ � 0.8, ξ � 0; and EDMC2 forρ∞ � 1.0, ξ � 0.2) algorithm.

Fig. 9 Rectangular thin shell.

Fig. 10 Deformed configuration of rectangular thin shell at different times.

Fig. 11 Linear momentum of rectangular thin shell.

Fig. 12 Angular momentum of rectangular thin shell.

Fig. 13 Kinetic energy K, strain energy S, and total energy K� S of

rectangular thin shell.

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During the process of nodal forces increasing linearly from zero toa constant value, the spherical cap shell experiences a dynamicdeformation process of prebuckling, buckling, and postbuckling.The deformed configurations at different times are shown in Fig. 15.The vertical displacement varying with time at point A for whichthe initial position locates at fR sin α1 0 R cos α2 g is shown inFig. 16. The results are in good agreement with the results by Branket al. [36], but a little phase difference is also observed. The phasedifference might be caused by the geometry of the shell element usedin the computations. Brank et al. [36] employed rectangular shellelements, whereas triangular shell elements were used in this paper.The vertical velocity against time at point A is shown in Fig. 17.

It can be seen that, in the postbuckling stage (about t > 2.2 s), the

responses obtained by using the EMC algorithm contained a lotof high-frequency components. With the introduction of numericaldissipation, the EDMC2 algorithm suppresses most of the high-frequency responses and gives a smooth curve. Unlike in the firstnumerical example, the numerical dissipation introduced by ρ∞ < 1in this example produces a dissipative effect, which confirms theuncertainty [25] of the numerical dissipation introduced by ρ∞ < 1. Inaddition, it is observed that the introduced numerical dissipations inboth algorithms almost have no effect on the low-frequency responses,which inflect in the displacement curves at pointA. At the prebucklingand buckling stages, the results given by the EDMC1 and EDMC2algorithms almost coincide with the results by the EMC algorithm. Atthe postbuckling stage, the displacement response gradually tends to a

Fig. 14 Top opened spherical cap shell.

Fig. 15 Deformed configurations of top opened spherical cap shell at different times.

Fig. 16 Displacements at pointA of top opened spherical cap shell.Fig. 17 Vertical velocity at point A of top opened spherical cap shell.

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constant value under the action of the constant external force. Theenergy curves of the structurevaryingwith time are shown inFig. 18. Itcan be seen that the introduced numerical dissipation obviouslydissipates a part of the total energy of the structure.

VII. Conclusions

A new formulation for corotational nonlinear dynamic analysis ofthin-shell structureswith finite rotations by introducing the kinematicdescription of the geometrically exact quasi-static shell model andapplying the generalized energy-momentum method was presentedin this paper. Three classic numerical examples were computed todemonstrate the performance of the time-integration algorithmsdeveloped in this paper and the capability of the present formulation.The following conclusions can be drawn:1) By neglecting the centrifugal force term, taking the value of a

corresponding node for the rotation matrix, and using a linear globalinterpolation, the new inertia force vector obtained in this paper canbe reduced to a traditional one adopted in the literature. The newformulation for the inertial part may give more accurate results fordynamic problems dominated by centrifugal forces. In addition,starting with a compatible kinematic description with respect to thecorotational formulation, the new formulation has clear physicalmeaning and can be further applied to flexible multibody systemdynamic analysis involving thin-shell structures.2) The energy-momentum-conserving algorithm inherits and

maintains the conservations of energy, linear momentum, andangular momentum for a Hamilton system. The energy-decaying/momentum-conserving algorithm possesses minimal numericaldissipation for low-frequency response and controllable numericaldissipation for high-frequency response. Relatively, the controllablenumerical dissipative effect introduced by a choice of ξ > 0 is largerthan that of ρ∞ < 1.3) The results obtained by the present formulation are in good

agreement with the results reported by the literature. They demonstratethat the present formulation can be used to accurately solve nonlineardynamic problems of thin-shell structures with finite rotations.

Acknowledgment

The authors gratefully acknowledge the support for this work fromthe National Natural Science Foundation of China (grantno. 51075208).

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S. PellegrinoAssociate Editor

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