Henry TY Yang, Chancellor, University of California

*Correspondence to: Henry T. Y. Yang, Chancellor, University of California, Santa Barbara, CA 93106, U.S.A.sProfessor and ChancellortProfessorAAssistant Professor

CCC 0029-5981/2000/010101}27$17.50 Received 26 January 1999Copyright ( 2000 John Wiley & Sons, Ltd.

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng. 47, 101}127 (2000)

A survey of recent shell "nite elements

Henry T. Y. Yang1,*,s, S. Saigal2,t, A. Masud3,A and R. K. Kapania4,t

1;niversity of California, Santa Barbara, CA, ;.S.A.2Civil and Environmental Engineering, Carnegie Mellon ;niversity, Pittsburgh, PA, ;.S.A.

3Civil and Materials Engineering, ;niversity of Illinois at Chicago, Chicago, I¸, ;.S.A.4Aerospace and Ocean Engineering, <irginia Polytechnic Institute and State ;niversity, Blacksburg, <A, ;.S.A.

SUMMARY

Since the mid-1960s when the forms of curved shell "nite elements were originated, including thosepioneered by Professor Gallagher, the published literature on the subject has grown extensively. The "rsttwo present authors and Liaw presented a survey of such literature in 1990 in this journal. ProfessorGallagher maintained an active interest in this subject during his entire academic career, publishingmilestone research works and providing periodic reviews of the literature. In this paper, we endeavor tosummarize the important literature on shell "nite elements over the past 15 years. It is hoped that this will bea be"tting tribute to the pioneering achievements and sustained legacy of our beloved Professor Gallagher inthe area of shell "nite elements. This survey includes: the degenerated shell approach; stress-resultant-basedformulations and Cosserat surface approach; reduced integration with stabilization; incompatible modesapproach; enhanced strain formulations; 3-D elasticity elements; drilling d.o.f. elements; co-rotationalapproach; and higher-order theories for composites. Copyright ( 2000 John Wiley & Sons, Ltd.

KEY WORDS: shell "nite element; composite shells; degenerated shell elements; hourglass control; enhanced strainformulation

1. INTRODUCTION

Since the time of mid-1960s when the forms of curved shell "nite elements were originated,including those developed by Professor Gallagher, the published literature on modelling of platesand shells in the linear and non-linear regimes and their application to dynamic or vibrationanalysis of structures has grown extensively. There has been a tremendous interest on the part ofresearchers with su$ciently large amount of resources devoted to the subject, and there continuesto be innovative activity in computational shell mechanics. In the last three decades, numeroustheoretical models have been developed and applied to various practical circumstances. It may befair to state that no single theory has proven to be general and comprehensive enough for theentire range of applications. At times, generality has been sacri"ced to obtain better performanceover certain class of physical problems with greater accuracy and e$ciency. At other times,accuracy of the schemes has been somewhat compromised to be able to address a wider range of

practical engineering problems. These needs have resulted in a signi"cant addition to theliterature published on the subject.

The computational implementation of shell elements has continued to challenge "nite elementresearchers. Several unresolved issues of the past have been settled over the years, and possibleexplanations of the strange behaviours of seemingly reasonable elements have been developed.Some of the pathologies that could not be explained mathematically eventually o!ered a rigoroussolution, while others being investigated via numerical experimentation still ba%e the researchcommunity.

In this paper, we endeavor to summarize the important milestones achieved by the "niteelement community over the last decade and a half in the arena of computational shell mechanics.Work previous to this period was summarized in Reference [1]. We have tried to provide anextensive survey of the literature, however, because of the sheer magnitude of the literatureavailable on the topic, this survey may not be exhaustive.

In order to organize the literature, we "rst outline the main ideas that have categorized thevarious approaches available in computational shell analysis. In general, they can be listed as:(i) the degenerated shell approach, (ii) stress-resultant-based formulations and Cosserat surfaceapproach, (iii) reduced integration techniques with stabilization (hourglass control), (iv) incom-patible modes approach, (v) enhanced strain formulations (mixed and hybrid formulations),(vi) elements based on the 3-D elasticity theory, (vii) drilling degrees-of-freedom elements,(viii) co-rotational approaches and (ix) higher-order theories for composites.

It is to be noted that any successful shell element is, in fact, a combination of more than one ofthe techniques outlined above. Consequently, in a general setting, these approaches are interre-lated and discussing any one in isolation from the others may not be thorough enough. However,in order to keep the discussions manageable, we choose to follow this organization, and use ourjudgmental discretion in designating an element to a particular category. We must clarify thatsuch designations are by no means to be viewed as rigid. We also seek the pardon of those authorswhose works were not mentioned here due to our negligence.

2. THE DEGENERATED SHELL APPROACH

Over the past two decades, computational shell analysis has been, to a large extent, dominatedby the so-called degenerated solid approach, which "nds its origins in the paper of Ahmadet al. [2]. The popularity of these elements is due, in part, to their simplicity of formulation bywhich the traditional classical shell theories are circumvented. The element is derived directlyfrom the fundamental equations of continuum mechanics. Besides, its implementation in the "niteelement procedures is straightforward. While the basic concept underlying the degeneratedelement is very simple, these elements are generally expensive in computation and, therefore, theirapplication to material non-linear problems, in particular, can be limited. The works of, amongothers, Ramm [3], Hughes and Liu [4, 5], Hughes and Carnoy [6], Bathe and Dvorkin [7],Hallquist et al. [8], and Liu et al. [9], constitute representative examples of this methodologycarried over in its full generality to the non-linear regime. The books by, for example, Bathe [10],Hughes [11], and Cris"eld [12], o!er comprehensive overviews of the degenerated solidsapproach and related methodologies which involve some type of reduction to a resultantformulation. Numerous modi"cations and generalizations of the degenerated shell approach canbe seen in References [13}112].

102 H. T. Y. YANG E¹ A¸.

Copyright ( 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 47, 101}127 (2000)

3. RESULTANT-BASED FORMULATIONS

There is an alternate point of view stating that thin bodies are best treated by replacing thegeneral set of three-dimensional governing equations by a set of, in some sense, equivalentequations leading to the construction of shell theories. Such theories enable an insight into thestructure of the equations involved independently, and prior to the computation itself. Based onthem, powerful "nite elements may be formulated. One of the "rst achievements in this directionwas due to Argyris and co-workers [113}116] in the development of the SHEBA family of "niteelements and, thereafter, their generalizations [117}121].

Eriksen and Truesdell [122] initiated the direct approach to the construction of shell theoriesby considering the shell as a surface with oriented directors. They were inspired by the concept ofa Cosserats [123] continuum by which, in addition to the displacement "eld and independent ofit, rotational degrees of freedom are assigned to every particle of the continuum. The resultingequations and strain measure chosen were quite di!erent from those proposed originally byCosserats [123]. The strain measures suggested in their studies were essentially based on thedi!erence of metrics. As far as the one-director formulation is concerned, it is equivalent to, and infact can be derived as the Green strain tensor of the three-dimensional theory of elasticity, if thedisplacement "eld is assumed to vary linearly over the shell thickness.

Working along similar lines, Simo et al. [124}130] proposed a stress-resultant-based geomet-rically exact shell model which is formulated entirely in stress resultants and is essentiallyequivalent to a one director inextensible Cosserat surface. The work by the research group ofSimo, in fact, represents a return to the origins of classical non-linear shell theory which, asmentioned, has its modern point of departure from the original work of the Cosserats [123],subsequently treated by Eriksen and Turesdell [122], and further elaborated upon by a number ofauthors; notably Green and Laws [131], Green and Zerna [132], and Cohen and DeSilva [133].Over the years, numerous papers have appeared in the literature that have provided sophistica-tion and generalization of these ideas. A list of the related notable works can be seen in References[134}149].

4. REDUCED INTEGRATION WITH STABILIZATION (HOURGLASS CONTROL)

Applications of "nite element methods to problems related to industrial applications, togetherwith the developments of numerical algorithms for non-linear and transient analysis, attracted"nite element researchers to develop elements that were simple and e$cient. This driving force ledto the emergence of a series of elements that used lower-order polynomial expressions, primarilyfor simplicity in mesh generation, and also for robustness in complicated non-linear problemswith multiple contacting surfaces. These elements used the concept of reduced and selectivereduced integration techniques for computational e$ciency. It was noted early on that in non-linear and transient problems, a plate element that requires only a single quadrature point isparticularly desirable since the evaluation of the constitutive equation and element kinematicsconsume a large share of the computer time. A vast portion of the literature has been devoted tothis topic from which we cite some of the most prominent ones. The development started witha pure application of reduced integration techniques. A quadrilateral element with bilinearde#ection and rotation "elds based on Mindlin plate theory with a single quadrature point wasintroduced by Hughes et al. [150] under the name U1. However, the element U1 turned out to be

A SURVEY OF RECENT SHELL FINITE ELEMENTS 103


rank de"cient: the rank of the sti!ness was less than its total number of degrees of freedom minusthe rigid-body modes. For some meshes and boundary conditions this rank de"ciency resulted insingularity or near singularity of the assembled sti!ness matrix, which manifested itself in solutionswith severe spatial oscillations, often called the hourglass patterns. Later Hughes and Tezduyar[151], using a scheme motivated by the work of MacNeal [152], corrected the rank de"ciency byusing 2]2 quadrature and re"ned the interpolation of the transverse shear so that locking could beavoided. However, these schemes resulted in the loss of an attractive potential of the bilinearelement, namely, the use of one-point quadrature. Another approach to this element was taken byTaylor [153], who explored the use of Koslo! and Frazier's [154] hourglass control scheme.

In a contemporary development, a four-node quadrilateral shell element with one quadraturepoint in the midsurface was described by Belytschko and Tsay [155]. This element was adoptedin DYNA3D, PAMCRASH and other commercial programs developed for crashworthinessstudies. The major objective in the development of the Belytschko}Tsay [155] element was toattain a convergent, stable element with the minimum number of computations. For this reason,the element employed bilinear isoparametrics with one quadrature point in the midplane whenthe material was elastic. For non-linear materials, several quadrature points were used throughthe thickness at a single midplane point. Since this element with one-point quadrature would berank de"cient, an hourglass control was added. Because of the emphasis on speed, severalshortcuts were made in formulating the element equations. On the whole, the element hasperformed quite well, but it has two shortcomings: (i) it performs poorly when warped and, inparticular, it does not correctly solve the twisted beam problem, and (ii) it does not pass thequadratic Kirchho!-type patch test in the thin plate limit. The latter shortcoming is shared byHughes}Liu [4] element and its importance was not realized until recently.

A uniform strain hexahedron and quadrilateral with orthogonal hourglass control was de-veloped by Flanagan and Belytschko [156]. They also proposed a treatment of zero-energymodes which arise due to one-point integration of "rst-order isoparametric "nite element. In theirwork, they studied two hourglass control schemes, namely (i) viscous and (ii) elastic. In addition,they also proposed a convenient one-point integration scheme which analytically integrated theelement volume and uniform strain modes. However, the use of one-point quadrature schemes forboth the volumetric and deviatoric stresses resulted in certain deformation modes remainingstressless. The reason lies in that if a mesh is consistent with a global pattern of these (and perhapsrigid body) modes, they quickly dominate and destroy the solution. These modes are calledkinematic, or zero energy modes in the "nite element literature, and hourglass modes forhexahedrons and quadrilaterals in the "nite di!erence literature. Belytschko and Tsai [155] hadproposed a stabilization procedure for controlling the kinematic modes of the four-node, bilinearquadrilateral element when single-point quadrature was used. These kinematics modes manifes-ted themselves by spatial oscillations or singularity of the total sti!ness. In their stabilizationprocedure, additional generalized strains were de"ned which were activated by the kinematicmodes. However, these generalized modes were not activated by rigid-body motion regardless ofthe shape of the quadrilateral. By using a scaling law the stabilization parameters were de"ned sothat they did not adversely a!ect the element's performance. In a contemporary development, thisde"ciency was eradicated in a series of papers by Belytschko and co-workers [157}161].

Working along similar lines, Liu and co-workers [162}165] showed that the stabilizationvectors could, in fact, be obtained naturally by taking partial derivatives with respect to thenatural co-ordinates. Their objective was to control the hourglass mode in the underintegrated"nite elements, to increase the computational e$ciency without adverse e!ects on accuracy, and

104 H. T. Y. YANG E¹ A¸.


to demonstrate that the resulting continuum element did not experience any lockingphenomenon when the material became incompressible. In comparison with the hourglass-controlled "nite elements developed by Belytschko and co-workers [155, 156], this element didnot require any stabilization parameters or numerical integrations.

Stabilization of the underintegrated elements continued to be of considerable importance andled various researchers to develop stabilization schemes based on the assumed strain method.Belytschko et al. [158] developed a projection operator, orthogonal to constant strain "elds onan eight-node hexahedral element with uniform reduced integration. It was shown that thestabilization forces depended only on the element geometry and material properties. The assumedstrain "eld was also used with four-point integration which did not require stabilization. Inaddition, two forms of the B-matrix were studied and it was shown that the mean form is moree$cient since it passed the patch test in a simpli"ed form. Despite its considerable success, theproblem with the assumed strain approach remains that these elements sometime show strangemodes in rather simple engineering problems. Some other researchers who worked along similarlines are listed in References [166}174].

5. INCOMPATIBLE MODES APPROACH

Numerous applications involve deformations which are associated with large strains. Further-more, problems undergoing large elastic strains are often constrained by the incompressibility ofthe material, as is the case for rubber. Due to their simple geometry, four-node quadrilateralelements are widely used in such applications. It is well known that the presence of incompressi-bility leads to the so-called &locking' phenomenon in case of a discretization with standarddisplacement elements. Several methods to circumvent this problem have been developed.Amongst these are the reduced integration techniques or the mixed methods. In some approachesrank de"ciency of underintegrated elements, which then leads to hourglassing, is bypassed bystabilization techniques. Lately, Simo and Rifai [175] in the linear case or Simo et al. [176] in thenon-linear case have developed a family of elements which are based on the Hu}Washizuvariational principle. These elements are extensions of the incompatible QM6 element developedby Taylor et al. [177]. They do not seem to have any rank de"ciency and perform well in bendingsituations as well as in the case of incompressibility. For geometrically non-linear analysis, Hueckand Wriggers [178}179] proposed a similar incompatible quadrilateral element that utilizesa second-order Taylor series expansion of element basis functions in the physical co-ordinates.The element is designated QS6. Later, Wrigger et al. [186] proposed a formulation of the QS6element for large elastic deformations. In their work, the basic Hu}Washizu principle is utilized toderive the underlying equations for the element construction.

Working on the stabilization of the rectangular four-node quadrilateral element, Hueck et al.[178], developed the standard bilinear displacement "eld of the plane linear elastic rectangularfour-node quadrilateral element, enhanced by incompatible modes. The resulting gradient oper-ators were separated into constant and linear parts corresponding to underintegration andstabilization of the element sti!ness matrix. Minimization of potential energy was used to generateexact analytical expressions for the hourglass stabilization of the rectangle. The stabilized elementwas shown to coincide with the element obtained by the mixed assumed strain method.

In a further generalization by Hueck et al. [179], the expressions for gradient operators wereobtained from an expansion of the basis functions into a second-order Taylor series in the



physical co-ordinates. The internal degrees of freedom of the incompatible modes were eliminatedon the element level. A modi"ed change of variables was used to integrate the element matrices.The formulation included the cases of plane stress and plane strain as well as the analysis ofincompressible materials. Some of the related works can be found in References [181}186].

Analysis of three-dimensional non-linear problems is certainly within the reach of the com-putational resources of today. Nevertheless, for rational use of the available computationalpower, the choice of element is a very important factor. In the non-linear Lagrangian computa-tions one would typically opt for solid elements with low-order interpolations: "rstly, becausethey have a more robust performance in the distorted con"gurations, and secondly, because theseelements facilitate more convenient manipulations in the adaptive h-type of mesh re"nement.However, it is well known that the standard trilinear brick elements exhibit rather poorperformance unless additional arti"ces are used.

The method of incompatible modes had been introduced by Wilson et al. [187] as an approachfor improving the behaviour of low-order elements in bending-dominated deformation patterns.The practical features of the method include higher-order accuracy for a coarse mesh, meshdistortion insensitivity and excellent performance in the analysis of nearly incompressible andnon-linear materials. However, the de"ciencies of the initial formulation of Wilson's elementswhen they assume the distorted con"guration, led the researchers to ignore even the desirablefeatures of the method of incompatible modes and not follow the approach. Taylor et al. [177]corrected the particular form of initial formulation by enforcing the patch test satisfaction.However, the method still did not receive wide acceptance. Instead, hybrid formulations whichconsidered the stresses and displacements as independent variables were developed as a successfulalternative. Ibrahimbegovic and Wilson [180] and Ibrahimbegovic and Kozar [181] presenteda geometrically non-linear version of the well-known eight-node Wilson brick element. Theelement was based on variational formulation and was modi"ed via the method of incompatiblemodes. It was shown that the incompatible modes formulation exhibited essentially the sameperformance as the hybrid methods. It is important to note that the displacement-based incom-patible modes formulation possesses de"nite advantages when it comes to non-linear constitutivematerial models. For example, many rate forms of constitutive equations are naturally integratedwith the displacement-driven algorithm, e.g. return mapping algorithm for J2 plasticity, orconstitutive equations directly given in the strain space.

The method of incompatible modes has recently been re-examined within the framework of thethree-"eld Hu}Washizu variational principle. In the work of Simo et al. [176], the originalincompatible mode concept is abandoned, and the enhanced strain "eld is constructed directlyinstead. In addition to the displacement and strain "elds the stress "eld is also constructed as anorthogonal complement to the enhanced strain "eld, so that it does not appear in the "nal form ofthe variational statement.

6. ENHANCED STRAIN APPROACH

Enhanced strain elements have also been an area of active interest. Since these elements performvery well in the incompressible limit as well as in bending situations, they have been applied tosimulate geometrically and materially non-linear problems. Several enhanced strain elementshave been developed over the last years [188}222]. These elements provide a robust tool fornumerical simulations in solid mechanics. Due to the construction of the elements with enhanced

106 H. T. Y. YANG E¹ A¸.


strains, these element formulations show a very good coarse mesh accuracy. Furthermore, theimplementation of inelastic material models is straightforward. Following the work of Simo andRifai [175], serveral other authors have also developed similar element formulations for smallstrain applications.

7. ENHANCED STRAIN METHOD FOR 3-D TYPE ELEMENTS

The search for 3-D type elements which provide a general tool for solving arbitrary problems insolid mechanics has a long history. This can be seen from the large number of papers which havebeen published on the subject. The main goal is to "nd a general element formulation which ful"llsthe following requirements: (i) no locking for incompressible materials, (ii) good bending behaviour,(iii) no locking in the limit of very thin elements, (iv) distortion insensitivity, (v) good coarse meshaccuracy, (vi) simple implementation of non-linear constitutive laws, and (vii) e$ciency. Theserequirements have di!erent origins. The "rst two result from the necessity to obtain acceptableanswers for the mentioned problems, especially the "rst point is essential for the analysis ofrubber-like materials or for classical J

2-elastoplasticity problems. The third point becomes

increasingly important since it enables the user of such elements to simulate shell problems bythree-dimensional elements, which is simpler for complicated structures. This spares the need forintroducing "nite rotations as variables in thin shell problems, results in simpler contact detectionon upper and lower surfaces and provides the possibility to apply three-dimensional constitutiveequations straight away. The fourth point is essential since modern mesh generation tools yield,for arbitrary geometries, unstructured meshes which always include distorted elements. Also,elements get highly distorted during non-linear simulations including "nite deformations. The"fth point results from the fact that many engineering problems have to be modelled asthree-dimensional problems. Due to computer limitations, quite coarse meshes have to be usedoften to solve these problems. Thus, an element which provides a good coarse mesh accuracy isvaluable in these situations. Point six is associated with the fact that more and more non-linearcomputations involving non-linear constitutive models have to be performed to design engineer-ing structures. Thus, an element formulation which allows a straightforward implementation ofsuch constitutive equations is desirable. Lastly, the e$ciency of the element formulation is ofgreat importance when "nite element meshes with several hundred thousands of elements have tobe used to solve complex engineering problems. To construct elements that ful"l most of theserequirements, and possibly all of them, di!erent approaches have been followed throughout thelast decade and a half. Among these are: (i) techniques of underintegration, (ii) stabilizationmethods, (iii) hybrid or mixed variational principles for stresses and displacements, involving theuse of complementary energy, (iv) mixed Hu}Washizu variational principles, (v) mixed varia-tional principles for rotation "elds, and (vi) mixed variational principles for selected quantities.References 223}243 provide a detailed exposition of the various approaches outlined above.

8. DRILLING DEGREES OF FREEDOM ELEMENTS

In recent years there has been a revival of interest in elements possessing in-plane rotationaldegrees of freedom (also called drilling degrees of freedom). Membrane elements of this kindpossess practical advantages in the analysis of shell structures and folded plates. For example,



combining a plate bending element with a membrane element possessing drilling rotations formsa shell element in which each node has six degrees-of-freedom, three displacements and threerotations. Typical membrane "nite elements do not possess the in-plane rotational degree offreedom, and so when combined with a plate element, they form a shell element with only "vedegrees-of-freedom per node. Although it is possible to work in a locally de"ned "ve-degree-of-freedom system at each node, numerous practical di$culties in programming and model con-struction must be overcome. Membrane "nite elements with drilling degrees of freedom circum-vent these problems [246, 247]. Thus, the presence of the sixth nodal degree of freedom is veryappealing from an engineering point of view. Numerous works have appeared in the engineeringliterature in the last decade in which successful approaches towards incorporating drillingrotations in membrane elements have been described [248}284]. It is interesting to note that mostof the elements proposed involve a variety of special devices. The simplest and most commonlyused remedy is the addition of a "ctitious torsional-spring sti!ness at each node. This, however,renders the numerical method inconsistent, possibly degrading its convergence properties. Therehave also been developments in the mathematics literature, where variational formulations thatemploy independent rotation "elds have been studied [244]. Ideas of this kind go back toReissner [268]. Hughes and Brezzi [244], and Hughes et al. [245}247] endeavored to pursue thissubject mathematically, with an aim at developing a theoretically sound and, at the same time,practically useful formulation for engineering applications. A number of variational formulationsfor linear elastostatics with independent rotation "elds were analysed and it was observed thatnumerical methods based on the conventional formulations are unstable when convenientinterpolations are employed. Consequently, several formulations based on modi"cations of theclassical variational framework were proposed and were shown to be convergent for all combina-tions of displacement/rotation interpolations. In particular, a displacement-type modi"ed varia-tional formulation was developed, and numerical assessments of membrane elements emanatingfrom this theory were presented in Hughes et al. [245, 246]. In a subsequent work, Hughes et al.[247] presented variational formulations for elastodynamics and for the corresponding time-harmonic problem. The issue of zero masses associated with the rotational degrees of freedomwas addressed and a novel method for consistently introducing rotational masses was introduced.

Working along similar lines, in a series of papers, Ibrahimbegovic et al. [272}274] presenteddrilling rotations in a stress-resultant-based geometrically non-linear shell model which hadfeatures in common with the approach proposed in Simo et al. [270].

9. COROTATIONAL FRAMEWORK FOR SHELL ANALYSIS

The requirement for more optimally designed structures in aerospace and other applicationsdemands that complex shell structures be analysed well into the non-linear regime. This, in turn,has motivated researchers to develop a number of improvements that permit the accuratemodelling of shells undergoing large rotations, e.g. during large de#ections or postbuckling.Traditionally, the implementation of most large rotation "nite element formulations has beencarried out in a single module where the constitutive law and the element kimematic descriptionsare tightly coupled. This approach renders many existing beam and shell "nite elements, based onmoderate rotation assumptions, ine!ective for large rotation problems. Moreover, there is nogeneral consensus as to which of these newly developed formulations is preferable, and often theanalysts resist parting with the reliable, yet more restrictive elements they have experience with.

108 H. T. Y. YANG E¹ A¸.


Hence, development of a more generic, element-independent approach to large rotations becamea topic of intense investigation. Belytschko and Hsieh [285] proposed a method based onconvected co-ordinates to develop a small strain, large rotation beam element. The use ofconvected co-ordinates, in e!ect, decomposes the motion into its deformational part and rigid-body component. Later, a procedure was developed that uses the above decomposition in aco-rotational co-ordinate frame to compute strains from arbitrarily large displacements androtations for any element. This approach may be used to construct a procedure that extracts,from a given displacement "eld, the pure deformations, i.e. displacement components that are freeof any rigid-body motion. An advantage of this procedure is that it can be implementedindependently of the element formulation. Thus, a set of algebraic operations can be described,and software utilities developed, which extend any implementation of a small displace-ment/rotation element formulation to that of a large displacement/rotation one. These operationsinvolve, in particular, a projection matrix that has a number of interesting properties. First, itconverts a non-equilibrating force vector associated with an element into a self-equilibrating onewhen multiplied with the transpose of the projection matrix. Second, the rigid-body componentsof an incremental displacement vector are eliminated when multiplied by the projector (largerigid-body rotation components are removed by a related projection matrix). Finally, it trans-forms an element sti!ness matrix to one with correct rigid-body properties. If the sti!ness matrixalready has the correct zero-energy modes, this transformation will have no e!ect on the sti!nessmatrix. In other words, the element is forced to have the correct invariance properties underrigid-body motion. These properties of the projection matrix can be used to extend the applica-tion range of many existing beam, plate and shell elements to account for large displacementbehaviour. Following the work of Belytschko, Liu et al. [286, 287] developed multiple quadratureunderintegrated elements.

Working along the lines of co-rotational framework, Moita and Cris"eld [288] developedenhanced lower-order element formulations for large strains where they showed that a more generalprocedure could be devised with the aid of mixed assumed strain procedures. A mathematicaldecomposition of motion into rotation and stretch was provided by Qin et al. [289]. In a sub-sequent work, Peng and Cris"eld [290] described an alternate approach that involves a form ofco-rotational technique. In a continuum context the co-rotational technique has very close linkswith Biot-stress formulation. In their work they showed that once the co-rotational technique isextended to large-strain plasticity, there are some advantages in considering the co-rotationalframework. A co-rotational, updated Lagrangian formulation for geometrically non-linear analy-sis of shells is proposed by Jiang and Chernuka [291, 292]. In their "nite element procedure,a standard updated Lagrangian formulation is employed to generate the tangent sti!ness matrix,and a co-rotational theory is used for updating element strain, stress and internal force vectorsduring the Newton}Raphson iterations. In a subsequent work, Wriggers and Gruttmann [293]and Gruttman et al. [294] developed thin shell formulation with "nite rotations based on theconcept of Biot stress. A set of examples using co-rotational procedure has been given in Jianget al. [292].

10. COMPOSITE SHELL FINITE ELEMENTS

Plate and shell structures made of laminated composite materials have often been modelled asan equivalent single layer using classical laminate theory (CLT), see for example the text by



Jones [295], in which the thickness stress components are ignored. Note that the CLT is a directextension of the classical plate theory in which the well-known Kirchho!}Love kinematichypothesis is enforced, i.e. plane sections remain plane and that a normal to the midplane beforedeformation remains straight and normal to the midplane after deformation. This theory isadequate when the ratio of the thickness to length (or other dimension) is small. However,laminated plates and shells made of advanced "lamentary composite materials are susceptible tothickness e!ects because their e!ective transverse modulii are signi"cantly small as compared tothe e!ective elastic modulus along the "bre direction. Reddy and Kuppusamy [296] have shownthat the natural frequencies predicted by the CLT may be as much as 25 per cent higher thanthose predicted by including the shear e!ects for a plate with side to thickness ratio of 10.Furthermore, the classical theory of plates under-predicts de#ections and overpredicts naturalfrequencies and buckling loads.

In order to overcome the de"ciencies in the CLT, re"ned laminate theories have beenproposed. A review of these theories along with the respective kinematic relations used in thesetheories is available in Reference [297]. These are single-layer theories in which the transverseshear stresses are taken into account. They provide improved global response estimates forde#ections, vibration frequencies and buckling loads of moderately thick composites whencompared to the CLT. A Mindlin-type "rst-order transverse shear deformation theory (FSDT)was "rst developed by Whitney and Pagano [298] for multi-layered anisotropic plates, and byDong et al. [299], and Dong and Tso [300] for multi-layered anisotropic shells. A description ofother available theories can be found, for example, in the review article by Kapania [301]. Bothapproaches (CLT and FSDT) consider all layers as one equivalent single anisotropic layer, thusthey cannot model the warping of cross-sections. Furthermore, the assumption of a non-deformable normal results in incompatible shearing stresses between adjacent layers. The latterapproach, because it assumes constant transverse shear stress, also requires the introduction of anarbitrary shear correction factor which depends on the lamination parameters for obtainingaccurate results. It is well established that such a theory is adequate to predict only the grossbehaviour of laminates. A higher-order theory overcoming some of these limitations was present-ed by Reddy [302] for laminated plates and by Reddy and Liu [303] for laminated shells. Notethat, because of the material mismatch at the intersection of the layers, the single-layer theorieslead to transverse shear and normal stress mismatch at the intersection. This renders thesetheories inadequate for detailed, accurate local stress analysis.

The exact analyses performed by Pagano [304] on the composite #at plates have indicated thatthe in-plane distortion of the deformed normal depends not only on the laminate thickness, butalso on the orientation and the degree of orthotropy of the individual layers. Therefore, thehypothesis of non-deformable normals, while acceptable for isotropic plates and shells, is oftenquite unacceptable for multi-layered anisotropic plates and shells that have a large ratio ofYoung's modulus to shear modulus, even if they are relatively thin. Thus, a transverse sheardeformation theory which also accounts for the warping of the deformed normal is required foraccurate prediction of the elastic behaviour (de#ections, thickness distribution of the in-planedisplacements, natural frequencies, etc.) of multi-layered anisotropic plates and shells.

In view of these issues, a variationally sound theory that accounts for the 3-D e!ects, allowsthickness variation, and permits the warping of the deformed normal, is required for a re"nedanalysis of thick and thin composites. A signi"cant contribution in this direction was presentedby Masud et al. [27]. A number of theories are available that can, short of a full-#edged three-dimensional analysis of plates and shells, accurately and e$ciently predict the stress distribution

110 H. T. Y. YANG E¹ A¸.


including the zig-zag variation of the inplane displacement components in the thickness direction.Two classes of theories are available: layerwise theories and the individual layer plate theory. Inthe layerwise plate theory, suggested by Reddy [305], the continuity of the transverse normal andshear stresses is not enforced. In the individual-layer plate models, see for example, [306}308], thetransverse shear stress continuity is enforced a priori. A recent review of the various availabletheories is given in [309]. For geometrically nonlinear theory, the reader is referred to the workby Librescu [310].

As is the case for isotropic shells, all types of shell elements have been used for the linear andnon-linear analysis of laminated shells. A review of earlier developments (1976}1988) in the "niteelement analysis of laminated shells is given by Kapania [301].

Various theories have been used for the development of "nite elements. Using CLT, the presentauthors [311] developed a 48 degrees-of-freedom "nite element to study geometrically non-linearresponse of imperfect laminated plates and shells. This element was successfully used by Byun andKapania [312] to study the impact response of imperfect laminated plates in conjunction witha reduced-basis approach [313]. A post-processor for this element that can accurately predict theinterlaminar stresses, by integrating the equilibrium equations of a laminated plate, wasdeveloped by Byun and Kapania [314]. In a subsequent study, Kapania and Stoumbos [315]performed impact response of laminated shells. The afore-mentioned element was derived usingthe tensor notation and a shell theory.

There still exists a considerable interest, mainly due to the simplicity of their formulation, inusing a large number of #at elements [316] to model curved shells. The #at &shell' element isobtained by combining a plate element with a membrane element. Often, the "nite elementdesigners use either the constant strain triangular (CST) or the linear strain triangular (LST)element to represent the membrane behaviour. As a result, the element lacks inplane rotationaldegree of freedom. This leads to a singular sti!ness matrix when all elements with a common nodeare coplanar and the local co-ordinate system coincides with the global co-ordinate system.A number of approaches have been suggested to avoid this singularity without overly constrain-ing the element. Zienkiewicz [317], for example, suggests the use of an arbitrary value of therotational sti!ness at that node. The approach is based on determining a unique normal at eachnode and ensuring that the attached elements produce no moments about it [318]. The originalapproach was found to give erroneous results in the case of, for example the linear analysis ofa hook problem, termed the Raasch Challenge [319] problem. This approach was subsequentlymodi"ed [318] and has been implemented in the commercial "nite element program NASTRAN.

Another approach, an obvious one, is to employ an element that has in-plane rotational degreeof freedom. Allman [320] suggested a membrane triangular element that has three degrees-of-freedom, two translations and a rotation, at each node. Ertas et al. [321] presented a three-nodetriangular element, termed AT/DKT, by combining an element similar to the Allman membranetriangular element with the discrete Kirchho! theory (DKT) for formulating the plate bendingelement to study laminated plates. The membrane element was obtained from the linear strainelement using a transformation suggested by Cook [253] and the formulation of the DKTelement is available in [321]. A computer program for this formulation was given by Jeyachan-drabose and Kirkhoppe [323]. Ertas et al. [321] compared their results for a cantilever #at platewith those given by STRI3, a three-node triangular faceted element in the commercial available"nite element program ABAQUS. Kapania and Mohan [324] tested the #at element developedby Ertas et al. [321] for static and dynamic response analyses of laminated shells to study itsaccuracy and convergence characteristics. They also extended the element to analyse shells



subjected to thermal loads. Since the DKT formulation suggested by Batoz et al. [322] does notemploy explicit interpolation functions for the transverse displacements, the determination ofconsistent mass is not straightforward. The mass matrix in the formulation of Kapania andMohan was determined using the cubic polynomial suggested by Specht [325] and the responseof laminated structures under both thermal and other induced strains was studied. The elementwas employed for static analysis, free vibration analysis, and thermal deformation analysis.A numerical example, previously solved by Jonalgadda [326], was also presented to study theresponse of a symmetrically laminated graphite/epoxy laminate excited by a layer of piezoelectricmaterial. The results in all cases were found to be in excellent agreement with those obtained byusing other "nite elements or the Ritz method. The element was used by Kapania and Lovejoy[327] to study the free vibration of point-supported skew plates, and by Kapania et al. [328] tostudy the control of thermal deformation of a spherical mirror segment to be used in next-generation Hubble-type telescope.

The ability of the DKT/AT to model the inplane rotation makes this element quite suitable tostudy large displacement analysis of laminated shells. Mohan and Kapania [329] extended theDKT /AT element to study such behaviour using an updated Lagrangian approach. Results werepresented for large-rotation static response, non-linear dynamic response, and thermal postbuck-ling analyses. The results obtained from the DKT/AT were found to be in excellent agreementwith those available in literature and/or those given by the commercial "nite element codeABAQUS. The element consistently performed better than STRI3, a combination of DKT andCST. Including the inplane rotational sti!ness is, thus, important for large displacement analysis.It is noted that Argyris and Tenek [118, 119, 330] presented geometrically non-linear analysis ofisotropic and composites plates and shells using the three-node #at shell element based on thenatural-mode technique.

Finite elements based on higher-order shear deformation theory have also been developed andemployed. Engelstad et al. [331] have employed a nine-node quadrilateral shell element, de-veloped by Chao and Reddy [332] to study the postbuckling and failure of graphite epoxy platesloaded in compression. Panels with holes were also studied and the results were compared withthe experimental data. A progressive damage model was applied that was successful in predictingthe experimentally observed failure of these panels. Geometrically non-linear response of sti!enedshells was performed by Liao and Reddy [333, 334].

A cylindrical shell "nite element using layerwise theory was developed by Gerhard et al.[335]. The element was employed to study buckling and "rst ply failure of geodesicallysti!ened cylindrical shells using the Tsai}Wu failure criterion. The sti!eners were modelled usinga layerwise beam "nite element allowing their sti!ness to be directly assembled with that of theshell element.

It is noted that the "nite elements developed using layerwise theory can provide more accurateresults, but at a price. The number of unknowns increase as the number of layers increase. Thismay make the use of such elements impractical, especially at the design stage. Individual layertheories, in which the continuity of the transverse stresses is enforced a priori, provide accuratestresses but without the drawbacks of the layerwise theory. Icardi [336], employing the third-order zig-zag theory of Di Sciuva and Icardi [337], developed an eight-node, 56 degrees-of-freedom, curvilinear plate "nite element. The nodal variables were: membrane displacement,transverse shear rotations, de#ections, slopes and curvatures for corner nodes, membranedisplacements and transverse shear rotations for mid-side nodes. The element was able toaccurately predict the transverse shear stresses using constitutive models. Cho [338] has

112 H. T. Y. YANG E¹ A¸.


developed a 40 degrees-of-freedom, eight-node "nite element using the zig-zag theory tostudy static and dynamic response of plates. To the best of our knowledge, the elements based onindividual layer theories have not yet been extended to shells, although it should be straightfor-ward to use these elements to analyse shells due to the presence of the membrane degrees offreedom.

Often, in the analysis of composite panels, the three-dimensional e!ects are importantin certain localized areas, such as those near a free edge. A local/global analysis provides ameans to reduce the CPU time and the storage requirements by using a global method, basedon a plate/shell theory to, determine the overall response and by modelling the regionwith noticeable three-dimensional e!ects using 3-D "nite elements. Such an approach wassuccessfully used by Kapania et al. [339] for composite plates with cut-outs and by Haryadi et al.[340] for composite plates with cracks. For modelling "bre-reinforced polymer-matrix com-posites, it is important to include the viscoelastic behaviour of the polymer matrix. Hammerandand Kapania [341] extended the capability of the AT/DKT [321, 324] element to performviscoelastic analysis of composite plates and shells. The viscoelastic properties are representedusing Prony series.

There is, presently, a considerable interest in modelling plates and shells that have piezoelectriclayers, either embedded or on top or bottom of laminated composites. These piezoelectriclayers act as both sensors as well as actuators [342]. For the most part, the "nite elementmethod is used to analyse these structures. Wang and Rogers [343] presented a laminate platetheory for spatially distributed induced strain actuators. Sophisticated "nite elements are beingdeveloped to analyse piezoelectric plates and shells. It is noted that, for these structures, theconstitutive relations relate stresses to strains and the so-called electric displacements, and theelectric "eld is related to the strain as well as the electric "eld. As a result, both mechanical andelectric quantities (electric potential) are used as nodal variables. Tzou and Ye [344] and Valeyand Rao [345] have performed analysis of shells with piezoelectric layers. A recent review ofapplication of the "nite element method to adaptive plate and shell structures is given by Sunarand Rao [346].

11. CONCLUDING REMARKS

In this paper, recent (last 15 or so years) advances in the "nite element technology forshells have been presented. Some additional recent papers address one or more aspects ofthe "nite element development for shells, for example [347, 348]. Chapelle and Bathe[349] discuss theoretical considerations that must be addressed when developing shell "niteelements that can be used for both bending and membrane dominated behaviours. They alsoprovided a list of test problems that are bending and membrane dominated, respectively. Bathe etal. [350], evaluate the MITC shell element for its performance in solving the test problemssuggested by Chapelle and Bathe [349]. MacNeal [348], provides his perspective on the "niteelement for shell analysis including some recent advances in the use of p-version "nite elementmethod.

Finally, it is noted that recent developments to analyse shells have also included both theboundary element methods and the element-free Galerkin methods. For the boundary elementmethods, the reader is referred to the recent work of Liu [351] and for the element-free Galerkinmethod to Krysl and Belytscko [352, 353].



DEDICATION*A SUMMARY OF PROFESSOR GALLAGHER'S WORKS ONSHELL FINITE ELEMENTS

This paper is dedicated to the memory of Professor Richard H. Gallagher in celebration of hislifetime achievements as an engineer, professor, higher education leader, and also his technicalcontributions to the development of the "eld of "nite elements, from its infancy to maturity.

As this is the journal issue dedicated to Professor Gallagher, there is no need in this particularpaper to account for his lifetime achievements. Rather, we will limit to an account of his researchcontributions in shell "nite elements, which are pertinent to the subject of this survey paper.Among the present authors, Yang was Gallagher's "rst Ph.D. student. Saigal and Kapania wereYang's Ph.D. students. Among the 379 papers surveyed, many authors were Gallagher's formerstudents and colleagues.

During Gallagher's early career, he spent 12 years (1955}1967) working at Bell AerosystemsCompany in Bu!alo, New York. During this period, his published works included the studies oflow aspect ratio wings with the e!ects of aerodynamic heating, optimum analysis and design ofintegral fuselage propellant tanks, elastic characteristics of airframes, laboratory simulationof non-linear static aerothermoelastic behaviour, minimum weight design of framework struc-tures, thermal stresses and buckling of sandwich panels, and some shell related works on elasticbuckling of isotropic cylindrical shells [354] as well as sandwich cylindrical shells [355]. Theseworks were done during the infancy period of the parallel developments of both electronic digitalcomputers and "nite element methods. Most of this work was done by computational methods,which was creatively original at the time and which shaped the earliest form of "nite elementmethods.

In a paper published in 1963, Gallagher and Padlog [356] introduced the concept of theformulation of incremental sti!ness matrix based on the minimum potential energy principle totreat buckling problems. In 1964, Gallagher [357] wrote one of the earliest textbooks on "niteelements, during a time when "nite element methods were neither widely accepted nor evenwidely known. In a report in 1966, Gallagher [358] was among the earliest researchers to developa 24 degree of freedom, doubly curved, thin shell "nite element. In a paper in 1967, Gallagher et al.[359] used #at plate "nite elements to model thin spherical cap to predict the buckling load. Thework in References [356}359] would appear rather primitive from the current point of view. Theywere, nonetheless, pioneering, original, and visionary during that period of time.

In 1968, Gallagher and Yang [360] published the work on shell buckling using a 24 degreeof freedom doubly curved thin shell "nite element developed earlier by Gallagher [358].The incremental sti!ness matrix was formulated using the minimum potential energy theoremand retaining the second-order terms in the strain-displacement equations. In 1969, Gallagher[361] presented a comprehensive paper summarizing the developments of "nite element methodsin the analysis of plates and shells. Later, Gallagher et al. [362] published the work on elasticbuckling of thin shells and extended it to the regime after buckling by including geometricnon-linearity.

In the subsequent few years, Gallagher [363}366] and his students published a series of papersre"ning the formulations for curved shell "nite elements and also progressively developed theprocedure to predict the buckling and postbuckling behaviours of plates and shells withinthe framework of "nite element methods. One notable application of these research works was theapplication to the buckling analysis of hyperbolic cooling towers [367]. During this period oftheir e!orts on the research of shell buckling analysis, Professor Gallagher and his students and

114 H. T. Y. YANG E¹ A¸.


colleagues also explored the e!ect of pressure sti!ness on shell instability [368], and theunsymmetric eigenproblem of shell buckling under pressure load [369]. Gallagher and Murthy[370] also used discrete Kirchho! theory to formulate an anisotropic cylindrical shell "niteelement.

Parallel to the early research of linear and non-linear analysis of thin shells, which included thecontinuously re"ned formulations of four-node doubly curved shell element and continuouslyimproved prediction procedures for pre- and post-buckling analysis, Gallagher and Thomas alsodeveloped a shell "nite element of triangular shape based on generalized potential energy[371, 372]. This triangular shell "nite element was successfully used in the instability analysis oftorispherical pressure vessel heads [373]. Gallagher and Murthy also developed a triangular thinshell "nite element based on discrete Kirchho! theory and performed patch test veri"cations[374}376].

One of Gallagher's numerous contributions in the development of "nite element methods, ingeneral, and the shell "nite elements in particular, was his education of hundreds (or perhapsindirectly thousands) of engineers through his regular classes, short courses, seminars, conferencepresentations, and research collaborations. In this regard, we would like to mention a few of hismost notable books and education papers on shells.

Gallagher's textbook [377] on the fundamentals of "nite elements has been translated into "velanguages, i.e. Japanese, German, French, Chinese, and Russian. The volume on thin shell andcurved member "nite elements edited by Gallagher and Ashwell [378] has been a fundamentalcontribution to the subject. In this book, Gallagher contributed two chapters*Chapter 1 sum-marized the problems and progress in thin shell "nite element analysis and Chapter 9 formulateda triangular thin shell element based on generalized potential energy [372]. One of Gallagher'snotable lecture papers on shell elements was given in Reference [379].

It is with great honor and deep appreciation, we dedicate this paper to the memory of ProfessorR. H. Gallagher.

REFERENCES

1. Yang HTY, Saigal S, Liaw DG. Advances of thin shell "nite elements and some applications*version I. Computersand Structures 1990; 35(4):481}504.

2. Ahmad S, Irons BM, Zienkiewicz OC. Analysis of thick and shell structures by curved "nite element. InternationalJournal for Numerical Methods in Engineering 1990; 2:419}459.

3. Ramm E. A plate/shell element for large de#ection and rotations. In Formulations and Computational Algorithms inFinite Element Analysis, Bathe KJ, Oden JT, Wunderlich W (eds). MIT Press: Cambridge, MA, 1977.

4. Hughes TJR, Liu WK. Nonlinear "nite element analysis of shells: Part I. Three-dimensional shells. ComputerMethods in Applied Mechanics and Engineering 1981; 26:331}362.

5. Hughes TJR, Liu WK. Nonlinear "nite element analysis of shells: Part II. Two-dimensional shells. ComputerMethods in Applied Mechanics and Engineering 1981; 27:167}181.

6. Hughes TJR, Cornoy E. Nonlinear "nite element shell formulation accounting for large membrane strains. ComputerMethods in Applied Mechanics and Engineering 1983; 39:69}82.

7. Dvorkin EN, Bathe KJ. A continuum mechanics based four-node shell element for general non-linear analysis.Engineering Computations 1984; 1:77}88.

8. Hallquist JO, Benson DJ, Goudreau GL. Implementation of a modi"ed Hughes}Liu shell into a fully vectorizedexplicit "nite element code. In Finite Element Methods for Nonlinear Problems, Bergan P et al. (eds). Springer: Berlin,1986; 283}297.

9. Liu WK, Law ES, Lam D, Belytschko T. Resultant-stress degenerated-shell element. Computer Methods in AppliedMechanics and Engineering 1986; 55:259}300.

10. Bathe KJ. Finite Element Procedures in Engineering Analysis. Prentice-Hall: Englewood Cli!s, NJ, 1982.11. Hughes TJR. ¹he Finite Element Method. Prentice-Hall: Englewood Cli!s, NJ, 1987.12. Cris"eld M. Finite Elements on Solution Procedures for Structural Analysis, I. ¸inear Analysis. Pineridge Press:

Swansea, UK, 1986.



13. Wang X, Jiang XY, Lee LHN. Finite deformation formulation of shell element for problems of sheet metal forming.Computational Mechanics 1991; 7:397}411.

14. Chaudhuri RA. A degenerate triangular shell element with constant cross-sectional wrapping. Computers andStructures 1988; 28(3):315}325.

15. Briassoulis D. Monomial test: testing the #exural behavior of the degenerated shell element. Computers andStructures 1988; 29(6):949}958.

16. Briassoulis D. The zero energy models problem of the nine-node Lagrangian degenerated shell element. Computersand Structures 1988; 30(6):1389}1402.

17. Teng JG, Rotter JM. Elastic-plastic large de#ection analysis of axisymmetric shells. Computers and Structures 1989;31(2):211}233.

18. Hsiao KM, Chen Y-R. Nonlinear analysis of shell structures by degenerated isoparametric shell element. Computersand Structures 1989; 31(3):427}438.

19. Farard M, Dhatt G, Batoz JL. A new discrete Kirchho! plate/shell element with updated procedures. Computers andStructures 1989; 31(4):591}606.

20. Aditya AK, Bandyopadhyay JN. Study of the shell characteristics of a paraboloid of revolution shell structure usingthe "nite element method. Computers and Structures 1989; 32(2):423}432.

21. Liu S. A "nite element analysis procedure using simple quadrilateral plate/shell elements. Computers and Structures1989; 32(5):937}944.

22. Yuan KY, Liang CC. Nonlinear analysis of an axisymmetric shell using three noded degenerated isoparametric shellelements. Computers and Structures 1989; 32(6):1225}1239.

23. Surana KS, Orth NJ. Axisymmetric Shell Elements for heat conduction with p-approximation in the thicknessdirection. Computers and Structures 1989; 33(3):689}705.

24. Lakshminarayan HV, Kailash K. A shear deformation curved shell element of quadrilateral shape. Computers andStructures 1989; 33(4):987}1001.

25. Naganarayan BP, Prathap G. Force and moment corrections for the wraped four-node quadrilateral plane shellelement. Computers and Structures 1989; 33(4):1107}1115.

26. Elangovan S. Analysis of functional shell by the isoparametric "nite element. Computers and Structures 1990; 34(2):303}311.

27. Masud A, Panahandeh M. A "nite element formulation for the analysis of laminated composites. ASCE Journal ofEngineering Mechanics 1999; 125(10):1115}1124.

28. White DW, Abel JF. Accurate and e$cient nonlinear formulation of a nine-node shell element with spurious modecontrol. Computers and Structures 1990; 35(6):621}641.

29. Ghosh B, Bandyopadhyay JN. Analysis of paraboloid of revolution type shell structure using isoparametric doublycurved shell elements. Computers and Structures 1990; 36(5):791}800.

30. Saetta AV, Vitallani RV. A "nite element formulation for shells of arbitrary geometry. Computers and Structures1990; 37(5):781}793.

31. Singh RK, Kant T, Kakodkar A. Coupled shell-#uid interaction problems with degenerate shell and three-dimensional #uid elements. Computers and Structures 1991; 38(5/6):515}528.

32. Surana K, Orth NJ. p-version hierarchical axisymmetric shell elements. Computers and Structures 1991; 39(3/4): 257}268.33. Ganasha N, Ramesh TC. Stress analysis of railroad wheel using a conical shell element. Computers and Structures

1991; 40(3):512}526.34. To CWS, Wang B. An axisymmetric thin shell "nite element for vibration analysis. Computers and Structures 1991;

40(3):555}568.35. Kbebari H, Cassell AC. Non-conforming modes stabilization of a nine-node stress-resultant degenerated shell

element with drilling freedom. Computers and Structures 1991; 40(3):569}580.36. Choi CK, Yoo SW. Geometrically nonlinear behavior of an improved degenerated shell element. Computers and

Structures 1991; 40(3):785}794.37. Kirkup SM. The computational modeling of acoustic shields by the boundary and shell element method. Computers

and Structures 1991; 40(5):1177}1183.38. Ben-Zvi R, Libai A, Perl M. A curved axisymmetric shell element for nonlinear dynamic elastoplastic problems*I.

Formulation. Computers and Structures 1992; 42(4):631}639.39. Ben-Zvi R, Libai A, Perl M. A curved axisymmetric shell element for nonlinear dynamic elastoplastic problems*II.

Implementation and results. Computers and Structures 1992; 42(4):641}648.40. Ericksson JL. On the thin shell element for non-linear analysis, based on the isoparametric concept. Computers and

Structures 1992; 42(6):927}939.41. Ganasha N, Ramesh TC. A "nite element based on a discrete layer theory for the free vibration analysis of cylindrical

shells. Computers and Structures 1992; 43(1):137}143.42. Ho PTS. Comparison of performance of a #at faceted shell element and a degenerated superparametric shell element.

Computers and Structures 1992; 44(4):895}904.43. Boisse P, Danel JL, Gelln JC. A simple isoparametric three-node shell "nite element. Computers and Structures 1992;

44(6):1263}1273.

116 H. T. Y. YANG E¹ A¸.


44. Liu JH, Surana KS. p-version axisymmetrical shell element for geometrically nonlinear analysis. Computers andStructures 1993; 49(6):1017}1026.

45. Kabir HRH. A shear-locking free robust isoparametric three-node triangular element for general shells. Computersand Structures 1994; 51(4):425}436.

46. Bhatia RS, Sekhon RS. A novel method of generation exact sti!ness matrices for axisymmetric thin plate and shellelements with special reference to an annular plate element. Computers and Structures 1994; 53(2):305}318.

47. Liu H, Dirr B, Popp K. Comparison of some shell elements. Computers and Structures 1994; 53(2):357}361.48. Kumbasar N, Aksu T. A "nite element formulation for moderately thick shells of general shape. Computers and

Structures 1995; 54(1):49}57.49. Chakravorty D, Bandopadyay JN, Sinha PK. Finite element free vibration of conoidal shells. Computers and

Structures 1995; 56(6):975}978.50. Bhatia RS, Sekhon GS. Generation of an exact sti!ness matrix for a cylindrical shell element. Computers and

Structures 1995; 57(1):93}98.51. Aksu T, A "nite element formulation for shells of negative Gaussian curvature. Computers and Structures 1995;

57(6):973}979.52. Ziyaeifar M, Elwi AE. Degenerated plate-shell elements with re"ned transverse shear strains. Computers and

Structures 1996; 60(6):1079}1091.53. Alaylioglu H, Alaylioglu A. A practicable and highly accurate #at shell hybrid element for engineering application

on PC. Computers and Structures 1997; 63(5):915}925.54. Briassoulis D. Analysis of the load carrying mechanisms of shells using the reformulated four-node C0 shell

element-the Scordelis}Lo barrel vault problem. Computers and Structures 1997; 64(1}4):253}273.55. Koh BC, Kikuchi N. New improved hourglass control for bilinear and trilinear elements in anisotropic linear

elasticity. Computer Methods in Applied Mechanics and Engineering 1987; 65:1}46.56. Argyris J, Balmer H, Doltsinis ISt. Implantation of a nonlinear capability on a linear software system. Computer

Methods in Applied Mechanics and Engineering 1987; 65:267}291.57. Briassoulis D. The C0 shell plate and beam elements freed from their de"ciencies. Computer Methods in Applied

Mechanics and Engineering 1989; 72:243}266.58. Chang TY, Saleeb AF, Graf W. On the mixed formulation of a 9-node Lagrange shell element. Computer Methods in

Applied Mechanics and Engineering 1989; 73:259}281.59. Vu-Quoc L, Mora JA. A class of simple and e$cient degenerated shell elements-analysis of global spurious-mode"ltering. Computer Methods in Applied Mechanics and Engineering 1989; 74:117}175.

60. Shi G, Voyiadjis GZ. Simple and e$cient shear #exible two-node arch/beam and four-node cylindrical shell/plate"nite elements. International Journal for Numerical Methods in Engineering 1991; 31:759}776.

61. Mang HA. On bounding properties of eigenvalues from linear initial FE stability analysis limits from geo-metrically non-linear pre-buckling analysis. International Journal for Numerical Methods in Engineering 1991;31:649}676.

62. Lakis AA, Sinno M. Free vibration of axisymmetric and beam-like cylindrical shells, partially "lled with liquid.International Journal for Numerical Methods in Engineering 1992; 33:235}268.

63. Vlachoutsis S. Shear correction factors for plates and shells. International Journal for Numerical Methods inEngineering 1992; 33:1537}1552.

64. Sorem RM, Surana KS. p-version plate and curved shell element for geometrically non-linear analysis. InternationalJournal for Numerical Methods in Engineering 1992; 33:1683}1701.

65. Buechter N, Ramm E. Shell theory versus degeneration*a comparison in large rotation "nite element analysis.International Journal for Numerical Methods in Engineering 1992; 34:39}59.

66. Phaal R, Calladine CR. A simple class of "nite elements for plate and shell problems II: An element for thin shells,with only translational degrees of freedom. International Journal for Numerical Methods in Engineering 1992;35:979}996.

67. Ravichandran RV, Gould PL, Sridharan S. Localizes collapse of shells of revolution using a local-global strategy.International Journal for Numerical Methods in Engineering 1992; 35:1153}1170.

68. Gebhardt H, Schweizerhof K. Interpolation of curved shell geometries by low order "nite elements-errors andmodi"cations. International Journal for Numerical Methods in Engineering 1993; 36:287}302.

69. Zhong Z-H. A bilinear shell element with the cross-reduced integration technique. International Journal forNumerical Methods in Engineering 1993: 36:611}625.

70. Cook RD. Further development of a three-node triangular shell element. International Journal for NumericalMethods in Engineering 1993; 36:1413}1425.

71. Allman DJ. A basic #at facet "nite element for the analysis of general shells. International Journal for NumericalMethods in Engineering 1994; 37:19}35.

72. Boisse P, Daniel JL, Gelin JC. A C0 three-node shell element for non-linear structural analysis. International Journalfor Numerical Methods in Engineering 1994; 37:2339}2364.

73. Onate E, Zarate F, Flores F. A simple triangular element for thick and thin plate and shell analysis. InternationalJournal for Numerical Methods in Engineering 1994; 37:2565}2582.



74. Kim YY, Kim JG. A simple and e$cient mixed "nite element for axisymmetric shell analysis. International Journalfor Numerical Methods in Engineering 1996; 39:1903}1914.

75. Briassoulis D. The four-node C0 shell element reformulated. International Journal for Numerical Methods inEngineering 1996; 39:2417}2455.

76. Keulen FV, Booij J. Re"ned consistent formulation of a curved triangular "nite rotation shell element. InternationalJournal for Numerical Methods in Engineering 1996; 39:2803}2820.

77. Poulsen PN, Damkilde L. A #at triangular shell element with loof nodes. International Journal for NumericalMethods in Engineering 1996; 39:3867}3887.

78. Bischo!M, Ramm E. Shear deformation elements for large strains and rotations. International Journal for NumericalMethods in Engineering 1997; 40:4427}4449.

79. Hueck U, Wriggers P. A formulation for the 4-node quadrilateral element. International Journal for NumericalMethods in Engineering 1996; 39:3007}3037.

80. Grosh K, Pinsky PM, Malhotra M, Rao VS. Finite element formulation for a ba%ed, #uid-loaded, "nite cylindricalshell. International Journal for Numerical Methods in Engineering 1994; 37:2971}2985.

81. Ettouney MM, Daddazio RP, Abboud NN. The interaction of a submerged axisymmetric shell and three-dimensional internal systems. International Journal for Numerical Methods in Engineering 1994; 37:2951}2970.

82. Jeans RA, Mathews IC. Elastoplastic analysis of submerged #uid-"lled thin shells. International Journal forNumerical Methods in Engineering 1994; 37:2911}2919.

83. Wunderlich W, Schapertons B, Temme C. Dynamic stability of non-linear shells of revolution under considerationof the #uid-soil-structure interaction. International Journal for Numerical Methods in Engineering 1994;37:2679}2697.

84. Okstad KM, Mathisen KM. Towards automatic adaptive geometrically non-linear shell analysis. Part I: Implemen-tation of an h-adaptive mesh re"nement procedure. International Journal for Numerical Methods in Engineering 1994;37:2657}2678.

85. Stein E, Seifert B, Ohnimus S. Adaptive "nite element analysis of geometrically non-linear plates and shells,especially buckling. International Journal for Numerical Methods in Engineering 1994; 37:2631}2655.

86. Tessler A. A C0-anisoparametric three-node shallow shell element. Computer Methods in Applied Mechanics andEngineering 1990; 78:89}103.

87. Kamoulakos A. A catenoidal patch test for the inextensional bending of thin shell "nite elements. Computer Methodsin Applied Mechanics and Engineering 1991; 82:1}32.

88. Providakis CP, Beskos DE. Free and forced vibrations of shallow shells by boundary and interior elements.Computer Methods in Applied Mechanics and Engineering 1991; 92:55}74.

89. A "nite element scheme based on the simpli"ed Reissner equations for shells of revolution. Computer Methods inApplied Mechanics and Engineering 1991; 93:111}124.

90. Oral S, Barut A. A shear-#exible facet shell element for large de#ection and instability analysis. Computer Methods inApplied Mechanics and Engineering 1991; 93:415}431.

91. Bernadou M. C1-curved "nite elements with numerical integration for thin plate and thin shell problems, Part 2:Approximation of thin plate and thin shell problems. Computer Methods in Applied Mechanics and Engineering 1993;102:389}421.

92. Keulen FV, Out AB, Ernst L. Nonlinear thin analysis using a curved triangular element. Computer Methods inApplied Mechanics and Engineering 1993; 103:315}343.

93. Destuyuder P, Salaun M. A mixed "nite element for shell model with free edge boundary conditions. Part 1. Themixed variational formulation. Computer Methods in Applied Mechanics and Engineering 1995; 120:195}217.

94. Destuyuder P, Salaun M. A mixed "nite element for shell model with free edge boundary conditions. Part 2. Thenumerical scheme. Computer Methods in Applied Mechanics and Engineering 1995; 120:195}217.

95. Key SW, Ho! CC. A improved constant membrane and bending stress shell element for explicit transient dynamics.Computer Methods in Applied Mechanics and Engineering 1995; 124:33}47.

96. Milford RV, Schnobrich WC. Degenerated isoparametric "nite elements using explicit integration. InternationalJournal for Numerical Methods in Engineering 1986; 23:133}154.

97. Lindgren LE, Karlesson L. Deformation and stresses in welding of shell structures. International Journal forNumerical Methods in Engineering 1988; 25:635}655.

98. Guan Y, Tang L. A geometrically non-linear quasi-conforming nine-node quadrilateral degenerated solid shellelement. International Journal for Numerical Methods in Engineering 1995; 38:927}942.

99. Parisch H. A continuum-based shell theory for non-linear applications. International Journal for Numerical Methodsin Engineering 1995; 38:1855}1883.

100. Tessler A, Spiridigliozzi L. Resolving membrane and shear locking phenomena curved shear-deformable axisymmet-ric shell elements. International Journal for Numerical Methods in Engineering 1988; 26:1071}1086.

101. Saleeb AF, Chang TY, Yingyeunyong S. A mixed formulation of C0-linear triangular plate/shell element-the role ofedge shear constraints. International Journal for Numerical Methods in Engineering 1988; 26:1101}1128.

102. Brockman RA, Lung FY. Sensitivity analysis with plate and shell "nite elements. International Journal for NumericalMethods in Engineering 1988; 26:1129}1143.

118 H. T. Y. YANG E¹ A¸.


103. Ausserer MF, Lee SW. An eighteen-node solid element for thin shell. International Journal for Numerical Methods inEngineering 1988; 26:1345}1364.

104. Briassoulis D. Machine locking of degenerated thin shell. International Journal for Numerical Methods in Engineering1988; 26:1749}1768.

105. Liao CL, Reddy JN, Engelstad SP. A solid-shell transition element for geometrically non-linear analysis of laminatedcomposite structures. International Journal for Numerical Methods in Engineering 1988; 26:1843}1854.

106. Szabo BA, Sahrmann GJ. Hierarchic plate and shell models based on p-extension. International Journal forNumerical Methods in Engineering 1988; 26:1855}1881.

107. Rhiu JJ, Lee SW. A nine node "nite element for analysis of geometrically non-linear shells. International Journal forNumerical Methods in Engineering 1988; 26:1945}1962.

108. Yang HTY, Wu YC. A geometrically non-linear tensorial formulation of a skewed quadrilateral thin shell "niteelement. International Journal for Numerical Methods in Engineering 1989; 28:2855}2875.

109. Yunus SM, Pawlak TP, Wheeler MJ. Application of the Zienkiewicz}Zhu error estimation for plate and shellanalysis. International Journal for Numerical Methods in Engineering 1990; 29:1281}1298.

110. Wagner W. A "nite element model for non-linear shells of revolution with "nite rotations. International Journal forNumerical Methods in Engineering 1990; 29:1455}1471.

111. Bergmann VL, Mukherjee S. A hybrid strain "nite element for plates and shells. International Journal for NumericalMethods in Engineering 1990; 30:233}257.

112. Mang HA. On special points on load-displacement paths in the pre-buckling domain of thin shells. InternationalJournal for Numerical Methods in Engineering 1991; 31:207}228.

113. Argyris JH, Scharpf DW. The SHEBA family of shell elements for the matrix displacement method. Part I. Naturalde"nition of geometry and strains. Journal of the Royal Aeronautical Society 1968; 72:873}878.

114. Argyris JH, Scharpf DW. The SHEBA family of shell elements for the matrix displacement method. Part II.Interpolation scheme and sti!ness matrix. Journal of the Royal Aeronautical Society 1968; 71:878}883.

115. Argyris JH, Scharpf DW. The SHEBA family of shell elements for the matrix displacement method. Part III. Largedisplacements. Journal of the Royal Aeronautical Society 1969; 73:423}426.

116. Argyris JH, Haase M, Malejannakis GA. Natural geometry of surfaces with speci"c reference to the matrixdisplacement analysis of shells. Proceedings of the Koninklijke Nederlandse Akademic van =etenschappen SerieB 1973; 76(5).

117. Argyris JH. An excursion into large rotations. Computer Methods in Applied Mechanics and Engineering 1982; 32:85}155.

118. Argyris JH, Tenek L. Linear and geometrically nonlinear bending of isotropic and multilayered composite plates bythe natural mode method. Computer Methods in Applied Mechanics and Engineering 1994; 113:207}251.

119. Argyris JH, Tenek L. High-temperature bending, buckling, and post-buckling of laminated composite plates usingthe natural mode method. Computer Methods in Applied Mechanics and Engineering 1994; 117:105}142.

120. Argyris JH, Tenek L. An e$cient and locking-free #at anisotropic plate and shell triangular element. ComputerMethods in Applied Mechanics and Engineering 1994; 118:63}119.

121. Argyris JH, Tenek L. A practicable and locking-free laminated shallow shell triangular element of varying andadoptable curvature. Computer Methods in Applied Mechanics and Engineering 1994; 119:215}282.

122. Erickssen JL, Truesdell C. Exact theory of stress and strain in rods and shells. Archive for Rational Mechanics andAnalysis 1958; 1(4):295}323.

123. Cosserat E, Cosserat F. Theorie des corps deformables. In ¹raite de Physique (2nd edn), Chwolson (ed.). Paris, 1999;953}1173.

124. Simo JC, Fox DD. On a stress resultant geometrically exact shell model. Part I: formulation and optimalparametrization. Computer Methods in Applied Mechanics and Engineering 1989; 72:267}304.

125. Simo JC, Fox DD, Rifai MS. On a stress resultant geometrically exact shell model. Part II: The linear theory;computational aspects. Computer Methods in Applied Mechanics and Engineering 1989; 73:53}92.

126. Simo JC, Fox DD, Rifai MS. On a stress resultant geometrically exact shell model. Part III: Computational aspectsof the nonlinear theory. Computer Methods in Applied Mechanics and Engineering 1990; 79:21}70.

127. Simo JC, Rifai MS, Fox DD. On a stress resultant geometrically exact shell model. Part VI: Conserving algorithmsfor non-linear dynamics. International Journal for Numerical Methods in Engineering 1992; 34:117}164.

128. Simo JC. On a stress resultant geometrically exact shell model. Part VII: Shell intersections with 5/6-DOF "niteelement formulations. Computer Methods in Applied Mechanics and Engineering 1993; 108:319}339.

129. Fox DD, Simo JC. A drill rotation formulation for geometrically exact shells. Computer Methods in AppliedMechanics and Engineering 1992; 98:329}343.

130. Simo JC, Tarnow N. A new energy and momentum conserving algorithm for the non-linear dynamics of shells.International Journal for Numerical Methods in Engineering 1994; 37:2527}2549.

131. Green AE, Laws N. A general theory of rods. Proceedings of the Royal Society of ¸ondon Series A, 1966; 293:145}155.132. Green AE, Zerna W. ¹heoretical Elasticity (2nd edn), Clarendon Press: Oxford, 1960.133. Cohen H, DeSilva CN. Nonlinear Theory of elastic directed surface. Journal of Mathematical Physics 1966;

7:906}966.



134. Mould MP. Comparison solutions for assessment of inextensional bending in shells of quadratic and cubicpolynomial representation. Computational Mechanics 1989; 4:31}45.

135. Poddar B, Mukherjee S. An integral equation analysis of inelastic shells. Computational Mechanics 1989; 4:261}275.136. Reissner E. On "nite axi-symmetrical deformations of thin elastic shells of revolution. Computational Mechanics

1989; 4:387}400.137. Bernadou M, Trouve P. Approximation of general shell problems by #at plate elements: Part 1. Computational

Mechanics 1989; 5:175}208.138. Stein E, Wagner W, Wriggers P. Nonlinear stability-analysis of shell and contact-problems including branch-

switching. Computational Mechanics 1990; 5:428}446.139. Basar Y, Eller C, Kratzig WB. Finite element procedures for the nonlinear dynamic stability analysis of arbitrary

shell structures. Computational Mechanics 1990; 6:157}166.140. Bernadou M, Trouve P. Approximation of general shell problems by #at plate elements: Part 3. Extension to

triangular curved facet elements. Computational Mechanics 1990; 7:1}11.141. Kollmann FG, Bergmann V. Numerical analysis of viscoplastic axisymmetric shells based on a hybrid strain "nite

element. Computational Mechanics 1990; 7:89}105.142. Holzapfel GA. Hermitian-method for the nonlinear analysis of arbitrary thin shell structures. Computational

Mechanics 1991; 8:279}290.143. Su FC, Taber LA. Torsional boundary layer e!ects in shells of revolution undergoing large axisymmetric deforma-

tion. Computational Mechanics 1992; 10:23}37.144. Basar Y, Ding Y. Finite-rotation shell elements for the analysis of "nite-rotation shell problems. International

Journal for Numerical Methods in Engineering 1992; 34:165}169.145. Chroscielewski J, Makowski IJ, Stumpf H. Genuinely resultant shell "nite elements accounting for geometric and

material non-linearity. International Journal for Numerical Methods in Engineering 1992; 35:63}94.146. Madenci E, Barut A. Dynamic response of thin composite shells experiencing non-linear elastic deformations

coupled with large and rapid overall motions. International Journal for Numerical Methods in Engineering 1996;39:2695}2723.

147. Keulen FV. A geometrically nonlinear curved shell element with constant stress resultants. Computer Methods inApplied Mechanics and Engineering 1993; 106:315}352.

148. Sansour C, Bednarczyk H. The casserat surface as a shell model, theory and "nite-element formulation. ComputerMethods in Applied Mechanics and Engineering 1995; 120:1}32.

149. Dvorkin EN, Pantuso D, Repetto EA. A formulation of the MITC4 shell for "nite strain elasto-plastic analysis.Computer Methods in Applied Mechanics and Engineering 1995; 125:17}40.

150. Hughes TJR, Cohen M, Haroun M. Reduced and selective integration techniques in "nite element analysis of plates.Nuclear Engineering and Design 1978; 46:203}222.

151. Hughes TJR, Tezduyar TE. Finite elements based upon mindlin plate theory with particular reference to thefour-node bilinear isoparametric element. Journal of Applied Mechanics 1981; 48:587}596.

152. MacNeal RH. A simple quadrilateral shell element. Computers and Structures 1998; 8:175}183.153. Taylor RL. Finite element for general shell analysis. 5th International Seminar on Computational Aspects of the Finite

Element Method, Berlin, 1979.154. Koslo! D, Frazier G. Treatment of hourglass patterns in low order "nite element codes. Numerical and Analytical

Methods in Geomechanics 1978; 2:52}72.155. Belytschko T, Tsay C. A stabilization procedure for the quadrilateral plate element with one-point quadrature.

International Journal for Numerical Methods in Engineering 1983; 19:405}419.156. Flanagan DP, Belytschko T. A uniform strain hexahedron and quadrilateral with orthogonal hourglass control.

International Journal for Numerical Methods in Engineering 1981; 17:679}706.157. Belytschko T, Wong BL. Assumed strain stabilization procedure for the 9-node Lagrange shell element. Interna-

tional Journal for Numerical Methods in Engineering 1989; 28:385}414.158. Belytschko T, Wong BL, Chiang HY. Advances in one-point quadrature shell elements. Computer Methods in

Applied Mechanics and Engineering 1992; 96:93}103.159. Belytschko T, Bindeman LP. Assumed strain stabilization of the eight node hexahedral element. Computer Methods

in Applied Mechanics and Engineering 1993; 105:225}260.160. Belytschko T, Leviathan I. Physical stabilization of the 4-node shell element with one point quadrature. Computer

Methods in Applied Mechanics and Engineering 1994; 113:321}350.161. Belytschko T, Leviathan I. Projection schemes for one-point quadrature shell elements. Computer Methods in

Applied Mechanics and Engineering 1994; 115:277}286.162. Liu WK, Ong JSJ, Uras RA. Finite element stabilization matrices*a uni"cation approach. Computer Methods in

Applied Mechanics and Engineering 1985; 53:13}46.163. Liu WK, Law ES, Lam D, Belytschko T. Resultant-stress degenerated-shell element. Computer Methods in Applied

Mechanics and Engineering 1986; 55:259}300.164. Liu WK, Hu YK, Belytschko T. Multiple quadrature underintegrated "nite elements. International Journal for

Numerical Methods in Engineering 1994; 37:3263}3289.

120 H. T. Y. YANG E¹ A¸.


165. Liu WK, Hu YK, Belytschko T. Three-dimensional "nite elements with multiple-quadrature-points.¹ransactions ofthe Eleventh Army Conference on Applied Mathematics and Computing, ARO Report 94}1, 229}246.

166. Fish J, Belytschko T. Stabilized rapidly convergent 18-degrees-of-freedom #at shell triangular element. InternationalJournal for Numerical Methods in Engineering 1992; 33:149}162.

167. Belytschko T, Plaskacz EJ. SIMD implementation of a non-linear transient shell program with partially structuredmeshes. International Journal for Numerical Methods in Engineering 1992; 33:997}1026.

168. Kim JH, Lee SW. A "nite element formulation with stabilization matrix for geometrically non-linear shells.International Journal for Numerical Methods in Engineering 1992; 33:1703}1720.

169. Vu-Quoc L, Ho! C. On a highly robust spurious-mode "ltering method for uniformly reduced-integrated shellelements. International Journal for Numerical Methods in Engineering 1992; 34:209}220.

170. Kebari H, Cassell AC. A stabilized 9-node non-linear shell element. International Journal for Numerical Methods inEngineering 1992; 35:37}61.

171. Fish J, Pan L, Belsky V, Gomaa S. Unstructured multigrid method for shells. International Journal for NumericalMethods in Engineering 1996; 39:1181}1197.

172. Vu-Quoc L. A perturbation method for dynamic analysis using under-integrated shell elements. Computer Methodsin Applied Mechanics and Engineering 1990; 79:129}172.

173. Zhu Y, Zacharia T. A new one-point quadrature, quadrilateral shell element with drilling degrees of freedom.Computer Methods in Applied Mechanics and Engineering 1996; 136:195}217.

174. Kamoulakos A. Understanding and improving the reduced integration of Mindlin shell elements. InternationalJournal for Numerical Methods in Engineering 1988; 26:2009}2029.

175. Simo JC, Rifai MS. A class of mixed assumed strain method of incompatible modes. International Journal forNumerical Methods in Engineering 1990; 29:1595}1638.

176. Simo JC, Armero F, Taylor RL. Improved versions of assumed enhanced strain tri-linear elements or 3D "nitedeformation problems. Computer Methods in Applied Mechanics and Engineering 1993; 110:359}386.

177. Taylor RL, Beresford PJ, Wilson EL. A non-conforming element for stress analysis. International Journal forNumerical Methods in Engineering 1976; 10:1211}1219.

178. Hueck U, Reddy BD, Wriggers P. One the stabilization of the rectangular 4-node quadrilateral element. Communica-tions in Numerical Methods in Engineering 1994; 10:555}563.

179. Hueck U, Wriggers P. A formulation for the four-node quadrilateral element. International Journal for NumericalMethods in Engineering 1995; 38:3007}3037.

180. Ibrahimbegovic A, Wilson EL. A modi"ed method of incompatible models. Communications in Numerical Methodsin Engineering 1995; 11:655}664.

181. Ibrahimbegovic A, Kozar I. Non-linear Wilson's brick element for "nite elastic deformations of three-dimensionalsolids. Communications in Numerical Methods in Engineering 1995; 11:655}664.

182. Sez KY, Chow CL. A mixed formulation of a four-node mindlin shell/plate with interpolated covariant transverseshear strains. Computers and Structures 1991; 40(3):775}784.

183. Gruttmann F, Taylor RL. Theory and "nite element formulation of rubberlike membrane shells using principlestretches. International Journal for Numerical Methods in Engineering 1992; 35:1111}1126.

184. Eberlein R, Wriggers P. A fully non-linear axisymmetrical quasi-Kirchho!-type shell element for rubber-likematerials. International Journal for Numerical Methods in Engineering 1993; 36:4027}4043.

185. Brank B, Peric D. On large deformations of thin elasto-plastic shells: implementation of a "nite rotation model forquadrateral shell element. International Journal for Numerical Methods in Engineering 1997; 40:689}726.

186. Wriggers P, Hueck U. A formulation of the QS6 element for large elastic deformation. International Journal forNumerical Methods in Engineering 1996; 39:1437}1454.

187. Wilson EL, Taylor RL, Doherty WP, Ghaboussi J. Incompatible displacement models. In Numerical and ComputerModels in Structural Mechanics, Fenves SJ, Perrone N, Robinson AR, Schnobrich WC (eds). Academic Press:New York, 1973; 43}57.

188. Rhiu JJ, Lee SW. A sixteen node shell element with a matrix stabilization scheme. Computational Mechanics 1988;3:99}113.

189. Reddy BD, Wriggers P. On the stabilization of the rectangular 4-node quadrilateral element. Communications inNumerical Methods in Engineering 1994; 10:555}563.

190. Basar Y, Ding Y, Kratzig WB. Finite-rotation shell elements via mixed formulation. Computational Mechanics 1992;10:289}306.

191. Cris"eld MA, Moita GF, Jelenic G, Lyons LPR. Enhanced lower-order element formulation for large strains.Computational Mechanics 1995; 17:62}73.

192. Lee SC, Yoo CH. A novel shell element including in-plane torque e!ect. Computers and Structures 1988; 28(4):505}522.193. Rankin CC, Nour-omid B. The use of projectors to improve "nite element performance. Computers and Structures

1988; 30(1/2):257}267.194. Kang DS, Pian THH. A 20-DOF hybrid stress general shell element. Computers and Structures 1988; 30(4):789}794.195. Liu Z, Ye R. Large deformation of thin elastic shell by a hybrid stress method. Computers and Structures 1988;

30(4):995}998.



196. Cheung YK, Chen W. Generalized hybrid degenerated elements for planes and shells. Computers and Structures1990; 36(2):279}290.

197. Omurtag MH, Akoz AY. Mixed "nite element formulation of eccentrically sti!ened cylindrical shells. Computers andStructures 1992; 42 (5):751}768.

198. Perry B, Bar-Yoseph P, Rosenhouse G. Rectangular hybrid shell element for analysing folded plate structures.Computers and Structures 1992; 44(1/2):177}185.

199. Peshkam V, Delpak R. A variational approach to geometrically nonlinear analysis of asymmetrically loadedrotational shell-II. Finite element application. Computers and Structures 1993; 46(1):1}11.

200. Basu D, Ghosh KK. Experimental validation of a generalized shell formulation by mixed "nite element approach.Computers and Structures 1993; 48(1):1}6.

201. Omurtag MH, Akoz AY. A compatible cylindrical shell element for sti!ened cylindrical shells in mixed "nite elementformulation. Computers and Structures 1993; 49(2):363}370.

202. Ish J, Guttal R. On the assumed strain formulation with selective polynomial order enrichment for p-version shells.Computers and Structures 1997; 63(5):899}913.

203. Loula AFD, Miranda I, Hughes TJR, Franca LP. On mixed "nite element methods for axisymmetric shell analysis.Computer Methods in Applied Mechanics and Engineering 1989; 72:201}231.

204. Jianqiao Y. Non-linear bending analysis of plates and shells by using a mixed spline boundary element and "niteelement method. International Journal for Numerical Methods in Engineering 1991; 31:1283}1294.

205. Sansour C, Bu#er H. An exact "nite rotation shell theory, its mixed variational formulation and its "nite elementimplementation. International Journal for Numerical Methods in Engineering 1992; 34:73}115.

206. Celigoj CC. A strain- and displacement-based variational method applied to geometrically non-linear shells.International Journal for Numerical Methods in Engineering 1996; 39:2231}2248.

207. Sze KY, Sim YS, Soh AK. A hybrid stress quadrilateral shell element with full rotational D.O.F.S. InternationalJournal for Numerical Methods in Engineering 1997; 40:1785}1800.

208. Lee CK, Hobbs RE. Automatic adaptive re"nement for shell analysis using nine-node assumed strain element.International Journal for Numerical Methods in Engineering 1997; 40:3601}3638.

209. Kemp BL, Cho C, Lee SW. A four-node solid shell element formulation with assumed strain. International Journalfor Numerical Methods in Engineering 1998; 43:909}924.

210. Omurtag MH, Akoz AY. Hyperbolic parabolic shell analysis via mixed "nite element formulation. InternationalJournal for Numerical Methods in Engineering 1994; 37:3037}3056.

211. Pinsky PM, Jasti RV. On the use of large scale multiplier compatible models for controlling accuracy and stability ofmixed shell "nite elements. Computer Methods in Applied Mechanics and Engineering 1991; 85:151}182.

212. Destuyuder P, Salaun M. A mixed "nite element for shell model with free edge boundary conditions. Part 3.Numerical aspects. Computer Methods in Applied Mechanics and Engineering 1996; 136:273}292.

213. Jang J, Pinky PM. Convergence of curved shell elements based on assumed covariant strain interpolations.International Journal for Numerical Methods in Engineering 1988; 26:329}347.

214. Rengarajan G, Aminpour MA, Knight NF. Improved assumed-stress hybrid shell element with drilling degrees offreedom for linear stress buckling and free vibration analyses. International Journal for Numerical Methods inEngineering 1995; 38:1917}1943.

215. Park H, Cho C, Lee SW. An e$cient assumed strain element model with six DOF per node for geometricallynon-linear shells. International Journal for Numerical Methods in Engineering 1995; 38:4101}4122.

216. Yeom CH, Lee SW. An assumed strain "nite element model for large deformation composite shells. InternationalJournal for Numerical Methods in Engineering 1989; 28:1749}1768.

217. Saleeb AF, Chang TY, Graf W, Yingyeunyong S. A hybrid/mixed mode for non-linear shell analysis and itsapplications to large-rotation problems. International Journal for Numerical Methods in Engineering 1990; 29:407}446.

218. Bergmann VL, Mukherjee S. A hybrid strain "nite element for plates and shells. International Journal for NumericalMethods in Engineering 1990; 30:233}257.

219. Parisch H. An investigation of a "nite rotation four node assumed strain shell element. International Journal forNumerical Methods in Engineering 1990; 30:127}150.

220. Yeom CH, Lee SW. On the strain assumption in a "nite element model for plates and shells. International Journal forNumerical Methods in Engineering 1991; 31:287}305.

221. Feng W, Hoa SV. A parcial hybrid degenerated plate/shell element for the analysis of laminated composites.International Journal for Numerical Methods in Engineering 1996; 39:3625}3639.

222. Surana KS, Orth NJ. p-Version hierarchical three dimensional curved shell elements for heat conduction. Computa-tional Mechanics 1991; 7:341}353.

223. Wriggers P, Korelc J. On enhance strain methods for small and "nite deformations of solids. ComputationalMechanics 1996; 7:413}428.

224. Yeom CH, Scardera MP, Lee SW. Shell analysis with a combination of eight node and nine node "nite elements.Computers and Structures 1988; 28(2):155}163.

225. Surana KS, Orth NJ. Three-dimensional curved shell element based on completely hierarchical p-approximation forheat conduction in laminated composites. Computers and Structures 1992; 43 (3):477}494.

122 H. T. Y. YANG E¹ A¸.


226. Gmur TC, Schorderet AM. A set of three-dimensional solid to shell transition element for structural dynamics.Computers and Structures 1993; 46(4):583}591.

227. Kant T, Kumar S, Singh UP. Shell dynamics with three-dimensional degenerated "nite elements. Computers andStructures 1994; 50 (1):135}146.

228. Liu JH, Surana KS. Piecewise hierarchical p-version axisymmetric shell element for laminated composites.Computers and Structures 1994; 50(3):367}381.

229. Buragohain DN, Ravichandran PK. Modi"ed three-dimensional "nite element for general and composite shells.Computers and Structures 1994; 51(3):289}298.

230. Surana KS, Sorem RM. p-version hierarchical three dimensional curved shell element for elastostatics. InternationalJournal for Numerical Methods in Engineering 1991; 31:649}676.

231. Ibrahimbegovic A, Wilson EL. Thick shell and solid "nite elements with independent rotation "elds. InternationalJournal for Numerical Methods in Engineering 1991; 31:1393}1414.

232. Yeh HL, Huang T, Schachar RA. A solid shell element for the shell structured eyeball with application to radialkeratotomy. International Journal for Numerical Methods in Engineering 1992; 33:1875}1890.

233. Mahe M, Sourisseau JC, Ohayon R. Explicit thickness integration for three-dimensional shell elements applied tonon-linear analysis. International Journal for Numerical Methods in Engineering 1993; 36:1085}1114.

234. Andel"nger U, Ramm E. Eas-elements for two-dimensional, three-dimensional, plate and shell structures and theirequivalence to hr-elements. International Journal for Numerical Methods in Engineering 1993; 36:1311}1337.

235. Buchter N, Ramm E, Roehl D. Three-dimensional extension of non-linear shell formulation based on the enhancedassumed strain concept. International Journal for Numerical Methods in Engineering 1994; 37:2551}2568.

236. Gruttmann F, Wagner W. Coupling of two- and three-dimensional composite shell elements in linear and non-linearapplications. Computer Methods in Applied Mechanics and Engineering 1996; 129:271}287.

237. Betsch P, Gruttmann F, Stein E. A 4-node "nite shell element for the implementation of general hyperelastic3D-elasticity at "nite strains. Computer Methods in Applied Mechanics and Engineering 1996; 130:57}79.

238. Wriggers P, Reese S. A note on enhanced strain methods for large deformations. Computer Methods in AppliedMechanics and Engineering 1996; 135:201}209.

239. Wasfy TM, Noor AK. Modeling and sensitivity analysis of multibody systems using new solid, shell and beamelements. Computer Methods in Applied Mechanics and Engineering 1996; 138:187}211.

240. Bathe K-J, Bolourchi S. Large displacement analysis of three-dimensional beam structures. International Journal forNumerical Methods in Engineering 1979; 14:961}986.

241. Dorfmann, Nelson RB. Three-dimensional "nite element for analysing thin plate/shell structures. InternationalJournal for Numerical Methods in Engineering 1995; 38:3453}3482.

242. Vlachoutsis S. Explicit integration for three-dimensional degenerated shell "nite elements. International Journal forNumerical Methods in Engineering 1990; 29:861}880.

243. Cofer WF, Will KM. A three-dimensional, shell-solid transition element for general nonlinear analysis. Computersand Structures 1991; 38(4):449}462.

244. Hughes TJR, Brezzi F. On drilling degrees of freedom. Computer Methods in Applied Mechanics and Engineering1989; 72:105}121.

245. Hughes TJR, Brezzi F, Masud A, Harari I. Finite elements with drilling degrees of freedom: theory and numericalevaluations. In Proceedings of the Fifth International Symposium on Numerical Methods in Engineering, vol. 1.Gruber R, Periaux J, Shaw RP (eds). Springer: Berlin, 1989; 3}17.

246. Hughes TJR, Masud A, Harari I. Numerical assessment of some membrane elements with drilling degrees offreedom. Computers and Structures 1995; 55 (2):297}314.

247. Hughes TJR, Masud A, Harari I. Dynamic analysis with drilling degrees of freedom. International Journal forNumerical Methods in Engineering 1995; 38(19):3193}3210.

248. Allman DJ. A compatible triangular element including vertex rotations for plane elasticity analysis. Computers andStructures 1984; 19:1}8.

249. Allman DJ. The constant strain triangle with drilling rotations: a simple prospect for shell analysis. ¹echnicalReport 86051, Material and Structures Department, Royal Aircraft Establishment, Farnborough, Hants, U.K.,August 1986.

250. Allman DJ. A quadrilateral "nite element including vertex rotations for plane elasticity analysis. InternationalJournal for Numerical Methods in Engineering 1988; 26:717}730.

251. Bergan PG, Elippa CA. A triangular membrane element with rotational degrees of freedom. Computer Methods inApplied Mechanics and Engineering 1985; 50:25}69.

252. Carpenter N, Stolarski H, Belytschko T. A #at triangular shell element with improved membrane interpolation.Communications in Applied Numerical Methods 1985; 1:161}168.

253. Cook RD. On the Allman triangle and a related quadrilateral element. Computers and Structures 1986; 22:1065}1067.254. Cook RD. A plane hybrid element with rotational d.o.f. and adjustable sti!ness. International Journal for Numerical

Methods in Engineering 1987; 24:1499}1508.255. Fox DD, Simo JC. A drill rotation formulation for geometrically exact shells. Computer Methods in Applied

Mechanics and Engineering 1992; 98:329}343.



256. Harder RL, MacNeal RH. A new membrane option for the QUAD4. MSC/NAS¹RAN;sers Conference, Pasadena,California, March 1985.

257. Ibrahimbegovic A, Wilson EL. A uni"ed formulation for triangular and quadrilateral #at shell "nite elements withdrilling degrees of freedom. Communications in Applied Numerical Methods 1991; 7:1}9.

258. Jaamei S. Numerical tests with JET shell elements. Part II: Plate tests. IREM Internal Report 88/5, D'epartment deG'enie Civil, Institut de Statique et Structures, 'Ecole Polytechnique F'ed'erale de Lausanne, April 1988.

259. Jaamei S. JET thin shell "nite elements with drilling rotations. IREM Internal Report 88/7, D'epartment de G'enieCivil, Institut de Statique et Structures, 'Ecole Polytechnique F'ed'erale de Lausanne, July 1988.

260. Jaamei S, Frey F, Jetteur Ph. Nonlinear thin shell "nite element with six degrees of freedom per node. SeventhInternational Symposium on Computing Methods in Applied Sciences and Engineering, Versailles, December 1987.

261. Jaamei S, Frey F, Jetteur P. Numerical tests with JET shell element. Part III: JET shell tests. IREM Internal Report88/8, D'epartment de G'enie Civil, Institut de Statique et Structures, 'Ecole Polytechnique F'ed'erale de Lausanne,August 1988.

262. Jetteur P. A shallow shell element with in-plane rotational degrees of freedom. IREM Internal Report 86/3,D'epartement de G'enie Civil, Institut de Statique et Structures, 'Ecole Polytechnique F'ed'erale de Lausanne,March 1986.

263. Jetteur P. Improvement of the quadrangular JET shell element for a particular class of shell problems. IREMInternal Report 87/1, D'epartement de G'enie Civil, Institut de Statique et Structures, 'Ecole PolytechniqueF'ed'erale de Lausanne, February 1987.

264 Jetteur P. Improvement and large rotations of the JET shell element in nonlinear analysis. IREM Internal Report 87/4,D'epartement de G'enie Civil, Institut de Statique et Structures, 'Ecole Polytechnique F'ed'erale de Lausanne, May1987.

265. Jetteur P, Frey F. A four node Marguerre element for non-linear shell analysis. Engineering Computations 1986;3:276}282.

266. MacNeal RH, A theorem regarding the locking of tapered four-noded membrane elements. International Journal forNumerical Methods in Engineering 1987; 24:1793}1799.

267. MacNeal RH, Harder RL. A re"ned four-noded membrane element with rotational degrees of freedom. Computersand Structures 1988; 28:75}88.

268. Reissner E. A note on variational theorems in elasticity. International Journal of Solids and Structures 1965; 1:93}95.269. Sabir AB. A rectangular and triangular plane elasticity element with drilling degrees of freedom. Proceedings of the

Second International Conference on <ariational Methods in Engineering, Springer: Berlin, 1985; 17}25.270. Simo JC, Fox DD, Hughes TJR. Formulations of "nite elasticity with independent rotations. Computer Methods in

Applied Mechanics and Engineering 1992; 95:277}288.271. Yunus SM. A study of di!erent hybrid elements with and without rotational degrees of freedom for plane

stress/plane strain problems. Computers and Structures 1988; 30:1127}1133.272. Ibrahimbegovic A. Stress resultant geometrically nonlinear shell theory with drilling rotations*Part I. A consistent

formulation. Computer Methods in Applied Mechanics and Engineering 1994; 118:265}284.273. Ibrahimbegovic A, Frey F. Stress resultant geometrically nonlinear shell theory with drilling rotations*Part II.

Computational aspects. Computer Methods in applied mechanics and Engineering 1994; 118:285}308.274. Ibrahimbegovic A, Frey F. Stress resultant geometrically non-linear shell theory with drilling rotations*Part III:

Linearized kinematics. International Journal for Numerical Methods in Engineering 1994; 37:3659}3683.275. Robinson J, Sheng LD. Cook's quadrilateral membrane with rotational degrees of freedom. Finite Element News

1988; 22}27.276. Taylor RL, Simo JC. Bending and membrane elements for analysis of thick and thin shells. In NUMETA 85 Numerical

Methods in Engineering: ¹heory and Applications, Middleton J, Pande GN (eds). A.A. Balkema: Rotterdam,1985;587}591.

277. Ernadou M, Trouve P. Approximation of general shell problems by #at plate elements: Part 2. Addition of a drillingdegree of freedom. Computational Mechanics 1990; 6:359}378.

278. Cook RD. Four-node &#at' shell element: drilling degrees of freedom, membrane-bending coupling, warped ge-ometry, and behavior. Computers and Structures 1994; 50(4):549}555.

279. Chinosi C. Shell elements as a coupling of plate and drill elements. Computers and Structures 1995; 57(5):893}902.280. Aminpour MA. An assumed-stress hybrid 4-node shell element with drilling degrees of freedom. International

Journal for Numerical Methods in Engineering 1992; 33:19}38.281. Aminpour MA. Direct formulation of a hybrid 4-node shell element with drilling degrees of freedom. International

Journal for Numerical Methods in Engineering 1992; 35:997}1013.282. Yeh JT, Chen WH. Shell elements with drilling degree of freedoms based on micropolar elasticity theory.

International Journal for Numerical Methods in Engineering 1993; 36:1145}1159.283. Jaamei S, Frey F, Jetteur P. Nonlinear thin shell "nite element with six degrees of freedom per node. Computer

Methods in applied mechanics and Engineering 1989; 75:251}266.284. Paramasivam V, Raj DM. Shear-deformable axisymmetric conical shell element with 6-DOF and convergence of

O(h4). Computer Methods in Applied Mechanics and Engineering 1994; 113:47}54.

124 H. T. Y. YANG E¹ A¸.


285. Belytschko T, Hseih BJ. Nonlinear "nite element analysis with convected coordinates. International Journal forNumerical Methods in Engineering 1973; 7:255}271.

286. Liu WK, Guo Y, Tang S, Belytschko T. A multiple-quadrature eight-node hexahedral "nite element for largedeformation elastoplastic analysis. Computer Methods in Applied Mechanics and Engineering, 1999, in press.

287. Liu WK, Hu YK, Belytschko T. Multiple quadrature underintergrated "nite elements. International Journal forNumerical Methods in Engineering 1994; 37:3263}3289.

288. Moita GF, Cris"eld MA. A "nite element formulation for 3-D continua using the co-rotational technique.International Journal for Numerical Methods in Engineering 1996; 39:3775}3792.

289. Qin Z, Chen Z. Large deformation analysis of shells with "nite element method based on the S-R Decompositiontheorem. Computers and Structures 1988; 30(4):957}961.

290. Peng X, Cris"eld MA. A consistent co-rotational formulation for shells using the constant stress/constant momenttriangle. International Journal for Numerical Methods in Engineering 1992; 35:1829}1847.

291. Jiang L, Chernuka MW. A Simple four-noded corotational shell element for arbitrarily large rotations. Computersand Structures 1994; 53(5):1123}1132.

292. Jiang L, Chernuka MW, Pegg NG. A co-rotational, updated Lagrangian formulation for geometrically nonlinear"nite element analysis of shell structures. Finite Elements in Analysis and design 1994; 18:129}140.

293. Wriggers P, Gruttmann F. Thin shell with "nite rotations formulated in biot stresses: theory and "nite elementformulation. International Journal for Numerical Methods in Engineering 1993; 36:2049}2071.

294. Gruttmann F, Stein E, Wriggers P. Theory and numerics of thin elastic shells with "nite rotations. Ingenieur Archiv1989; 59:54}67.

295. Jones RM. Mechanics of Composite Materials (2nd edn). Taylor and Francis: Philadelphia, 1999.296. Reddy JN, Kuppusamy T. Natural vibrations of anisotropic plates. Journal of Sound and <ibrations 1984; 94:63}69.297. Kapania RK, Raciti S. Recent advances in analysis of laminated beams and plates, Part I: shear e!ects and buckling.

AIAA Journal 1989; 27:923}934.298. Whitney J, Pagano N. Shear deformation in heterogeneous anisotropic plates. Journal of Applied Mechanics 1970;

37:1031.299. Dong SB, Pister KS, Taylor RL. On the theory of laminated anisotropic shells and plates. Journal of Aerospace

Sciences 1962; 29.300. Dong SB, Tso FKW. On a laminated orthotropic shell theory including transverse shear deformation. ASME

Journal of Applied Mechanics 1972; 39:1091}1096.301. Kapania RK. A review on the analysis of laminated shells. ASME Journal of Pressure <essel ¹echnology 1989;

111:88}96.302. Reddy JN. A General third-order nonlinear theory of plates with moderate thickness. Journal of Nonlinear

Mechanics 1990; 25:677}686.303. Reddy JN, Liu CF. A higher-order shear deformation theory of laminated elastic shells. International Journal of

Engineering Sciences 1985; 23:319}330.304. Pagano NJ. Exact solutions for rectangular bidirectional composites and sandwich plates. Journal of Composite

Materials 1970; 4:20}34.305. Reddy JN. A generalization of two-dimensional theories of laminated composite plates. Communications in Applied

Numerical Methods 1987; 3:173}180.306. Di Sciuva M. An improved shear-deformation theory for moderately thick multilayered anisotropic shells and

plates. ASME Journal of Applied Mechanics 1987; 54:589}596.307. Bhaskar K, Varadan TK. Re"nement of higher-order laminated plate theories. AIAA Journal 1989; 27:1830}1831.308. Cho KN, Bert CW, Striz AG. Free vibrations of laminated rectangular plates analyzed by higher-order individual

layer theories. Journal of Sound and <ibration 1991; 145:429}442.309. Reddy JN, Robbins Jr. DH. Theories and computational models for composite laminates. ASME Applied Mechanics

Reviews 1994; 47:147}169.310. Librescu L. Re"ned geometrically nonlinear theories of laminated shells. Quarterly Journal of Applied Mathematics

1987; 1}27.311. Saigal S, Kapania RK, Yang TY. Geometrically nonlinear "nite element analysis of imperfect laminated shells.

Journal of Composite Materials 1986; 20:197}214.312. Byun C, Kapania RK. Nonlinear impact response of thin imperfect plates using a reduction method. Composites

Engineering: an International Journal 1992; 2:391}410.313. Kapania RK, Byun C. Reduction methods based on eigenvectors and Ritz vectors for nonlinear transient analysis.

Computational Mechanics: an International Journal 1993; 11:65}82.314. Byun C, Kapania RK. Prediction of interlaminar stresses in laminated plates using global orthogonal interpolation

polynomials. AIAA Journal 1992; 30:2740}2749.315. Kapania RK, Stoumbos TJ. Nonlinear transient response of thin imperfect laminated shells under sudden loads.

Composites Engineering: an International Journal 1996; 4:343}357.316. Meek JL, Tan HS. A faceted shell element with loof nodes. International Journal for Numerical Methods in

Engineering 1986; 23:49}67.



317. Zienkiewicz OC. The "nite element method (3rd edn). McGraw-Hill: London, 1977.318. MacNeal RH, Wilson CT, Harder RL, Ho! CC. The treatment of shell normals in "nite element analysis. Finite

Elements in Analysis and Design 1998; 30:235}242.319. Knight Jr. NF. The Raasch challenge for shell elements. AIAA Journal 1997; 35:375}381.320. Allman J. A compatible triangular element including vertex rotations for plane elasticity analysis. Computers and

Structures 1984; 19:1}8.321. Ertas JT, Krafcik, Ekwaro-Osire S. Performance of an anisotropic allman/DKT 3-node thin triangular #at shell

element. Composites Engineering: an International Journal 1992; 2:269}280.322. Batoz JL, Bathe KJ, Ho LW. A study of three noded triangular plate bending elements. International Journal for

Numerical Methods in Engineering 1980; 15:1771}1812.323. Jeyachandrabose, Kirkhope J. An alternative explicit formulation for the DKT plate element. International Journal

for Numerical Methods in Engineering 1985; 21:1289}1293.324. Kapania RK, Mohan P. Static, free vibration and thermal analysis of composite plates and shells using a #at

triangular shell element. Computational Mechanics: an International Journal 1996; 17:343}357.325. Specht. Modi"ed shape functions for the three node plate bending element passing the patch test. International

Journal for Numerical Methods in Engineering 1988; 26:705}715.326. Jonnalgadda KD, Blandford GE, Tauchert TR. Piezothermoelastic composite plate analysis using "rst-order shear

deformation theory. Computers and Structures 1994; 51:79}89.327. Kapania RK, Lovejoy AE. Free vibration of thick generally laminated quadrilateral plates having arbitrarily located

point supports. Journal of Aircraft 1998; 35:958}965.328. Kapania RK, Mohan P, Jakubowski A. Control of thermal deformations of a spherical mirror segment. Journal of

Spacecraft and Rockets 1998; 35:156}162.329. Mohan P, Kapania RK. Updated Lagrangian formulation of a #at triangular element for thin laminated shells.

AIAA Journal 1998; 36:273}281.330. Argyris J, Tenek L. Postbuckling of composite laminates under compressive load and temperature. Computer

Methods in Applied Mechanics and Engineering 1995; 128:49}80.331. Engelstad SP, Reddy JN, Knight Jr. NF. Postbuckling response and failure prediction of graphite-epoxy plates

loaded in compression. AIAA Journal 1992; 30:2107}2113.332. Chao WC, Reddy JN. Analysis of laminated composite shells using a degenerated 3-D element. International Journal

for Numerical Methods in Engineering 1984; 20:1991}2007.333. Liao CL, Reddy JN. Continuum-based sti!ened composite shell element for geometrically nonlinear analysis. AIAA

Journal 1989; 27:95}101.334. Liao CL, Reddy JN. Analysis of anisotropic sti!ened composite laminates using a continuum-based shell element.

Computers and Structures 1990; 34:805}815.335. Gerhard S, Gurdal Z, Kapania RK. Finite element analysis of geodesically sti!ened cylindrical composite

shells using a layerwise theory. CCMS-94-08, Center for Composite Materials and Structures, VPI&SU, Blacksburg,1994.

336. Icardi U. Eight-noded zig-zag element for de#ection and stress analysis of plates with general lay-up. Composites,Part B: Engineering 1998; 29B:425}441.

337. Di Sciuva M, Icardi U. A geometrically nonlinear theory of composite plates with induced-strain actuators.Proceedings of XIII AIDAA Congress, Rome, Italy, 1995; 309}318.

338. Cho Y-B. New zig-zag composite plate theories and three-dimensional "nite elements for static and explicit dynamicanalysis of laminated composite and sandwich panels. Ph.D. Dissertation, Michigan State University, East Lansing,MI, 1998.

339. Kapania RK, Haryadi SG, Haftka RT. Global/local analysis of composite plates with cutouts. ComputationalMechanics: an International Journal 1997; 19:386}396.

340. Haryadi SG, Kapania RK, Haftka RT. Global/local analysis of composite plates with cracks. Composite Part B:Engineering 1998; 29B:271}276.

341. Hammerand D, Kapania RK. Thermo-viscoelastic analysis of composite structures using a triangular #at shellelement. 39th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, LongBeach, CA, 1998. AIAA Journal 1999; 37:238}247.

342. Chee CYK, Tong L, Steven GP. A review on the modelling of piezoelectric sensors and actuators incorporated inintelligent structures. Journal of Intelligent Material Systems and Structures 1998; 9:3}17.

343. Wang T, Rogers CA. Laminate plate theory for spatially distributed induced strain actuators. Journal of CompositeMaterials 1991; 25:433}452.

344. Tzou HS, Ye R. Analysis of piezoelastic structures with laminated piezoelectric triangle shell element. AIAA Journal1996; 34:110}115.

345. Valey, Rao SS. Two-dimensional "nite element modelling of composites with embedded piezoelectrics. 35thAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference 1994; 2629}2633.

346. Sunar M, Rao SS. Recent advances in sensing and control of #exible structures. Applied Mechanics Reviews 1999;52:1}16.

126 H. T. Y. YANG E¹ A¸.


347. Bucalem M, Bathe KJ. Finite element analysis of shell structures. Archives of Computational Methods in Engineering1997; 4(1).

348. MacNeal RH. Perspective on "nite elements for shells. Finite Element in Analysis and Design 1998; 30:175}186.349. Chapelle D, Bathe KJ. Fundamental considerations for the "nite element analysis of shell structures. Computers and

Structures 1998; 66:19}36.350. Bathe KJ, Iosilevich A, Chapelle D. An evaluation of the MITC shell elements. Computers and Structures, to appear.351. Liu Y. Analysis of shell-like structures by the boundary element method based on 3-D elasticity. International

Journal for Numerical Methods in Engineering, submitted.352. Krysl P, Belytschko T. Analysis of thin plates by the element-free Galerkin method. Computational Mechanics 1995;

17:26}35.353. Krysl P, Belytschko T. Analysis of thin shells by the element-free Galerkin method. International Journal of Solids

and Structures 1998; 33:3057}3080.354. Gallagher RH. Elastic instability of cylindrical shell under arbitrary circumferential variation of axial stress. Journal

of Aerospace Sciences 1960; 27(11).355. Gellatly RA, Gallagher RH. Sandwich cylinder instability under nonuniform axial stress. AIAA Journal 1965; 2:398}400.356. Gallagher RH, Padlog J. Discrete element approach to structural instability analysis. AIAA Journal 1963;

6:1437}1439.357. Gallagher RH. A Correlation Study of Matrix Methods of Structural Analysis. Pergamon Press: Oxford, 1964.358. Gallagher RH. The development and evaluation of matrix methods for thin shell structural analysis. Report No.

8500-902011, Bell Aerosystems, Bu!alo, NY, 1966.359. Gallagher RH, Gellatly RA, Padlog J, Mallett RH. A discrete element procedure for thin shell instability analysis.

AIAA Journal 1967; 1:138}145.360. Gallagher RH, Yang HTY. Elastic instability predictions for doubly-curved shells. Proceedings of Second Conference

on Matrix Methods in Structural Mechanics, Dayton, OH, 1968.361. Gallagher RH. Analysis of plate and shell structures. Proceedings of Conference on Applications of Finite Element

Methods in Civil Engineering, Vanderbilt University, 1969; 155}206.362. Gallagher RH, Lien SY, Mau ST. Finite element plate and shell pre- and post-buckling analysis. Proceedings of ¹hird

Air Force Conference on Matrix Methods in Structural Mechanics, Dayton, OH, AFFDL TR71-160, 1971; 857}880.363. Gallagher RH, Mau ST. Finite element procedure for nonlinear pre-buckling and initial post-buckling analysis.

NASA CR-1936, 1972.364. Gallagher RH, Mau ST. Finite element method in shell stability analysis. Computers and Structures 1973; 3:543}557.365. Gallagher RH, Thomas GR. The "nite element method in plate and shell instability analysis. Proceedings of fourth

Australian Conference on Structures and Materials, Brisbane, 1973.366. Gallagher RH. Finite element representations for thin shell instability analysis in buckling of structures. In

Proceedings of I;¹AM Symposium on Buckling of Structures, Budiansky B (ed.). 1974.367. Gallagher RH, Mang H, Cedolin L, Schwinden W. Finite element instability analysis of hyperbolic cooling towers.

2nd ASCE/EMD Specialty Conference, North Carolina State University, 1977.368. Gallagher RH, Loganathan K, Change SC, Abel JF. Finite element representation and pressure sti!ness in shell

stability analysis. International Journal for Numerical Methods in Engineering 1979; 14(9):1413}1420.369. Gallagher RH, Mang HA. On the unsymmetric eigenproblem for the buckling of shells under pressure loading.

Journal of Applied Mechanics 1983; 50:95}100.370. Gallagher RH, Murthy S. Anisotropic cylindrical shell element based on Discrete Kirchho! theory. International

Journal for Numerical Methods in Engineering 1983; 19:1805}1823.371. Gallagher RH, Thomas GR. A triangular thin shell "nite element linear analysis. NASA Contractor Report,

CR-2483, 1975.372. Gallagher RH. A triangular element based on generalized potential energy. In Finite Elements for ¹hin Shells and

Curved Members, Chapter 9. Gallagher R, Ashwell DG (eds). Wiley: New York, 1976.373. Gallagher RH, Kanodia V, Mang H. Instability analysis of torispherical pressure vessel heads with triangular

thin-shell "nite elements. ASME Journal Pressure <essels ¹echnology, 1977.374. Gallagher RH, Murthy S. A triangular thin shell element based on Discrete Kircho! theory: Some patch test

solution results. In Proceedings of International Conference on Finite Elements in Computational Mechanics, Kant T(ed.). Bombay, India, 1985.

375. Gallagher RH, Murthy S. A triangular thin shell element based on Discrete Kircho! theory. Computational Methodsin Applied Mechanics and Engineering 1986; 54:197}222.

376. Gallagher RH, Murthy S. Patch test veri"cation of a triangular thin shell element based on Discrete Kircho! theory.Communications in Applied Numerical Methods 1987; 2:83}88.

377. Gallagher RH. Finite Element Fundamentals, Prentice-Hall: Englewood Cli!s, NJ, 1975.378. Gallagher RH. Finite elements for thin shells and curved members. (Editor with D. Ashwell; author of two chapters)

Wiley: New York, 1976.379. Gallagher RH. Shell elements, Invited Feature Lecture First =orld Conference on Finite Elements in Structural

Mechanics, Bournemouth, England, 1975.



Henry TY Yang, Chancellor, University of California

Documents

Transcript of Henry TY Yang, Chancellor, University of California