2000_Yang_A Survey of Recent Shell Finite Elements

8/9/2019 2000_Yang_A Survey of Recent Shell Finite Elements

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*Correspondence to: Henry T. Y. Yang, Chancellor, University of California, Santa Barbara, CA 93106, U.S.A.Professor and ChancellorProfessorAAssistant 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. Yang*, S. Saigal, A. MasudA and R. K. Kapania

;niversity of California,Santa Barbara, CA, ;.S.A.Civil and Environmental Engineering,Carnegie Mellon ;niversity, Pittsburgh, PA , ;.S.A.Civil and Materials Engineering, ;niversity of Illinois at Chicago, Chicago, I, ;.S.A.

Aerospace and Ocean Engineering,


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

literature published on the subject.

The computational implementation of shell elements has continued to challenge "nite element

researchers. Several unresolved issues of the past have been settled over the years, and possible

explanations 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 rigorous

solution, while others being investigated via numerical experimentation still ba%e the research

community.

In this paper, we endeavor to summarize the important milestones achieved by the "nite

element 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 an

extensive survey of the literature, however, because of the sheer magnitude of the literature

available on the topic, this survey may not be exhaustive.

In order to organize the literature, we "rst outline the main ideas that have categorized the

various 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 surface

approach, (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 of

the 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 our

judgmental discretion in designating an element to a particular category. We must clarify that

such designations are by no means to be viewed as rigid. We also seek the pardon of those authors

whose 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, dominated

by the so-called degenerated solid approach, which "nds its origins in the paper of Ahmad

et al. [2]. The popularity of these elements is due, in part, to their simplicity of formulation by

which the traditional classical shell theories are circumvented. The element is derived directly

from the fundamental equations of continuum mechanics. Besides, its implementation in the "nite

element procedures is straightforward. While the basic concept underlying the degenerated

element is very simple, these elements are generally expensive in computation and, therefore, their

application to material non-linear problems, in particular, can be limited. The works of, among

others, Ramm [3], Hughes and Liu [4, 5], Hughes and Carnoy [6], Bathe and Dvorkin [7],

Hallquistet 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 solids

approach and related methodologies which involve some type of reduction to a resultant

formulation. Numerous modi"cations and generalizations of the degenerated shell approach can

be seen in References [13}112].

102 H. T. Y. YANGE A.

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


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3. RESULTANT-BASED FORMULATIONS

There is an alternate point of view stating that thin bodies are best treated by replacing the

general set of three-dimensional governing equations by a set of, in some sense, equivalent

equations leading to the construction of shell theories. Such theories enable an insight into the

structure of the equations involved independently, and prior to the computation itself. Based on

them, powerful"nite elements may be formulated. One of the "rst achievements in this direction

was due to Argyris and co-workers [113}116] in the development of the SHEBA family of"nite

elements and, thereafter, their generalizations [117}121].

Eriksen and Truesdell [122] initiated the direct approach to the construction of shell theories

by considering the shell as a surface with oriented directors. They were inspired by the concept of

a Cosserats [123] continuum by which, in addition to the displacement "eld and independent of

it, rotational degrees of freedom are assigned to every particle of the continuum. The resulting

equations and strain measure chosen were quite di!erent from those proposed originally by

Cosserats [123]. The strain measures suggested in their studies were essentially based on the

di!erence of metrics. As far as the one-director formulation is concerned, it is equivalent to, and in

fact 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, Simoet al. [124}130] proposed a stress-resultant-based geomet-

rically exact shell model which is formulated entirely in stress resultants and is essentially

equivalent to a one director inextensible Cosserat surface. The work by the research group of

Simo, in fact, represents a return to the origins of classical non-linear shell theory which, as

mentioned, 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 of

authors; 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, together

with 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 led

to the emergence of a series of elements that used lower-order polynomial expressions, primarily

for simplicity in mesh generation, and also for robustness in complicated non-linear problems

with multiple contacting surfaces. These elements used the concept of reduced and selective

reduced 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 is

particularly 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 to

this topic from which we cite some of the most prominent ones. The development started with

a pure application of reduced integration techniques. A quadrilateral element with bilinear

de#ection and rotation "elds based on Mindlin plate theory with a single quadrature point was

introduced by Hugheset al. [150] under the name U1. However, the element U1 turned out to be

A SURVEY OF RECENT SHELL FINITE ELEMENTS 103



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rank de"cient: the rank of the sti!ness was less than its total number of degrees of freedom minus

the rigid-body modes. For some meshes and boundary conditions this rank de"ciency resulted in

singularity or near singularity of the assembled sti!ness matrix, which manifested itself in solutions

with 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 by

using 22 quadrature and re"ned the interpolation of the transverse shear so that locking could be

avoided. However, these schemes resulted in the loss of an attractive potential of the bilinear

element, namely, the use of one-point quadrature. Another approach to this element was taken by

Taylor [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 quadrature

point in the midsurface was described by Belytschko and Tsay [155]. This element was adopted

in DYNA3D, PAMCRASH and other commercial programs developed for crashworthiness

studies. The major objective in the development of the Belytschko}Tsay [155] element was to

attain 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 when

the material was elastic. For non-linear materials, several quadrature points were used through

the 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, several

shortcuts were made in formulating the element equations. On the whole, the element has

performed quite well, but it has two shortcomings: (i) it performs poorly when warped and, in

particular, it does not correctly solve the twisted beam problem, and (ii) it does not pass the

quadratic Kirchho!-type patch test in the thin plate limit. The latter shortcoming is shared by

Hughes}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-energy

modes which arise due to one-point integration of"rst-order isoparametric "nite element. In their

work, 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 the

element volume and uniform strain modes. However, the use of one-point quadrature schemes forboth the volumetric and deviatoric stresses resulted in certain deformation modes remaining

stressless. The reason lies in that if a mesh is consistent with a global pattern of these (and perhaps

rigid body) modes, they quickly dominate and destroy the solution. These modes are called

kinematic, or zero energy modes in the "nite element literature, and hourglass modes for

hexahedrons and quadrilaterals in the "nite di!erence literature. Belytschko and Tsai [155] had

proposed a stabilization procedure for controlling the kinematic modes of the four-node, bilinear

quadrilateral 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 stabilization

procedure, additional generalized strains were de"ned which were activated by the kinematic

modes. However, these generalized modes were not activated by rigid-body motion regardless of

the shape of the quadrilateral. By using a scaling law the stabilization parameters were de"ned so

that 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 stabilization

vectors could, in fact, be obtained naturally by taking partial derivatives with respect to the

natural 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




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to demonstrate that the resulting continuum element did not experience any locking

phenomenon when the material became incompressible. In comparison with the hourglass-

controlled"nite elements developed by Belytschko and co-workers [155, 156], this element did

not require any stabilization parameters or numerical integrations.

Stabilization of the underintegrated elements continued to be of considerable importance and

led various researchers to develop stabilization schemes based on the assumed strain method.

Belytschkoet al. [158] developed a projection operator, orthogonal to constant strain "elds on

an eight-node hexahedral element with uniform reduced integration. It was shown that the

stabilization forces depended only on the element geometry and material properties. The assumed

strain "eld was also used with four-point integration which did not require stabilization. In

addition, two forms of the B-matrix were studied and it was shown that the mean form is more

e$cient since it passed the patch test in a simpli"ed form. Despite its considerable success, the

problem with the assumed strain approach remains that these elements sometime show strange

modes in rather simple engineering problems. Some other researchers who worked along similar

lines 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 of

the material, as is the case for rubber. Due to their simple geometry, four-node quadrilateral

elements 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 standard

displacement elements. Several methods to circumvent this problem have been developed.

Amongst these are the reduced integration techniques or the mixed methods. In some approaches

rank de"ciency of underintegrated elements, which then leads to hourglassing, is bypassed by

stabilization techniques. Lately, Simo and Rifai [175] in the linear case or Simo et al. [176] in the

non-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 developed

by Tayloret al. [177]. They do not seem to have any rank de"ciency and perform well in bending

situations as well as in the case of incompressibility. For geometrically non-linear analysis, Hueck

and Wriggers [178}179] proposed a similar incompatible quadrilateral element that utilizes

a 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 QS6

element for large elastic deformations. In their work, the basic Hu}Washizu principle is utilized to

derive 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 rectangular

four-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 generate

exact analytical expressions for the hourglass stabilization of the rectangle. The stabilized element

was 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 were

obtained from an expansion of the basis functions into a second-order Taylor series in the




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physical co-ordinates. The internal degrees of freedom of the incompatible modes were eliminated

on 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 of

incompressible 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 computational

power, 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, because

they have a more robust performance in the distorted con"gurations, and secondly, because these

elements 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 poor

performance unless additional arti"ces are used.

The method of incompatible modes had been introduced by Wilsonet al. [187] as an approach

for 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, mesh

distortion insensitivity and excellent performance in the analysis of nearly incompressible and

non-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 desirable

features 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 which

considered the stresses and displacements as independent variables were developed as a successful

alternative. Ibrahimbegovic and Wilson [180] and Ibrahimbegovic and Kozar [181] presented

a geometrically non-linear version of the well-known eight-node Wilson brick element. The

element was based on variational formulation and was modi"ed via the method of incompatible

modes. It was shown that the incompatible modes formulation exhibited essentially the same

performance 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 constitutive

material models. For example, many rate forms of constitutive equations are naturally integratedwith the displacement-driven algorithm, e.g. return mapping algorithm for J2 plasticity, or

constitutive equations directly given in the strain space.

The method of incompatible modes has recently been re-examined within the framework of the

three-"eld Hu}Washizu variational principle. In the work of Simo et al. [176], the original

incompatible mode concept is abandoned, and the enhanced strain "eld is constructed directly

instead. In addition to the displacement and strain "elds the stress "eld is also constructed as an

orthogonal complement to the enhanced strain "eld, so that it does not appear in the "nal form of

the variational statement.

6. ENHANCED STRAIN APPROACH

Enhanced strain elements have also been an area of active interest. Since these elements perform

very well in the incompressible limit as well as in bending situations, they have been applied to

simulate geometrically and materially non-linear problems. Several enhanced strain elements

have been developed over the last years [188}222]. These elements provide a robust tool for

numerical simulations in solid mechanics. Due to the construction of the elements with enhanced




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strains, these element formulations show a very good coarse mesh accuracy. Furthermore, the

implementation of inelastic material models is straightforward. Following the work of Simo and

Rifai [175], serveral other authors have also developed similar element formulations for small

strain 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 in

solid mechanics has a long history. This can be seen from the large number of papers which have

been published on the subject. The main goal is to"nd a general element formulation which ful"lls

the 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 mesh

accuracy, (vi) simple implementation of non-linear constitutive laws, and (vii) e$ciency. These

requirements have di!erent origins. The "rst two result from the necessity to obtain acceptable

answers for the mentioned problems, especially the "rst point is essential for the analysis of

rubber-like materials or for classical J-elastoplasticity problems. The third point becomesincreasingly important since it enables the user of such elements to simulate shell problems by

three-dimensional elements, which is simpler for complicated structures. This spares the need for

introducing"nite rotations as variables in thin shell problems, results in simpler contact detection

on upper and lower surfaces and provides the possibility to apply three-dimensional constitutive

equations 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 as

three-dimensional problems. Due to computer limitations, quite coarse meshes have to be used

often to solve these problems. Thus, an element which provides a good coarse mesh accuracy is

valuable in these situations. Point six is associated with the fact that more and more non-linear

computations involving non-linear constitutive models have to be performed to design engineer-ing structures. Thus, an element formulation which allows a straightforward implementation of

such constitutive equations is desirable. Lastly, the e$ciency of the element formulation is of

great importance when "nite element meshes with several hundred thousands of elements have to

be used to solve complex engineering problems. To construct elements that ful"l most of these

requirements, and possibly all of them, di!erent approaches have been followed throughout the

last decade and a half. Among these are: (i) techniques of underintegration, (ii) stabilization

methods, (iii) hybrid or mixed variational principles for stresses and displacements, involving the

use 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 rotational

degrees of freedom (also called drilling degrees of freedom). Membrane elements of this kind

possess practical advantages in the analysis of shell structures and folded plates. For example,




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combining a plate bending element with a membrane element possessing drilling rotations forms

a shell element in which each node has six degrees-of-freedom, three displacements and three

rotations. Typical membrane "nite elements do not possess the in-plane rotational degree of

freedom, and so when combined with a plate element, they form a shell element with only "ve

degrees-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 very

appealing from an engineering point of view. Numerous works have appeared in the engineering

literature in the last decade in which successful approaches towards incorporating drilling

rotations in membrane elements have been described [248}284]. It is interesting to note that most

of the elements proposed involve a variety of special devices. The simplest and most commonly

used 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. There

have also been developments in the mathematics literature, where variational formulations that

employ independent rotation "elds have been studied [244]. Ideas of this kind go back to

Reissner [268]. Hughes and Brezzi [244], and Hugheset 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 formulations

for linear elastostatics with independent rotation "elds were analysed and it was observed that

numerical methods based on the conventional formulations are unstable when convenient

interpolations are employed. Consequently, several formulations based on modi"cations of the

classical 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 emanating

from this theory were presented in Hughes et al. [245, 246]. In a subsequent work, Hugheset 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 freedom

was addressed and a novel method for consistently introducing rotational masses was introduced.Working along similar lines, in a series of papers, Ibrahimbegovicet al. [272}274] presented

drilling rotations in a stress-resultant-based geometrically non-linear shell model which had

features 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 applications

demands 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 accurate

modelling 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 descriptions

are tightly coupled. This approach renders many existing beam and shell "nite elements, based on

moderate rotation assumptions, ine!ective for large rotation problems. Moreover, there is no

general consensus as to which of these newly developed formulations is preferable, and often the

analysts resist parting with the reliable, yet more restrictive elements they have experience with.




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Hence, development of a more generic, element-independent approach to large rotations became

a topic of intense investigation. Belytschko and Hsieh [285] proposed a method based on

convected co-ordinates to develop a small strain, large rotation beam element. The use of

convected 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 a

co-rotational co-ordinate frame to compute strains from arbitrarily large displacements and

rotations 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 free

of any rigid-body motion. An advantage of this procedure is that it can be implemented

independently 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 operations

involve, in particular, a projection matrix that has a number of interesting properties. First, it

converts a non-equilibrating force vector associated with an element into a self-equilibrating one

when multiplied with the transpose of the projection matrix. Second, the rigid-body components

of an incremental displacement vector are eliminated when multiplied by the projector (large

rigid-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 matrix

already has the correct zero-energy modes, this transformation will have no e!ect on the sti!ness

matrix. In other words, the element is forced to have the correct invariance properties under

rigid-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 displacement

behaviour. Following the work of Belytschko, Liuet al. [286, 287] developed multiple quadrature

underintegrated elements.

Working along the lines of co-rotational framework, Moita and Cris"eld [288] developed

enhanced lower-order element formulations for large strains where they showed that a more general

procedure could be devised with the aid of mixed assumed strain procedures. A mathematical

decomposition 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 links

with Biot-stress formulation. In their work they showed that once the co-rotational technique is

extended to large-strain plasticity, there are some advantages in considering the co-rotational

framework. 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 vectors

during 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 the

concept of Biot stress. A set of examples using co-rotational procedure has been given in Jiang

et al. [292].

10. COMPOSITE SHELL FINITE ELEMENTS

Plate and shell structures made of laminated composite materials have often been modelled as

an equivalent single layer using classical laminate theory (CLT), see for example the text by




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Jones [295], in which the thickness stress components are ignored. Note that the CLT is a direct

extension of the classical plate theory in which the well-known Kirchho!}Love kinematic

hypothesis is enforced, i.e. plane sections remain plane and that a normal to the midplane before

deformation remains straight and normal to the midplane after deformation. This theory is

adequate 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 to

thickness e!ects because their e!ective transverse modulii are signi"cantly small as compared to

the e!ective elastic modulus along the "bre direction. Reddy and Kuppusamy [296] have shown

that the natural frequencies predicted by the CLT may be as much as 25 per cent higher than

those 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 natural

frequencies and buckling loads.

In order to overcome the de"ciencies in the CLT, re"ned laminate theories have been

proposed. A review of these theories along with the respective kinematic relations used in these

theories is available in Reference [297]. These are single-layer theories in which the transverse

shear stresses are taken into account. They provide improved global response estimates for

de#

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 by

Donget al. [299], and Dong and Tso [300] for multi-layered anisotropic shells. A description of

other available theories can be found, for example, in the review article by Kapania [301]. Both

approaches (CLT and FSDT) consider all layers as one equivalent single anisotropic layer, thus

they cannot model the warping of cross-sections. Furthermore, the assumption of a non-

deformable normal results in incompatible shearing stresses between adjacent layers. The latter

approach, because it assumes constant transverse shear stress, also requires the introduction of an

arbitrary shear correction factor which depends on the lamination parameters for obtaining

accurate results. It is well established that such a theory is adequate to predict only the gross

behaviour 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 theories

lead to transverse shear and normal stress mismatch at the intersection. This renders these

theories inadequate for detailed, accurate local stress analysis.

The exact analyses performed by Pagano [304] on the composite #at plates have indicated that

the in-plane distortion of the deformed normal depends not only on the laminate thickness, but

also on the orientation and the degree of orthotropy of the individual layers. Therefore, the

hypothesis of non-deformable normals, while acceptable for isotropic plates and shells, is often

quite unacceptable for multi-layered anisotropic plates and shells that have a large ratio of

Young's modulus to shear modulus, even if they are relatively thin. Thus, a transverse shear

deformation theorywhich also accounts for the warping of the deformed normal is required for

accurate prediction of the elastic behaviour (de#ections, thickness distribution of the in-plane

displacements, 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, allows

thickness variation, and permits the warping of the deformed normal, is required for a re"ned

analysis of thick and thin composites. A signi"cant contribution in this direction was presented

by 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




11/27

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. In

the layerwise plate theory, suggested by Reddy [305], the continuity of the transverse normal and

shear stresses is not enforced. In the individual-layer plate models, see for example, [306}308], the

transverse shear stress continuity is enforced a priori. A recent review of the various available

theories is given in [309]. For geometrically nonlinear theory, the reader is referred to the work

by Librescu [310].

As is the case for isotropic shells, all types of shell elements have been used for the linear and

non-linear analysis of laminated shells. A review of earlier developments (1976}1988) in the "nite

element analysis of laminated shells is given by Kapania [301].

Various theories have been used for the development of"nite elements. Using CLT, the present

authors [311] developed a 48 degrees-of-freedom "nite element to study geometrically non-linear

response of imperfect laminated plates and shells. This element was successfully used by Byun and

Kapania [312] to study the impact response of imperfect laminated plates in conjunction with

a reduced-basis approach [313]. A post-processor for this element that can accurately predict the

interlaminar stresses, by integrating the equilibrium equations of a laminated plate, was

developed 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 using

the tensor notation and a shell theory.

There still exists a considerable interest, mainly due to the simplicity of their formulation, in

using a large number of #at elements [316] to model curved shells. The #at &shell' element is

obtained by combining a plate element with a membrane element. Often, the "nite element

designers 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 rotational

degree of freedom. This leads to a singular sti!ness matrix when all elements with a common node

are 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 the

rotational 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 original

approach was found to give erroneous results in the case of, for example the linear analysis of

a hook problem, termed the Raasch Challenge [319] problem. This approach was subsequently

modi"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 degree

of 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-node

triangular element, termed AT/DKT, by combining an element similar to the Allman membrane

triangular element with the discrete Kirchho!theory (DKT) for formulating the plate bending

element to study laminated plates. The membrane element was obtained from the linear strain

element using a transformation suggested by Cook [253] and the formulation of the DKT

element is available in [321]. A computer program for this formulation was given by Jeyachan-drabose and Kirkhoppe [323]. Ertaset al. [321] compared their results for a cantilever #at plate

with 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 developed

by Ertas et al. [321] for static and dynamic response analyses of laminated shells to study its

accuracy and convergence characteristics. They also extended the element to analyse shells




12/27

subjected to thermal loads. Since the DKT formulation suggested by Batoz et al. [322] does not

employ explicit interpolation functions for the transverse displacements, the determination of

consistent mass is not straightforward. The mass matrix in the formulation of Kapania and

Mohan was determined using the cubic polynomial suggested by Specht [325] and the response

of laminated structures under both thermal and other induced strains was studied. The element

was employed for static analysis, free vibration analysis, and thermal deformation analysis.

A numerical example, previously solved by Jonalgadda [326], was also presented to study the

response of a symmetrically laminated graphite/epoxy laminate excited by a layer of piezoelectric

material. The results in all cases were found to be in excellent agreement with those obtained by

using 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] to

study 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 to

study large displacement analysis of laminated shells. Mohan and Kapania [329] extended the

DKT /AT element to study such behaviour using an updated Lagrangian approach. Results were

presented 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 agreement

with those available in literature and/or those given by the commercial "nite element code

ABAQUS. The element consistently performed better than STRI3, a combination of DKT and

CST. 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 of

isotropic and composites plates and shells using the three-node #at shell element based on the

natural-mode technique.

Finite elements based on higher-order shear deformation theory have also been developed and

employed. 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 plates

loaded in compression. Panels with holes were also studied and the results were compared with

the 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!ened

shells 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 geodesically

sti!ened cylindrical shells using the Tsai}Wu failure criterion. The sti!eners were modelled using

a layerwise beam "nite element allowing their sti!ness to be directly assembled with that of the

shell element.

It is noted that the "nite elements developed using layerwise theory can provide more accurate

results, but at a price. The number of unknowns increase as the number of layers increase. This

may make the use of such elements impractical, especially at the design stage. Individual layer

theories, in which the continuity of the transverse stresses is enforceda priori, provide accurate

stresses 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, membrane

displacements and transverse shear rotations for mid-side nodes. The element was able to

accurately predict the transverse shear stresses using constitutive models. Cho [338] has




13/27

developed a 40 degrees-of-freedom, eight-node "nite element using the zig-zag theory to

study static and dynamic response of plates. To the best of our knowledge, the elements based on

individual 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 of

freedom.

Often, in the analysis of composite panels, the three-dimensional e!ects are important

in certain localized areas, such as those near a free edge. A local/global analysis provides a

means to reduce the CPU time and the storage requirements by using a global method, based

on a plate/shell theory to, determine the overall response and by modelling the region

with noticeable three-dimensional e!ects using 3-D "nite elements. Such an approach was

successfully used by Kapaniaet 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. Hammerand

and Kapania [341] extended the capability of the AT/DKT [321, 324] element to perform

viscoelastic analysis of composite plates and shells. The viscoelastic properties are represented

using 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 piezoelectric

layers act as both sensors as well as actuators [342]. For the most part, the "nite element

method is used to analyse these structures. Wang and Rogers [343] presented a laminate plate

theory for spatially distributed induced strain actuators. Sophisticated "nite elements are being

developed to analyse piezoelectric plates and shells. It is noted that, for these structures, the

constitutive relations relate stresses to strains and the so-called electric displacements, and the

electric"eld is related to the strain as well as the electric "eld. As a result, both mechanical and

electric quantities (electric potential) are used as nodal variables. Tzou and Ye [344] and Valey

and Rao [345] have performed analysis of shells with piezoelectric layers. A recent review of

application of the "nite element method to adaptive plate and shell structures is given by Sunar

and Rao [346].

11. CONCLUDING REMARKS

In this paper, recent (last 15 or so years) advances in the "nite element technology for

shells have been presented. Some additional recent papers address one or more aspects of

the "nite element development for shells, for example [347, 348]. Chapelle and Bathe

[349] discuss theoretical considerations that must be addressed when developing shell "nite

elements that can be used for both bending and membrane dominated behaviours. They also

provided a list of test problems that are bending and membrane dominated, respectively. Bathe et

al. [350], evaluate the MITC shell element for its performance in solving the test problems

suggested by Chapelle and Bathe [349]. MacNeal [348], provides his perspective on the "nite

element for shell analysis including some recent advances in the use ofp-version "nite elementmethod.

Finally, it is noted that recent developments to analyse shells have also included both the

boundary element methods and the element-free Galerkin methods. For the boundary element

methods, the reader is referred to the recent work of Liu [351] and for the element-free Galerkin

method to Krysl and Belytscko [352, 353].




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DEDICATION*A SUMMARY OF PROFESSOR GALLAGHER'S WORKS ON

SHELL FINITE ELEMENTS

This paper is dedicated to the memory of Professor Richard H. Gallagher in celebration of his

lifetime achievements as an engineer, professor, higher education leader, and also his technical

contributions 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 particular

paper to account for his lifetime achievements. Rather, we will limit to an account of his research

contributions 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 were

Yang's Ph.D. students. Among the 379 papers surveyed, many authors were Gallagher's former

students and colleagues.

During Gallagher's early career, he spent 12 years (1955}1967) working at Bell Aerosystems

Company in Bu!alo, New York. During this period, his published works included the studies of

low aspect ratio wings with the e!ects of aerodynamic heating, optimum analysis and design of

integral fuselage propellant tanks, elastic characteristics of airframes, laboratory simulation

of 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 elastic

buckling of isotropic cylindrical shells [354] as well as sandwich cylindrical shells [355]. These

works were done during the infancy period of the parallel developments of both electronic digital

computers 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 element

methods.

In a paper published in 1963, Gallagher and Padlog [356] introduced the concept of the

formulation of incremental sti!ness matrix based on the minimum potential energy principle to

treat buckling problems. In 1964, Gallagher [357] wrote one of the earliest textbooks on "nite

elements, during a time when "nite element methods were neither widely accepted nor even

widely known. In a report in 1966, Gallagher [358] was among the earliest researchers to develop

a 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. The

work in References [356}359] would appear rather primitive from the current point of view. They

were, 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 degree

of freedom doubly curved thin shell "nite element developed earlier by Gallagher [358].

The incremental sti!ness matrix was formulated using the minimum potential energy theorem

and retaining the second-order terms in the strain-displacement equations. In 1969, Gallagher

[361] presented a comprehensive paper summarizing the developments of"nite element methods

in the analysis of plates and shells. Later, Gallagher et al. [362] published the work on elastic

buckling of thin shells and extended it to the regime after buckling by including geometric

non-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 the

procedure to predict the buckling and postbuckling behaviours of plates and shells within

the framework of"nite element methods. One notable application of these research works was the

application to the buckling analysis of hyperbolic cooling towers [367]. During this period of

their e!orts on the research of shell buckling analysis, Professor Gallagher and his students and




15/27

colleagues also explored the e!ect of pressure sti!ness on shell instability [368], and the

unsymmetric eigenproblem of shell buckling under pressure load [369]. Gallagher and Murthy

[370] also used discrete Kirchho! theory to formulate an anisotropic cylindrical shell "nite

element.

Parallel to the early research of linear and non-linear analysis of thin shells, which included the

continuously re"ned formulations of four-node doubly curved shell element and continuously

improved prediction procedures for pre- and post-buckling analysis, Gallagher and Thomas also

developed 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 of

torispherical pressure vessel heads [373]. Gallagher and Murthy also developed a triangular thin

shell "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, in

general, and the shell "nite elements in particular, was his education of hundreds (or perhaps

indirectly thousands) of engineers through his regular classes, short courses, seminars, conference

presentations, and research collaborations. In this regard, we would like to mention a few of his

most notable books and education papers on shells.Gallagher's textbook [377] on the fundamentals of"nite elements has been translated into "ve

languages, i.e. Japanese, German, French, Chinese, and Russian. The volume on thin shell and

curved member "nite elements edited by Gallagher and Ashwell [378] has been a fundamental

contribution 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 formulated

a triangular thin shell element based on generalized potential energy [372]. One of Gallagher's

notable 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 Professor

R. H. Gallagher.

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