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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].
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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
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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
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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
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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
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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
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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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