Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
Developing an Innovative Finite Element Analysis for Free-form Shells
RAJI, S.A., Civil Engineering Department,
University of Ilorin, P.M.B. 1515 Ilorin, Nigeria.
e-mail address : saraji@unilorin.edu.ng
Abstract
Different finite elements have been developed for the analysis
and optimization of free-form shells. Ahmad et.al.[3] first
developed a degenerated , Mindlin-Reissner type, curved shell
element which is quite efficient and simple. In the degenerated
shell element formulation of the isoparametric element concept
is extended to shells. While the original degenerated shell
elements are capable of dealing satisfactorily with thick plates
and shells, their performance deteriorate as the plate or shell
become thin. This phenomenon has been attributed to shear
and membrane locking. Several researchers modified and
extended the original degenerated shell element to improve its
behavior for both thick and thin shell situations.
In this paper, historical review of shell elements that have been
developed for accurate analysis of curved shell structures is
made and the formulation of an improved nine-node
Lagrangian shell element with special emphasis on selective
and reduced integration is presented and it is hoped that this
would assist in overcoming some of the inherent problems of
shear and membrane locking in thin shells.
Keywords: degenerated shell element, Mindlin-Reissner element,
Lagrangian
1. Introduction
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Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
Although shell structures have been featuring in building construction
over a long period of time, like the mosque of Santa Sophia in
Istanbul built in 538 A.D. [ 1 ] , it was not until the 1920s that the thin
shell roof emerged as a practical means for spanning large distance.
The first general theory of thin shell structures was established by
Love [2] in 1888. However, analytical solutions to thin shell structural
problems are limited in scope and are not applicable to arbitrary
shapes, load conditions, irregular support conditions, cut –outs other
aspect of practical designs. These limitations are however overcome
by the development of finite element method in the twentieth century
[ 3]. This method is widely used in various engineering and structural
mechanics problems including the equilibrium in which the
parameters within the system do not change with time; Eigen-value
problems in which critical values of certain parameters must be
determined; propagation problems in which the time dimension has to
be determined. During the last two decades there has been extensive
research on developing efficient, accurate and reliable shell elements
for analysing structures with complex geometries and loadings.
Consequently, the various types of elements described in the next
section are now available for the analysis of thin shells.
2. Finite Elements for Thin Shells
Several types of element exist for the analysis of thin shells. These
include :
 Flat plate (facet) elements with quadrilateral and triangular
shapes.
 Curved elements of quadrilateral and triangular shapes derived
from classical shell theory.
 Elements degenerated from three-dimensional solid elements.
2.1 Flat Plate (Facet) Elements
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Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
Various authors [4 to 9] have studied the advantages and
disadvantages of the various types of these elements. Greene et al. [6]
first suggested the concept of the use of 'faceted' form flat element in
shell analysis. However, the success of such analysis was delayed
until formulations for plate elements in bending became well
grounded. The historical account of the formulations and
developments for the flat elements are obtainable in textbooks [7
to10] and in survey papers[l1,12]. Meek and Tan [13] developed a
triangular flat element with 'Loof nodes' with good performance and
gave an interesting survey of developments in flat elements for shell
analysis. Some of the major attractive features of this modeling are
[8]:
(a) it is simple to formulate; (b) it is simple to define the geometry; (c)
it is easy to mix with other types of element; (d) it is capable of
modeling rigid body motion without including strains; and (e) the
requirement of using a relatively large number of elements provides
the advantages of convenience in incorporating complex loading and
boundary conditions.
In general, adequate performance can be obtained when the sought
response is predominantly either membrane or bending [14].
However, when the membrane and bending stiffness are strongly
coupled, the performance is extremely poor. For example, in an
attempt to model the torsional behavior of a slit cylinder (one of
Morley's [15] 'sensitive' solution) the flat elements failed completely.
The use of flat modeling of shell has been a major feature in almost
all the popular finite element code, such as ABAQUS [16], AD1NA
[17], ANSYS [18], MARC [19], NASTRAN series [20] and SAP
series[21]. When only approximate solution is sufficient, flat plate
elements have been used to solve shell problems with static [22, 23],
dynamic [24,25], buckling [26] and other physical phenomena
especially for practical problems in the industry.
Some of the problems encountered in the application of the flat
elements in the analysis of shells include [7]:
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Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
(a) The actual coupling of stretching and bending in the element is
neglected in this formulation. Such coupling is a major contributor
towards load carrying mechanism in shells and other curved
members;
(b) The junctions where elements are coplanar presents problems of
null stiffness corresponding to rotation about the axis normal to the
plane (  z )
(c) The restriction to triangular shapes when general shells are to be
treated.
(d) The influence of the geometric approximation upon the solution
for imperfection-sensitive structures.
(e) When the stiffness equations from the 'faceted' form are used to
analyse curved shell structure, 'discontinuity' bending moments which
are not present in the continuously curved structure at the element
juncture lines were observed.
2.2
Curved Elements Derived From Classical Shell Theories
The limitations of flat elements in shell analysis arise from the
flatness of the elements and the discontinuity in geometry caused by
these elements in representing a curved surface. The singly curved
elements have been developed with a view to overcoming such
limitations. They are divided into singly and doubly curved shell
elements.
The singly curved elements have gone through various improvements
by various researchers [27 to 49], each improving on the previous
researcher's work. The singly curved elements were first developed in
the axisymmetric form for the analysis of shells of revolution. The
conical segment element for the analysis of shell of revolution was
introduced by Grafton and Strome[27]. Improvements in the
derivation of element stiffness were presented by Popov et. al [28].
Percy et al [29] extended this formulation for orthotropic and
laminated materials. Dong [30] presented results for laminated
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Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
spherical caps which were in good agreement with analytical results.
However, Navaratna [31] showed a significant discontinuity in stress
at the transition point when using two conical segments of different
sizes for a spherical cap. He also showed that the stress discontinuity
disappeared when curved rather than conical segments, which are a
special case of flat elements, were used. Fulton et al [32] and Beitch
[33] recognized the need for curved elements for modeling shell
structures. Gradually, researchers have increased the sophistication of
these elements by improving the geometric approximations [34] and
the displacement interpolation functions [35] and by extending the
applicability to include anisotropic materials [36], dynamic response
[37, 38], elastic stability [39] and nonlinear effects [40]. In practical
applications, shell structures have been most commonly designed in
axisymmetric shape [41, 42] because elements of axisymmetric shape
are more efficient than elements of general shape.
Gallagher [9] presented a 24 degree of freedom conforming element
using the complete bicubic Hermite interpolation functions. Connor
and Brebbia[43] developed a 20 degree of freedom, non-conforming
rectangular cylindrical shell element with a complete bilinear
expansion for middle surface displacement u and v, and an incomplete
bicubic expansion for transverse deflection w. Cantin and Clough [44]
modified the Gallagher's element[5] to a cylindrical shell element and
added the six rigid body modes explicitly by introducing the
appropriate trigonometrical terms into the displacement functions.
This 24 degree of freedom element was conforming and well
behaved, with monotonic convergence of the results towards correct
solutions with mesh refinement. Their displacement functions did not
contain the constant circumferential strain mode since the constant
term was missing from the w-function and the inter-element
continuity was violated. Sabir and Lock [45] and Ashwell and Sabir
[46] showed that if certain terms were omitted from the displacement
function of [44], a nonconforming element with smaller stiffness
resulted but without any apparent loss in accuracy. Fonder and
Clough[47] explicitly added four missing rigid body modes in a
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Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
cylindrical shell element with 24 degree of freedom and found
significant improvement over the perfomance of the same element
without this addition.
Jones and Strome [48] and Stricklin et al .[49] were among the first to
apply doubly curved elements to the shells of revolution. They used
curved meridional elements rather than conical segments. Attempts
was made by Utku [50] to develop a general curved shallow shell
element by accounting for the shear deformation with linear variation
of strain across the thickness in the 15 degree of freedom triangular
element with nonzero Gaussian curvature. Thus, the displacement
fields could be represented with simple linear functions. As the
distribution of the transverse displacement component was not
compatible with the assumed linear distribution of rotations, the
element experienced considerable strain under rigid body
rotation[57]. Brebbia and Deb Nath [51] reviewed the shallow shell
elements and compared their relative convergence rate with an
established finite difference solution for a clamped hypar shell under
uniformly distributed normal load. Yang [52] also developed a high
order 48 degree of freedom shallow shell finite element with two
principal radii of curvature and a twist radius.
2.3
Elements Degenerated from Three-Dimensional Solid
Elements
Although, the hypothesis underlying the degenerated shell element
and the classical shell theory are essentially the same, the reduction to
resultant form is typically carried out numerically in the former, and
analytically in the latter. The isoparametric concept for three
dimensional elements has strong appeal for shell application since
relatively few isoparametric elements can be used to model curved
shell structures. Also, complex mathematical expressions and
assumptions of shell theories could be avoided [53].
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Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
In the use of a 20-node brick element, the basic assumption made is
that no strain occurs across the thickness and that the direct strain
vary linearly in that direction. However, the displacement pattern for
pure bending reveal a curved upper and lower surfaces, rather than
straight as derived from the above assumptions. Wilson et. al[46] adds
displacement modes called 'bubble modes' to describe the curvatures
of the upper and lower surfaces but difficulties have been encountered
with non-rectangular forms of this element. A number of curved shell
elements have been developed using the three-dimensional
formulation. The SHEBA family developed by Argyris and
Scharpf[6] are series of complex elements based on the displacement
method. The inadequate membrane representation in 'faceted’ flat
elements were overcome, but the simplicity of the elements was lost.
This is because analytical functions describing the shell geometry and
additional degrees of freedom (first- and second-order derivatives)
were necessary.
Ahmad [60] developed a shell element for moderately thick shells by
applying the so-called 'degeneration' process to the three dimensional
element. The 3-dimensional formulation was degenerated by
introducing the assumption that originals normal to the middle surface
are inextensible and remain straight (which could be linked with the
Winkler's theory of thick curved beams [6l]), and that the elastic
modulus in normal direction is zero. This procedure permitted a
measure of the transverse shear deformation since the mid-surface
normal are not necessarily to be kept normal during deformation, it
may make angles of other than right angle with the deformed surface,
thus, accommodating an important feature in thick shell situations.
Although the Ahmad's shell element is attractive due to its simplicity
and ability of reproducing shear deformation, unfortunately the results
obtain worsened and finally completely failed as the thickness of the
element was reduced [53]. In addition, the element was too stiff and
showed a very slow convergence rate. Too [54] observed that 2x2
Gaussian integration gives remarkably improved results in the
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Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
quadratic stack, provided the final stresses are calculated at the same
2 x 2 points. This prompted the improvement of the Ahmad's element
by means of the totally reduced integration method by Zienkiewicz el.
al. [55] which greatly improve the results in shell applications.
Another developed family of isoparametric shell elements is the
SemiLoof shell elements by Irons[58].These elements are general
curved triangles and curved quadrilaterals with three degrees of
freedom u,v, and w at each of four corners and four midsides and
normal rotations at the two Gauss points (the so-called 'Loof’ nodes)
along each side. The discrete Kirchhoff hypothesis is imposed on the
32-degrees of freedom element at the 'Loof’ nodes and a reduced
integration technique is applied. The resulting shell element obtained
is capable of representing shells with junctions and discontinuities to
a great extent [56].
MacNeal [59] developed a simple 4-node quadrilateral element,
QUAD4, which is basically a thick shell element in which the shear
stiffness is improved by adjusting the shear modulus of rigidity
according to the actual plate thickness and by using a special form of
the reduced integration technique for the shear terms. The membrane
component of the element stiffness is computed by applying selective
integration since the shear component is integrated using only one
integration point at center of the element. The resulting shell element
can be used for both thick and thin shell problems with high accuracy.
Structured and unstructured finite-element meshes based on bilinear
quadrilateral and linear triangular elements were used in discretizing
assumed gradient algorithm in solving the level set equation [64,65].
Several other researchers had modified and extended the original
degenerated shell element to improve its behaviour for both thick and
thin shell situations. Among them is the element formulated by Huang
and Hinton [62]. This element has exhibited excellent general
behaviour and eliminated several drawbacks associated with earlier
degenerated shell elements. The basic finite element formulation for
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Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
the new proposed nine-node degenerated shell element is described in
the next section.
3.
FORMULATION OF THE
LAGRANGIAN SHELL ELEMENT
IMPROVED
9-NODE
The Huang-Hinton element has been used successfully in series of
applications and the general performance has been found to be better
compare to Lagrangian and Heterosis elements, especially for shear
force distribution. However, the element performed poorly for skewed
plate when very high span/thickness ratios were considered with
parallelogramic methods.[62].
The proposed new 9-Node Lagrangian element seeks to use a recent
and more efficient method to rectify the locking and mechanism
problems. The method uses the sampling points at which no locking
effect exists and the new shear strains constructed from these
sampling points are used instead of the original displacement fields.
3.1
Element Displacement Field
As far as transverse shear effect is concerned, the Mindlin plate(Fig.1)
and shell elements(Fig.2) are quit similar. Hence, the Mindlin plate
bending theory is used to determine the sampling points for transverse
shear strain of the shell element. For a typical point (x, y, z) on the
Mindlin plate, the displacement components due to bending could be
expressed as
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Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16






 =-1


=+1

11
 =+1
Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
u = z x (x, y) v = z y (x, y) w = w(x, y)
(1)
where u and v are the in-plane displacement components and w is the
transverse deflection.
Ѳx and Ѳy are the normal rotation in x z and y z planes respectively.
For conventional n-node Mindlin plate element, the displacement and
rotations are expressed as
w 
 wi 
n
 
 
( 2)
  x    N i   xi 
i

1
 
 
 y
 yi 
where, wi,  xi and  yi are unknown displacement values at node i
Ni is the shape function associated with node i
As mentioned earlier, in order to eliminate shear locking, the assumed
strain method which required appropriate sampling points for
transverse shear strain will be used. Hence, only transverse shear
strain component will be discussed.
The moment curvature and shear force-shear strain relationship for
plate are as for degenerated shell element in which the in-plane
stresses are
 p'  ( x' , y' , xy' )T
(3)
and the transverse shear stresses are
 s'  ( xz' ,  yz' ) T
(4)
The shear strain components in the global coordinate system are
s  (u,z '  w, x ' ), (v,z '  w, y ' )
T
(5)
The above equation can be rewritten in the view of Equation (1) as
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s  ( x  w, x ' ), ( y  w, y ' )
T
(6)
When we use the same shape function for normal rotations and
transverse deflection w, the order of the two right-hand side terms of
Equation (6) are not equal. The desirable displacement field will be
introduced to establish the consistency of the two terms.
The natural transverse shear strain can be written as,
   
 x ,  xz 
  

( 7)
 y ,  yz 
   
Substituting into Equation (6) into Equation (7) yields
   
 x ,  x  w 
  

(8)
 y,   w 
 
  y

  

giving
   
 
   
   w 

 
  w 
( 9)
The real strain components in the global coordinates could be
obtained by inversion of equation (7) as,
 , x   
  xz 

(10)
   
  yz 
 , y   
As shown in equation (7), natural strain tensor can be expressed as
polynomial terms in the natural coordinates. Through this relation, the
natural strain components will be used instead of the real strain
components.
Transverse shear will vanish if the span/thickness ratio is extremely
large.
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Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
   
 
   
   w 

 
  w 
0
(11)
     w 
   

(12)
    w 
For a nine-node element, the polynomial term for   , w, and  ,
or
w, are [2]
     1,  , , ,  2 ,  2 ,  2,  2 ,  2 2 




w,  w, 1,  , , ,  2 ,  2
   1,  , , ,  2 ,  2 ,  2,  2 ,  2 2 
w,  w, 1,  , , ,  2 ,  2
(13)
It is clear that the polynomial of term   and  of equation (13) are
generally one order higher than that of the expression of w, and w, .
Two ways of resolving this inconsistency are:
1. The coefficients of higher order term in   and  have to be
enforced to be zero.
2. All terms presented in the expression   and  have their
counterpart in the expression of w, and w, .
In adopting the second approach, a desirable deflection field w and
w .are chosen to express the transverse shear strain in Equation (13)
such that that the polynomial terms of w, and w, should include all
terms of the   and  .
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Therefore, inconsistency of both terms of equation 13, hence, the
shear locking problem, can be solved by the above definition and the
desirable displacement field will express the realistic relationship
between the deflection and the rotational fields.
3.2 Sampling Points for Transverse Shear Strain
For nine-node Lagrange element, Fig. 3, the rotation and deflection
fields are





w 
 
 x 
 
 y
 wi 
 
  N i   xi 
i 1
 
 yi 
9
(14)
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Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
where N i , is the quadratic Lagrange shape function associated with
node i.
The desirable deflection field for expression of   may be expressed
as
w
3
c

pq
 pq
(15)
p ,q  0
Let the biquadratic approximation of deflection be
9
 N w
w 
i
i 1
(16)
i
where wi is nodal desirable displacement value at node i
Enforcing the equality of derivatives.
w,

w ,
(17)
yields the six sampling points of   :
(ε,η)i , i=1,6
1
 1
  1
3, 1 1:
3, 0
 : 1
 

3 ,1 2 : 1 3, 0
4

(18)
3

3 ,  1, :  1
3 , 1
5
6
(19)
The above six points are suitable for expression of the shear strain  
in an element of arbitrary element geometry.
Another set of six points for the shear strain   can be found by same
procedure.
Location of sampling points of   : (ε,η)i , i=1,6 are
1, 1
 1,  1
3 1:
 
3 2 : 0, 1 3

0,  1 3 : 1, 1 3 :1,  1 3
4
(20)
3
5
6
3.3 Evaluation of Strain Energy Components
16
(21)
Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
The membrane, bending and shear strain energy components for a
Mindlin-Reissner degenerated shell element can be evaluated from
the membrane, bending and shear stress resultants


 M , M , M 
 Q , Q 
 'm  N x ' , N y ' , N x ' y '
 'b
 's
T
T
x'
y'
x'y'
T
x'
(22)
y'
where x’,y’,z’ is a locally directed coordinate system in which the
x’,y’ plane is tangential to the shell mid-surface. These stress
resultants for a degenerated Mindlin-Reissner shell element can be
obtained by appropriately integrating the corresponding stress
components with respect to the thickness coordinate[63]. For a typical
element the bending, membrane and shear strain energies can be
evaluated by the expressions,
^
2
wb 
1
   Db 
'
b
T
'
b
dA
A
^
2
wm
1
   Dm 
'
m
T
'
m
dA
A
^
2
ws 
1
   Ds  dA
'
s
T
'
s
(23)
A
where for surface integration
 x /    x / 

 

dA   y /   x y /  dd


 
 z /    z /  
(24)
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Centrepoint (Science Edition ) (2006) Vol.12 No.1 10-16
^
Note that
Db
^
^
,D
m
, D are the matrix of flexural, membrane and
s
shear rigidities respectively where,
^
^
t /2
Dm 

Dp z 'dz ' ,
 t /2
^
t /2
Db


Dp ( z ' ) 2 dz ' ,
 t /2
Dz
t /2

 D dz
s
'
(25)
 t /2
The accumulated contributions to the bending, membrane and shear
energies are obtained by summing the contributions from each
element. The total strain energy of the finite element solution is then
computed using the expression
2
w 
2
2
2
wb wm ws
(26)
4. Conclusions
An historical review of different finite element has been presented
with a view to develop an improved nine-node Lagrangian shell
element that would behave well in the analysis and optimization of
plates and shells. Benchmark tests including plates and shells of
different geometries and loading would be carried out in order to
study the behaviour of this element. The bending, membrane and
shear strain energies would be evaluated to observe the contribution
of each form of energy. It is hoped that this element would solve the
spurious mechanism problem which produces singular element
stiffness matrix and causes shear and membrane locking in thin shells.
Acknowledgements
The author wishes to thank University of Ilorin for the Senate
Research Grant used for this research work.
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REFERENCES
1. Ashwell, G.H. and Galagher, R.H.(eds), Finite element for thin
shells and curved members, Wiley, New York.
2. Love , A.E.H. On the small free vibrations and deformations of thin
shells Phil. trans. Royal Soc. (London), 17A, 1888, pg. 491.
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