COMPUTER
METHODS
NORTH-HOLLAND
IN APPLIED
MECHANICS
AND
ENGINEERING
73 (1989) 259-281
ON THE MIXED FORMULATION OF A 9-NODE LAGRANGE
SHELL ELEMENT
T.Y. CHANG,
A.F. SALEEB
Department of Civil Engineering,
Revised
and W. GRAF
University of Akron,
Akron,
Received 25 November
1987
manuscript
received 24 October
OH 44325, U.S.A.
1988
A 9-node Lagrange shell element is examined using a strain-based
mixed method. Starting from a
modified Hellinger-Reissner
principle,
finite element
equations
are derived by assuming both the
displacement
and strain fields independently.
The strain functions are carefully chosen in conjunction
with several considerations
discussed
in the paper. The resulting element
is free from shear and
membrane
locking, hence it can be used for modeling of either thin or moderately
thick shells. All
kinematic deformation
modes have been systematically
suppressed.
Further, a Jacobian transformation
for the strain functions is employed between the natural and lamina coordinates
to reduce the element
sensitivity to geometric distortions.
Six examples are given to illustrate the numerical performance
of
the proposed
element.
1. Introduction
Ever since the work published by Ahmad, Irons and Zienkiewicz [l], the class of
degenerated isoparametric shell elements has played a significant role in the history of finite
element development. Numerous discussions can be found in the literature, e.g. [l-7], on the
desirability of such elements for applications to plates and shells. It is also recognized that
degenerated plate or shell elements exhibit overstiffening effect due to inadequate modeling of
transverse shear and membrane actions [&lo]. Consequently,
several schemes or new
formulation
approaches have been proposed to alleviate the aforementioned
numerical
problems. A comprehensive literature survey has been given in [7], hence no duplication of
the same effort will be attempted in this paper.
From the computational standpoint, it often appears that low order, simple elements are
preferred by analysts, especially for solving transient dynamic problems. Nevertheless, 8- or
9-node quadratic elements are more advantageous for capturing high deformation gradients or
modeling shells of curved geometries. Moreover, motivated by the fact that the numerical
behavior of 9-node Lagrange elements is generally superior to &node serendipity elements
under both static and dynamic situations [ll-131, our discussion will therefore be focused on
specifically the 9-node Lagrange element.
Within the spirit of degenerated
shell formulation,
an either selectively or uniformly
reduced integration scheme was recommended
to improve the numerical behavior of the
9-node Lagrange element for linear elastic analysis of plates and shells [ll, 12,141. Following
this effort, several authors extended the same methodology to nonlinear applications [3,1500457825/89/$3.50
0
1989, Elsevier
Science
Publishers
B.V. (North-Holland)
260
T.Y.
Chang et al., Mixed formulation of a 9-node Lagrange shell element
201. It was subsequently found that the use of selectively reduced integration may not resolve
the locking difficulty for highly constrained problems and the uniform reduced integration will
cause spurious zero energy modes, and hence unreliable solutions. Consequently, a number of
new formulations have been suggested. These include the use of stabilization or projection
matrices to control spurious zero energy modes resulting from the uniform reduced integration
[5,21], assumed strain methods to enhance the element performance in tranverse shear and
membrane responses [22-251, and a mixed formulation to remove locking difficulty [26,27].
More detailed discussions of the aforementioned
methods and other approaches for the
formulation of plate and shell elements are given in [29-311.
In our recent papers, [28-311, we have focused on the use of a general methodology, based
on the mixed method, for the development of simple quadrilateral and triangular plate/shell
elements free from locking problems. The work reported here is an extension to the case of a
curved 9-node Lagrange shell element. Although a similar mixed method has been reported in
[26], our main objective differs from the previous work in that we are seeking specific
guidelines for the selection of strain functions in order to achieve a trouble-free shell element.
It is noted that, in the mixed method, one has the freedom, on the basis of a modified
(strain-based) Hellinger-Reissner
principle, to assume both the displacement and strain fields
independently. This freedom, when it is appropriately utilized, can provide the leverage to
avoid locking. However, no specific guidelines are currently available for the right choice of
strain functions. Moreover, the treatment of element distortion effects, which has not been
considered previously, is addressed in the present formulation.
In our previous papers, i.e. formulation of curved beams [28] and low order bending
elements for plate and shell analysis [29-321, selection of strain parameters (or functions) was
made on the basis of three guidelines: (1) use of natural coordinates and balanced polynomial
terms for all strain components in order to preserve invariant element properties, (2) all
kinematic deformation modes are suppressed, and (3) strain terms are chosen in a way to
achieve the most favorable constraint index. Based on these guidelines, it was possible to
remove locking problems, and the resulting elements performed exceedingly well for the
benchmarks tested. For example, in the case of linear and quadratic curved beams, we were
able to show that the constraint indices of these elements are always greater than one, hence
no locking is expected. For low order triangular and quadrilateral bending elements, favorable
constraint index was arrived at by considering the number of independent shear constraints.
Therefore, no shear locking was experienced when the elements were applied to thin plate
problems. Since these elements are flat facet plates, membrane and bending stiffnesses are
uncoupled, hence no membrane locking is expected either. In the formulation of the 9-node
curved element, we are able to select a set of strain functions free from shear locking by
following the same guidelines as mentioned before. However, there does not seem to be a
clear-cut guideline for removal of membrane locking, especially for doubly curved shells. For
this purpose, additional considerations have to be given in order to enhance the element
performance in bending and membrane responses. This point will be further elaborated in
Section 4.
Included in the paper are a brief description of the Lagrange element and the reference
coordinate systems adopted, an outline of the mixed finite element formulation, and the
selection of strain functions. Finally, six numerical examples are presented to illustrate the
performance of the element.
T.Y.
Chang et al., Mixed formulation
of a 9-node Lagrange shell element
261
2. Element description
The element considered herein is a typical 9-node degenerated
shell having curved
geometry as depicted in Fig. 1. The element may have variable thickness and such variation is
defined by specifying the positions of the two opposite nodes on the upper and lower surface
of the shell. This will facilitate the use of fiber coordinate system at each node to maintain
displacement continuity when the element is applied to problems with geometric discontinuities. For derivation of element equations, the middle surface of the shell is taken as the
reference.
In the present formulation three different coordinate systems will be referred for reasons of
convenience. These are defined as below:
(1) Global coordinates-The
global reference frame is a rectangular Cartesian (x, y, z)
system with the corresponding base vectors designated by (e,, e2, e3). It is used to define the
element geometry and translational degrees of freedom at each node.
(2) Fiber coordinates-A
used as the reference frame
associated orthonormal base
with the fiber direction at a
ei=(_rL
local fiber coordinate system is constructed at each node, and is
to define the rotational degrees of freedom at the node. The
vectors are (ei, e:, e:). The direction of e: is chosen to coincide
given node as follows:
-xi)lhk,
(1)
and
(2)
h, = 11~: - x; II ,
where xi is the position vector of the kth node on the top surface of the shell, xi is the
position vector of the kth node on the bottom surface of the shell, h, is the thickness of the
shell at the kth node, and 11. (( d enotes the Euclidean norm (or length) of a vector. There are
Fig. 1. A 9-Node
Lagrange
shell element.
262
T.Y.
Chang et al., Mixed formulation of a 9-node Lagrange shell element
severai ways to define the directions of e&, a = 1 or 2, see, for example, [3]. In this paper, the
following convention is adopted:
4 = (4 X 4 / 114X e3 II ,
(3)
e: = e: X ef .
(4)
In the case that ei is parallel to es, ei is taken to be in the direction of e,.
(3) Lamina coordinates-A
local Cartesian system is defined at each integration point in an
element such that two of its axes are tangent to the Iamina (or middle) surface of the shell.
This system is specified by three orthonormal base vectors et, i = 1,2,3, of which e\ is always
perpendicular to the lamina. Detailed definition of lamina coordinates adopted herein can be
found in [33]. Since the lamina system varies from integration point to integration point, an
orthogonal transformation relationship exists between the global and local systems according
to
where T is a transformation matrix consisting of direction cosines between the global and
lamina systems, x and X’ denote the position vectors of a generic point of the shell referring to
the global and the lamina systems, respectively.
Following the isoparametric formulation, the initial geometry of the element,
Fig. 1, can be described by a set of natural coordinates (r, s, t),
x(r, s, t) =
$,Nkxk + i $
k=l
k-l
h,N,ei
depicted in
,
where x = (x, y, z) are the Cartesian coordinates of a generic point of the shell, and Nk
denotes the usual two-dimensional
shape function in terms of (v, s) for node k. Specific
expressions of the shape functions for the element can be found, for instance, in [34].
For a fixed pair of (r, s) values, the line obtained from (1) is called a fiber [33]. In general,
fibers are not perpendicular to the laminae. Since the element geometry is defined through the
nodal coordinates situated at the top and bottom surfaces of the shell, compatibility between
two adjacent elements, such as shell-to-shell intersections or folded lines, can be easily
satisfied.
At each node in the element, five degrees of freedom are considered: three translations
(u, u, w) along the global axes (x, y, z) and two rotations (f3,, 6,) about mutually perpendicular axes e: and e:, normal to the fiber direction at the node (see Fig. 1). Thus, the element has
a total of 45 degrees of freedom. To derive the kinematic relations, we adopt the usual
assumptions for a degenerated
shell element [14]: (1) MindlinIReissner’s
plate theory
prevails, (2) only small rotations are considered, and (3) nodal fibers are inextensible. Thus,
the displacement vector u = [u, u, W] at a generic point of the shell is given by
T.Y.
u= i
Chang et al., Mixed formulation
Nkuk + i 5
k=l
h,N,[-O:ei
of a 9-node Lagrange shell element
+ @e:] ,
263
(7)
k-l
or in symbolic form
u=Nq,
(8)
where N represents a matrix of modified shape functions by combining the terms in (7), and q
is a nodal displacement vector of the element, which is defined by
qt = [z$, u,, W,) ey,
where the notation
(- )’
e’,l’ ) . . . 7 u9,
denotes the transpose
7J9, w,,
el”‘,
g9’],
(9)
of a matrix.
3. Finite element equations
In our present study, attention is focused on shells with linearly elastic, isotropic materials
undergoing small deformation. The finite element equations are derived from a modified form
of the Hellinger-Reissner
variational principle, which is expressed in terms of strains and
displacements. Based on this principle, both the strain and displacement fields are approximated independently.
The strain-based assumption, although essentially equivalent to the
stress assumption in the case of linear analysis, offers distinct advantages for future extension
to shell analysis with material nonlinearity, such as the case of plasticity or viscoplasticity. For
example, in the situation of nonlinear materials, the bending strains still vary linearly in the
thickness direction of the shell whereas the stresses do not. Another advantage of the strain
assumption is that all the material models designed for displacement based elements in a finite
element code can be utilized for mixed elements without major coding modifications.
To derive finite element equations via the variational principle, an immediate question
arises as to which reference frame the strain components are referred. An appropriate choice
of reference frame will facilitate finite element implementation and yield improved numerical
results. Although several options have been exercised by other authors [3,23-251, we adopted
the lamina system to define the strain components for the following reasons: (1) it is relatively
easy to impose the assumption of zero lamina-normal stress, (2) it is a natural system to define
the constitutive relations, especially for extension to composite materials, and (3) it is easy to
satisfy the invariance requirement in the assumption of strain functions.
Referring to the lamina coordinates, the strain components at a generic point of the shell
are partitioned into three parts: membrane, bending and transverse shear strains, respectively,
.G= [E”, ib, T] )
(10)
and can be expressed in terms of the nodal displacement
membrane
vector q:
strains
Em = [L,
222, ?nl
= B;q >
(11)
264
T.Y.
Chang et al., Mixed formulation
of a 9-node Lagrange shell element
bending strains
and transverse shears
(13)
where the strain transformation
matrices BL, BL and Bb are referring to the lamina
coordinates and are functions of isoparametric coordinates (r, s). These matrices will be
defined later.
For linear elastic deformation of a degenerated shell element, the Hellinger-Reissner
principle is expressed in the following functional form:
7TR
=
-;E~DE+E~LI~
1
dv-w,
(14)
where
E is the independently assumed lamina strain vector,
g is the lamina strain vector derived from assumed displacements u defined in (7)-(13),
u = [u, v, W] is the vector of global displacement components at a point,
D is the material stiffness matrix,
W is the potential due to externally applied loads,
dv = dxdydz = ]J(drdsdt,
]J] is the determinant of the Jacobian matrix J.
In finite element approximation, the displacements u are interpolated in terms of nodal
displacements q as indicated in (8). The strain components, including the membrane, bending
and transverse shear strains, are approximated by
8 =
[Em,Eb,y] = P/3
)
(15)
where P = [P,, P,, P,,] is the strain interpolation matrix, and p is the generalized strain vector.
with respect to p and q, an (averaged)
By invoking the stationarity condition on 7rTTR
compatibility condition is given by
,O=H-lGq.
(16)
The stiffness matrix for the mixed shell element
following component matrices:
(k), = (G’H-‘G),
,
is expressed in terms of the sum of the
(17)
where
(G)i = (IPtDB1
dV)i 7
(18)
T. Y. Chang et al., Mixed formulation
of a 9-node Lagrange shell element
265
(19)
Q is the element nodal vector, and i = m, b or y.
Note that the matrices k, G and H are partitioned into membrane, bending and transverse
shear parts for later discussions. Detailed derivations of the above equations are omitted,
since it can be found from [26,27,30].
4. Strain assumption
In our previous work, i.e. formulation of low order (linear) elements for plates and shells
[28-321, we have followed three guidelines in the selection of strain (or stress) functions.
These are: (1) all kinematic modes must be suppressed, (2) natural (or local) coordinates must
be used to preserve invariant element properties, and (3) the element must have a favorable
constraint index. The third guideline was the one responsible for elimination of shear locking.
Membrane locking was not present since the bending and membrane actions of the linear
elements are uncoupled. However, in the case of a 9-node curved shell element, elimination of
membrane locking is not as apparent as in the case of linear elements. In this case, additional
strategies have to be considered. The essence of our approach consists of: (1) membrane and
bending strains are interpolated separately in the local coordinates, (2) strain functions have
complete linear polynomial terms in r and s so that uniform convergence is ensured to the
linear order, and (3) the number of strain parameters is kept minimal. The procedures in our
selection of strain functions are outlined as follows.
The foremost important requirement in our selection of strain functions is the suppression
of kinematic deformation modes. In this regard, a necessary condition for the stiffness matrix
to be of sufficient rank is that the number of strain parameters should be greater than or equal
to IZ- r, where rz is the total number of displacement degrees of freedom of the element, and r
is the number of rigid body modes. Based on the consideration of deformation energy, it was
suggested in [35] that the total number of strain parameters must be kept minimal while
simultaneously
suppressing all kinematic deformation
modes. This can be achieved by
choosing the strain parameters in such a way that at least one strain p-term corresponds to
each of the strain terms obtained from the strain-displacement
relations. For example,
consider the displacement components
(u, U, w, 8,, 0,) at a point in a regular (flat and
rectangular) element, which are written in polynomial form in terms of r and S, equivalent to
(8), except that the variable t in the thickness direction is ignored for clarity. Then, one has
u=
1
I1
u
u
w
=
%
where
*s
N’ = [l r s
N’I
0
0
0
0
rs
0
N’I
0
0
0
0
0
N’I
0
0
0
0
0
N’I
0
r2 s* r’s rs* r2s2] ,
0
0
0
0
N’I
1
a,
(20)
(21)
266
T. Y. Chang et al., Mixed formulation
of a 9-node Lagrange shell element
(22)
and I is the 9 x 9 identity matrix.
The strain-displacement
relations in the natural coordinates
are given by
(23)
(24)
2, = [f?,, ri,,k,,] =
[!$-~(!tp)],
(2%
Substituting (20) into (23), one can obtain the expressions of strains in terms of (Y’s. To
suppress all the kinematic modes, at least one of the assumed strain terms, or p’s, must
correspond to the terms derived from the above relations. Since there are 39 basic deformation modes for the element, the strain functions involve 39 independent p’s; 15 p’s for the
membrane action and 24 p’s for the bending action. After identifying the required polynomial
terms in r and s by suppressing all the kinematic modes, the distribution of p-terms among the
various strain components still remains somewhat arbitrary. That is to say, each of the basic
deformation degrees of freedom associated with the assumed displacements may activate more
than one strain mode. It is indeed this freedom which allows us to assign various p-terms to
different strain components so that undesirable strain modes derived from the straindisplacement relations can be controlled, thus the locking problems are alleviated.
To tackle shear locking, we utilized the concept of constraint index [28-301. The constraint
index (CI) is defined as: CI = NK - NC, where NK is the number of kinematic degrees of
freedom brought by an element, when added to an existing finite element mesh, and NC is the
number of independent constraint conditions of the element which must be enforced when
used in a limiting case such as a thin plate or thin shell in the present context. A favorable
value for the constraint index, i.e. CI > 0, suggests that the element is free from shear locking.
Following this guideline, a minimum number of 9 p’s (out of 24 p’s) is assigned to the
transverse shears. For example, consider a thin plate problem: NK = 12, NC = 9 (when T~(+ 0
and rrr + 0), thus CI = 3, and no shear locking is indicated for the present element when used
to model a thin plate problem.
However, for situations involving both shear and membrane constraints in shells, the use of
index CI does not seem to be as successful as the case of a plate or other constrained media
problems such as incompressible materials. To gain some insight on the extent of difficulty
when applying the constraint index to shells, we consider a situation as follows. At first, as
originally conceived in [33], the index CI is intended as a quick measure to determine the
locking property of an element. An equivalent measure for the same purpose may be used,
i.e. a constraint ratio CR = NK/NC, with the premise that as the number of equations
corresponding to kinematic variables (e.g. displacements and rotations) approaches to infinity,
CR should ideally approximate the ratio of equilibrium equations to the constraints for the
T. Y. Chang et al., Mixed formulation
of a 9-node Lagrange shell element
267
governing system of partial differential equations. For a doubly curved thin shell, there are 5
equilibrium equations and 5 constraints (corresponding to the zero transverse shear and
membrane strains) for the case of inextensional bending. In this situation one has: CR =
5/5 = 1, which represents an ideal value for CR, and it is certainly different from other
constrained media problems, e.g. CR = 3/2 for a thin plate problem and CR = 2/l for a 2D
incompressible continuum element. Further, it is an implied assumption that in using the
standard mesh for calculation of CR, or similarly CI, the finite element mesh must be
sufficiently refined. This will in turn relax (at least) some of the ostensible membrane
constraints. Based on these agruments, we conclude that the calculation of CI or CR to
account for all membrane constraints, i.e. demanding E,+ 0 or 15 - &+ 0 (see (27) and
(28)), may be too pessimistic.
In view of the above discussions, apparently the question of membrane locking is a much
more difficult issue to deal with. From the standpoint of finite element modeling, a shell
element (whether thin or thick) must have the same ability in representing the membrane as
well as bending actions of a structure. In fact, many shell problems of practical importance are
dominated by membrane actions. Then the assumed strain functions for either action should
not be biased. For this reason, each part of the membrane and bending strains is approximated by 15 - p terms, i.e.,
where
p,
&= [ 0
Pm=
0
Pb
P,
0
(27)
1
p,
1
r
s
rs
s2 rs2
0
0
0
0
r2
0
0000
1 r
s
rs
[0
0
0
0
0
0
0
0
0
0
P, = tP, )
p Y = [ 01
00000
r’s 0
0
1
0
r
0
s
0 ,
rs I
(28)
(29
0r
0s
0
rs
rs2
r’s
01
0r
0s
rs
0
1’
(30)
Moreover, the polynomial expressions for the laminar strains given in (27) are useful for a
rectangular (or nondistorted)
element, i.e., r (constant lines) and s (constant lines) are
mutually orthogonal. For a distorted element (i.e. skew in the local r, s coordinates), we
introduce the following covariant coordinate transformation
E’=jt&j
(31)
)
where
j = [dx)/dr,]-’
nates,
is a Jacobian transformation
and i,j=l,2
with
performed in the (r,
element. The reason
test for convergence
matrix between the lamina and natural coordi-
rl=r
and r2 = s. It is noted that the coordinate transformation
is
s)-plane only, and the Jacobian matrix is evaluated at the centroid of the
for doing this is to ensure that the resulting element satisfies the patch
[36]. If evaluated otherwise, the order of strain polynomials would be
268
T. Y. Chap
et al., Mixed formulation
of a 9-node Lagrange shell element
altered. This in turn may either trigger the kinematic deformation modes or induce the locking
problem again.
It is noted that (31) represents a planar transformation on the (r, s)-plane for the strain
functions, analogous to a stress assumption for a quadrilateral plate element [29] in terms of
global bending and shear stress components. The strain components of the present shell
element, which are defined with respect to the lamina system, can satisfy only the invariance
condition. The above Jacobian transformation process is found to be useful to reduce the
element sensitivity to geometric distortions. This point is to be illustrated later in Section 5.3.
It seems that such a treatment was not considered in other mixed formulations, e.g. [26]. We
note that the effect of element distortion has also been dealt with in the assumed strain
formulation by Park and Stanley [24], in which the local Cartesian strain components at each
integration point are obtained directly by a tensor transformation of the natural-coordinate
strains.
Some remarks regarding the selection of strain functions are given.
REMARKS
(1) The lamina strains for the membrane, bending and transverse shear parts are interpolated separately in the local coordinates. That is, the corresponding structural actions are
decoupled in the local sense.
(2) All the strain component are interpolated to the complete linear polynomial expansion.
This will provide an improved convergence property, especially for shells experiencing a
significant amount of high deformation gradients (e.g. compared to a bilinear quadrilateral
element [29]).
(3) All kinematic deformations are suppressed. This is accomplished by matching each
assumed strain (or /3) term with the corresponding term derivable from the strain-displacement relations. This effect was further verified by an eigenvalue solution of the element
stiffness, that only six rigid body modes were obtained.
(4) In the membrane and bending parts, the normal (or curvature) strains contain higher
order terms as compared to the inplane shear (or twisting).
(5) In the matrix P,, the terms s2 and r* share the same &. This is necessary to suppress
the kinematic modes and keep the number of p-terms minimal simultaneously.
(6) All strain components are symmetric with respect to the permutation of I and s
variables, thus ensuring element invariance.
(7) A Jacobian transformation between the natural and laminar systems is performed for
the assumed strain components to account for the effect of element distortion, e.g. skewness
on the shell surface.
It is noted that both the assumed strain E and the strain E derived from the straindisplacement relations, appearing in the functional rR, i.e. (14), are referring to the lamina
coordinate system ei. The components of E are already assumed in the directions of lamina
coordinates. Since the components of i are related to the derivatives of the global nodal
displacement, an appropriate coordinate transformation must be performed. In this connection, the components of shell strains referring to the global axes are calculated first according
to
;g=Bg. 9,
(32)
T.Y.
Chang et al., Mixed formulation
of a 9-node Lagrange shell element
269
where Bg is a strain-displacement
transformation matrix referring to the global coordinate
system, which involves the derivatives of the shape functions # with respect to the natural
coordinates and the inverse of the Jacobian matrix [33,34]. Then the lamina strains are
obtained from the familiar strain transformation law as outlined in [37], i.e.,
E=B’-q,
(33)
B’=T,-Bg,
(34)
where
and T, is a strain transformation
the components of e! vectors.
5. Numerical
matrix from the global to the lamina system, which contains
tests
The 9-node degenerated shell element discussed in the previous sections was implemented
into a research version finite element program [38] to test the numerical performance of the
element. For later discussion, this element is designated as SHELM9. A wide variety of test
problems have been analyzed. The purpose of these test problems is to determine the
numerical performance of the element with respect to (i) shear locking, (ii) membrane
locking, (iii) sensitivity to element distortion, and (iv) accuracy in stress calculations.
Presented in this paper are the analysis results of six test problems, which are outlined below.
5.1.
A patch
test
Considered here is a cantilever plate subjected to a line-bending moment along the free
edge as shown in Fig. 2. The purpose of this test problem is to determine the sensitivity of
..
..
l--t--l
1 = T = 1
1 ? ! : 1
(a) Ragular
Distortion
Fig. 2. A square
1
L :I
1
AZ!
(b) Llnaar
Mash
plate
tc) Curved
Dlstortion
for patch
tests.
T.Y. Chang et al., Mixed formulation of a g-node Lagrange shell element
270
Table 1
Deflection
and bending stress in a cantilever
plate
Bending stress in element
Deflection along
free edge
1
2
3
4
Regular mesh
1.0
17.3
17.3
17.3
17.3
Distorted mesh A
1.0
17.3
17.3
17.3
17.3
0.99
17.4
17.0
16.7
17.0
Distorted
mesh B
element distortion under pure bending, A bending moment was applied to achieve unit
deflection at the free end and the corresponding maximum bending stress in the plate is equal
to 17.3 (nondimensionalized).
The plate was considered to be linearly elastic and isotropic
with Young’s modulus E = 30 x lo4 ksi, and Poisson ratio Y = 0.3. At first, the plate was
modeled by a 2 x 2 square (or regular} mesh as shown in Fig. 2a. Then, two types of distorted
elements are used: linear distortion (Fig. 2b) and curved distortion (Fig. 2~).
For both the regular and linearly distorted meshes, the deflections at all nodes along the
free edge is equal to 1.0 and uniform bending stresses were obtained at the centroids of all
four elements as indicated in Table 1. For the curved distortion, a slight stiffening effect was
noticed in the solution of deformations. Further, the element bending stresses are no longer
uniform. Nevertheless, the maximum deviation in stress calculation is on the order of 3.5%.
We may also apply linear stress distribution patch tests to examine the element’s performance. In fact, similar behavior was found for the case of linearly varying curvature. That is,
exact solution was obtained for distorted meshes with straight edges, but not for curved edges.
Nevertheless, the solution does converge when the element size is being reduced.
5.2. Cook’s
panel
To determine the mesh convergence of the present element when subjected to membrane
action, a plane stress panel due to Cook [37] was analyzed. The panel is subjected to an end
shear as shown in Fig. 3. Depicted also in Fig. 3 are the convergence curves of the present
shell element in conjunction with four other types of shell element for the purpose of
comparison:
SHELMB: Shell element of the present paper,
element
with uniform
Q9-URI:
9-node displacement
integration order [ 191,
Shell element by Park and Stanley [24],
9-ANS:
Shell element by Jang and Pinsky [25],
9-ACS:
Heterosis
element 1391.
9-HET:
reduced
integration,
2x 2
It is seen from Fig. 3 that the present element gives fairly accurate results for coarse
meshes, although the deflection for a one-element model yields a slighly lower value than the
9-ANS element. As the mesh is refined, rapid solution convergence is obviously shown in the
plots. All stiffness calculations were made using full integration order (3 x 3 Gauss quadrature
in the lamina system) and no locking difficulty was experienced for the element.
T. Y. Chang et al., Mixed formulation
of a Y-node Lagrange shell element
271
1.2
1.1
iii
<
3
1.0
0.9
,
A’
El
Q9-URI
0.8
Q SHELMS
0
0.7
ACS
ih HET
Number of Nodes per Side
Fig. 3. Mesh convergence
for Cook’s panel.
In addition, the normal stress distribution at a central section AB is shown in Fig. 4 for a
4 x 4 mesh. Excellent stress distribution was obtained from the present element as compared
with the reference solution, which was obtained by using a 6 x 6 mesh of the Q9-URI
elements.
5.3. A
rhombic
thin pIate
Considered in Fig. 5 is a simply supported rhombic plate. The plate is subjected to a
uniform pressure. The purpose of this problem is to test the sensitivity of the element to
geometric distortion under transverse bending action. The plate was modeled by N x N
skewed SHELM9 elements. This is a rather challenging problem since the bending moments
M, and MY at the obtuse corner of the plate possess a singular character [40]. To demonstrate
the importance of the planar Jacobian transformations outlined in (31), two cases of analysis
were considered: Case A-No Jacobian transformation was included and Case 3-A Jacobian
transformation was included.
272
T. Y. Chang et al., Mixed formulation
of a g-node Lagrange shell element
0.20
0.10
d
h
0
3
k
fii
-0.10
-
Reference
I
&
-0.20
A
Solution
Solution
From SHELLMS
Fig. 4. Normal stress distribution
on section AB of Cook’s panel.
Young's
Modulus:
E
=
30 Y lo4
Poisson
ratio:
"
=
0.3
Length:
L
=
10
Thickness:
t
=
1.0
Uniform
P
=
10
Pressure:
Fig. 5. A rhombic thin plate.
T. Y. Chang et al., Mixed formulation
Table 2
Normalized
maximum
deflection
and bending
Normalized
moment
maximal
SHELM9
273
of a 9-node Lagrange shell element
of a skewed
plate
Normalized
deflection
maximal
SHELM9
moment
A
B
Bathe and
Dvorkin [23]
4x4
0.759
0.920
0.879
0.842
0.938
0.873
8X8
0.844
0.915
0.871
0.897
0.947
0.928
16 x 16
0.897
0.946
0.933
0.935
0.967
0.961
24 x 24
0.921
0.963
-
0.951
0.976
-
Mesh
A
B
Bathe and
Dvorkin [23]
The normalized maximum deflection and bending moment of the plate obtained for Cases
A and B together with the solution given in [23] are listed in Table 2. It is apparent that
without the inclusion of coordinate transformation between the distorted natural system and
the lamina system, the element yielded much stiffer results. When such transformation was
included, both the predicted maximum deflections and maximum bending stresses in the plate
are somewhat better than those obtained in [23] using a 4-noded bilinear element.
For comparative purpose, the normalized maximum deflection versus the mesh refinements
obtained from the present element was also plotted in Fig. 6 in conjunction with three other
biquadratic shell elements: the 8-SER (the 8-node serendipity element [l, ll]), the 9-ANS and
the 9-HET elements. It is seen from the figure that the present element performs quite
favorably for this problem as compared with other quadratic elements shown.
01
5
I
I
I
I
I
J
IO
15
20
25
30
35
NUMBER
Fig. 6. A comparison
of mesh convergence
OF NODES
of various
PER
SIDE
biquadratic
shell elements
for a rhombic
plate
T. Y. Chang et al., Mixed formulation
274
5.4.
of a 9-node Lagrange shell element
A cylindrical roof
This problem is often referred to as the Scordelis-Lo cylindrical roof (Fig. 7) which was
used to demonstrate the numerical performance of a concrete shell element [41]. The shell is
subjected to a uniform gravity load, i.e. g = 90 (nondimensionalized)
per unit surface area. It
is a moderately deep shell (40= 40”); both membrane and bending actions are equally
important in its structural response. To determine mesh convergence property of the element,
the roof was modeled by 1 X 1, 2 X 2, 4 X 4 and 6 X 6 meshes of SHELM9 elements. The
normalized vertical deflection at the midpoint of the free edge of the roof versus the number
of nodes used in the finite element models is shown in Fig. 8. For comparison purpose,
numerical results obtained from other similar elements are also shown in the figure. These
elements are:
Q9-URI:
Q9-SRI:
Q9-y:
9-node displacement element, defined previously,
9-node displacement element with reduced integration (2 X 2 order) for transverse shear only,
9-node displacement element with 2 x 2 integration rule and stabilization matrix
SHELMS:
5-node mixed shell element
PI,
[30].
As seen in the figure, the present element shows fairly rapid convergence rate, similar to
the Q9-URI element, and no locking is indicated. In fact, the solution of one element model
is already very close to the reference result.
Geometry
and Material
:
+=
400
R=25
L/R=2
R/h = 100
E =4.32
~10s
v =o.o
Rigid
diaphra
:
Loodina
Vertical
g =90
Diaphragm
”
Fig. 7. Scordelis-Lo
cylindrical
=
roof.
shell
(per
weight
unit
support
v =a,=0
surface
:
oreal
T.Y.
Chang et al., Mixed formulation
275
of a 9-node Lagrange shell element
1.2
A
t\
-8-
Analytical
QO-SRI
.....x..... QQ_"R,
A
OB-Y
a
SHELM5
SHELMQ
-O-
0.4
0.2
q
’
!
/
8’
ot
’
,
3.
6
v
9
Number
Fig. 8. Displacement
5.5.
A pinched
12
of Noder
convergence
per
16
16
Side
curves for the cylindrical roof.
cylinder
A thin-walled cylinder, with rigid diaphragms at the two ends shown in Fig. 9, is subjected
to two concentrated forces, pointing towards each other. This problem portrays two main
features in terms of deformation behavior of a structure: inextensional bending action, and
membrane response around the central section of the cylinder. Using symmetry conditions,
only one-eighth of the shell was modeled by various N x N uniform meshes, where N is the
Rigid diaphragm
Data.
R =300
L/R=2
R/h
= 100
E =3x104
Y -0.3
Rigid diaphragm
Fig. 9. A pinched cylinder with end diaphragms.
T. Y. Chang et al., Mixed formulation
276
of a P-node Lagrange shell element
1.0
0.9
-+---.--..
0.6
-o-
--A--
0.4
Analytical
QO-3x3
QQ_
y
SHELF45
SHELMS
0.2
a
w
3
6
9
12
18
21
of Nodes per Side
Number
Fig. 10. Displacement
15
convergence
curves for the pinched cylinder.
RN-g
P
A
-2
AA
0.003
M-0
0.002
-
0.001
-
P
0
-0.001
-0.002
Fig. 11. Membrane
-0
I
I
I
shear and twistmg moment dtstrmutions
,
along section AD.
T. Y. Chang et al., Mixed formulation
of a 9-node Lagrange shell element
277
number of nodes along an edge. Nondimensionalized
vertical deflection directly underneath
the applied load versus N are shown in Fig. 10. In addition, the calculated membrane stresses,
bending and twisting moments are given in Figs. 11 and 12.
Referring to Fig. 10, the solution obtained from a 4 X 4 mesh (or N = 9) is about 97% of the
reference solution [42]. When the grid was refined to a 6 x 6 mesh, i.e. N = 13, the present
element gives almost identical results to the reference solution. The performance of SHELM9
is somewhat similar to that of Q9-r. For comparative purpose, solutions obtained from
SHELMS and Q9 with full integration order were also plotted on the same figure. Rapid mesh
convergence of the present element is apparent for this problem.
Shown in Figs. 11 and 12 are the stress predictions of the present element. The overall
accuracy is quite good, except that some deviations are noticed for the mininum value of
inplane shear and the maximum value of twisting moment as compared to the reference
solution given in [42]. The present solution was nevertheless verified by two independent
cases: (i) doubling the mesh of SHELM9 elements, and (ii) using a 16 X 16 mesh Q9-URI
elements. In both cases, almost identical stress distributions were obtained.
-
Fig. 12. Membrane
stresses
Reference
and bending
Solution
moment
distribution
along section
DC.
278
T. Y. Chang et al., Mixed formulation of a 9-node Lagrange shell element
L
Radius
Thickness
= IO
~0.04
-cY
F = 1.0 (on quadrant)
/
Fig. 13. A hemi-spherical
-b
-El--Q--
shell.
Analytical
09-v
QUAD0
SHELM9
I
6
Number
Fig. 14. Displacement
12
6
of Nodes
convergence
15
18
per Side
of curves of the hemi-spherical
shell.
T. Y. Chang et al., Mixed formulation
5.6.
of a 9-node Lagrange shell element
279
A spherical shell
Considered herein is a spherical shell with openings at the top and bottom, shown in Fig.
13. The shell is subjected to two pairs of concentrated forces, opposite in directions in the
diametrical plane. This is a challenging test problem for determining the element’s ability in
representing:
(i) doubly curved deep shell action, (ii) inextensional bending modes with
almost no membrane strains, and (iii) the shell experiences predominantly rigid body rotations
about normals to the shell surface.
By symmetry, only one-eighth of the sphere was modeled by various mesh refinements, i.e.
N x N mesh, where N is the number of nodes along one edge. The convergence curves for
normalized displacement in the direction of applied load was plotted against N in Fig. 14. For
comparison, the results of two similar elements, i.e. Q9-r, and QUAD8 are also included in
the plots. In the figure, the coarse mesh (N = 5) did not seem to give favorable results as
compared with those of the e9-r element. However, as the mesh is refined, quick convergence of the solution becomes apparent.
6. Conclusion
In this paper, a 9-node Lagrange shell element based on a mixed formulation method is
presented for the linear elastic analysis of plates and shells. Within an element, both the
displacement and strain fields are independently
assumed. The displacement functions are
those of a typical 9-node isoparametric element. The strain functions are selected on the basis
of a set of guidelines discussed in Section 4.
From the theoretical considerations and numerical results of test problems obtained, several
conclusions can be made:
(1) The 9-node Lagrange element is useful for both thin and moderately thick shells.
(2) The element does not exhibit any kinematic deformation modes.
(3) From the argument of favorable constraint index, the element is free from shear locking.
(4) With the enhanced considerations in the assumption of bending and membrane strains,
the element is also free from membrane locking.
(5) The element is relatively insensitive to geometric distortions.
(6) Based on the numerical results, the element appears to give fairly accurate stress
predictions.
Acknowledgment
This work is supported by NASA Lewis Research Center, Cleveland, Ohio under a grant
number NAG 3-307. The program manager of this project is Dr. Robert L. Thompson.
References
[l] S. Ahmad, B.M. Irons and O.C. Zienkiewicz, Analysis of thick and thin shell structures
elements, Internat. J. Numer. Methods Engrg. 2 (1970) 419-451.
by curved finite
280
T. Y. Chang et al., Mixed formulation
of a g-node Lagrange shell element
[2] R.H. Gallager, Problems and progress in thin shell finite element analysis, in: D.G. Ashwell and R.H.
Gallagher, eds., Finite Element for Thin Shell and Curved Members (Wiley, New York, 1976) l-14.
[3] T.J.R. Hughes and W.K. Liu, Nonlinear finite element analysis of shells: Part I. Three-dimensional
shells,
Comput. Methods Appl. Mech. Engrg. 26 (1981) 331-362.
[4] A. Tessler and T.J.R. Hughes, An improved treatment of transverse shear in the Mindlin-type four-node
quadrilateral element, Comput. Methods Appl. Mech. Engrg. 39 (1983) 311-335.
[5] T. Belytschko, W.K. Liu, J.S.-J. Ong and D. Lam, Implementation and application of a 9-node Lagrange shell
element with spurious mode control, Comput. & Structures 20 (1985) 121-128.
[6] G.R. Heppler and J.S. Hansen, A Mindlin element for thick and deep shells, Comput. Methods Appl, Mech.
Engrg. 54 (1986) 21-47.
[7] T. Belytschko, A review of recent developments in plate and shell elements, in: A.K. Noor, ed., Computational Mechanics-Advances
and Trends, ASME publication, AMD-75 (1986) 217-231.
[8] I. Fried, Shear in Co and C’ bending finite elements, Internat. J. Solids and Structures 9 (1973) 449-460.
[9] H. Stolarski and T. Belytschko, Membrane locking and reduced integration for curved elements, J. Appl.
Mech. 104 (1982) 172-176.
[lo] H. Stolarski and T. Belytschko, Shear and membrane locking in Co curved elements, Comput. Methods Appl.
Mech. Engrg. 41 (1983) 279-296.
[ll] E.D. Pugh, E. Hinton and O.C. Zienkiewicz, A study of Quadrilateral plate bending elements with reduced
integration, Internat. J. Numer. Methods Engrg. 12 (1978) 1059-1078.
[12] E. Hinton and N. Bicanic, A comparison of Lagrangian and serendipity Mindlin plate elements for free
vibration analysis, Comput. & Structures 10 (1979) 483-493.
[13] M.A. Crisfield, A quadratic Mindlin element using shear constraints, Comput. & Structures 18 (1984)
833-852.
[14] H. Parisch, A critical survey of the 9-node degenerated shell element with special emphasis on thin shell
application and reduced integration, Comput. Methods Appl. Mech. Engrg. 20 (1979) 323-350.
[15] E. Ramm, A plate/shell element for large deflection and rotations, in: K.J. Bathe, J.T. Oden and W.
and Computational
Algorithms in Finite Element Analysis (MIT Press,
Wunderlich, eds., Formulations
Cambridge, MA, 1977).
[16] H. Parisch, Geometric nonlinear analysis of shells, Comput. Methods Appl. Mech. Engrg. 14 (1978) 159-178.
[17] K.J. Bathe and S. Bolourchi, A geometric and material nonlinear plate and shell element, Comput. &
Structures 11 (1980) 23-48.
[I81 T.J.R. Hughes and W.K. Liu, Nonlinear finite element analysis of shells: Part II. Two-dimensional shells,
Comput. Methods Appl. Mech. Engrg. 27 (1981) 167-181.
[19] T.Y. Chang and K. Sawamiphakdi, Large deformation analysis of laminated shells by finite element method,
Comput. & Structures 13 (1981) 331-340.
[20] T.Y. Chang and K. Sawamiphakdi, Large deflection and post-buckling analysis of shell structures, Comput.
Methods Appl. Mech. Engrg. 32 (1982) 311-326.
[21] T. Belytschko, J.S.-J. Ong and W.K. Liu, A consistent control of spurious singular modes in the 9-node
Lagrange element for the Laplace and Mindlin plate equations, Comput. Methods Appl. Mech. Engrg. 44
(1984) 269-295.
[22] H.C. Huang and E. Hinton, A nine-node Lagrangian Mindlin plate element with enhanced shear interpolation, Eng. Comput. 1 (1984) 369-379.
[23] K.J. Bathe and E.N. Dvorkin, A formulation of general shell elements-The
use of mixed interpolation of
tensorial components, Internat. J. Numer. Methods Engrg. 22 (1986) 697-722.
[24] K.C. Park and G.M. Stanley, On curved Co shell elements based on assumed natural-coordinate
strains, J.
Appl. Mech. 53 (1986) 278-290.
[25] J. Jang and P.M. Pinsky, An assumed covariant strain based 9-node shell element, Internat. J. Numer.
Methods Engrg. 24 (1987) 2389-2411.
[26] S.W. Lee, S.C. Wong and J.J. Rhiu, Study of a nine-node mixed formulation finte element for thin plates and
Shells, Comput. & Structures 21 (1985) 1325-1334.
[27] J.J. Rhiu and S.W. Lee, A new efficient mixed formulation for thin shell finite element models, Internat. J.
Numer. Methods Engrg. 24 (1987) 581-604.
T. Y. Chang et al., Mixed formulation
of a 9-node Lagrange shell element
281
[28] A.F. Saleeb and T.Y. Chang, On the hybrid mixed formulation
of Co curved beams, Comput. Methods Appl.
Mech. Engrg. 60 (1987) 95-121.
[29] A.F. Saleeb and T.Y. Chang, An efficient quadrilateral
element
for plate bending analysis. Internat.
J.
Numer. Methods Engrg. 24 (1987) 1123-1155.
[30] A.F. Saleeb, T.Y. Chang and W. Graf, A quadrilateral
shell element using a mixed formulation, COmput. 62
Structures 6 (1987) 787-803.
[31] A.F. Saleeb, T.Y. Chang and S. Yingyeungyong,
A mixed formulation
of Co linear triangular
plate/shell
element-The
role of edge shear constraints,
Internat.
J. Numer. Methods Engrg. 26 (1988) 1101-1128.
[32] T.Y. Chang and A.F. Saleeb, On the selection of stress or strain polynomials
for hybrid/mixed
elements, in:
G. Yagawa and S.N. Atluri, eds., Proceedings
of the Internat.
Conference
on Comput. Mech. (Springer,
Berlin, 25-29 May 1986) 103-110.
[33] T.J.R. Hughes, The finite Element Method (Prentice-Hall,
Englewood
Cliffs, NJ, 1987).
[34] K.J. Bathe, Finite Element Procedures
in Engineering
Analysis (Prentice-Hall,
Englewood
Cliffs, NJ, 1982).
(3.51 T.H.H.
Pian and D.P. Chen, On the suppression
of zero energy deformation
modes, Internat.
J. Numer.
Methods Engrg. 19 (1983) 1741-1752.
[36] B. Iron and S. Ahmad, Techniques
of Finite Elements
(Ellis Horwood,
Chichester,
England,
1980).
[37] R.D. Cook, Concepts and Applications
of Finite Element Analysis, 2nd edition (Wiley, New York, 1981).
[38] T.Y. Chang, NFEP-A
Nonlinear Finite Element Program, a user manual, Department
of Civil Engineering,
University of Akron, OH, June 1987.
[39] T.J.R. Hughes and M. Cohen, The ‘Heterosis’
finite element for plate bending,
Comput.
& Structures
9
(1978) 445-450.
[40] L.S.D. Morley, Skew Plates and Structures
(Pergamon,
Oxford, 1963).
[41] A.C. Scordelis and K.S. Lo, Computer analysis of cylindrical shells, J. Amer. Concr. Inst. 61 (1969) 539-561.
[42] G.M. Linberg, M.D. Olson and G.R. Cowper, New developments
in the finite element analysis of shells, Q.
Bull. Div. Mech. Eng. and Nat. Aeronaut.
Establishment
4, Nat. Res. Council of Canada (1969) I-38.