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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
SHEAR FLEXIBLE FIELD-CONSISTENT CURVED SPLINE
BEAM ELEMENT FOR VIBRATION ANALYSIS
B. P. PATEL, M. GANAPATHI* AND J. SARAVANAN
Institute of Armament ¹echnology, Girinagar, Pune-411025, India
Centre for Aeronautics System Studies and Analyses, Bangalore-560 003, India
SUMMARY
In this paper, an e$cient curved cubic B-spline beam element is developed based on the "eld consistency
principle, for vibration analysis. The formulation is general in the sense that it includes anisotropy,
transverse shear deformation, in-plane and rotary inertia e!ects. The element is based on laminated re"ned
beam theory, which satis"es the interface transverse shear stress and displacement continuity, and has
a vanishing shear stress on the top and bottom surfaces of the beam. The lack of consistency in the shear and
membrane strain "eld interpolations in their constrained physical limits causes poor convergence and
unacceptable results due to locking. Hence, numerical experimentation is conducted to check these
de"ciencies with a series of assumed shear/membrane strain functions, redistributed in a "eld-consistent
manner. The performance of the element is assessed by studying the free vibration behaviour of a variety of
problems ranging from a straight beam to a circular ring. Copyright 1999 John Wiley & Sons, Ltd.
KEY WORDS: spline; frequency; locking; consistency; shear and membrane strains
1. INTRODUCTION
The application of spline functions in the analysis of problems concerning structural mechanics
has emerged to be an exciting area of research in the recent past. Due to their piecewise form,
smoothness, capacity to handle local phenomena and higher-order continuity, spline functions
o!er distinct advantages such as computational e$ciency, #exibility to model di!erent boundary
conditions, good accuracy and convergence characteristics, and versatility, etc. Among the spline
functions, the functions based on B-spline basis, which can have di!erent orders of polynomials, is
more in usage for the structural analysis.
The study of static and dynamic behaviour of various structural elements, using di!erent
methods considering B-spline functions, have recently been carried out by many researchers.
Some of the important contributions are cited here. Cheung et al. [1], and Cheung and Fan [2],
have studied static problems by employing the spline "nite strip method whereas the "nite
element technique adopting spline functions has been used in the work of Shik [3] and Gupta
et al. [4]. The dynamic characteristics of beams/plates and shells have been analysed based on
spline "nite strip procedure by Sheikh and Mukhopadhyay [5] while Zhou and Li [6] have
introduced the spline "nite point method. Furthermore, "nite element procedure using spline
* Correspondence to: M. Ganapathi, GM Faculty, Institute of Armament Technology, Girinagar, Pune-411025, India.
E-mail: gana@iat.ernet.in
CCC 0029-5981/99/270387}21$17.50
Copyright 1999 John Wiley & Sons, Ltd.
Received 27 May 1998
Revised 20 January 1999
388
B. P. PATEL, M. GANAPATHI AND J. SARAVANAN
functions has been attempted for vibration study by Leung and Au [7], and Fan and Luah [8].
It may be observed from the existing literature that most of the available works applying various
methods incorporating spline functions for structural analysis are based on the classical theory.
It is further apparent from the available research works that the development of spline-functionbased curved beam element has not been attempted, although several works report about straight
beam, plate, and shell elements having interpolation functions based on spline bases.
It is a well-known fact that the classical theory is not suitable for analysing either thick
isotropic structures or even thin composite laminate cases, and therefore, it is more appropriate to
analyze such structures by including the e!ect of shear deformation. Some of the important
contributions for straight/curved beam are the work of Chandrasekhara et al. [9], Abramovich
[10], Nabi and Ganesan [11], Qatu [12], and Qatu and Elsharkawy [13]. However, the
application of spline function in conjunction with shear deformation theory has been sparsely
treated in the literature [14}18]. Patlashenko and Weller [14, 15] have adopted a spline
collocation procedure for solving static problems whereas Dawe and Wang [16], Wang and
Dawe [17], and Wang [18] have analysed the vibration characteristics of straight beams and
plates using spline functions in the Rayleigh}Ritz approach. It has been brought out from these
studies that these methods yield results of good accuracy for moderately thick beams, plates and
shells, but that the accuracy of the results deteriorates signi"cantly for the thin structures due to
the shear locking phenomenon. The occurrence of the shear locking phenomenon with respect to
problems dealing with shear deformation theory in conjunction with B-spline functions has been
eliminated by choosing di!erent order of spline functions for the constrained "eld variables [16],
by constructing a B-spline displacement "eld based on the inspection of the Timoshenko beam
mode functions [18], which is the function of thickness parameter, and by introducing modi"ed
shear moduli [14, 15]. The shear locking behavior in the "nite element analysis of beams, plates
and shells has been studied by many authors [19}23] in the context of elements based on
conventional polynomial functions for the "eld variables. A more versatile approach, among the
available techniques for alleviating locking such as the reduced/selective integration scheme and
assumed strain "elds, etc., is the "eld-consistent formulation [22, 23]. It involves systematically
eliminating spurious constraints causing shear/membrane locking in shear #exible "nite elements.
With such an approach, the order of integration required is freed and therefore, exact numerical
integration schemes can be employed to evaluate all the strain energy terms. The performance of
the element based on such a technique has been proved to be excellent for both thick and thin
situations. An attempt is made here to develop a "eld consistent shear-#exible curved beam
element, adopting spline basis functions for the dynamic analysis.
In this paper, we examine the cubic B-spline element from the point of view of "eld-consistency
to "nd the optimal assumed membrane/shear strain functions that will eliminate locking for all
boundary conditions. The performance of the element is tested for vibration analysis of both thick
and thin beams considering a number of problems. The results are compared, wherever possible,
with the available analytical solutions.
2. FORMULATION
2.1. ¸aminated beam theory
A laminated composite beam, having radius of curvature R, is considered with the co-ordinates
x along the axis of the beam and z along the thickness direction, as shown in Figure 1. The
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
SHEAR FLEXIBLE FIELD-CONSISTENT CURVED BEAM ELEMENT
389
Figure 1. Laminated curved beam co-ordinate system
displacements in the kth layer, uI and w at point (x, z) from the median surface are expressed as
functions of mid-plane displacement u and w and independent rotation h of the normal in the xz
M
plane, as
uI(x, z, tN )"u (x, tN ) (1#z/R)!z w (x, tN )#[ f (z)#gI(z)] (w (x, tN )#h(x, tN ))
M
V
V
w(x, z, tN )"w(x, tN )
where w "*w/*x and tN is the time. The functions f (z) and gI(z) are de"ned as
V
f (z)"(t/n) sin(nz/t)!(t/n) b cos(nz/t)
gI(z)"aI z#bI
(1)
(2a)
(2b)
where t is the thickness of the laminate.
In equation (2) coe$cients bI are determined such that displacement component uI is continuous at the interface of the adjacent layers and zero at the mid-point of the cross-section. Finally,
the coe$cients b and aI in equation (2) are computed from the requirement that the transverse
shear stress pI is continuous at the interface of the adjacent layers and vanishes at the top and
VX
bottom surfaces of the beam. The details of the derivations of constants, b , aI and bI can be
found in References 24 and 25.
The strains for the kth layer can be written as
+e,"
eN I
e@ I
#
0
eQ
(3)
The mid-plane strain +e,, bending strain +e, and shear strain +e, in equation (3) are written as
+e,"u # w/R
(4a)
MV
+e,I"!zw #[ f (z)#gI(z)] (w #h )#zu /R
(4b)
VV
VV
V
MV
+e,I"( f #gI ) (w #h)
(4c)
X
X V
where the subscript comma denotes the partial derivative with respect to the spatial co-ordinate
succeeding it. For a composite laminated beam of layers of thickness t (k"1, 2, 3, . . . ), and the
I
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
390
B. P. PATEL, M. GANAPATHI AND J. SARAVANAN
ply-angle j (k"1, 2, 3, . . . ), the necessary expressions for computing the sti!ness coe$I
cients, available in the literature [26], are used. The stress}strain relation for the kth layer is
written as
+p,I"
Q Q
I +e,I
Q Q
(5)
where Q (i, j"1, 6) are the reduced sti!ness coe$cients of the kth layer.
GH
The strain energy function ; for a curved beam with length l is given as
;(d)"( )
J
RI>
[+p,I]2+e,I b dz dx
I RI
(6)
where d is the vector of the degrees of freedom associated with the displacement "eld in a "nite
element discretization and b is the width of the beam.
The kinetic energy ¹ of the beam is written as
¹(d)"( )
J
RI>
I R
oI+uR I wR I, +uR I wR I,2 b dx dz
(7)
I
where the dot over the variable denotes the partial derivative with respect to time and oI is the
mass density of the kth layer.
3. ELEMENT DESCRIPTION
3.1. Curved cubic B-spline beam element
A beam element is assumed to be having q equal sections. The spline function adopted to
represent the three "eld variables u , w and h is the cubic B-spline of equal section length (h), and
M
is given as (Figure 2)
O>
O>
O>
u " a u, w" b u, h" c u
(8a)
M
G G
G G
G G
G\
G\
G\
in which each local cubic B-spline u has non-zero values over four consecutive sections with the
G
section-knot x"x as the centre, and is de"ned as
G
1
u"
G
6h
x(x ,
G\
(x!x ),
x )x)x
G\
G\
G\
h#3h(x!x )#3h(x!x )!3(x!x ), x )x)x
G\
G\
G\
G\
G
h#3h(x !x)#3h(x !x)!3(x !x), x )x)x
G>
G>
G>
G
G>
(x !x),
x )x)x
G>
G>
G>
0,
x (x
G>
0,
Copyright 1999 John Wiley & Sons, Ltd.
(8b)
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
SHEAR FLEXIBLE FIELD-CONSISTENT CURVED BEAM ELEMENT
391
Figure 2. (a) Typical cubic B-spline; (b) basis of cubic B-spline expression
1
uM "
GV
6h
x(x ,
G\
3(x!x ),
x )x)x
G\
G\
G\
3h#6h(x!x )!9(x!x ),
x )x)x
G\
G\
G\
G
!3h!6h(x !x)#9(x !x), x )x)x
G>
G>
G
G>
!3(x !x),
x )x)x
G>
G>
G>
0,
x (x
G>
0,
(8c)
a , b , c are the spline parameters. The strain}displacement as well as the stress}strain relationG G G
ships can be rewritten in terms of spline parameters, after substituting equation (8) in (4) and (5).
Finally, the incorporation of strain energy and the kinetic energy, as de"ned by equations (6) and
(7), in Lagrange's equation of motion leads the governing equation for the free vibration of the
beam as
[M] +dG ,#[[K ]#[K ]#[K ]] +d,"+0,
Copyright 1999 John Wiley & Sons, Ltd.
(9)
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
392
B. P. PATEL, M. GANAPATHI AND J. SARAVANAN
where [M] is the consistent mass matrix of the beam; [K ], [K ], [K ] are sti!ness matrices,
concerned with membrane, bending and shear, respectively. These matrices are evaluated with
respect to +d,, where +d, is the generalized vector based on spline parameters only.
The displacements u and w, and rotation h at the ends of the spline beam element can be
M
expressed in terms of the spline parameters as
1
u " (a #4a #a ),
M 6 \
1
u " (!a #a )
MV 2h
\
and
1
1
u " (a #4a #a ), u " (!a #a )
MO 6 O\
O
O>
MOV 2h
O\
O>
1
1
w " (b #4b #b ), w " (!b #b )
6 \
V 2h
\
and
1
1
w " (b #4b #b ), w " (!b #b )
O
O>
OV 2h
O\
O>
O 6 O\
1
h " (c #4c #c ),
6 \
(10)
1
h " (!c #c )
V 2h
\
and
1
1
h " (c #4c #c ), h " (!c #c )
O 6 O\
O
O>
OV 2h
O\
O>
If the element is divided into q equal sections, there will be, all together, [3;(q#3)] spline
parameters to de"ne the displacements (u , w) and rotation (h) functions. Instead of the spline
M
parameters a , a , b , b , c and c at the ends of the element and a , a , b , b , c and
O O O
\ O> \ O> \
c
outside the element, one can use the displacements, rotation and their derivatives at the ends,
O>
i. e. u , u , w, w , h and h . Thus, within the element, for the present case, there are [3;(q!1)]
M MV
V
V
interior spline parameters which are included in the generalized vector +d,, after the transformation based on equation (10). In this way, the original generalized vector of spline parameters +d,
will now consist of physical co-ordinates vector (displacements and their derivatives) pertaining
to the ends of the element and vector of interior spline parameters. In view of the modi"cation in
the original vector of spline parameters, mass and sti!ness matrices shown in equation (9) are to
be updated with respect to the modi"ed generalized vector. This procedure enables us to assemble
the element level matrices and introduce the desired boundary conditions, as followed in the
standard "nite element procedure.
Here, we shall experiment with a series of assumed membrane/shear strain functions for
a displacement type cubic B-spline element to derive the optimal membrane/shear strain de"nitions that leave the element free of all problems for most of the boundary conditions. The
systematic procedure adopted here, as per the "eld consistency principle, is by establishing
the consistency of membrane/shear strain "elds by simply smoothing the w and h to the order of
the functions for u and w (Level 1 functions below*the Q-element), by smoothing memMV
V
brane/shear strain "elds to linear functions (Level 2 functions*the L-element), and by smoothing
these strain functions to constant form (Level 3 functions*the C-element).
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
393
SHEAR FLEXIBLE FIELD-CONSISTENT CURVED BEAM ELEMENT
3.2. O-element2¹he original element ( ,eld-inconsistent element2ISIM)
In a conventional approach, the interpolating functions u based on B-spline basis, as de"ned
G
by equation (8b), are cubic and associate four constants within intervals, and are used for de"ning
the "eld variable interpolations, see equation (8a). Thus, the membrane and shear strains can be
derived from
0,
a #a x#a x#a x,
a #a x#a x#a x,
u" G
a #a x#a x#a x,
a #a x#a x#a x,
0,
0,
b #b x#b x#b x,
b #b x#b x#b x,
w" G
b #b x#b x#b x,
b #b x#b x#b x,
0,
0,
c #c x#c x#c x,
c #c x#c x#c x,
h" G
c #c x#c x#c x,
c #c x#c x#c x,
0,
as
x(x
G\
)x)x
G\
G\
x )x)x
G\
G
x )x)x
G
G>
x )x)x
G>
G>
x (x
G>
x
x(x
G\
)x)x
G\
G\
x )x)x
G\
G
x )x)x
G
G>
x )x)x
G>
G>
x (x
G>
x
(11b)
x(x
G\
)x)x
G\
\
x )x)x
G\
G
x )x)x
G
G>
x )x)x
G>
G>
x (x
G>
x
(11c)
x(x
0,
(a #b /R)#(2a #b /R)x#(3a #b /R)x#(b /R)x,
(a #b /R)#(2a #b /R)x#(3a #b /R)x#(b /R)x,
eN"
G
(a #b /R)#(2a #b /R)x#(3a #b /R)x#(b /R)x,
(a #b /R)#(2a #b /R)x#(3a #b /R)x#(b /R)x,
0,
Copyright 1999 John Wiley & Sons, Ltd.
(11a)
G\
)x)x
G\
G\
x )x)x
\
G
x )x)x
G
G>
x )x)x
G>
G>
x (x
G>
(12a)
x
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
394
B. P. PATEL, M. GANAPATHI AND J. SARAVANAN
eQ"( f #gI )
G
X
X
x(x
G\
(b #c )#(2b #c ) x#(3b #c ) x#(c )x,
x )x)x
G\
G\
(b #c )#(2b #c ) x#(3b #c ) x#(c )x,
x )x)x
G\
G
(b #c )#(2b #c ) x#(3b #c )x#(c )x,
x )x)x
G
G>
(b #c )#(2b #c ) x#(3b #c ) x#(c ) x, x )x)x
G>
G>
0,
x (x
G>
(12b)
0,
If the element is to model a situation in which in-extensional bending and the Kirchho! limit of
classical thin beam predominate, the requirements that the membrane and shear strain should
vanish, produce in e!ect the following conditions:
Membrane case:
(a #b /R)"0, (2a #b /R)"0, (3a #b /R)"0, (b /R)"0
(a #b /R)"0, (2a #b /R)"0, (3a #b /R)"0, (b /R)"0,
(a #b /R)"0, (2a #b /R)"0, (3a #b /R)"0, (b /R)"0,
(a #b /R)"0, (2a #b /R)"0, (3a #b /R)"0, (b /R)"0,
x
)x)x
G\
G\
x )x)x
G\
G
x )x)x
G
G>
x )x)x
G>
G>
(13a)
Shear case:
(b #c )"0, (2b #c )"0, (3b #c )"0, (c )"0,
(b #c )"0, (2b #c )"0, (3b #c )"0, (c )"0,
(b #c )"0, (2b #c )"0, (3b #c )"0, (c )"0,
(b #c )"0, (2b #c )"0, (3b #c )"0, (c )"0,
x )x)x
G\
G\
x )x)x
G\
G
(13b)
x )x)x
G
G>
x )x )x
G>
G>
Using the element, based on the original interpolation functions employed directly in deriving the
constrained strain "elds, our studies here show that this element locks (i.e. spurious oversti!ening that increases inde"nitely with reduction in beam thickness). As per the "eld consistency
paradigm, the derivative functions for u , and w, do not match that for the w and h in the
MV
V
membrane and shear strain de"nitions in precisely these cubic terms. Field consistency requires,
in a simple sense, that the membrane/shear strain must be interpolated in a consistent manner.
Thus, w term in e, and h in e must be consistent with the "eld functions for u and w ,
MV
V
respectively.
3.3. Q-element2¹he element based on level 1 consistency (CMCS21)
At the simplest level of "eld consistency, we consider the use of "eld redistributed substitute
interpolation functions which include only those speci"c terms that must be made "eld-consistent, as outlined in References 22 and 23. This is achieved here by smoothing the original
interpolation function in a least-squares accurate fashion to the desired form i. e. the functions
that are consistent with the derivative functions. Here, we need smoothed functions for w and
h which are consistent with the interpolations u , , and w for the use in the membrane and shear
MV
V
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
SHEAR FLEXIBLE FIELD-CONSISTENT CURVED BEAM ELEMENT
395
strains de"nitions, respectively. It can be noted that u , and w, are of the quadratic form,
MV
V
as seen from equation (8c) in de"ning the function u which is involved in evaluating the
GV
derivative of the "eld variables, as per O-Level description. This means that the smoothed
functions we must derive for w, and h are, designated as u, obtained by smoothing the original
G
functions u to be of least-squares form consistent with the derivative functions u . This
G
GV
operation is simple and the substitute interpolation functions obtained, in this way, are given as
follows:
0,
x(x
1/120!1/10(x!x
x
)/h#(1/4) (x!x )/h,
G\
G\
17/120#(4/5) (x!x )/h!(x!x )/4h,
G\
G\
u"
G
83/120!(36/120) (h!x #x)/h!(30/120) (h!x #x)/h,
G>
G>
19/120!(2/5) (h!x #x)/h#(1/4) (h!x #x)/h,
G>
G>
0,
G\
)x)x
G\
G\
x )x)x
G\
G
(14)
x )x)x
G
G>
x )x)x
G>
G>
x (x
G>
The functions u is retained for u and w in the membrane and shear strains computation.
GV
MV
V
However, we shall see below that this element can still lock for certain sets of boundary
conditions, even though consistent de"nitions for the membrane and shear strain "elds have
been assured within the element domain. This suggests that an inconsistency in the
shear/membrane strain de"nitions have been introduced for some set of boundary conditions.
It is therefore necessary to examine lower levels of consistency based on further sets of
assumed membrane/shear strain functions to see if these locking mechanisms/de"ciency can be
removed.
3.4. L-element2¹he element based on level 2 consistency (CMCS22)
It is logical now to consider smoothed interpolation functions based on the linear form for
w and h pertaining to membrane and shear strains, respectively. This will necessitate that the
derivative functions de"ning u and w will also have to be smoothed to the linear form. The
MV
V
shape functions required here can be denoted as u for w and h, and u for u and w ,
GV
MV
V
G
respectively. The interpolation functions u are obtained by smoothing the original derivative
GV
forms u in a least-squares accurate operation over the element domain. These functions are
GV
then
x(x
G\
!1/30#(3/20) (x!x )/h,
x )x)x
G\
G\
G\
11/60#(11/20) (x!x )/h,
x )x)x
G\
G\
G
u"
G
11/15!(11/20) (h!x #x)/h, x )x)x
G>
G
G>
7/60!(3/20) (h!x #x)/h,
x )x)x
G>
G>
G>
0,
x (x
G>
0,
Copyright 1999 John Wiley & Sons, Ltd.
(15a)
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
396
B. P. PATEL, M. GANAPATHI AND J. SARAVANAN
x(x
G\
!1/(12h)#(1/2h) (x!x )/h,
x )x)x
G\
G\
G\
3/4h!(1/2h) (x!x )/h,
x )x)x
G\
G\
G
u "
GV
!1/4h!(1/2h)(h!x #x)/h,
x )x)x
G>
G
G>
!5/12h#(1/2h) (h!x #x)/h, x )x)x
G>
G>
G>
0,
x (x
G>
0,
(15b)
Numerical experiments below show that this element is identical in behaviour as seen in the case
of Level 1, and it locks for most of the practical boundary conditions for a beam. Thus smoothing
to level 2 has failed to remove the locking mechanism.
3.5. C-element2¹he element based on level 3 consistency (CMCS23)
We now go down to the lowest level (a constant form for representing the w and h) and
correspondingly, the derivative functions de"ning u , and w will also have to be smoothed to
MV
V
the constant form. The interpolation functions for this case is assigned as u for w and h, and
G
u for u and w , respectively. The functions u are obtained by smoothing the original
GV
MV
V
GV
derivative forms u in a least-squares accurate operation over the element domain. It can be
GV
shown that these functions to be used are
x(x
G\
1/24, x )x)x
G\
G\
11/24, x )x)x
G\
G
u "
G
11/24, x )x)x
G
G>
1/24, x )x)x
G>
G>
0,
x (x
G>
(16a)
x(x
G\
1/6h,
x )x)x
G\
G\
1/2h,
x )x)x
G\
G
u "
GV
!1/2h, x )x)x
G
G>
!1/6h, x )x)x
G>
G>
0,
x (x
G>
(16b)
0,
0,
The numerical analysis below shows that this element does not lock for the boundary conditions
considered here, i.e. its accuracy is insensitive to a very large variation in the thickness parameter
of the beam. It can be noted that this can be achieved by using a reduced integration rule, i.e.
a Gaussian one-point integration of membrane and shear energy will produce the same results.
However, for the thick beam, the numerical study given below brings out di!erent results
compared to the actual one. The procedure adopted by the "eld consistency approach permits
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
SHEAR FLEXIBLE FIELD-CONSISTENT CURVED BEAM ELEMENT
397
greater #exibility in the choice of integration order for the energy terms, as these can now be
determined by other considerations (such as variable thickness, more accurate integration of 1/R
terms, etc.) and not dictated by the need for "eld consistency.
4. NUMERICAL EXPERIMENTS
Numerical computation is carried out for the rank of the element and it has three proper zero
values, and thus, does not produce any spurious mode. Further, for the critical evaluation of the
present formulation, a series of test examples with di!erent boundary conditions, are considered
to check the convergence properties and locking behaviour. The element variations chosen for the
study are the di!erent functions for the redistributed strain "elds through the "eld consistent
strategy.
¹est case 1: Clamped}clamped circular arc with sector angle "1203
The numerical results for the free vibration analyses are evaluated using the element developed
in the earlier section and are shown in Tables I and II. These results are related to the "rst three
modes of vibration of arch with slenderness ratio of ¸/t"10, 100 and 10, and are in the form of
convergence studies of frequencies ) ("u [oA¸/D], where u is the frequency, A is the
G
G
cross-sectional area and D is the #exural rigidity) with the number of B-spline sections q. The
solutions obtained based on the original functions for the "eld variables (u, w, h), which results in
"eld inconsistency approach for the constrained strains (membrane and shear), and also employing di!erent levels of "eld consistent representation for the constrained strain functions (quadratic, linear, constant) are indicated in these tables. The comparative results quoted in these tables
are calculated using a one-point numerical reduced integration scheme (RI) for the evaluation
of membrane and shear energy terms, and the analytical solutions within the con"nes of
Euler}Bernoulli beam theory [27] (CL). It can be noted here that the numerical experiments with
various combinations of reduced integration techniques for estimating the membrane and shear
energy terms were conducted. The best accuracy was attained for one-point quadrature scheme
for both membrane and shear energy terms. For the sake of brevity, the detailed study is not
presented here. It is clear from Tables I and II that the convergence is good for fairly thick and
thin cases (¸/t"10, 100), irrespective of the level of consistency whereas the consistency
approach with constant level (CMCS-3) yields a good convergence even for an extremely thin
situation (¸/t"10). However, the reduced integration scheme yields erroneous results for
a fairly thick beam ¸/t"10, but it gives acceptable results for fairly as well as extremely thin
situations as highlighted in Tables I and II. For the reasonable number of spline sections q to
model the beam adequately, it can be seen from Table II that the element exhibits severe locking,
when the full integration scheme is employed. With the lowering of the order of the level for the
"eld consistency requirement in de"ning the constrained strain terms from quadratic to linear
functions and then to constant one, shear locking gets freed faster than the membrane counterpart and it is more so with the increase of spline sections. However, applying the consistent
strategy either for shear or membrane alone cannot completely free the spurious energy locked in
the element. It may be further inferred from these tables that the consistent way of representing
the constrained strains (membrane and shear) at constant level produces remarkable recovery
from the locking syndrome, and the results agree well with the classical solutions for very thin
beams. This study, in general, suggests that for the beam thickness parameter ¸/t of the order 100,
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
Mode 1
¸/t
q
2
SO
SQ
SL
SC
MQ
ML
MC
MO
MQ
ML
MC
MO
MQ
75)857
75)787
75)726
75)568
75)724
75)654
75)593
75)434
75)724
75)654
75)593
75)434
63)577
63)494
63)418
63)230
124)74
124)74
124)74
124)65
124)74
124)74
124)74
124)65
113)91
113)91
113)91
113)82
113)91
113)91
113)91
113)82
680)463
680)408
653)810
312)468
680)463
680)408
653)809
312)468
10)042
46)507
46)462
45)803
45)517
46)460
46)415
45)754
45)468
44)782
44)781
44)675
44)222
44)781
44)780
44)675
44)221
44)573
44)573
44)540
44)309
44)573
44)573
44)540
44)309
RI
8
SO
SQ
SL
SC
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
16
SO
SQ
SL
SC
24
32
RI
CL
61)754
61)719
61)395
61)264
44)776
44)775
44)669
44)215
44)737
44)736
44)632
44)180
58)816
58)816
58)737
58)524
58)816
58)815
58)736
58)523
44)572
44)572
44)538
44)307
44)554
44)554
44)521
44)290
58)236
58)236
58)212
58)078
58)236
58)236
58)212
58)078
44)509
44)509
44)492
44)348
44)509
44)509
44)492
44)348
44)508
44)508
44)491
44)347
44)481
44)481
44)471
44)371
44)480
44)480
44)470
44)371
24)420
51)975
59)883
59)854
59)594
59)491
122)153
121)793
117)696
115)211
122)020
121)660
117)558
115)070
58)668
58)668
58)588
58)376
57)896
57)895
57)829
57)635
100)601
100)596
100)250
98)768
100)598
100)594
100)247
98)765
58)046
58)046
58)034
57)947
58)046
58)046
58)034
57)947
58)171
58)171
58)146
58)013
57)771
57)771
57)749
57)621
99)942
99)942
99)873
99)171
99)942
99)942
99)873
99)171
57)953
57)953
57)946
57)885
57)953
57)953
57)946
57)885
32)085
103)668
680)463
680)408
653)803
312)467
121)333
120)975
116)910
114)436
116)393
116)002
111)469
108)800
100)534
100)529
100)182
98)701
100)198
100)193
99)831
98)333
99)914
99)914
99)845
99)142
99)739
99)739
99)668
98)961
99)739
99)739
99)707
99)311
99)623
99)623
99)589
99)192
99)661
99)661
99)642
99)382
99)573
99)573
99)555
99)294
43)267
46)930
58)003
58)003
57)991
57)904
57)738
57)738
57)727
57)642
99)758
99)758
99)725
99)330
31)774
44)473
44)473
44)463
44)364
680)463
680)408
653)809
312)467
41)338
31)188
23)552
44)481
44)481
44)471
44)371
61)121
61)084
60)732
60)584
29)666
44)498
44)498
44)481
44)337
MC
261)756
28)711
21)958
RI
SO
SQ
SL
SC
61)764
61)729
61)407
61)277
18)141
RI
SO
SQ
SL
SC
46)218
46)172
45)506
45)216
13)576
RI
10
46)259
46)213
45)553
45)266
36)950
ML
99)758
99)758
99)725
99)330
49)153
57)922
57)922
57)915
57)853
57)724
57)724
57)718
57)657
99)675
99)675
99)656
99)396
99)675
99)675
99)656
99)396
50)484
187)474
B. P. PATEL, M. GANAPATHI AND J. SARAVANAN
4
Mode 3
MO
RI
SO
SQ
SL
SC
Mode 2
398
Copyright 1999 John Wiley & Sons, Ltd.
Table I. Values of ) for moderately thick and fairly thin clamped}clamped circular arcs with sector angle 1203 (M: Membrane, S: Shear, O: Original,
G
Q:Quadratic; L: Linear, C: Constant, RI: Reduced integration, CL: Classical theory)
729)03
728)09
728)09
725)87
727)64
726)70
726)69
724)47
RI
4
SO
SQ
SL
SC
8
121)57
117)05
113)87
94)021
119)55
114)95
111)71
91)396
16
SO
SQ
SL
SC
52)736
52)635
52)573
52)268
52)668
52)568
52)506
52)201
51)885
51)884
51)882
51)880
51)884
51)883
51)881
51)879
32
RI
CL
94)749
88)748
84)424
54)010
514)64
507)26
486)97
334)39
51)884
51)883
51)881
51)879
51)876
51)876
51)875
51)874
51)876
51)876
51)875
51)874
51)876
51)876
51)875
51)874
52)422
52)321
52)260
51)956
110)13
109)31
108)57
105)21
51)875
51)875
51)875
51)873
51)875
51)875
51)874
51)873
51)766
51)975
67178)9
64729)7
57447)5
3174)49
513)73
506)25
485)69
331)00
109)91
109)09
108)34
104)99
51)883
51)881
51)879
51)877
103)07
103)06
103)05
103)03
103)06
103)05
103)04
103)02
501)82
493)89
472)17
309)20
481)21
469)65
436)91
125)48
839)035
828)067
804)889
730)009
103)01
103)01
103)00
103)00
103)01
103)00
103)00
103)00
109)61
108)79
108)05
104)69
108)65
107)81
107)07
103)68
235)984
231)758
225)896
199)604
103)00
103)00
103)00
102)99
103)00
103)00
103)00
102)99
99)838
103)668
67178)9
64729)7
57447)5
3174)49
837)477
826)538
803)448
729)107
835)264
824)551
801)973
729)017
734)937
724)263
703)244
650)309
235)410
231)169
225)297
198)965
234)032
229)758
223)864
197)433
229)511
225)090
219)078
191)980
187)782
187)727
187)668
187)581
187)764
187)709
187)650
187)563
187)470
187)466
187)455
187)443
187)469
187)464
187)454
187)441
187)447
187)446
187)441
187)428
187)446
187)445
187)440
187)426
161)503
103)06
103)05
103)04
103)02
103)06
103)04
103)03
103)02
187)798
187)743
187)684
187)596
187)788
187)734
187)674
187)587
182)321
103)01
103)00
103)00
103)00
103)00
103)00
103)00
102)99
187)472
187)467
187)457
187)444
99)640
51)875
51)875
51)874
51)873
67178)9
64729)7
57447)5
3174)49
118)607
99)271
51)876
51)876
51)875
51)874
67178)9
64729)7
57447)5
3174)49
2679)589
98)655
51)764
51)875
51)875
51)875
51)873
1136)8
1136)8
1136)8
1136)8
98)642
51)760
RI
SO
SQ
SL
SC
52)632
52)531
52)469
51)164
1136)8
1136)8
1136)8
1136)8
187)471
187)466
187)456
187)443
184)933
103)00
103)00
103)00
102)99
103)00
103)00
102)99
102)99
187)448
187)447
187)441
187)428
187)448
187)447
187)441
187)428
185)466
187)474
399
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
24
114)68
109)98
106)50
85)001
1245)3
1245)3
1245)3
1245)3
298)621
51)800
RI
SO
SQ
SL
SC
1245)3
1245)3
1245)3
1245)3
52)240
RI
10
599)51
598)37
598)36
595)66
72)645
RI
SO
SQ
SL
SC
727)64
726)70
726)69
724)47
SHEAR FLEXIBLE FIELD-CONSISTENT CURVED BEAM ELEMENT
Copyright 1999 John Wiley & Sons, Ltd.
2
SO
SQ
SL
SC
Mode 1
¸/t
q
2
SO
SQ
SL
SC
MQ
ML
MC
MO
MQ
ML
MC
MO
MQ
ML
MC
7E#06
7E#06
7E#06
7E#06
7E#06
7E#06
7E#06
7E#06
7E#06
7E#06
7E#06
7E#06
6E#06
6E#06
6E#06
6E#06
1E#07
1E#07
1E#07
1E#07
1E#07
1E#07
1E#07
1E#07
1E#07
1E#07
1E#07
1E#07
1E#07
1E#07
1E#07
1E#07
6)7E#12
6)5E#12
5)7E#12
3)2E#12
6)7E#12
6)5E#12
5)7E#12
3)2E#07
6)7E#12
6)5E#12
5)7E#12
3)2E#07
6)7E#12
6)5E#12
5)7E#12
3)2E#07
722093)9078
1E#06
1E#06
1E#06
774146
1E#06
1E#06
984997
741993
93637
87363
84884
58515
89613
82982
80452
52476
9923)3
9096)5
8950)1
5954)6
9383)4
8502)7
8354)4
5128)9
RI
8
SO
SQ
SL
SC
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
16
SO
SQ
SL
SC
24
32
RI
CL
5E#06
5E#06
5E#06
3E#06
87429
80589
78024
48723
71466
62518
59614
52)056
387263
360711
341724
189641
380612
353465
334161
176837
9309)5
8421)0
8271)4
4962)2
7705)3
6589)0
6425)1
51)974
36032
32555
31452
16522
35051
31453
30332
14522
2827)0
2573)1
2550)3
1674)5
2664)9
2394)6
2371)6
1424)2
2654)9
2383)5
2360)3
1397)7
1107)4
991)58
985)74
586)67
1105)0
988)89
983)03
579)65
51)969
51)975
5E#06
5E#06
4E#06
126)45
8330794
8220856
7992036
7255197
8315338
8205725
7977843
7246576
372218
344275
324570
159135
334581
302147
280135
104)3
140 4226
1325244
1228126
563595
1394364
1314692
1216774
540267
34752
31117
29987
13744
31673
27577
26371
103)6
111852
101391
96058)2
41033)1
110197
99524)9
94135)0
36852)6
9889)3
8811)9
8632)9
4466)7
9589)2
8472)2
8290)0
3843)2
9548)1
8425)5
8242)5
3722)0
4050)3
3584)7
3538)4
1818)2
3922)1
3439)1
3392)0
1549)6
3912)2
3427)8
3380)5
1517)9
103)576
103)668
1371498
1290209
1190467
482833
1281822
1192501
1083720
193)106
109318
98535)9
93106)9
34137)9
103410
91784)9
86104)4
188)531
188)681
8714)0
7459)4
7264)2
103)58
29326)2
26050)6
25153)5
10560)9
103)577
916)10
772)41
766)02
51)970
7289286
7184328
6981791
6475306
194)8775
103)600
2199)8
1862)3
1836)9
51)970
8294356
8187045
7964249
7246004
1125284)34
104)2217
51)968
1176)0
1067)2
1061)4
693)2
5E#06
5E#06
5E#06
3E#06
26803770)76
107)2506
510971
RI
SO
SQ
SL
SC
5E#06
5E#06
5E#06
3E#06
52)041
RI
SO
SQ
SL
SC
783694
708851
654435
54)126
52)5667
RI
10
1E#06
964224
925850
662194
2977875)208
28808)6
25459)8
24551)7
9213)59
28685)5
25319)7
24407)3
8787)40
27178)8
23575)7
22623)6
188)390
188)412
3571)4
3031)6
2981)2
103)58
11777)4
10347)6
10110)5
4210)00
11554)3
10091)4
9851)16
3621)13
11524)2
10056)8
9815)86
3509)10
188)373
187)474
10921)9
9355)06
9103)51
188)368
B. P. PATEL, M. GANAPATHI AND J. SARAVANAN
4
Mode 3
MO
RI
SO
SQ
SL
SC
Mode 2
400
Copyright 1999 John Wiley & Sons, Ltd.
Table II. Values of )G for an extremely thin clamped}clamped circular arc with sector angle 1203 (M: Membrane, S: Shear, O: Original,
Q: Quadratic; L: Linear, C: Constant, RI: Reduced integration, CL: Classical theory)
SHEAR FLEXIBLE FIELD-CONSISTENT CURVED BEAM ELEMENT
401
Figure 3. Locking studies for a simply supported isotropic circular arc with sector angle "1203
the original spline element does not seem to su!er much from locking problems whereas it is
susceptible to membrane and shear locking for the extreme thin case. Furthermore, one can
conclude that unlike elements employing the reduced integration scheme, which may not be
suitable for the problems wherein a higher order of integration is needed (for instance tapered
beams), consistent element frees the order of integration and thus, is applicable for both thick and
extreme thin situations, as expected [22, 23]. Finally, it can be opined that the consistency
approach with constant level (CMCS-3) alone produces good results for fairly thick to extremely
thin beams.
¹est case 2: Simply supported circular arc with sector angle "1203
Natural frequencies obtained for the simply supported circular arch are given in Figure 3 by
varying the aspect ratio of the beam from a fairly thick one to an extremely thin situation. The
di!erent results calculated considering full and reduced integration schemes are highlighted along
with those of the "eld-consistent formulation at the level of constant function strategy for the
membrane and shear strain "elds. The results based on "eld consistency manner once again
reveal that it is in very good agreement with those of the full integration scheme for the thick
situation and they match well with the classical solution for the thin cases. It is also amply clear
from Figure 3 that for the higher frequencies, the introduction of spurious energy is seen early
with respect to the aspect ratio i.e. higher the frequency, the occurrence of locking phenomena can
be noticed with low value of aspect ratio.
¹est case 3: Clamped}clamped circular arc with sector angle "3153
The results for the natural frequencies are shown in Figure 4. Similar observations as brought
out in test problem 2 are made here. The introduction of excessive membrane and shear energy
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
402
B. P. PATEL, M. GANAPATHI AND J. SARAVANAN
Figure 4. Locking studies for a clamped}clamped isotropic circular arc with sector angle "3153
Figure 5. Locking studies for an isotropic full circular ring
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
SHEAR FLEXIBLE FIELD-CONSISTENT CURVED BEAM ELEMENT
403
appears to in#uence the results for the higher values of aspect ratio in comparison with the case of
"120 (Figure 3).
¹est case 4: Complete circular ring
Free vibration analysis of the circular ring is carried out. The variation of natural frequencies
) ("u [oAR/D]) with the aspect ratio is described in Figure 5. The element behaviour for
G
G
this problem, in general, is the same as those of earlier test problems. However, it is apparently
visible from Figure 5 that the results evaluated for a thin ring adopting the reduced integration
scheme is di!erent from the classical solutions and this di!erence decreases with the increase in
the order of frequency. But, the "eld consistent element based on constant level approximation
for the spurious energy terms behaves very well even for very thin beams while obtaining the
lower modes and the results are in good agreement with the analysis using Euler}Bernoulli beam
theory [27].
¹est case 5: straight beams
The frequencies ) ("u [o¸/Gt], where G is the shear modulus, Poisson ratio l"0)3125)
G
G
pertaining to simply supported beam and clamped}clamped beam cases are calculated by
deducing the present formulation for the straight one and are displayed in Figure 6 and Table III,
respectively. It is again clear that the convergence for consistent model with constant level
approach is good with respect to spline sections q, for both thick and thin straight beams. The
results are in very good agreement with the existing results [27].
Test case 6: <ibration analysis of composite laminates
To see the e$cacy of the element formulation, with reference to the locking mechanism for the
composite laminates, free vibration problems of simply supported straight and curved symmetric
Figure 6. Locking studies for a simply supported isotropic straight beam
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
404
SO
¸/t
10
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
1 000 000
SL
q
Mode 1
Mode 2
Mode 3
Mode 1
Mode 2
Mode 3
2
4
6
8
10
10)63
9)8894
9)8665
9)857
9)8512
50)9063
26)1518
25)5653
25)4958
25)4583
105
50)8999
47)0623
46)6035
46)4775
10)523
9)888
9)8664
9)857
9)8512
50)9063
26)1276
25)563
25)4953
25)4582
105
50)8999
47)0506
46)6012
46)4768
Ref. 18
100
SQ
Mode 1: 9)801,
2
4
6
8
10
33)529
11)11
10)513
10)466
10)46
509)063
50)5892
30)6053
29)0746
28)8633
2
4
6
8
19
317745
36736
10370
4241
2130)3
5090625
412234
99164)3
37669
18114)3
CL
1050
281)621
77)2829
59)4646
57)0509
Mode 2: 25)1008
28)732
10)94
10)5
10)464
10)459
1)1E#07 266469
2758113 31055
517117
8809
177949
3600)1
81032)1 1804)3
Mode 1: 10)46,
509)063
1050
47)4768 270)4768
30)2832 74)7362
29)0243 59)0403
28)8505 56)9501
5090625
372829
87683)3
32974)1
15738)6
Mode 1 Mode 2
10)412
9)8633
9)856
9)8505
9)8465
50)906
25)749
25)46
25)444
25)424
SC
Mode 3
105
50)8999
46)4621
46)3713
46)3447
RI
Mode 1 Mode 2 Mode 3 Mode 1 Mode 2 Mode 3
10)164
9)8199
9)8148
9)818
9)8204
50)906
25)56
25)169
25)187
25)216
105
50)246
45)754
45)477
45)575
10)508
9)9914
9)9044
9)8796
9)8674
50)906
27)013
26)001
25)709
25)592
10)84
10)486
10)462
10)459
10)458
509)06
29)707
28)939
28)836
28)813
1050
65)913
57)925
56)747
56)513
10)844
10)515
10)465
10)459
10)458
509)06
20)696
29)056
28)88
28)832
Mode 3: 45)3394
28)698
10)926
10)496
10)463
10)459
509)06
1050
45)237 244)6609
30)106 72)0915
28)991 58)6215
28)843 56)8501
1)1E#07 266177 5E#06 10500000 10)847 5E#06 1E#07
2645893 30791 344185 2377397)1 10)494 29)761 66)17
476378
8671)4 82864 435281)86 10)469 28)987 58)113
160723
3542)6 31431 149022)12 10)466 28)883 56)916
72288
1779)2 15099 67646)632 10)465 28)859 56)675
Mode 2: 28)84,
105
50)9
49)56
47)664
47)055
Mode 3: 56)55
1050
67)096
58)897
57)151
56)691
10)847 5E#06 1E#07
10)52
29)728 67)233
10)471 29)094 59)027
10)466 28)919 57)285
10)464 28)872 56)828
B. P. PATEL, M. GANAPATHI AND J. SARAVANAN
Copyright 1999 John Wiley & Sons, Ltd.
Table III. Values of )G for a clamped}clamped straight beam (S: Shear, O: Original, Q: Quadratic; L: Linear, C: Constant, RI: Reduced
integration, CL: Classical)
SHEAR FLEXIBLE FIELD-CONSISTENT CURVED BEAM ELEMENT
405
Figure 7. Locking studies for a simply supported laminated straight beam (03/903/03)
Figure 8. Locking studies for a simply supported laminated circular arc (03/903/03) with sector angle "1203
cross-ply (0/90/0) laminated beams are considered. The material properties used here are:
E /E "25, G /E "0)5, G /E "0)2, G /E "0)5, l "0)25
The frequencies ) ("u [oA¸/D ], where D is the bending sti!ness coe$cient) for straight
G
G
and curved beams are plotted in Figures 7 and 8, respectively. The element performance is quite
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
406
B. P. PATEL, M. GANAPATHI AND J. SARAVANAN
Table IV. Frequency parameter )* ("u ¸ [12 o/E t]) for a simply supported
G
G
two-layered cross-ply [03/903] laminated arc
100
¸/t"20
Sector
angle, 5)729583
57)29583
180
Mode
Present
Reference 12
Present
Reference 13
1
2
3
4
4)64630
17)9626
38)4482
64)5268
*
*
*
*
4)69811
18)8138
42)3390
75)2691
4)6939
18)796
42)294
75)274
1
2
3
4
4)05676
17)6089
38)4293
64)8491
3)9109
17)089
37)667
64)041
4)04220
18)1587
41)6581
74)4162
4)0115
18)030
41)432
74)608
1
2
3
4
*
12)7163
34)2522
61)7930
*
*
*
*
*
12)7430
36)0239
68)9281
*
12)482
35)258
68)616
similar to those of isotropic cases. It is observed from these "gures that the general characteristics
of laminated beams are similar to that of isotropic cases. Furthermore, it is seen from Figure
8 that unlike the isotropic cases, the occurrence of locking here is noticed for a high value of
aspect ratio. This is because of reduced transverse shear rigidity for the laminates.
¹est case 7: <ibration of laminated composite arch
Free vibration analysis of a simply supported two-layered cross-ply [0/90] laminated arc is
carried out considering three values for sector angle (
) and two values for ¸/t. The material
properties used are:
E /E "15, G /E "0)5, G /E "0)5, G /E "0)5, l "0)25
The frequency parameters )* ("u ¸ [12 o/E t]) obtained here are compared in Table IV with
G
G
the available analytical solutions given in References 12 and 13, and are found to be in good
agreement.
5. CONCLUSIONS
The shear-membrane locking phenomena in the shear #exible curved element based on cubic
B-spline functions for the displacement "elds have been analysed using the "eld-consistency
approach. The "eld consistent redistribution of membrane and shear strain "elds at the lowest
level yields accurate results for both fairly thick and extremely thin beams by using the full
integration scheme for the evaluation of all the strain energy terms. The capabilities and
e!ectiveness of the element have been demonstrated here for the vibration analysis of a wide
range of problems without membrane or shear locking e!ects emerging.
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
SHEAR FLEXIBLE FIELD-CONSISTENT CURVED BEAM ELEMENT
407
REFERENCES
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5. Sheikh AH, Mukhopadhyay M. Transverse vibration of plate structures with elastically restrained edges by the
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Computers and Structures 1996; 59:257}263.
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Sound and <ibrations 1995; 179:763}776.
9. Chandrasekhara K, Krishamurthy K, Roy S. Free vibration of composite beams including rotary inertia and shear
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Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 387}407 (1999)
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