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A conoidal shell analysis by modified isoparametric element
Compurers & Structures Vol. 18, No. Pnnted in Great Britain.
5, pp. 921-924,
004s7949/84 Pergamon
1984
$3.00 + .xl Pnss Ltd.
A CONOIDAL SHELL ANALYSIS BY MODIFIED ISOPARAMETRIC ELEMENT GANG-KDON CHOIR Department of Civil Engineering, Korea Advanced Institute of Science and Technology, Seoul, Korea (Received
11 March 1983; received for publication
10 May 1983)
Abstract-The quadratic isoparametric element is modified for applications in thin shell analysis. Four extra nonconforming displacement modes are added to transverse displacement then, later, they are eliminated through static condensation. A conoidal shell under uniform pressure is analyzed using the new element and the results compared with previous works. INTRODUCTION
some of the additional modes that were originally introduced into the quadratic element in Ref.[4]. While the modes added to the transverse displacement component for the quadratic element are retained, the additions to the in-plane displacement components have been dropped because the quadratic element already has second order terms present in the in-plane displacement. The addition of nonconforming modes to in-plane displacements has been found to have negligible effect. Thus, the quadratic isoparametric element has been further modified in this study so as to have four extra displacement modes added to the out-of-plane displacement component only. The addition of the extra displacement modes tends to destroy displacement compatibilities along the interelement boundaries. It is noted, however, that strict conformity along the interelement boundaries is not essential in order to assure convergence[3,4].
in a truncated form, is most commonly used for factory type buildings, cantilevered canopies and dams. A conoid is a shell of complicated shape, i.e. doubly curved nondevelopable shell whose Gaussian curvature is negative. Depending upon the curve used as the directrix, a parabolic, circular or catenary conoid is generated by a straight line moving parallel to a vertical plane with one of its ends on a plane curve and the other on a straight line (Fig. l)[lO]. The structural behavior of the conoid is also complicated. While the surface near the straight directrix loses all the curvature and functions much like a flat plate, the part of shell near the curved directrix, where the sharp curvature exists, behaves more like a shell and membrane behavior is predominant. Some exceptions occur near the boundaries where the disturbances penetrate some distance into the interior of the shell. Because of these complexities, only limited work has been done on this type of shell. The isoparametric element due to its versatility ought to be one of the more suitable tools for complicated shell analysis. However, unless some modifications are made to the element, the direct application of this type of element to shell analysis can lead to unrealistic results. This stems from the fact that the element is too stiff against flexure. Furthermore, too large a number of variables have to be included[2, 1I]. The conoidal
shell, usually
ELEMENT STIFFNESS The displacement field for the element is formed from the displacement definitions of Ref. [4] by dropping the extra displacement modes added for the in-plane displacements. Thus,
or expressed in a more compact form:
MODIFICATIONS OF ISOPARAMETRIC ELEMENT
U = zN,U, + CNjoj
modifications of the isoparametric element to adapt it to shell analysis are its conversion to an ordinary shell element which has 5 basic degrees of freedom at each mid-surface node by Ahmad and Irons[l], the softening of the shear stiffness by the use of a reduced integration technique as described by Zienkiewicz et al.[6, 121 and the restoration of a more realistic displacement shape by the addition of extra displacement modes by Choi [4] and Wilson [9]. Because the addition of extra displacement modes requires a significant increase in computations, it is desirable to minimize that addition whenever possible. This has been done in this study by dropping Among
the significant
(2)
where j = 1,2,3,4 and &=4(1-5*) N2=1(1
-v2)
(3) %
=
410
-V)
fi4
=
cttu
-
1*1
N, and Vi have usual definitions. The first two modes in eqn (3) are to eliminate the
TAssociate Professor. 921
922
CHANG-KOON 0101 b =49 5’ t=05” L = h, = 18” h2= 9”
72”
E =562 x lO-6 p =o 15 q =60 ps
Curved D~rectnx
PSI
(pressure)
Fig. 1. Geometry and material properties of conoidal shell.
transverse displacement constraints which are present in the original isoparametric element. The third and fourth modes contribute to the softening of the twisting stiffness of the element. The additional degrees of freedom, fj in eqn (I), (or 0 in eqn 2) do not need to represent displacements of any physical node. They are interpreted merely as amplitudes of the added displacement modes. In order to express the strain components in terms of nodal displacements and rotations, two transformations have to be performed. These are (1) the transformation of the global derivative matrix into a local derivative matrix by the standard axis rotations and (2) expressing the global displacement derivatives by the curvilinear derivatives and the inverse of Jacobian. Thus, the local derivative matrix is expressed by
With the stress-strain
relations expressed by
{c’} = [D’]{t’}.
(8)
The stiffness matrix is computed by the direct application of a minimization procedure[8]
K’=
I
1 1
-1 -I SH
-I
[~‘lT[~‘IP’l~ JI d5 dq dS (9)
For the thin shell element, which has the Kirchoff hypothesis in operation, an explicit integration thru the thickness, i.e. in the form of j!, f(l;) d[ can be easily performed and numerical integration is needed in only two directions (r, q). Rearranging the order of displacement components in the equations, the element stiffness matrix K' can be partitioned into conforming and nonconforming parts as follows. K&x44,=
*4) 1. (10)
K %a X4) K ~bX40) K22tq [ KG4 I40)
With the proper combinations of the terms in eqn (4), the strain components can be expressed in terms of the local coordinates. Thus
Then load-deflection equations may be written in the form
(11)
au’
ax -ad
1
=
Because there is no loads corresponding to the internal degrees-of-freedom, 0, eqn (11) can be condensed to
w
E+E
ay* aw’
GfF awf
K'U=R
ax'
ad
(12)
where
ad
K = K,,- K,2K;2'K;2.
ar’+az’
(13)
The enlarged element stiffness matrix K’ (44 x 44) with the extra modes added to displacement definition (eqn 1 or 2), is now condensed back to K' (40 x 40) which has the same order as an ordinary 8-node shell element.
or
In more compact form (7)
NUMERICALEXAMPLE A truncated conoidal shell subjected to a uniform pressure of 60 psi was analyzed using the modified isoparametric element. The dimensions and material properties of the shell were given in Fig. 1. This shell was previously analyzed analytically and
A conoidal shell analysis by modified isoparametric element -0
H- Meth -
L -lOr .-2
I l/4
251 0 (a)
at
WI
This study
y=O
(b)
w* at
(d )
MY), ot y=o
I 314
I l/2
x=
923
;,
-6 c
T e
-4
r;
-2
I c
\
_
k
\ 15-
20
0 (cl
\9
\
\
\
/
at
y$-
/’
0
P’
b-d
l/4 W’
P
/’ I
l/2
3/4
IO
x i
0 ( e
l/4
1 M,,,
at j; = +
I/2
3/4
IO
f
Fig. 2. Displacements of simply supported conoidal shell under uniform load.
tested experimentally by Hadid[5]. The results obtained from this study are compared with those presented earlier in Ref. [5] in Figs. 2 and 3. Good agreements between the results from different analyses are evident. The results from this study are generally in closer agreement with Hadid’s analytical solution than with his experimental solution.
However, the general profiles of the displacements (Fig. 2) and stress resultants (Fig. 3) from the experimental solution are similar to those of analytical solutions. Therefore, it is concluded that if the curves of experimental study are shifted a little at supports, the results of experimental study would be reasonably close to the analytical solutions.
924
CHANG-KOONCHOI
YP
f
-H-MethI,_, -o-
This
( a ) Nxc ,at
‘\
‘o,\
study
I
I I /4
0
c
I/2
IO
y= $ -02
1
4 PC ) N,,
-06-
l/2
314
Ii
at’:=0
I
,L
I
I
-04-
\ \
I/4
0
(b)
N,,,ati=$
l/2
3/4
IO
I/4
0
l/2 r
(d)
N,,at
ji=k
314
b
E
Fig. 3. Stress resultants of simply supported conoidal shell under uniform load.
REFERENCES 1 S. Ahmad, B. M. Irons and 0. C. Zienkiewicz, Analysis
2.
3.
4.
5.
6.
of thick and thin shell structures by curved finite elements. Int. J. Num. Meth. Engng 2, 419451 (1970). C. A. Brebbia and J. J. Conner, Fundamentals of Finite Element Techniques for Structural Engineers. Wiley, New York (1974). G. P. Bazeley, Y. K. Cheung, B. M. Irons and 0. C. Zienkiewicz, Triangular elements in plate bendingconforming and non-conforming solutions. Proc. Co& on Matrix Methods in Structural Mechanics, AFFDDTR-66-80 (1965). C. K. Choi and W. C. Schnobrich, Use of nonconforming modes in finite element analysis of plate and shells. Civil Engineering Studies, SRS-401, University of Illinois, Urbana, Illinois (1973). H. A. Hadid, An analytical and experimental investigation into the bending theory of elastic conoidal shell. Ph.D. Dissertation, University of Southampton (1964). S. F. Pawsey, The analysis of moderately thick to thin
shells by the finite element method. SESM Rep. 70-12, Structural Engineering Laboratory, University of California, Berkeley, California (1970). 7. D. A. Pecknold and W. C. Schnobrich, Finite element analysis of skewed shallow shells. Civil Engineering Studies, SRS-332, University of Illinois, Urbana, Illinois (1968). 8. T. H. H. Pian and P. Tong, Basis of finite element methods for solid continua. Int. J. Num. Meth. Engng 1, 3328 (1969). 9. E. L. Wilson, R. L. Taylor, W. P. Doherty and I. Ghaboussi, Incompatible displacement models. Int. Symp. on Num. Comput. Meth. in Structural Mech.,
University of Illinois, Urbana, Illinois (1971). 10. G. S. Ramaswamy, Design and Construction of Concrete Shell Roofs. McGraw-Hill, New York (1968). 11. 0. C. Zienkiewicz, The Finite Element Method, 3rd Edn. McGraw-Hill, London (1977). 12. 0. C. Zienkiewicz, R. L. Taylor and J. M. Too, Reduced integration technique in general analysis of plates and shells. Int. J. Num. Meth. Engng 3, 275-290 (1971).
5, pp. 921-924,
004s7949/84 Pergamon
1984
$3.00 + .xl Pnss Ltd.
A CONOIDAL SHELL ANALYSIS BY MODIFIED ISOPARAMETRIC ELEMENT GANG-KDON CHOIR Department of Civil Engineering, Korea Advanced Institute of Science and Technology, Seoul, Korea (Received
11 March 1983; received for publication
10 May 1983)
Abstract-The quadratic isoparametric element is modified for applications in thin shell analysis. Four extra nonconforming displacement modes are added to transverse displacement then, later, they are eliminated through static condensation. A conoidal shell under uniform pressure is analyzed using the new element and the results compared with previous works. INTRODUCTION
some of the additional modes that were originally introduced into the quadratic element in Ref.[4]. While the modes added to the transverse displacement component for the quadratic element are retained, the additions to the in-plane displacement components have been dropped because the quadratic element already has second order terms present in the in-plane displacement. The addition of nonconforming modes to in-plane displacements has been found to have negligible effect. Thus, the quadratic isoparametric element has been further modified in this study so as to have four extra displacement modes added to the out-of-plane displacement component only. The addition of the extra displacement modes tends to destroy displacement compatibilities along the interelement boundaries. It is noted, however, that strict conformity along the interelement boundaries is not essential in order to assure convergence[3,4].
in a truncated form, is most commonly used for factory type buildings, cantilevered canopies and dams. A conoid is a shell of complicated shape, i.e. doubly curved nondevelopable shell whose Gaussian curvature is negative. Depending upon the curve used as the directrix, a parabolic, circular or catenary conoid is generated by a straight line moving parallel to a vertical plane with one of its ends on a plane curve and the other on a straight line (Fig. l)[lO]. The structural behavior of the conoid is also complicated. While the surface near the straight directrix loses all the curvature and functions much like a flat plate, the part of shell near the curved directrix, where the sharp curvature exists, behaves more like a shell and membrane behavior is predominant. Some exceptions occur near the boundaries where the disturbances penetrate some distance into the interior of the shell. Because of these complexities, only limited work has been done on this type of shell. The isoparametric element due to its versatility ought to be one of the more suitable tools for complicated shell analysis. However, unless some modifications are made to the element, the direct application of this type of element to shell analysis can lead to unrealistic results. This stems from the fact that the element is too stiff against flexure. Furthermore, too large a number of variables have to be included[2, 1I]. The conoidal
shell, usually
ELEMENT STIFFNESS The displacement field for the element is formed from the displacement definitions of Ref. [4] by dropping the extra displacement modes added for the in-plane displacements. Thus,
or expressed in a more compact form:
MODIFICATIONS OF ISOPARAMETRIC ELEMENT
U = zN,U, + CNjoj
modifications of the isoparametric element to adapt it to shell analysis are its conversion to an ordinary shell element which has 5 basic degrees of freedom at each mid-surface node by Ahmad and Irons[l], the softening of the shear stiffness by the use of a reduced integration technique as described by Zienkiewicz et al.[6, 121 and the restoration of a more realistic displacement shape by the addition of extra displacement modes by Choi [4] and Wilson [9]. Because the addition of extra displacement modes requires a significant increase in computations, it is desirable to minimize that addition whenever possible. This has been done in this study by dropping Among
the significant
(2)
where j = 1,2,3,4 and &=4(1-5*) N2=1(1
-v2)
(3) %
=
410
-V)
fi4
=
cttu
-
1*1
N, and Vi have usual definitions. The first two modes in eqn (3) are to eliminate the
TAssociate Professor. 921
922
CHANG-KOON 0101 b =49 5’ t=05” L = h, = 18” h2= 9”
72”
E =562 x lO-6 p =o 15 q =60 ps
Curved D~rectnx
PSI
(pressure)
Fig. 1. Geometry and material properties of conoidal shell.
transverse displacement constraints which are present in the original isoparametric element. The third and fourth modes contribute to the softening of the twisting stiffness of the element. The additional degrees of freedom, fj in eqn (I), (or 0 in eqn 2) do not need to represent displacements of any physical node. They are interpreted merely as amplitudes of the added displacement modes. In order to express the strain components in terms of nodal displacements and rotations, two transformations have to be performed. These are (1) the transformation of the global derivative matrix into a local derivative matrix by the standard axis rotations and (2) expressing the global displacement derivatives by the curvilinear derivatives and the inverse of Jacobian. Thus, the local derivative matrix is expressed by
With the stress-strain
relations expressed by
{c’} = [D’]{t’}.
(8)
The stiffness matrix is computed by the direct application of a minimization procedure[8]
K’=
I
1 1
-1 -I SH
-I
[~‘lT[~‘IP’l~ JI d5 dq dS (9)
For the thin shell element, which has the Kirchoff hypothesis in operation, an explicit integration thru the thickness, i.e. in the form of j!, f(l;) d[ can be easily performed and numerical integration is needed in only two directions (r, q). Rearranging the order of displacement components in the equations, the element stiffness matrix K' can be partitioned into conforming and nonconforming parts as follows. K&x44,=
*4) 1. (10)
K %a X4) K ~bX40) K22tq [ KG4 I40)
With the proper combinations of the terms in eqn (4), the strain components can be expressed in terms of the local coordinates. Thus
Then load-deflection equations may be written in the form
(11)
au’
ax -ad
1
=
Because there is no loads corresponding to the internal degrees-of-freedom, 0, eqn (11) can be condensed to
w
E+E
ay* aw’
GfF awf
K'U=R
ax'
ad
(12)
where
ad
K = K,,- K,2K;2'K;2.
ar’+az’
(13)
The enlarged element stiffness matrix K’ (44 x 44) with the extra modes added to displacement definition (eqn 1 or 2), is now condensed back to K' (40 x 40) which has the same order as an ordinary 8-node shell element.
or
In more compact form (7)
NUMERICALEXAMPLE A truncated conoidal shell subjected to a uniform pressure of 60 psi was analyzed using the modified isoparametric element. The dimensions and material properties of the shell were given in Fig. 1. This shell was previously analyzed analytically and
A conoidal shell analysis by modified isoparametric element -0
H- Meth -
L -lOr .-2
I l/4
251 0 (a)
at
WI
This study
y=O
(b)
w* at
(d )
MY), ot y=o
I 314
I l/2
x=
923
;,
-6 c
T e
-4
r;
-2
I c
\
_
k
\ 15-
20
0 (cl
\9
\
\
\
/
at
y$-
/’
0
P’
b-d
l/4 W’
P
/’ I
l/2
3/4
IO
x i
0 ( e
l/4
1 M,,,
at j; = +
I/2
3/4
IO
f
Fig. 2. Displacements of simply supported conoidal shell under uniform load.
tested experimentally by Hadid[5]. The results obtained from this study are compared with those presented earlier in Ref. [5] in Figs. 2 and 3. Good agreements between the results from different analyses are evident. The results from this study are generally in closer agreement with Hadid’s analytical solution than with his experimental solution.
However, the general profiles of the displacements (Fig. 2) and stress resultants (Fig. 3) from the experimental solution are similar to those of analytical solutions. Therefore, it is concluded that if the curves of experimental study are shifted a little at supports, the results of experimental study would be reasonably close to the analytical solutions.
924
CHANG-KOONCHOI
YP
f
-H-MethI,_, -o-
This
( a ) Nxc ,at
‘\
‘o,\
study
I
I I /4
0
c
I/2
IO
y= $ -02
1
4 PC ) N,,
-06-
l/2
314
Ii
at’:=0
I
,L
I
I
-04-
\ \
I/4
0
(b)
N,,,ati=$
l/2
3/4
IO
I/4
0
l/2 r
(d)
N,,at
ji=k
314
b
E
Fig. 3. Stress resultants of simply supported conoidal shell under uniform load.
REFERENCES 1 S. Ahmad, B. M. Irons and 0. C. Zienkiewicz, Analysis
2.
3.
4.
5.
6.
of thick and thin shell structures by curved finite elements. Int. J. Num. Meth. Engng 2, 419451 (1970). C. A. Brebbia and J. J. Conner, Fundamentals of Finite Element Techniques for Structural Engineers. Wiley, New York (1974). G. P. Bazeley, Y. K. Cheung, B. M. Irons and 0. C. Zienkiewicz, Triangular elements in plate bendingconforming and non-conforming solutions. Proc. Co& on Matrix Methods in Structural Mechanics, AFFDDTR-66-80 (1965). C. K. Choi and W. C. Schnobrich, Use of nonconforming modes in finite element analysis of plate and shells. Civil Engineering Studies, SRS-401, University of Illinois, Urbana, Illinois (1973). H. A. Hadid, An analytical and experimental investigation into the bending theory of elastic conoidal shell. Ph.D. Dissertation, University of Southampton (1964). S. F. Pawsey, The analysis of moderately thick to thin
shells by the finite element method. SESM Rep. 70-12, Structural Engineering Laboratory, University of California, Berkeley, California (1970). 7. D. A. Pecknold and W. C. Schnobrich, Finite element analysis of skewed shallow shells. Civil Engineering Studies, SRS-332, University of Illinois, Urbana, Illinois (1968). 8. T. H. H. Pian and P. Tong, Basis of finite element methods for solid continua. Int. J. Num. Meth. Engng 1, 3328 (1969). 9. E. L. Wilson, R. L. Taylor, W. P. Doherty and I. Ghaboussi, Incompatible displacement models. Int. Symp. on Num. Comput. Meth. in Structural Mech.,
University of Illinois, Urbana, Illinois (1971). 10. G. S. Ramaswamy, Design and Construction of Concrete Shell Roofs. McGraw-Hill, New York (1968). 11. 0. C. Zienkiewicz, The Finite Element Method, 3rd Edn. McGraw-Hill, London (1977). 12. 0. C. Zienkiewicz, R. L. Taylor and J. M. Too, Reduced integration technique in general analysis of plates and shells. Int. J. Num. Meth. Engng 3, 275-290 (1971).