and Flight Sciences, George Washington University, Hampton, Virginia. _ ~ _. . 16. ..... in the [5(j  1) + J]th column of the element stiffness matrix. Similarly, the ...
N A S A TECHNICAL NOTE
NASA TN 08067
:LOAN COPY: RETURN TO A F ~ L.TECHNICAL L~BRARY K I R T ~ N D A m , w. Ma
A COMPUTERIZED SYMBOLIC INTEGRATION TECHNIQUE FOR DEVELOPMENT OF TRIANGULAR A N D QUADRILATERAL COMPOSITE SHALLOWSHELL FINITE ELEMENTS C. M . Andersen and ArSmed K . Noor
Ldngley Resemch Center Humpton, Va, 23665 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON,) D.' c. . !. i ' '
DECEMBER 1975 \
J.
/
TECH LIBRARY KAFB, NM
1I11110133753 811IIll~ l1 l m l1
.
3. Recipient's Catalog No.
2. Government Accession No.
1. Report No.
NASA TN D8067 5. Report Date
4. Title and Subtitle
December 1975
A COMPUTERIZED SYMBOLIC INTEGRATION TECHMQUE FOR DEVELOPMENT OF TRIANGULAR AND QUADRILATERAL COMPOSITE SHALLOWSHELL FINITE ELEMENTS
6. Performing Organization Code
1
7. Author($)
I
C. M. Andersen and Ahmed K. Noor (See block 15)
.._
8. Performing Organization Report No.
L10354
[:.Work
9. Performing Organization Name and Address
Unit No.
506259906
NASA Langley R e s e a r c h Center Hampton, Va. 23665
11. Contract or Grant No.
13. Type of Report and Period Covered 12. sponsoring Agency Name and Address
Technical Note 
National Aeronautics and Space Administration Washington, D.C. 20546 15. Supplementar
~
14. Sponsoring Agency Code
Notes
C. & Andersen: I. Senior R e s e a r c h Associate in Mathematics and Computer Science, College of William and Mary, Williamsburg, Virginia. Ahmed K. Noor: R e s e a r c h P r o f e s s o r of Engineering, Joint Institute f o r Acoustics and Flight Sciences, George Washington University, Hampton, Virginia. ._ _ _ ~ _ 
16. Abstract
Computerized symbolic integration is used in conjunction with grouptheoretic techniques t o obtain analytic expressions f o r the stiffness, geometric stiffness, consistent mass, and con sistent load m a t r i c e s of composite shallow s h e l l s t r u c t u r a l elements. The elements are s h e a r flexible and have variable curvature. A stiffness (displacement) formulation is used with the fundamental unknowns consisting of both the displacement and rotation components of the refer ence s u r f a c e of t h e shell. Both triangular and quadrilateral e l e m e n t s are developed. The t r i angular elements have six and ten nodes. The quadrilateral elements have four and eight nodes and can have internal d e g r e e s of freedom associated with displacement modes which vanish along the edges of t h e element (bubble modes). The stiffness, geometric stiffness, consistent mass, and consistent load coefficients are e x p r e s s e d as l i n e a r combinations of integrals (over the element domain) whose integrands are products of shape functions and t h e i r derivatives. The evaluation of the elemental m a t r i c e s is thus divided into two s e p a r a t e problems  determination of the coefficients in the linear com bination and evaluation of the integrals, The l a t t e r problem can be computationally the m o r e t i m e consuming and the p r e s e n t study focuses on simplifications and m e a n s of reducing the effort involved in t h i s task. The integrals are p e r f o r m e d symbolically by using the symbolicandalgebraic manipulation language MACSYMA. The efficiency of using symbolic integration in the element development is demonstrated by comparing the number of floatingpoint a r i t h m e t i c operations required in t h i s approach with those r e q u i r e d by a commonly used numerical quadrature technique.
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17 Key Words (Suggested by Author(s))

Finite elements Composite m a t e r i a l s Shells Symbolic integration Computers An isotropy 

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19. Security Classif. (of this report)
Unclassified .~
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18. Distribution Statement
Group theory MACSYMA Stiffness
1
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For sale b y the National Technical Information Service, Springfield. Virginia 22161
I
A COMPUTERIZED SYMBOLIC INTEGRATION TECHNIQUE FOR DEVELOPMENT OF TRIANGULAR AND QUADRILATERAL COMPOSITE SHALLOWSHELL FINITE ELEMENTS C. M. Andersen* and Ahmed K. Noor** Langley Research Center SUMMARY Computerized symbolic integration i s used in conjunction with grouptheoretic tech niques to obtain analytic expressions f o r the stiffness, geometric stiffness, consistent m a s s , and consistent load matrices of composite shallow shell structural elements. The elements are s h e a r flexible and have variable curvature. A stiffness (displacement) formulation i s used with the fundamental unknowns consisting of both the displacement and rotation components of the reference surface of the shell. Both triangular and quadrilat eral elements are developed. The triangular elements have six and ten nodes. The quadr i l a t e r a l elements have four and eight nodes and can have internal degrees of freedom associated with displacement modes which vanish along the edges of the element (bubble modes). The stiffness, geometric stiffness, consistent mass, and consistent load coefficients a r e expressed as linear combinations of integrals (over the element domain) whose inte grands are products of shape functions and t h e i r derivatives. The evaluation of the ele mental matrices is thus divided into two separate problems  determination of the coef ficients in the linear combination and evaluation of the integrals. The latter problem can be computationally the more t i m e consuming and the present study focuses on simplifi cations and means of reducing the effort involved in this task. The integrals are performed symbolically by using the symbolicandalgebraic manipulation language MACSYMA. The efficiency of using symbolic integration in the element development is demonstrated by comparing the number of floating point a r i t h metic operations required in t h i s approach with those required by a commonly used numerical quadrature technique
.
_
_
_. *Senior R e s e a r c h Associate in Mathematics and Computer Science, College of William and Mary, Williamsburg, Virginia. **Research P r o f e s s o r of Engineering, Joint Institute for Acoustics and Flight Sciences, George Washington University, Hampton, Virginia. ~
INTRODUCTION Although considerable literature h a s been devoted t o the analysis of isotropic shells using doubly curved finite elements, investigations of the finiteelement analysis of lami nated composite shells are r a t h e r limited in extent. The reliable prediction of the re sponse of composite shells often r e q u i r e s the u s e of highorder s h e a r flexible elements with the consequent increase in the developmental effort of the elemental matrices (the stiffness, geometric stiffness, and m a s s matrices). In most of the published work, numerical integration was used f o r the evaluation of the element stiffness matrices. Although numerical integration is both simple and adapt able t o computer programing, it can become computationally v e r y expensive, particularly f o r higher o r d e r elements. This drawback has been recognized and improvements have been suggested (refs. 1 and 2); however, the difficulty has not been overcome. This r a i s e s the question as t o whether an alternative approach such as using analytic (closed form or symbolic) integration in the element development is computationally m o r e efficient. The present study a d d r e s s e s this question. More specifically, the objective of this paper i s to demonstrate the computational advantages resulting f r o m the use of computerized sym bolic integration in conjunction with grouptheoretic techniques (or s y m m e t r y transforma tions) in the development of triangular and quadrilateral shallow shell elements. The analytical formulation is based on a f o r m of the shallowshell theory modified such that the effects of s h e a r deformation and r o t a r y inertia are included. Indicia1 nota tion is used throughout the development, since it is particularly useful in identifying the symmetries. The integrals are performed symbolically by using the symbolicand algebraicmanipulation language MACSYMAl (refs. 3 and 4). This is done in o r d e r to re duce the tedium of analysis and to d e c r e a s e the likelihood of e r r o r s . The u s e of group theoretic techniques (or symmetry transformations) r e s u l t s in considerable reduction in both the amount of analytic computation needed and the size of the p r o g r a m s for numeri cal computation.
SYMBOLS .ijkt
ijk
ij
,Bo ,Cap
an,dn,en
basic integrals defined in equations (14) t o (16)
representations of dihedral groups defined in equations (56), (60), (61), and (69) ~
~
~
~
l T h e MACSYMA system is being developed by the Mathlab group at Massachusetts Institute of Technology under the support of ARPA (Advanced Research P r o j e c t s Agency of the U.S. Department of Defense) through Office of Naval Research Contract No. NO0014 70A0362001. 2
I
8 by 8 matrix of shell stiffnesses
[CI C,pyp,C,3p3
extensional and t r a n s v e r s e s h e a r stiffnesses of shell, respectively
(k) (k) Capyp7Ca3p3
extensional and t r a n s v e r s e s h e a r stiffnesses of kth layer of shell
CCJ
portion of shell boundary over which tractions are prescribed
D,pyp
bending stiffnesses of shell
EL,ET
elastic moduli in direction of fibers and normal to fibers, respectively permutation symbol
F,pYP
stiffness interaction coefficients of shell
GLT,GTT
s h e a r moduli in plane of f i b e r s and normal t o plane of fibers, respectively
ij
ij
&po,Q,p1
functions of coordinates of corner nodes of elements
He
weighting coefficients f o r numerical quadrature
hk,hkl
distances from reference (middle) s u r f a c e t o top and bottom s u r f a c e s of kth layer, respectively
[KI
element stiffness matrix
K g
stiffness coefficients of shell element
3
[Kl

geometric stiffness matrix
K;$
geometric stiffness coefficients of shell element
ka!p
curvatures and twist of shell reference surface
kLp
nodal values of kap
L1,L2,z;1,z2,z [MI
MYJ
.3a!p
logarithmic functions defined in equations (33), (38), and (42)
consistent m a s s matrix
consistent m a s s coefficients of shell element
prescribed bending stress resultants on shell boundary
mo,ml,m2 density p a r a m e t e r s of shell
Ni
shape o r interpolation function
4Ya
prescribed extensional (inplane) stress resultants on shell boundary
G p
relative magnitudes of initial s t r e s s resultants ( p r e s t r e s s components)
"a!
unit outward normal to shell boundary
(PI
consistent 1oad vector
Pi
consistent load coefficients
Pa!,P
external load intensities in coordinate directions
PL,P'
nodalvalues of pa! and p
4
..
%
. . .
.
prescribed t r a n s v e r s e s h e a r stress resultants on shell boundary
ijkf RijkP,Rfkf R1 ' 2
coefficients associated with A integr als
*y,QF,Q'j.,Q,'
integers associated with representative Aintegrals
r
number of shape functions associated with an element
ijk ijk ijk s 1 9s2 $3
coefficients associated with Bintegrals
integers associated with representative Bintegrals S T , S ~ , S ~ , S ~ S
ratio,

S
a quantity defined in equation (39)
T
kinetic energy of shell
T
transpose symbol
Tij,Tij,Tij3 , Tij

coefficients associated with Cintegrals
integers associated with representative C integrals
FTqm,FJF t
U2/U1
ratio,
UQ/UI
t
a quantity defined in equation (39)
U
s t r a i n energy of shell
U0
strain energy due to p r e s t r e s s
U~,U~,UQ functions of nodal coordinates
5
displacement components in coordinate directions (see fig. 1 )
UQ!,w
linear functions of coordinates of c o r n e r nodes
W
work done by external f o r c e s
XQ!9x3
orthogonal coordinate system (see fig. 1 )
?&
values of x,
x
inplane load p a r a m e t e r
P
number of numerical quadrature points
v~~
Poisson's ratio measuring s t r a i n in T direction due to uniaxial normal stress in the L direction