analysis of laminated anisotropic plates and shells using symbolic computation

International Journal of Mechanical Sciences 41 (1999) 397—417

Analysis of laminated anisotropic plates and shellsusing symbolic computation

I.A. JonesDepartment of Mechanical Engineering, University of Nottingham, Nottingham NG7 2RD,U.K.

Received 30 June 1998

Abstract

The extension of classical isotropic plate and shell solutions and finite element formulations to cope withorthotropic/monoclinic laminated and shear deformable structures often involves very complex intermediatestages and final results within the derivations. This paper examines four case studies covering the use ofsymbolic computation to manage this complexity. These case studies comprise the derivation of a catalogueof solutions to orthotropic circular plate problems, the formulation of two axisymmetric shell finite elements(respectively using Flugge’s shear-rigid shell assumptions and the shear-flexible assumptions of Soldatos) andthe derivation of the eighth-order governing differential equation for a laminated monoclinic or orthotropicshell. The emphasis is placed upon the techniques required to achieve these derivations using symboliccomputation, and the considerable effort involved in putting the results into publishable form is noted.( 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Symbolic computation; Composites; Orthotropic materials; Laminated shells

Notation

Symbolsa radius of cylindrical shellA

o, B

o, C

oconstants in equation for deflection of outer region of circular plate (similarlyA

i, B

i, C

irelate to inner region)

[D] shell section modulus matrixEh, E

relastic moduli in circumferential and radial directions

F, G load terms relating to plateh shell thickness[J

k] matrix relating to contribution to [M] of kth ply of laminate

K1,2, K

30constants in simultaneous differential equations for u, v and w

[Kk] matrix relating to contribution to [D] of kth ply of laminate

0020-7403/99/$ — see front matter ( 1998 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 0 - 7 4 0 3 ( 9 8 ) 0 0 0 7 4 - 5

k integer ply index within laminatel meridional length of elementM

x, Mh, M

xh moment resultants (moment per unit length)N number of plies in laminateN

x, Nh, N

xh stress resultants (force per unit length)[M] shell inertia matrix[Q]

kmaterial modulus matrix for kth on-axis orthotropic ply of laminate

[Q1 ]k

material modulus matrix for kth off-axis or monoclinic ply of laminater radial position of point on shellri, r

m, r

oinner, intermediate and outer radii of plate (refer to Fig. 1)

rs, rh principal radii of curvature of axisymmetric shell

s arc distance along meridian of shellt time¹ kinetic energyº strain energyu displacement in axial or meridional directionu1

shear strain csz

at the shell reference surfacev displacement in circumferential directionv1

shear strain chz at the shell reference surfacew displacement normal to plate or shellx axial positionz through-thickness position within plate or shella1,2, a

12, b

1,2, b

3coefficients in partial differential equations

*ij

partial differential operatore, c direct and shear strainsMe0Np, q

set of amplitudes of ‘‘pseudostrains’’ (displacements and their derivatives)direct and shear stresses

h non-dimensional circumferential position ("s/a)t1(z), t

2(z) shape functions for interpolation of u

1and v

1/ angle of shell normal to axis of symmetryu circular frequency of vibration

Subscriptsr relating to radial directions relating to meridional directionx relating to axial directionz relating to displacement at position z within shellh relating to circumferential directionb JEh/Er

Operators( )@ aL( )/Lx( )f L( )/Lh

(N.B. the meanings of u, v, u1

and v1

have been chosen to be self-consistent throughout this publication; itshould be noted that Mu, u

1N and Mv, v

1N are transposed in other references relating to the thin and thick shell

axisymmetric elements).

398 I.A. Jones / International Journal of Mechanical Sciences 41 (1999) 397—417

1. Introduction

A considerable amount of work has been carried out worldwide in recent years on the analysis ofanisotropic and laminated plates and shells for stiffness, stress, natural frequencies and modeshapes. This has included both the extension of classical shell theory to cope with anisotropy andlaminated construction, and the relaxation of the classical (Kirchhoff—Love) assumption of non-deformable normals following the approaches of Soldatos and Timarci [1] and of Reddy and Lui[2]. A significant obstacle to the practical application of laminated shell theories is the complexityof the algebra involved, for example, in the application of practical boundary conditions or thederivation of finite element formulations based upon these theories. A practical and effectivesolution to this problem lies in symbolic computation (also known as computerized symbolicalgebra or symbolic manipulation), yet only recently have publications appeared regularly on thisapplication of the technique and these have mainly concentrated upon the results rather than thedetails of the processes used to obtain them.

The derivation of almost any plate or shell solution involves a governing differential equationderived from a set of strain—displacement (compatibility) assumptions and a set of equilibriumequations. These are linked via a plane-stress version of Hooke’s law (also incorporating transverseshear stiffnesses for thick plate/shell models). For isotropic materials, the Young’s modulus andPoisson’s ratio are the only independent material constants, and for homogenous shells variousterms cancel when through-thickness integrations are performed between the limits of $h/2. Forcomposite shells lacking isotropy the equilibrium and compatibility equations are unchanged butthere are now at least four and (for off-axis or monoclinic layers in a shear-deformable shell) up tonine independent material constants. This clearly means that terms will not simplify to the sameextent; furthermore, the lack of symmetry of a general laminate means that many additional termspersist in the final solution rather than cancelling.

This increase in complexity often means that an error-free solution is almost impossible toachieve manually and in a reasonable timescale. Symbolic computation provides the opportunityto make these derivations feasible. While the technique eliminates the possibility of direct humanalgebraic errors, the task of accurately programming the derivation (via a command batch file) isoften non-trivial; it requires not only a very clear understanding of the derivation procedure butoften considerable ingenuity to enforce rules and identities not implicit in the computerized system.

Symbolic computation has been available for many years, during which time papers have beenpublished rather infrequently on its application to composite plates and shells. For example,Wilkins [3] described the static analysis of cylindrical shells using this technique, includinga stage-by-stage commentary on the FORMAC coding of the solution method. Noor andAndersen [4] surveyed the limited volume of literature on the technique’s application to structuralmechanics, including the vibration analysis of laminated composite elliptical plates. In addition tothe advantages which included its reliability in evaluating integrals and derivatives, these authorsidentified several problems including the large size of intermediate expressions. Elishakoff andTang [5] discussed the analysis of the buckling of polar orthotropic circular plates with the aid ofthe REDUCE system and briefly reviewed other applications of the technique, but gave no furtherdetails of how the technique was applied.

More recently, a greater number of relevant publications have appeared. Li et al [6] describework on cylindrical shells which utilized computerized symbolic algebra, and Han and Petyt [7]

I.A. Jones / International Journal of Mechanical Sciences 41 (1999) 397—417 399

mention its application to the vibration analysis of laminated rectangular plates, but neither papergives many details of the technique’s implementation. However, Argyris and Tenek [8] present theformulation of a finite element for the study of temperature fields in laminated shells, and include intheir publication the MACSYMA input file used in its derivation. Of particular relevance is a paperby Webber and Stewart [9] who describe their use of MACSYMA to derive and solve the differentialequations governing the buckling of sandwich panels, and also present the MACSYMA input file.Their paper shares with the fourth example in the present work the need to solve three simulta-neous differential equations, although Webber and Stewart [9] used MACSYMA’s own differentialoperators in contrast to the present author’s use of multiplicative constants to represent them.

2. Case studies

This paper considers four plate/shell problems, most of which would have been difficult orimpossible to solve without computational tools. The emphasis of the following descriptions isplaced not on the symbolic computation software itself nor on the results obtained (which will bepublished elsewhere) but on the derivation routes and the problems overcome in encoding thesolution methods.

In all cases, the REDUCE system was used, together with the TAYLOR package for powerseries expansion and the RLFI postprocessor for generating output in LaTeX typesetting language(both supplied with REDUCE). REDUCE was initially chosen as the only system readily availableto the author. It proved to be reasonably well suited to this kind of problem and has therefore beenretained for the subsequent work. In common with other similar packages, it may be usedinteractively or using batch files of commands which for practical purposes constitute programswritten in a Pascal-like high-level structured language.

2.1. Polar orthotropic circular plates

This example relates not to laminated shells but to a range of homogeneous orthotropic circularplates, and was originally intended to form part of a Roark-like catalogue of solutions to assist inthe design of composite components. It is presented as an example of a problem which would bemessy and tedious to solve manually but which involves few pitfalls in its solution using symboliccomputation. The plate is assumed to be uniform and thin (i.e. shear-rigid) and to have radial andcircumferential principal material directions. A variety of load cases were considered; a typical oneis shown in Fig. 1. The solution method [10] is broadly similar to that presented in standard textse.g. that of Rees [11] for isotropic problems, and leads to expressions such as that for the deflectionw at radius r for the loaded region of the plate in Fig. 1:

w"Fr4!Gr2#A

or1`b

1#b#

Bor1~b

1!b#C

o(1)

where F and G are constants dependent upon the pressure load and the plate’s radial flexuralrigidity, A

o, B

oand C

oare constants of integration which depend upon boundary conditions, and

b"(Eh/Er)1@2 where Eh and E

rare, respectively, the circumferential and radial moduli. The

corresponding equation for the unloaded region omits the first two terms and contains three


Fig. 1. Typical loadcase for flat circular polar orthotropic plate.

different constants Ai, B

iand C

i. In principle, it is straightforward to find the constants from the

boundary conditions; in practice this means enforcing continuity of deflection, slope and momentat radius r

mand imposing three further edge boundary conditions. This involves the solution of six

linear simultaneous equations and is a straightforward task for the SOLVE facility withinREDUCE. The numerator and denominator of each of the six constants were typeset directly intoLaTeX using the RLFI postprocessor, although much manual formatting was then required andsome simplifications were possible. A total of 24 variations on loadcase and boundary conditionswere examined. The resulting list of constants (notably the set of denominators) was rationalized toremove duplicate constants. The LaTeX code was then edited back to produce FORTRAN andREDUCE files, the latter being used for checking that the original differential equations andboundary conditions were satisfied. The results were presented in catalogue form [12].

2.2. ¸aminated orthotropic axisymmetric Fourier shell element

This example was actually the author’s original application of computerized symbolic algebra tolaminated shells, and involved extending an existing commercial implementation of a thin shellfinite element based upon Flugge shell theory [13] to cover laminated orthotropic shells. Theexisting element is available as element 42130 within the PAFEC system, and applies only tohomogeneous isotropic shells. The element (Fig. 2) is assumed to form a thin surface of revolutionwith constant meridional radius of curvature; the version shown has three nodes although otherelement variants (with two or four nodes) are permitted. Each node has three displacement degrees offreedom together with a rotational freedom (rotation in the generator plane). The deformations of theelement may be uniform around the circumference (corresponding to harmonic order m"0) or mayvary sinusoidally, the harmonic order m of the variation being specified for a given finite element run.For m"0, the extended element is assumed to consist of a shell made up of N generally- orthotropiclayers; however, for mO0, the material properties must coincide with the principal (meridional andcircumferential) directions of the shell since only real Fourier terms are considered.

The derivation of the extended element is based upon the following integration (derived fromWebster [14]) of strain energy over the element volume of an N-layered element with each kth ply


Fig. 2. Geometry of axisymmetric shell elements.

made from a material with off-axis orthotropic modulus matrix [Q1 ]k:

º"P2n

0P

l

0

N+k/1P

zk

zk~1C12MeNT[Q1 ]

kMeND

sin/rs

(rs#z) (rh#z) dzdsdh (2)

If the rotational inertia within the shell is neglected, the inertia matrix is trivial, but inclusion ofrotational terms requires the integration of the kinetic energy over the volume of the element usingthe following expression (also extended from Webster [14]):

¹"P2n

0P

l

0

N+k/1

Pzk

zk~1

ok

2 CALu

zLt B

2#A

Lvz

Lt B2#A

Lwz

Lt B2

Dsin/

rs

(rs#z) (rh#z) dzdsdh. (3)

The displacement field (uz, v

z, w

z) at a through-thickness position z within the shell is given by the

following expressions obtained from Flugge [13]:

uz"u

rs#zrs

!

LwLs

z,

vz"v

rh#zrh

!

LwLh

zr

, (4)

wz"w,

where r is the radial position and u, v and w are respectively the meridional, circumferential, andnormal displacements which vary sinusoidally with harmonic order m and may also be assumed tovary sinusoidally with time t. The set of strains MeN for each harmonic order m are expressed in


terms of the local meridional, circumferential, and normal displacements and their derivativesusing the following strain—displacement relations derived from Flugge [13, 14]:

es"

LuLs

#

wrs#z

!

zrs#z Ars

L2wLs2

#

urs

drs

ds!

drs

dsLwLsB, (5a)

eh"1r

LvLh

#

urs

cot/Ars#z

rh#zB#w

rh#z!

zrh#zA

1r sin/

L2wLh2

#cot/LwLsB, (5b)

csh"A

rh#zrs#zBA

rs

rh

LvLs

!

rs

r2hcot/vB#A

rs#z

rh#zB1

rssin/

LuLh

!zA1

rh#z#

rs

rh (rs#z)BA1

sin/L2wLsLh

!

cot/rh sin/

LwLhB . (5c)

In practice, only the integrals through the shell thickness and around the circumference areevaluated explicitly, since the integral with respect to the meridional position s is evaluatednumerically at run-time within the finite element software. The shell section modulus matrix for anN-layered laminate in this application

[D]"N+k/1

[[K]k]zkzk~1

(6)

is obtained by equating the strain energy calculated in Eq. (2) to the following:

º"Pl

0

12

Me0NTN+k/1

[[K]k]zkzk~1

Me0Nds (7)

where Me0N is the vector of seven so-called ‘‘pseudostrains’’, which are the amplitudes of thesinusoidally varying displacements u, v and w and their derivatives with respect to s. Each kthlayer’s contribution to [D] consists of [K]

kevaluated at the top of the layer (at z"z

k) minus [K]

kevaluated at the bottom of the layer (at z"z

k~1). Similarly, the complete inertia matrix

[M]"N+k/1

[[J]k]zkzk~1

(8)

may be obtained by equating the kinetic energy from Eq. (3) to the following:

¹"u2 Pl

0

12

Me0NTN+k/1

[[J]k]zkzk~1

Me0N ds (9)

This kinetic energy calculation incorporates the set of displacement and slope amplitudes of theshell mid-surface Mu0, v0, w0, Lw0/LsN, whose four members form a subset of the seven ‘‘pseudos-trains’’ Me0N.

The derivation of the stiffness terms is given in detail by Jones [15] and closely follows thatdescribed in the literature (notably Henshell et al. [16]) for the existing isotropic element. Thederivation process, as coded within the REDUCE batch file, is illustrated by the flowchart in Fig. 3.Flugge’s strain—displacement expressions (Eqs. (5a)—(5c)) were entered into Eq. (2), the through-thickness integration was performed and the TAYLOR package was used to expand the result as


Fig. 3. Derivation procedure for thin axisymmetric shell element.

a power series in the through-thickness co-ordinate z. In order to optimize the derivation process,some bracketed terms in Eqs. (5a)—(5c) were defined as constants which were not expanded until theseries expansion had been performed. The circumferential integration was straightforward, butinsertion of the integration limits involved the explicit definition of values of cos 2nm and sin 2nm(m3I) since these are not implicit in REDUCE. The result of the integrations was an expression forstrain energy per unit of meridional length s.

The individual elements of [K]kwere extracted from the strain energy expression using a three-

stage approach. Firstly, each of the leading diagonal terms Kii

was extracted directly as thecoefficient of the second power of the ith pseudostrain. A subexpression was then found as thecoefficient of the first power of each pseudostrain; the terms K

ijwere then found by halving

the coefficient of the jth pseudostrain in the ith subexpression. A virtually identical procedure to theabove could be followed for the inertia terms J

ijused to obtain the shell inertia matrix [M].

However, in the version of the element used in published natural frequency verification results [17],


Fig. 4. Derivation procedure for thick axisymmetric shell element.

the rotational inertia terms were neglected and a trivial diagonal inertia matrix was used. Thisfollows the precedent set by the coding of the isotropic version of the element.

While it was straightforward to output the elements of [K]kin the form of accurate and useful

FORTRAN code, the grouping of the terms was somewhat verbose and unsuitable for publicationin conventional mathematical notation. A further REDUCE program was therefore written to


rearrange the results and the corresponding LaTeX code produced via the RLFI postprocessor wasfurther simplified manually. This was then edited back to FORTRAN and REDUCE format forverification against the original solution.

Although the derivation process for this application is straightforward and requires few tricks toencode using REDUCE, initial manual attempts at the derivation achieved little progress becauseof the very large amount of manipulation involved. Indeed, the impracticality of performing thetask manually seems a plausible explanation of why this element was not previously extended tocover orthotropic laminates.

2.3. Laminated thick axisymmetric shell element

Out of the four case studies, the greatest challenges in the application of symbolic computationto shells arose in the extension of the element described in Section 2.2 to take account of transverseshear deformations. The modified element is also applicable to axisymmetric and non-axisymmet-ric (Fourier) loadcases, but has two extra degrees of freedom per node corresponding to transverseshear strains at the shell mid-surface. These are interpolated quite separately from the otherdisplacements, in contrast to most other thick shell elements (e.g. the well-known Ahmad element[18]) which simplify the displacement field following a uniform-shear (Reissner—Mindlin) ap-proach. The present approach to thick shell modelling was based upon that of Soldatos andTimarci [1], whereby an additional displacement field representing shear deformations is superim-posed onto the displacement field from a thin shell theory such as that of Flugge [8]. Thedisplacements due to transverse shear (representing distortion of the shell normal) are extrapolatedfrom the transverse shear strains u

1and v

1at the shell reference surface using shape functions t

1(z)

and t2(z). These are left undefined except that their values are zero at z"0 and their first

derivatives are unity at z"0 and (if the condition of zero free-surface shear stresses is enforced)zero at z"$h/2. In order to satisfy continuity of displacements and interlaminar shear stresses,functions such as piecewise cubics or piecewise hyperbolic functions are chosen. The resultingin-surface strain field was derived manually from the displacement field:

es"ec

s#

rs

rs#z

t1(z)

Lu1

Ls, (10a)

eh"ech#1

(rh#z) sin/t

2(z)

Lv1

Lh#

1rh#z

t1(z)u

1cot /, (10b)

csh"cc

sh#rs

rs#z

t2(z)

Lv1

Ls!

1rh#z

t2(z)v

1cot/#

1(rh#z)sin/

t1(z)

Lu1

Lh, (10c)

where ecsetc. are given by Flugge’s classical strain—displacement relations (Eqs. (5a)—(5c)). Further-

more, by differentiating the displacement field with respect to z, the following expressions fortransverse shear strain were obtained:

csz"u

1

dt1

dz,u

1t@

1(z), (11a)

chz"v1

dt2

dz,v

1t@

2(z). (11b)


The [K]kand [J]

kmatrices used in evaluating the element’s shell section modulus and inertia

matrices were derived using an extension of the approach used in Section 2.2, although somechanges were made to the derivation sequence in order to minimize the processing time andmemory required by REDUCE to perform the derivation. The set of REDUCE batch files used toderive the [K]

kand [J]

kmatrices is too lengthy to reproduce in this publication, but is summarized

by the derivation procedure illustrated in Fig. 3.The particular challenge of the work lay in the need to manipulate t

1(z) and t

2(z), together with

their squares and products, without ever specifying the actual form of these functions, in order to beable to use the resulting code for trials involving different versions of t

1(z) and t

2(z). Moreover,

explicit through-thickness integration was retained in order to allow evaluation of the variousoptions for t

1(z) and t

2(z) without any unwanted influence of the choice of numerical integra-

tion method. The following ‘‘tricks’’ were therefore employed in order to perform the requiredderivation.

(1) In order to prevent the expansion of t1(z) and t

2(z) as terms in the power series, REDUCE was

not informed of the dependency of these functions upon z until the series expansion of theirmultiplicands had taken place. In practice, the dependency on z of the functions representingt

1(z) and t

2(z) (along with t

1(z)t

2(z), t@

1(z)t@

2(z), etc.) did not take place until much later,

following the circumferential integration and the identification of the squares and productsdescribed below.

(2) Squares and products such as [t1(z)]2, t

1(z)t

2(z), [t@

1(z)]2, etc. were separated out, noting that

at this stage each of t1(z), t@

1(z), etc. was represented as a variable not a function. Each of these

products was later redefined as a function of z in its own right.(3) Multiple integrals (ignoring constants of integration) and the first derivatives (all with respect to

z) of t1(z), t

2(z) and the products identified in stage 2 were each defined as functions of z, and

the integral and differential relationships between these were explicitly defined. Furthermore,integrals of quantities such as znt

1(z), znt

1(z)t

2(z), etc. (n"1, 2,2 , 4) were each defined in

terms of the multiple integrals mentioned above. The definitions were obtained manually byrecursive use of integration by parts, leading to identities such as:

Pb

a

z2[t1(z)]2dz"Cz2P[t1

(z)]2dzDb

a

!C2zPAP[t1(z)]2dzBdzD

b

a

#2Pb

aAPGP [t

1(z)]2dzH dzBdz. (12)

It was verified that the constants of integration vanish either from all the expressions such asEq. (12) or from the overall solution for the shell section modulus and inertia matrices [D](Eq. (6)) and [M] (Eq. (8)) and hence their omission from the integrals is valid in thisapplication.

(4) Having established the integration rules governing the shape functions and their squares andproducts with each other and with z, it is straightforward to perform the through-thicknessintegration (i.e. that with respect to z). The earlier use of the series expansion is of influence here:by converting all multiplicands of t

1(z), t

2(z), etc. into power series in z, it becomes possible to

carry out manipulations such as integration of the strain energy expression, having specified


only the integration rules described in stage 3 and without prescribing the actual form of t1(z)

and t2(z).

(5) In practice, the result of the integration involves functions of z which include t1(z), t

2(z), their

products and various multiple integrals and first derivatives. In order to output these inmachine-readable form (FORTRAN code), it was necessary to assign each of these to a simplevariable. Each of these variables is evaluated numerically within the finite element implementa-tion at the limits (z

k~1and z

k) of the definite integral corresponding to the top and bottom of

the kth layer, prior to the variable’s use within the machine-generated FORTRAN code.

In order to allow implementation of the thick shell element, various modifications to the originalelement were carried out by Pafec Limited to include the shear deformations as additional degreesof freedom and hence allow incorporation of the new, larger [K]

kmatrix.

Excellent agreement with alternative FE and experimental results has been obtained for a varietyof moderately thick laminated cylinders [19]. These included a Perspex (PMMA) sandwichcylinder with an extremely shear-flexible (silicone rubber) core, which presented an extreme test ofthe shear-deformable capabilities of the element.

2.4. Differential equation for laminated orthotropic or monoclinic cylinder

A different set of challenges lay in the derivation of the eighth-order governing differentialequation for a laminated monoclinic cylindrical shell, enabling a pinched-cylinder benchmarksolution for orthotropic shells to be obtained. This derivation was based upon a method presentedby Schwaighofer and Microys [20] and took as its starting point Flugge’s strain-displacementrelations for a cylindrical shell [13] at through-thickness position z:

ex"

u@a!z

w @@a2

, (13a)

eh"vf

a!

za

wff

a#z#

wa#z

, (13b)

cxh"

uf

a#z#

a#z

a2v@!

w @f

a Aza#

za#zB, (13c)

where u, v and w are, respectively, the displacements in the longitudinal (axial), circumferential andradial directions, respectively, and the operators ( )@ and ( )f represent differentiation with respectto the dimensionless axial and circumferential coordinates x/a and h, respectively. Eqs. (13a)—(13c)are then inserted into the following expression for the system of stresses, assuming for illustrativepurposes that the material is orthotropic with principal material directions coincident with theshell’s principal directions:

Gpx

phqxhH" C

Q11

Q12

0

Q12

Q22

0

0 0 Q66Dk

GexehcxhH . (14)


These stresses, together with their moments, are integrated through the thickness h of the shellusing expressions such as

Nx"P

h@2

~h@2

px A

z # aa B dz. (15)

A problem encountered was that performing such an integration literally would have resulted inthe omission from the solution of shell coupling terms such as that for B

11:

B11"P

h@2

~h@2

Q11

z dz (16)

since Q11

, etc. vary with position through the laminate thickness. The approach actually taken wasto omit from the derivation the explicit through-thickness integration of stresses and stress couples,and instead to pick out of the integrands for N

x, etc. the integrands for shell stiffness terms such as

B11

. This is illustrated in Appendix A.For a shell under axial, circumferential and radial distributed loads p

x, ph and p

r, the equilibrium

relations are given by Flugge [13]:

N@x#Nf

hx#pxa"0, (17a)

aNf

h#aN@xh!Mf

h!M@xh#ph a2"0, (17b)

Mff

h #M@fxh#M@fhx#M@@

x#aNh!p

ra2"0, (17c)

aNxh!aNhx#Mhx"0. (17d)

Inserting the stress and moment resultants obtained from integrals such as Eq. (15) into Eqs.(17a)—(17c) leads to the following simultaneous differential equations (noting that Eq. 17(d) isredundant and leads to an identity):

K1uA#K

3uff

#K5v@f#K

7w@!K

9w@@@!K

11w@ff

#pxa2"0, (18a)

K5u@f#K

13vA#K

15vff

#K17

wf!K

19wAf

!K21

wfff#ph a2"0, (18b)

K7u@!K

9u@@@!K

11u@ff

#K17

vf!K

19vAf

!K21

vfff

#K22

w!K23

wA!K25

wff#K

26w@@ @@#K

28wAff

#K30

wffff!p

ra2"0. (18c)

The apparently discontinuous numbering sequence is chosen to be consistent with Ref. [21], inwhich a similar analysis with many more terms is presented, for the more general case of a laminatewith monoclinic layers. The facilities within REDUCE for dealing with partial derivatives werejudged not to be appropriate for this problem, so in contrast to the approach of Webber andStewart [9], the solution of the three simultaneous differential equations was achieved by treatingthe differential operators aL/Lx and L/Lh throughout the whole problem as multiplicative constants(termed DASH and DOT, following the shorthand notation ( )@ and ( )f used by Flugge [13] andSchwaighofer and Microys [20] for these differential operators). It was then noted that the threedifferential equations can be represented in the form:

*11

u#*12

v#*13

w#pxa2"0,

*12

u#*22

v#*23

w#pha2"0, (19)

*13

u#*23

v#*33

w!pra2"0,


where the differential operators *ij

each consist of expressions containing various powers of DASHand DOT. It is therefore a relatively straightforward matter of linear algebra (as shown in AppendixB) to obtain an eighth-order partial differential equation from which u and v have been eliminated,containing only w and the load terms and their derivatives. The coefficients of the individualdifferential operators on w were found by picking out the coefficients of the various powers ofDASH and DOT. Once again, a special case may be considered for illustrative purposes byassuming that only normal distributed forces are applied to the shell, i.e. p

x"ph"0. The resulting

eighth-order equation is of the following form:

a1w

@@@@@@@@#a

2w

@@@@@@ff

#a3w

@@@@ffff#a

4w

@@ffffff

#a5w

ffffffff

#a6w

@@@@@@ #a

7w

@@@@ff#a

8w

@@ffff

#a9w

ffffff

#a10

w @@@@#a11

w @@ff#a

12w ffff

!

a4

D11Ab1

pr@@@@#b

2pr@@ff

#b3prffffB"0. (20)

This equation contains only even derivatives of w and the subscript numbering system is chosen tobe consistent with that used in the governing equation for homogeneous orthotropic cylinderspresented by Schwaighofer and Microys [20]. The complete eighth-order equation for a laminatedmonoclinic cylinder is much more complex, as is each of the larger number of coefficients(a

1,2 ,a

21, etc.). Such an equation is presented in a separate publication [21], and also includes

terms for the in-surface loads px

and ph. This equation is presented alongside two ancillaryequations also including u and v. The coefficients in Eqs. (18a)—(18d) and (20) may be simplifiedfrom the ones presented in Ref. [21] by setting to zero the coupling terms Q1

16and Q1

26which would

occur if Eq. (14) were to be expressed for the monoclinic material using the fully-populatedmodulus matrix [Q1 ]

k. The simplified coefficients were presented in Ref. [22]. For the simplest case

of a homogeneous orthotropic material, these coefficients reduce to the ones presented bySchwaighofer and Microys [20].

In practice a two-stage approach was taken to the derivation of the coefficients in Eq. (16). OneREDUCE batch file (presented in simplified form in Appendix A) was used to derive the differentialoperators *

ij, from which the constants K

1,2,K

30were extracted manually. A second batch file

(simplified in Appendix B) was then used to obtain coefficients including a1,2 , a

12and b

1,2, b

3in terms of K

1,2,K

30. Further coding (not presented here) was then used to verify that these

values of coefficients did indeed satisfy Eq. (20) and its ancillary equations. It will be appreciatedthat while the present orthotropic version of the analysis is for illustrative purposes and givesresults which would be achievable (with care) without the use of computational tools, themonoclinic case is much more involved and the results [21] are rather voluminous.

For the specially orthotropic laminated case considered above (i.e. a cross-ply cylinder), thedifferential equation was solved manually using an approach similar to that of Yuan and Ting [23]to give a variety of benchmark solutions [21, 22]. Virtually identical results for test problems wereobtained as from the element described in Example 2.

3. Discussion and conclusions

The problems described have covered the range from the very straightforward (if mathematicallymessy) to the rather complex. While no originality is claimed for the application of symbolic


computation to this kind of problem, it has been demonstrated that a good deal of ingenuity issometimes necessary to achieve the desired results. In addition to the actual results obtained fromthese analyses (which are presented in other publications, and are generally too voluminous tosummarize here), the following conclusions can be drawn.

1. The use of symbolic computation can make feasible the algebraic derivation of plate and shellsolutions which would be extremely tedious or quite impractical to achieve manually. Out of thefour examples described, the author believes that only the first could reasonably be attempted bya patient and methodical analyst, the second and fourth examples would at best be exceedinglydifficult and tedious to undertake manually and the third would probably be quite impracticalto achieve without computer assistance. Thus the method makes it possible to obtain resultssuch as finite element formulations and analytical benchmark solutions which would nototherwise be available.

2. Notwithstanding the usefulness of symbolic computation in this application, the difficulty ofpresenting the output is worthy of mention. The machine-readable output (e.g. FORTRAN) cangenerally be used directly, and the facilities exist (via the RLFI postprocessor) to generatetypesetting output (LaTeX code) directly. However, the resulting output often requires consider-able manipulation (either manually or by judicious use of commands within REDUCE prior topost-processing) and reformatting to make it acceptable for publication using reasonablyconcise mathematical notation. The author’s experience is that this stage is generally more timeconsuming than the actual generation of the results in machine-readable form.

Acknowledgements

The author wishes to thank Dr R.D. Henshell and Dr J. Platt (PAFEC Limited) for their adviceand practical assistance in implementing the orthotropic shell elements.

Appendix A: REDUCE input file for derivation of differential operators

COMMENT******************************************************************************************************;COMMENT **** REDUCE INPUT FILE TO CREATE TERMS IN THREE SIMULTANEOUS ****;COMMENT **** P.D.E.’S FOR CYLINDRICAL SHELL, BASED UPON FLUGGE THEORY. ****;COMMENT **** USES LAMINATED ON-AXIS ORTHOTROPIC MATERIAL PROPERTIES ****;COMMENT ******************************************************************************************************;COMMENTCOMMENT ******************************************************************************************************;COMMENT **** LOAD PACKAGE FOR EXPANSION OF AWKWARD INTEGRANDS AS TAYLOR ****;COMMENT **** SERIES. NOTE THAT TAYLOR SERIES IS EXPANDED ABOUT ZERO. ****;COMMENT ******************************************************************************************************;LOAD—PACKAGE TAYLOR;

COMMENT ******************************************************************************************************;COMMENT **** ARRAYS CONTAINING INTEGRANDS OF THROUGH-THICKNESS INTEGRALS ****;COMMENT **** FOR UNIT RESULTANTS, BEFORE AND AFTER TAYLOR EXPANSION. ****;COMMENT ******************************************************************************************************;


ARRAY INTTHICK (8);ARRAY UNITRESULT (8);

COMMENT ******************************************************************************************************;COMMENT **** ARRAYS CONTAINING EQUILIBRIUM EXPRESSIONS AND THE ****;COMMENT **** DIFFERENTIAL OPERTATORS ON U, V AND W WHICH APPEAR IN THEM ****;COMMENT ******************************************************************************************************;ARRAY EQUILEQN (4);ARRAY DIFFOPS (3, 3);

COMMENT ******************************************************************************************************;COMMENT **** DEFINE U, V AND W AS ACTUAL DEFLECTIONS IN AXIAL, ****;COMMENT **** CIRCUMFERENTIAL AND NORMAL DIRECTIONS OF CYLINDER ****;COMMENT ******************************************************************************************************;COMMENT **** NOTE THAT DASH AND DOT HAVE SAME MEANING AS IN TEXT AND ****;COMMENT **** CORRESPOND TO A D/DX AND D/DPHI. THEY ARE TREATED AS ****;COMMENT **** MULTIPLYING CONSTANTS SO AS TO BE ABLE TO DO LINEAR ALGEBRA ****;COMMENT **** ON THEM WITHIN REDUCE. ****;COMMENT ******************************************************************************************************;COMMENT **** STRAIN-DISPLACEMENT RELATIONSHIPS:THESE ARE EQUATIONS 5 A—C ****;COMMENT **** ON PAGE 212 OF FLUGGE’S STRESSES IN SHELLS. ASSUME THAT ****;COMMENT **** GAMMAXPHI"GAMMAPHIX. ****;COMMENT ******************************************************************************************************;EPSILONX:"U*DASH/A!Z*W*DASH**2/A'2;EPSILONPHI:"V*DOT/A!(Z/A)*(W*DOT'2)/(A#Z)#W/(A#Z);GAMMAXPHI:"U*DOT/(A#Z)#(A#Z)*V*DASH/A'2-(W*DASH*DOT/A)*(Z/A# Z/(A#Z));GAMMAPHIX:"GAMMAXPHI;

COMMENT ******************************************************************************************************;COMMENT **** FOR ORTHOTROPIC MATERIAL FORMING A CROSS-PLY LAMINATE USE ****;COMMENT **** MODULUS MATRIX ASSUMING THAT THERE IS NO COUPLING BETWEEN ****;COMMENT **** TENSION AND SHEAR ****;COMMENT ******************************************************************************************************;SIGMAX:"Q11*EPSILONX #Q12*EPSILONPHI;SIGMAPHI:"Q22*EPSILONPHI#Q12*EPSILONX;TAUXPHI:"Q66*GAMMAXPHI;TAUPHIX:"Q66*GAMMAPHIX;

COMMENT ******************************************************************************************************;COMMENT **** FOR LAMINATED ORTHOTROPIC SHELL, USE A ‘‘DODGE’’ TO AVOID ****;COMMENT **** LOSING B TERMS IN SHELL STIFFNESS MATRIX. DO NOT INTEGRATE ****;COMMENT **** STRESSES BUT DEFINE INTEGRANDS IN EQNS 7 A—H P. 213 OF FLUGGE ****;COMMENT ******************************************************************************************************;INTTHICK(1):"SIGMAX*(1#Z/A);INTTHICK(2):"SIGMAPHI;INTTHICK(3):"TAUXPHI*(1#Z/A);INTTHICK(4):"TAUPHIX;INTTHICK(5):"!SIGMAX*(1#Z/A)*Z;INTTHICK(6):"!SIGMAPHI*Z;INTTHICK(7):"!TAUXPHI*(1#Z/A)*Z;INTTHICK(8):"!TAUPHIX*Z;

COMMENT ******************************************************************************************************;COMMENT **** EXPAND INTEGRANDS AS SERIES TO 2ND OR THIRD POWER OF Z ****;COMMENT **** (SECOND IF IGNORING ‘‘F’’ TERMS, THIRD TO INCLUDE THEM) ****;COMMENT **** THEN CONVERT TAYLOR SERIES FORMAT TO POLYNOMIAL ****;COMMENT ******************************************************************************************************;TAYLORORDER:"3;


FOR IRESULT:"1:8 DO((

EXPAN:"TAYLOR(INTTHICK(IRESULT),Z,0,TAYLORORDER);UNITRESULT(IRESULT):©TAYLORTOSTANDARD(EXPAN);

'';

COMMENT ******************************************************************************************************;COMMENT *** AS A ‘‘DODGE’’, DEFINE SHELL MATRIX TERMS IN TERMS OF INTEGRANDS;COMMENT ******************************************************************************************************;LET Q11"A11; MATCH Q11"A11; LET Q12"A12; MATCH Q12"A12;LET Q22"A22; MATCH Q22"A22; LET Q66"A66; MATCH Q66"A66;LET A11*Z "B11; MATCH A11*Z "B11; LET A12*Z "B12; MATCH A12*Z"B12;LET A22*Z "B22; MATCH A22*Z"B22; LET A66*Z"B66; MATCH A66*Z "B66;LET B11*Z "D11; MATCH B11*Z "D11; LET B12*Z "D12; MATCH B12*Z "D12;LET B22*Z "D22; MATCH B22*Z "D22; LET B66*Z "D66; MATCH B66*Z "D66;LET D11*Z "F11; MATCH D11*Z "F11; LET D12*Z "F12; MATCH D12*Z " F12;LET D22*Z "F22; MATCH D22*Z "F22; LET D66*Z "F66; MATCH D66*Z "F66;

COMMENT ******************************************************************************************************;COMMENT **** GIVE THE UNIT RESULTANTS MEANINGFUL NAMES ****;COMMENT ******************************************************************************************************;NX:"UNITRESULT (1); NPHI:"UNITRESULT (2);NXPHI:"UNITRESULT (3); NPHIX:"UNITRESULT (4);MX:"UNITRESULT (5); MPHI:"UNITRESULT (6);MXPHI:"UNITRESULT (7); MPHIX:"UNITRESULT (8);

COMMENT ******************************************************************************************************;COMMENT **** EQUILIBRIUM EQUATIONS FOR SHELL GIVEN ON PAGE 209 OF FLUGGE ****;COMMENT **** EQUATIONS 2 A—D. EQUATION D IS NOT NECESSARILY CONSISTENT ****;COMMENT **** AND IS NOT USED IN REST OF ANALYSIS BUT IS GIVEN HERE ANYWAY ****;COMMENT ******************************************************************************************************;EQUILEQN (1):"NX*DASH#NPHIX*DOT#PX*A;EQUILEQN(2):"A*NPHI*DOT#A*NXPHI*DASH!MPHI*DOT!MXPHI*DASH#PPHI*A'2;EQUILEQN(3):"MPHI*DOT'2#MXPHI*DASH*DOT#MPHIX*DASH*DOT#MX*DASH'2#A*NPHI - PR*A'2;EQUILEQN(4):"A*NXPHI - A*NPHIX #MPHIX;

COMMENT ******************************************************************************************************;COMMENT **** BY NOW THE APPROPRIATE TERMS FOR NX... MPHIX HAVE ALREADY ****;COMMENT **** BEEN CALCULATED AND HAVE BEEN SUBSTITUTED INTO ABOVE EQUNS. ****;COMMENT **** THE DIFFERENTIAL OPERATORS ON U, V & W CAN THEREFORE BE ****;COMMENT **** PICKED OUT, ALONG WITH THE COEFFICIENTS OF THE LOAD TERMS ****;COMMENT ******************************************************************************************************;FOR IEQN:"1:3 DO ((

DIFFOPS(IEQN,1):"COEFFN(EQUILEQN(IEQN),U,1);DIFFOPS(IEQN,2):"COEFFN(EQUILEQN(IEQN),V,1);DIFFOPS(IEQN,3):"COEFFN(EQUILEQN(IEQN),W,1);

'';

COMMENT ******************************************************************************************************;COMMENT **** IN PAPER BY S & M, THE FIRST OF THE THREE P.D.E.’S HAS BEEN ****;COMMENT **** MULTIPLIED BY a SO THAT THE MATRIX OF DIFFERENTIAL OPERATORS ****;COMMENT **** IS SYMMETRIC. DO THIS NOW SO THAT IT DOES NOT CAUSE ****;COMMENT **** DISCREPANCIES LATER. NOTE ALSO THAT DEFLECTIONS ARE ****;COMMENT **** NON-DIMENSIONALISED IN S&M BY DIVIDING BY a SO THERE IS ****;COMMENT **** IMPLICITLY A FACTOR OF a DISCREPANCY IN ALL EQNS, VISIBLE ****;COMMENT **** WHEN COMPARING WITH THE CORRESPONDING EQUATIONS IN FLUGGE. ****;COMMENT ******************************************************************************************************;FOR ITERM:"1:3 DO ((


DIFFOPS(1,ITERM):"DIFFOPS(1,ITERM)*A'';

COMMENT ******************************************************************************************************;COMMENT **** OUTPUT THE DIFFERENTIAL OPERATORS THEN EXTRACT K’S MANUALLY ****;COMMENT ******************************************************************************************************;OUT ‘‘SPEC—ORTH—CYL.OPS’’;FOR IEQN:"1:3 DO ((

FOR ITERM:"1:3 DO ((

WRITE DIFFOPS(IEQN,ITERM):"DIFFOPS(IEQN,ITERM);''

'';QUIT;;END;

Appendix B: REDUCE input file to obtain constants in eighth-order equation

COMMENT ******************************************************************************************************;COMMENT **** REDUCE INPUT FILE TO EXTRACT TERMS IN EIGHTH-ORDER P.D.E. ****;COMMENT **** FOR CYLINDRICAL SHELL, BASED UPON FLUGGE SHELL THEORY. ****;COMMENT **** NOTATION AND SUBSCRIPTS CONSISTENT WITH SCHWAIGHOFER ****;COMMENT **** AND MICROYS, TRANS. ASME, J. APP. MECH, VOL. 46 (1979), P358 ****;COMMENT **** RESULTS ARE EXPRESSED IN TERMS OF CONSTANTS K(1)... K(30) ****;COMMENT ******************************************************************************************************;

ARRAY K(30); COMMENT **** CONSTANTS INVOLVING [A], [B], ETC. ****;ARRAY DIFFOPS(3,3); COMMENT **** DIFFER’L OPERATORS IN EQN 19 ****;ARRAY DIFFANCIL(3); COMMENT **** ANCILLARY DIFFER’L OPERATORS ****;ARRAY DASHDOTSALPHA(9,9); COMMENT **** COEFFS OF POWERS OF DASH&DOT ****;ARRAY DASHDOTSANCIL(3,9,9); COMMENT **** USED IN MAIN AND ANCIL. EQNS ****;ARRAY SUBEXPRS(9); COMMENT **** INTERMEDIATE SUBEXPRESSIONS ****;

COMMENT ******************************************************************************************************;COMMENT **** DEFINE TERMS IN K AS CONSTANTS FOR LATER CONVENIENCE ****;COMMENT ******************************************************************************************************;K(1):"KX1X; K(3):"KX3X; K(5):"KX5X; K(7):"KX7X; K(9):"KX9X;K(11):"KX11X; K(13):"KX13X; K(15):"KX15X; K(17):"KX17X; K(19):"KX19X;K(21):"KX21X; K(22):"KX22X; K(23):"KX23X; K(25):"KX25X; K(26):"KX26X;K(28):"KX28X; K(30):"KX30X;

COMMENT ******************************************************************************************************;COMMENT **** DEFINE DIFFERENTIAL OPERATORS IN TERMS OF K(1)2K(30) ****;COMMENT ******************************************************************************************************;DIFFOPS(1,1):" K(1)*DASH'2#K(3)*DOT'2;DIFFOPS(1,2):" K(5)*DASH*DOT;DIFFOPS(1,3):" K(7)*DASH - K(9)*DASH'3 - K(11)*DASH*DOT'2;DIFFOPS(2,1):" K(5)*DASH*DOT;DIFFOPS(2,2):" K(13)*DASH'2# K(15)*DOT'2;DIFFOPS(2,3):" K(17)*DOT - K(19)*DASH'2*DOT - K(21)*DOT'3;DIFFOPS(3,1):" K(7)*DASH - K(9)*DASH'3 - K(11)*DASH*DOT'2;DIFFOPS(3,2):" K(17)*DOT - K(19)*DASH'2*DOT - K(21)*DOT'3;DIFFOPS(3,3):" K(22) - K(23)*DASH'2 - K(25)*DOT'2

# K(26)*DASH'4# K(28)*DASH'2*DOT'2# K(30)*DOT'4;

COMMENT ******************************************************************************************************;COMMENT **** USE LINEAR ALBGRA (LOOSELY BASED UPON THE METHOD OUTLINED ****;


COMMENT **** BY DONNELL) TO ELIMINATE U AND V FROM MAIN P.D.E. AND OBTAIN ****;COMMENT **** ANCILLARY P.D.E’S INVOLVING U&W AND V&W RESPECTIVELY ****;COMMENT ******************************************************************************************************;

EXPANALPHA:"(DIFFOPS(2,1)*DIFFOPS(3,2)-DIFFOPS(3,1)*DIFFOPS(2,2))*DIFFOPS(1,3)

#(DIFFOPS(3,1)*DIFFOPS(1,2) - DIFFOPS(1,1)*DIFFOPS(3,2))*DIFFOPS(2,3)#(DIFFOPS(1,1)*DIFFOPS(2,2) - DIFFOPS(2,1)*DIFFOPS(1,2))*DIFFOPS(3,3);

DIFFANCIL(1):"(DIFFOPS(2,1)*DIFFOPS(3,2)-DIFFOPS(3,1)*DIFFOPS(2,2));DIFFANCIL(2):"(DIFFOPS(3,1)*DIFFOPS(1,2)-DIFFOPS(1,1)*DIFFOPS(3,2));DIFFANCIL(3):"(DIFFOPS(1,1)*DIFFOPS(2,2)-DIFFOPS(2,1)*DIFFOPS(1,2));

COMMENT ******************************************************************************************************;COMMENT **** PICK OUT INDIVIDUAL PAIRS OF POWERS OF DASH AND DOT TO GIVE ****;COMMENT **** FIRST OF ALL A SUBEXPRESSION FOR EACH POWER OF DASH, THEN ****;COMMENT **** INDIVIDUAL TERMS CONTAINING EACH POSSIBLE COMBINATION OF ****;COMMENT **** POWERS OF DASH AND DOT. NOTE THAT ARRAY SUBSCRIPTS ****;COMMENT **** IDASH AND IDOT ARE 1#POWERS OF DASH AND DOT RESPECTIVELY. ****;COMMENT ******************************************************************************************************;FOR IDASH:"1:9 DO ((SUBEXPRS(IDASH):"COEFFN(EXPANALPHA,DASH,IDASH-1)'';FOR IDASH:"1:9 DO ((

FOR IDOT:"1:9 DO ((DASHDOTSALPHA(IDASH,IDOT):"COEFFN(SUBEXPRS(IDASH),DOT,IDOT-1);

''

'';

FOR IANCIL:"1:3 DO ((

FOR IDASH:"1:9 DO (( SUBEXPRS(IDASH):"COEFFN(DIFFANCIL(IANCIL),DASH,IDASH-1)'';FOR IDASH:"1:9 DO ((

FOR IDOT:"1:9 DO ((DASHDOTSANCIL(IANCIL,IDASH,IDOT):"COEFFN(SUBEXPRS(IDASH),DOT,IDOT-1);

''

''

'';

COMMENT ******************************************************************************************************;COMMENT **** GIVE COEFFICIENTS THE NAMES USED BY SCHWAIGHOFER & MICROYS ****;COMMENT ******************************************************************************************************;ARRAY ALPHA (12); COMMENT **** COEFFICIENTS USED IN MAIN EIGHTH-ORDER EQN ****;ARRAY BETA (8); COMMENT **** COEFFICIENTS USED IN EQN CONTAINING U & W ****;ARRAY GAMMA (7); COMMENT **** COEFFICIENTS USED IN EQN CONTAINING V & W ****;

ALPHA (1):"DASHDOTSALPHA (9,1); ALPHA (2):"DASHDOTSALPHA (7,3);ALPHA (3):"DASHDOTSALPHA (5,5); ALPHA (4):"DASHDOTSALPHA (3,7);ALPHA (5):"DASHDOTSALPHA (1,9); ALPHA (6):"DASHDOTSALPHA (7,1);ALPHA (7):"DASHDOTSALPHA (5,3); ALPHA (8):"DASHDOTSALPHA (3,5);ALPHA (9):"DASHDOTSALPHA (1,7); ALPHA (10):"DASHDOTSALPHA (5,1);ALPHA (11):"DASHDOTSALPHA (3,3); ALPHA (12):"DASHDOTSALPHA (1,5);

BETA (1):"DASHDOTSANCIL (3,5,1); BETA (2):"DASHDOTSANCIL (3,3,3);BETA (3):"DASHDOTSANCIL (3,1,5); BETA (4):"DASHDOTSANCIL (1,4,1);BETA (5):"DASHDOTSANCIL (1,2,3); BETA (7):"DASHDOTSANCIL (1,6,1);BETA (8):"DASHDOTSANCIL (1,4,3); BETA (6):"DASHDOTSANCIL (1,2,5);

GAMMA (1):"DASHDOTSANCIL (3,5,1); GAMMA (2):"DASHDOTSANCIL (3,3,3);GAMMA (3):"DASHDOTSANCIL (3,1,5); GAMMA (5):"DASHDOTSANCIL (2,3,2);


GAMMA (4):"DASHDOTSANCIL (2,1,4); GAMMA (6):"DASHDOTSANCIL (2,5,2);GAMMA (7):"DASHDOTSANCIL (2,3,4);

OFF EXP;OFF ECHO;

COMMENT ******************************************************************************************************;COMMENT **** NON-DIMENSIONALISE THE VALUES OF ALPHA, BETA AND GAMMA ****;COMMENT ******************************************************************************************************;FOR IALPHA:"1:12 DO ((

WRITE ALPHA(IALPHA):"ALPHA (IALPHA)*A'2/(A11*A66*D11)'';FOR IBETA:"1:8 DO ((

WRITE BETA (IBETA):"BETA (IBETA)/(A11*A66)'';FOR IGAMMA:"1:7 DO ((

WRITE GAMMA (IGAMMA):"GAMMA (IGAMMA)/(A11*A66)'';OFF EXP;OFF ECHO;

COMMENT ******************************************************************************************************;COMMENT **** WRITE OUT THE VALUES OF ALPHA, BETA AND GAMMA IN FORTRAN FORMAT; ****;COMMENT ******************************************************************************************************;ON FORT;OUT ‘‘SPEC—ORTH—CYL.f’’;FOR IALPHA:"1:12 DO ((

WRITE ALPHA (IALPHA):"ALPHA (IALPHA)'';FOR IBETA:"1:8 DO ((

WRITE BETA (IBETA):"BETA (IBETA)'';FOR IGAMMA:"1:7 DO ((

WRITE GAMMA (IGAMMA):"GAMMA (IGAMMA)'';SHUT ‘‘SPEC—ORTH—CYL.f ’’;OFF FORT;QUIT;;END;

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analysis of laminated anisotropic plates and shells using symbolic computation

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