NGN~LINEAH BEHAVIOR OF CYLINDRICAL SHELLS

Thesis for the Degree of Ph. D.
MEGHIGAN STATE UNIVERSITY
Cary KawKei Mak
1965

t r tit“

LIBRARY

Michigan State
University

       

This is to certify that the

thesis entitled

I NON-LINEAR BEHAVIOR OF CYLINDRICAL SHELLS i ‘

 

presented by I

 

Cary Kau-Kei Mak ‘ ‘

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Civil Engineering

QKZVQW w

Major professor

 

Dam View“; /7,, W65

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ABSTRACT v'i

  

NON-LINEAR BEHAVIOR OF CYLINDRICAL SHELLS
by Cary Kau-Kei Mak 1

An analytical method has been developed to study the non-
linear behavior of elastic thin circular cylindrical shells under- ;
. v I
~ going large displacements. The shells are supported by flexible i
' ..‘ beams on the longitudinal edges and rollers on the curved edges, l
r or by rollers on all the edges. Three types of loading are con- i
sidered: a uniform radial pressure, a uniform-live load (vertical 1
.3. load distributed over the horizontal projection of the shell), and l
.a uniform dead load (vertical load distributed over the curved i
:7‘ surface of the shell). , i
t The method of analysis is based on a large deflection theory i
|

'._‘ _of shells by including the quadratic terms ( 95%. )2 and ( 2% )2

       
  
 

1
.in the strain tensor. The variational problem resulting from an 1
application of the principle of stationary potential energy is solved i
approximately, by the method of Rayleigh—Ritz. The radial dis- 1

. p |
placement function w, with two undetermined parameters, is chosen

to represent a fir st harmonic approximation of the deflection of

.the shell. The longitudinal and circumferential displacement

 

  
 

p
I Q
. .'

 
  
  
  
  

Cary Kau-Kei Mak

functions 11 and v, are considered to consist of two parts: up, v

P

and uh, vh. The functions u and vp are chosen to be the parti-

P
cular solutions of the equatiOns of equilibrium in the longitudinal
and circumferential directions, respectively, and uh and vh are
homogeneous solutions of 'V‘lu}1 = Vévh = 0, so that the sums
u = up + uh and v = vp + Vh satisfy approximately the geometric
and natural boundary conditions. By applying these approximating
functions to the Rayleigh-Ritz procedure, a set of two simultaneous
algebraic cubic equations are obtained. With the use of a high speed
digital computer, these equations are solved by the iteration scheme
of Newton-Raphson. For a given shell and loading type, a load-
deflection curve is obtained from a series of solutions corresponding
to a range of loading intensity. The curve, in general, is non-
linear. It is indicated that after a certain range of essentially linear
behavibr,, the stiffness of the shell decreases. In many cases the
shell "buckles, " i. e. , the displacement would increase substantially
with little change in load.

By a repeated application of the above procedure for different
values of shell parameters, a number of load deflection curves are

obtained. From these numerical results, the principle findings may

,be summarized as follows:

Among the three loading conditions considered, the shell has

the lowest stiffness (or buckling load) under the dead load. The

,-‘- 1:,‘g,_. . .

——4“. -m.

i.

 

~ .Yyr'rr'v‘ ’r'vv- v—n— 1' V's?)

Cary Kaua Kei Mak

     
   

”fl-have lower stiffnesses or‘buckling loads for smaller values
“(the opening angle i‘k’ smaller values of the radius to length
. 'fiarameter’s, larger values of the radius to thickness parameter Z,

p—rfifld for smaller edge beams.

it“!

 

  

t O

- ‘3‘,

p
n

NON-LINEAR BEHAVIOR OF CYLINDRICAL SHELLS

BY

Cary Kau-Kei Mak

A THESIS

Submitted to
-Michigan State University
.in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
v.3 ‘
' And 1a at, ffflepartrnent of Civil and Sanitary: Engineering

1965

 

  

  
  
  
  
  

ACKNOWLEDGMENTS

This report constitutes the author's doctoral dissertation, written
under the direction of Dr. Robert K. L. Wen, Associate Professor

of Civil Engineering, whose valuable guidance and stimulating suggestions
during the course of this study is gratefully acknowledged.

The author wishes to express his thanks to Dr. C. E. Cutts,
Chairman of the Department of Civil Engineering, for his encouragement
and interest during the course of the author's graduate studies, to
Dr. L. E. Malvern, Professor of Metallurgy, Mechanics and Material
Science, and Dr. J. S. Frame, Professor of Mathematics, for their
valuable suggestions in solving the non-linear equations. Also,
special thanks are extended to the National Science Foundation and
to the Division of Engineering Research, Michigan State University
for their financial support which made this work possible.

In addition, the author wishes to extend his sincere appreciation
to Dr. B. N. Beyleryan, former Research Assistant in Civil Engineering,
for'his assistance in checking some of the numerical computation and
in obtaining computer solutions from the computer laboratory during
the author's. absence from Michigan State University.

And last, but not least, go thanks to the author's wife, Leonora,

~ whose encouragement during many difficult periods is genuinely appre-

ii

 

 

3,
'l
'1.

.-

A 3:.LE‘...

TABLE OF CONTENTS

     
  

' ACKNOW LE DGMEN TS
‘msr OF FIGURES
I. INTRODUCTION

1. 1. Object and Scope
1. 2. Review of Literature
1. 3. Notations

METHOD OF ANALYSIS

2.1. Shell Structure Considered x.

2 2. Assumptions and Limitations

2. 3. Potential Energy of Shell System

2 4. Principle of Stationary. Potential Energy
of Shell System

5. Rayleigh-Ritz Method

. 6. Approach Used in Choosing Approximate
Displacement Functions

.2 7. Choice of w

2.8. Choice of“1 and VP to Satisfy Equilibrium

Equations

2. 9. Choice of Eh and 3h to Satisfy Boundary Conditions

2 l . Dimensionless Form of Potential Energy

.2. ll. ' Derivation of Algebraic Equations

V 2 12. Solution of Non-Linear, Simultaneous Algebraic

_ ‘ Equations

. y-"

fig. NUMERICAL RESULTS

-. 3. 1. Effect. of Types of Load
' v 3-. 2. Effect of Shell Geometry
‘ i :3. 3. Comparison of Results

L iii

 

Page

ii

ooasr‘

14

14
15
17
20

21
21

28
28

3 3
41
42
45

49

49

51
55

 

P.‘ hi.

 
   
  
  
    

a
,.
v

4‘“. SUMMARY AND CONCLUSION
In.'i,‘J-4.l. Summary

4. 2. Concluding Remarks

l
L '34. 3. Suggested Future Work

-" ~v. BIBLIOGRAPHY
v1. FIGURES

APPEN DIX

iv

 

LIST OF FIGURES

Page

Cylindrical Shells Supported by Flexible Beams 65

on Longitudinal Edges and by Rollers on

the Curved Edges

Cylindrical Shells Supported by Rollers on 66

A11 Edges

Effect of Types of Load 67

(Roller Supported Shell)

Effect of Z ' 68

(Z = t , Beam Supported Shell)

Effect of y! 69

(Beam Supported Shell)

Effect of S ‘ ,70

(S =%—, Beam Supported Shell)

Effect of V 71

(V = 7?», Beam Supported Shell)

Effect of W 72 .
(W =35: , Beam Supported Shell) '
Comparisonof Nx and N7; 13

(Beam Supported Shell)

Comparison of Results of This Report with 74

Reference 11 2.,
(Roller Supported Shell) . "1

     

    
  

LIST OF APPENDICES

Page

 
    

p
a.

  

«11? Comparison of Two Choices of Displacement Functions ‘75

‘. 'fif 1" Coefficients of Equations (2.49a-b) 77

. f-

“1'
.} 1
Li: ,- ..
. - v

.‘ Danl.‘..

en 9‘“;

_‘.

I. INTRODUCTION

   
  
  
  
  
  
  
  
  
  
  

l. 1. Object and Scope

The purpose of this study is to investigate analytically the non-
linear‘behavior of thin circular cylindrical shell panels undergoing
large deflections. The objectives of this investigation were:

(1) To develop a procedure of analysis to solve the large
deflection problem of cylindrical shell panels with certain
boundary and loading conditions that have not been considered
thus far.

(2) To apply the procedure to investigate the influences of

 

different types of loading and of various shell parameters

on the behavior of the shell structure.
This study is based on a large deflection theory advanced by
‘Donnell(4), (1934) who developed the non- -linear equations of cylindri-
cal shells by including certain quadratic terms in the strain- 1
displacement relations. This approach leads to three non-linear ll
\ partial differential equations of equilibrium in terms of the displace-
ment'u (in the longitudinal direction, x ), v (in the circumferential
direction,,¢i), and w (in the radial direction, 2).

In the particular case when the loads are applied in the radial '

 

" , *Numbers in the first and second parentheses refer. to reference ",5;
_;-number and year of publication, respectively, as listed in the ' l
bibliography.

 

   

rr—v" , _

   
  
  
  
  
  
  
  
  
   
 
  
  
  
  
  
  
  
   
  

'9 direction and/or on the boundary only, the equilibrium equations in x
‘ and p directions can be satisfied identically by the introduction of an
Airy Stress function 1". Thus the problem becomes simplified
appreciably as the three equilibrium equations are reduced to one
equilibrium equation in the z direction and a compatibility equation
in terms of w and ’V". These two equations, being non-linear in w,
are usually solved approximately either by means of the Rayleigh-
Ritz or the Galerkin-Bubnov method. Because of the inherent
difficulty in solving non—linear boundary-value problems, all the
published work on shell stability from the large deflection point of
view has been limited to the previously mentioned types of loading,
which made the above simplification possible.

In this study, three types of loading conditions are considered;
namely: radial pressure, live load (a vertical load distributed over
the horizontal projection of the shell), and dead load (a vertical load
distributed over the surface of the shell). It is noted that the latter
two types-of loading have a component in the circumferential
direction, so that the simplification mentioned above is not applicable.
Consequently the problem is treated in terms of all three displace-
ment components 11, v and w, and is solved approximately by the

‘ Rayleigh-Ritz method. In applying this method, w is chosen to be

I ”a first harmonic approximation of the shell deflection, while u and

"fivxare chosen not only to be the particular integrals (up, VP) of the

affimfm‘. ' ..

its“
ninth.“ ‘ "arc-l. " '

  
  
  
   
  
  
   
  
  
   
  
  
  
  
   
  
  

filibrium equations (as was done in Ref. 9) but : to contain also
homogeneous solutions (uh, vh) so that the sums u = 111) + uh, v-= vp‘+ V‘h
approximately satisfy the geometric and natural boundary conditions.
Therapproximations involved are twofold. First, in considering the
'natural boundary, conditions only the membrane forces Nx and Ng are
taken into account. (For "long" shells, these are the dominant
forces.) Second, certain trigonometric functions describing the

force distribution along the boundary are approximated by polynomials.
. $11221; approximations are necessary because it seems impossible to
find a set of u and v so that the governing differential equations and

all the associated boundary conditions are simultaneously; and

rigorously satisfied. With 11 and v chosen as described, even with
‘only'two undetermined parameters. in the assumed radial deflection
function w, physically meaningful results are obtained. These

_ results depict the non-linear behavior of the shells considered.

This thesis also deals with shells supported by flexible, rectangular
edge beams, which represent more realistic boundary conditions for
.zwconcrete roof shells than that assumed by other researchers.

Briefly, the contents of this study are arranged as follows:

Chapter 11 gives an outline of the basic assumptions, as well

. titartanexpression of the total potential energy of the shell and the edge

‘gfigama. A detailed discussion of the choice of the approximate de-

 

  

  

 

4
from an'application of the Rayleigh-Ritz procedure.

The numerical results obtained in this investigation are pre-
sented in Chapter III, in which the influences of the types of loading
andthe properties of the shell on its behavior are considered.
Then,the accuracy of the analysis is evaluated by comparing the
solutions with a linear problem solved in the ASCE Manual No. 31 (l)
(1952) and with a non-linear problem investigated by Kornishin and
Mushtari (11), (1959).

In Chapter IV, in addition to a short summary of the work,
some remarks are made concerning the relation of this study, to
the present practice of stability consideration in concrete roof
shell design. Finally, some suggestions are offered for possible

futur e wo rk.

l. 2. Review of Literature

 

In 1934, Donnell, (4) (1934) making use of von‘Karman's (22)
(1910) approach to the problem of large deflections of plates, for—
mulated the non-linear governing differential equations for cylin-
drical shells by including in the strain tensor the quadratic terms
in 5%? and g?) . Later, von Karman and Tsien (9), (1941) used
the same formulation to. investigate the bucklingproblem of
cylinders under axial compression with the Rayleigh-Ritz procedure.

In this celebrated-work, stable po st-buckling equilibrium configurations

were found, corresponding to axial loads as low as 25% of the critical V ‘

 

5
leads predicted by the classical small deflection theory. These'im-
rportant results partly explained the large discrepancies which existed
between experimental and earlier theoretical results, and demonstrated
dramatically the inadequacy of the classical small deflection theory-in
predicting the buckling load of thin shells.

Since that time, research efforts devoted to the investigation cf
the stability problem of shells have been very intense. The emphasis,
however, has been restricted to axially symmetric, closed shells,
suchas complete circular cylinders, truncated conical shells and
spherical shells. Very little work has been done on open shell panels,
such as circular cylindrical roof panels of rectangular planform.

In the American Society of Civil Engineers Manual No. 31 (l), (1952),

. it is suggested that the buckling stress of a long circular cylindrical roof
shell could be approximated by the critical stress of a long cylinder

under axial compression obtained by Timoshenko (21), (1936), using
the- classical small deflection theory. But Karakas and Scalzi (8), (1961),
who test-loaded a cylindrical shell panel made of reinforced plastic,
showed that such an approximation over simplifies the problem and leads
to erroneous and even unsafe design. The shell was found to buckle
at 32% of the critical stress calculated according to the suggestions
of the ASCE Manual No. 31.
, Koiter (10), (1956). investigated the po stbuckling behavior of a

narrow cylindrical panel, such as that occurring in stiffened cylindrical .

 

6

  

a very narrow curved panel in the advanced po stbuckling stage would

approach the behavior of a flat plate panel of the same width.
Soderquist (18), (1960) investigated experimentally the buckling

) strength of a series of curved panels with rectangular stiffeners.

} The load was applied in compression axially, and measurements were

made of the initial buckling stress. The ultimate strength of the

panels was found to increase markedly with curvature, and the rate

of increase to depend on the ratio of stiffener spacing to shell

thickness.
i Finkel'shtein (5), (1956) studied the buckling problem of a

i cylindrical panel under the combined action of axial compression

 

   
  
  
  
  
  
  
  
  
  
  
  
  

and uniform transverse radial pressure. Considering the panel to

be simply supported along the edges of the shell, he assumed that

no moment would appear in the shell so long as the loads were below
their critical buckling value. However, when buckling took place,
large deformations were produced. Thus, the problem was reduced '
to a system of two non-linear differential equations. The unknown
functions were the radial displacement w and the stress function V
The radial displacement was assumed to be the same as in the case
of small displacements and was substituted into one of the differential
equations which was solved for )0‘ . Substituting both w and V into
the second equation, the author obtained a function f which contained
the maximum deflection and the loading as its arguments. The

function é was expressed as a Fourier series and, by equating its

  

 

  

7
coefficients to zero, the conditions of buckling were obtained.
Kornishin and Mushtari (11), (1959) presented an algorithm

applicable to the solution of nonlinear problems of the theory of shallow
shells. They applied the algorithm to the buckling problem of a cir-
cular cylindrical panel of rectangular planform supported by"'rollers“
on all sides and loaded transversely by a uniform radial pressure.

(At a roller support, w = 0, and the forces vanish.) As mentioned
earlier, the simplicity of loading enabled them to express the
problem in the form of an equilibrium equation in the radial direction
and a strain compatibility equation in terms of a stress function, ’V’ ,
and w. They were both non-linear 4th order partial differential
equatiOns. After choosing a set of appropriate trigonometric functions
containing a total of six arbitrary undetermined parameters for w and
30‘ , the differential equations were solved approximately using the
Bubnov-Galerkin (method. In this way the problem was reduced to a
set 0f 6 CUbiC algebraic equations to be s'olved 'simultaneously. The
authors then proposed an algorithm to solve approximately these
non-linear algebraic equations. The results were presented in the
form of a set of load-deflection curves for different parameters.

Sunakawa and Uemura (20), (1960) solved a problem similar

to that of Kornishin and Mushtari (11) except that the straight edges
were assumed to be clamped while the curved edges were simply
. supported. Using techniques similar to those employed by Kornishin

and Mushtari, Sunakawa and Uemura approximated w and 31’ by a

 

  

  

polynominal containing only one arbitrary undetermined parameter.

The numerical results of the last two references, (11) and (20)
will be further referred to in the later chapters of this thesis.

This brief review has included materials on the large deflection
or buckling of cylindrical panels only, as they are of primary concern
in the present study. A more comprehensive survey of published
literature on the general theory of elastic stability of closed shells
may be found in reviews by'Langhaar (.12), (1958), Nash (16), (1960)

and Fung and Sechler (6), (1960).

l. 3. Notation
The symbols used in this study are defined as they first appear in
the text. They are summarized here in alphabetical order for
convenient reference:
A = cross sectional area of edge beam;
Ai; Bj = coefficients relating to up, vp, defined by‘Eqs. (2.21a-e)
and (2. ZZa-h). i varies from 1 to 7, and j from 2 to 11;

a,b

depth and width of edge beams;
Cij- dij = coefficients of Eq. (2.49a-b) as listed in Appendix It
i and j vary from 1 to 3;

L " flexrl ' 'dit'
12“.”)2), - u a. r1g1 y,

II

A
v
(D
ll

quantity to be evaluated'at the junction of the edge beam
and the shell;

' = Young's Modulus;

' 109,)

ii

M,' M‘

fo " Mpx

 

 

undetermined parameters defining the center deflection

 

of edge beam and shell and the center deflection of shell, 3,
respectively;

-§- . 33,-, :respectively; 1 ‘1
horizontal centroidal axis of edge beam; _ ii:
moment of inertia of the beam section about the

horizontal principal axis, Ho;

 

J acobian matrix of a system of equations evaluated at Xn;
Et
1_ V2 = extensional rigidity;

 

coefficients of Eq. (2. 50), defining the total potential
of the shell system;

longitudinal length of shell;

bending moments per unit of longitudinal and circum-
ferential length, respectively; A

circumferential and longitudinal twisting moments per
unit of circumferential and longitudinal length,
respectively;

longitudinal and circumferential normal forces per
unit of circumferential and longitudinal length,
respectively;

circumferential and longitudinal shearing forces per
unit of circumferential and longitudinal‘length,

r e spe ctively;

 

 

-
-—

II

N

 

10

the superscript m indicates the number of the load
increment applied, and the subscript n indicates

the number of iterations performed by the computer;
intensities of load components in the longitudinal,
circumferential, and radial directions, respectively.
Their positive senses are orientated in the
directionsof positive x, y and z;

intensity of radial load;

intensity of live .' load;

intensity of dead load; 5 "

dimensionless load parameter. in longitudinal,

circumferential, and radial direction, respectively;

dimensionless load parameter of radial, live and

dead load, respectively;

radius of shell;

curved length of shell;
—‘%—, dimensionless parameter of shell;

transverse shearing forces per unit of circumferential

and longitudinal length, respectively;

A

 

 

T

1:

U

U

u,v,w

u :3
t

.7 =1!

- W

“1:?

311,311

us, uB

 

lateral thrust acting on the shell edge;

thickness of shell;

total potential energy of the shell system;

2 Z
EtR¢kL

displacement components in the longitudinal,

circumferential and radial direction, respectively;

dimensionless displacement components in the

longitudinal, circumferential and radial directions;

dimensionless displacement functions that satisfy
the biharmonic equations of 1‘1 and v respectively;
dimensionless displacement functions that satisfy

; the equilibrium equations in the axial direction,
and the circumferential direction, respectively;
longitudinal displacement of the shell at Y1 = if. 1/2 9
and of the edge beam, respectively;

7a., dimensionless she'll'parameter;

strain energy of edge beams;

strain energy of shell;

vertical centroidal axis of edge beam;

% , dimensionless shell parameter;

shell coordinates defining the mid-surface of the shell;

 

,5
1 In}.

6

a, 5,, 5,,

 

 

12

-%, dimensionless shell parameter;
horizontal displacement component of the edge beams
along the centroidal axis Ho’ positive to the right;
vertical displacement component of the edge beams
along the centroidal axis V0, positive upward;
vertical displacement of the shell at 7 = ~:.l;-é-_- , and
of the edge beam, respectively;
axial strain of the edge beam;
longitudinal, circumferential and shear strain at
the middle surface of shell, respectively;
quantity is to be evaluated at S = ii» 7 and 11: iJz: ,
respectively;

dimensionless coordinates on middle surface;
Jé—k- ," ris‘e angle of shell;

Poisson's Ratio;

potential energy of the loads;

shell coordinate in the circumferential direction;
opening angle of shell;

longitudinal, circumferential curvature change,
and twist of the middle surface, respectively;

Airy Stress Function;

 

),m+%r< was Mm
)mss +figt( )MHE'ZK Matt

1
.‘.
a

,.. , - 1 'r, . \.;‘.
m amalmumw -'§.- $3.1 "

 

 

 

II. METHOD OF ANALYSIS

2. 1. Shell Structure Considered

The shells considered in this investigation are shown in Fig. 2.1
and Fig. 2. Z. The shell, with its mid-surface defined by the co-
ordinates x and y‘, is cut from a perfect circular cylindrical shell
of constant thickness by two pairs of planes containing the principle
radii of curvature. Fig. 2. la depicts a shell supported by two
identical rectangular flexible beams along the longitudinal edges
and by rollers along the curved edges. The shell shown in Fig. 2. 2
is supported by rollers along all edges.

The cross—section of the edge beams is shown in Fig. 2.1c
in which V0 and “0 denote the vertical and horizontal centroidal
axes, and ,6 and pd the corresponding displacements.

The external load applied on the shell is to be represented by
the three components: Px, P55, and Pr’ denoting load intensities in
the longitudinal, circumferential, and radial directions, respectively.
The loading types considered are, as mentioned in Chapter I,
radial load, live load, and dead load.

As usual, the symbols Nx’ N75, Nx¢, Nyix denote the normal and
shearing membrane forces; Mx' M¢, Mxyi’ M¢x are the bending and

twisting moments; and SK and 8,5 are the transverse shearing forces

 

14 7. r"

 

 

acting on the shell. The positive directions of these internal forces

are indicated in Fig. 2.1b.

2. 2. Assumptions and Limitations

The analysis is based on a large deflection theory first
advocated by Donnell (4), (1941). Associated with this theory are
the following basic assumptions:

(1) The problem is restricted to small strains, 1. es, the
strains are small in comparison with unity.

(Z) The problem is restricted to geometrical non-linearity.
The material which forms the shell, however, remains
linearly elastic so that Hooke's Law for a homogeneous
and isotropic material may be applied.

(3) The shell under investigation is assumed to be thin;
that is, éf o: 6 in which t and R are the thickness and
radius of the shell, respectively, and 6 is the strain
in the x or ¢ direction. This assumption reduces the
shell to a two dimensional problem and justifies the
use of a simplified expression for strain energy of the
shell by neglecting quantities having the same order of
magnitude as {3 in comparison with unity. It becomes
possible to apply the Kirchhoff—Love hypothesis

that vectors perpendicular to the mid- surface of the

 

 

7——.—‘ .~ —— — — , . ‘_' ’

l6

shell before bending remain perpendicular after bending.
At the same time normal stresses perpendicular to the
mid-surface are considered to be small in comparison
with the stresses tangential to the mid-surface. This
hypothesis leads to an error of at mostfi in comparison

with unity. (17) .

    
  
 

(4) In addition, the shell is assumed to be limited to
"medium bending, " that is, the maximum deflection is
of the same order of magnitude as that of the thickness,
but is small in comparison with other linear dimensions.
(5) The shell is also assumed to be shallow; that is,

(is): = (.2472): << / in which 8 is the curved length
of the shell and pk is the opening angle. Except forone
case, the maximum value of 55k considered in this study

is limited to 55k = 0. 632 (approximately 36°) so that

(6) Furthermore, the shell considered is assumed to be

long, i. e. ,
¢kR
...__. <
L .. 0. 5
in which L = longitudinal length of shell. In this way,
the deformed shape of the shell might be closely apprOxi-
mated by a half co sine wave in both the longitudinal and

. j- eircumferential directions, and the dominant internal

. 2". 7‘
,f i “'7 '1.
I_ V ‘

 

"VVWva ' "M‘" v v—A '."

l7

    
   
  
  
  
  
  
  
  

_ the analytical procedure used in this study. Generally, these
assumptions are applicable to reinforced concrete roof shells

[provided the opening angle is not too large.

2. 3. Potential Energy of the Shell System

As mentioned earlier, the method of Rayleigh-Ritz is used
Linsthis study. It is therefore necessary to have the expression

4%. i of the potential energy of the shell system.

2. 3. l. Strain-Displacement Relations of the Shell: Based

I ‘ upon assumptions outlined in section 2. 2. , the strain of the mid-
surface of the shell can be related to displacements u, v and w by

the following expr e s sions:

6x 1'- ll); + i" (”2’02

5+ = '1E~“’2¢+% + 2%.sz

Zéxf = .45; + fiufl + kw”, «54,

(2.1a-f)
xx = — )XX

M = is»

1,, =, 12””

 

 

 

 

.in which 6,.) 64,) 6x4: = longitudinal, circumferential

and shear strain in mid-surface,
respectively.
fix, $11); Add = longitudinal , circumferential
curvature change, and twist
of mid-surface, respectively.
The notation has been adopted that a comma followed by a

subscript indicates a partial derivative. Thus M,x— - "37' etc.

2. 3. 2. Strain Energy of Shell: The strain energy of the
shell, Vs’ can be expressed in the following form if quantities of
the order of magnitude of —Rt— in comparison with unity are neglected '

(13). (1962)
Vs=-'-S: sf <K((e,+ e,)— 2(I-D>(6x颗 e,,)}+ (“J S}:
7 ”i“ f:

+ D ((10% Ma)- 2( (PVXM M; - Lg?)}>Rd¢ dx "

in which

K = extensional rigidity = fig

D = flexural rigidity = 75%) ,-
9 = Poisson's ratio . -I
E = Young's modulus .

If the strain-displacement relations Eqs.(2. la-f) are substituted

into the strain energy expression of the shell, Eq. (2. 2) becomes

.~

r‘

LL ’-

 

19

l % ‘9‘ I 7- ' w I 1. 7'
Vs =f§ %<K{[M,x+%(w}x)+fi45¢ +'§_+§'§1(w}¢)]_
“with”iW][sawing-dawn}
‘ilfinimh fiw,mr,¢]z}§+ (2.3.)

+ D{[w,xx+ #:Mflflz—

'2( WW? WWW '- 7:3 («cx+)‘]}> Mick

   
  
  
  
   
   
   
   
  
 
 
 

.2. 3. 3. Strain Energy of Edge Beam: Assuming that the

displacements ofthe edge beams are small in comparison with their
sfcross usec-tional , dimensions, the elementary beam theorvaill

be used. Furthermore, in accordance with the accepted procedure
of shell designl(l), it is assumed that the edge beams have zero
rigidity against bending in the horizontal plane and against twisting,
and the strain energy due to shear deformation is negligible. Thus

‘ .the strain energy of the two edge beams is:

. i .

vlr=j { EA(€°)2+ E1H(/5xx)z} ax (2.4.)
J;

a (sub); a is the depth of the beams and b the width;
5%}?! pmoment of inertia of the beam cross section about
. the horizontal principal axis, H0;
=7 (axial strain of the edge beam.

The deformations of the edge beams are relatedto those of . “ ,. I

“:11 by, the following expressions:

I
v
" .m,‘

,. gar-9' .

 

 

  

20
fi-‘WL 009% "MAM?

 

69: (6:3; + ‘S'i/fipok». (2- 5-)

= <u,.>.+%<w,.):+%{<«5xxhwv% —(4;..).mt=}

The subscript e indicates the quantity to be evaluated at the junction

of the edge beam and the shell; 1. e. , at y! = Q‘k

Substituting Eqs.(2. 5) into Eq. (2. 4) the total strain energy of

the two edge beams becomes

i- 1 7- .1 ' 2
Vb =3 <EA((M,x)z+EW5xL+2[(Mxxltmi“(45“)sm%]i +
+ EIH{[w¢mgh ~11}. M25],xx}z>dx (2. 6')

M"

2. 3. 4. Potential Energy of Loads olthe Shell: The potential

energy of the loads acting on the shell domain is

L
9- =_STS%(RM+BV+ROT)Rd¢JK (2'7"
4 4r

2. 4. Principle of Stationary Potential Energy of thie Shell System

The total potential energy of the shell system is:

U = Vs+vb+fl . (2.8.)
in which st Vband .0. are given by Eqs. (2. 3), (2.6) and (2.7),
respectively.

If the shell system is in equilibrium the variation of the total
potential 5U must vanish for any arbitrary virtual displacement, i. e. ,

8U = 6(vs+vb+.mso . (2.9.)

 

   
  
  
  
  
  
   
 
  
  
  
  
  
   
  
  
    
  

7);? '~ Rayleigh-Ritz Method
Instead of solving the variational equation, Eq. (2. 9).
.dir'ectly, the approximate method of Rayleigh-Ritz is applied.
‘The procedure of solution is outlined as follows:
(1) A set of displacement functions, u, v, and w with
, n undetermined parameters are assumed and substi—
" .tuted into Eq. (2.8). I
_ . (2) The total potential of the shell system is made stationary
' with respect to the n undetermined parameters, i. e. ,
, : the partial derivative of the total potential, , U , with
respect to each of the 11 parameters is obtained and set
equal to zero.
' . (3) ‘ After carrying out the integration, the result is a
set of n simultaneous non-linear algebraic equations
from which the n parameters can be determined.
" ' ‘- The' above procedure yields the deflection of the shell for
1.- {‘1 'a—given load intensity. By repeating this process for different
f‘ values of the load, a load-deflection curve, which depicts the

i x" hehaviorof the shell, can be obtained.

: (.i 2. 6. 1. General: The method of Rayleigh-Ritz has been
“~' gh‘lLLl ‘1
- cessfully applied to stability,» problems of shell structures when

 

22

 

the deflection functions chosen actually approximate the real deformed

shapes of the shells observed in experiments. Such was the case of

a circular cylinder in compression; the deflection functions used by

various investigators approximated the diamond shape deflection

pattern observed in laboratory tests. In the case of cylindrical

panels under transverse loads, however, no definitive experimental

data are available. However, one might suspect that the radial

deflection w will be close to the fir st harmonic in both the longitu- .
dinal and circumferential directions. This approximation was found

1 .
to be satisfactory by Kornishin and Mushtari (11), (1959) for '.“ ‘

I ‘ ‘I
‘1 ’$
cylindrical panels supported by rollers on all sides and loaded ‘2‘) i
w
12,.
transversely, by radial forces, provided that the assumptions listed ”1"
in section 2.2 are satisfied. It is far more difficult, however, to ‘.;i
'. '83}-
estimate by physical intuition alone the forms of the displacement 1'."

  
  
  
  
  
  
   
  

functions u and v. Chung and Veletsos (3), (1962), in solving the

 

linear equilibrium problem of a cylindrical roof shell by means of

the Rayleigh—Ritz method, used orthogonal trigonometric

functions. By using all harmonics up to and including the 4th in
each of the deflection functions, u, v,and w, they found the solutions ‘ ‘..
converged to those given in ASCE Manual No. 31 (1). However,

to investigate the non-linear behavior of the shell, these approxi-

mating functions with 15 arbitrary. undetermined parameters may

not be accurate enough as they do not satisfy the equilibrium equations

 

Z3

  
  
  
  
  
   
  
  
   
     
  
   
  
  
  
   
  
  
     

éfithe shell in its interior, or the forced and natural boundary condi-
: . ' tions.

In this investigation, the function w is chosen essentially on

.3 an intuitive basis and is limited to a first harmonic approximation.
The functions, u and v, however, are not chosen arbitrarily.
Rather,they are made consistent with the choice of w in that they
satisfy exactly the equilibrium equations in the x and 9‘ directions
and approximately the associated boundary conditions. The dis-
placement functions u and v may be considered to be composed

of two parts:

(.3 u = up + uh ' (2.10a-b)

i v = vp + vh

. in which up and vp are the particular solutions of the equilibrium
equations in the x and y! directions (Eqs.(2. lla-d), to be given
later). .In general, up and vp do not satisfy all the natural and
geometric boundary conditions. Therefore, the additional ex-
pressions, uh and vh, which are solutions of the homogeneous
equilibrium equations, are obtained in such a manner that the sums
up'i- uh and vp + vh satisfy approximately the boundary conditions.

. In general, for each particular case of load type and boundary

_ conditions, u and v must be derived individually.

. 2. 6. 2. Equilibrium Equations of Shell: The equilibrium

Sasquations for the x and ,5 directions of the shell can be obtained

.
. ‘1
k‘hr.

 

24

 

. from a consideration of the equilibrium of a differential element

of the shell. (14), (1961). They are given as follows:

Mufififlim +94”): M.I>¢+%Wx+fu+—" = .

(2.11a-b)
+9 - I B _
‘éz-WWV—liu “Ixé’rIer—Ntxfiiwiwfz“ I? * 0
in which
I I) z" _
f. = [flunxlz’r 2.12 MI) ],.+('—22) 7%an “540,49 .
‘- - (2.11c-d)
1 -
s = flaw.) +%<«cx>‘],,+ (12% (mam
Eqs. (2.11a-b) can be expressed in terms of w only by the use of
the following relations:
I ,%1— {5%(2 IIaI} ”+11%? I IaI} ,, -}—‘:%-.‘.~{I:3(z-II 9})“,= (62.12343)

[ Rz-{quw I~lb)})+¢+'—-2. ,{Eqm III») ,0, ~‘,.” as“: IIa-)}x,,= o

Then‘Eqs. (2.11a-b) are transformed into the following:

I;
M

<

9
"E W} xxxl' ‘9'} 440} x¢¢ - ‘FIfitx— (radii fifi’i’ + (%)-&‘F2,X+ “

‘kifl‘i‘ RIM-k Pm + I? i‘é’iii: P+Ix+

a. Q .' (2.13a-b)
. II.

V‘U' ="I'i4 ”EM? " (2+9)‘;'{zw}u¢ " (T33)+2,xx”é1+2,¢+ +

 

+(I—»)g+a,x¢‘ K—KIP?)¢¢- K(T"-»)Rl>,xx+x “HT ”LBW

‘
r
' 7 v‘
r: ,
, _\

 

t—rg'l-i 'W-‘_ . '-

25

V“ = ( ),xxxx + fiftii )fixN +_I%‘( ))¢¢<H’

   

,.-:$.ubstituting the derivatives of f1 and f2 (Eqs. (2. llc-d))into
'Eqs.g(2.l3a-b), then Vita and V311: become functions of w, the
4 applied .load, and their derivatives,as follows:

4 _ _2 'Lb
VM " RMXXX-fk Mx¢¢-M1;x‘w}xxxx-3M’}M ”fix!"

. ' ‘-
- z.( 2 +2) 7.. w,» as... 4;. “has... 42.11%. «as» -
_J. '_ I _
”We” aw.» «4wa has» «5m +

R‘ )
+%. «5,... am — fih—E—y) Pu,» ‘1'? R,“ + #(Jfi’irmwr -
I (2.14a-Ia)

V‘” = ”I? Mm ‘ 2‘3 “5"? "IE? “H ”In” ‘ 42—31 “infini-
‘fliflw 4% aw... new Jaw,“ as... —
“£1457.“ “5*? "k3 “3+ “6% ‘ i" “724’? “5M +

+ is “5») 445m - in PM» - .4. (15?) R,“ +

.L +9 J.
+K(I-y RES"? ‘7' '-
i

 

26

2. 6. 3. Dimensionless Coordin_ates and Dimensionless Parameters:
In the following, the x and )5 coordinates are expressed in
terms of the dimensionless coordinate E, and 7 ;
in which
g A
L
2 = ii:

Furthermore, the properties of the shell system will be expressed

in terms of 525k, 9 as well as the following dimensionless

parameter s:

-E
S‘L
z_—__R_

t

—i

V‘t
b

W=T{-
Q

Qx’ Q¢, r = 17:23, .13} , 1'12: , respectively.
At the same time, the displacements u, v, and w are expressed

in the following dimensionless form:

' _ 11.
u ‘ t
- _ .‘L
V ' t
v7, = E
t

In terms of dimensionless coordinates and parameters and
setting the Poisson ratio 1) = 0, the equilibrium equations,

Eqs. (2. l4a-b),are transformed into:

 

  
 

‘74- ‘53—'53 warm *‘i‘Qs‘Thss's'séflssQnS‘ . _
“fill-55': “Cam" 3%); (Ea-mm “titsflmd‘ ‘
féi'z'fit (’33: 475mm + 55225521: + 255$!) @eoo)‘
- gfi me - *3 (>1qu + flaw
. * (2.15a—b)
,3 475m $5,471.er '5 s 3‘ 43:2 533m “

I
=' 23?
' S?‘ (4‘3)QO *2 wjfivz wind) "

K

V447

." 5‘5. (W70 min“ + Eisiwism + Z wimwn'l) "
"sisfif (11),, mini}? + 5513qu EM?)

‘51:: QM “Eit- QI’II; + giant?

in which

V‘—-—( ),,,“+-§§-;.( '),mo+v}z‘< ”mm

 

28

 

2. 7. Choice of w
The displacement function w used in this investigation is given

as follows:

A» = 3m¢m¥~ + hung: coil?— (2.16)

i in which the parameter h accounts for part of the deflection at
l the center of the shell, while the other parameter g accounts
| for part of the deflection at the center of the shell and the de-
i flection of the edge beams. Eq. (2.16) may be reduced to a

I dimensionless form as follows:

W‘ = swim mm; + Hmm) mung (2.17)
inwhich
_ g
G - T
H = _h_
t

2. 8. Choice of up. and VP to Satisfy, Equilibrium Equations

2. 8. 1. Radial Load Case: For the case of a uniformly

distributed radial load pRL:

Q = 0

x
Qd = 0 ‘ ’ . (2. 18a-c)
Q = PRL = q

r E RL

 

Substituting Eqs,(2.l7) and (2.18a-c) into Eqs.(2. 15a-b), after

several transformations, the following partial differential equations

 

29

offi and \7 in terms of G and H are obtained.

V47

 

 

7447. =

6%. WWI MW; t H323; we“? mng + (61+ H‘}3§1Mngomv§+
+ 6H<fzn1l+,[ [Hg-)1] Haw): WIKQWWMM mug _

— eH<4n1+5LJ I + (II‘Difg’; mm Wm WM, mm, +
+:lr:<%(1r"++ gz)1>m2#7w~n$mng +

211-5 To +§lq::) )z>,oau.2nq Mngmns
(2.19a-b)

= —G('sl#‘z—'i' 51) AIM/inn mwg— H (r34+;—::) Awnr) wing +

+ Gigi; Minn m$uz+eH [I + Babjiim Wham/«g +

+ 6H [sdfiflag—Zfi mm 000an + H‘fg‘iré anmnq +
+ 6H" + 3’1] (52; —)M~4>Idzm¢uz m2ng+

+6H([2"+é' 2- +4? + 31;}mmi’fllqumzng-I-
+ GH {n‘[zIaE+—S'; 2+55— + £65352»! Mtg) mm m2rr<5+

‘ [I +El§tizg anmnrz mvré,

It is noted that Eqs. (2.19a—b) do not involve any loading term
because the consideration of equilibrium condition in x andp directions
does not involve the radial load. Thus, a particular solution. of

Eqs. (2. l9a-b) is found to be the following:

 

 

 

30

47/49 = GIAI mimmvng + HA2 mnqwng + (61+ H1)”A3mn‘5mw§ +
+ GIN Camiia'zAa‘wng went + &H A5 mtg? mug Mgwong +
+ 6-H AbM+Kan2Aiwn$mn§ + HZA7 mznva/vang Mug

(2. ZOa-b)
Alt? = (a BZM¢qunls + H 153,anng +

+

G‘s, mm mm + &H s, was.) AMmz-l-

4.

GH 15;, Mi’fllm‘n? ‘1' H157 Mwymuan-I

+

(1" Be Wfigqmflq mus + 6H Bq wufiszM/nq mm) +

+

6H 5,0 Mflqmiflpmzmfi + H13” .4»)wa Mm? mini,

 

inwhich
A1 = 7%?”
A2 = 7.8%.:
A3 = A4 = A7 ="4g‘5 (2.21a-e)
A5 _ (2.1:: :22},
A = mat—0.10;

 

 
 
  
    
   
   
  

31

[mean «I: + n‘)7'+ 4n‘ié'5‘] «+15
(41r‘+.3'$" + Ir2 + eIvI3‘.‘)7'+ 4n‘4a‘;

a, = (41% 23+ «2+ 4%) 21.4.2;

a4 .= (4nzits‘+rr‘+ iaEMn‘ig

B 3 __ (Max:151)

‘l . 2 (it“s‘I- I )‘L

l‘ I‘ I .
‘ '.".‘ s = _ H 2453‘)
O; ., 3 Tr I+ 49.3.5:

. “if; B _ . Sh.
' 8

 

4‘ 33’

[TI-[71- 11's i)?
—'—'—T—

I)?" it" I‘ (2' 223‘h)

s ,= Jnhzhlzs.
6 . Irz-I:

B7 = Bu = ‘S-
lflrb'lflf
W

   

“2+ ”9121 : b. = [sass-4M1 I
(“‘4’ Hafiz} ; b 4 .= 4,14,41,25,! +5) ’3‘, .f
[M 2.21.. m. was. But] ~11“;

afZ ; b8 = a4 . iii-2.1L
[11% Z+Ifsz+ I)"+ 41r‘ehf6‘ + 3 4:3] 1'ng

32

2. 8. 2. Live Load Case: For the shell system subjected to

 

- a uniform vertical live load pLL’ the force components can be

expressed in terms of pLL as follows:

I Ox = °
QI‘ = %Mi’m+ = $11 Minym+eq .: . .(2‘.23a-c)
' Qir = ‘g‘mfii’ = im- “’6' 4141

If Eqs. (2.17) and (2. 23a-c) are substituted into Eqs. (2.15aI-b),
the following equations are obtained:
7% = Right hand side of Eq. (2.19a) :5. . ;
_ (2. 24a- b)
v4”. = Right hand side of Eq. (2.19b) + %LL —$—Z:M2+‘rl
Then a. particular solution of Eqs. (2. 24a-b‘) for the uniform

live load Case : is found to be

Tip = Right hand side of ‘ Eq. '(2. 20a)
I it (2. 25a-b)
vp = min hand Slde of Eq. (2. 20b) + 9m? M2414
2. 8. 3. Dead Load Case: For a shell subjected to a uniform
E dead load pDL' the force components can be expressed as follows:
Qx = 0
Q9; ___ %M+ := $DLW+HI ’ . (2.26a-c)
= . 19m. _ _
Qr "é‘ m+ _ (gawk?

    
 

 

Substituting Eqs. (2. l7) and (2. 26a-c) into Eqs. (2.15a-b), the
following equations are obtained:
V45. = Right hand side of Eq. (2.19a)

(2. 27a-b)
V4,}; = Right hand side Eq (2.191)) + %9L'$'T Mil

fir

A

 

’33

  

Then h- particular solution of Eqs. (2. 27a-b) for the dead

load case it found to be:

fip = Right hand side of Eq. (2. 20a) (2. 28a-b)
— . . Z- ‘
vp = Right hand Side of Eq. (2. 20b) + $1,01- M‘i’K'L

2. 9. Choice of Tl}. ar_i_d 5h to Satisfy Boundary cop! ditions

2. 9.1. General: It has been pointed out that the parti-
cular solutions of tip and VP given in Section 2. 8 generally will
not satisfy the geometric and natural boundary conditions of
the shell. In passing, it may be mentioned that when W, fip and
"iip alone are applied to the Rayleigh-Ritz procedure, the load-
deflection response of the shell is very ‘stiff' and exhibits only

mild nonlinearity even at large deflections. If, however, the

1‘ assumed functions of fip and VP are modified by Eh and 3h so

     
   
  
   

that the geometric and the natural boundary conditions are

 

approximately satisfied, ~(the same shell shows a marked decrease
of Stiffness wl'en the deflection becomes sufficiently large (See
Fig. A. l in Appendix I).
In the following, the procedure for obtaining ab and Vb will
be discussed in detail for shell systems
(a) with all edges supported on rollers,and
' . (b) with the longitudinal edges supported by flexible beams

while the curved edges .are supported 'by rollers.

 

34

 

2. 9. 2. Shells Supported by Rollers on All Sides: Along the
curved edges, the boundary conditions corresponding to roller supports

are expressed in dimensionless form as follows:

{E} $:ié = 0

{Mxlgdzi = — fizlfiiiig=ii = O
__ 5 _ __I z ' - (2.29a—d)
{Mxi§,ii = ’i—{Mr}g+2%( )1) }§=iJ{ = O

5 _.. _ ._ .—
(NW}E=£J?:= '{il‘fflts MIQI‘ZIitestwii} 0

fizii

in which i 331% indicates that the quantity is to be evaluated at fixi‘z ,
Along the longitudinal edges, the boundary conditions for roller

support are as follows:

{midi = o

I , miles. = ‘ m {4759?} 944, = 0

(2.30a-d)

_ __ - _ m — z =
E {Minna ‘ J9; HE”? J" ”H new?) ha: 0
Whig; 33%? + 13‘7” + if: 47;, 415$;ng 0

In general, it seems impossible to find a in and ‘7h such
that 1.1 = GP + Eh and ‘7 = VP + Vh satisfy exactly the boundary condition

‘Eqs. (2. 29a-d) and (2. 30a-d) simultaneously. In the particular

case of roller supports on all edges, G in Eqs. (2.1?) and (2. ZOa-b)

    

 

is set equal to zero. The resulting equations reduce to functions

 

 

'35

of H alone. When these simplified expressions for .17, T1 ,and VP
P
are substituted into the boundary condition equations, it is found that

only Eqs. (2. 29a-b) and (2. 30a-b) are satisfied, while the membrane

forces

{Edgié ' imilvi-‘i ' {fittiqeigi and {flaps}:
do not vanish on the boundary. The additional terms Th and 5h are
then chosen so that only Eqs. (2. 29c) and (2. 30c) are also satisfied;
i.e., {lelicfli = 0 and {fi+}Q=i'i = O. Ajustification for
this procedure may be, as indicated earlier, that the dominant
internal forces for long shells ( —% (i0. 5) are Nx and N55. These
forces probably contribute more to the non—linear behavior of
shells than any other stress resultant (l).

2. 9. 2.a. Radial Load Case: For a shell supported by
rollers on all edges, G = 0. Under the action of radial pressure,

the deflection functions are reduced to the following:

it = H mnr) 04:01:47 ‘ (2.3la-c)
Hp = H A2 count) Aiwng + H"(A3+ A7m2nq)b)~tr4)mms
VP - H B3 mmmnp H‘( 57+ bumzm Mmgmm)

After substituting Eqs. (2.10a-b) into Eqs. (2. 29c) and (2. 30¢), the

following relations are obtained:

W: =1 41 +471 +1034)z = 0
flak-iii M in". 2-2 2 Lug: (2.323?“

I - I - _— z _
Lift”+ifim+w+2—%e<wm’lq-.i °
- 1

 

 

 

 

36

- Eqs. (2. 32a-b) imply that

in“)?! as = ’ if I: + $5591}an

{4’}qu1%=‘147+»7+ IKE)? gigfl‘xizlliqng

(2. 33a-b)

when substituting the values of fip , 3p and wp, taken from Eqs.(2. 3laac)
intoEqs. (2. 33a-b) the following relations are obtained:

lui,gl§=,,z = H201 (Ar-3:9“ + wuznq)
{”fi’qkh I = H2 (TI By “4%) )(I +WDZTT5) (2' 34a-b)

Since 5'11 and vhrhave to satisfy the biharmonic equations
0

V45“ = i V = 0

$I

the trigonometric expressions (1+c032nq) and (1+COSan) are
replaced by approximating polynomials 2(1-2Yl ) and 2(1-2‘5 ),

respectivelyzfor all 11 defined In 0‘ V4 .. z , and 2 defined in 02% 5.42- .

ThenTy1 a d 3h may be written as the follwoing:
Eh = H2(2)(nA,-ilԤ)g(u-2v))
vh = H2(2)("57_¢T1g3) YL(|-Z,fl (2. 35a—b)

It is noted that the original trigonometric functions are even
4L .
over the intervals —i_ g- 2 , -— fi- 5. I? 5.2% . Since the
approximating polynomial expressions are defined only over the
intervals 0 5.3 E. )5: , 0 S Q 4.- J2.- , the energy integrals,

in changing to the approximating polynomials and the corresponding

new limits of integration, must be multiplied by 2.

 

 

 

vv

 

 

37

It should be pointed but also that the polynomials are close
approximations of the trigonometric functions. Over the interval
of definition, the polynomials have approximately the same shape
as the trigonometric expressions; and in this particular case, have
identical “'area‘” under the curves. Since these expressions
essentially represent the distribution of Nx and Ny‘ on the boundaries,
the "same area" aspect implies .that the replacement is ":statically
equivalent” to the original trigonometric functions. In short,
this replacement of the trigonometric functions by the polynomials
physically means that instead of Nx and N?‘ strictly vanishing on
the boundary, i there willrbe a small residual distribution of these
forces equal to the differences between the trigonometric and
polynomial functions.

2. 9. 2.b. Live Load Case: In a procedure similar
to that used in the radial load case, the deflection functions are

found to be as follows:

w = Right hand side of Eq. (2. 31a)

Up = Right hand side of Eq. (2. 31b) (‘2 36a e)
- _ - . . 2%; v ‘

VP —- Right hand Side of Eq. (2. 31c) + QLL?(W7$‘Q)

1111 = Right hand side of Eq. (2. 35a)

- E" '

Vh = Right hand side of Eq. (2. 35b)— %LL% (mlhgy’l

 

case,

.9.

38

2. 9. 2. c. Dead Load Case: Similarly, in this

 

Right hand side of Eq. (2. 31a)
Right hand side of Eq. (2. 31b) '

(Z. 37a-e)
Right hand side of Eq. (2. 31¢) + $mitcddflv4’n'z)

Right hand side of Eq. (2. 35a)

Right hand side of Eq. (2. 3513)" %9LEL+K<W%) ‘2

3. Shells Supported by Rollers on Curved Edges,

 

and Rectanfllar Beams on the Logitudinal Edges:

Along the curved edges, the boundary conditions are given by

Eqs. ”(2. 29a-d) . Along the longitudinal edges, however, if the

edge beams are assumed to have zero rigidity against lateral bending

and twisting as indicated in Section 2.3.3, the boundary conditions

are as follows:

{Tiqgti- =
{M's-t;

u _
8

J55 =

in which

Mrs-it

us’ uB

(2.38a-d)

7%

lateral thrust acting on shell edge 4 at- Yi=ili ;

longitudinal displacement of shell at 72 2.132: 9

and of the edge beam, respectively;

39

25;, IEB = vertical displacement of the shell at r] siJi,
and of the edge beam, respectively.

Eq. (2. 38a) can be expressed in terms of the following boundary

.forces:

{T}n=igi={fif unfit-+3? Mii‘hdiéo ; . 7 (2.39)

For thin and shallow shells, Nygcos? >> .51, M%.

Eq. (2. 39) is then simplified to the following:

{Th-xii); = {Nim%3'l=ii E O ’. . 'I (2.40)

In terms of dimensionless displacements:

i‘fit-mi‘iqgt%='i’m%i#sflh+ w)“ iii—fiw—T’Vflrij': (2°41) '

2. 9. 3. a. Dead Load Case: For shells acted

upon by dead load, the deflection functions {5, "it, and 3" expressed

P
by Eqs. (2.17) and (2. 28a-b), respectively, do not satisfy all the
boundary condition equations (2. 29a-d) and .(2. 38a-d). Again it will
be impossible to find a 'dh and a Vh so that all these boundary condi-
tions are rigorously fulfilled simultaneously. However, it can be
shown that Eqs.(2. 17) and (2. 28a-b) satisfy the boundary conditions

Eqs.(2. Z9a-b) identically. In addition, lih and 27h are chosen in

such‘a way that Eqs. (2. 29c) and (2. 41) are satisfied; i. e. ,

{Hawk = o and {mung} ._. o,

na‘z

40

A posmble Justification of such a choice is Similar to that stated

in the preVious section that is, both {Nxhg‘i‘k and {N¢Wfi}7-t%

are the dominant boundary forces

.When Eqs. (2 10a b) are substituted into Eqs. (2. 29c) and
(2 41), the following relations are obtained

04%;}

%{Mi”3 + flhig+2i(w}5)z§} fist-$0

€2.42a-b)
ID?
mix-4,, = $053th n issues ’Lif
qs. (2.42a-b) imply that
-— a 5-(’ T}
{M ”IVS," 1: i fi3+25 W}; $1.1. . .
- (2.43a-b)

_ . _, _. -— z

{W»iiq.if'i%m*i~”*“'§ze (M5?) L1 ,,
Substituting the values of i-av u

, —p,and VP from Eqs. (2. 17) and

(2 28a b) mtO Eqs (2. 43a-b), the following'relations are obtained
{514,3}: :2,” (“Agra :1)“, i + LUV-24M” (1H MA; t)wo41qwung.+
+&H “A6 Mn Aim/n1) + HZ(NA3-§E§)(I +mzmp}

(2. 4435b)
'lr-i M: i“ [Wang‘- G (¢+ b,¢n)canifi mm;—

- Gr [Bfidmét -.W$—)+% W%](l+m2n3
+ 5H [(fibgi’ neg-£5) mg: +

+ mm 15.5%)MW95m2Wi] +

iii [“57 -§%¢K](I+wuzwg)}
in which Bp=%DL 2

41

In order to have uh and 3,, satisfy the biharmonic equations, the
trigonometric expressions in g and r] are replaced by approxi-
mating polynomials as follows:

(1+coszhv) c: Z—(tl-LOSthJZr]

(mhnmnyw I—ZQ

(WMWW-e amid?

('1+m2rlq)’-= 2(I—-2q)

for all values of )2 defined in the interval 057 s 5— , and

coSrrg a ( I — 445‘)

(1+ mung) 0- 2(f—zs)

an. Zn‘g 0: “—40

for all values of 3 defined in the interval 0 sg .4. 15.
Then, Eh and vb are found to be as follows:

(11,) ={ot(rA,-§’—;-)[z —<u —mtdzi]i +
+GH[n(As-“{—)(I-2q) + A,2n (Mg—99% +
+H2(nA3-34:—l§)(2)(1-Z|7)§} (2.45a-b)

(47,.) = {-13, Movie)? 6: (4>+ brhxmigxn—ii‘flz -

GZ[54¢K(wo_iL~j “5) +i— Arm/$5)“ l- (297+
+GH [(4% P25 + “ Bea—ii) W}? + (i459 + “Blo'zswz'w '43)]?
+H2 [na-fiiyzw-zm}
2.10. Dimensionless Form 'Of the Poiflent'iahz:Ener‘gy
In terms of the dimensionless coordinates and parameters used

in the foregoing, the equations for the total potential energy of the

42

shell system, Eqs. (2. 8), (Z. 3), (2. 6) and (Z. 7) are reduced

to the following:

17 = (Vs + figflv, 4' 2555-) (2.46)
in which
U = Tat—T U

Vs: 4) RU [SR/3+Zii(w}i)+fi4-5q+w+—1W(w}f)]
O —2[$u,;+2§(517,3 HEB}? +m+fi7€flmd ] ‘
[tawsrl‘bf-z'ri
{[5 MggI'fi‘wn'l] ’2?thsz intend} ”>44?
v.,= zS<{s<a‘,s>e+§-;(wt\r +55: [WbsQLw‘fl‘ (4772i) ““5953“

4.
y—%z{[we cao?‘ damn] 2H),, }>d€

2.11. Derivation of the Algebraic Equations

For the various loading and support conditions, '5 and V can
be obtained by substituting Tip, 3p, Eh,and vh derived in the preceding
sections. The resulting expressions, Ti' and \7, together with '55
expressed by Eq. (2. 17) or Eq. (2. 31a), are then substituted into

the total potential energy expression ‘1'] of Eq. (2. 46). The latter

quantity is then made stationary with respect to the undetermined

43

parametersG and H, i. e. ,

a (2. 48a-b)
r0
7F? = 0

After carrying out the integration, Eqs. (2. 48a-b) are transformed
into two simultaneous non-linear algebraic equations of G and H
which have the following form:
c3OG3+c21G2H+c12GH2~+c03H3+czoGz+anH+cosz+c10G+c01H+c00= o
£12.49a-b)

d3oG3+dZIGzH+d12GH2 +d0 3H3+dzoGz+duGH+dozH2 +d10G+d01H+d00= o
The coefficients Cij and dij: related to the variables GiHj, are
very complicated expressions, containing the shell and loading
parameters, and are listed in Appendix II.

It may be pointed out that some of the coefficients of the
simultaneous non-linear algebraic Eqs. (2. 49a-b) are related.
This is due to the fact that the equations are derived from making
the total potential of the shell system stationary. Since the strain
is a quadratic of displacements, the total potential of the shell
system a must be a 4th degree polynomial of displacement parameters
C and H in the following form:
fie, H): k4OG4+k3lG3H+kzszHz+k13GH3+kO4H4+

+ .k3OG3+kZIGZH+k12GHZ+k03H3+kzoGz+kuGH+k02H2+ (2. 50)

+k10Cv+k01H+koo

44

in which ki' is the coefficient for term GiHj.

J

By makin ’g the. total potential:energylstationary:

gig; e

9.9..
2H

4k4oo3+3k3lozH+2kzchZ +k13H 3+
+3k30G2+2k21CrH+k12H2+

+2k20G+an+k10 = O (2. Sla-b)
k31G3+2kzzGZH+3k13GH2+4kO4H3+
+kZIGZ+2k12GH+3k0 3142+

+kuG+2kozH+k01 = 0

When the coefficients of Eqs. (2. Sla-b) are compared with Eq5.(2. 49a-b)

it becomes obvious that some of the coefficients in Eqs. (2. 49a-b)

are related in the following manner: :

c:21 ‘

c12 -

3C03

C311 '

ZCOZ

C01 -

3d30

d21

(112 (2. 52a-f)
zdzo

dn

(110

Even though the realization of these relations, Eq$.(2. SZa-f),

has no great theoretical value, yet, in practice, it is of some

importance. Since the generation of the coefficients Cij and dij into

a form adaptable to the computer involves very tedious computations,

recognition of Eqs.(2.52a~f) will save labor or serve as a check

when the coefficients are derived independently. This becomes

45

even more important when a larger number of undetermined
parameters are used.
Making use of the relations shown in Eq$.(2. 52a-f),
Eq. (2. 49b) may be written as:
'(-‘3-’)CZIG3+c1_zG2H+3c03GHZ+dO3H3+(‘ZL)CHGZ+2cozGH+dOZH2+
+c01G+d01H+dOO = 0 (Z. 53)

2. 12. Solution of the Non-linear, Simultaneous Algebraic Equations

 

2.12. 1. General: For a given set of load and geometric

parameters, the coefficients Ci'

J and dij are simply constants.

The resulting set of cubic equations are then solved by the Newton-
Raphson iteration scheme programmed for the CDC 3600 computer
at the Michigan State Computer center. In passing, it may be
mentioned that the Gauss-Seidel method was tried but it failed

to converge in some cases. The computer generates the numerical
values of the coefficients as well as solves the equations. The
method of solution, is described in the following section:

2.12.. 2. Newton-Raphson Iteration: A normal system of

 

algebraic equations,

f1(x1, x2, X3, . . . . Xi) = O (2. 54)

Ha

'i(x1, x2, X3, xi) = 0
can be expressed in matrix notation as

F(x) : o (2.55)

46

The solution of Eq. (2. 55) by the Newton-Raphson iteration pro-
cedure is as follows:

xn+1 = x,1 - (J(Xn))'1F(Xn) (2.56)
in which J(Xn) is the Jacobian matrix of the system of Eqs. (2. 55)
evaluated at Xn. The subscript n indicates the number of iterations.
As pointed out by Henrici (7), (1962) and Zaguskin (23), (1961),
the above procedure converges to the real solutions provided

that X the initially guessed solutions, are sufficiently close to

o,
the true solution and that J(Xn) is non-singular. Thus the
solutiOns of‘Eqs. (2. 49a) sand-(2. 53), when eicpressed in
the form of the iterative Eq. (2. 56), become:
2
(C3)?“ = (Carril—(DH x CC — CH x DD)?/ (CG if DH —-CH )nr:
(2.57a-b)
(Hffl‘lil = (H)‘;1-(—CH x cc + CG x mung/(CG x DH - CH2)“;
The subscript 11 indicates the number of iterations while the
superscript m indicates the number of load increments applied, and
we)“; = { Left-hand side of Eq. (2.4%))": (2. 58a-b)

mo)“: = { Left-hand side of Eq. (2. 53)}“111

m

with G and H evaluatedat or: and H2, while we)“;1 , (CH): , (DG) n

and (DH): are partial derivatives of (CC): and (DD): with respect

to or: and Hmn. They are given as follows:

47

2
(C0)“; =—-

m _ — 2 2 m
76 {(CC) n} - (3630C +2C21GH+C12H +ZC20G+C11H+C10) n

(2. 59a-c)
m m 2 2 m
(DG) n =5%‘{(DD) n} '-' (CZIG +2C12GH+3CO3H +C11C1+2C02H+C01) n
(DH): =5%{$DD)’:} = (c12G2+6c03GH+3d03H2+2cOZG+ZdOZH+d01f§

=ii<cc>ml= W} =

For a given shell system, under the first small increment
of load, the shell behavior will be essentially linear, therefore,
the solution of the linearized equations, GI]; and H]: , obtained by
setting all the non-linear terms of Eqs. (2. 49a) and (2. 53) equal
to zero, will be very close to G1 and H1 , the real solutions of
Eqs. (2. 49a) and (2. 53). Therefore the linearized solutions, GIL

and Hi are .usedas a~ first approximation applied in the iteration 1

Scheme outlined by" Eqs'. ('2. 57’a-b):

1. e. , 1 1
C3'1 = GL
H1 — HL ’ ’ .

After substituting Eq$.(2. 60a-b) into EqS.(Z. 57a-b) and starting the

iteration, new values (121 and H1 are obtained, which are in turn

2
substituted back into Eqs.(2. 57a-b) to obtain G13 and H13. This

iterative process will continue until G1 +1 and Hh+l reach the value
n

A A I ..
of (:11 and H1, such that '6— 6:141 x 10'9 and £31 - H341 x10 9,

 

A
simultaneously. Cl1 and fi1 are considered to be the solutions to

- Eqs. (2. 49a) and (2. 53) with load increment equal to one unit. When

48

the next increment of load is added, the initial guessed solutions

A
G: and H: will be extrapolated linearly from 61 , I-Il and Go , f-Io.

/\ A
The latter quantities, Go , H0 are equal to zero as they correspond

to the case of no load on the shell. It can be shown that.)

A A A

of) = 2(<31 - 6°) + (3° (2.61a-b)
A A A

H: = 2(H1- HQ) +H°

In general, when the “(m+l)th increment of lOad is added to the shell
system, the initial guessed solutions Gr?“ and Hem+1 can be

expressed as

A A _ A ..
a?“ = 2mm - Gm 1) + Gm 1 (2. 62a-b)
A
Han“ = 203‘“ - Hm'l) + fim'l

and from the existence of a continuous solution of the problem,
G?“ and H?“ will be very close to the real solutions, Gm+1
and Hm+1 , provided that the load increment chosen is sufficiently
small. The iterative procedure of Newton-Raphson therefore
converges rapidly to em“ and fimH. However, if the load increment
used happens to be not small enough so that the iterative procedure
diverges, it is halved and the guessed solutions will also be reduced

accordingly. The halving process will continue until a solution is

obtained.

III. NUMERICAL RESULTS

3.1. Effect. of Types of Load

 

As pointed out earlier, because of the inherent difficulty in
dealing with loading that has a component in the circumferential
direction, all the research work done on the large deflection be-
havior of cylindrical shell panels has been concerned with radial
pressure only. However, for shell structures in civil engineering,
such as cylindrical roof shells, the dead load and live load are the
more common types of loading considered in design. It is
therefore of interest to compare the behavior of a cylindrical
shell with different types of loading.

Fig. 3.1. presents the load-deflection behavior of shells
supported by rollers on all the edges and subjected to the three
types of loading: radial load, live load and dead load. All
shells have ¢k = 0. 632 and S = O. 791. Three sets of curves are
shown for Z = 100, 125, and 150. These curves are plotted with
the load parameters qLL’ qRL or qDL as the ordinate, and the
dimensionless deflectionlgfl as the abscissa, in which we
is the deflection at the center of the shell.

It can be seen from the figure that for small deflections,

(say Eto- 4 O. 5, ) the load-deflection behavior is essentially linear.

49

50

With increasing load, however, the characteristics of non-linear
behavior become evident. The slopes of the curves decrease over
a large range of deflection -- indicating a loss of stiffness. After
that, within the range of deflections considered, the stiffness may
continue to decrease or begin to increase, depending upon the
values of parameters used. For example, in the case of Z = 150,
the stiffness of the shell continues to decrease. In fact, these
curves all have a large "flat" portion. (For convenience of
discussion, the loading corresponding to this flat portion of the
curve will be referred to as the "buckling load." ) However,

for Z = 100, the curves begin to regain stiffness after some initial
loss.

From Fig. 3.1. it is seen that for Z = 150, the buckling load
for radial pressure is about 5% higher than that for live load and
10% higher than that for dead load. This may be explained
qualitatively by noting that the radial component of load tends to
keep the shell circular in shape while the tangential component tends
to flatten the shell, and therefore contributes more to instability.
A simple analysis shows that, by integrating the load functions of
the three types of loading over a half section of the shell, the total
resultant force in the radial direction for the three loading conditions
are nearly the same. However, the dead load has a larger re-
sultant of tangential component than that of the live load while the

radial load has a zero tangential component.

51

It can be seen also from Fig. 3. 1. that the differences in
the stiffness of the shell subjected to different types of loads
decrease as the value of Z is increased. This may be explained
by noting that Z is essentially a curvature parameter, and the
difference in the three types of loading is essentially due to the
curvature of the shell surface. When Z becomes very large, the
shell approaches a flat plate and the three types of loading
become identically the same.

It might be pointed out also, that the effectsof different
types of load on the large deflection behavior are not great, because
the shells considered have relatively small 55k. .When ¢k is large,

the effects might be more pronounced than those shown in Fig. 3.1.

3. 2. Effect of Shell Geometry

 

The following discussion is concerned with the effects of the
geometric parameters on the behavior of shells supported by flexible
beams on the longitudinal edges, and by rollers on the curved edges.
Only the dead load case is considered. As before, the behavior of
the shells is described in terms of load-deflection curves.

3. 2.1. Effect of Z (radius/thickness ratio): From the figure

 

presented in the preceding section, it can be seen that the shell is
stiffer for smaller values of Z. Additional data on the influence of
Z are presented in Fig. 3. 2. in which the shells are supported by

edge beams (V = 10, W = 0. 025). Five values of Z, ranging from

52

75 to 175 are considered. As before, it is seen that the buckling load
is lower for higher values of Z. This general result agrees with
the physical intuition that the thinner the shell (or flatter the cur-
vature), the smaller would be the buckling load.

In order to better relate the results to practical cases, the
shell considered in. Fig. 3. 2. may be interpreted as having the
following dimensions:

R = 60'-0“, L = 76'-0", a = 41" and b =18".

For the case of Z = 175, t is equal to 4.1", and E = 3 x106psi;

the buckling load is then equal to 340 psf. If Z =100, and t is equal
to 7. 2", then the buckling load is 880 psf. Whereas,if Z = 75 so
that t becomes 9. 6", the shell becomes very stiff, and does not
buckle even when the load has been increased to four times the
buckling load for t = 4. 1". This nonlinear phenomenon is different
from the linear relationship between the buckling load and Z im-

plied in the ASCE Manual No. 31.

 

3. 2. 2. Effect ogk : Obviously, the size of the opening angle
of a shell, 515k, influences the buckling strength of the shell.
Fig. 3. 3. presents the effect of 55k on the buckling strength of shells
having the following properties:

S = 0. 791, V =10, W = 0. 025, Z = 125 and 150.
Three values of 513k are considered: 0. 5, 0. 632 and 0. 8. It is seen

that the buckling strength of shells increases with an increase of 95k.

53

(For 75k = 0. 8, the shell did not buckle at all). For the case Z = 150,
the shell could be interpreted to be one having L = 63'-4“, R = 50'-0'.',.
t = 4", a = 40", b =15" and E = 3 x106psi. For this shell,if 55k = 0. 5
(roughly 29°), the buckling dead load is 200 psf. If 95k = 0. 632
(roughly 36°), the buckling load becomes 430 psf. When 95k = 0. 8
(roughly 46°), the shell becomes so stiff that even when. pDL is

equal to l, 250 psf. it is still stable.

3. 2. 3. Effect of S :(radius/length ratio): The effect of S on
shell behavior is presented in Fig. 3. 4. in which the shells con-
sidered have the following properties:

95k = 0. 632, Z =125, W = 0. 025, V =10,
and 5 takes on five different values. It is seen that the buckling
load increases with an increase of the value of S.

If t is again assumed to be 4", then the shells considered in
Fig. 3. 4. correspond to those having the following dimensions:

$51. = 0. 632. R = 41'-8", a = 40", b = 15".

If L is equal to 88'-0" (corresponds to 5 0.475) and E = 3 x106psi, the
buckling load is pDL. = 70ps‘f. 'If L is decreased to 66'-0" (corresponds
to S = 0. 632), the buckling dead load is 300 psf. If L is decreased

to 53'-0 (corresponding to S = 0. 791), the buckling dead load becomes
580 psf. These results simply indicate that if all other parameters

are held constant, a decrease in the span length of the shell would

result in an increase of the buckling strength.

54

3. 2. 4. Effect of Edge Beams: The role of the edge beams

 

is represented by the depth parameter V (= f") and the width parameter
W(='E). The influence of V and W are shown in Fig. 3. 5. and Fig. 3. 6. ,
respectively. In these two figures, the following shells are considered:
ylk = 0. 632, S = 0.791, Z = 100, 125,and 150.
In Fig. 3. 5. , W is held constant at 0. 025 and V takes on
the values of: 5, 10, 15 and 20. It is seen that for a given Z,the
initial deflection is essentially independent of V. However, as the
deflection increases to a certain value (depending on the value of Z),
the influence of V becomes more conspicuous; it is more pro-
nounced for smaller values of Z. Furthermore, as V increases
in value, the shell becomes stiffer.
In Fig. 3. 6. , V is held constant at 10, and three values of
W are assumed: O. 0125, 0. 025 and 0. 05. The behavior pattern
is similar to that just discussed for Fig. 3. 5. That is, the influence
of W becomes apparent only after the deflection assumes a substantial
magnitude. This influence is also larger for smaller values of Z.
This case may be interpreted as indicating that the influence of the
edge beam is greater for thicker shells. This behavior might be
explained by the fact that for thinner shells, the stiffness of the edge
beam is probably not called on to play its part, even when the shell

is undergoing large deflections.

55

3. 3. Comparison of Results

The method of analysis used in this study is an approximate
one and involves a number of assumptions. It is, therefore,
natural to question the accuracy of the results obtained. In general,
the assessment of the accuracy ‘of an approximate method of this
type is to compare results with known exact solutions. As dis-
cussed in the Introduction, for the case of nonlinear behavior of
cylindrical shell panels, available solutions are extremely scarce;
besides, they are all approximate solutions of the Rayleigh-Ritz
type. In fact, so far as is known to the author, Ref. (11) contains
the only existing data that may be used for comparison in order
to give some indication of the accuracy of the results of this
study. Before presenting this comparison, however, a linear
problem will be examined.

Consider a concrete shell simply supported by edge beams
with the following dimensions: R = 33I—4“, L =111'-o", 55k = 30°,
t = 4", a = 60" and b = 8". The load is derived from a live load
of 25 psf. and the weight of the shell itself. The solutions of this
structure in terms of Nx and N56 at the mid-span of the shell for
different values of 9‘ are plotted in Fig. 3. 7. It might be pointed
out that in this case, the linear version of the solution (by

dropping out the non-linear terms of G and H in Eqs. (2. 49a) and

(2. 53)) is very close to the non-linear solutions. This is, of

56

course, to be expected since the deflections are small. The linear
response of the same shell has also been discussed in ASCE Manual
No. 31 (1) (page 60). For all practical purposes, the solutions
therein may be considered as exact7and they are also graphed in
Fig. 3. 7. It can be seen that results corresponding to the present
analysis differ from the ASCE Manual solution only by about 1% at
the crown. However, the agreement is not as good for points
closer to the edge of the shell. Nevertheless, in view of the
gross approximation used in the present analysis, the differences
indicated in Fig. 3. 7. should not be considered as being large.
For a comparison involving a non-linear problem, consider
a shell loaded radially and supported by rollers on all its edges.
Limiting to 75k 4 0. 2, and ka 1' 0. 5, the load-deflection curves
for different values of deZ are calculated and presented as solid
curves in Fig. 3. 8. , in which the load parameter (qRLZ455k4)
is plotted againstlzl. Also, shown as dotted lines in Fig. 3. 8.
are the results obtained by Kornishin and Mushtari, (11) for the
same shell. As mentioned in the Introduction, the latter results
were obtained by applying the method of Bubnov and Galerkin to
the compatibility equation and the radial equilibrium equation. It
is seen that the solutions obtained by the two procedures seem to
differ appreciably. Depending on the value of kaZ, the buckling

loads corresponding to the present analysis are approximately

57

10% to 70% higher than those indicated by Ref. (11), the discrepancy
being smaller for smaller values of silk Z.

It should be noted that the results of Ref. (11) were obtained
employing six undetermined parameters in the assumed functions,
while in this study, only two undetermined parameters have been
used. Therefore, it is probably reasonable to assume that for
the problem considered the numerical results of Ref. (11) would
be more accurate. It may appear that the difference between the two
are substantial. It should be bOrne in rnind, however,‘that.the
procedure used herein is devised to handle more realistic problems
(particularly from the point of view of concrete shell structures)
to which the technique used in Ref. (11) cannot be applied. Further-
more, against the background of the present state of knowledge of
large deflection behavior of shells, as discussed in the Introduction
and later in the Conclusion, this difference may not be as signi—

ficant as it seems at first glance.

IV. SUMMARY AND CONCLUSION

4.1. Summary

A method has been developed to study analytically the non-
linear behavior of elastic thin cylindrical shells. The shells are
supported by rollers on all the edges or by rollers on the curved
edges and flexible beams on the longitudinal edges. Three types
of loading are considered: a uniform radial pressure, a uniform
live load, and a uniform dead load.

The method of analysis is based on a large deflection theory
of thin shells by including the quadratic terms (%r)2 and (210).?
in the strain tensor. The variational problem resulting from an
application of the principle of stationary potential energy, is
solved approximately by the method of Rayleigh-Ritz. A first
harmonic approximation with two undetermined parameters, is
chosen to represent the radial displacement function w. The longi-
tudinal and circumferential displacement functions u and v are
considered to consist of two parts: up, vp and uh, vh. The functions
uP and vp are chosen to be the particular solutions of the equation of
equilibrium in the longitudinal and circumferential directions,
respectively. The functions uh and vh are homogeneous solutions

to V411}; = V4Vh = 0, sothatthe sumsuzup+uhandv=v +vh

P

satisfy approximately the geometric and natural boundary conditions.

58

59

By applying these approximating functions to the Rayleigh-Ritz
procedure, a set of two simultaneous algebraic cubic equations
are obtained. Using a high speed digital computer, these equations
are solved by the iteration scheme of Newton-Raphson. For a
given shell and loading type, a load-deflection curve is obtained
from a series of solutions corresponding to a range of load
intensity. The curve is, in general, non—linear. It is indicated
that after a certain range of essentially linear behavior, the
stiffness of the shell decreases. Depending upon the values of the
parameters of the system, the shell may or may not buckle.
(Buckling is considered to have occurred if the shell undergoes
substantial displacement with little change in load magnitude. )

By a repeated application of the above procedure for different
values of shell parameters, a number of load deflection curves are
obtained. From these numerical results, the principal findings may
be summarized as follows:

Among the three loading conditions considered, the shell has
the lowest stiffness (or buckling load) for the dead load case. The
shells have lower stiffness or buckling loads for: smaller values
of the opening angle, 55k, smaller values of the radius to length
parameter, S, larger values of the radius to thickness parameter,

Z, and for smaller edge beams.

60

4. 2. Concl'udinLRemark s

 

In the past, the elastic stability of thin shells, treated either
as a linear eigen-value problem or a non-linear large deflection
problem, had been formulated in such a way that the boundary con-
ditions were assumed not to play an important role in the behavior
of the system. (2), (1947) and (9), (1941). Recently it was pointed
out that the degree of constraint offered by the boundary could
be a significant factors (15), (1961) and (19), (1962). This is further
demonstrated by the following comparison.

If one considers a shell loaded radially and having the following
parameters: S = O. 91, 55k = 0. 632, Z = 100, and the shell is simply
supported on the curved edges and clamped along the longitudinal
edges, the radial buckling load has been found by Sunakawa and

Uemura (20), (1960) to be:

__ 10, 800 sf -.
qRL — TL . However, if the

boundary conditions are changed to roller supports on all edges, the
buckling load reduced to qRL = 329%B—S—f (obtained by the procedure
used herein). Thus, it is noted that the different boundary conditions
lead to a difference in buckling load of 500%!

The aspect of boundary condition on shell buckling has not been
emphasized in the discussions of Stability of Roof Shells in the
ASCE Manual No. 31 (1). In fact, the manual stated that for a long
roof shell, Nx being the predominant force, the buckling characteristics

are analogous to those of a curved panel stiffened at the edges sub-

61

jected to axial compression; thus, the actual character of the
boundary supports never enters into consideration. Such an
assumption obviously over simplifies the problem, as it should be
clear from the preceding. That is, the buckling strength of a
roof shell depends significantly on the degree of restraint offered
by the supports.

Therefore, in the design of a roof shell, if the buckling problem
is to be investigated, the actual boundary conditions should be
duly taken into account. The method described in this thesis,
admittedly approximate, may be used for that purpose.

4. 3. Suggested Future Work

 

As a possible extension of the present work, it is natural to
consider the use of the present approach by including higher harmonics
, in the assumed displacement functions. However, it is emphasized
that the amount of labor involved in the analysis is immense.
Therefore, before making such an effort, it seems desirable to
conduct an experimental investigation of the problem. The results
of such an investigation may provide a more definite idea about the
accuracy of the present approach. Furthermore, observations
on the actual physical behavior may suggest a more intelligent
choice of the assumed deflection functions for the Rayleigh-Ritz

method.

10.

V. BIBLIOGRAPHY

American Society of Civil Engineers, ”Design of Cylindrical
Concrete Shell Roofs, " Manuals of Engineering Practice
No. 31, 1952.

Batdorf, S. B. , "A Simplified Method of Elastic-Stability
Analysis for Thin Cylindrical Shells, " NACA Rep. 874, 1947.

Chung, K. P. and Veletsos, A. S. , "A Study of Two Approxi-
mate Methods of Analyzing Cylindrical Shell Roofs, "
Structural Research Series No. 258, University of Illinois,
October 1962.

Donnell, L. H. , "A New Theory for the Buckling of Thin
Cylinders under Axial Compression and Bending, " Trans.
ASME, Vol. 56, p. 795, 1934.

Finkel'shtein, R. M. , "On a Problem in the Statics of Thin
Cylindrical Plates" (in Russian), Izv. Akad. Nauk SSSR. ,
Otd. Tekb. Nauk No. 5, pp. 130-140, May 1956.

Fung, Y. C. and Sechler, E. E. , "Instability of Thin Elastic
Shells, "Structural Mechanics, Proceedings of the First
Symposium on Naval Structural Mechanics, Pergamon Press,
New York, pp. 115-168, 1960.

Henrici, Peter, "Lecture Notes on Elementary Numerical
Analysis, " John Wiley and 'Sons, 1962.

Karakas, J. and Scalz, M. , "Test Load of Shell Fails at
Design Load, " Civil Engineering, March 1961.

Karman, Th. von, and Tsien, H. S. , "The Buckling of Thin
Cylindrical Shells under Axial Compression, " Journal of
the Aeronautical Sciences, Vol. 8, No. 8, p. 303, June 1941.

Koiter, W. T. , ”Buckling and Post-Buckling Behavior of a

Cylindrical Panel under Axial Compression, " Nat. Lucht Lab. ,
Amsterdam Rap. 5, p. 476, 1956.

62

11.

12.

13.

14.

l5.

l6.

l7.

18.

19.

20.

63

Kornishin, M. S. and Mushtari, Kh. M., "A Certain
Algorithm of the Solution of Non-Linear Problems of the
Theory of Shallow Shells, " (English Translation of Prikl.
Mat. Mekh.) 23, 1, Pergamon Press, New York, pp. 211-218,
1959.

Langhaar, H. L. , "General Theory of Buckling, " Applied~
Mechanics Reviews, Vol. 11, No. 11, Nov. 1958.

Langhaar, H. L. , "Energy Methods in Applied Mechanics, "
John Wiley and Sons, New York, 1962.

Mushtari, Kh. M. and Galimov, K. Z. , "Non-linear Theory
of Thin Elastic Shells, " (English Translation). Published
for the National Science Foundation and the National
Aeronautics and Space Administration by the Israel Program
for Scientific Translation. 1961.

Nachbar, W. and Hoff, N. J. , "On Edge Buckling of Axially-
Compressed, Circular Cylindrical Shells, " SUDAER No. 115
(NsG 93-60), Dept. Aero. Eng., Stanford University, Nov. 1961.

Nash, W. A. , “Recent Advances in the Buckling of Thin Shells, "
Applied Mechanics Review, Vol. 13, No. 3, March 1960.

Novozhilov, V. V. and Finkel'shtein, R. M., "On the Error
of Kirchhoffs' Hypothesis in the Theory of Shells, " Prikl.
Nat. Mekh. Vol. 7, No. 5, 1943.

Soderquist, A. B. T. , "Experimental Investigation of Stability

, and Po stbuckling Behavior of Stiffened Curved Plates, "

University of Toronto, Inst. AerOphys. , TN 41, 1960.

Stein, M. , ”The Effect on the Buckling of Perfect Cylinders
of Prebuckling Deformations and Stresses Induced By Edge
Support, ” NASA Technical Note D 1510, 1962.

Sunakawa, M. and Uemura, M. , "Symmetrical Buckling of
Cylindrical Shells under External Pressure, " Aeronautical
Research Institute, University of Tokyo, Report No. 356,
Mat 1960.

21.

22.

23.

64

Timoshenko, S. P. and Gere, J. M., "Theory of Elastic
Stability, ” McGraw Hill Book Company, 1961.

Timoshenko, S. and Woinowsky—Krieger S. , "Theory of
Plates and Shells, " 2nd ed. , McGraw Hill, New York, 1961.

Zaguskin, V. L. , "Handbook of Numerical Methods for the
Solution of Algebraic and Transcendental Equations, "
(English Translation), Pergamon Press, 1961.

65

190”: r
Jupporfcd

Ed7e Jean?

 

 

4‘s .
"I

(a) Shell System Considered

 

 

 

 

 

(b) Force System Considered (c) Edge Beam Cross Section

Fig. 2.1. Cylindrical Shells Supported by Flexible Beams on
Longitudinal Edges and by Rollers on the Curved Edges

Roflz f" .
3’ julojoorfea/

 

 

Fig. 2. 2. Cylindrical Shells Supported by Rollers on All Edges

67

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1000 I
7" ’
\OA
9.
X
2’» 300
531
L...)
O /
£2 / /
g / ‘2: ‘2....—-5 d I
I /
c: 600 ,II: ' ___ —— ’
I: / /,r:= ’ "' L/
A / ' If
D. //
0* / ____.,
H / __——-‘"" ""' ‘-
o / /’ .4: " —' — — — — - - -
3400 / ,/ """= \z=l50 ‘
0" I
g ————— ' Live Load
0* ,/ —— -— Radial Load
200 ,2/ —'————— Dead Load _1
¢k = 0. 632
S = 0. 791
0 . s
l. 0 Z. 0 3.
W
7°
Fig. 3.1. Effect of Types of Load (Roller Supported)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I 100
Z=|00
f? ————s ‘
” /
800 ’V
‘0; //
2
>4 600 Z=|25 .__i
N .
‘8
s / / Z=l50
'a‘ 400‘ / I
:3
.5 // 2:175
Q .
0" / // . ¢ = O 632
200 V / sk = 0.791
/ v = 10.0
w = 0.025
0 ,
1.0 2.0 3

Fig. 3. 2.

Effect of z (z :%)'

(Beam Supported)

69

100

 

800

 

 

600

 

 

 

400

 

 

 

 

 

qDL (in units of (432 x 106)-l)

200

 

 

 

 

 

 

 

 

 

 

 

Fig. 3. 3. Effect of dk. (Beam Supported)

70

1000

 

 

 

 

4-=: *
fl0349
d = 0.632
zkz 125
W = 0.025
800 V = 10.0 /

 

600

 

 

 

, 5:0.79l
/ fl

 

 

qDL (in units of (432 x 106) '1)

 

 

 

 

 

 

 

 

 

 

 

 

400 / /
/ 530.632
//
0
1 0 2.0

 

Fig. 3. 4. Effect of S (S =-BI:). (Beam Supported)

6) -1)

qDL (in units of (432 x 10

71

1000

 

 

 

 

 

800 4 ‘

 

 

 

 

 

 

 

 

0 v 420
600 _ ’\/ A {v='o
ab V=5J
1’4 o v :‘5
/ V52
400) / .607 7‘ V“ 5 V = IO

/ /

 

 

 

 

 

 

 

 

/
dk = 0.632
S ==07m
W = 0.025
0 I
o 1.0 “’0 z 0

Fig. 3. 5. Effect of V (V =-:’—). (Beam Supported)

 

72

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1000
’00?)
‘" W: 0025;
//
r 800 A I w: 0.0I25/—-‘
‘3.
X
S 600
I",
“5
_ i3
'5'
5 400 l j
V s \50 w=o.0125
A /
o .
0" /
200 /
dk = 0. 632
s :0. 791
v = 10.0
0 .
0 l 0 2.0 3.

 

Fig. 3. 6. Effect of W (W 2% ). (Beam Supported)

Force kip/ft

Compression

[‘

-——>~

 

4—--

Force Lip/ft

‘lension

”n

73

 

-50

 

\--42-1\\~ \

 

/ {NAM

\

 

 

I

to
g
/'

 

 

+37< -31 8 \
4_- __ ._ ‘- ~'— ".
_ 3.2 T N

\

 

 

 

0 "'7 /l —('4 A
{Ni}, O \

 

 

 

 

 

+1“

_ ..._. __ — Results of ASCE Manual No. 31

-F20——

 

Results of this report

1 l

 

 

 

 

 

+3
0° 10° , 20
(I? in Degrees

Fig. 3. 7 Comparison of NX and N36 (Beam supported shell)

74

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

70
Qtl‘bo
60
50
40 /
/ / / 4 ‘ A¢£-2_E__4.—O—J/’
/ ‘_______...
‘/ ‘V'
’ 1. ob: /
J/
— -— -— Results of Refer. 11
Results of This Report
561,5 = 0. 5
1 75k < 0. 2
2.0 3 0
2’2
1’.

Fig. 3. 8. Comparison of Results of This Report with Reference 11
(Roller Supported)

APPENDIX I

COMPARISON OF TWO CHOICES OF DISPLACEMENT FUNCTIONS

This appendix gives the comparison of the solutions obtained by
a set of approximating displacement functions which do not satisfy the
natural boundary conditions, and by those which satisfy the natural '.
boundary conditions approximately. ’ The comparison is shown in
Fig. A. l. in the form of load-deflection curves for shells roller

supported)loaded radially.

75

76

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1009 e
/ [f k/ /
/ // /
800 ..+_ ,‘Q __ O__. -/ r/
I:

.r N ' /

I N 0

" ‘3
‘03 1’4\ / 15125

x /

E l

“5 z=n50

a)

+3

:5 400 /

a

VA — — — u, v not satisfying boundary

m conditions
0"
200 u, v approximately satisfying
boundary conditions '
95k = 0.632
S =' 0. 791
0 .
l. 0 Z. 0 3. 0
2‘2
1’.

Fig. A. 1. Comparison of Load-Deflection Relations for Two
Choices of Displacement Functions (Roller Supported)

APPENDIX II.

COEFFICIENTS OF EQUATIONS (2. 49a-b)

A. 2. 1. General

The coefficients of Eqs. (2. 4.9a-b) are given below in terms
of the Fortran language (see, for example, ‘ ‘McCracken7D. D. ,
"A Guide to Fortran Programming," John Wiley and Sons’ 1961). The
definitions of the Fortran variables used are given in Section A. 2. 2. ,
and then followed by the presentation in Section A. 2. 3. of the
coefficients Cij and dij‘ It is noted that the materials presented
subsequently are direct printouts from the original computer program.
This is done in order to avoid possible errors in transcribing these

lengthy expressions, as well as for convenience of reproduction.

77

000000000 ,0

0000

78

A2.2 %**** LIST OF FORTRANVARlAsLEs. *s***

*iPARAMETERS OF SHELL0**

stR-RADlUs TO LENGTH RATIO. 5.

21 sRADlus TO THICKNESS RATIO. 2.

P :OPENING ANGLE.

v IEDGE BEAM DEPTH TO THICKNESS RATlO.

WBARssDGE BEAM wlDTH To RADIUS RATIO.

U tpOISSONoS RATlo

DLs LOAD INTENSITY OF RADIAL RRESSURE. LIVE LOAD. OR DEAD'LOAD.
QR..leMENSIONLEss LOAD PARAMETER. DLis

E svouNcs moouLus

ssFORTRAN VARIBLES USED IN COMPUTER RROGRANss

DDL sLOAD INCREMENT.

DOADiNaLOAD INCREMENT COUNTER. THAT ls THE NUMBER OF LOAD lNCREMENT

ACCUMULATEDO

ORaDL/E .

2812.*P*P*21

Y310/Zl/p

X=XBAR*P

WBV*V*V*WBAR

Fanpl/P

F3=PI§X

F4=Plip

FSSP/x

F6-Pl/x

Flchlipl/P/P

F13aPl*PI*X*X

F143Pl§PI§PiP

F23=PI*PI*X*PI*X*X

F31‘005/(F2‘100)**2

F3280.5/(F2+1o0)**2

F33=0.5/<F12-1.ol ‘

Al=((P*P -U*s13 )ssa *X)/(P*P +F13 )sse

A23¢<l.O-U*X*X 1*Pix1/(PI*(1.0+X*X 34*2)

A38(Pt**2*x**2-U*P**2)/(8.0*P1*X1

A48((X**2~U)*pI)/(BoO*X)

A5a<f¢<4.o*Fla )+¢P1*PI+P*P.))**2+4.0*F14 )*F3 *((2
.0*F13 +PI*PI+P*P )**2+2.0*F13 *(2.0*F13e
0*(PI*PI+P*P )))-8oO*F3*F4*F4 *(t4.0*F13 , +PI*PI+P*P )*

(2.0*(2.0+U)*F13 ‘ +pip +PI*P1)’)/(((4.0*Fl3 +PI*PI
+P*P )**2+4.0*F14 )**2~(4.0*F4 *(4.0*F13 +Pl*Pl+
P*P.))**2)

79
A6=( (F3 *(4.0+F4 *(4.0*F13 +PI*PI+P*P 111*! (2.0*F13
+PX*P{+P*P 1**2+2.0*F13 *(2.0*F13 -U*(P*P ~91
*91111-2.0*F3*F4 ~ *((4oO*F13 +PI*PI+P*P )**2+4.0*F14
)*(2.0*¢2c0+U1*F13 +PI*PI+P*P 11/(((4.0*F13
+PI*PI+P*P 1**2+4.0*F14 )**2-(4.0*F4 *(4.0*F13 +
P1*PI+P*P 11**21
A78Pl4X/800,
A20=a.o*(A3*F3-o.25*F131
Ban-((P*P +(2.0+U1*F13 1#R*R )/(P*P +F13 1**2
BSs-ttlo0+(2o O+U1*X*X 1*P/(PI*(1oO+X*X 1**211
848(Pip ~U*F13 1/(8.0*P1 T
BSatl(RI*R1+R*R 1**2+4.0*F14 1*(P*P *(3.o* PI*P1+P*P )~U*(
#13 - )*(PI*PI+3.0*P*P 11*(91/2o01-(2.0*P1*F14 *(PI*P1+P* . i
P 11*(P1*P1+3.0*P*P -U*X*X *(3.0*PI*P1+P*P 111/(((P1*P1+P*P
1**2+4.0*F14 1**2~(4.0*F4 *(PI*PI+P*P 11**21

86¢(I(PI*PI+P*P )**2+400*PI*PI*P*P 1*(PI*PI*(PI*PI+3.0*P*P 1'U*F13 i
*4300*P1*91+P*P ))*005-(200*PI*PI*(PI*PI+P*P 11*iP*P

*(3.0*PI*PI+P*P )—U*F13 *(PI*PI+3cO*P*P 11)*P/(((P1*PI+P*P ,
1**2+4.0*F14 . , 1**a~(4.0ss4 *(PI*PI+P*P ))**21 . i
87: Pl*(1.0~U*x*x s/s.o
BBHP/BOO
39=(((4.0*F13 +RI*RI+R*R 1**2+4.0*F14 1*(12.0*F13
+P*P 1**2+P*P *(4.0*F13 +3oO*Pl*PI1-U*F13 *
(Pl*P1-P*P 11*PI/2.o-(4.0*F4 *(4.0*F13 +PI*PI+P*P 11*
(<2.0*F13 +Pl*PI1**2+PI*PI*(4.0*F13 +3.0*P*P 1+
U*F13 *(Pl*PI-P*P 11*P/2.01/(((4.0¥F13 +Plibl+b
*P 1**2+4.0*F14 )**2~(4.0*F4 *(4.0*F13 +R1*R1+
P*P 11**21
810=((¢4o0*F13 +PI*PI+P*P 1**2+4.0*F14 ;)*(<2.0*F13
+PI*PI1**2+P1*Pl*(4o0*F13 +3.0*P*P 1+U*F13
*(PI*PI~P*P )1*P/2¢0~(4.0*F4 *(4.0*F13 +PI*PI+P*P ))*
((2.0*F13 +P*P 1**2+P*P *(4.0*F13 ' +3.0*PI*PI1~U*
F13 *(PI*PI~P*P 1)*Pl/2.01/((<4so#F13 +PI*P1+P*P
1**2+4.0*Fl4 1**2~(4.0*F4 *(4.0*F13 +PI*PI+P*P
11*R21
Bll‘pI/BOO
8208000

Bait-(82+1.01*C1*Cl
822:2.*(-.25*R1+s71*F2#c1

c1-c05F<R/2.01

ca-COSF(R1

CBaCOSFtp/2.01/(F12 «1.0)

C48CO$F(P/200)/(1F12 “1001*( 900*F12 “1001)
CSBCO$F(P)/((Pl/P1**2-4.O1

C63CO$F(105*P)/(F12 “900)

C108400*F31*F32*Cl

80

01a SINF(P/2.01
D2:SINF(R1
63:51NF(1.5*R1
D43$INF(200*pI
D5251NF¢R>/¢F12 f.01
06-51NF(R/2.01/< 4.0%?12 $1.01
DIO=F31*DI

011-F32*DI

0128F33*01
DXl-2.0*A2/R/F12
oxeaAa/R/Rla/X/x
oxaasa/R/RlE/x
0x4283¥X/F1?
staAZ/Fla
onaa.0¥sa/Fla/x
Dx7=1.980/F2/R12/x
E103100“C2

E11=DZ~P*C2 ~
ale-(F12+l.0)*c10-R*Dl
E13=F2*Dla-F33
ElAsF2*F33~Dlz
E15=R1*012~2.0*F2*c10
EI6‘DI“O§5*p*C1
El7‘IoO‘Cl

A203 ***** COEFFICIENTS 0F ALGEBRAIC EQUATIONS. *****

CALCULATION OF C30

C30-CA30+CB3O

1N WHICH f

CA308tA3*A3*PI*FS*P1+2.O*A3*A7*PI*F3 *DZ+A7*A7*PI*F3 *(P+D4*0o5)
*OcS-(A3*PI*F13 *(P+D£11*O.25~(A7*PI*F13 *(R*O.5+02+DA#O.
2511*Oo125+(A7*A7*P*F5*(P~D4*.S11*Oc25+(A7*BB*F4 *(P-Dé*o.5)1*
Oo5~IA7*F4*P *(P~04*0.5)1*0o0625+(88*88*PI*F3 *(P-D4*0.511*0
.25-(ss*F3*R4 *(P-D4*0.5)1*0.125+(B4*84*P*P /x*(R+D4*0.sh)+ss
*BB*P*P /x*(P+Da*0o51*0.5+B4*P**3/X*(DZ-P*Oo5-D4*O.251*O.5+88*P
sss/x*0.25 *(DZ-P*0.5-04*0.251-A7*PI*F13 ‘ '*(P*O.5+D£+D4*Oo251
*0.125+3.0*P1*F23 /32.0*<0.75*R+Da+04*.1251+F14 sxs
(P-D4*O.51/64.0-A7*F4*P *(P~D4*Oo51*060625

GJQIDUlbIJhJH

u. -3 'K‘I.!';

81

C83080c09375*P**4/X* (0.75*P~DZ+04*O.1251+

1 w/v/v *(2.0*A3*A3*P1*F3 +4.O*A3*A7*P1*F3 *C2+2.0*A7*A7*PI*F3

2 *C2*C2-A3*PI*F13 *Cl*C1-A7*PI*F13 *C1*C1*C2+3.0*PI*F23
3. /8.0*C1**41+w*Y/V*(4oO*A3*BB*PI*Fl3 *01*DI*C1+8.0*A7*BB*PI
4 *F13 *DI*DI*C1*C2«2.0*BB*PI*F23 *DI*D1*Cl**3+4.0*A3*88*P1*
5 F13 *Dl*Dl*Cl1+W*Y*Y *(32.0*BB*BB*PI*F23 23.O*01**4*c1*c11

CALCULATION OF C21
cal=CA21+cs21+ccal+CD21+CE21
IN WHICH
CA21=3.0*A3*A5*PI*FZ*F3 *C3+3oO*A3*A6*PI*F3 *C3+1.5*A5*A7*PI*F2*
F3*(C3+C6)+1.5*A6*A7*PI*F3~ *l3.0*C6-C31-A3*PI*F2*F13 1*C38A5*
PI*F2*F13 *Oc0625*(3o0*C3+C61-A6*PI*F13 *0.0625*(C3+3.O*C6
1~A7§PI*F2*F13 40.5 *(C3+C61+O.375*A5*A7*F4 /x#(C3+3.0¥c61+O.3
7S*A5*A7*FA /x*(c3~C6)~0.375*A6*A7*RI*R1/X*IC3-C61sO.375*Ao#A7
*P*P /x*(C3+3. O*C61+88*(A5-A6*F2 1*RI*RI*O.125*(C3~CO1+ss*(As*
F2 ~Ao)*F4 *0.125*(C3+3.0*C61+A7*89*F4 #(C3+3. 0*C61
+A7*slo*RI*PI*Ic3~col-IA5*F2 ~A61*0c03125*F4*P , *(C3+3. 0*C6)“(
A5-A6*PI/P1*0.03125*PI*F4 *(Ca-C61-A7*PI*F4 *O.25*tc3uce1
C8218~A7*PI*F4 *0.25*(C3+3.0*C61+1.5*BB*B9*PI*F3 *(C3+3.0*C61+
1 1o5*BB*BlO*PI*F3*F2 *(CBdC61+(A5*FZ ~A60*BB*F4 *0.25*(C3+3.0*
2 C61+(A5-A6*F2 1*BB*PI*PI*0c25*(C3~C61+A7*89*F4 *0.5*(C3+3.0*C61
3 +A7*BIO*PI**2*O.5*(C3~C6)-(2.0*88+Bio1*0o125*P1**3*X*(C3~C61F(2¢0
4 *ssssa +891*O.125*F3*F4 *(C3+3.0*C61+3.0*B4*85*P*P /k*((F12
5 +1.01*C3+(F12 ~3.01*C61+o.0*34*soss4 2x*IC3sc51+l.s%ss#s
69*P*P /X*((F12 +1. 01*C3+(F12 ~3.01*C61+3. Osssssio*FA 2
7 x*(c3- C61+B4*F4*P /x*( -C3+3.0*C61+Oo125*85*P*43/X*(F12 .
s 3.01*(C3-C61+O.25*86*F4* F5*(-C3+C61
cceluo.5*ss*F4*Fs *(—C3+3. 0*co1+o. 0625*B9*P*P*F5*(F12
-3.0)*(C3-C6)+0.125*810*F4*F5 *(«C3+C6)-O.125*A5*RI*F13*F2
*13.0*C3+C61-0.125*A6*P1*F13 *(C3+3.0*C61-O.5*A3*P1*F13*F2
*C3-0.25*A7*PI*F13*F2 *Ic3+csl+o.28125*PI*F23*F2 *(3.0*C3+C6
1+PI§F4§F3 *0.03125*(C3-C6)+P1*F4*F3 *0.0625*(C3+3.0*C6)~BIO¥
PI*PI*F3*o125*(C3~C61~0c125*(BB*F2 +891*F3*F4 *IC3+3.0*C61~(
A5*F2 ~A6)*F4*P *0.03125*(63+3.0*C61-(A5-A6*F2 1*0o03125*PI*F
4 *(C3-C6.-A7*0.125*RI*F4 .*(C3+3o0*C61+Oo84375*F4*P*F5 *IC3~C6
1+Oo25*BS*P*P*F5*( F12 -3.0)*Ic3~col
CD21=0o5*86*F4*F5 .*(~C3+C6)+Oc5*84*F4*F5 *(~C3+3.0*C61+
0.125*89*P*P*F5*(F12 . ~3o01*(C3~C61+0.25*BlO*F4*F5 *(~
C3+Col+0.25*ss*R4*F5 *(~C3+3.0*C61+0.0625*PI*F3*F4 *(c3-C61+
P1*F3*F4 *0.03125*(C3+3.0*C6)-B9*F3*F4 *O.lzs*(C3+3.0*col-Iss+s
101*Pl**3*x*0o125*(C3-C6)~(A5*F2 ~A6)*F4*P *0.03125*(C3+3.0*C61
-(A5~A6*F2 1*PI*F4 *Oo03125*(C3-C61~A7*PI*F4 *Oo125*(C3~C61+w2
V/v *(3.0*A3*A6*PI*F3’ *Dl+3oO*A6*A7*PI*F3 *Dl*C2-O.75*A6*PI*F13
*01*C1*C11+W*Y/V*(6.0*A3*B9*PI*F13 *Dl*C1+6.0*A6*BB*Pt*F1
3 *01**3*C1+6.0*A7*B9*P1*F13 *Dl*CI%c21

mqomaom»

mqounsumw

m~100¥&UN“

1

C

C

IOGJQOLR-bUNO-e 0340011§UNH OOQO‘W-DUNM

omqombmm»

82

CE21=W*Y/v*(-l.5*89*PI*F23 *Dl*C1**31+W*Y*Y *(Io.0*ss*39¥RI*F23

*Dl**3*C1*Cl1

CALCULATION OF C12

C128CA12+CBI?+CC12+CDIZ+CE12

IN UHICH
CA128A3*A4*PI*F3*P+O.125*A5*A5*PI*F3 *(P+02+051

+0.125*A6*A6*PI*F3 *(P-DZ+DS)+A¢*A7*PI*F3 *02+A7*A7*PI*F3 *
Ds-o.125*A3*F13*Fa' , ~0.0625*A5*PI*F13 *lR+Da+051~0.0625*A6
*PI* F13*F2 *DS-O.125*A7*PI*F13*F12 *DS+0.015625*A5*A5*RI*F6

*(P+DZ-051+0.03125*A5*A5*P1*F6 *DS-Oo03125*A5*A6*PI*F2*F6 *
DS~0.03125*A5*A6*F4 /X*(P+DE—051+0.015625*A5*A5*P*F5 *(R-D2«DS)
-0c03125*A5*A6*F4 /x*(P—DZ-DS1-0o03125*A5*A6*F4 /x*os+0.0l5625*
A6*A6*PI*F6 *(P-DS-DZ1+0o03125*A6*A6*PI*F6 *DS+0.015625*A6*A6*
P*F5 I*(P+D£-DS)+0.5*A7*A7*PI*F6 *05+0.0625*A5*(B9+610*F2 1*Pl
*PI*DS+0.0625*810*(A5-A6*F2 1*F4 *(R—02~05)

C812=+0.0625*B9*(A5*F2 ~A61*F4 *IR+Da—DSI-o.0625*A6*I89*F2 +8

101*P1*P1*DS+0c5*A7*Bl1*PI*PI*DS-0.015625*(A5*F2 -A6)*PI*F4 *
05-0.015625*(A5*F2 -A6)*RI*F4 *¢R+02-D5)—0.015625#(A5~A6*F2 )
*F4*P *(R-Dz-DSI~0.015625*IAs-Aossa -1*PI**3*05-0o125*A7*PI**3
*DS+0.25*B9*B9*PI*F3 *(P+02-DS)+0.2S*BIO**2*P1*F3 *(R-Dé-D51+O
o5*(BB*Bl1+89*8101*PI*F3*F2 *DS+O.5*A7*88*PI**3/P*DS+0.0625*(AS*
PI/P-A6)*89*F4 *(P+DZ-DS1+Oc0625*((A5*F2 ~A6)*BlO+(A5-A6*F2 1
*B91*PI*PI*DS+0.0625*(A5-A6*F2 1*810*F4 *(P~DZ~DS)-000625*89*P1
**3*X*(P+D£~DS1-Oc0625*(2.0*88*F2 +B9+BIO*F2 1*RI**3*X*DS

CC12fl~00625*810*F3*F4 *(P-DE-DS)+2.0*84*B7*F4 /X*DS+0¢25*BS**

2*P*FS *((F12 +1.01*P+(F12 ~2.01*DZ1+85*86*F4 2x*(R
~0.5*D21+0.25*86*86*P*F5 *(IFla +l.O)*R-F12 *02)+88*B
11*F4 /X*DS+0.125*89*89*P*F5 *((F12 +1.01*P+(F12 ~2.
01*D2)+0.5*B9*810*F4 -/X*<P«O.S*DZ1+0o125*810**2*P*F5 #(IFIe

+1.01*P—F12 *DZ)+0.25*B4*F4¥F6 *(DZ-DS1-0o125*BS*F4*
F5 *(P-Z.O*D21+Oc125*86*F4*F6 *(DE~P1+0o125*88*F4*F6 4
(DE-051+0o0625*89*F4*F5 *t2o0*02-P1+0.0625*BIO*F4*F6 *(02
-P1-0o125*A4*PI*F13 *(P+021-0.125*A7*PI*F13 *DS-O.0625*A5*
PI*F13 *(P+DZ+DS1~0.0625*A6*PI*F13*F2 *DS ,

C012800140625*PI*F23 *(P+DZ+DS1+0003125*PI**4*X*DS+00015625*PI

**4*X*<R+02e051-O.0625 *89*RI**3*X*IR+02-Ds)«0.0625*(BIO*RI2R+B
111*RI**3*x*05~0.0625*A7*RI**3*05-0.015625*IA5*RI2R-A61*RI*F4 4
(R+Da-DS)-o.o15625*(A5-A6*F2 1*Pl**3*05+0o140625*F14 /x*(R
-DZ+D51~O.25*BT*F4*E5 *DS+O.125*BS*F4*F5 *(2.0*02—R1+O.125
*B6*F4*F6 *(DZ-P1-0.125*811*F4*F5 *05+0.0625*B9*F4*F5 . *
(2.0*DZ—P1+0.0625*BIO*F4*F6 *(02-P1+0o015625*F14 *X*(R~
02-05)+0.03125*Pl**4*X*D5-Oo0625*(89+8111*Pl**3*X*DS-Oo0625*810*
F3*F4 *(P-DB~DS1~0.0625*A7*PI**3*DS-Oc015625*(A5*F2 ~A61*PI*
F4 *DS-0.015625*(A5~A6*F2 1*F4*P *(P-Dad051

83

cslzsw/v/v *(2.0*A3*A4*PI*F3 ~2.0*A3*A7*PI*F3 +A6*A6*P1*F3 *01
I *DI+2.0*A4*A7*RI*R3 *C2-2.0*A7**2*P1*F3 *CZ-0.5*A4*PI*F13 *
2 C1*C1+0.5*A7*PI*F13 *C1*Cl1+w*Y/V*(2oO*A6*B9*PI*F13 *Dl*Dl
3 *C1+4.0*A4*BB*PI*F13 *Dl*Dl*C1-4.0*A7*BB*PI*F13 *Dl*Dl¥C1+
4 2oO*A6*B9*PI*F13 *Dl*Dl*Cl1+U*Y*Y *(16.0*B9*B9*PI*F23 23.0
5 *DIsolsc1sc1) +0.25*A5*A6*PI*F3*F2 *Ds ,

CEIE-A20*(PI*PI*X*(O.375*P~Oo25*C2/P1~0.25*P/X*E10+A7/F2/X/X*El0+
1 BB/X*ElO1+BZZ*(-o5*P*X*E11+.125*P*P*FS*(PI*PI+4oO)/Pl/PI*

2 IR-D£1+A7*2.O/F2*EII+2.0*ss*x*EI1+84*R*F5*02+4.0*ss/F12/x*02I

CALCULATION OF C03

cos-CA03+cs03+cc03+CE03

IN WHICH

CA03-A4*AS*PI*F3*F2 *C3+A5*A7*PI*F3*F2 *(12.0*F12 *ca«c3)+A4
*A6*PI*F3 *C3+A6*A7*PI*F3 *(4.0*F12 *C4-C31-1.5*A5*PI**6*
x**2/P**3*C4~0.5*A6*P1*?!3*F12 *C4+0.5*A5*A7*PI*F2*F6 *13.
0*F12 -1.01*C4+A5*A7*PI*F2*F6 *Ca-A6*A7*PI*F12*F6 *c4

F2 ~A61*PI**3*(3.0*F12 ~1-0)*C4-0.25*(A5-A6*F2 1*Pl**4/P*
C4+2.0*B9*811*PI*F3*F2 *(3.0*F12 ~1.01*C4+4.0*810*Bl1*PI*F3
*FIZ *C4+2.0*A7*B9*P1**3/P*(3.0*F12 ~1.01*C4+4.0*A7*810*P
1*PI*F12 *C4~O.5*89*RI**4*X/R*(3.0*F12 ~1.01#C4
C8038~BIO*PI*PI*F3*F12*C4+2.O*85*B7*F4 /X*(3.0*F12*F12 +6.0%F12
-1c01*C4+16.0*86*B7*PI*F12*F6 *C4+B9*811*F4 /x*(3.0*
F12*F12 +6.0*F12 ~1o01*C4+8-0*810*811*PI*F12*F6 *C4+O.5*8

omqombumw

*F6*C4+0025*B9*F4*F6 *(3.0*F12*F12 -~8¢O*F12 +1oO)*C4-Bl

*F12 +1.O1*C4+2.25*P1**7*X**3/P**3*C4+0o125*Pl**5*X/P*(300*
F12 -loO1*C4-O.5*Bll*PI**4*X/P*(3.0*F12 -I.OI*C4~0.5#A
7*PI**4/P*(3.0*F12 ~1.0)*C4+0.375*PI*F4*F6 *(7.0*F12 -
1.01*C4+B7*F4*F6 *(4.0*F12 *C4ac3l

CC03=Oo5*Bll*F4*F6 *(4.0*F12 *C4~C31+O.25*PI**5*X/P*C4~Bl

I l*PI**4*X/P*C4-A7*PI**4/P*C4+W/V/V *(A4*A6*PI*F3 -*Dl-A6*A7*PI*F

2 3 *Dl1+W*Y/V*(2.0*A4*B9*PI*F13 *Dl*Cl~2o0*A7*B9*PI*Fl3 *0

3 1*c11 ‘

CE03=A20*I2.0*F3*F2*IF31+DII+F32~Dlo)-F6*EI4~Fs*Els+A5/X/X*EI4+
A5/F2/X/X*El3-A6/X/X*E13-A6/F2/X/X*E14+B9*4o0/X*El4+4c0*810/
X*EIB1+822*(~2.0*F3*E12+O.5*F5/F2*(PI*PI+4.0)*C3-2-0*P*X*EIS
'+A5*250*E12+A5*200*(P*012~2.0*C10)-A6*2.0*E15-2.0*A6/F2*El£+
BoO*B9*X*EIZ+8oO*Bl0*X*E15+BS*P*F5*(Fla-1.01*C3+4.0*B9/F12/X
*(F12~1.01*531

10014 om #019»

“hum“

-Oo5*A6*A7*PI*F6 *(3.0*F12 , ~1.0)*C4+Bll*(A5-A6*F2, 1*PI**3/
P*C4+0.5*Bl1*tA5*PI/P-A6)*PI*PI*(3.0*F12 -I.OI*c4-0.125*(As*

5*F4*F6 *(3.0*F12*F12 ~800*F12 +1001*C4-200*86*P1*PI*F12

O*PI*PI*F12*F6*C4-0o5*A4*P1*F13*F2 *C3-O.S*A7*PI*F13*F2 *(3oO

84

CALCULATION OF C2?

czosCA20+Cseo+CE20
IN wHICH
CA20=1.02Y*<4.0*A1*A3*F3 *Dl+2. 0*A1*A7*F3 *(Dl+03/3.01+A1*F13
I *0.s*13. 0*01+03/3.01+ 2. O*A1*A7*F5/F2 sIDIs03/3.01
2 +2.0/3.0*A1*88*P*(3. 0*Dl~031+2. 0
3 *A7*82*P*(DI-03/3.1-1.0/6.0*A1*P*P *I3.0*DI-03)+2.O/3.0*82*ss
4 *F3 *(3. 0*Dl~031 . .
5 -1o0/6.0*BZ*F4 *x*13.0*01~031+o.0*s2ssasss/22 *(Dl
6 +D3/3o01+2.0*82*BB*F5/F2 *(01+D3/3o01+6c0*84*F5/F2 *(D1+03
7 /3.01+2.o*ss*FS/F2-- r*(Dl+D3/3.01+82/3.0*P*F5/F2 *(3.0*Dl-D31+
a P*F5/3.0/F2 *(3.0*DIe0311+w/v/vxv *(4.0*A1*A3*F3 *C1+4.0*A1*A7
9 *F3 *C1*C2+2.0*A1*F13 *Cl**31 ‘
C8208W/V*(-2o0*A3*F13 . *Cl*Cl-2.0*A7*F13 *C1*C1*C2+8o0
1 *A1*BB*F13 *Dl*D1*C1*Cl+2oO*A3*BZ*F13 *D1*Dl+2o0*A7*
2 82*F13 ~ *Dl*Dl*C2-F23 *C1**4+82*F23 *Dl*Dl*C1*C
3 11+wsvsc-Io.023.0*ss*F23 *Dl*Dl*C1**3+16.0/3.0*82*BB*F23
4 *Dl**4*C11

CEZO=4oO/Y*(821*(BB*0075*X*E11+A7*0. 75/F2*E11-300/16.0*P*X*Eli+

I. B4*0.5*P*P/><*DZ+1 .5*88/F12/X*DE+1.0/16.0*P*(PI*PI+3.01/F-‘12/
2 X*(Pa02111

CALCULATION OF C11

CIIRCA11+CBII+CC11+CE11
IN WHICH ,
CA1181o0/Y*(4oO/3.O*A1*AS*X*P*ClcO+C2+4.0*C51+8.O/3.0*A1*A6*PI*X*
C5+8o0/3oO*A2*A3*X*P+8.0/3.0*A2*A7*F3*F2 *CS+4.0/3.0*A1*PI*P*
x*x *(1o0+C2+4oO*C51+4.0/3.0*Al*A5*P/X*C5+2oO/3oO*A1*A5*P/x*(15/
F12 -C51~2.0/3oO*A1*A6*PI/X*(lc/FIZ ~cs1~4.0/3.0*A1*A6*R*R2
PI/X*C5+16o0/3.0*A2*A7*P/X*CS+8.0/3.0*A1*89*P*C5+8c0/3.0*A1*Bl0*
PI*(1./F12 ~C5)+BZ*(A5~A6*PI/P)*PI/3.0*(lo/Flz eC51+2.O/3.0*
82*(A5*PI/P-A61*P*C5+16.0/3.0*A7*83*P*C5-2c0/3.0*A1*PI*P*(lo/FIE
~C51~2oO/3.0*A1*Pl*P*C5+16o0/3o0*82*89*P1*X*CS+8.0/3oo*82*810
*F3*F2 *II./212 ~C51+16.0/3.0*83*BB*PI*X*C5+2.o/3.0*Ba*(A5*
PI/P-A61*P*C5+82*(A5~A6*P1/P1*PI/3.0*(lo/Flz ~cs11
C8118!.0/Y*(8.0/3o0*(2cO*A2*BB*PI/P+A1*891*P*C5-2.0/3o0*82*PI*F3
*C5-2c0/3.0*82*PI*F3 *(1.2F12 ~ce1+s.o*sa*sa*RI/x*cs+4.0*82*
85*F5/F2 *(1.0+C2+2.0*C51+4.0*82*86*F5/F12 ‘ *(I.o+c2+(4.0
-2.0*F12 1*C51+4¢0/3.0*82*B9*F5/F2. *Ils0+C2+2.o#C51+a.02
3.0*82*810*F5/F12 *(loO+C2+(4.0-2.0*F12 I*C51+s.O/3.0*
BB*BS*PI/X*C5+8o0*B4*P/X*C5+4.0*BS*FS/F2 *lloO+C2+2.0*C51+4o0
*86*F5/F12 *(1.0+C2+(4c0~2o0*F12 1*C51+8oO/3.0*88*P/X*
C5+4oo/300*89*F5/F2 ' *11.0+C2+2.0*C51+4.0/3.0*Bl0*F5/F’12' ‘
*(1.0+c2+(4.0 0-2. 0*F12 1*C5)+16.0/3.0*82*P*F5 *CS+16.023.0#
Rst *cs+2. O/3.0*A2*PI*P*X*X'*(l.O+C2+4.0*C511

1003400190000"

omqombumu

85

CCl1:1oO/Y*(~2o0/3.O*(BZ*PI/P+831*PI*P*X*C5-2.O/3.O*(A1+A2)*PI*P*C5

l +4.0/3o0*83*Pl*P/X*(lo/F12 ~C5)+4.0/3.0*P*F5 #(1./F12 ~C51

2 -2o0/300*83*Pl*P*X*C5-2.O/3.O*A2*Pl*P*C5)+W/Y/V/V *(s.O/3.0*AI
3 *A6*PI*X*DI*C1)+U/V*(-4.0/3.0*A6*Fl3 *DI*C1*C1+16.0/3oO*A1
4 *B9*F13 *D1*C1*Cl+4.0/3.0*A3*83*F13 *Dl+4.0/3.0*A6*8
5 2*?13 *Dl**3+4.0/3.0*A7*83*F13 , *Dl*C2+2.0/3.0*83*F23
6 *D1*C1*C1)+W*Y*(-32o0/9o0*89*F23 ' *Dl*Cl**3+32oO/9o0*8
7 2*B9*F23 *Dl**3*C1+32.0/9o0*83*BB*F23 , .*Dl**3*C11

CEl1:821*4./Y*I4.0*89*X*E12+4.0*810*X*E15+A5*E12+A5*(P*012~2.*FTo;
1 ~A6*ElS-A6/F2*El2+BS*2.0/3.0*P*F5*(Fl2-1.01*C3+2o0*89/X*C

2 1.0-1oO/F121*C3+l.0/3c0/F2*F5*(PI*PI+3cO)*C3+F3*(1.0“F121

3 *C101

CALCULATION OF C02

C02=CA02+cs02+cco2+CE02
INewHICH
CAOZIIoO/Y*(4.O/3.O*A1*A4*F3 *D1+4.0/3.0*A1*A7*F3 *Do+s.0/3.0¥A2
*A5*F3*F12 *Dé+4.0/3.0*A2*A6*F2*F3 *06+4.0/3.0*A1*F13*F12
*06+8.0/3.0*A1*A7*F6 *D6+1.0/3.0*A2*A5*F6 *(Dl~061+2
o0/3.0*A2*A5*F6 *06-2.0/3.0*A2*A6*F2*F6 *06-1.0/3o0*A2*A6*F5
*(Dl-Dé)+8.0/3.0*A1*Bl1*PI*06+1o0/3o0*83*(A5*F2 *A6)*P*(D1~D
61+2.0/3.0*83*(A5-A6*F2 1*Pl*06~2.0/3.0*A1*PI*PI*06+4.0/3o0*83
*89*F3 *(Dl~061+8.0/3.0*(82*Bl1+83*8101*F3*F2 *06+8.0/3oO*A
7*82*PI*F2 *D6+4.0/3.0*A2*89*P1*(DI-D61+8c0/3.0*A2*B10*PI*F2
*Dé-2.023.0*82*RI*F3*22 *Do+4.0*sa*ss*so *(2.0*F12 -1.0)*
Dé+4o0*83*86*F2*F6 *Do+4.0*82*87*F5 *06+4oO/3.0*82*811*F5 *061
CBOZRIo0/Y*(4o0/3.O*83*B9*F6 *12.0*212 -1.01*Do+4.0/3.0*s3
*BIO*F2*F6 *D6+4.0*BS*F5 *(2.0*F12 ~1.01¥06+4.0*86*F6
*Dé+4.0/3.0*BQ*F5 *(2.0*F12 ~1.01*Dé+4.0/3.0*810*F6 *06+2
oO/3.0*82*F4 /X*(Dl~Dé1+4.0*B7*FS *06+4.0/3o0*811*F5 *06+2.0/3
.o*F4 /X*(Dl~D61+8.0/3.0*A2*F13*F12 *06-83/3.0*PI*F3 *
(DI-DoI-A2/3.0*RI*PI*IDleDoI+s.0/3.0*s3*RI*Fo *Do+8oO/3.0*F4
/x*Do-2.0/3.o*53*RI*F3 *06-200/390*A2*PI*PI*061+W/Y/V/V *(4.o/
3.0*A1*A4*F3 *Cl-4.0/3.0*A1*A7*F3 *CII+w2v*(2.0/3.0*A6*B3*F13
*Dl*Dl-2o0/3.0*A4*Fl3 *C1*C1+2.0/3o0*A7*F13 *C
1*C1+2.0/3.0*F13*DI*DI *82*(A4~A711
CC028W*Y*(16.0/9.0*83*89*F23 *Dl*Dl*C11
CE02=1oO/Y*(BZI*(BII*P*X/F2+A7*P/F2~O.25*P*P*X+P*P/12oO/X* (91*914
1 3.011+8.0*822/F12/X*Dl+A20*A1*El7*(8./P+4.0*P/F131+822*E16*(A
2 1/F2+82*X1*4o0+821*822/6.0*P*P*P*X+A20*BZ*4oO/F3*E17+A20*821/6oo
3 *P*F5+82*822*8o0/F12/X*Dl+821*822*500/12o0*P*P*F5)

1001401515070”

1061405015111“).-

86

CALCULATION OF DOB

0038003 +0603
IN WHICH
DO3=A4*A4*PI*F3*P+0.5*A7*A7*F3*F4 ~0o125*A4*F4*F13 ~0.o
625*A7*F4*F13 +0.25*A7*A7*F4*F6 +0.25*A7*BII*PI*F4 ~0.
0625*A7*PI**3*R+O.25*BI1*81I*F3*F4 +0.25*A7*811*PI*F4 ~o.0625
*BII*PI*F3*F4 +B7*87*F4*F6 +0.S*BII**2*F4*F6 ~0.125*87*RI*
F4*F6 -0.0625*811*PI*F4*F6 ~0.I25*A4*F4*F13 ~0.0625*A7*
F4*F13 +0.0703125*F4*F23 +0.0078125*PI**4*P*X-Oo03125*81I
*PI*F3*F4 ~0.03125*A7*PI**3*P+Oo0703125*PI**4*P/X~O.125*B7*PI*F4
.*F6 ~0o0625#Bll*PI*F4*F6 +0.0078125*PI**4*P*X-O.03125*BI1*PI*F3
,*F4 -0.03125*A7*PI**3*P+W/V/V *I2.0*A4*A4*PI*F3 -4.0*A4*A7*P1
*F3 +2.0*A7*A7*PI*F31
06038200*(A20*(Io/3o0*A20*P/X+PI*PI*X*(Oo125*P+0o5/P/F12)+2o0*A7/
1 F2/X/x+2.o*sll/F2/x~o.5*R/x1+522*(2.o*sl1*R*X/F2+2.O*A7*R2F2
2 ~0.5*R*R*x+o.125*R*R2x*IRI*RI+4.O)I+A20*(A20*R/6./x/xxx+822
3 *0.25*F5*P1+822*822*P*P*P/3.0*(X/2o0+1oO/X)1

omqombum»

CALCULATION OF 002

002=DoaN+DE021+DE022

IN wHICH
DOZNtI.0/Y*(4.0*A2*A4*P*X+4oO/B.O*A2*A7*P*X+4.0/9.*A2*F4*x*x
+8.0/3.0*A2*A7*P/x+8oO/9.0*A2*8II*P+I6o0/9.0*A7*83*P~2o0/9o0*A2
*Fa +8.0/9.0*B3*Bll* P*X+16.0/9.0*83*811*P*X+8.0/9oO*A
7*83*P+16o0/3o0*A2*Bl1*P»2oO/9.0*BB*F3 *P+4.O*83*87*P/X+4o0/300
*83*BI1*P/x+8.0/3.0*B7*F5/F2 +8.0/9.0*BII*FS/F2 +4.0/9.0*
83*F4 /x+4o0/3.0*B7*F5/F2 +4.0/9.0*BII*F5/F2 +4.0/9.0*P*
F5 +8.0/9.0*A2*F3*P*x ~200/9o0*83*F4 *x~2.0/9.0*A2*F4 +8.0
/9.0*83*F4 /X+8.0/9.0*P*FS ~2.0/9o0*83*F4 *X~2.0/9oO*A2*F4 1
+W/V*(2.0*A4*83*F13 *Dl~2.0*A7*83*F13 *Dl)

05021312o/Y*822*IDX4+DX5+DX6+DX71

DEOZEtIZ.0/Y*IDXI+DX2+DX31*A2O

CALCULATION OF C10

mqombmm»

CIOaCA10+cs10+CEIO

IN WHICH

CAIOtIoO/Y/Y *(A1*A1*PI*F3 /4.0*(P+DZ1+AI*A1*P*F5*.125 *(R~D2)+
A1*82*F4 /s.o*IR~021+82*82*RI*F3*.I250*¢R~021+O.125*AI*82*FA
*(P~021+2.0*Q*84*P*F5 *(Dl+03/3o01+82*P*F5*.5 *(P+021+P*F5
*.25 *(P+021+.25*82*82*P*F5 *(P+D£1+Q*P**3/(1200*X1*(3o0*01~
0311+lo0/12.0*(0.25*PI*F23 *(P+021+0.25*P**4/X*IP+02)+0.5*F14

*X*(P=DZ))+W/(Y*Y *v*v 1*A1*AI*PI*F3 *CI*CI+W/(Y*V1*(~

0.5*AI*PI*F13 *C1**3+0.5*A1*BZ*PI*F13 *DI*DI*CI~O.5*AI*PI
*F13* *CI**3+O.5*AI*82*PI*F13 *DI*DI*C1)+W*(PI*F23 23.0*
Cl**4~2.0*82*PI*F23 /3oO*DI*Dl*C1*C11

mqombumw

 

C

C

87

“1.2.5!,u-;;_%,,-,‘,rl,;.5§,,;‘~.fr.y-j.$_-zs‘je,.. 3.... , (“I i Mi" , W3 4,31...“- I {gr _\-_ J 5 ~.. . ',_ . , .‘. ‘~:.: .. ‘. .1
C8108W*(BZ*82*PI*F23 /3¢0*DI**4)-O*2*Z /I44oO*B4/(3¢0*X1*I300*
I Dl-D3)
CE!084.0/Y/Y*(821*(82*8o0*X/PI*516+AI*8.0*P/PI/PI*516+(16.0/F12/

I F3*DI1*(BZ+I.0)+821*P*P*P*X/18.0+821*P*P*P/x*2.0/15.01+BZO*
2 P*P*P/16oO/X*(P-DB)+B4*BZO*O.5*P*P/X*D21

CALCULATION OF C01

COIaCOI +CE01

IN wHICH

C01 x1.0/v/v #(AI*A2*PI*F3*F2 *C3+0o5*A1*A2*F4 /X*C3+O.5*AI*83*
F4 '*C3+0.5*82*B3*PI*F3 *C3+0.5*A2*BZ*PI*PI*C3+Q*BS*P*FS *11.0
+c2¥2.0*C51+o*86*R*F5/F2 - *11.o+c2+14.o-2.0*F12 1*C51+82
*83*PI*F6 *C3+82*F4 /X*C3+F4 /X*C3+83*PI*F6 *C3+Q*F4*F5
*C51+1o0/12.0*(PI*F23*F2 *C3+PI*F4*F6 *C3+2.0*PI*F3*F4 *c3)+
W/(Y*V1*¢O.5*AI*BB*PI*F13 *DI*C1)+W*(-83*PI*F23 /3.0*DI*
CI*C1+82*83*PI*F23 /3.0*Dl**31~0*2*2 /I44o0*(2o0*85/X*C5+86
*P/FB *(1.0-F12 *C511

C50184o0/Y/Y*(821*1B3*4o0*X/PI/F12+A2*4.O/F12/PI+83*8o0/F12/F3+8o0

1 *R/RI/Fia/FaI+220*O.25*RI*R*F5*C3+ss*820*P*R/x*0.5*IF12~I .01*

2 C31

dOlflbUN—F

CALCULATION OF C00

COOIQR/Y**4*400/F3*DI

CALCULATION OF 001

0018 DOI +DEOI

1N wHICH -

DOIaloO/Y/Y *(O.25*A2*A2*F3*F4 +0.125*A2*A2*F4*F6 +0.25#A2
1 *83*PI*F4 +0.125*83*B3*F4*F3 +4.0*O*87*F4 /X*06+0.25*83*83
2 *F4*F6 +0o5*83*F4*F5 +0.25*P**3/X+0.5*Q*F4*F6 *(DImDéII
3 +1.0/12o0*(0.25*F4*F23 ~+0.25*RI*RI*F4*(F6+2.*F31I +U*B3*
4 83*PI*F23 /3.0*DI*DI~Q*Z*Z /144oO*4.0*B7*P1/(P*X)*Dé
DEOI-IcO/Y/Y*(Q*B£2*P*P*F5*CI+0.25 *BZO*F14/X+Bao*822 *F5*P*P)

CALCULATION OF 000

000 84.0/Y/Y/Y*I820*P/F12/X+820*83/F2*F5 )+QR/Y**4*4o O/FE/FB

l
3145 0699 .

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