Computer programme for the determination of discontinuity and couples at the junction of a laminated fibre reinforced cylinder and domed end closure

College of Aeronautics Memo 7501 October I974

CRANPIELD INSTITUTE OF TECHNOLOGY

COMPUTER PROGRAMI.IE FOR THE DETERMINATION OF

DISCOOTIINTUITY FORCES AND COUPLES AT THE JUNCTION OF A

LAMINATED FIBRE REINFORCED CYLINDER AND DOMED END CLOSURE .by

R.C. Pews M.Sc.

7 DEC. m

R. Tetlow M.Sc.

SUr-ïïvlARY

A computer programme is developed for the analysis of discontinuity forces and couples experienced at the junction of a cylindrical pressure vessel with a domed end closure. Continuum analysis and anisotropic shell bending theory is used to give solutions where both cylinder and end closure are constructed from laminated fibre reinforced composite materials, Transient functions are developed from the shell differential equations and used to investigate the decay of discontinuity forces and couples.

Work carried out as part of M.O.D. (P.E.) Agreement No. AT/2028/059GC 'Study of the Application of Modern Materials

Page

NOTATION

1.0 INTRODUCTION 1

2.0 PROGRAMI'E DESCRIPTION AND OPERATION 2

2.1 General description 2

2.2 Sign conventions j 2

2.3 Programme input data 2

2,h,

Programme output 5

3.0 FUTURE WORK 5

4.0 REFERENCES 6

APPENDICES

A Theory of laminated anisotropic cylindrical shells

B Theory of laminated anisotropic shells of revolution

C Solution of redundancies at the end closure

FIGURES

1. B'llament wound pressure vessel

2. i''orces at the junction of the elements

3. Forces and moments on a cylindrical shell element 4. Typical laminated structure

5. Axis system

6. Curvilinear co-ordinate system

7. Co-ordinate system for the surface of revolution 8. Elemental forces and moments (surface of revolution) 9. Alternative equilibrium configuration

10. V/orked example 1 11. Worked example 2

a

Direct membrane stress

N Membrane force

T In plane shear stress

€ Membrane strain

Y In plane shear strain

M Bending moment

X Curvature

Q,

Transverse shear force

h Distance of laminates from reference surface (see flg.ii)

u

V

CD Radial displacement

a Radius of cylinder

P Internal pressure

Y Component of external load in direction parallel to a

meridian

Z Component of external load in the direction of a parallel

circle

^1

^2

Orthogonal displacements in 'x' and 'y' directions

respectively

Radii of curvature

r Radius of a parallel circle

o

S Distance between reference surfaces (see fig.2)

|J.^.j, Poisson's ratio (strain effect in-fc direction due to load

applied in

i

direction)

[A] In plane flexibility matrix

[B] In plane stiffness matrix

[D] Bending stiffness matrix

[G] [A] -1

M Ü3]

[A]

-1

SUBSCRIPTS

X Parameter has a direction parallel to the 'x' axis y Parameter has a direction parallel to the 'y' aais z Parameter has a direction parallel to the 'z' axis x,y Parameter is referred to an x,y plane

I Parameter has a direction in line with the fibres

t Parameter has a direction at right angles to the fibres It Parameter is referred to an It plane

k Parameter is referred to the kth layer in a laminated composite

0 Parameter is referred to the reference surface

1 Parameter has 'x' direction on the reference surface 2 Parajneter has 'y' direction on the reference surface 5" Refers to steady state conditions

e n

\. Parameters referred to respective curvilinear ^ J co-ordinates (surface of revolution see fig.8)

MATRIX NOTATION

{_ J Column matrix n x 1 [ ] Square matrix n x n

1.0 INTRODUCTION

With the use of filament and polar winding techniques composite fibre reinforced materials allow themselves to be very readily used in the manufacture of pressure vessels. The type of design of interest in this report is that of a basic pressure containing cylinder closed by a continuously wound domed end cap. (see flg.l). Both dome and cylinder

are composed of a number of layers of quasi-homogeneous continuum, which may be isotropic or unidirectional fibre reinforced composite. The governing constitutive equation for such structure is the relation between in plane stress and moment and in plane strain and curvature, (see Appendix A) where material coefficients are expressed as functions of

the properties of individual layers and lamination parameters. Computation of global elastic constants and membrane laminate stress levels for such construction is handled in Ref.4 and it will be the purpose of this report to extend

that work to include the secondary stress distributions

associated with the discontinuity at the junction of cylinder and dome. At Appendices A and B, laminated anisotropic shell bending theory is discussed in order to establish Influence coefficients in slope and displacement for cylindrical shells and shells of revolution. The influence coefficients, which are derived for end shear force, end couple and pressurisation loading conditions are used in Appendix C to satisfy

compatibility and equilibrium requirements at the junction and to thereby evaluate the redundant shear force and couple active at the end of each shell element.

Failure due to discontinuity loading is considered to be most likely in the cylindrical shell element, as the hoop stresses in the dome are inherently lower than those in the cylinder and the method of construction necessitates an

increasing wall thickness in the dome element. Discontinuity forces and couples are computed for all elements at the

junction line but in view of the above, decay functions have only been evaluated for the main cylindrical vessel (see Appendix A ) ,

For some applications, i.e. rocket motor cases it may be required to add a further cylindrical shell element beyond the end closure of the main pressure vessel. This cylinder will contribute to the discontinuity situation and its effects are considered at Appendix C.

2

-2.0 PROGRAM!^ DESCRIPTION AND OPERATION 2.1 General Description

The programme is capable of analysing pressure vessels comprising two or three shell elements at the end closure main cylinder junction (see fig.2 (a) and (b)). Each shell element can be constructed from laminated material and a maximum of twenty laminations of differing thicknesses, if required, is catered for. The fibres in each lamination

can take up any orientation and an arbitrary mixture of three differing materials may be used throughout each shell thickness.

Cross plied laminations may be fed in layer by layer specifying layer thickness and orientation, i.e. +6 or - 0 , or in blocks, where the total thickness of the block is specified. In this case each block will be idealised into six equal thickness layers, i.e. three layers at +© and three layers at -© (see worked example 2 Appendix D ) ,

2.2 Sign Conventions

Forces and bending moments output from the programme conform to the sign convention Indicated at fig,3.

+Ve Bending moments give compression on the outer surface +Ve Membrane forces are tensile

+Ve Radial displacements are inwards

+Ve Transverse shear stress as shown at fig.3

Fibre angles of orientation are measured relative to the cylinder axis and are +Ve as shown in fig.5

Lamination locations relative to the reference surface are -Ve of below the reference surface (In fig.4(a) h^, h2 and h-j are -Ve)

2.3 Programme Input Data

Integer variables must be punched without a decimal point and real variables with a decimal point. All data cards must commence at card column one and where more than one variable is required per card each variable must be separated from the next by leaving one space.

To follow is a list of input cards in the order required by the programme. The actual variable names used in the

programme are given in order to show the number and type of variables required per card.

a) I^MAT Single integer variable

NMAT = The number of separate materials used in the construction (3 maximum i.e. 1, 2 or 3)

^) ^J' ^J ^Y^ H U ^ ^ ' I N M A T Repetitions; four real variables

EL ET GLT UMLT >• ^g^, ^^rd EL ET GLT Ur/ILT_J ^^^ ^^^^

EL = Longitudinal Young's Modulus ET = Transverse Young's Modulus

GLT = Modulus of rigidity in the longitudinal-transverse plane

UMLT = Poisson's ratio for strains in the 't' direction Induced by a strain in the 'I' direction

c) NEL Single integer variable

NEL = Number of shell elements employed in the construction (2 or 3)

d) RA RHO S AN3 P Five real variables

RA = Radius of the main pressure cylinder

RHO = Meridional radius of the end closure (RA if hemispherical)

S = Distance between reference surfaces (l) and (2) (see fig.2)

AN3 = Membrane force applied to element (3) P = Internal pressure

If NEL is 2 the construction comprises only two shell elements and (S) and (AN3) should be punched as zero.

e) INPT Single integer variable

INPT = Marker for 'blocked' cross ply input

The following block of cards varies according to the value of lOTT

If the number of layers is relatively small with the

orientation and location of each layer in the material thickness known, then INPT should be punched as 1. The remaining data should be punched as follows (see fig.4a)

4

-f,) N Single integer variable N = Total number of layers g-^) MT(I) I = 1 to N

M(I) = An integer array of (N) variables describing the material of each layer i.e. If MT(l) is punched as

(2) then this means that layer (1) is composed of material (2) having elastic constants identified by the second of cards (b)

hi) LR(I) I = 1 to N

LR(I)= An integer array of (N) variables describing the orientation of each layer, i.e. if LR(l) is punched as 45 then the orientation of layer (1) is +45^

(see fig.5 for orientation sign convention)

±l) H(I) I = 1 to N + 1

H(l) = A real array of(N + 1) variables describing the location of each layer within the shell thickness relative to the reference surface (see fig.4a), hi, h2, and hj, must be punched with a negative sign, hj^ is zero and h5 is a positive number, If the shell thickness is comprised of cross plies i.e. alternative layers at +© and -Q where the number of Individual layers is large and the exact location of each layer in the material thickness Is difficult to assess, then the cross ply layers may be input in 'block' and INPT should be punched as

zero. Data cards fj to i^^must then be replaced by the following, f2) NS Single integer variable

NS = Number of layer types. For the layup of fig.4b NS would be punched as three

gg) MARK(I) I = 1 to NS

MARK(I) = An i n t e g e r a r r a y of NS v a r i a b l e s d e s c r i b i n g t h e t y p e of l a y e r b e i n g c o n s i d e r e d . I f M n K ( l ) i s punched a s z e r o t h e n l a y e r one i s t a k e n a s a crosB p l i e d l a y e r . I f lAk]lK{l) i s punched as one then l a y e r one i s a s i n g l e l a y e r of Qormiiii-hK f,-fi^ï'tih'i-^i^4tf For t h e l a y u p of f l ; / . / | i 3 , m/iK vimlö ^j*i uatïtihi^Q 0 0 1

hg) MMT(I) I = 1 t o NS

MIVIT(I) = An integer array of NS variables describing the material of the layer typej similar to MT(I) (card Si)

ig) LLR(I) I = 1 to NS

LLR(l) = An Integer array of NS variables describing the orientation of each layer type. If the layer under consideration is a cross ply layer the

positive orientation only should be punched. For the layup of fig.4b LLR should be punched 60 45 90 jg) HH(I) I = 1 to NS + 1

HH(I) = A real array of NS + 1 variables and is similar to array H(l) (see description for card 1^). For the layup of fig,4b HH would be punched as the numerical values of -hi 0-0 h^ hj^

2.4 Programme Output

The following information is output from the programme 1) Elastic constants in the hoop and meridional directions

for each shell element.

2) Redundant loads and couples at the junction of the elements 3) Decay of loads and couples in the main cylindrical element

(i.e. element 1)

For typical programme output see the worked examples of Appendix D,

3.0 FUTURE WORK

In the work of this report all shell elements are assumed to be long,so that the transient effects at one end are not affected by conditions at the other end. This is a perfectly reasonable assumption for the types of construction considered in this report since it can be seen from the worked examples of Appendix D that the effects of discontinuity tend to be extremely

local.

A different type of structure of interest in this programme of v/ork is the pressure cylinder with stiffening rings and in this case shell elements are likely to be sufficiently short as to render theory inaccurate. The subject of future work will

therefore be to modify Appendix A to give theoretical relationships for short cylinders and thereby to create a programme capable

of handling the discontinuity situation at a stiffening ring. Future work must also be directed at creating a programme capable of using the output from the current programme and the proposed stiffening ring discontinuity programme to predict individual laminate stress levels and strength factors.

6 -4 . 0 1 , 2 , REFERENCES

3.

4.

DONG, S.B. PISTER, K.S, TAYLOR, R.L. PADOVEC, J,

On the Theory of Laminated Anisotropic Shells and Plates

Journal of the Aerospace Sciences August 1962

Stress Analysis of Shell Junctions Fabricated by the Filament Winding Method

PI-RPG Conference, Filament Winding 11, 15 and 16 March 1972

TIMOSHEITKO, S, Theory of Plates and Shells WOINOWSKY-KRIEGER, S, McGraw-Hill Book Company ENG, TAYLOR, P,T, Computer Programmes for the

Determination of Elastic Properties and Strength of Thin Laminated Fibre Reinforced Composite Plates

Cranfield Institute of Technology Cranfield Memo 92,

APPENDIX A

THEORY OF LAMINATED ANISOTROPIC CYLINDRICAL SHELLS

1.0 INTTIODUCTION

The relationships developed in this appendix are derived for the specific case of a circular anisotropic cylindrical shell submitted to axisymmetric loading. The theory is listed with a view to the derivation of the Influence coefficients required for the solution of redundant shear forces and couples experienced at the junction of a domed end closure with a

cylinder. A linear theory of shells is used throughout and the usual assumptions conducive to linear shell theory are applicable to this work.

For a more general treatment of the theory of laminated anisotropic plates and shells the reader should consult Ref.l. 2,0 CO-ORDINATE SYSTEM

Use is made of an arbitrary reference surface within the shell thickness. The general curvilinear coordinate system adopted (x,y,z) is shown in Fig.6, x being parallel to the

shell generator, y falling on the circumference of the reference surface and a being +Ve outward,

3.0 GEi^^RAL THEORY

Consider a shell element as shown in Fig.6b, consisting of an arbitrary number of matrix/reinforcement layers. The thickness and orientation of each layer may differ from the rest.

3.1 Stress Strain Relationships

Assuming each layer Irran element to be In a state of pure stress, the stress-strain relationship for a general

(kih) layer may be stated as in Pig.4 i.e.

k xy ^ k ^ k k °11 "^12 ^13 k k k °12 °22 °23 k k k °13 23 °33 y xy

A2

-Where the symmetric matrix c. . represents the elastic coefficients associated with the kvhlamina. For a particular lamina these coefficients depend on both the elastic properties of the material referred to a set of elastic axes and the

orientation of these axes with respect to the surface coordinate lines (x,y). Equations of this type are discussed in detail in Ref.4.

3.2 Stress Resultants and Stress Couples

In terms of the reference surface strains and curvatures the stress resultants and stress couples for the element are given by

r """

"x

" y & L

px

M y

h^-= = — * i i * 2 1 ^ 3 1 B l l B j l

bi

—— ^12 ^13 ^22 ^ 2 3 A A

32 ''32J

^12 ^13 ^22 ^23 ^32 ^33_ xil ^ 1 1 ^12 ^21 ^ 2 2

531 ^"32

\ l O12 ^ 2 1 D22 D31 032 ^ 1 3

Bajl

^ »

^7

° 2 3

°33

X X

Xx

X y

2 X • • ^ ^ y 2 ^

1 xy

y ...(A2) The symmetric matrices [A1 , [ B ] and LD] are defined in Ref.l as follows n _(k)

^ij

(\ -

Vi)

k=i 1 ^T (^) 2 2

^j = i L ^ij ^^k - Vi)

_k=l n D ij

4 I

\ - l ) k=l ...(A3)

Equations (Al) and (A2) may be summarised as follows

ory = tB] {,} - M {^^-} ...(,5)

From (A4)

O o 3 = [ A ] " ' W * [ A ] " ' [ B ] { X J ...(,6)

Substituting in (A5)

{If} = [B]

[A] "^{"NI

+

[B][A]

"^ [B]{XJ}- [D]{X^1

g i v i n g

where - 1 r - 1 ->. T [b] = LB] [A] ={LA] CB]J>

[d] = CD] - [B]LA] ' [ B ]

3.3 Strain Displacement Equations

The strain displacement equations using the reference surface strains and the changes in curvature appropriate to

the X, y, z coordinate system chosen and incorporating Donnell's approximations are as follows

^1 = "ST"

^^o , w

^2 ^•S^-' I

èv èu «, o , o •^12 ~ "SF" ^y~ ...(A8) 1 i 7 ' 2 ^y2' ^ ^ ^ y ...(A9) w

- A 4

. * . Expanding ( A 6 ) and (A7) w r i t i n g L A ] a s KJ

bu >2 N 2 1 ÖX 11 X 1 2 y 1 3 x y 1 1 ^ ^ 2 21 ^ y 2

^ 2 = ^ - 1 = Gl2^^x -^ ^22Ny + Gg^^^y ^ ^ 2 0 ^ ^22 f ^

^

^^32

J %

i ?

'23

i 7

'^12

= ^ ^ ^ =

^I3^x

-^

^ 2 3 \

-^ ^33 V " ^ 3 7 7 " ^23

+ 2^22 ÏÏF^ . . . ( A I O )

^x =

^l^x

+ ^ 2 ^ ^ ^ 3 V " ^11 $ " ^12 0 -

2^13

^

My = bg^N^ + bg^Ny + b^yj^y - d g i - - ; ^ - ^22 --;2 " ^^23 ^STS?

a ? • 22 ayi

2 2 ^2 Ö w , è W o , Ó w

^ V = ^33^^x ^ ^ 2 \ -^ ^ 2 3 ^ " ^31 ^ " ^32 T T " 2d3^ 3 3 ^ ^

Sx'^ -^"^ èy^

...(All)

3.4 Equilibrium Equations

Prom the shell element diagrams shown in Pig.3 the

follov;ing equiliblrum equations can be v/ritten

^Sr"" ~5F" " ° ...(A12a)

ON èN

J x + _ X + i N - P

dx öy a y =- O öM öl-i X ^ __2£Z: - Q • = o cix oy ^ x . . . ( A 1 2 c ) . . . ( A 1 2 d ) ó\\ èM

5 / -^ - ^ - S = °

3.5 Internal Pressure ...(A12e)

For an axisymmetrically loaded cylindrical shell (i.e. shell subjected to internal pressure) the stress resultants, stress couples and displacements must be independent of (y).

.'. Substituting (A12d) and (A12e) into (A12c) and neglecting all (y) dependent terms.

è2K^ i\

ax

2 - ^ 1 ^ - ^ = 0 . . . ( A 1 3 )

è^

Differentiating the first of equations (All) twice Ing that N,

1 = i N _ - P

w..r.t.(x) and noting that N and N are constants yields xy X

è2N

W y^

^11 öx^ " 12 ^^2 ^ y ...(AI4)

Differentiating the second of equations(AlO) twice w.r.t gnoring terms dc

are constants yields

(x) ignoring terms dependent on (y) and noting that N and N

X xy 1 a2w . „ - ax^ ' "22 From (AI4) ax2 b^2 Prom (AI6) ax2 O22

L

1 a2w

L" ^ 3x5

-dx*

Ï

'V - p_

^12 a / J

...(A15) ...(A16) . . . ( A 1 7 )

A6

->rn (Al6) ar.d (Al?) ö w

èx^ ^11 ^22 ^ ^ 2 ' ax2

^

- ^22? +

^22 ^'y

From the second of equations(AlO)

..,(A18)

'22

- ï - 012NX - ° 2 3 V - ^ 2 a^*

= N.

Substituting for N in (AI8) yields

4

èx^ ^ 1 1 ^ 2 2 •" ^ 2

2b

_{12afw w p _ ^ M ^23 .j}

a ~ ^7? •" ;;2 - - ^22^ - ^ ^x a \ y

Assume that b,p is negligible compared with other terms then:

è^7 _W

bx^ a.\^

G22

HP d _{11 ^ ^22 "11} '12 "ÏÏ: G, ^x a G^^ d, N. '22 "11 xy _...(A19a) Let

P =

4a d-Li G22 Then

^*" 1

K B V

- ^ °12 1 ^

ax" *^ <»ii °22 ^"^n ^

3.6 Homogeneous Solution ^23 1 G22 ad^3^

N

_{xy ...(A19b)}

The homogeneous solution to (A19b) is given in Ref.3 as the following

w = e^^(C-L cos Px + C2 sin px) + e"^^(C, cos Px + C. sin px) Since any effects on slope and deflection are known to be local C-, and C2 must be zero and the solution reduces to

w = e ^ (C, cos Px + C^ sin Px) •px

Prom (A20)

aw

r

y - « - pe"'^^<jc^(cos px + sin px) + C^(sin px - cos Px) >

ax

a^w

^ = p^e ^^ <; 2C, sin Px - 2C^ cos Px

= 2p^e~^^< C,(cos Px - sin px) + C^(cos Px + sin Px) r

.(A21) 3.7 Boundary Conditions

a2w

Prom (A21) At x = 0 , ^ = - 2C„p2; | ^ = 2p^(C3 +

ax

Prom the first of equations (All)

ax^ %^

M^ = b , T N „ + b _{11-x • -12^y -^ ^13^'xy "11 ^^} + b , , N . . . - d, a2w , * . Assuming b^2 ^^ s m a l l

""^ ^ 2p2-

i-r(^^x)o-hl^^^x)o- W o

_{dii L J}

aM^

"5F

.(A22) Prom (A12d) assuming - ^ = 0 f o r a x i s y m m e t r i c l o a d i n g

aM„ 'ST = Q, a^w • ' • ^x " " ^11 ^ ^^12 ^^^^'^> \ ^^^ \ y c o n s t a n t )

ax

• • • ( ^ x ) o - = - ^ l l 2 P ^ ( C 3 + C , ^ ) S u b s t i t u t i n g f o r C,, from (A22) 4 C , = -2^11^" '_ (^x^o + P^^Sc^^o - P ^ l ^ V o - ^ ^ 3 ^ ^ l x y ) _{x y o} .(A23) S u b s t i t u t i n g (A22) and (A23) i n t o (A20) y i e l d s

A8

-^^«'"x'o - e ''llC'x'o - P ^15("xy'o !»=1" P^ . *. At X = O

(v^). = - ^

...(A24)

O _{2d,, P}

11'

-(Q ) - P(M ) + p b, ,(N ) + p bT,(N ) _{^^X''o ^^ x'o ^ 11^ x'o ^ 1;)^ xy'o} ...(A25)

(i) =

_{o 2d^^p} 2 (Q^)^ + 2P(M^)^ - 2P"x'o ""^^"x'o '"" 11^ x'o '"'" '^13^"xy''o ^ T ( N ^ ) ^ - 2P b,^(N^y), ...(A26) 3.8 Decay Functions

Prom the second of equations (AlO) w.. \ = G 22 - G, oN - G ^ ^ - b 12 X 23^xy small ...(A27) ,*, Substituting for w^ from (A24) and noticing that N is constant and can be replaced by (N )

N„ = - 2a p2e"^^ iK)r. (sin Px - cos Px) + 2a p e'^^(Q^)^ cos px

- 2a p2e~^^ (N^)^ (cos px - sin Px)

X J. ^ o

ötf 'Vo

Prom (A21)

^ . A 2 „ - P X

,,(A28)

ax^

= P e ^ { ^ ( ^ x ) o - P(^^x)o - P ^l(Nx)o - P ^ 3 ^ ^ ^ ^

- ^ ^ ^ x ) o - ^l(N,)o - ^ 2 ( V o - ^ 3 ( ^ ^ 0 1 Prom the first of equations (All)

s m a l l

a2w

a2w

. •. substituting for —;? and setting N„„ to zero

ax2

^

M^

= ^ll(Nx)o •*• t (^x^o ^^" Px + e ^''(M^)^ (sin Px + cos px)

- ^^""(Vo ^l^^i" P^

•"

°^^ ^^^ ...(A29)

awy

Prom (A12d) Q^ = - ^

.'. differentiating the first of equations (All) w.r.t. x Ignoring

all 'y' dependent terms and substituting the resulting expression

aM

for - T — in the above

«X = - ^-^^ 2 P ( M ^ ) Q sin Px - ( Q ^ ) ^ (cos Px - sin px)

- 2P

h.,{N)^

sin Px

'11 ^"x'o

3.9 Steady State Parameters

,.,(A30)

The expressions shown in 3.7 and 3.8 for w„, N , M , and

jQ y^ j \ ,

0 are all defined from the homogeneous solution to the shell

differential equation for axisymmetric loading. In order to

get the particular solutions the steady state criteria must be

added to these expressions.

a) Steady State Radial Deflection

The steady state radial deflection for an anisotropic

cylinder is given by the following

For the basic pressure cylinder

AIO

-b) Steady State Hoop Load per Unit Length

From (A27)

«y = G

22

"x

- i^ - ^12 \

As is mentioned previously w„ in the above expression is the

homogeneous solution to the shell equation.

The total radial deflection is however w - w

••• ^ = G

22

-

T-

^ ^22^V

•" ^

1 2 \

-

C^12^^x]

w.

_X

y -22

G..a + Ny

3.10 Final Equations for Membrane and Bending Forces

The following relationships are defined for simultaneous

apolication of end shear (Q ) , end moment (M^) and membrane

forces (N^)^ and (N J,.

x'o

X' s •y's

The equations are obtained by summing the relationships

of 3.8 and 3.9 with the applied in plane shear force (N^^J

set to zero.

xy'

w,

^ 2d,^P^

"^^x^o

"

"

" ^^^'^0 '• ^ ^^(N^)« L °°^ P^

11^ x's

M ^ d g o - p ^ i ( V s > ^ i ^ ^ ^

- G22a

G

₁₂

'22

y's G^^ ^"x's

N.. =

- 2ap2e"^^(M^)^ (sin px - cos Px) + 2ape"^^(Q^)^ cos Px

'•'x =

M =

_X

^-p- (Q^)Q sin Px + e '"^(M^)Q(sln px + cos px)

.Px

°12

«y = 0 ^ '"x

^ X 2 P ( M ^ ) Q sin Px - ( Q ^ ) Q (COS px - sin px)

BI

-APPEITOIX B

THEORY OF lAMINATED ANISOTROPIC SHELLS OF REVOLUTION

1.0 INTRODUCTION

A surface of revolution is obtained by rotation of a

plane curve about an axis lying in the plane of the curve.

This curve is called the meridian and its plane is the meridian

plane. The intersection of the surface with planes perpendicular

to the axis of rotation are parallel circles and are called

parallels.

For such shells the lines of principal curvature are its

meridians and parallels. A convenient selection of co-ordinates

for the surface are the angle / between the normal to the

middle surface and the axis of rotation and the angle ö

determining the position of a point on the corresponding parallel

circle. See Fig.7.

2.0 BASIC THEORY

The load case to be considered in the theory to follow is

that of internal pressure, which is Inherently symmetric v/ith

respect to the axis of the surface of revolution. It can be

concluded from the condition of symmetry therefore that there

will be no transverse shear stresses on the surfaces of a shell

element lying in the meridian planes. The overall force and

moment situation on an element in the surface of the shell is

thus as shown in Fig.8.

2.1 Equilibrium Equations

Prom consideration of the equilibrium of forces and couples

acting on the shell element of Fig.8 it is possible to derive

the following three equations.

I? ( V o ) -

\''l

^ ° ^ i^ - ^o ^^ ^ ^o^l ^ = 0 ...(Bla)

V o -^ V l ^i^ ^ ^ - ^ | —

^

2 ^l^o = ° ...(Bib)

h

^ V o ^ - V l °°^ ^ - V l ^ o = 0 ...(Blc)

Equation (a) is derived by considering balance of forces in

direction parallel to the meridians, equation (b) from balance

of forces in the Z direction and equation (c) from balance of

moments with respect to the tangent to the parallel circle.

It is possible to derive equation (a) in a different form by considering the equilibrium of the portion of the shell above a parallel circle as defined by the angle / (see Fig.9). Then: 2Trr N/ s i n / + 27rr„ Q,/ cos j6 = 0 _{o j6 o p ,}

or in terms of x,y co-ordinate system consistent with Appendix A.

2Trr^ N^ sin / + 27rr^Q^ cos ;i = 0 ^^^^32) 2.2 Reference Surface Strains and Curvatures

Strains and curvatures are given in Ref .3 as follov^s ^1 = r i__ av w_ 1 ^ ' ^ 1 ^2

=

r- cot

/ -

--^

rg rg T,2

= 0

X =

X o

=

1 a j v aw I

r^ "3? > r^ r^dpj

{

v_ aw i__ 1 c o t ^

^1 "^ ^ij ^2

...(B3) X^ = 0 2.3 stress-strain Relationships From equations (B3) and (AlO)

"x = ' 1 1 ^ 1 1 ^ 2 2 " ^ 1 2 22 1 _ ^ 1 1 ^ 1 ^ ' "^ G 12 1 ^ 1 1 ^ 2 . . . ( B 4 ) v c o t /Z$ - w G N„ = 11 2 U-L2- _11^22 ^ 1 2 1 ^ 1 1 ^ 1 av _{- VI r - — '( V c o t fè - VI}

B3

-E l i m i n a t i n g v cot / - w

t^

Eliminating -^ - w

...(B5)

V cot / - w = G22

r^

I^V ^ G ^ ^^x

Eliminating w from (B5) and (36)

...(B6)

av

^ - V CO. p = ^22^'^^^ ïït; ^1 - G ^ ^ 2 ^ ^

- V cot /$ = GooN,

^ 2 2 ^ ^ G ^ 3 ; G

12

'22

'22

22

^1 - ^2

...(B7)

Differentiating equation (B6) v;,r.t./ and using the result

together with equation (B7) to eliminate av yields

fe ^1 - fe ^2 r

"

^ 2 2 \ (ïï~ïï^ ^

G22^'x °°^ ^

'22

'22

'11"22

^ - rgL + V cot X$

V aw _ a

s m p ^ ^

^2^22 <;^ + G ^ ^'^x

..,(B8)

From (32)

N^ = - Q^ cot

_...(B9)

Substitution of (39) into the second of equations (Bl) with

Z = 0 yields

^y = • ^ I? ^ ^x ^2

_...(310)

Prom (B8)

V + -^ = cot /i G22J y c — -1 Q —

'"^11 "^12 A

- r, - — ^ r^ N + '

^2J

^^x • VGV-.G00 '1 ^2 r v

f

^12

^'^

r-, - r^ N.

d

22

2V'y"4l">='

'22

'11"22

...(311)

^ =

^S-^

Substituting (39), (BIO) and (312) into (Bil) yields

...(B12) 22 ! ; L . Ö ! U , i_ r 2 , Q 2 r, ^12

Y..,, V , ^2 a

au

1_ Tl 11 ^1 r.r.'r^A ^ 12 ^22 2 ^22 ...(B13)

The second equation for u and v is obtained by substituting expressions (B4) into (All) to give expressions for (M^) and

(M ) which are substituted into (Blc) i.e.

X ' - u '11 r v2 2 a V 1 _ 2 :vv2 r , r , op 1 1 2 ^11 cot / -

121

^11 cot

^

av

i l ^^^2^ _ l l ^ ^^^2^

^11 ^2 ^11

...(314)

2.4 Approximate Solution

Consider conditions at the equator of the dome where

r = a and r, = P^J (a) and p are considered as being invariant with ^ over the short region of the dome at the equator.

In the case of very thin shells if the angle ^ is not small the quantities Q, / and v are damped out rapidly as the distance from the edge increases. Tirnoshenko (Ref.3) suggests

that a satisfactory approximation is achieved if terms involving V, -TT- , ^ and u are neglected from (313) and (BI4)

1 .e. a2 S \ 2 :^v2 Po ^^ a a2v Pn^ ^^^ " V

°22

...(315)

B5 -E l i m i n a t i n g v from (B15)

2 a V

^.

r a- ' -X , Let 2 ,4 Po 4d^3_G22a2 Then 'X

0

= 0 .,.(B16) 2,5 Homogeneous Solution

The homogeneous solution to (Bl6) is given in (Ref,3) as being of the form

Q^ = C^e^^ cos X/tJ + Cge^^ sin X/^ + C , e ~ ^ cos T^fi + C^e"'^^ sin A/ Since any effects of discontinuity are known to be local the above equation reduces to

Q^ = C ^ e ^ cos "hfi + CgS sin X^ ,..(B17)

C-, and Co are arbitrary constants and depend on the

boundary conditions of the problem concerned. Internal forces and moments are given by the following relations,

N^ = - ^x °°^ ^

^^x

a

^^x

= -

^11 ^ = - ^11 r a2 V = G22 2 Po

1 av ^12 V cot

^

Po

^^ \ l ^

1

^22

V

cot üJ

,

^21 av

1

^11

^

^11 ^^ Po

1

a2^

,..(Bl8a) ...(Bl8b) ..,(Bl8c) ,,,(Bl8d) ...(Bl8e)

The radial'displacement of a parallel circle is obtained from: VI = a sin ^ G aq _{X a '12 ^ „^. /} - ^--- Q^ cot ;i 22*1 • ^ r P^ .22 2.6 Boundary Conditions ...(BlBf)

Consider the case where the shell is loaded by an edge shear force Q i.e. / = ^2 ; r-'x = 0 ; Q^ = - Q From (317) C-, < | c o s X / - s i n ?v/ A + ^ 2 ^ ^ ° ^ ^ ^ •*" ^^^•^ ^^ a2Q aiz{' ^ . A2e^^ - 2C^ s i n A^ + 2C2 c o s A/

a\

^ ^ 2A^e^^

a>!$^

From ( 3 l 8 e ) - C, ( c o s 7sj6 + s i n A/zJ) + Cg ( c o s }\fi - s i n X/z$) av _{= G} 2 a^Q

"5? ~ ''22 7 T

X

P . "

a^-Prom ( B l 8 c )

"x = - ^ l l t ^ = °

1 1 p _ a ^ 7^-K^f.^l^

° Po

- C, (cosAjd + sinA;z$) + C^(COGAJZ$ - slnKjó) \ =0

']-. C - ( c o s A^ - s i n A/j) ^

• • ^ 1 c o s 7^p + s i n A/$ :

37

-Prom (317)

-Q = e'^^(C^ cos A/ + Cg sin A/$) substituting for C,

•Q 2 2

c o s Ax$ - C O S A / s l n A ^ + cos7\/!5 slnhfi + s i n A/ • Q Cge A/ *. Cp = —^<^cosA)zJ -i- s l n A / C-, = —j^ <cosA;ii - s l n A /

^ L_

ao

ubstltutlng for - ^ C^ and C2 in (Bl8b), (Bl8c) and (Bl8f) yields I = 2qA-^ y Po

v = -Q^^f 2A2G22

V; = 2Qa'^AG22 o ...(320)

No;v consider the shell under the action of a bending moment M uniformly distributed along its edge. The boundary conditions

for this case are as follov/s

16 = ^2 } %^ = ^ i \ =

M

Using equations (317), (319) and (BI8 b,c and f) gives:

APPEITOIX C

SOLUTION OP THE REDUNDANCIES AT AN END CLOSURE

1 . 0 GENERAL

From the relationships derived at appendices A and B it is possible to collate influence coefficients in slope and deflection for anisotropic domes and cylinders under specified loading conditions,

In this appendix the required Influence coefficients are

specified and used in conjunction with equilibrium and compatibility criteria to evaluate equations for the redundant shears and

bending moments at the junction of a cylinder with a domed end closure,

The theory to follow discusses conditions at the junction of three shell elements but in the computer programme of Appendix D the reinforcing cylindrical end ring may be omitted if desired,

2,0 NOTATION

In this Appendix the following notation is used to describe respective Influence coefficients and overall slopes and

deflections. For the definition of shell elements 1 - 3 see Fig,2.

fi^ = deflection influence coefficient p s

„©„ = slope influence coefficient p s A = total deflection P = total slope

Prefix

P

1

= 2

3

Suffix

S

q

= m

n

P

shell element 1 shell element 2 shell element 3, see Pig.2

influence coefficient due to shear force Influence coefficient due to bending moment influence coefficient due to end load

C2

-3.0 SIGN CONTVEOTION

End shears and bending moments are -ve as shown in Pig.2. Radial deflection is +ve Inwards,

4.0 INFLUENCE COEFFICIENTS

4.1 Relationship Between Dome and Cylinder Influence Coefficients

5,

From equation (A25) the deflection influence coefficient

y-1

for a cylinder due to an applied moment (M ) is: •^ 2d,,p2

for a dome the similar parameter from the first of equations (B21) Isi

•"

2^\^

But P^ - 1 . ^4 ^ 4 4a-d^^G22 4a-d^3_G22 . •. ?^ = P Q P

Therefore the Influence coefficients for the deflection of a dome or cylinder under the action of an applied end moment are the same.

Similarly 5 dome = 5 cylinder

b^ dome = 5j^ cylinder etc,

4.2 Summary of Relevant Influence Coefficients

Prom equations (A25) and (A26) the relevant influence coefficients can be listed as follows.

a) 5^ = l/(2d^^p2) e^ = l/(d^^P) 5q = l/(2d^^P^) e^ = l/(2d^^p2)

,2

b)

Pressure Influence Coefficients l^p = ^^22 2^p = ^^22 Pa + -—^ N, G22 1 (cylinder) (dome)

c) Membrane Force on Element 3

Provision is made for the application of a membrane force N^ to the cylindrical end ring (element 3 Fig,2), The force will have a 'Polsson' effect which must be added to the general compatibility conditions,

i,e,

Sn3 = ^^12^^3

5 . 0 SOLUTION OP REDUITOANCIES

5.1 Squilibrium of Forces

Prom consideration of the forces acting on the three elements of Fig.2 the following equilibrium equations may be written,

Ql + Q2 + S = °

-N^ + Ng + N, = 0

M2 + M, - M^ - N,S = 0 _,.,(C1)

5,2 Compatibility of Deflection and Slope

In order that slopes and deflections may be compatible at the junction of the elements the following conditions must apply,

1 ^ = 2 ^ =^^

C4

-5.3 Slope and Deflection Equations Prom element 1 From element 2 25q^2 + 2 V 2 -^ 2ÖnN2 "^ 2 V = 2 ^ 2 V 2 -^ 2 V 2 -^ 2®nN2 =23 From element 3 3 q 3 3 m 3 3 n 3 n3 > 3 q 3 3 m 3 3 n 3 JP

But^A =^A (from C2)

• • • l ^ ^ l - 2Öq^2 ^ i V l - 2 V 2 = ( 2 ^ - i V ^ ^ 2SnN2 " i V l • • • ( C 3 ) Alsoj^^ =-jf^ (from C2)

i q l I m i i n l I p 3 q 3 3 n i 3 3 n 3 n3 But M, = M^ - Mg + N,S and Q = - Q^ - Qg (from CI)

• ' • ^l^l^q •*• 3 V ^ ^=^2 3'q ^ ^l^l'm " 3^m) ^ ^2 3'm

= - i V " V 3 ^ n ^ 3 V ) - ^ 3 - l V l ...(C4)

Prom (C2)^P = - , p

. - . Q i ( i e q ) + Q2(2^q) ^ ^l(l®m) ^ M 2 ® m ) = " i V l " 2 ^ 2

05 -A l s o , P , = - P^ 3 Q 3 3 n i 3 3 ^ 3 I q l I m l i n l But from (Cl) M, = M^ - Mg + N,S j p , = - p^ - Pg • • • « l ( l « q - 3®q> * « 2 ' - 3 9 q ) * " 1 ( 1 % * ^%^ * "s'-'^^m''

= N j ' - ^ V - A ' - i V l ...(06)

Equations (03)* (04)> (05) and (C6) may be written in matrix form as follows. CA] < M, M,

= ®

y

where

w =

_{" i\}

l^q ^ 3*q

l « q

1^

-

?«q

- 2 ^

3«q 2*q

-A

i 5 „ 1 m l^m • J^m 1% 1 % * J^m -2Sm' J^n, 2«m -3«m ^^ It X It It X 1

{B> r (26p - i6p)P * 2 ^ 2 - i V i

- 1 V * " J S ^ * 5'm-S) - i V l - «„5

- l«n"l - 2 V 2

"jf-J^m-S - 3 % ) - l^nNl

The redundant forces and moments Q,, Qp, M, and Mp can now be calculated from the matrix product,

-1 M^

[Al -'•

{_B}

Dl

-APPEJIDIX D

PROGI^M'-IE LIST AND WORKED EXAMPLES

1 . 0 GEII5PAL

The theory expanded at Appendices A, B and C has been Incorporated in a digital computer programme, thus making the process of discontinuity analysis, within the context of this report fairly automatic. The method of operation of the

programme together v;lth its capabilities and limitations are discussed in the main body of this report and will not be reproduced here.

It is the purpose of this Appendix to highlight the types of structure that can be analysed and to demonstrate the function of the programme in the form of providing actual solutions.

Two worked examples are discussed and solved by the programme; relevant input and output data being given, together with a full programme list.

2.0 WORKED EXAMPLE 1 2.1 General

A six inch diameter cylinder and compatible hemispherical end closure form the boundaries of a pressure vessel designed to v;ithstand 10001b/in2. it is required to investigate the discontinuity effects at the junction of the hemisphere with the cylinder v;hen both elements are constructed from Aluminium Alloy and both have a total thickness of 0.1".

2.2 Material Properties E^ = 10000000.0 lb/in2 E^ = 10000000.0 lb/in2 G^^ = 3850000.0 lb/in2 u,^ = 0.3

D.2.3 DATA FOR WORKED EXAMPLE 1 i u O ( i G i O U . Ü 1 0 0 0 0 0 0 0 . 0 5 8 5 0 0 0 0 . 0 0 . 3 ? 3 . 0 S.a 0 . 0 0 . 0 l o O O . O I 2 I -I - i . i . v i 5 0 , 0 O . o S ^ 'I 1 9 " 9 0 - u . u 5 0 . 0 o.o:>

n^i

-D3

D2 A WORKED EXAMPLE 1 PROGRAMME OUTPUT

SHELL ELEMENI ELASTIC P R O P E R T I E S

ELFME'^T 1 E L E M E N T 2 E L E M E N T 3 A X I A L M O D U L d b (, EX ) M O O P M U D U L U S (,EY) SHfeAk ^.oouius ( G X Y ; P O i S S o N ' 6 RAflO M Ü ( X , Y ) k'OlSSOK'b KA t 10 M U C Y f X ) Ü . 1 0 0 " O 0 0 E 08 0 O . I O O o O O O t 08 0 0.3«5i,u00fc 0^ 0 0.3uOviOO0E 00 Ö 1 0 0 0 0 0 0 E l O O O o O O E 3 6 5 0 O 0 0 E 3 0 0 0 0 0 0 E 08 O . O O O O O O O E 00 08 O.OOOOOOOfc 00 0? O . O O O O O O O E uO 00 O.OOOOOOOfc 00 0.3oüi'0ö0£ 00 0 . 3 0 0 0 0 0 U E 00 O . O O O O O O O E Ü0

LOADS AND C O U P L E S AT THE J U N C T I O N OF THE E L E M E N T S •X' D T H E C T I O N P A K A M E I B B S A R E Q U O l E O PER UNIT

ClWCOMf-feRfcWTlAL SHELL LENGTH

'Y' D l H E c i l n N P A K A M E I E B S A K E Q U O I E D P|R UNIT

M E R I D I Ü N A L S H E L L LtNOTH

ENo M O M E N T S AKF iJUOTtD R E L A T I V E TO E L E M E N T

N E U T K Ü L A X É ^ E L E M E ^ T 1 ELEMENT 2 E L E M E N T S L O N G I TUOi>JA| fiUOP M ; A I ' M É K i D i ü N A L PI M Ü i P P . M . LOAD N ( A ) N ( Y ) M. N)(X) M ( Y ) IRANSVfcKSfe SHEAR U ( > ) 0 , 0 0 0 I b O t ' O O O E 2 ^ 5 v ^ 0 0 0 l : 0 J O u w O O t OvO H Ü O É - 0 . ! > 3 2 ó 3 5 3 t OA o . i S o o o o o F 04 o . o o o n o o o E 00 QC 0.<:<i5ÜOOOE 0 4 0.00000001; 00 00 O.OOOOi'OOE 00 O . O O O O O O O E 00 00 O . O O O O O O O E 00 O.OOOOOOOfc 00 02 0.33i(ó3S3E 02 O . O O O O O O O E 00

WORKED EXAMPLE 1 OUTPUT CONTINUED

DECAY OF LOADS A " 0 COUPLES"FOR ELEMENT 1

LOADS Arc COUPLES ARE QUOTED PER INCH OF SHELL LENGTH MERIDIONAL OR CIRCUMFERENTIAL, WHICHEVER IS APPLICABLE

MERIDIONAL LOAD N(X) IS CONSTANT AT, 0 . 1 5 0 0 0 E 04

' X ' DISTANCE 0. OOOOE 0. 2677E 0. 5355E 0 . 8 0 3 2 E C. 107 I E C. 133SE 0. 1606E 0. i e 7 4 E 0. 2142E 0 . 2 4 1 0 E 0 . 2677E 00 00 00 00 01 01 01 01 01 01 01 HOOP LOAD M(Y) 0 . 2 2 5 0 E 0 . 2 6 7 6 E 0 . 2 9 3 4 E 0 . 3 0 3 5 E 0 . 3 0 4 9 E 0 . 3 0 3 2 E 0 . 3 0 1 4 E 0 . 3 0 0 3 E 0.299ÖE 0 . 2 9 9 o E 0 . 2 9 9 9 E 04 04 04 04 04 04 04 04 04 04 04 MERIDIONAL B.M. M(X) 0.OOOOE 00 - 0 . 7 1 1 7 E 01 - 0 . 6 1 4 3 E 01 - 0 . 3 2 7 7 E 01 - 0 . 1 0 8 1 E 01 0 . 3 9 9 6 E - 0 9 0 . 3 0 7 6 E 00 0 . 2 6 5 5 E 00 0 . 1 4 1 6 E 00 0 . 4 6 7 0 E - 0 1 - 0 . 3 1 0 0 E - 1 0 HOOP B . H . M(Y) 0.OOOOE 00 - 0 . 2 1 3 5 E 01 - 0 . 1 8 4 3 E 01 - 0 . 9 8 3 2 E 00 - 0 . 3 2 4 2 E 00 0 . 1 1 9 9 E - 0 9 0 . 9 2 2 7 E - 0 1 0 . 7 9 6 4 E - 0 1 0 . 4 2 4 9 E - 0 1 0 . 1 4 0 1 E - 0 1 - 0 . 9 3 0 1 E - 1 1 TRANSVERSE SHEAR Q(X) - 0 . 5 3 2 6 E 02 - 0 . 6 2 8 6 E 01 0 . 9 7 3 3 E 01 0 . 1 0 1 9 E 02 0 . 6 0 2 6 E 01 0 . 2 3 0 2 E 01 0 . 2 7 1 7 E 00 ~ 0 . 4 2 0 6 E 00 - 0 . 4 4 0 4 E 0 0 - 0 . 2 6 0 4 E 00 - 0 . 9 9 4 7 E - 0 1 RADIAL DEFLECTION - 0 . 5 4 0 0 E - 0 2 - 0 . 6 6 7 9 E - 0 2 - 0 . 7 4 5 2 E - 0 2 - 0 . 7 7 5 6 E - 0 2 - 0 . 77 97 E-02 - 0 . 7747 E-02 - 0 . 7692E-C2 - 0 . 7 6 5 9 E - 0 2 - C . 764 5 E-02 - 0 . 7 6 4 4 E - 0 2 - 0 . 7 6 4 6 E - 0 2

D5

-5 . 0 V/QRKED SXAMPLE 2 5 . 1 G e n e r a l

Fi^ii.ll shows the boundary of a pressure vessel consisting of a laminated anisotropic cylinder and hemispherical end

closure. Por the purpose of attaching the pressure vessel

to surrounding structure a continuation cylinder is addedjV/hich is also of laminated anisotropic construction.

The cylinders and hemisphere are nominally 6" dla and the layups and thicknesses of each element are as shov/n in Fig,lib. It is required to investigate the discontinuity effects at the junction of the elements when an internal pressure of 2000 lb/in is applied to the vessel.

5.2 Material Properties E^ = 30000000.0 Ib/in^ E^ = 1100000.0 Ib/in^ G^^ = 700000.0 Ib/in^

1 30000000.n iinnoon.n 7onoon.o n.3 3 3.0 3.0 0.06 n.n pnnn.n 0 3 0 0 1 1 1 1 60 60 90 -0.04 0.0 n.n4 o.n6 0 2 0 0 1 1 60 6n -0.04 0 . 0 0 . 0 4 0 2 0 0 1 1 45 45 r'

D.3.A.WORKED EXAMPLE 2

D 7 -PROGRAMME OUTPUT

SHELL ELEMENT ELASTIC PRDPEKllFS

ELEMENT 1 FLFMENl P ELEMENT 3

AXIAL MODULUS (EX) HOOP MODULUS (EY> SHEAK MODULUS (GXY1 POISSON'S KATIO MUfX,Y) POISSON'S RATIO MUCY,X)

n . ) 7 3 9 ^ 3 ? E 0 7 n . 1 P 7 7 R 5 9 F OR n . 1 4 1 f t R R O E n 7 n . 7 9 3 5 6 4 n E n 7 n.?573R36E 01 n.P573R36E 07 0 . 4 8 6 1 1 I R E 0 7 n . S 9 n i 3 9 7 E 0 7 n . 7 6 3 5 1 9 6 F 0 7 n.ppin594E nn 0.3134731E nn n.R3R4545E no n.l63R499E ni n.l755696E ni n.R384545E 00

LOADS AND COUPLES Al THE JUNCTION OF THE ELEMENTS

'X' DIRECTION PARAMETERS ARE QUOIED PER UNIT CIRCUMFERENTIAL SHELL LENGTH

•Y' DIRECTION P A R A M E T E K S A K F QUOTED PER UN 11 MERIDIONAL SHELL LENGTH

END MO^^ENTS ARF (jUOTED RELATIVE 10 FLEMF.N 1

N E U I K A L AXES LONGITUDINAL LOAD HOOP LOAD MERIDIONAL R.M. HOOP R.M. IRANSVERSF SHEAR N(X) N ( Y •) M C X ) M( Y ) Q( X ) = = = = = ELEMENT 1 o.3nnononE n.39^nRR6F 0.77 146R0E - 0 . IPft/iOSOF -n. ipnP734E 04 n4 ni np 0 3 E L F M F N l ? o . 3 n n n n n n E 0 . 4 7 R 7 R 6 n E - n . 1 4 7 6 4 6 P E 0 . P S 9 P P 1 9 F n . 1 3 7 9 0 1 7 E 04 04 n? OP 03 F L F M F N l 3 0 . o n n n n o n F - n . 7 4 6 9 R P 1 F n . 4 7 P n 5 6 5 E - n . 3 9 S 7 9 7 9 E - n . 1 7 6 P R 3 n F on np ni ni np

fïECAV OF L O A D S A N D C')^JPL6S F O R E L E M E N T 1 L O A D S A N D C O U P L E s A B H Q U O T E O O E R I N C H O F S ^ E L L L E N G T ^ M E P I O l O N A L O R C l H C U M F K R g N T l A L , W H l c H E V g » IS A P P L I C A B L E M E R I D I O N A L L O A D N ( X > IS C O N S T A N T A T , 0 , 3 0 0 0 0 E 0 4 • X ' D I S T A N C E 0 . 0 0 0 0 f. 0 . 1 6 8 3 F 0 . 3 3 6 6 E 0 . 5 0 4 8 6 0 . 6 7 3 1 6 0 . « 4 1 A 6 0 . 1 0 1 0 6 0 . 1 1 7 8 6 0 . 1 3 4 6 6 0 . 1 5 1 5 E 0 . 1 6 « 5 E 0 0 0 0 UÜ 0 0 uO 00 01 01 01 01 01 HOOP LOAO N ( Y ) 0 . 3 9 S 1 p U . 4 9 1 3 E 0 . ^ 6 4 5 6 U . 6 0 0 3 E 0 . 6 1 0 4 6 Ö,60H9E 0 . 6 0 A 7 E 0 . 6 0 1 5 F 0 , 6 0 il 0 6 Ö . ^ 9 9 6 F 0 . 5 9 V 6 6 i i A 04 04 OA 0 4 OA OA OA uA u4 OA MERIDIONAL B . M . M ( X ) 0 . 7 7 1 5 ^ - O . A 3 Ï . 2 F - 0 . 5 9 b 3 E - 0 . 5 9 0 0 E - 0 . 1 6 7 2 F - 0 . 3 3 3 4 E 0 . 1 8 M 1 E 0 . 2 5 7 2 6 0 . 1 6 Ö 5 E 01 01 01 01 01 00 00 0 0 0 0 0 . 7 2 2 5 E - 0 1 0 , 1 AA1 E'.OI MOOP B . M . M ( Y ) 0 . 1 1 « « p " 0 . 6 7 0 1 F - 0 , 9 1 6 5 E - 0 . 6 0 0 4 E - 0 . 2 5 7 4 E - 0 . 5 1 3 3 E 0 . 2 8 9 6 F 0 . 3 9 6 1 F 0 . 2 5 9 5 E 02 01 01 01 01 00 00 00 00 0 . 1 1 1 2 E 00 0 . 2 2 1 8 E - 0 1 TRANSVERSE SMFAR OCX) - 0 . 1 2 0 3 E " 0 . 3 2 2 6 E 0 . 6 3 8 4 E 0 . 1 4 6 9 E 0 . 1 0 8 7 6 0 . 5 1 9 7 E 0 . l 3 9 A e - 0 . 2 7 5 9 P - 0 . 6 3 4 9 F - 0 . A 6 9 5 E - 0 . 2 2 4 6 E 03 02 01 02 02 01 01 00 00 00 00 RADIAL OEFLfCTION 0 . 2 2 6 5 F - 0 2 0 . 5 2 6 7 E - 0 5 - 0 , 1 7 1 3 6 - 0 2 - 0 . 2 5 5 3 E - 0 2 - 0 . 2 7 8 9 E - 0 2 - 0 . 2 7 5 A E - 0 2 - 0 . 2 * 5 6 F « 0 2 - 0 . 2 5 f l 2 E - 0 2 - 0 . 2 5 4 6 E - 0 2 - 0 . 2 5 3 6 E » 0 2 - 0 . 2 S 3 7 E - 0 2

D 9 -D.A. PROGRAMME LIST

MASTER CLOSURE DIMENSION A(3#3),B(3»3),D(3*3)«E(3,3)*F<3*3),G(3,3). I W2<3,3)*ANS<4,1)*T(3)*REDN(3)*REDM(3) COMMON /COMA/EL<3)*ET(3)*GLT(3)*UMLT(3)/COMC/MTC20)*LR(20) l/CaMD/H(21)/CDME/C(2n*3*3)/CaMl/ BB(3#3*3)#DD(3*3*3).6G(3*3*3) 2/CQM5/EX(3)#EyC3)*GXY(3)*AMUYX(3)#AMUXY(3)/C0M6/AB(3) 3/CQM2/HL(20)# W(20)#AMX(20),QQ(20)*AMY(2n)*DIST(20) 4/CÜM7/ANX(3)«ANY(3)*ABMX(3)*ABMY<3)*AQX(3) CALL MATPROP READ( I, n NIÜL READ (1,2) RA,RHa,S,AN3,P 2 FORMAT (SFO.O) 1 FORMAT(IO) DO 40 L=1*NEL READ (1*1) INPT IF (ÏNPT.NE.1) GO TO 33 CALL LAYUP(NA,T*L) GO TO 44

33 CALL LAYUP2 (NA,T,L) 44 CONTINUE DO 30 1=1,NA IN=MT(I) CALLCMAT(EL(IN),ET(IN),GLT(IN),UMLT(IN),LR(I)#I) 30 CONTINUE DO 10 1=1,3 DO 10 J=l,3 A(I,J)=0.0 B(I,J)=n.O D(i,j)=n.n DO 20 K=1,NA AH=(H(K*1)-H(K)) BH=H(K+1)**2 - H(K)**2 DH=H(K+1)**3 - H(K)**3 A(I,J)=A(I,J)+C(K,I,J)*AH B(I,J)=B(I,J)*C(K,I,J)*BH 20 D(I,J)=D(T,J)*C(K,1,J)*DH B(I,J)sB(I,J)/2.n D(I,J)=D(I,J)/3.0 10 CONTINUE CALL INVER(G,A) CALL MULT(B,G,E) CALL MULT(E,B,W2) NAS=1

DO «^1 1 = 1,3 DO 8 1 J=),3 BB(L,I,J)=-E(I , J) DD(L, I,.J) = F(I, J) GG(L,I,J)=G(I,J) P?l CONTINUE 40 CONTINUE 90 FORMAT (IHl) WRITE (2,90)

CALL SOLVE (NEL,RA,RHO,S,AN3, P, ANS) DO 4 1=1,NEL

4 CALL ELASPROP (I,T) CALL OUTE(^(NEL) HEDM(1)=ANS(3,1) REDM(2)=ANS(4,1) REDM(3)=REDM( 1 ) •^AN3*S-REDM(2 ) REDN(1)=ANS(1,1) REDN(2)=ANS(2, 1) REDN(3)=-REDN(1)~REDN(2) DO 3 1=1,NEL M=l AN=P«RA/2.O IF (I.EQ.3) AN=AN3 AM=REDM(I) QX=REDN(I)

CALL DECAYLONG (M.RA,RHO,I,AM,QX,AN,P) A N Y d ) = HL( 1 )

A N X d )=AN

ABMX(1)=AMX(1) A B M Y d )=AMY( 1 ) 3 AQX(I)=QQ(1)

CALL OUTLOAD (NEL) M=l 1

AM=REDM(1) AN=P*RA/2.O QX=REDN(1)

CALL DECAYLONG (M,RA,RHO,1,AM,OX,AN,P)

CALL OUTDECAY (AN) STOP OK

D 1 1

-Sljnp.OUTl'F. OECAYLONQ (M, R, RHO, K, AM, OX, AN,P)

CQ;;MON' / C O : : 2 / H L ( 2 0 ) , w ( 2 0 ) , A M X ( 2 0 ) , Q Q ( 2 0 ) , A M Y ( 2 0 ) , X ( 2 0 ) co:'.MON / C C ; M / F:F)(3, 3 , 3 ) , n D ( 3 , 3 , 3 ) , Q G ( 3 , 3 , 3 ) / C O M 6 / A B ( 3 ) BETA=AF(K) ^M1 = r'E;(K, 1 , 1 ) r , 1 2 = G Q ( K , 1 , 2 ) Q22=GG(!':,2,2) D 1 1 = n D ( K , 1 , 1 ) D 2 1 = n D ( K . 2 , 1 ) F l = 3 . 1 4 1 5 5 2 6 5 4 A L = 2 . 0 * P I / C E T A - n T = o . o A 1 N C = A L / 1 0 . 0 DO 1 1 = 1 , i-^ EX=EXF(EETA*BlT) E X = 1 . 0 / E X 5=S11!( BETA* B I T ) C=COS(BETA*BIT) DUr-n=-2.n«R«BETA*BETA*EX D U M 2 = D U M 1 / R E T A SS=P*R IF(K.EQ.2) S S = P * R / 2 . 0 IF (K.EO.3) SS=0.0 H L ( 1 ) = D O M 1 * A K * ( S - C ) - D U M 2 * Q X * C + D U M 1 * B 1 1 * A N * ( C - S ) + S S D U M 3 = E X / ( 2 . 0 * D 1 1 * B E T A * B E T A * B E T A ) P1 = (-QX-DETA*At!+BETA*B11*AN)*C P2=(BETA*A:'I-BETA*B11*AN)*S SS = Q22*P.*(P*R + G12/Q22*AN) IF ( K . E Q . 2 ) SS=SS-P*R*R*R*Q22/(2.0*RHO) IF ( K . E n . 3 ) s s = 0 . 0 W ( 1 ) = D U M 3 * ( P 1 + P 2 ) - ' ^ S P2=EX*9X*S/BETA P3=EX*Ar':«(s+c) P4=B11*EX»An*(S+C) A:^.X(I)=P2+P3-P4 A M Y ( ! ) = D 2 1 » A M X ( 1 ) / D 1 1 P1=2.0*BETA*AM*S P 2 = 0 X * ( C - S ) P 3 = 2 . 0 * B E T A * B 1 1 * A N * S n o ( l ) = - E X * ( P 1 - P 2 - P 3 ) X ( 1 ) = B 1 T 1 BIT=F'1T+A1MC RETURN END

SUBROUTINE LAYUP2 (L,T,K) DIMENSION HH(5),LLR(4),MMT(4),MARK(4),T(3) COMMON /COMC/MT(2n),LR(20)/COMD/H(21) READ (1,1) NS READ (1,1) (MARK(I),I=1,NS) READ (1,1) (MMTd ), I = 1,NS) READ (1,1) (LLRd ), I = 1,NS) READ (1,2) (HHd),I=1,NS+1) T(K)=-HH(1)+HH(NS+l) 1 FORMAT (2010) 2 FORMAT (lOFO.O) L=n DO 3 1=1,NS IF (MARKd ).EQ. 1 ) GO TO 4 TH = HH(I + 1 )-HHd ) AlNC=TH/6.O BTTsO.O DO 5 J=l,6 L = L*1 H(L)=HH(I)+BIT MT(L)=MMT(I) MIK=LLR(I)•(-!)**L LR(L)=-MIK 5 BIT = BIT-»-AINC GO TO 3 4 L=L+1 H(L)=HH(I) MT(L)=MMT(I) LR(L)=LLR(I) 3 CONTINUE H(L+1)=HH(NS*1 ) RETURN END

-D13-::junKni)TiNF S O L V E (K,KA,RMn, S , A N 3 , P , C ) DIMFNISION A(4, 4),R(4, 1 ),C(4, 1 ) , WKSP(4),D1 1 ( 3 ) , R22 ( 3 ) , R 1 1(3) 1,B12(3),BT1(3),BT2(3),RT3(3),ON(3),RN(3) , AN ( 3) , PD ( 3) COMMON /CQMl/ BR(3,3,3),DD(3,3,3),GG(3,3,3)/CnM6/AB(3) AN(3)=AN3 DO 2 1=1,3 Dl 1 d ) = Dn{1,1,1) 82?(I)=GG(I,2,2) Bl1(I)=-BB(1,1,1) ? B 1 2 d ) = G G d , 1 ,2) RAA=KA+b SRA=RA DO 20 1=1,3 20 AB(1 ) = n.n DQ 10 I = 1 , K IF(I.EQ.3) BKA=RAA in AB(I)=l.n/((4. n* BRA**?* Dl l d ) * R P P d ) ) * * n . 2 5 ) DO 21 1=1,3 B T i d ) = n. n BT2(I)=n.O 21 BT3(I)=n.O DO 11 I = 1 , K B T l ( I ) = 1 . n / ( A B ( I ) * D 1 1 d ) ) BT2(I)=l.n/(?.n*AR(I)**2*Dll(I)) 11 B T 3 ( I ) = l . n / ( 2 . n * A R d ) * * 3 * D l l ( I ) ) A ( 1 , 1 ) = R r 3 ( 1 ) A( 1,2) = -BT3(2) A(1,3)=BT2(1) A(1,4)=-BT2(2) A(2,1)=A(1,1)+BT3(3) A(2,2)=BT3(3) A(P,3)=A(1,3)-BT2(3) A(2,4)=BT2(3) A ( 3 , 1 ) = A( 1,3) A(3,2)=-A(l,4) A(3,3)=BT1(1) A(3,4)=BT1(2) A(4,1)=A(2,3) A(4,2)=-BTP(3) A(4,3)=BT1(1)+BTl(3) A(4,4)=-BTl(3)

DO 2 3 1 = 1 , 3 D N ( I ) = O . O 2 3 B N ( I ) = 0 . 0 DO 12 I = 1 , K D N d ) = B 1 1 ( l ) * B T 2 d ) 12 B N d ) = B 1 1 d ) * B T 1 d ) A N ( 1 ) = ( P * R A / 2 . 0 ) • A N ( 3 ) A N ( 2 ) = P * R A / 2 . 0 P D ( 1 ) = P * R A * * 2 * B 2 2 ( 1 ) • R A * B 1 2 ( 1 ) « A N ( 1 ) P D ( 2 ) = P * R A * * 2 * B 2 2 ( 2 ) + R A + B 1 2 ( 2 ) • A N ( 2 ) - P * R A * * 3 * B 2 2 ( 2 ) / ( 2 . n * R H O ) P D ( 3 ) = - R A A * B 1 2 ( 3 ) * A N ( 3 ) M D 3 S = A ( 2 , 4 ) * S M B 3 S = - A ( 4 , 4 ) * S B ( I , l ) = P D ( 2 ) - P D ( l ) + D N ( 2 ) * A N ( 2 ) - D N ( l ) * A N d ) B ( 2 , 1 ) = P D ( 3 ) - P D ( 1 ) - D N d ) * A N ( 1 ) + D N ( 3 ) * A N ( 3 ) + M D 3 S * A N ( 3 ) B ( 3 , 1 ) = - B N ( 2 ) * A N ( 2 ) - B N ( l ) + A N d ) B ( 4 , 1 ) = - B N ( 3 ) * A N ( 3 ) - B N ( 1 ) * A N ( 1 ) - M B 3 S * A N ( 3 ) I F ( K . E Q . 2 ) GO TO 500 I F A I L = 1 CALL F n 4 A A F ( A , 4 , B , 4 , 4 , 1 , C , 4 , WK S P , I F A I L ) I F d F A I L . E Q . O ) GO TO 50 W R I T E ( 2 , i o n )

100 F0RMAT(23H F A I L U R E I N MATRIX ROUTINE) STOP GO TO 50 1 500 C 1 = A ( 1 , 1 ) - A ( 1 , 2 ) C 2 = A ( 1 , 3 ) + A ( 1 , 4 ) C 3 = A ( 3 , 1 ) - A ( 3 , 2 ) C 4 = A ( 3 , 3 ) + A ( 3 , 4 ) C ( 3 , 1 ) = ( C 3 * B ( 1 , 1 ) - C 1 + B ( 3 , 1 ) ) / ( C 2 * C 3 - C 1 * C 4 ) C ( 1 , 1 ) = ( B ( 1 , 1 ) - C 2 + C ( 3 , 1 ) ) / C l C ( 2 , 1 ) = - C ( 1 , 1 ) C(4, 1 ) = C(3, 1 ) 50 CONTINUE 50 1 CONTINUE RETURN END

D 1 5 -SUBROUTINE C M A T C E L , E T , G L T , U M L T , N T H E T A , N ) COMMON / C L J M E / C ( 2 n , 3 , 3 ) T H E T A = N T H F T A * 3 . 1 4 1 5 9 2 6 5 4 / 1 B n . O U M T L = ( E T / E L ) * U M L T A M = 1 - ( U M L T * U M T L ) CA=CÜS(THE1A) S = S I N ( T H E T A ) P = 2 . n * E L * U M T L Q = 4 . n * A M * G L T K = C A * * 2 U = S * * 2 V = C A * * 4 U = S * * 4 X = C A * * 3 Y = S * * 3 C ( N , 1 , 1 ) = ( E L * V + ET*W + ( P + Q ) * R * U ) / A M C ( N , 2 , 2 ) = ( E T * V + EL*W + ( P + Q ) • R + U ) / A M C ( N , 3 , 3 ) = ( ( E L + E T - P ) * R * U + ( U / 4 . n ) + ( R - U ) * * 2 ) / A M C ( N , 1 , 2 ) = ( ( E L - » - E T - Q ) * R * U + ( P / 2 . O ) • ( V+W) ) / A M C ( N , 2 , 1 ) = C ( N , 1 , 2 ) C ( N , 1 , 3 ) = ( ( E T - ( P + Q ) / 2 . n ) * Y * C A - ( E L - ( P * Q ) / 2 ) « X ^ S ) / A M C ( N , 3 , 1 ) = C ( N , 1 , 3 ) C ( N , 2 , 3 ) = ( ( F T - ( P + 0 ) / 2 . 0 ) * X * S - ( E L - ( P + Q ) / 2 . O ) * C A * Y ) / A M C ( N , 3 , 2 ) = C ( N , 2 , 3 ) RETURN END SUBROUTINE A D D S U B T ( A , R , W , N A S ) DIMENSION A ( 3 , 3 ) , B ( 3 , 3 ) , W ( 3 , 3 ) DO 10 1 = 1 , 3 DO 10 J = 1 , 3 I F ( N A S . E Q . O ) GO TO 20 W d , J ) = A ( I , . J ) - B d , J ) GO TO 10 2n W( I , J ) = A ( I , , J ) + B ( I , J ) 10 CONTINUE RETURN END SUBROUTINE MATPROP COMMON /C0MA/EL(3),ET(3),GLT(3),UMLT(3) READ(1,2) NMAT 2 FORMATdO) DO in 1=1,NMAT

RFAD( 1,1) E L d ),ETd ),GLT(I ),UMLTd ) 10 CONTINUE

1 FORMAT(4F0.n) RETURN

SUBROUTINF MULT(A,B,W) DIMENSION A(3,3),B(3,3),W(3,3) DO in 1=1,3 DO 10 J=1,3 iMd, j) = n.n DO 10 N=l,3 in W(I,J) = W d , J ) + A ( I , N ) * B ( N , J ) RETURN END SUBROUTINE INVER(A,R) DIMENSION A(3,3) ,B(3,3) S=B(1,1)*R(2,2)*B(3,3) -R(1,1)*B(2,3)**2 1 +2.n*B( 1 ,2)*B(2,3)*B( 1,3) -B(3,3)•B(1,2)••P 2 -B(2,2)*B(l,3)+*2 A(1,1)=(B(2,2)*B(3,3)-B(P,3)**2)/S A(2,2)=(B(1,1)*B(3,3)-B(1,3)**2)/S A(3,3)=(B(1,1)*B(2,2)-B(1,2)**2)/S A(1,2)=(B(1,3)*B(3,2)-B(1,2)*B(3,3))/S A(1,3)=(B(1,2)*R(2,3)-B(1,3)*B(2,P))/S A(2,3)=(B(1,3)*B(1,2)-R(1,1)*B(2,3))/S A(2,1)=A(1,2) A(3,1)=A(1,3) A(3,2)=A(P,3) RETURN END SUBROUTINE ELASPROP (K,TH) DIMENSION TH(3) COMMON /COMl/ BB(3,3,3),DD(3,3,3),GG(3,3,3) COMMON /C0M5/ EX(3),FY(3),GXY(3),AMUYX(3),AMUXY(3) T=TH(K) EX(K)=l.0/(GG(K,1,1)*T) EY(K)=1.n/(GG(K,2,2)*T) GXY(K)=1.n/(GG(K,3,3)*T) AMUYX(K)=-GG(K,1,2)/GG(K,2,2) AMUXY(K)=-GG(K,1,2)/GG(K,1,1) RETURN END

D 1 7 -1 2 SUBROUTINE L A Y U P ( N , T , K ) DIMENSION T ( 3 ) COMMON / C O M C / M T ( 2 0 ) , L R ( 2 0 ) / C O M D / H ( 2 1 ) R E A D ( 1 , 1 ) N R E A D ( 1 , 1 ) ( M T ( I ) , 1 = 1 , N ) R E A D ( 1 , 1 ) ( L R ( I ) , I = 1 , N ) R E A D ( 1 , 2 ) ( H d ) , I = 1 , N + 1 ) T ( K ) = - H ( 1 ) + H ( N + l ) F O R M A T ( 2 0 i n ) F O R M A T ( l O F O . O ) RETURN END SUBROUTINE 0 COMMON / C 0 M 5 10 FORMAT ( / / / / 1 2 X , 3 2 H 2 0 FORMAT ( 2 9 X , 30 FORMAT ( 2 X , 2 40 FORMAT ( 2 X , 2 50 FORMAT ( 2 X , 2 60 FORMAT ( 2 X , 2 70 FORMAT ( 2 X , 2 WRITE ( 2 , i n ) WRITE ( 2 , 2 0 ) I F ( N E L - 3 ) 1 , 2 , 2 1 E X ( 3 ) = n . n E Y ( 3 ) = n . n G X Y ( 3 ) = 0 . 0 A M U Y X ( 3 ) = 0 . 0 A M U X Y ( 3 ) = 0 . n 2 WRITE ( 2 , 3 0 ) WRITE ( 2 , 4 0 ) UTEP ( N E D / EX ( 3 ) , E Y ( 3 ) , G X Y ( 3 ) , AMUYX ( 3 ) , AMUXY ( 3 ) , 2 X , 3 2 H S H F L L ELEMENT E L A S T I C P R O P E R T I E S , / , , / / ) 9HELEMENT 1 , 6 X , 9 H E L E M E N T 2 , 6 X , 9 H E L E M E N T 3 , / ) 5 H A X I A L MODULUS ( E X ) 5HH00P MODULUS ( E Y ) 5HSHEAR MODULUS ( G X Y ) SHPOISSON'S RATIO M U ( X , Y ) SHPOISSQN'S R A T I O M U ( Y , X ) = , 3 ( 1 X , E 1 4 . 7 ) ) = , 3 ( 1 X , E 1 4 . 7 ) ) = , 3 ( 1 X , E 1 4 . 7) ) = , 3 ( I X , E l 4 . 7 ) ) = , 3 ( U , E 1 4 . 7 ) ) WRITE WRITE WRITE RETURN END ( E X d ) , I = 1 , 3 ) ( E Y ( I ) , I = 1 , 3 ) ( 2 , 5 0 ) (GX Y d ) , I = 1 , 3 ) ( 2 , 6 0 ) ( A M U X Y d ) , 1 = 1 , 3 ) ( 2 , 7 0 ) ( A M U Y X d ) , 1 = 1 , 3 )

SUBROUTINE OUTLOAD ( N E L )

COMMON / C 0 M 7 / A N X ( 3 ) , A N Y ( 3 ) , A B M X ( 3 ) , A B M Y ( 3 ) , A Q X ( 3 ) i n FORMAT ( / / / / 2 X , 3 3 H L 0 A D S AND COUPLES AT THE J U N C T I O N ,

1 1 6H OF THE E L E M E N T S , / , P X , ? n H , 2 2 9 H , / / )

20 FORMAT ( 2 X , 4 4 H ' X ' D I R E C T I O N PARAMETERS ARE QUOTED PER U N I T , / , 1 2 X , 2 B H C I R C U M F E R E N T I A L SHELL L E N G T H , / )

30 FORMAT ( 2 X , 4 4 H ' Y ' D I R E C T I O N PARAMETERS ARE QUOTED PER U N I T , / , 1 2 X , 2 3 H M E R I D I 0 N A L SHELL L E N G T H , / ) 40 FORMAT ( 2 9 X , 9 H E L E M E N T 1 , 6 X , 9 H E L E M E N T ? , 6 X , 9 H E L E M E N T 3 , / ) 50 FORMAT ( 2 X , 2 5 H L 0 N 6 I T U D I N A L LOAD N ( X ) = , 3 ( 1 X , E 1 4 . 7 ) ) 60 FORMAT ( 2 X , 2 5 H H 0 0 P LOAD N ( Y ) = , 3 ( 1 X , E 1 4 . 7 ) ) 70 FORMAT ( 2 X , P 5 H M E R I D I Q N A L R . M . M ( X ) = , 3 ( 1 X , E 1 4 . 7 ) ) 80 FORMAT ( 2 X , 2 5 H H 0 0 P B . M . M ( Y ) = , 3 ( 1 X , E 1 4 . 7 ) ) 90 FORMAT ( 2 X , 2 5 H T R A N S V E R S E SHEAR 0 ( X ) = , 3 ( 1 X , E 1 4 . 7 ) )

100 FORMAT ( 2 X , 4 2 H E N D MOMENTS ARE QUOTED R E L A T I V E TO E L E M E N T , / , 1 2 X , 1 2 H N E U T R A L A X E S , / / ) WRITE ( 2 , 1 0 ) WRITE ( 2 , 2 0 ) WRITE ( 2 , 3 0 ) WRITE ( 2 , i n n ) WRITE ( 2 , 4 0 ) I F ( N E L - 3 ) 1 , 2 , 2 1 A N X ( 3 ) = 0 . 0 A N Y ( 3 ) = n . n A B M X ( 3 ) = 0 . 0 A 8 M Y ( 3 ) = 0 . n A Q X ( 3 ) = n . 0 2 WRITE ( 2 , 5 0 ) ( A N X ( I ) , I = 1 , 3 ) WRITE ( 2 , 6 0 ) ( A N Y d ) , 1 = 1 , 3 ) WRITE ( 2 , 7 n ) ( A B M X d ) , 1 = 1 , 3 ) WRITE ( 2 , 8 0 ) ( A B M Y d ) , 1 = 1 , 3 ) WRITE ( 2 , 9 0 ) ( A Q X ( I ) , I = 1 , 3 ) RETURN END

-D19-SUPRÖUTINE OUTDECAY (AN)

C 0 - : M 0 N / C 0 M 2 / H L ( 2 0 ) , W ( 2 0 ) , A M X ( 2 0 ) , 0 Q ( 2 0 ) , A M Y ( 2 0 ) , X ( 2 0 ) 10 FCRHAT ( / / / / , 2 X , 4 0 H D E C A Y OF LOADS AND COUPLES FOR ELEMENT 1 ,

1 / , 2 X , 4 0 H . / / ) 20 FORMAT (2X,40HLOADS AND COUPLES ARE QUOTED PER INCH OF,

11X,12HSHELL L E N G T H , / , 2 X , 30HilERlDIONAL OR CIRCUMFERENTIAL,, 21X,23HWH1CHEVER IS A P P L I C A B L E , / / )

3C FORMAT ( 4 X . 3 H ' X ' , 7X,9HHOOP LOAD, 2X, 22HMER I Dl ONAL HOOP B. M. , 12X,20HTRANSVERSE RAD 1 AL, / , 2 X , 8HD1STANCE,6X, 4 H N ( Y ) , 6 X ,

29HB.M. H ( X ) , Ó X , 4 H M ( Y ) , 4 X , 2 2 H S H E A R Q(X) D E F L E C T I O N , / ) 40 FORMAT ( 6 ( 1 X , E 1 1 . 4 ) )

50 FORMAT (2X,36HMERID10NAL LOAD N(X) IS CONSTANT A T , E 1 2 . 5 . / / ) WRITE ( 2 , 1 0 ) WRITE ( 2 , 2 0 ) WRITE ( 2 , 5 0 ) AN WRITE ( 2 , 3 0 ) DO 1 1 = 1 , 1 1 1 WRITE ( 2 , 4 0 ) X ( l ) , H L ( l ) , A M X ( l ) , A M Y ( l ) , Q Q ( I ) . W ( l ) RETURN END F I N I S H

Hoop Winding

Element 3 M-1^3 & N2 Ni / ' M , Element 2 Element 1 Reference Surtoces /

z w ^x

Mxy Qy

M , , . ^ ^ x

ay

( a ) ^

h

h j

N .

r ( b ) h2 .•.••/90 l A 5 Reference Surface • 60

z

t

h z

Reference Surface

Axis Of Revolution

N(|)*aN^d(() d(|)

Q(j)*dQ^d0

ad)

% ^

Reference Surface Element ^ Element 2 Reference Surface

Reference Surfaces

Element 1 t 6 0 , 20% Hoop(90) Element 2 ±60

ELEMENT 1 90 tSO Reference Surface 160 ELEMENT 2 ELEMENT 3 Reference Surface