Computer programme for the determination of discontinuity forces and moments at a reinforcing ring on a circular anisotropic pressure cyliner

November 1974

CRANPISLD INSTITUTE OF TECHNOLOGY

COLLEGE OP AERONAUTICS

JECHNIS

LUCHTVAAR FECHfiiiEK

KIuyverweg 1 - DELFT

COMPUTER PROGRAr.ïï-lE FOR THE DETERMINATION OF

^g.

«77

DISCOOTINUITY FORCES AND MOMEOTS AT A REIIiFORCING RING

ON A CIRCULAR ANISOTROPIC PRESSURE CYLINDER

by

R.C, Fex^s M.Sc.

R. Tetlow M.Sc.

SUJv'J^lARY

A computer programme is presented for the evaluation of

discontinuity forces and moments at a reinforcing ring on a thin

walled cylindrical pressure vessel. Both ring and pressure

cylinder may be composed of a number of layers of uni-directional

fibre reinforced material. Layer thicknesses and fibre

orientations may vary throughout the total shell thickness and

up to three differing layer materials may be used.

V/ork carried out as part of M.O.D. (P.E.) Agreement No,

AT/2028/059GC 'Study of the application of modern materials

CONTENTS P a g e 1. 2. 2.1 2.2 2.5 2.4 2.5 2.6

3.

4.

NOTATION INTRODUCTION

PROGRAMTffi DESCRIPTION AND OPERATION

Type of Construction

Co-ordinate System

Sign Conventions

Programme Input Data

Programme Output

Programme Use

CONCLUDING REMARKS

REFERENCES

APPEI^JDIX A Short Cylinder Influence Coefficients

APPENDIX B Worked Examples and Programme List

1

2

2

2

2

3

5

6

6

7

Al Bl

1. Typical Component

2. Axis System

3. Typical Laminated Structure

4. Curvilinear Co-ordinate System

5. Sign Convention

6. Redundancy Sign Convention

7. Worked Example 1

8. Radial Displacements (Worked Example 1)

9. Transverse Shear and Meridional Bending Moment Distributions for Element 1 (Worked Example 1)

10. Transverse Shear and Meridional Bending Moment Distributions for Element 2 (Worked Example 1)

NOTATION

E Youngs Modulus

G Modulus of Rigidity

\i

Poisson's Ratio

N In plane force per unit length

M Bending moment per unit length

Q Transverse shear force per unit length

w Radial Displacement

R Radius of cylinder (internal)

P Internal pressure

P Constant for cylindrical shells p = /—

^^-^

V 4R ^^1^22

L Length of shell element

6 Influence coefficient for radial displacement

e Influence coefficient for slope (•3—)

A Total radial displacement

n

See Ref.4 for definition of

tij

h^y i

n

k^

n

=i/"'

CHC'

-

Vi')

See Pig.3

til]= f'ij - tlljtlj' tlJ

SUBSCRIPTS X y xy o 1 2 l t It T S k

Parameter has direction parallel to the 'x' axis 1

Parameter has direction parallel to the 'y' axis

^i

See Fig.4 Parameter is referred to an 'x,y' plane

Parameter is applicable to conditions at the end of the cylinder

Parameter is referred to shell element 1 Parameter is referred to shell element 2

Parameter is referred to the longitudinal direction Parameter is referred to the transverse direction Parameter is referred to an (l>t) plane

Total radial displacement

Parameter referred to steady state conditions Parameter referred to kth layer

1

-1. INTRODUCTION

The computer programme presented in this report forms part of a study concerned with the application of fibre reinforced materials to Guided Weapons.

Fibre reinforced composites normally consist of two materials bonded together so that the gross properties of the composite are superior to those of any one of the materials if used on its own. These composites are used to best advantage when constituent high modulus fibres can be accurately aligned in the directions of applied structural loading. Convenient manufacturing processes allowing

accurate fibre orientation are those of filament and polar winding, both of which involve the automated winding of resin rovings around an accurately formed mandrel. Cylindrical vessels can be produced with relative ease by these methods and have obvious application to the manufacture of guided weapons in the form of bodies and rocket motor cases.

Such structures are normally subjected to internal pressure and may require end closure to contain the pressure and reinforcing rings for the input of external loads (e.g. wing loads). Both end closure and reinforcing rings constitute discontinuity to the basic pressure shell and as such Initiate local stress distribution.

These local stress distributions may cause premature failure and therefore some method of Investigating their magnitude, direction and decay, is essential if a sound lightweight design is to be achieved.

Force and moment distributions at the Junction of a domed end closure with a pressure cylinder are dealt with in Ref.l. This report and programme is therefore concerned with the computation of similar forces and moments experienced at the

junction of a pressure cylinder with a reinforcing ring. At

Appendix A of Ref.l influence coefficients in slope and deflection are developed for long circular cylinders, where the effects at one end are not influenced by effects at the other. For short cylindrical rings however this long cylinder assiimption may no longer apply and the shell bending theory of Ref.l has to be modified to take account of the close proximity of the two ends. The relevant theoretical considerations are developed at Ref.2

for isotropic materials and the same basic method in conjunction with the relationships of Refs, 3 and 1 are employed at Appendix A of this report to deal with short laminated anisotropic cylinders.

As was the case in Ref.l the method of influence coefficients is used to satisfy equilibrium and compatibility requirements at the junction of the elements and to thereby evaluate the redundant shear forces and moments at the joint. The programme, which is listed together with two worked examples at Appendix B, contains several subroutines that are common to the End Closure Discontinuity Programme of Ref.l. In fact the general treatment of laminated

anisotropic shell structure together with much of the shell bending theory used in this report is common to that established at Ref.l and before reading this report in detail the reader should therefore be fairly familiar with the contents of Ref.l.

2. PROGRAMME DESCRIPTION AND OPERATION

2.1 Type of Construction

The programme evaluates membrane and bending loads in the vicinity of a reinforcing ring on a pressure cylinder (See Pig.l). Solutions are geared to laminated construction and both pressure shell and reinforcing ring may contain up to twenty individual laminations. Each leunination is considered as an orthotropic continuum and may be orientated with its ajces of directionality at any angle to the global axes of the composite. Materials used in the construction are specified by four elastic constants referred to the longitudinal-transverse plane and up to three differing

materials may be used throughout the cross-section.

The location, material, thickness and orientation of each layer is identified by use of an arbitrary reference surface from which all thickness, or (h) dimensions relate (See Fig.3). Where the niAmber of laminations is small, information can be fed

into the programme relating to each Individual layer. In the case of cross plied laminates (i.e. alternative layers set at +6 and -e to the global axes) however, it may be difficult to specify the location of individual layers and a facility exists whereby information can be input in 'block'. In such cases each block (three maximum) is idealised in the programme to contain six layers? three at +0 and three at -0, and data relating to each layer is generated in a similar fashion to the manual choice of Fig.3a (See Fig.3b). This facility ensures the correct stiffness distribution throughout the composite thickness and considerably reduces the amount of input data required by the programme when either shell element contains cross plied iMiinates.

2.2 Co-ordinate System

Both ring and cylindrical shell are identified by an x, y, z curvilinear co-ordinate system such that the 'x' axis follows the axis of the cylinder, the 'y' axis a cylinder circumference and the z axis a cylinder radius (See Pig.4)

2.3 Sign Conventions

Positive bending moments give compression on the outer surface and positive transverse shears are as shown in Fig.5.

Positive membrane forces are tensile and shell radial displacements are positive inwards. The fibre orientation in uni-directional laminates can vary from +90° (Hoop direction) through 0° (axial) to -90° (hoop) according to the convention shown in Pig.2.

Laminate locations ((h) dimensions see Pig.3) input to the programme must be punched as negative when referring to laminates inward of the reference surface.

3

-2.4 Programme Input Data

Data input to the programme comprises both real and integer variables. Integer variables should always be punched without a decimal point and real variables with a decimal point. All data should commence at card column one and where more than one variable is called for per card, each variable should be separated from the next by leaving one space. In the following description of the input data required by the programmci the actual variable names are stated in order to show the number and type of variables required per card.

a) NI'lAT Single integer variable

NiviAT = The number of separate materials being used in the construction (1,2, or 3)

Four real variables per card NMAT repetitions

b) EL(I) ET(I) GLT(I) UMLT(l)

EL(I) ET(I) GLT(I) UMLT(l)

EL(I) ET(I) GLT(I) UMLT(l)

EL = Longitudinal Young's Modulus

ET = Transverse Young's Modulus

GLT = Shear modulus in the l,t plane

UMLT = Poisson's Ratio (M-I^^)

c) RA AN AL(1) AL(2) P Five real variables

RA = Internal radius of the pressure cylinder

AN = Applied membrane force per unit length (usually longitudinal pressure load (PxRA)/2.0

AL(1) = The length of shell element 1 (see Fig.6)

AL(2) = The length of shell element 2. If either AL(l) or AL(2) is greater than 4'n'/p for the respective shell element the programme will set the

parameter to 4'n"/P

P = Internal pressure applied to the cylinder.

d) INPT Single integer variable

INPT = Marker for 'blocked' cross-ply input. INPT can be punched as zero, or one and remaining data depends on the value chosen. Punch INPT as one for explicit data input, or zero for 'blocked' cross-ply data.

If INPT is punched as one remaining data Is as follows

e) N Single integer variable

N = The total number of laminations in shell element 1

f) MT(I) 1 = 1 to N

MT = An Integer array of (N) variables describing the material of each layer. If MT(l) is punched as one then the material of layer one will be that identified by the first of cards(b)

g) LR(I) I = 1 to N

LR = An Integer array of N variables describing the orientation of each laminate. For the layup of Fig.3a LR would be 6oA-60A60 A9QA,whereA denotes a space.

h) H(I) I = 1 to N + 1

H(l) = A real array of N + 1 variables describing the location of each layer in the material thickness, For the layup of Pig,3a H(l) would be the

numerical equivalents of -h, A - h p A-h, A^ii^ ^c;

Data cards (d) to (h) must now be repeated to describe the layup of shell element, 2 (see Fig.6)

If INPT is punched as zero data cards (e) to (h) must be replaced by the following

e2) NS Single integer variable

NS = The number of layer types in the shell cross-section. For the layup of Pig.3t) NS would be punched as 3

f2) MARK(I) I = 1 to NS

MARK(I) = An integer array of NS variables describing the type of layer under consideration. If MARK(l) is punched as zero then layer one is a cross plied layer. If MARK(l) is punched as zero then layer one is hoop, axial or isotropic. For the layup of Pig.3b MARK would be punched O A o A l ,

g2) I4MT(I) I = 1 to NS

r^T(l) = An integer array of NS variables describing the material of each layer type. Similar to card (f)

5

-h2) LLR(I) I = 1, NS

LLR(I) = An integer array of NS variables describing the orientation of each layer type. For cross plied laminates, only the positive angle need be entered, For the layup of Fig.3b MI^IT would be punched

6 O A 4 5 A 9 0 .

12) HH(I) I = 1 to NS + 1

HH(I) = A real array of NS + 1 variables describing the location of each layer type in the material thickness. For the layup of Pig.3b HH would be punched as the numerical equivalents of

- h ^ A h g A h ^ A h ^ .

2.5 Programme Output

The first data output bv the programme are the section constants in the global (x,y) axis system for both pressure cylinder and ring i.e.

Axial Youngs Modulus E

Hoop Youngs Modulus E

Shear modulus G

Poisson's Ratio u „ and u

xy ^yx

This output is followed by two tables describing the magnitude and decay of the discontinuity forces and couples

induced at each shell element. For the ring element (element 1) data is output at eleven stations covering the half width of the ring. All loads and couples are symmetric about the ring centre line and therefore this output is applicable to the entire ring width. For the cylindrical element (element 2) output is again given at eleven equally spaced stations but this time the increment length is set to match the total decay length of the parameters under consideration. For both elements 1 and 2 the origin of the distance 'x' is at the joint of the ring edge with the cylinder (see Fig.6). Information is given regarding the effect of the discontinuity on the following parameters.

Hoop load per unit circumferential shell length N

Meridional load per unit meridional shell length N

Hoop bending moment per unit meridional shell length M

Meridional bending moment per unit circumferential shell length M

Transverse shear force per unit circumferential shell length Q,

Radial displacement of the shell boundary w

2.6 Programme Use

Typical applications of the programme are shown in the two worked examples at Appendix B. It is important to note that although the examples are evaluated in lb.in.units any consistent system of units may be used.

3. CONCLUDING REMARKS

As is mentioned in the Introduction to this report there is an obvious application of laminated fibre reinforced materials in the production of cylindrical pressure vessels. Before

these 'modern' materials can be effectively used for this purpose however, theoretical methods of stress analysis must be readily available and it is the purpose of this suite of

programmes to highlight the techniques required and to facilitate the subsequent design process. The programmes of Ref.4

demonstrate methods of evaluating membrane stiffness and strength characteristics of laminated structures and cover the primary load situation experienced in pressure cylinders. With the use of the current programme and the programme listed in Ref.l means are now available for the Investigation of secondary load

distributions associated with two types of structural discontinuity. Both end closure and reinforcing ring discontinuity load

distributions cause local bending of the shell structure and hence an additional stress distribution to that used in the strength predictions of Ref.4. The next phase in the work programme must therefore be to provide a method of evaluating individual lamina stress levels due to simultaneous bending and membrane loading and to thereby predict shell strength in the region of discontinuities.

7

-4.

1, 2.

3.

4.

REFERENCES FEWS, R.C. TIMOSHENl^O, S. WOINOWSKY-KRIEGER, S DONG, S.B. PISTER, K.S. TAYLOR, R.L. TAYLOR, P.T.

Computer Programme for the Determination of Discontinuity Forces and Couples at the Junction of Laminated Fibre Reinforced

Cylinder and Domed End Closure, Cranfield Institute of Technology, College of Aeronautics Memo 7501

Theory of Plates and Shells. McGraw-Hill Book Company Inc.

On the Theory of Laminated Anisotropic Shells and Plates, Journal of Aerospace Sciences August 1962,

Computer Programmes for the

Determination of Elastic Properties and Strength of Thin Laminated

Fibre Reinforced Composite Plates, Cranfield Institute of Technology, Cranfield Memo-No.92

APPENDIX A

SHORT CYLIITOER INFLUENCE COEFFICIENTS

1, GE^JERAL

At Appendix A Ref,l a simplified shell theory using Love's first approximation,is used to develop the differential equation relating radial displacements to axlsymmetrically applied

loading parameters for laminated cylindrical shells. The solution to this equation is given at Ref,1 to the case where conditions at one end of the cylinder can be assumed as being independent to conditions at the other» i.e. for cases where the cylindrical shell is relatively long. For short cylindrical rings this criterion no longer applies and in this appendix

the solution will be re-stated so that due account can be taken of the close proximity of the two ring ends. On substitution of the shell boundary conditions into this short cylinder

solution a modified equation for radial displacement is obtained, The influence coefficients required for redundant force and

moment calculation at the ring-cylinder joint are deduced from this equation and expressions are established to describe the decay of forces and moments at stations in the vicinity of the

joint.

2. THEORY OP SHORT LAMINATED CYLINDERS

2,1 Differential Equation and Solution

The differential equation relating radial displacement to applied ajcisyrametrlc loading parameters is defined in Ref.l ast

ax^ ^^ ^11 ^22 ^^11 ^ ^22 ^ ^ ^ y

The homogeneous solution to this equation is given in Ref,2 as the following

w = e^''^(C^ cos px + Cg sin f3x) + e"^"'^(C, cos Px + C. sin Px) ,..(A1) where C, , Cp, C^ and C^ are arbitrary constants,

If the origin of co-ordinates is taken at a mid length position in the cylinder the expression for (w) must be an even function of x and C^ and C^ are zero. Re-writing (A)

with the exponential functions replaced by hyperbolic equivalents gives:

A2

-Therefore

èw

^ = p C-j^ «rsin px cosh px + sinh px cos pxj'+ C^ -Tcos px sinh Px - cosh Px sin Px>

•^-^ = p^ 120, cosh Px cos Px - 2C,. sinh Px sin px Bx^ • -^ ^

...(A3)

...(A4)

^ = 2 P ^ èx^

C, «T- cosh Px sin px + cos Px sinh Px}"

-{^

- C,, <sinh Px cos px + sin Px cosh P x ^

...(A5) 2,2 Particular Solution For Transverse Shear

Consider the cylinder under the action of a uniformly distribution end shear force Q^

Boundary conditions

at X = L/2J Qx = - %i ^x = °

But - a _X - d,, ^ - 4 See Ref.l

Let |ii = a

Therefore from (A5) and the boundary condition ^x ~ ~ "^o

-So = 2P <»11

+ sin a cosh

-, •< cosh a

a-}

•1

siho^ - cos a sinhoC ^+ C. J^sinh a cos a ,..(A6)

^2

Also M^ = - d,, ^-% See Ref.l

Therefore from A4 and the boundary condition M„ = 0

.,,(A6b)

2P^d 11

1

C^ sinh a sin a - C, cosh a cos a C, sinh a sin a

* n — _ii

• • •• cosh a cos a

= 0

Substituting for C, in (A6) yields

=4 =

-cosh a cos a

sinh a cosh a + sin a cos a

Q,

2P^d 11

And substituting for C^ in (A7)

^1 =

-sinh a sin a

sinh a cosh a + sin a cos a

Q,

U 2p-"d ₁₁

Therefore from (A2)

V7 =

-2p-'d 11

sinh a sin a

sinh a cosh a + sin a cos a • s i n Px s i n h Px +

. . . n h r. . n ^ h ^ ' ' ^ ' ' c ? r . " ^ on. r. )>COS PX COSh P x sinh a cosh a + sin a cos a f

At the ends of the cylinder where x = L/2

w = V^ = -2P^d 11 cosh 2a + cos 2a sinh 2a + sin 2a .,.(A8) ...(A9)

2,3 Particular Solution For End Moment

Boundary conditions

at X = |, M^ = M^; Q^ = 0

substituting Q, = 0 = o into (A6) yields

X. • o

°i = - s

sinh a cos a + sin a cosh a cosh a sin a - cos a sinh a

,.,(A10)

...(All)

Prom (A6b), (A4) and (AlO)

% = 2P d^^ C. Sinh a sin a - C, cosh a cos a

•1

...(A12) Substituting (All) into (A12) and solving for C^ yields

M. 2p^d 11 cosh g sin cosh a sinh a - sinh a cos a , a + sin a cos a ...(A13)

- A4 ~

And from (All)

M

C^ = - o

2p^d 11

sinh g cos a + sin a cosh a cosh a sinh a + sin a cos a

...(A14)

SubstitutiP,'5 (A13) and (I4) into (A2) yields

M.

w = sinh g cos g + sin a cosh g _{cosh g sinh g + sin g cos a}

2P

a^^ i^

cosh g sin a - sinh g cos g

.sin Px sinh Px +

'COS Px cosh px cosh g sinh g + sin g cos g

At the ends of the cylinder x = L/2 and (AI5) reduces to

M_ ...(15) v/ = w, 2 P % 11 sin 2g - sinh 2g sinh 2g + sin 2g ...(AI6)

2.4 Particular Solution For Force N.

On at least one of the shell elements it is likely that the membrane tension N has a line of action in a plane other than the neutral axis of the shell element concerned. In such cases a bending moment of magnitude b,, N is induced and the boundary conditions become:

When X = L/2j M^^ = - b^^ N^j 0,^ = 0

This is identical to the previous case (2.3) with the applied end moment M. replaced by -b-,, N

O J- X -^ w, ^ 1 \ 2P''d 11 sin 2g - sinh 2g sinh 2g + sin 2g ...(AI7) 2.^ :> Influence Coefficients

In the description of shell influence coefficients tine following notation is introduced.

6 = Influence coefficient for radial displacment 5

èw 9 - Influence coefficient for slope -^

The subscript 's' can be any of the following

M Parameter is a result of an applied end bending moment (M^)

Q Parameter is a result of an applied end transverse shear

N Parameter is a result of applied membrane force (N^)

P Parameter is a result of internal pressure

Therefore from equations (A9), (Al6) and (A17)

z. " 2 ^ " 2d,,s2 ''l 5 - • • - • •, ^ 2d^^P^ . "^11 ^2 Where , cosh 2g ^1 sinh 2g , sinh 2g ^2 ~ sinh 2g . cosh 2g ^J> - sinh 2g 6L g - ^ + + "+"

"?"

cos sin sin sin cos sin 2g 2g 2g 2g 2g % \ % - ^ ^ - d,,P ^2 2d,,p2 k^ h^^ - d,,P

Internal pressure also causes dilation of the shells and in

the notation of this report an influence coefficient for pressure loading can be written as follows.

t"

A6

-3. SOLUTION OF REDUNDANCIES

3.1 Equilibrium Relationships

With reference to Pig.6 the following equilibrium relationships can be establish

Q^ + Qg = 0

Mg - M^ = 0

Ng - N^ = 0

3.2 Compatibility of Slope

Since shell and ring are rigidly connected to each other ( ÖW '

they must share a common slope Z' öw \ at the intersection

• ® = -®

• • 2 1

3.3 Total Slopes and Deflections

Using the loads and sign convention of Pig.6 the following equations can be written for the total slopes and deflections of each element at the joint line.

l^Q^l ^ 1 V l ^ 1 V l ^ l^pP = ^ 1

l^Q^l ^ i V l ^ l^N^l = ®1

2ÖQ^2 -^ 2¥^2 ^ 2 V 2 + 2h^ = \

2 V 2 ^ 2 V 2 ^ 2 V 2 = ®2

Substitution of the equilibrium and compatibility conditions into the above equations yields the following

^ l \ ' 2 ® Q ) ^ 1 + ^2% ^ l^M^f^l = - l ^ N ^ l - 2®N^2

( I 5 Q + 2 6 Q ) Q I + (i5j^ - 25j.^)M^ = (gSp - ^b^)P + 2 V 2 - i V l From these two relationships the redundant quantities Q-j^ and M, can be evaluated.

4. DECAY FUNCTIONS

4.1 Radial Displacement (w)

Due to shear force Q, (equation A8)

W = — r r

2P'^d

11

A sin Px sinh Px + B cos Px cosh Px

•]

Due to bending moment M (equation A15)

w =

2P^d

11

C sin Px sinh Px + D cos Px cosh px

•]

Due to end load N.

w = -

2

2P d

11

E sin Px sinh Px + P cos Px cosh Px

.•. By superposition

w^ = — 3

-•^ 2p^d

₁₁

A + p(C - E ) > sin px sinh px +

where

A =

-B + p(D - P) )*cosh px cos Px

sinh g sin

ct Q, ~

sinh g cosh g + sin

a

cos g

cosh g cos g Q,^

B = - --r

C =

-D =

sinh g cosh g + sin g cos g

(sinh g cos g + sin g cosh g)M

sinh g cosh g + sin a cos a

(cosh g sin g - sinh g cos g)M

sinh g cosh g + sin g cos g

...(A18)

E =

P =

c

D ' ' l l •^0 ^ 1

"x

"x

M,

A8

-4.2 Meridional Bending Moment M,

X

K

= ^ n i (see Ref.l)

,..(a)

w

2P^d

11

C^ sin Px sinh Px + C2 cosh Px cos px

...(b)

where

C3, = A + P(C - E) ; Cg = B + P(D - F) ' .,.(A19)

.•, differentiating (b) twice w,r,t x and substituting into

(a)

'•'x = ÏÏ C2 sinh Px sin px - C, cosh Px cos px

4,3 Transverse Shear Force Q^

^x = ^11

h\

hx

(see Ref,l) ,*, Prom (A18) and (AI9)

^x 1 cosh Px sin Px - cos Px sinh Px + C, sinh Px cos Px

+ sin Px cosh Px

4,4 Hoop Load Per Unit Length N

0

K

= -

W^ ' r ^ x

(S®® Ref.l)

y i\^22 22 Prom (A18) 1 N.. = y " 2P^d^^RG22

C sin Px sinh Px + Cg cosh px cos px ^12

Go^ ^^x But P^ = ^ ^ 1 1 ^ 2 2 N = - 2RP y

C^ sin Px sinh px + Cg cosh Px cos px • ^^12 N

5.

STEADY STATE FUNCTIONS

The functions for loads and couples developed at para,A4 are all derived from the homogeneous solution to the shell differential equation and as such represent only the transient parts of the total distributions. The shell steady state

conditions must now therefore be evaluated and added to the distributions of (A4) to give the complete solution to the

shell differential equation and therefore the final expressions for shell membrane loads and load couples.

5 . 1 Nov;

Steady S t a t e Radial Displacement

ÜXS _ l V 3 c s _ .!ls

^^y ^^x " ^

,•, In the notation of this report

I — G. w^ = - G22 R 5.2 Nov/ "y = Nys * G ^ «xs

Steady State Hoop Load Per Unit Length

w,

_T

G

RG

₂₂

12 '22

N.

v/rp in the above expression represents the total radial displacement experienced at the cylinder wall as defined by the homogeneous

solution to the shell differential equation for the particular boundary conditions of end shear force and end moment,

v/m must now be replaced by w where: w = w^ + w^

S

= RG

22

- w^ + G22 R | N y 3 -^ ^ N

'22 xs

G

12 22

N

xs

N,B. N^ is constant and will be equivalent to N^^ in any problem.

w„ N =

-y

RG

22

+ N For a pressurised cylinder N = PR ys ^ "^ ys

B l

-WORKED EXAMPLES AND PROGRAMME L I S T

1 . GENERAL

In this appendix two typical programme applications are

demonstrated in the form of worked examples.

The programme is developed for application to laminated

construction where each layer is orthotropic and can be

identified by its orthotropic elastic constants. This does not

limit the programme to orthotropic materials however, as

isotropy is merely a special case of orthotropy and the use of

the programme in the solution of an isotropic construction is

shown in worked example 1.

The second example demonstrates how the programme is used

in its specialised role and a carbon fibre reinforced composite

construction is analysed. In both examples only one material

is used but hybred designs involving mixed

orthotropic-isotropic laminations may be analysed if desired, provided no

more than three differing materials are used throughout the

construction.

N.B. An orthotropic layer of high modulus fibre in shear

resistant matrix is specified by E|_, E^, G,

P^i^

and is

designated as one material, i.e. upto three differing

fibre-matrix systems may be used, or one fibre-matrix

system and two isotropic materials, or three isotropic

materials etc.

2. WORKED EXAMPLE 1

A 6 in. diameter pressure cylinder has 1.5

^^

wide reinforcing

rings located at six inch intervals along its length. The total

thickness of the rings is 0.2 in., whilst the wall thickness of

the cylinder is 0.1 in. and both cylinder and ring are constructed

from light alloy. It is desired to investigate the discontinuity

situation at the ring-cylinder junction when an internal pressure

of 1000 lb/in2 is applied to the cylinder.

A diagramatic representation of the problem is given at

Fig,7.

2.1 Material Properties

Longitudinal Youngs Modulus (E^) = 10,000,000 Ib/in^

Transverse Youngs Modulus (E^) = 10,000,000 Ib/in^

Modulus of Rigidity (G,^) = 3,850,000 Ib/in^

Poisson's Ratio |J,,^

=>

0.3

2 2 INPUT DATA 1 1 0 3 . 1 2 1 1 n 0 - 0 . 0 5 0 . 0 0 . 1 5 1 ?. \ 1 0 0 - 0 . 0 5 0 . 0 0 . 0 5 n o o o n n . o l o o o o o o o . o 3 « 5 0 o o n . o 0 . 3 0 1 5 0 0 . 0 1 . 5 6 . 0 1 o n o . 0

B 3

-2.3. PROGRAMME OUTPUT

STiFFENlKGr Fl! DISCONTI Ü'JITY PROGRAMME

SHELL ELEMENT ELASTIC PROPERTIES

ELEMENT 1 ELEMENT 2

AXIAL MODULUS (EX) HOOF MOrULUS ( F Y ) SHEAR MODULUS (QXY) F - O l S S n i ' S RATIO ;-lU(X,Y) = p O j c c o M ' c PATIO MU(Y,X) = 0 , 1 0 0 0 0 0 0 E 0Ö 0 , 1 0 0 0 0 0 0 E 0Ö 0 . 3 8 5 0 0 0 0 E 0 7 0 . 3 0 0 0 0 0 0 E 0 0 0 , 3 0 0 0 0 0 0 E 00 0 . 1 0 0 0 0 0 0 E 08 0 . 1 0 0 0 0 0 0 E Od 0 . 3 8 5 0 0 0 0 E 07 0 . 3 0 0 0 0 0 0 E 00 0 . 3 0 0 0 0 0 0 E 00

DECAY OF LOADS AüD COUPLES FOR ELEMENT 1

LOADS AND COUPLES ARE QUOTED PLR INCH OF ^:ERlDIO^!AL OR CI''.'CUriFERENTl AL, WHICHEVER

SHELL LENGTH I S APPLICABLE KEPlOir^r'AL ' X ' ClSTAMCt

c. crooE

0. 7 5 r r E -0. 1500E 0 , 2 2 ' ; OF 0. 3,r^rL ^ . 375OF n . /i^rnv: r , 5 2 5 O F r,e<^'^''r. r . É 7 5 ^ F 0. 7 5^^F -00 - 0 1 00 on no 00 on '^n 0^ no 00 LOA'^ ! ' ( x ; ) IS HOOP LOAD : ' ( Y ) 0 . 4 1 7 4 E 0 , 1 8 6 b E 0.35V1E 0 , 3 3 6 ? E 0,'^161E 0 , / 9 9 4 E o.r'oCoE 0,.''756E 0 , 2 o ó 4 E 0.2C40E 0 . 2 6 2 6 E 04 04 04 04 04 04 04 0 / 04 04 04 CONSTANT AT ^ERiDior;. ^L P.M. M(X) 0.Ó798E 0 . 6 9 4 1 E 0 , 6 9 2 0 E 0 . 6 78 7 E 0.658'3E 0 . 6 3 5 3 E 0 . 6 1 2 1 E 0 . 5 9 1 5 F 0 . 5 7 5 5 E 0 . 5 6 5 3 E 0.561ÖE 02 02 02 02 02 02

o:'

02 02 02 02 , 0 . 1 5 0 0 0 E 04 HOOP B.M, M(Y) 0 , 2 0 3 9 E 02 0.20Ö2E 02 0 . 2 0 7 6 E 02 0 , 2 0 3 6 E 02 0 , 1 9 7 6 E 02 0 , 1 9 0 6 E 02 0 , 1 6 3 6 E 02 0 . 1 7 7 5 E 0? 0 . 1 7 2 6 E 02 0 . 1 6 9 6 E 02 O.1605E 02 TRANSVERSE SHEAR Q(X) 0 . 3 2 3 8 E 02 0 . 6 9 2 6 E 01 - 0 . 1 1 3 1 E 02 - 0 . 2 3 2 3 E 02 - 0 . 2 9 6 S E 02 - 0 . 3 1 5 6 E 02 - 0 , 2 9 6 7 F 02 - 0 , 2 4 8 I E 02 - 0 . 1 7 7 4 E 02 - 0 . 9 2 3 3 E 01 - 0 . 1 1 4 7 E - 0 8 RADIAL DEFLECTION - 0 . 5 5 8 6 E - 0 2 - C . 512 7E-02 - 0 . 4 7 2 1 E - C 2 - 0 . 4366E-C2 - 0 . 4C67E-02 - 0 . 3616E-C2 - 0 . 3 6 1 4 E - 0 2 - 0 . 3460E-C2 - 0 . 3 3 5 0 E - 0 2 - 0 . 3 2 6 5 E - 0 2 - 0 . 3 2 6 4 E - 0 2

PROGRAMME OUTPUT CONTINUED

DECAY n\- LOADS A:'n COUPLES FOR ELEMENT 2

LOADS A;;n C O U P L E S A R E Q U O T E D P E R I N C H OF S H E L L LENGTH MER1D10:'AL OR Cl-^CUMFEREKTl AL, WHICHEVER I S APPLICABLE

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

' X ' DIST.^'NCE 0 . CO^^^F r . 2 6 7 7 E r . 5 3 ^ ^ F 0. ec32E 0 . 1 ^ : 7 1 ^ r, 133SE 0.16C6E 0.1&74E r * 2 1 4 2 E r;2A1^E

czeii^

00 O'^ nr no 01 n i n i 01 n i 01 01 HOOP Lni N(Y) 0 . 2 3 1 2 E 0 . 2 7 7 6 E 0.30r.2E 0 . 3 0 ü 6 E 0 . 3 0 5 ü E 0 . 3 0 3 ^ E 0 . 3 0 1 0 E 0 . 3 0 0 0 E 0 . 2 9 9 7 E 0 . 2 9 9 7 E 0 . 2 9 9 7 E

\n

04 04 04 04 04 04 04 04 04 04 04 NERIDIONAL B.M, M(X) - 0 , 7 0 1 9 E 01 - 0 . 9 5 5 7 E 01 - 0 , 6 2 5 2 E 01 - 0 , 2 6 7 6 E 01 - 0 , 5 3 0 4 E 00 0 . 3 0 3 9 E 00 0 . 4 1 2 0 E 00 0 . 2 6 5 2 E 00 0 . i n 4 0 E 00 0 , 5 1 0 6 E - 0 2 - 0 . 2 6 2 1 E - 0 1 HOOP B.M. M(Y) - 0 , 2 1 0 6 E 01 - 0 , 2 8 6 7 E 01 - 0 . 1 8 7 6 E 01 - 0 . 8 0 2 8 E 00 - 0 . 1 5 9 1 E 00 0 . 9 1 1 6 E - 0 1 0 . 1 2 3 6 E 00 O . 7 9 5 5 E - 0 1 0 . 3 1 2 0 E - 0 1 0 . 1 5 3 2 E - 0 2 - 0 . 7 3 6 4 E - 0 2 TRANSVERSE SHEAR 0 ( X ) - 0 . 3 2 3 8 E 0.D509E 0 . 1 4 8 4 E 0 . 1 0 9 5 E 0 . 5 2 3 3 E 0 . 1 3 9 7 E - 0 . 2 9 1 0 E - 0 . 6 6 1 5 E - 0 . 5 0 1 0 E - 0 . 2 3 8 3 E O.OOOOE 02 01 02 02 01 01 00 00 00 00 00 RADIAL DEFLECTION - 0 . 5 5 8 6 E - 0 2 - 0 . 6S7 7E-C2 - 0 . 7657E-C2 - 0 . 7 6 4 7 E - 0 2 - 0 . 7618E-C2 - 0 . 7739E-C2 - 0 . 767SE-02 - 0 . 7650E-C2 - 0 , 764 I E - 0 2 - 0 . 7 6 4 1 E - 0 2 - 0 . 7 6 4 2 E - 0 2

B5

-2.3 Continued

Radial displacement, meridional bendir^ moment and transverse shear distributions from the above output are shown graphically at Figs, 8,9, and 10,

3 . WORKED EXAMPLE 2

A thirty inch long,six inch diameter pressure cylinder is reinforced by a 1.5 inch wide ring at a point where external loads are to be applied to the cylinder. Both ring and cylinder are constructed from laminated carbon fibre-epoxy composite

material with geometric orientations as specified at Fig.11. It is desired to investigate the discontinuity loads and moments induced at the joint between cylinder and ring when an internal pressure of 1000 Ib/in^ is applied to the cylinder.

3.1 Material Properties

Longitudinal Youngs Modulus (E,) = 30,000,000 Ib/in^

Transverse Youngs Modulus (E.) = 1,100,000 lb/in

Modulus of rigidity (G^^) = 700,000 Ib/in^

3.2. INPUT DATA 1 30000000.0 1100000.0 700000.0 o 3.0 1500.0 1.5 30.O 1 000.0 O 3 0 O 1 1 1 1 60 60 90 -0.05 0.0 0.05 0.15 O 2

n n

1 1 60 60 -0.05 0.0 0.05

B7 -3 -3 PROGRAMME OUTPUT

STlFFE^:l^:c. Rn;o P I S C O N T I N U I T Y PROQRAMME

SHELL ELFi'ENT ELASTIC PROPERTIES

AXIAL MODULUS i-iOOF MODULUS ' SHEAR MODULUS FOI ssr*' ' s :^ATi o F ' O I S S O M ' S " A T i n (EX) (EY) (QXY) MU(X, :'!U(Y, Y) X) = = = = = ^ ELEMENT 1 0 . 1 7 6 7 3 7 5 E 0.19Ö3385E 0 . 3 3 0 0 6 9 9 E 0.122ö01Öi-: 0.137Ö107E 07 Ob 07 00 01 ELEMENT 2 0 . 14l6üöOE 0 . 7 9 3 5 6 4 0 E O.5901397E 0 . 3 1 3 4 7 3 1 E 0 . 1 7 5 5 6 9 6 E 07 07 07 00 01

L E C ; Y OF LOADS A::O COUPLES FOR ELEMENT 1

LOADS AND COUPLES ARE OUOTED PER IMCH OF SHELL LENGTH MEPlDlO'-'AL OR CIRCUMFERENTIAL, WHICHEVER IS APPLICABLE

r.ERlDIOMAL ' X ' CI STANCE r. r o ' " ^ E C. 7500E-r . 1 5 0 C E 0. 22^0E ^ . 3r'^rE 0 . 3 7 ^ ^ F r . / j r r r ^ f ^ . 5 2 ^ ^ E CirrC': r,c7^:rz 0. 7^.0^E on • 0 1 QO o r 00 nn C 00 r o nr no LOAD N(X' H^'OP L o ; ' ! ( Y ) 0 . 4 1 0 2E 0 . 3 9 6 ^ E 0 . 3 1 2 3 E 0.2Ö35E 0.26C5E o.::^r/9E 0 . 2 5 4 9 E 0.::^^>0E 0 . ? 5 6 4 E 0 . 213 70E 0.23;33E IS \D 04 04 04 04 04 04 04 04 04 04 04 CONSTANT AT MERIDION. ^L B.M. M(X) 0 . 3 2 4 6 E 0.2Ö4 3E 0 . 2 3 31 E 0 . 1 7 9 4 E 0.12Ö6E 0.O393E 0 . 4 7 1 4 E 0.1Ö71E - 0 . 1 3 1 1 E - 0 . 1 3 1 b E - 0 . 1 7 0 7E 02 02 02 02 02 01 01 01 00 01 01 , 0.1500C ^E C HOOP P.M. M(Y) 0 . 4 6 2 5 E 0 . 4 0 5 1 E 0 . 3 3 2 2 E 0 . 2 5 5 7 E 0 . 1 3 3 3 E 0 . 1 1 9 7 E 0 . 6 7 1 7 E 0 . 2 6 6 6 E - 0 . 1 8 6 7 E - 0 . 1 Ö 7 5 E - 0 . 2 4 3 3 E 02 ,02 02 02 02 02 01 01 00 01 01 )4 TRANSVERSE SHEAR 0 ( X ) - 0 . 4 1 6 3 E 02 - 0 . 6 3 1 6 E 02 - 0 . 7 1 4 1 E 02 - 0 . 7 0 6 1 E 02 - 0 . 6 4 1 5 E 02 - 0 . 5 4 5 S E 02 - 0 . 4 3 5 7 E 02 - 0 . 3 2 2 6 E 02 - 0 . 2 1 1 6 E 02 - 0 . 1 0 4 7E 02 - 0 . 1 1 9 4 E - O Ö RADIAL DEFLECTION - 0 . 1 6 0 7 E - 0 2 - 0 . 1 1 3 3 E - 0 2 - 0 . 7 9 8 9 E - 0 3 - 0 . 5 6 0 6 E - 0 3 - 0 . 4 5 1 S E - 0 3 - 0 . 3 6 7 I E - 0 3 - 0 . 3 6 4 4 E - 0 3 - 0 . 36F3E-C3 - 0 . 3 7 5 9 E - 0 3 - 0 . 3 8 6 2 E - 0 3 - 0 . 3903E-C?

3-3- CONTINUED

DECAY OF LOADS Ar:D C^.'UPLES FOR ELEMENT 2

LOAF.'S Af:r COUPLES AF:.-; nijnxED P E R ir:cH OF S H E L L L E N G T H

! ' E R I D I 0 " A L 0 " CI RCU.'lFERENTl AL, WHICHEVER I S APPLICABLE

N E F i r i O : A L L - ' . f ' ; ( X ) i s CONSTANT AT, 0 . 1 5 0 0 0 E 04

' X ' HOOP LOA[; NERIDI0?'AL HOOP B.M. TRANSVERSE RADIAL f ' l STANCE N(Y) F . M . M(X) M(Y) SHEAR 0 ( X ) DEFLECTION 0. r c o r p r - 0 . ' n 5 9 E ^4 - 0 . 1 2 0 5 E 02 - 0 . 2 2 0 5 E 02 0 . 4 1 6 3 E 02 - 0 . 1607E-02

r. 207^'E or, n.-3C45P n/, - o . 5 l 2 f i E 01 -0.8936E 01 0.2900E 02 - 0 . 2310E-02 ^.4139E •"•^ 0.3195E 04 -n,ü260E 00 -0.1439E 01 0.1319E 02 - 0 . 2121E-02 .^. 62^SE ^o 0.309b'-: 04 n.7466E 0^ 0.1301E 01 n . 3 l 3 0 E 01 - 0 . 1757E-02

r.i.ZlcE rr n.3^29E 04 O.Ö799E 00 0.1533E 01 -0.1050E 01 - 0 . 1497E-02

"^. 1C3!:E 0-1 r>.2397E 04 0.5477E 00 0.9545E 00 - 0 . 1 7 9 5 E 01 - 0 . 1376E-02 c. 1242E •'"I 0.29e9E 04 0.2232E 00 0.3S9OE 00 -0.1251E 01 - 0 . 1346F-02 0. i44Cfr n^ o.:->99?E 04 0.3708E-01 0.6462E-01 -0.5757E 0''> - 0 . 1354E-02 0. 16E6E 01 0.299ÓE 04 - 0 . 3 3 ' i 1 E - 0 1 - 0 . 5 6 9 1 E - 0 1 -0.159ÖE 00 - 0 . 1371E-02 0. 1fc63E ^1 0.2099E 04 - 0 . 4 7 5 8 E - 0 1 - 0 . 8 2 9 2 E - 0 1 - 0 . 8 6 8 8 E - 0 2 - 0 . 1382E-02 0. 207f'E ^>1 O.'^/OOOE 04 - 0 . 4 7 2 5 E - 0 1 - 0 . 8 2 3 4 E - 0 1 0.1621E-11 - 0 . 1386E-02

39

-A-O PROGRAMME LIST

I I S T ( L D ) P R O G R A M ( D O B A ) t N P U T 1 « C R O O U T P U T ? « L P 0 MASTER W l N G K l N G r i M E N S l O N A ( 3 , 3 ) » B ( 3 , 3 ) , D ( i , J ) , E ( 3 , 3 ) , F ( 3 . 3 ) . G ( 3 . 3 ) , 1 u 2 ( 3 , 3 ) » A N S f 4 , 1 ) , T ( 3 ) , R f D N ( 3 ) , R E 0 M ( 3 ) , A L ( ? ) rOMMOKi / C O M A / E L (3) , E T ( 3 ) , G L T ( ^ ) , U M L T ( 5 ) / C 0 M C / M T ( 2 ( O » L R < 2 o ) (/rOMO/H(2l)/COMfc/C(20.3,3)/(:üM1/ B B ( 3 , 3 , 3 ) , D D ( 3 , 3 , 3 ) f G G { 3 , 3 , 3 ) < : / t O M b / F X ( 3 ) . E Y ( 5 ) , G X Y ( 3 ) r A M U Y X ( 5 ) , AMUXY ( 3 > / C 0 M 6 / A B ( 3 > 3 / C O M 3 / H L ( 2 0 ) , W ( 2 0 ) ,AMX(?-0) , U Q ( 2 0 ) , A M Y ( 2 0 ) , 0 I S T ( 2 0 ) 4 / C O M 7 / A N X ( 3 ) , A N V < 3 ) , A R M X ( 3 ) , A B M Y ( 3 ) , A Q X < 3 ) CALL M A T P R O P NEL = 2 READ ( 1 , 2 ) ^ A , A N , A L ( 1 ) » A L < 2 ) , P 1 FORMAT ( 1 0 ) 2 FORMAT ( 5 F 0 . 0 ) no '•0 |«1.Nf:L PEAD ( 1 , 1 ) INPT TF ( lNpT.»iE.1 ) GO TU 33 CALL L A Y U P ( N A . T , L ) GO TO 44 53 CALL L A Y U P ? ( N A , T , L ) 4 4 C O N T I N U E 0Ü 3 0 l a l f N A T N « M T ( 1 ) C A L L C M A T ( E L ( 1 N ) , F T ( I N ) , G L T ( 1 N ) , U M L T ( I N ) , L R < I ) . I > "{} C O N T I N U E f>0 1 0 1 « 1 , 3 no 1 0 j « 1 , 3 » ( I , J ) 8 0 . U n ( I , J ) « ( ' . 0 n ( I , J ) > 0 . 0 n o <?o i f B l , N A A H a ( M ( K + 1 ) . H ( K ) )

MASTER CONT R M B H ( K * 1 ) * * 2 - H ( K > * * 2 t>H»H ( K * 1 ) * * 3 - H ( ( ( ) * * 3 A U # J ) » A ( I . J ) * C ( K , I , J ) * A H R( I r J ) « B ( I ' J ) 4 ' C ( K , I , J ) * B H ^0 n ( I , J ) B n ( I , J)<».C(K,I , J ) * D H n ( I , J ) « B ( I , J ) / 2 . 0 n ( l , J ) . 0 ( I , J ) / 3 . 0 iO CONTINUE CALL I N V E R ( G , A ) CALL M U L T ( B , 6 , E ) CALL » * U L T ( E , B , W 2 ) NAS»1 CALL A n D S U B T ( 0 , W 2 , F ' N A S > DO «1 I « 1 , 3 no 81 J«1 , 3 « B ( L / I , J > » - E ( I , J ) n n ( L , I , J ) « F ( I , J ) G G C C ' I , J ) » 6 ( I # J ) «1 C O N T I N U E 40 C O N T I N U E 90 FORMAT ( 1 H 1 , 2 X , 3 9 H S T I F F E N 1 N G RING D I S C O N T I N U I T Y P R O G R A M M E , 1 / , 2 X , 3 9 H - - - - - - - , / / ) W H I T E ( 2 , 9 0 ) CALL S0LVF1 < N E L # R A , A N , P , A N S , A L ) oO U 1.1,NEL 4 CALL E L A S P R O P ( J , T ) CALL O U T E P ( N E L ) P E 0 M ( 1 ) « A N S ( 3 , 1 ) P E D M ( 2 ) « A N S ( 4 , 1 ) p E O N d >»ANS(1 ,1 ) B E 0 N ( 2 ) « » A N S ( 2 , 1 ) Ms6 no 3 lal,2 4 M « R E D M ( I ) Q X . P E p N U ) CALL O p C A Y S H R T ( M , R A , I , A M , Q X , A N , A L # P ) CALL O U T D E C A Y ( A N , 1 ) 5 C O N T I N U E STOP OK F NO

B i l -< ; U 8 R 0 U T I N E LAYUP2 ( L , T , K ) M M E NS ION H H ( 5 ) , L L R ( 4 ) , M M T ( 4 ) , M A R K ( 4 ) , T ( 3 ) COMMON / C r i M C / M T C 2 0 ) , L R ( 2 0 ) / C Ü M D / H ( 2 1 ) READ ( 1 , 1 ) NS PEAO ( 1 , 1 ) ( M A R K ( I ) , I « 1 , N S ) "EAO ( 1 , 1 ) ( M M T d ) , 1.1 , N S ) PFAO ( 1 , 1 ) ( L L R ( l ) , I » 1 , N S ) PEAD ( 1 , 2 ) ( H H ( I ) , 1 = 1 ,NS + 1 ) T ( K ) « - H H ( 1 ) * M H ( N S + 1 ) 1 FORMAT ( 2 ü I o ) 2 FORMAT ( 1 0 F 0 . 0 ) I »0 no 3 l«l,NS IF ( M A R K d ) .EQ.1 ) GO TO 4 THsHM ( 1*1 )-HH( I ) A I N C S T H / 6 . 0 R I T = 0 . 0 no b J « 1 , 6 l=Li.1 H ( L ) » H H < 1 ) * H I T V T ( L ) « M M T ( I ) M I K » L L R ( I ) * ( - 1 ) * * L I R(L)«-Mi|r 5 B I T m B I T ^ A I N C GO TG •% 4 I5L*1 M ( L ) » H M ( I ) M T ( L ) « M M T ( 1 ) I K { L ) » L L R ( I ) 3 C O N T I N U E H ( L * 1 ) » H M ( N S * 1 ) B E T U R N FND

SUBROUTINE S0LVE1 (K,RA,AN,P,C.AL) niMENSTON C(4,1),D11(3),B?2(3),B11(3),B12(3) DIMENSION BTi(2),BT2(2>'BT3<2>'PD(2)'DNC2),BN(2),AL(2) COMMON /C0M1/ BB(3,3,3) ,t)D(3,3.3) ,G6(3,3,3)/C0M6/A8(3> PI«3.141592654 no 2 !«1 ,K 1)11 (I)«DD(I,1 ,1) R 2 2 ( I ) « G G ( I , 2 , 2 ) nl1(I)«-8B(1,1,1) ? m 2 < I )«GG<I ,1 ,2) no 10 1*1 , K A6(I)s1.Ü/((4.0*RA**2.0*P11(l)*B22(I))**0,25) nUM«4.0*PI/AB(I) IF (AL(I).GT.nUM) AL(I)»OUM 10 CONTINUE 00 11 l«1 , K A L F A s A B d )*AL( I ) nUMI»ExP(ALFA) nUM2«1 0/DUMl CSH«0. 5*<r>UMl*DOM2) «;HI«0.5*(DUMl-DliM2) CS»COS(ALFA) S I • S I N ( A L F A ) B O T S S H I * S I F l a ( C S H * C S ) / B O T F 2 » ( S H I - S I ) / B 0 T F 3 » ( C S H - C S ) / B 0 T BT1(Ï)«f3/(AB(I)*D11(1)) n T 2 d ) « F 2 / ( 2 . 0 * A B d ) * * 2 . 0 * D 1 1 ( D ) RT3(I)«Fl/(2.0*ABd)**3.0*D11(l)) PD(I)«RA*B22CI)*(P*RA*Bl2d)*AN/B22<I>) nN(I)a611 d ) * 8 T ? d ) ,1 BN(I)«8l1(I)*BT1(1) C 1 » B T 3 d ) * B T 3 ( 2 ) C 2 « B T ? ( 1 ) - B T 2 ( 2 ) C3«C2 C 4 « B T 1 d > * B T 1 < 2 ) R H S 1 « P n < 2 ) - P D d ) * D N ( 2 ) * A N - D N d > * A N B H S 2 " - B N < 2 ) * A N - B N ( 1 ) * A I H C(5,1)«(C3*ftHS1-C1*RHS2>/(C2*C3-C1*C4) C < 1 d )«<RHS1-C2*C(3,1))/Cl C(2,1 ) « - C d ,1 ) C U d )"C<3d ) RETURN FND

B13 -S U B K Ü U T l N E C M A T C E L , b T , 6 L T , U M L T , N T H Ê T A , N ) COMMON / C O M t / C < 2 0 , 3 » 3 ) T H E T A « N T H F T A * 3 . 1 4 1 5 9 2 6 5 4 / 1 8 0 . O I J M T L ^ ' F T / E D ^ U M L T A M a 1 - ( U M L T * l ) M T L ) CA«COS(THETA) < ; » S I N ( T H E T A ) P » 2 . 0 * F L * U M T L Q « 4 . 0 * A H * G L T P » C A * * 2 ti = S * * 2 V » C A * * 4 W » S * * 4 y « C A * * 3 V « S * « 3 CCN,1 , 1 ) : C ( N , 2 . 2 ) C ( N , 3 , 3 ) C ( N » 1 » 2 > C ( N , 2 , 1 ) » ( E L * V • ( E T * V « ( ( F L * « ( ( F L * » C ( N , 1 * ETi EL' E T - P ) ^ F T - Q ) ' , 2 ) • R« * R ' 1 * \ * »U »U ( P * ü ) * R * U ) / A M ( P * Q ) * R * U ) / A M • ( Ü / 4 . 0 ) * ( R - U ) * * 2 ) / A M • ( p / 2 . 0 ) * < V * W ) ) / A M C ( ; ; r , 3 ) . ( ( ^ T : ( p ; Q ) / 2 . 0 ) * Y * C A - ( E L - C P * Q ) / 2 ) * X * S ) / A M

?i!i;2;3U5iE;';piö)/2.o)*x.s -

( E L . ( P * Q ) / 2 . O ) * C A * Y ) / A M C ( N , 3 , 2 ) - C ( N , 2 , 3 ) PETURN FND

^0 .0 «tUBROUTlNF ADDSIJBT( A , B , W . N A S ) D I M E N S I O N A ( 3 , 3 ) . B ( 3 , 3 ) , U ( 3 , 3 ) nO 10 1=1r3 nO 10 J"1,3 I F ( N A S . E Q . O ) G O TO 20 W(I , J ) « A d , J ) - B d ,J) RO TO 10 W ( I , J ) " A d , J ) * B d ,J) C O N T I N U E RETURN FND

i

- B15 " SUBROUTINE MATPkOP COMMON / C O M A / E L ( 3 ) , E T ( 3 ) , G L T ( 3 > , U M L T ( 3 ) R E A D ( 1 , 2 ) NMAT 2 F O R M A T ( I O ) «0 10 I » 1 , N M A T R E A O d . D E L ( I ) . e T ( l ) , 6 L T ( I ) , U M L T d ) . 0 CONTINUE 1 F O H M A T ( 4 F 0 . 0 ) RETURN END < ; U B R O U T I N F M U L T ( A , B , W ) nIMENSlON A ( 3 , 3 ) , 8 ( 3 , 3 ) , W ( 3 / 3 > DO 10 I " 1 » 3 nO 10 J - 1 , 3 U d , J ) . U . 0 00 10 N«1 , 3 ^0 W d , J ) B U d , J ) 4 . A d , N ) * 8 ( N , J ) RETURN FND •SUBROUTINE I N V E R ( A , B ) DIMENSION A ( 3 , 3 ) , 8 ( 3 , 3 ) « ! « B ( 1 , 1 ) * R ( « ^ , 2 ) * B ( 3 , 3 ) - B ( l d > * B ( 2 ' 3 ) * * 2 1 • 2 . 0 * 8 ( 1 , 2 ) * B ( 2 , 3 ) * 8 ( 1 , 3 ) - 8 ( 3 , 3 ) * B ( 1 , 2 ) * * 2 I - B < 2 , 2 ) * B ( 1 , 3 ) * * 2 A ( 1 , 1 ) « ( B ( 2 , 2 ) * B ( 3 , 3 ) - B ( 2 , 3 ) * * 2 ) / S A ( 2 , 2 ) « ( B ( 1 , 1 ) * B ( 3 , 3 ) - B ( 1 , 3 > * * 2 ) / s A < 3 , 3 ) » ( 8 ( 1 , 1 ) * B ( 2 » 2 ) - B ( 1 » 2 ) * * 2 > / s A d , 2 ) - ( B ( 1 , 3 ) * B ( 3 , 2 ) - B d , 2 ) * B < 3 , 3 ) ) / S A d , 3 ) a ( B ( 1 , 2 ) * H < 2 , 3 ) - B d , 3 ) * B ( 2 , 2 ) ) / S A ( 2 , 3 ) . ( 8 ( 1 , 3 ) * ? d , 2 ) - B d , 1 ) * B < 2 , 3 ) ) / S A ( 2 , 1 ) « A ( 1 , 2 ) A ( 3 , 1 ) » A ( 1 , 3 ) A ( 3 , 2 ) » A ( 2 , 3 ) RETURN FND «SUBROUTINE L A Y U K N , T , K ) D I M E N S I O N T ( 3 ) rOMMON /COMC/MT(20),LR(20>/COMO/H(21) R E A D d ,1 > N P E A D d ,1 ) <MT( I ) , 1*1 ,N) P E A D d , 1 ) (LR(I) ,I«1 ,N) P E A 0 d , 2 ) ( H d ) ,1-1 ,N*1 ) T(K)«-H<1)*H(N*1) 1 F O R M A T ( 2 0 I O ) 2 F O R M A T ( I O F O . O ) RETURN END

SUi ROUT i :'E D.•;0'^ YSüRT (X , R, K, AM, Q, AN, AL, F) ni;;Er:siON A L ( 2 ) OOXXO.v /C0M3/ H L ( 2 0 ) , W ( 2 0 ) , A M X ( 2 0 ) , 9 n ( 2 0 ) , A H Y ( 2 0 ) , X ( 2 0 ) cn;;;!or' / C 0 ' ; i / BF ( 3 , 3 , 3 ) , no( 3 , 3 , 3 ) , 0 G ( 3 , 3 , 3 ) / C O M 6 / A B ( 3 ) FETA = Ar.(K) r i i = n n ( K , i , 1) 012 = GQC<,1,2) 0 2 2 = G r i ( K . 2 , 2 ) r . ' 1 1 = 0 0 ( K , 1 , l ) r ' - 1 = D D ( K , 2 , 1 ) AFA = E . E T A * A L ( K ) / 2 . 0 F lT = A L ( K ) / ' ; . n AirO = ' " l T / 1 0 . 0 r.'OM=EXP(AFA) C'j':i = i.o/ni.j;i o<:i;=o.3*(noi:+DUMi) r = S!':(AFA) C=OOS(AFA) :.OT=0HI •OSH+S»0 r = - n « c s ; i » C / r ' 0 T CC=—A:XSlil»C+S»CSH)/BOT r : = A : ' » ( G £ ! l » S - S ; ! l * 0 ) / B O T E=CC»A-'»ni1/AM F = D»A'»:" 11/'01 CO"'.01=A+r;ETA»(00-E) r;:;-- F;2=I+OETA»(D-F) 00 1 1 = 1 , 1 1 AF;'^ = : E T A * ^ ^ 1 T D Ü ; ' = E X F ( AFA) D'J:-:I = I . ^ / C U " I ^ÜUO.^'CDUK-OUnl) ?'!I=0.3^(DUK-DUM1) C5;H = ^.:>(nijM+ouM1) S-J:;! " ( A F A ) 0 = 0OS(A[-A) HL(l )=-2.0«r.*r.ETA*(COMS1*S*SHl+CONS2*C*CSFl)+F»R W( I ) = (CO'.'S1»S*S;il+C0MS2»C»CSH)/(2.0*ni1»BETA»EETA*LETA) W( I )=!.'.'( I )-i'.2r »R»(P»R+Q12/G22»AN) A;:X( I ) = (oor;s2#siii • s - c o r i s i » c s i i * c ) / n E T A A : ' Y ( I )=r 2i*Ai;x(i ) / n i i

no( 1 )=orOS1»(0«SHl-OSH*S)-CO*!S2»(SHI *C + S*OSH) X( I ) = A L ( K ) / Z . 0 - ^ ' I T

r i - - = B i T - A i ; ' c PFTURF

- BIT -SUBROUTINE ELASPROP (K,TH) nlMENSION TH(3) COMMON /C0M1/ B B < 3 , 3 , 3 ) , D D ( 3 , 3 , 3 > , G G < 3 , 3 , 3 ) COMMON /C0M5/ EX(3)»EY(3),GXY(3)»AMUYX(3),AMUXY<3> T«TH(K) E X ( X ) « 1 , Ü / ( G G ( K , 1 , 1 ) * T ) E V ( K ) » 1 . Ü / ( 0 6 ( X , 2 , 2 ) * T ) C,XY(X)«1 . 0 / ( G 6 ( K , 3 # 3 ) * T ) AMUYX(lf)«-GG(X,1 ,2)/6G(K,2,2) A M U X Y ( K ) » - G & ( X , 1 , 2 ) / G G ( K , 1 , 1 ) RETURN END SUBROUTINE OUTEP < N E L ' COMMON /C0M5/ EX(3) , E Y ( 3 ) , G X Y ( 3 ) , A M U Y X ( 3 ) . A M U X Y ( 3 ) .0 FORMAT (////,2X,32HSHELL ELEMENT ELASTIC P R O P E R T I E S , / ,

1 ? X , 3 2 H - ,//)

/(, FORMAT (29X,9HELEMENT 1 ,6X ,9HELEMENT ?,/)

^0 FORMAT (2X,25HAXIAL MODULUS (EX) •,2(1X,El 4.7)) 40 FORMAT (2X,25HHOOP MODULUS (EY) • , 2 d X , E l 4.7))

bO FORMAT (2X,25HSHEAR MODULUS (GXY) • , 2 d X , E l 4.7>) 60 FORMAT (2X,25HPOISSON'S RATIO MU<X,Y) • , 2 d X , E l 4.7>) 70 FORMAT (2X,25HP0ISS0N'S RATIO MU(Y,X) •,2(1X,El 4.7))

WRITE (2,10) WRITE (2,20) WRITE ^2,30) (EX(I),1*1,2) WRITE f2,40) ( E Y d ) , I * 1 , 2 ) WRITE (2,50) (GXY(I),I«1,2) WRITE (2,60) ( A M U X Y d ) ,I«1 ,2) WRITE (2,70) ( A M U Y X d > » 1*1'2) RETURN FND

SU ^'ROUT 1 '••:'€ OUTDECA Y ( A'•', K)

COMMO'' / C r : ; : 3 / H L ( 2 0 ) , W ( 2 0 ) , A H X ( 2 0 ) , Q 0 ( 2 0 ) . A M Y ( 2 0 ) , X ( 2 0 )

10 F 0 R : : A T ( / / / / , 2 X , 3 8 H 0 E C A Y O F . L O A D S A N D C O U P L E S FOR ELEMENT, 1 2 ,

1 / , 2 X , 4 ' ^ H , / / )

2^ F O R X A T (2X,4nHLOADS AND COUPLES ARE QUOTED PER INCH OF,

11X,12liSHELL L E ' 0 T H , / , 2 X , 3^iHMERlD10NAL OR CIRCUMFERENTIAL,, 21X,23HUi:iCH'-VER IS A P P L I C A B L E , / / )

3^ FORMAT ( 4 X , - 3 ! - l ' X ' , 7X,9MH00P LOAD, 2X, 22!-iMER I Dl ON AL HOOP P . M . , 12X, 20HTR'^"SYER.0E RADl AL, / , 2X, 8HD I STANCE, 6X, 4HN (Y) . 6X,

2 ' , : : B . " i . M(X),6X,41IM(Y),4X,22HSHEAR 0 ( X ) DEFLECTION,/) 40 FOt^-'AT ( ( . ( I X , E l l . 4 ) )

^o FooMAT (2X,36Hr.ERlD10NAL LOAD N(X) IS CONSTANT A T , E 1 2 . 5 , / / )

V ; R I T E ( 2 , 1 0 ) K V - I T E ( 2 , 2 ^ 0 U'-^ITE ( 2 , 5 0 ) AM W^ITE ( 2 , 3 0 ) C"' 1 1 = 1 , 11 1 W'MTE ( : : , 4 " ) X ( 1 ) , H , L ( 1 ) , A M X ( I ),AMY(1 ) , 0 0 ( l ) , V'd ) RETURN E'T; F I - I S H

Reinforcing Ring (Element 1)

Pressure Cylinder ( Element 2)

( a ) •^ • *^ h2

h

• ;:-.•:^;.v.v•i'^••:-••^•i•-V Reference Surface ( b ) h2 ^ 1 hi

h

' 90

Re teI ene/. Suiiac^ t 60

z

+

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.. \ i •i

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A

uy

1

•

1

h . z

D

G 4 CURV ILINEAF J (

f

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Reference Surface

'X' Datum

Element 2 To Next Ring Element 1 ELEMENT 2 Reference Surface ELEMENT 1 Reference Surface

8 0007 15) 3 0 0 0 6 • - • - 0 0 0 5 Q ^ 0 0 0 A 0003

Shell Cross- Section

1

.. 0-0

0 75 , | _{Length (Inches)} 1-0 2 0

f

."570 ^ 6 0

S

c 50 -0.1 0-2 0 3 OA 0-5 Distance From Datum (Inches)

0 6 0 7 ü_ 30 20 .^10 -en 10 20 -30 Element 1 Element 2

FIG 9 TRANSVERSE SHEAR AND MERIDIONAL BENDING MOMENT DISTRIBUTIONS FOR ELEMENT 1 (WORKED EXAMPLE 1)

c dl E5 CT6 c •D c: 7 CD 8 9

1

1 . 0 / 1 5 2 0 2-5 Distance From Datum (Inches)

Da urn Element 2 Element 1 20 10 10 20 30 / 1

ƒ

0-5 - / - / /

1 ^ - ^

l

1 0 1 5 2 0 2 5 Distance From Datum (Inches)

-1

3-0

FIG 10 TRANSVERSE SHEAR AND MERIDIONAL BENDING MOMENT DISTRIBUTIONS FOR ELEMENT 2 (WORKED EXAMPLE 1)

Element 2 _{Element 1} eference 5u Reference Surface nace 015 ELEMENT 2 005 0 0 5

K M

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ELEMENT 1 005 1 005 .. :^ .: i-i' - . . • '.^ . [ OOOxA Layer 2

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