Computer programme for the stress and strength analysis of axisymmetrically loaded laminated anisotropic cylinders

UNLIMITED

C. of A. Memo 7505

COMPUTER PROGRAMME POR THE STRESS AND STRENGTH ANALYSIS OP AXISYMMETRICALLY LOADED

LAMINATED ANISOTROPIC CYLINDERS

TECHNISCHE HOGESCHOOL DELFT LUCHTVAART- EN RUIMTEVAARTTECHNIEK R.C. FEWS MSc BiSLfOTg^EEK

Kluyverweg 1 - DELFT R. TETLOW MSc

CRANPIELD INSTITUTE OP TECHNOLOGY CRANPIELD

BEDFORD

SEPTEMBER 1975

Prepared for Procurement Executive Ministry of

Defence

Contract Number ATGW/2028/059 GC

All recipients of this report are advised that it must not be copied in part or in whole or be given further distribution outside the Ministry of Defence without the written approval of the Assistant Director Research Guided Weapons Procurement

ColleKo of Aeronautics Memo 7505 rieptember' 1975

CRANPIELD INSTITUTE OF TECHNOLOGY

COMPUTER PROGRAMME FOR THE STRESS AND STRENGTH

ANALYSIS OF AXISYMMETRICALLY LOADED, IJ^MINATED, ANISOTROPIC CYLINDERS

by

R.C. FEWS M.Sc. R. TETLOW M.Sc.

SUMMARY

A computer programme is presented for the evaluation of stress distributions through the wall thickness of a general axisymmetrically loaded laminated composite cylinder.

The programme is intended for use in conjunction with the discontinuity programmes of Refs 1 and 2. These programmes define shell loads and moments in regions of structural

discontinuity and as such, generate the input for the programme being discussed. With input biaxial tension or compression, together with biaxial bending, individual in plane lamina stress vectors are evaluated and analysed in a 'yield type' failure criterion to give an assessment of the overall

structural strength.

For the types of structural discontinuity discussed at Refs 1 and 2 axisymmetric transverse shear forces are normally induced. These shear forces ai-e catered for in the programme in so far as I'esulting interlaminar shear stresses are

evaluated, but no strength factors are quoted for interlaminar shear.

CONTENTS

Page NOTATION

1. INTRODUCTION 1

1.1 General 1 1.2 Co-ordinate Systems and Sign Conventions 1

1.3 Programme Function 1 1-4 Types of Shear Stress 2 2. THEORETICAL RELATIONSHIPS (SUMMARY) 2

2.1 General 2 2.2 In Plane Stresses at the kth Lamina 3

2,Ji Interlaminar Shear Stresses 3

2.4 Strength Criterion 4 :5. STRUCTURAL DEFINITION 4 4. PROGRAMME INPUT DATA 4 5. PROGRAMME OUTPUT DATA 7 5.1 Applied Shell Forces and Couples 7

5.2 Layer Stresses in the (x,y) Co-ordinate System 7

5.3 Local Stresses and Strength Factors 8

5.4 Interlaminar Shear Stresses 8

6. GENERAL COMMENTS 8 7. REFERENCES • 9

APPENDICES

A Prediction of Lamina Strains, Stresses and Strength Factors

FIGURES

1. Curvilinear Co-ordinate System 2. Local Axis System

3. Positive Forcer; and Couples ^1 . Types of Shear Stress

5. Typical Construction

6. Typical Laminated Structure 7. Shear Stress Free Body Diagram 8. Worked Example

SUBSCRIPTS

l,t Filamentary axis system for layer x,y,z Global axis system

k Parameter referred to the kth layer ss Steady state

o Parameter referred to the reference surface

MATRIX NOTATION

C] Square matrix n x n

NOTATION

E Youngs modulus P Strength factor G Shear modulus

h Distance from reference surface M Bending moment

N Membrane Force

Q Transverse shear force

•I

Q Allowable stress - longitudinal tensile Q Allowable stress - longitudinal compressive

R Allowable stress - transverse tensile and compressive R Cylinder radius

S Allowable stress - shear

u Displacement in the (x) direction V Displacement in the (y) direction w Radial displacement

' 1

y > Orthoginal curvilinear co-ordinate system z —^

€ Direct strain y Shear strain

a Direct stress T Shear stress

\i,. Poisson's ratio (strain effect in 't' direction due to load applied in 'I' direction)

© Angle of ply orientation relative to 'x' axis _X

P =

Change in curvature 4

1

-1 . I^JTRODUCTION 1 . 1 G e n e i ' a l

This work forms par-t of M.O.D. (P.E.) Agreement No. AT/2028/059GC, which is concerned with the application of modern materials to the design of Guided Weapons.

The term 'Modern Material' in the context of this study refers to laminated fibre reinforced composites and in Refs 1 and 2, typical Guided Weapon and allied structures are discussed for which these materials may be particularly well suited. Such structures include rocket motor cases and missile bodies, which are normally of cylindrical shape and as such lend themselves to efficient manufacture by helical or polar winding methods.

1.2 Co-ordinate Systems and Sign Conventions

For global definition of the shell a cartesian x,y,z curvilinear co-ordinate system 1 B used (see fig.l). Use is made of an arbitrary reference surface located at some point through the thickness of the shell,such that the (x) axis follows the axial,or meridional direction along the reference surface, the (y) axis a circumferential direction along the reference surface and the z axis a direction normal to the reference surface.

Individual laminathave their own local co-ordinate system designated by subscripts 'I' and 't' which refer to longitudinal and transverse directions respectively. The longitudinal direction co-incides with the fibre direction, which is orientated at ©° to the global (x) direction. (See

fig.2)

Bending moments, membrane forces and transverse shear forces are +ve as indicated at fig.3.

1.3 Programme Function

The main function of this programme is to handle the stress analysis at a particular section in a thin walled laminated cylinder,when loads are applied consistent with internal pressure and local structural discontinuity. The load vectors required are readily available from the

computerised discontinuity solutions presented in Refs 1 and 2. For biaxial direct and bending load input, in plane

direct and shear stresses are evaluated in global co-ordinate directions at each lamina through the cylinder wall thickness. The stress vector so derived Is then transformed to the local axes of the unidirectional lamina concer-ned, whei'eupon a

failure criterion suggested by Hill(Ref.4) is used to evaluate a reserve factor for the layer. Each layer thi'oughout the

2

-and loading variations,each layer will have a different local stress system and therefore a different reserve factor. Thus a table of reserve, or strength factors referring to each

layer in the construction is assembled and structural failure is assumed if any one of these strength factors falls below unity.

1.4 Types of Shear Stress

In a laminated fibre reinforced material three types of shear stress are possible. These stresses are defined by

Berg (Ref.5) as Intralamlnar, Interlaminar and Short Transverse or 'Log Rolling' shear stresses. (See fig.4). Of these three shear stress components the intralamlnar or 'in plane' shear stress is the most commonly encountered and shear strength in this plane can be varied by varying the fibre orientations. Intralamlnar shear stresses are Induced in all angle ply layups when subjected to either direct in plane loads or bending moments. Where no overall in plane shear stress is applied to the composite, as is the case for pressure cylinders these shear stress components will sum to zero when all layers in the construction are considered. The computer programme

takes the applied loading system and evaluates this shear stress component for each layer, transforms it to the local axes of the layer, and applies it together with the relevant direct stress system to the Hill (Ref.4) strength criterion to give a strength prediction for the layer concerned.

Interlaminar or Log Rolling shear stresses (depending on local fibre orientation) are primarily dependant on

transverse shear load input. As can be seen from Refs 1 and 2 transverse shears are generated in regions of structural discontinuity such as end closures, or reinforcing rings. The programme evaluates interlaminar shear stresses for this type of loading but since there is very little strength data

available in this mode, it does not make any strength predictions. Berg (Ref.5) quotes Interlaminar and Log Rolling shear strengths

for graphite/epoxy composites of 16000 Ib/ln^ and 3000 Ib/in^ respectively.

Both interlaminar shear modes are primarily resin

dependent and are therefore little effected by fibre orientation. The general strength levels in these modes are significantly

lower than structural metals but it must be appreciated that interlaminar shear failure does not usually constitute overall structural failure and for lots of applications these low

properties may therefore not result in weight penalty. 2. THEORETICAL RELATIONSHIPS (SUMMARY)

2.1 General

The governing differential equations for a laminated anisotropic shell can be expressed in terms of two dependant variables; namely the Airy Stress Functions (U) and the

3

-equations are derived relating stresses and strains at a general kth lamina in a fibre reinforced composite to the two shell dependant variables. A solution to the shell governing equations for axisymmetrically loaded cylinders is given at Ref.l and used in Appendix A to establish

expressions for the relevant Airy Str-ess Function and radial displacement parametei-s appearing in the stress equations.

Thus explicit relationships for the stresses at a general kth layer- are available in terms of the applied loading parameters. The final relationships so derived are summarised in the two paragraphs to follow.

2.2 In Plane Stresses at the kth Lamina

X y 'xy. (k) ^11 ®12 ®13 ^21 ®22 ®23 ®31 ^32 ®33

i^).

\ ^ o j + f f f ^ 1 1 12 13 r f f ^ 2 1 22 23 f f f

b l 32 '33

(k)

1°

0

X ^11 ^ - z _k °11 ^^12 ^13 r> o C ^21 22 23 "^31 ""32 ""33 (k)

See Appendix A for matrix definitions. 2.3 Interlaminar Shear Stresses

k+l,k _n zx

- I

k=l g (k) 12 2P^R ( Q ^ + 2PM (k) x ) ^ (^11 (k)v Q^ " Pll ) d ^ k+l,k _n zy

= -I

k=l (K)

'b2 ^P

" " ( \ ^ 2PMJ. (h

(k) 31 - P (k) X Q 13

jd,

_iJ See Appendix A

4

-2. /| Strength Criterion

The pror.i'amme uses a strength criterion suggested by Hill (Ref .V) . i.e.

2 2 2

'U

'W

,

_\

, '^It 1

( Q " ) ^ " (Q")''^ • ( R ) ' ' s^ " F 2

II

In the above formula Q is used if o is tensile and Q if a, is compressive.

A brief description of the derivation of this formula is given in Appendix A.

3. STRUCTURAL DEFINITION

The structure to be analysed by the programme is defined by providing the thicknesses relative to an arbitrary

reference surface of a maximum of five layer blocks (See fig.5). Each layer block may consist of unidirectional or cross plied, (alternate layers orientated at ^© to the 'x'

or axial direction) fibre reinforced material, or any matei'ial that can be defined by elastic constants E,, E, , G,. , V^^^- Each

layer block specified will be idealised in the programme to six individual layers, an automatic device setting

alternate layers at ±© should a cross ply block be required. In plane and interlaminar shear stresses are output for

each individual layer, such that a minimum of six stations or a maximum of thirty stations throughout the material thickness are considered according to the number of input layer blocks selected by the user.

4. PROGRAMME INPUT DATA

All data used by the programme must be input in card images with the first record starting in card column one. Where more than one parameter is required per card adjacent parameters must be separated by leaving one space. Real numbers must be punched with a decimal point and integer numbers without a decimal point.

To follow is a description of a typical set of data required by the programme. In order to show the number and types of variable appearing on individual cards the actual programme names will be used. Reference to the structure shown at Fig.5 will be made from time to time throughout the description of the data list.

5

-a) NMAT Single integer variable.

NMAT = The number of separate materials appearing in the cross section. (NMAT would be punched as 2 for the structure of Fig.5)

b) EL(I) ET(I) GLT(I) UMLT(I) H ^^^^ ^^^, variables per c) SLT(I) SLC(I) STTC(I) SS(l)_j ^^^'^

Cards b) and c) will be repeated 'NMAT' times to cover the 'NMAT' materials appearing in the structure.

EL(l) = Longitudinal Youngs Modulus ET(I) = Transverse Youngs Modulus

GLT(I) = Shear modulus in the l,t plane

UMLT(I)= Poisson's Ratio (strain effect In 't' direction due to load applied in 'I' direction)

SLT(l) = Strength of a single unidirectional lamina in the longitudinal, or fibre direction.

SLC(l) = Compressive strength of a single unidirectional lamina in the longitudinal direction.

STTC(l)= Tensile strength of a single unidirectional lamina in the transverse direction.

SS(l) = Shear strength of a unidirectional lamina in the l,t plane.

d) NEL Single integer variable

NEL = The number of separate shell elements for which output is required. This parameter simply

'loops' the programme so that a number of separate shell elements can be considered without the need to re-complle.

e) AM(1,1) AN(1,1) AN(2,1) QX R Five real variables AM(l,l)= Applied meridional bending moment per unit

circumferential length (M see fig.3)

AN(1,1)= Membrane force per unit circumferential shell length (N see fig.3)

AN(2,l)= Membrane force per unit meridional, or axial shell length (N see fig.3)

QX = Transverse shear force (Q see fig.3) R = Inside radius of cylinder

NS Single integer variable

NS = The number of layer types appearing in the structure cross section. (NS would be input as 3 for the structure of fig.5)

MARK(I) I = 1 to NS NS Integer variables.

MARK = An array of (NS) variables describing each layer type used in the construction. Two layer types are possible and are identified in 'MARK' by inputing either zeros or ones. If zero is punched the layer in question will be treated as a cross plied layer and will be split into six laminae of equal thickness with alternate lamina set at plus and minus ©° (the fibre orientations '©' are input on card image i). If one is punched in 'MARK' then the

layer in question is isotropic, or has all its fibres orientated in one direction '©'. This layer is split into six laminae of equal

thickness, with all the fibre orientations at +©°. For the construction shown at fig.5

'MARK' would be input O i l , the first layer is cross plied at ±60°, hence '0', the second layer has all its fibres aligned in the hoop direction (90°), hence 'l', and the third layer is Isotropic, hence '1'.

MMT(I) I = 1 to NS NS Integer variables

MMT = An array of (NS) variables describing the

material of each layer- type. If '1' is punched for a specific layer then the material of that layer is that as identified by the first pair of cards previously input at (b) and (c). If

'2' is punched then the material is as

identified by the second set of cards (b) and (c) etc. A maximum of three differing materials is allowed in any particular construction so that 'MMT' will be a series of ones, twos and threes. For the structure shown at fig.5 MMT would be punched 1 1 2 (Assuming the same

fibre-resin system is used for layer blocks 1 and 2 ) . LLR(I) I = 1 to NS NS Integer variables

LLR = An array of (NS) variables describing the angular orientation of the fibres in each of the (NS) layer blocks. If a cross plied block is being used only the positive orientation need be input, (angles are quoted relative to the axial direction) For the structui^e shown at fig.5, 'LLR' would be punched 60 90 0.

7

-j) HH(I) I = 1 to N S a ^ N S + 1 Real variables

HH = An array of (NS+l) variables describing the location of each layer block relative to the reference surface. For the structure of Fig.5*

HH would be punched -0.05 0.0 0.015 0.125

(i.e. stations below the reference surface must be input as -ve)

Cards (e^ to (j) must now be repeated (NEL) times to cover the (NEL) separate constructions for which output is required.

5. PROGRAMME OUTPUT DATA

5.1 Applied Shell Forces and Couples

The first set of output from the programme identifies the elemental forces and couples on which subsequent stress level output is based. The following parameters are included.

a) 'x' direction force N per unit axial length)

per unit circumfei-ential length) per unit circumferential length) per unit circumferential length) per unit axial length)

The above forces and couples are positive as shown at b) 'y' direction force N

c) Transverse shear force Q d) Moment about 'y' axis M e) Moment about 'x' axis M

Fig.3.

5.2 Layer Stresses in the (x,y) Co-ordinate System

Each input layer block is idealised by the programme to six individual laminae and the following information is output at each of the lamina so defined.

a) Fibre orientation.

b) Location in the total shell thickness relative to the reference surface.

c) 'x' direction stress. d) 'y' direction stress.

8

-5.3 Local Stresses and Strength Factors

This block of output is similar to that described at (5.2) but in this case stresses are quoted relative to the local axes of respective laminae and an overall strength factor for the lamina is given. Failure of a particular lamina is indicated if the strength factor for that lamina falls below unity. It must be remembered however that the strength factors output in this section arc derived for in plane stresses only and therefore interlaminar shear failure must be assessed separately. The table of data is entitled

'Layer Stresses Referred to the Axes of the Laminate' and includes the following.

a) Direct stress in the 'I' direction (direction of fibres) b) Direct stress in the 't' direction (right angles to

fibres)

c) Shear stress in the 'l,t' plane (interlaminar shear) d) Strength factor (Hill criterion see Appendix A)

The above information is output for each lamina in the structural idealisation.

5.4 Interlaminar Shear Stresses

The following data is output under this heading a) Interlaminar shear stress in the 'z,x' plane

(See Fig.4)

Stresses are quoted at all layer junctions throughout the material thickness.

6. GENERAL COMMENTS

Axisymmetric loading applied to axisymmetric structure results in zero change in hoop curvature X . This observation

t/

is used to evaluate the hoop bending moment M appearing in

0'

the output (5.1). M is therefore the bending moment required to give hoop curvatures equal and opposite to the antlclastlc curvatures resulting from M^. The application of M with M

X y X

therefore gives the zero change in 7^ as required.

The programme is written to perform the stress analysis for the discontinuity solutions presented at Refs. 1 and 2 and as such will only give accurate results for balanced laminates.

REFERENCES

9

-1. FEWS R.C. TETLOW R.

Computer programme for the determination of discontinuity forces and couples

at the junction of laminated fibre reinforced cylinder and domed end closure. C of A. Memo 7501

Cranfield Institute of Technology. 2. FEWS R.C.

TETLOW R.

Computer programme for the determination of discontinuity forces and moments

at a reinforcing ring on a circular^

anisotropic cylinder. C. of A. Memo 7502 Cranfield Institute of Technology.

3.

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 November 1972.

4.

HILL R. The Mathematical Theory of Plasticity Clarendon Press Oxford 1950.

5.

BERG K.R, Shear Loading In Fibrous Composites. Paper presented at the 32nd annual conference of the Society of Allied Weight Engineers.

Whlttaker Corporation, San Diego, California, Paper No.993 June 1973. DONG S.B.

PISTER K.S. TAYLOR R.L.

On the Theory of Laminated Anisotropic Shells and Plates.

Journal of Aerospace Sciences August 1962.

7.

CALCOTE L.R. The Analysis of Laminated Composite Structures

Al

-APPENDIX A

PREDICTION OF LAMINATE STRAINS. STRESSES AND STRENGTH FACTORS

1. IN PLANE .STRESSES AND STRAINS 1.1 Strain Displacement Relationships

Employing the Kirchoff-Love hypothesis (plane sections remain plane) the deflection of a point located at distance (z) from the reference surface (see Fig.l) must be as follows u = V = VJ = For 'y-V o ^o small ÖU

'- by

èw èw ÖW

defleo tion theory

. . . ( 1 )

y . lü + |V ^ ^

xy ^ "Sx _y^ = r zx xz dz ox V - bv bvi _ y

^Y7. "SI ^ ^ ~ "^zy ...(2)

.*. Assuming strains in the (z) direction are negligible (l) and (2) give the following

A2 -öu _ o 'X bx bv - z o -.,2 ó w ., 2 o x . 2 o w ' y

" ^ "

% y 2 T bu by o , o öv rix d x o y - C"7. N 2 o v;

x y c3y rix " öxöy . . . (3) See Refs (6) rmd (7) E q u a t i o n s (3) can be w r i t t e n i n m a t f i x form a s f o l l o w s

toi;'

- oio - ^ w ,

_xy 1.2 T o t a l S t r a i n s i n t h e k,. Laminate From R e f . l Appendix A

Q^ = W b>\^ +[b] {x>

x y - 1 where CG] = [ A ] ( k ) ' x y Lc] (h^^ - h^_^) k=l ( 4 ) (5) LbJ = [ B ] C G ] n ( k ) LB] = I ^ [c]

(h^^

2 , 2

- Vi ,

k=l [ C ] ( k ) ( k ) - 1 (k) £ J } See R e f . 3 ' x y S u b s t i t u t i n g (5) i n t o (4) y i e l d s (k) X N = [ G J / N \ I f b . . - z 5 . .1 )v xy_ N ^ y . . . . ( 6 )

A3

-where 5. • is the 'Kronecker Delta' 1.3 Total Stresses in the k,. Lamina Now ( k ) ( k )

tOxv ^ ^'^ oy

xy ( k ) ( S e e R e f . 3 ) . • . From ( 4 ) ^ ^ ( k ) k ^ -, k _ ._

{g} - Lc] O}

- z, Lc] {X}

^ -'xy ^ - - ' o k ' 'xy S u b s t i t u t i n g f o r {_^\ i n ( 7 ) from ( 5 )

La J = lc]

LG]

{

NJ-

+

*- -"xy ^ -'xy ( k ) ^ T ( k )

Lc] Lb] - z, Lc]

_^k . . . ( 7 )

] {:^>xy

o r

ioX. = Lej

C N I -H 'xy 'xy

Lf]

( k ) - Z , L C ] ( k )

{.>:}

xy ( k ) ( k ) w h e r e L e ] = L c ] L G ]

...(8)

(k) (k) T

Lf] = Lc] Lb]

2. INTERLAMINAR SHEAR STRESSES 2.1 General Equations

To determine the shear str-ess in the bond layers between laminae, (or the interlaminar' sheéiring stresses) It is

necessary to consider an approach similar to that used to determine the horizontal shearing str'esses in simple beam theory.

Suppose it is required to determine the interlaminar shearing stresses between laminae 2 and 3 in Pig.6. The free body diagram of the first two laminae is shown at Fig.7 and from equilibrium considerations the following relationships can be established for the desired quantities (23) and (23)

"^ zy "^ zx x'^^'dx dy = Z X "^ b-v dx + — ^ dy dz

...(9)

T^^'-'^dx dy = zy -^ bx Ö X dz ..(10)

A 4

-Since dx and dy ar-e arbitrary lengths let dx = dy = 1, then substituting the relevant parts of (8) into (9) and introducing the Air-'y Stress Functior; (U)

.(23) • z x

-^J-^

^j

le (^^ e ^J) e (^^l

Lll 12 13 J

•^'yyx

u,

_{X X X} U, xyx

w,.

| _ r , , ( ^ ) r , , ( J ) . , 3 ( J ) j / -^11 ^12 ^13

w,

w,

xyx

{2W, X X X

yyx

xyx<

+

e ^J) e ^J^ e ^^^

_{^31 32}

_^y^

U,

u,

u,

yyy

xxy

xyy

+

(j) (J) (J)

f f 31 32 ^33

w,

w,

2w,

xxy

yyy

xyy

- z . (J) c (>5) r (j '13 23 33

where (,) denotes partial differentiation and

(11)

b^

^x = t r =

_by'

"'yy

N _{y bx^} = ^ = U,

N

è^u

xy

S T ^ = U'

xy

Equation 11 can be simplified by introducing the following

expressions

Lg]

( k ) \ + l ( k )

[ e ] dz

_{( \ . l}

_{K) Le]}

( k ) k

V

A5 -[ h ] ( k ) j^ik+l (k) (k) Cf] dz = (h^^^ - h^^) [ f ] h. L P j ( k ) k + l ( k ) . ^ p p ( k ) z ^ L c ] dz = i ^ h ^ ^ ^ - - h ^ ' ^ ^ L c ] hT _{. . . ( 1 2 )} S u b s t i t u t i n g e q u a t i o n s (12) i n t o (11) and g e n e r ' a l i s i n g such t h a t t h e bond l i n e l i e s between laminae j+1 and j w i t h j laminae below t h e p l a n e i n q u e s t i o n (j+l,j) _ ^^^

'zx " 2_

1=1 U, ^11 ^-^12 ^13 ) ' yyx X X X U, xyx +

(^n<" - Pll'Vs'^» - Pia'^V,.*^' - P.j'^^)

W , W , [2w, X X X yyx xyx ^31 ^32 ^33 U, U, U, yyy xxy xyy

?>j,(i) - p,/H(hj2*^) - P^jt^V^j'^' - pJ'] ",

w, 2w, xxy yyy xyy_^ ...(13) Similarly equation (lO) can be generalised to

(J+1,1) 'zy

1=1

'21 (i) . (i) . (i) '23

"'yy

U, > + xxy

Ö ,

A6 -]^2l (i) 1 (_ £-'. cc. C-J C-IJ w _'xxy w, yTJ : 'w, xyy =31

(i) . (i) . (i) ^32 '33 U, yyx

u,

u,

X X X xyx w (^31 (i) ^13 ''^32 ^23 ^^33 ^'j'j '

2.2 Solution to Shell Differential Equations

'xxx w yyx 2w,

xyx ..(14) The governing shell partial differential equations in the two dependent variables 'U' and 'w' are derived at Ref.6 for a general shell element. For the special case of

axisymmetrically loaded cylindrical shells, these equations reduce to two coupled ordinary differential equations, since stress resultants and stress couples are independent of the

'y' variable. These two equations can be combined to yield a single fourth order differential equation relating radial displacement (w), to the applied loading parameters. The following solution is given at Ref.l to the final fourth order differential equation so derived.

-Px _{(C^ cos Px + C, sin Ox) + w}

^ 3 ^1 '^ ss ...(15)

v/here C-^ and C. a r e a r b i t r a r y c o n s t a n t s t o be d e f i n e d by t h e boundar'y c o n d i t i o n s .

2 . 3 Boundar-y C o n d i t i o n s

At X = O; Q = (Q ) ; M = (M ) ; N = 0 _{X X ^ X X Q x y}

The above boundary conditions when substituted into (15) together with the relevant relationships for Q, and M (see Ref.l) give the following expressions for C-, and C. .

A7 -'3 _2d^^B'

(^x)o" '(^x). - '^-^l^V

..(16)

c

'' 2n2d 11 (M ) - b , (N ) '' X'' M ^ x' _{o ' o} ..(U) Ref.l.

The following i'elat ionships are also avail.-ible from

^x = - ^11 ^'xxx w

\ ' -

RO

'12 N . + N 22 ^22 ^ ^ ^ ^ M. - d,, w, 11 XX ...(18)

2.4 Radial Displacement Vectors

For axisymmetric loading radial displacements ar-e

independent of the 'y variable, ther'efore with i-eference to equations (13) <->-nd (l^i) the only displacement derivative with a value is

-w, b-^ w

Q,

XXX

ex

3 _^11

2.5 Airy Stress Function Vector

As for radial displacements N , N and N are all ^ X' y xy independent of the 'y' var-iable for axisymmetric loading therefore

U, = U, = U, = 0 yyy xxy 'xyy

Also N is a constant and N = 0

X xy •. U, c3N yyx

= 0 ; u,

:yx ^N - ^ = 0 _ox

i . e the only Airy S t r e s s Function par-amoter to liave a value

i s U, jfrom (18)

X X X ^ '

IJ. _{RG^,~, G,} v;

°jsW.

+ N y s j

A8

-• -• "5x '^'xxx RG^p L ^ J ,^gx From (15)

f'e"^ ^^C^(cos Ox + sin px) + C,(sin px - cos Px)'\

dx

b^'vj o2 ^-Px

I = p^ e "^ /"2c., sin Px - 2C. cos PxA

= p^ e"^'^/2C^(cos px - sin Px) + 2C^ (cos px + sin Px)~l.

bx

b \ _

«3 ^-Px

bx^~ ...(20) From (18) Q + 2pM = - d,, fw, + 2Pw, 1 X ' X 11 L XXX 'xxj .•. From the last two of equations (20)

(Q^ + 2pM^) = - 2d^^p^ e"^^ J^C^ (cos px + sin px) + C.(sin Px - cos Px)y

.'. From the first of equations (20)

^ ^ i i p . . . ( 2 1 )

öw / \

Substituting for Y ~ ^^'om (19) and remembering that 4

P =

22^11

U'xxx = - ^^''^ i'^-x ^ 2P^^x}

2.6 Final Interlaminar- Shear Equations

Substituting the displacement and 'Airy Stress Function' vectors from para's 2.4 and 2.5 into the interlaminar shear

A9 -j + 1 , J zx 1^1 L j + l , j •zy F.y/'^ 2 P ^ ' R ( Q ^ . 2PM^) . ( h ^ ^ / i ) - P l 3 ^ ' ^ ) ï ï 7 7 . . . ( 2 2 ) 3 . FAILURE CRITERIA 3 . 1 G e n e r a l

Once the layer by layer stress distribution in a

structural laminate is known, it becomes necessary to establish a method of assessing the overall composite strength. In

general terms the state of stress in a laminated composite is extremely complex and it is unlikely that accurate, or even representative stress distributions could be achieved through simple experimental technique. It is possible to experimentally determine the fundamental strengths Q", Q' R' and S of a uni-directional composite however- and it is these parameters that will be used to establish a general failure criterion for laminated structures.

3.2 Yield Theory

Generalising the von Mises yield criterion for isotropic materials to include anisotropic mater-ials Hill (Ref.4)

assumed a quadratic in the stress components I.e,

(G+H)a^^ + (F+H)at^ + {P+0)a^'^ 2Ho^CT^ - 2Ga;a^

- 2Pa^a^ + 2LT,., + 2 M T , o ^NT, = 1

't^z ' '^^tz ' '-"'Iz ' ""'It ~ ^ ...(23) Where F, G, H, L, M, N are material anisotropic coefficients

Now suppose the only none zero stress component acting is T,. then 211 = - ^ Similar'ly 1 G+H = (Q') ..\-2 ^ ° ^ ^^l ^ °' ^^t = "7. = '^tz = "^Iz = '^It = 0

A] o

-^^"

-

-^TP ^°^' ^t ^ °' ^l = ^z = -^tz ^ -^iz = -^It = °

p^.G = ^ f o r a^ ^ 0, a^ = a^ = T^^ = T^^ = -^it = ^ ^^^^^^^^ where Z is the strength normal to the plane of lamina

Prom equations (24) 2 n = ^ • ' "• (Q")' (H') 2G = ^^—T;1 • 1_ i -I _{2 ~ R} (Q")" Z' 2F = — + —Q ~ ^5 (R')''- Z^ (Q

y

...(25)

3.3 Plane Stress

Composite laminates are usually in the form of thin plates and a state of plane stress in which

% " "^tz " "^Iz " ° ...(26) .*. Substituting (26) into (23)

(G+H)a^^ H- (F+H)a^^ - 2Ha^a^ + 2V1T^^^ = 1 /p^x

s u b s t i t u t i n g the r e l e v a n t f)afts (24) and (25) i n t o (27) yieldj

2 ' ^ ' 2

""^

'"l-^'t , j V _ ^ lit .

+ ' • r\ + A — = 1

(Q")^ ( Q " ) ' ( R ' ) ' o

or Introducing the strength factor ( F ) 2 2 2

2l „ ^l^t , ^t , ""it 1

( 0 ^ "

(Q")2 (R')2 S2

% 2

lamina failure is indicated by a value of P-<^1.

31

-APPENDIX B

PI{0GR/.^'U'1E LIST AND WORKIÏD EXAMPLES 1. ' WORKED EXAMPLE

1.1 Problem DcGcription

It is desired to investigate the stress distributions and strength reserves of three cylindrical pressure vessels

designated Vessel 1, Vessel 2, and Vessel 3. Vessel 1 is of isotropic design being constructed from light alloy, whilst vessels 2 and 3 are laminated constructions in carbon

fibre-epoxy composite, with fibre stacking and orientation as indicated in Pig.8.

1.2 Applied Loading

The loads applied in the programme are those consistent with the structural discontinuity at hemi-spherical end

closures, when the vessels form 0.152|j-m dia. containers, subjected to an internal pressure of 6.895 MN/m^.

A total of four sections will be considered, which are designated by the programme as Element 1, Element 2, Element 3 and Element 4. Elements 1, 2 and 3> refer to the section

formed at the point of tangency between end closure and cylinder for vessels 1, 2 and 3 respectively. Element 4 refers to a

section located 5.882 mm beyond the tangency point of Vessel 3^ where significant stress couple discontinuities exist as well as

the discontinuity transverse shear.

The applied discontinuity stress couples and stress resultants are derived from the programme and theory of Ref.l and can be summarised as follows.

- B2 - , Vessel 1 2 3 •

3

X location(mm) 0.0 0.0 0.0 5.82 MN/m \ 0.2627 0.2627 0.2627 0.2627

^V

0.39'^ 0.394 0.394 0.4687 MN. m/rn \ 0.0 0.0 0.0

-23.68

M

y

0.0

0.0

0.0

- 7 . 4 2

MN/m

^x

-0.00828 -0.008099 -0.008068 -0.0009522 1.3 Material Properties

a) Uni-directional Carbon Pibre-Epoxy Composite (approx 6 0 ^ V.P.) E^ = 206500.0 MN/m^ E^ = 7580.0 MN/m^ G^^ = 4820.0 MN/m"^ Q " = 1065.0 MN/m^ Q' = 827.0 MN/m^ R' = 41.4 MN/m^ S = 5 5 . 1 MN/m^ ^ /

b) Extrusion Type Al. Alloy spec DTD 5074 E = 68950.0 MN/m^ E^ = 68950.0 MN/m^ G^^ = 26550.0 MN/m'^ ix^^ = 0.3 Q " = 5 4 0 . 0 Q' = 5 4 0 . 0 R' = 5 4 0 . 0 S = 4 8 1 . 0 MN/m MN/m"^ MN/m^

I A Programme Input Data 2 6 « 9 5 o , 0 6 a 9 ' i 0 . 0 2 ^ ^ S o ; 0 0 . 3 5 4 0 . 0 * 5 4 0 . 0 5 ^ 0 . 0 4 a i | o 2 O 6 5 0 ' . Ü 7 5 8 0 . 0 4.'^20.0 0.,^ 1 0 6 5 . '• 8 2 ^ . 0 41 . 4 -iS.I 4 0.0 0.2627 0.394 -0.008277 0.0762 2 1 1 1 1 O O -0.001 0.0 0.1)01 0.0 0.2627 0.394 -0.00809*^ 0.(»762 2 O O 2 2 54 -54

-O.OOl O.U O oOI

0.0 0.2627 0.394 -0.00806H 0.(^761 4' 0 1 1 0 2 2 2 2 50 90 90 -30 -0.001 -O.ÜOOb 0.0 0.0005 0.001 •0.00.02368 0,2627 0.4687 -0.0009522 0.0762 4 0 1 1 0 2 2 2 2 30 90 90 - 3 0 - 0 . 0 0 1 -0.0(»0& 0 . 0 0 . 0 0 0 5 0 . 0 0 1

« v j — o > c a e - ^ o u i ^ M IV»-» I I I 1 I I c c - c - c o o o c o o c-=> o-^vt9~T<oat,OBfsi^vi -^ o J v O M 9 « 0 « d i 4 l M O O . M c v -O ^ -O M t ^ ' -O M ' . M -O ^ M 0 < ^ -<« ^ V » - ».- t « •" ~<» wr ~w f¥i nu ïn fn ir* iiï iri ÏTI n^ ïrt .TI m

i i l l l l l l l l l l O O O O O O O O O O o o '^ 4^ •_-. V^ l.-- * • f \f^ \ft VA '.A. '«K >•% 5C O O O O O O O O O O O O ^ - » - » - » - » - » - » - » - » - » - * - » - » o M M M M M M M M M M M M - «

j ü t ^ o * i--. U^ * ^ i^- Vr^ » ^ l.^ >» irt TT" •

' T > r r . - n - » > m m " T ' m r » i n r m i » -t ^ M M M M M M M M M M M Z C C' O O O O O O c c- o o - < f -» < m s O x; M .T-z - 1 » — »• _'O z .«^ •>. .1^ 1— o r j -i» - t >« _O ^ o —' _a r n . - ï — >» - i X m «/> V» » »-X * • - 2 » — r - Z •» O - 1 • - m ja j n c o » -K~* - < m C 2-Z - • » • . ^ - t m o z - «» X - » _{JC .} j » .tl X *- o _{Ji "Z.} >-' o -m O » _m 1— » — -• < ! T | -* O H X T« 1— > 2 M « Z :> _{- I} m —i O -i X T » •»1 m xt m z o m </> ^ X I • ^ . • » o m ^^ !>i ^^ O —* z m z •Ji • • ^ O z ^*' 3> 1';. O *^ M -« _{> •} Z f -m i/> T - , TO O X -* _X i n : £ • •-• ; ? V r— » z nn O »-j » -< m » C/l - 1 JD nn. c/> L/» m V —• z o r— O SB > r-.-« X • <. ^^ rs O 1 c X I o •— z > -« - T M> < _{y j} -« !TI z o o o o o o o o o o o o XIV» o o o o o o o o o o o o m ?n f n f n en ii> ffï f¥i tïï% T l m f n tn f > 0 3 0 0 0 0 0 0 0 0 3 3 - ^ O O O O O O O O O O O O «.^x O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o fT» fT" rf^ iTï ft^ n\ tT\ fTï n^ .ft fT% n i o o o o o o o o o o o o o o o o o o o o o o o o o z •J> c X m » » < XI •o -* r- x> » 1» z v» z z o o z z m m z z -< -« » » B 19 c c c c -« - » • X -< • • > > ~* XJ > z «/> < f n X (/» _{j r .} v» 3 : .-T > x: T -X x o * . ^ f^ !/> (/> z z ^ ^ ^ K < X >- ^-• H X n m o ^ • V > M 1 O O O • m o o O O c c o o • •m < •m O «^ _{X I} m o -* »-« O z T ï o X ,' _m z ^ v <. ^^ H O « g» M fv; • N ^ o o o o o •n m O rr. 1 O t^ o o o m • • X — O »^* _{X I} m r> -« • - • O z •n O X I v " " j m z ^^ _X >^ II o • _l\> c^ rv. •-^ o o -T! i > _T3 •O f -*-1 1 O (/> X m — f— •n O X ;-m •J-I > z o -T O X n --^ o c • o 1— m v f f » — m Z m z -< — I » I < I (tl I .13 I i - * \ X I m I Vi I </> # r* I y i I I » I ._ 1 3 t I <rt 1 - I I XI I rr I z I I -n I 3» I n I - * I O « x> V5 0 1 1 - 1 O) 3 3 ft> O c •O c D 21 5 J o o fVi o 3

L A Y E R S T H F S S F b R E V E R E D TO T H E A X E S OF T H E L A M I N A T E •R A Y E R 1 2 10 11 12 D I R E C 'T S ( L ) D I R E C T I O N 0 . 1 3 1 3 5 E 0 . 1 3 1 3 5 E 0 . 1 3 1 3 5 E 0 . 1 3 l i 5 E 0 . 1 3 1 3 5 E 0 . 1 3 l i 5 E 0 . 1 3 1 3 5 E 0 . 1 3 l i 5 E 0 . 1 3 1 3 5 E 0 . 1 3 1 ^ 5 6 0 . 1 3 l i 5 E 0 . 1 3 l i 5 E o3 0 3 w5 o3 -J 3 o3 .^3 03 u3 .J 3 o3 u3 jRESSrS (T) DIRECTION 0.1^700^ 0.1'>'700fc 0.1'^700fc 0.lV700fe 0.1'^700fc 0.1''700E 0.1'^700t 0.1*^7001: 0.1^700fc 0.1<<'7006 0.1'/700t 0.1V700C 03 03 03 03 03 03 03 03 03 03 03 03 SHEAR SjRESS (L.T) PLANE O.OOOOOE O.OOOOOE O.OOOOOE O.OOOOOE O.OOOOOE O.OOOOOE O.OOOOOE O.OOOOOE O.OOOOOE O.OOOOOE O.OOOOOE O.OOOOOE 00 0 0 oO 00 0 0 00 00 00 00 00 00 00 S j R E N G ^ H F A C T U R 3.108 3.108 3.108 3.108 3.108 3.108 3.108 3.108 3.108 3,108 3.108 3.108 I N T E R L A M I N A R S H E A R S T R ^ S .

LAYER SHEAR STRESS JUNCTION <Z,A> PLANE

1- 2 2- 3 3- 4 4- 5 5- 6 6- 7 7- 8 8- 9 9-1 . 10-1 1 11-12 -0.18V68E 01 -0.34^rt7E 01 -0.46!)58E 01 -o,55180E 01 -u,60353E 01 -Ö.62078E 01 -U.60353E 01 -0.55180E 01 -0.46i58E 01 -0.34488E 01 -0.18V68E 01

> < T l X I O X •-• m Z -i X J» — Z > - • • - z x> o -« " < T l x : r- c o x-- « x-- • IfcIVJ Z « -• M z c » J» - • -« z m m z - « V ! O — D r r * X -> ^«. >-y i ^^ O z V ï > X • n c d O -< T I • Ï 7 -n T C X I >-• rrvi z -« o > m 2 r> vï m Z.'-/i X I T l - n > X o o m z M - » o «O OB - > 4 * \ ^ « ^ M I V - * i l I I I I Z . ^ - 4 X X I • I I I I ^ O O o 3 o O O O O O O O Ni

- » iw/1 3C - » ^ J » X i/l - » 3S 'JI - » /— > o V/^ 3 - O M J < O > »^ O O O

>> o vw > o M M O » 1^4 O : ^ n

*^ —• w*» ^ c w<o»- r "^^ ^ . * ^ >

i i l l l l l l l l l l - •

O O O O O o o o o o o o o M M 0« M M ^ ^ . ^ U>i «y.J ' i ^ L N 'oi Z

X o o o o o o o o o o o o ^ • • • • • • • • • • • • X» - » - > - * - > - > - > - > - > - > - > - * - * . ^ rr -« M M M M M M 0 4 M 0 4 M M M > - < |— Z T i m m m r r r n i f i m r n r n m T — I T I < >^ •— ci m c» o o o o o O O O O O O O O - « M M M M M M M M M M M 0 « Z - 4 T l y> O i— ^ - I » O O O O O O O O O o . o o < x -» z • • • • • • • • • • • • * ^ m X T» •O ' O .O <0 O o - o < 0 C ' 0 > 0 < 0 C 7 < / > o >J >J N > J >J - ^ ^ ' ^ > J ^ - > J ^ t - i r r . T» O O O O O O O O O O O O X I -y) o o o o o O O o o o o o m - 4 f ^ f n i n m m f n m m m m / n r n o z - « • » • o o o o o o o o o o o o -^ M M M M M M M M M M M M O Z I I I I I I t/* O O O O O o o o o o o o >^ X • . . X t l ^^ r- *> *.- *> * ^ * ^ *" *^ *^ r» *^ -< » o o o o o o o o o o o o ^^

iM »v» ro iv* IM ruiv» I M iv» f\) i v r o (/> v/l «JI U I w« ^ IJIVM VM v« VM ( ^ ;«i x i -4 fn m fn m m r n / n *n / n iri n i nt r" x o o o o o o o o o o o o z iy> M M M M M M M M 0 « M M M m V) -< rn X I (/» -t X r r c/> y» f— O X -< r> O I X z -< z o z m z -« 3 * V ^. c H • X -s>. X 1 - ^ c/l 3 y*fc < Z O z m z -« » O l r^ C - 1 « -< -> « -X < » -z (/> o < •—• _{t l X} X m c/> o r r H .-^ irt o X z m > - n X O X -rr r . X O T l t—t V> z >-x X X r> z m ^^ < c ^ ^ • ^ -X -o -• X m o -t »-H O z T l 3 X r: rt^ Z j ' x X \^ R • N H H I o o o o o o o O S U i IVI o o o < ! > o o < > * ' iv; o o o o -g o o o o o o o 3 3 o m m rïi m :TI I o o o o C: o o ^w^ o o » •D •O r— ^ 1 4 . T l • 3 V> i m f— T l O X — _m y ' > ? • ^ ^ c X n m o o cz T3 f— t l y-z t l z I I r -I » I -< I f n I X I I v> I -< I X-I rr I :•> I i/) I .T, I </l I I > I i -1 C7 I 1 X y )

LAYFR S T R h S S F S RE^fcRFD TO THE AXES OF THE LAMINATE YER 9 0 1 2 DiRFf T S (L) DIRECTION 0.3o77oE 0.3077nF 0.3f)770E 0.30770F 0.307*'OF 0.30770E 0.30770E 0.30770E 0. 3(i7^üE 0.307/^06 0.307/'OF 0.307/0^ 1.3 (i3 03 0 3 0 3 o3 03 0 3 o3 0 3 0 3 ('3 T R F S S ^ S ( T ) D I R E C T I O N 0.?')649F 0.2(I649F; 0.20649E 0.20649 E 0.2O649F 0.20649 E 0.20649E 0,20649 F: 0.20649E 0,2<t640E 0.2O649E 0.20649E 02 02 02 02 02 02 02 02 02 02 02 02 SHEAP S T R E S S < L , T ) f' L A N F 0 . 1 2 1 2 0 F - 0 , 1 ? 1 2 0 E 0 . 1 2 1 2 0 E - 0 . 1 2 1 2 0 E 0 . 1 2 1 2 0 E -0.12120E -0.12120E Ü.12120E -0.12120E 0.12120E -0.12120E 0.12120F 02 02 02 02 02 02 02 02 02 02 02 02 S J R F N G T H FACTOR 1 .633 1 .633 1 .633 1 .633 1 .633 1 .633 1 .633 1 .633 1 .633 1 .633 1 .633 1 .633 INTER L A M I N A R SHFAR S T R E S S E R L A Y E R SHEAR STRESS' J U N C T I O N ( 2 , X ) P L A N E 1 - 2 2 - 3 3 - 4 4 - 5 5- 6 6- 7 7- 8 8- 9 9-1n 1 0-1 1 11-12 -0.18S60E 01 -0.33746F 01 -0.45iS7E 01 -0.53V93E 01 -0.S9055F 01 -0.60«'4?E 01 -O.S9'.S'5E 01 -O.S3'''93E 01 -0.45i>57E 01 -0.33/46F 01 -U.18:>60E 01 L

L A Y E R S T R E S S E S A N D S T R F N Ü T H F A C T O R S E L E M E N T APPLIED S H E L L P O R I ; E S A N D F O R C F C O U P L E S 'X' 'Y' D I R E C T I O N D I R E C T I O N T R A N S V E R S A M O M E N T M O M E N T A B O U T A B O U T FORCfc N < X ) FORCE N < Y ) SHEAR FORCE Q ( X ) •Y' AXIS M ( X ) •X« AXIS M ( Y ) 0 . 2 6 2 7 0 0 0 E 00 O . 3 9 4 O O O O F 00 • 0 . 8 0 6 8 0 0 0 E - 0 2 O . O O O O O O O F 00 O . O O O O O O O E 00

LAYER S T R E S S E S IN G L O B A L <X,Y) CO'-ORDINATF SYSTEM

(Z) D I M E N S I O N S ARE D I S T A N C E S FROM THE M I D PLANE OF T^E L A M I N A T E TO THE R E F F R E N C F S U R F A C E L A M I N A T E O R l F N T A T i O N S ARE Q U O T E D R E L A T I V E TO THE A X I A L D I R E C T I O N (IE 'X' A X I S ) LAYER O R I E N T A T I O N CZ) L O C A T I O N D I R E C T S T R E S S E S SHEAR S T R E S S (X) D I R E C T I O N (Y) D I R E C T I O N ( X , Y ) PLANE 1 2 3 4 5 6 7 3 9 10 11 12 13 14 15 16 17

'2

19 20 21 22 23 24 30 -30 30 -30 30 -30 9 0 90 9 0 9 0 9 0 90 90 9 0 9 0 9 0 90 9 0 - 3 0 30 -30 30 -30 30 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0. 0 0 0 0 0 0 0 (1 0 0 0 9 5 8 3 3 E - 0 3 8 7 5 0 Ö E - 0 3 7 9 1 6 7 E - 0 3 , 7 0 8 1 3 6 - 0 3 6 2 5 0 0 6 - 0 3 5 4 1 6 7 E - 0 3 4 5 8 3 3 6 - 0 3 3 7 5 0 0 E - 0 3 2 O 1 6 7 E - 0 3 2 0 8 3 3 E - 0 3 1 2 5 0 O E - 0 3 4 1 6 6 7 E - 0 4 4 1 6 6 7 E - 0 4 1250OE-0.3 2 0 8 3 3 6 - 0 3 2 9 1 6 7 6 - 0 3 3 7 5 0 0 É - 0 3 4 5 8 3 3 6 - 0 3 5 4 1 6 7 E - 0 3 6 2 5 0 0 E - 0 3 7fl833E-03 7 9 1 6 7 E - 0 3 8 7 5 0 0 6 - 0 3 9 5 8 3 3 6 - 0 3 0, 0 0 0 0 0 0 0 0 0 0 0 0, 0 0

§

t

0 0 0, 0 0 2 4 7 3 6 F 2 4 7 3 6 F 2 4 7 3 6 E ,247366 ,247366 ,247366 1 5 3 3 7 E ,15337E 1 5 3 3 7 E 1 5 3 3 7 E 1 5 3 3 7 E 1 5 3 3 7 E 1 5 3 3 7 E 1 5 3 3 7 E 1 5 3 3 7 E 1 5 3 3 7 E 1 5 3 3 7 E 1 5 3 3 7 E 2 4 7 3 6 E 2 4 7 3 6 E 2 4 7 3 6 E 2 4 7 3 6 E 2 4 7 3 6 E 2 4 7 3 6 E 0 3 0 3 03 03 0 3 03 02 02 02 0 2 0 2 02 02 02 0 2 02 02 02 03 0 3 03 0 3 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , 9 1 6 1 6 E .91616E .91616E ,91616E ,91616E ,916166 ,50238E ,502586 ,502586 5 0 2 5 8 6 ,50258E ,502386 5 0 2 5 8 E 5 0 2 3 8 E ,50258E 5 0 2 3 8 E ;50238E ,50258E ,916i6E .91616E ,916166 9 1 6 1 6 E ,916166 .916166 02 02 02 02 02 02 05 05 05 05 03 05 03 03 05 05 03 03 02 02 02 02 0? 02 -0 0 -0 0 -0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0 0 -0 0 -0 1 3 5 6 0 E 05 1 3 5 6 9 E 05 1 5 5 6 9 6 05 ,155696 05 .15569E 05 ,13569E 05 6 0 5 8 2 E - 0 7 , 6 0 5 8 2 E - 0 7 6 0 5 8 2 6 - 0 7 6 0 5 8 2 E . 0 7 6 0 5 8 2 E - 0 7 6 0 5 8 2 E - 0 7 6 0 5 8 2 E - 0 7 6 0 5 8 2 6 - 0 7 6 0 5 8 2 6 - 0 7 6 0 5 8 2 6 - 0 7 , 6 0 5 8 2 E - 0 7 6 0 5 8 2 E - 0 7 .15569E 05 .13569E 05 1 3 5 6 0 E 05 1 3 5 6 0 E 05 .15569E 05 ,135696 05

LAYER STRHSSE'i REFERED T O T H E AXES OF yHE LAMlNAjE

LAYER DiRFCy S^RESSKS SHEAP STRESS S^RENG^H <L) DIRhCTION ( T ) DIRECTION (L.T) PLANE FACTOR 1 0.324-'1F ()3 2 0.324tlF 03 3 0.3?4c'1E 03 4 0.32421F (-3 5 0.324/lF. o3 6 0.32421F ,)3 7 0.30238F 03 8 0.3O238F 03 9 0.30238P 03 10 0.30238E o3 11 0.30238E 03 12 0.30238E 03 13 0.30238F 03 14 0.30238F o3 15 0.3O238E u3 16 0.30238F 03 17 0.30238F 03 18 Ü.30238E o3 19 0.32421F 03 20 0.32421E 03 21 0.32421E o3 22 0.324,^1F 03 23 0.32421E o3 ?4 0.32421F 03 O.1.»770H 02 0.1'*770E 02 0.14770E 02 0.14770E 02 0.1'*77oE 02 0,14770E 02 0.l5337fc 02 0.15337E 02 0.i:)337E 02 0.1^3376 02 0.1.S337E 02 0.15337E 02 0.1i337E 02 0.1S337E 02 0.1S337E 02 0.1''>337E 02 0.1S337E 02 n.l^337E 02 0.14770E 02 0.14770E 02 0.1'*770E 02 0,14770E 02 0,1^770E 02 0.14770E 02 0.592H4F 00 -0.592H4E 00 Ü.59284E 00 -0.S9284E 00 0.59284E 00 -0.592H4E 00 0.64513E-08 0 . 6 4 5 1 3 E - 0 8 0 . 6 4 5 1 3 F - 0 8 0 . 6 4 5 1 3 E - 0 8 0 . 6 4 5 1 3 E - C 8 0 . 6 4 5 1 3 E - 0 8 0 . 6 4 5 1 3 E - 0 8 0 . 6 4 5 1 3 E - 0 8 0 . 6 4 5 1 3 E - 0 8 0 . 6 4 5 1 3 E - 0 8 0 . 6 4 5 1 3 E - 0 8 0 . 6 4 5 1 3 E - 0 8 - 0 . 5 9 2 8 4 E 00 0 . 5 9 2 8 4 E 00 - 0 . 5 9 2 8 4 E 00 0 . 5 9 2 8 4 E 00 - 0 . 5 9 2 8 4 6 00 0 . 5 9 2 8 4 E 00 1 5 2 1 5 2 1 5 2 1 5 2 1 5 2 1 5 2 1 6 3 1 6 3 1 6 3 1 6 3 1 6 3 1 6 3 2 . 1 6 3 2 . 1 6 3 1 6 3 1 6 3 1 6 3 1 6 3 1 5 2 1 5 2 1 5 2 1 5 2 1 5 2 1 5 2 I N T E R LAMINAp S^^EAR S T R E S :

LAYER SHEA.R STRESS JUNCTION ( Z , X ) PLANE 1 2 3 4 5 6 7 8 9 -2 3 4 5 6 7 8 9 • 10 1 0 - 1 1 1 1 - 1 2 1 2 - 1 3 1 3 - 1 4 1 4 - 1 5 1 5 - 1 6 1 6 - 1 7 1 7 - 1 8 1 8 - 1 9 l 9 - 2 ( *

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