CONTRACT No. NUN 629 (39)

PROJEcT No. NR 064-167

I I

APPROXIMATE ELASTICITY SOLUTION FOR ORTHOTROPIC

CYLINDER UNDER HYDROSTATIC PRESSURE AND BAND LOADS

by

A. P. Misovec and Joseph Kempner

VON0

POLYTECHNIC INSTITUTE OF BROOKLYN

DEPARTMENT

ofAE1ROSPACE ENGINEERING

and

APPLIED MECHANICS

MAY, 1968

DImTImUTION or THISDOCUMENT IS UNLIMITED CL f N' H 0 (1 P NEAL REPORT No. 68-11

Contract No. Nonr 839(39)Project No. NR 064-167

APPROXIMATE ELASTICITY SOLUTION FOR ORTHOTROPIC CYLINDER

UNDER HYDROSTATIC PRESSURE AND BAND LOADS

by

A.P. Misovec and Joseph Kempner

Polytechnic Institute of Brooklyn

Department of

Aerospace Engineering and Applied Mechanics

May 1968

PIBAL Report No. 68-11

Reproduction in whole or in part is permitted for any purpose of the UnitedStates Government. Distribution of this document is unlimited.

g-

ABSTRACT

An approximate solutioi to the Navier equations of the three-

dimensional theory of elasticity for an axisymmetric orthotropic circular

cylinder subjected to internal and external pressure, axial loads, and

'closely spaced periodic radial loads is introduced. Numerical comparison

with the exact solution for a transversely isotropic cylinder subjected

to periodic band loads shows that very good accuracy is obtainable.

When the results of this approximate solution are compared with

previously obtained results of a Flugge-type shell solution of a ring-

reinforced orthotropic cylinder, it is found that the shell theory gives

a fairly accurate representation of the deformation except in the neighbor-

hood of discontinuous loads. The addition of transverse shear deformations

does not improve the accuracy of the shell solution.

When Nill's orthotropic yield criterion is applied, it is found

that yielding could begin rather early at the inner surface of the shell

adjacent to the frame. It Is noted that the transverse shearing stress

has no great effect on the initial yield pressure.

II

I

LIST OF SYMBOLS

A Ij elastic coefficients in stress-strain relations,

Eqs. (3.3b)

A1,2 arbitrary constants of integration in solution to

generalized plane strain problem, Eqs. (5.8ap'b)

Bc b~Wad load, Eq. (2.1a)

Be axially varying portion of band load (load

for first problem), Eq. (2.2)

Bj elastic constants In generalized plane strainI J

elastic law, Eqs. (5.2c)

ijconstants, Eqs. (4.13)

D half width of band load, Fig. I

D 1 nondimensional elastic constants appearing In

Navler equations, Eqs. (3.5c)

Er, Eq, Ez, G radialp circumferential) lal and transverse

shear elastic moduli, respectively, Eqs. (3.2)

E elastic modulus of an isotropic material, Eq. (.7)

E'~J exponential functions of shell geometry and

Fourier index, Eqs. (4.13)

Gn arbitrary constants of integration of asymptotic

solution to first shell problem, Eqs. (4.12)

Nil nondimensional elastic constants relating asymp-

totic displacement solutions, Eqs. (4 .12g)

r ~ ______________________

N, Hnondimensional Fourier coefficients of externally

applied loads and radial band load, respectively,

Eqs. (4.13) and (2.1b)

constants appearing in plane-strain solutions,

Eqs. (5.8)

Ut. center-to-center distance between successive bandloads, Fig. I

Srn, , Szn P Srzn respective radial, circumferential, axiAl aid

transverse shear stress coefficients in tri-

gonometric series

U, W nondimenslonal axial and radial displacements,

respectively, Eqs. (4.4)

X, so Z, S radial, circumferential, axial and transverse

shear yield stresses, respectively, Eq. (6.1)

a radius of datum surface (usually mean radius),

Eqs. (.2)

:2: :23 nondimensional elastic coefficients, Eqs. (4.8e)

Or: eO), ez classical radial, circumferential and axial

strain parameters, respectively, Eqs. (3.1)

• Naperian logarithmic constant, Eq. (4.2c)

II

f plastic potential function of the stresses,Eq. (6.1)

fn' gn radially varying functions In the solution

for the displacements in the first problem,

Eqs. (4.4)

fni' gnt asymptotic approximations to fn and gnlprespectively, Eqs. (4.10)

kn " nra/,4 Eq. (4.4c)

n Fourier series index integer, Eqs. (2.1)

po, Pi uniform external and internal pressures,

respectively, Fig. I

magnitude of band load, Fig. I

P1 Inner lateral surface pressure for plane

strain problem, Eq. (5.7b)

Poy (r,z) pressure at which yielding begins at the

position (r,z), Eq. (6.2)

r radial coordinate, Fig. 1

ri , ro internal and external surface radii, respectively,

Fig. I

u, w axial and outward radial displacements, respectively,

Fig. I

x) y nondimensional radial and axial transformation

coordinates, Eqs. (4.2)

Xo 0 Xvalues of x at outer and inner shell surfaces,

respectively, Eqs. (4.2)

z axial coordinate, Fig. I

roots of characteristic equation of asymptotic

Navier equations, Eqs. (4.12)

<O, P1, 4, D} constants appearing In generalized plane strain

solution, Eqs. (p.6)

1,2,3,4 normalized exponents appearing in asymptotic

solution to first problem, Eqs. (4.12)

external-to-internal radius ratio, Eq. (5.81)

7 rz) 7rr 7ez classical shearing strains, Eqs. (3.1)

a n &/r1anw, Eq. (4.7c)

* cylindrical coordinate, Eqs. (3.1)

% nondimensional exponent appearing in ortho-

tropic generalized plane strain solution, Eq.(5.6f)

Vrz' "ii' Ver Poisson ratios, Eqs. (3-.2a) and (3.7)

VJzrl VzSO' Vez' 'j

nondimensional radial transformation coordinate,

Eq. (4.6)

IF

got C values of [ at outer and Inner shell surfaces,

respect ively

ar, 09, 0z' Trz raoial, circumferential, axial normal stresses

and transverse shear stresses, respectively

i.

I. INTRODUCTION

To date, Investigators of pressurized rirg-reinforced circular

cylindrical shells have employed shell theory in order to describe the

behavior (see, e.g., Ref. I). For thin-walled, isotropic shells, in which

the collapse mechanism is primarily a buckling instability, the results of

shell theory are generally quite satisfactory. However, in short, thick-

walled) filament wound composite cylinders, the collapse mechanism is quite

complicated (as yet no single description has been agreed upon). Examina-

tion of the ruptured composite test models indicate that failure is prob-

ably due to a transverse shear build up at the frame (Ref. 2), a phenomena

not predicted by shell theory.

Because of the complicated failure mechanism of composite shells,

the need is obvious for an analysis which would yield a more accurate des-

cription of the stress istriuution through the thickness than is presently

available through existing shell theories. The work presented by Klosner

and Levine (Ref. 3) on Isotropic shells, and later extended by them (Ref. .)

to include transversally Isotropic shells, consisted of an elasticity analy-

sis in which the st,ess functions suggested by Leknitskii (Ref. 5) were used

to satisfy exactly the classical equations of elasticity. Their results

demonstrated clearly that, although the commonly used Donnell shell theory

gave excellent results throughout most of the shell, it could not accurately

predict the stress distributions at the frame. However, the Leknitskii

stress functions are useful only for the case of a transversally iqotropic(or isotropic) material. The cylindrically orthotropic nature of the filament

Va

2.

wound composite shells can only 5e accounted for by seeking a solution to the

more complicated equations of elasticity for such a material. In the present

study an approximate solution for the displacements is used to solve the

axisymmetric Navier equations of equilibrium for an infinitely long, ortho-

tropic, hollow cylinder under external uniform pressure and closely spaced

periodically varying internal band loads. It is indicated how more accuracy

may be obtained by use of either a perturbation or iteration method.

It is found that the asympto 'c solution is quite accurate and

that the shell solution presented in Ref. I is in very good agreement with

the asymptotic solution, except in the vicinity of the frame. When trans-

verse shear deformations are considered in the shell theory no better agree-

ment is obtained.

III.t

1~3

"4

' . FORMULATI ON

The shell of infinite length is subjected to an external hydrostatic

pressure load po and internal, axisyrimetric, periodically spaced band loads.

These band loads are described as (e.g., see Ref. 4)

Bc = p [ H Cos -Z A] (2.la)c n=l n

with

H mr (_,)n+lsin nTA (2.1b)

and

A = (2.1c)

where pc is the magnitude of the band load, D is half the width of the band

load, t. is half the distance between the centers of two successive band

lo ....; and z is the axial coordinate (see Fig. I).

When the classical linear equations of three-dimensional elasticity

are used to describe the shell deformation, the solution may be taken as the

-_ superposition of the following two solutions:

1. The solution of the shell subjected to an internal load des-

cribed by [see Eqs. (2.1)]

B =p E H cos ny (2.2)cI c n=I n

-a

corresponding to which the displaicements can be taken as trigonometric series

in z. In the ensuing analysis the classical elasticity equations are used

to find the coefficients in the latter series as functions of the thickness

coordinate r.

2. A generalized plane strain solution which includes uniform

lateral pressure loadings on the inner and outer surfaces [the inner surface

load must, of course, include the term - p A appearing in Eq. (2.la)] as

well as a constant axial force.

V

5.1

3. BA;IC EQUATIONS

The classical strain-displacement equations used to describe the

"axisymmetric deformations of a circular cylindrical shell are

er W , e = w/r (3.1a,b)rr

ez =U,z r =0 (3.Ic,d)

7r ur +w = 0 (3.le,f)

in which a comma followed by a subscripted variable denotes differentiation

with respect to that variable. 'r, 6 and z are the radial, circumferential

and axial coordinates, respectively; er eq and ez are the corresponding

normal strains; 7re' 7rz and 79z are the shear strains. w and u are the

displacements In the radial and axial directions, respectively. It should

be noted that both r and w are taken positive outward (Fig. I).

The generalized Hooke's law for a cylindrically orthotropic,

homogeneous material is taken as

Ere /1 -VrO ar

Eee = Ver ''ez Il

E Ee -Vz -V 6 I az z zr ze z

(3.2a)

6. 4

T rz G , I r=O 0 Tz 0 (3.2b, cd)

where

V V V V V Vre 9 r rz _z r ez VzeEr r z 6z

follow from symmetry of the stress and strain tensors. [ denotes a column

matrix and ( ) denotes a square matrix. ar, a and az are the normal stresses

in the radial, circumferential and axial directions, respectfvely; Trz is the

transverse shear stress. Er) E. and Ez are the elastic moduli in the radial,

circumferential and axial directions; G is the transverse shear modulus

and V r, Ver VrzI Vzr , Vez and Vzg are Poisson ratios. It can be seen

that Eqs. (3.2) contain 7 independznt elastic constants.

The inverse relation is given by

a A11 -A12 -A13 r

a A21 A22 --A23 e

A z A 31 -A32 A 33 e z (3.3a)

where

All = (l-vzovz) (Er/N) , A22 = ('-Vrzvzr)(Eo/N) , A,, = (1V reVer) (Ez/N) f(continued on next page)

------ ------

7. 1Al A = r(ve +Vrzze)(Ee/) , Al, = Al = . (Vrz + VreVez)(Ez//n)

A23 A32 -- (vez + Ve rvrz) (E2/N)

N =I VzeVz - Vorlvre + V rzVzo) Vzr(VrOvOz + Vrz) (3.3b)

The axisymmetric equilibrium equations are

(rar),r- aO + r-rz,z = 0 (3.4a)

raz,z + (r'zr) ,r 0 • (3.b)

Equations (3.1), (3-.3) and (3.4) are combined to yield the well- Iknown Navier equations for an orthotropic cylinder undergoing axisymmetric

deformations (Ref. 5). i

2 2 21r w + rw r+ D r w z - D2 w + D~r u rz Druz = 0 (3.5a)

and

Dur Drw +Dru +rU + rur =0 (3.5b)

where

, ,uI

LI

8.

D= G/A1 , D= (A23 - A13 )/A1 i , D=A/G

D2 = A22/A| , D= (G -AI)/G

(G - A, D6 = (A2 - G)/G (3. c)

It is noted that only five of the constants D,, ", are independent, since

D = DID and DI, + D(3. d)

Equations (3.5a) and (3.5b) are a set of second order, linear, homogeneous

partial differential equations with variable coefficients.

If the material is transversely isotropic,

Er= E Vre =Ver , 'rz 'ez Vzr Vze

A 13 = A31 = A32 Ali1 A22

D2 = I D= (3.6)

It follows that the number of independent elastlc constants reduces to

five.

If the material is isotropic,

Er =E =E z = E Vr V6r Vrz =Vz Vzr Vz V

(continued on next page)

* 9.1G E" Aij =A i

D2 .1 , D°s ="D6 1-2i

D~-D

= 1v ( )

1-2v1DI 2(1-v(3-T

10.

4. AXIALLY VARYING LOADS

The first problem to be investigated is that of an infinitely long

circular cylinder with an internal load described by Eq. (2.2). For such a

loading the solution is any solution to the Navier equations, Eqs. (3.),

which satisfy the condit;ons that:

on the outer surface of the shell

a =0 .a)

Tr

Trz =0o (Ii..lb)

and on the inner shell surface

ar =p E Hncos L (4.lc)nI L

T rz =0 (I4..ld)

The major difficulties In finding such a solution are caused by the fact

that the coefficients in the Navier equations are variable In r. In order

to reduce the number of variable coefficients encountered (and at the same

time nondimensionalize the equations) the following well-known transformation

is introduced:

x tri(r/a) (4.2a)

ii I

~~I1.

- z/a (4.2b)

with the inverse rlation

x 'r = ae ( . c)

z= ay (4.2d)

in which a is the radius to any selected datum surface such that ri < a : ro.

Hence, for relatively thin-walled cylinders x = (r/a) - 1, which is small

compared to unity. The Navier equations now become

x x xx

w~x + De2 w .Dw+D Xu + D0XU,x 1 ,yy 2 eU,xy = 0CU~

D ew,xy - D6 eXw,y + De u + u =0 (4.3b)5e6 ,, yy ,x

The separation of variables method is used to find a solution.

The assumed from of the displacements must reflect the symmetry of the load-$

ing (Fig. I). Thus,

W A l w oW " Z gn(x)cosk ny (4.4a)aPc n=l

A11uU =-= E fn(x)sink y (4.4b)apc n= I

~12.

nwhere

k (44

and W and U are nondimensional displacements. Hence, the Navier equations

yield

gnxx (kD e + D2)gn + knDe fn'x +k n efn (4.5a)

-k ex 2x nxknD Xg + k D eXn knDe f + f =0 (4.5b)

A significant insight is gained into the nature of the functions g and f

upon the stretching of the x-coordinate through the introduction of the trans-

format ion

= knX = (nwa/. )x (4.6)

The Navier equations finally yield

gn . " (D 18 e i + 6 g n + D 3e 5n tfn, + n D4e n 9fn = 0 (4.7a)I 8 ,6 8

n D n +8 fna D 7 n (4.7b)n 6)g n D nn

where

=1 /k n=1 ra (4.7c)

n n

13. 1Equations (4.Ta,b) are second order, linear, simultaneous,

ordinary differential equations with variable coefficients. These

equations can be uncoupled with the aid of integrating factors. How-

ever, the resulting relations have complicated variable coefficients

and are singular for the not uncommon case o! a transversely isotropic

cylinder. The solution obtained for transversely isotropic cylinders

In Ref. 4 (which, as mentioned previously, is formulated in terms of

stresses and solved with the Lehknitskii functions) is in terms of

Bessel functions. Accordingly, if Eqs. (3.6) are introduced into

Eqs. (Q.7a,b) and the resulting relations are uncoupled, the displacement

solution is also in terms of Sessel functions, which correspond to those

found in Ref. 1.

The success of Klosner and Levine (Ref.4 ) in using Bessel

functions for the transversely isotropic case tempts one to use a Frobenius

or other type of power series solution for the more complicated orthotropic

equations. However 9 grows in proportion to n [Eq. (4.6)] and unless (as

in classical shell theory) the thickness-to-length ratio is very small

and the load converges quickly (i.e., is continuous and varies slowly with

y) t becomes large as n increases. The large values of kn (and hence of 9)

which evolve from the short shells of interest here make a power series

solution impractial in that many terms are necessary for convergence.

Also, such solutions may give rise to small-difference terms, unless

appropriate asymptotic expansions can be obtained.

14.

On the other hand, when kn is large, 8n is small [Eq. (4.7c)).

This suggests that a perturbotion or iteration scheme may yield a reason-

able approximate solution. Any solution which capitalizes on the size of

8n might also be thought of as an asymptotic expansion.

Once gn and fn have been found, the stresses can be determined by

combining Eqs. (4.4) and (3.3). Hence,

P E [e nn [ ,(g a12nn - al fncoskny (4.8a)I1PC n=l 5n n,t 12 n n 13nJ UnY

1-8 n(al 2 gn, - a2 2bgn) - a23 fn]coskny (4-8b)

n=1 n

z e " n (a 13 gn + a 5 + a33f ]coskny (48c)c n=1 n ~ n a23 n n n"'(4.c

T"rz 1 8n~f g ]sinkny (4.8d)r E 9 - [e n, n nc n=l n

where

a12 =12 /A 11 a 13 - D3 a22 =D 2

a2 =D + D- D3 a33 = DI (4.8e)23 4 D1 D, 7D

15.

The boundary conditior.s are satisfied by [see Eqs. (4.1) and

Eqs. (4.8a,d)]

-xe 0( n,- a2gn) aif= 0 (14.9a)

e-X 0f " g n = 0 (4.9b)

at the shell outer surface x = xo g = o 0

and

e fr n = 0 (4.9c)

e (gn, " al.5n a 3 f H (4 9d)n, n 1 n

at the shell inner surface x = x1, P = <0

If a perturbation solution is used in which the functions 9n and

f are considered to be perturbed about about the solution of Eqs. (4.7)n

with 8 set to zero, 9n and fn may be expressed as

gn = iI 9ni M)'n (4.1]Oa)

f = 1: f .(0)6' (414.lb)n i=l ni n

It =i Ii n () in (.lb

16.

For i = I (zeroth perturbation) the differential equations are

gni,- - D19nl + D3fnl,, =0 (4.1la)

a-d

D D59nl, + fnli - D7fnI = 0 (4.11b)

Equations (4.11), in which 8 = /nra = 0, correspond to the equa-

tions of a doubly infinite flat plate (for which a .-. co) with a thickness

coordinate = ax. The solution is written as

4 Pngni i=I G (4. 12a)

i=1

4 G n

f = H Gnie (4.12b)n] i=l I i

where

n = a, 2(g - o) (4.12c)

n34 = " a2 1 (g " (4.12d)

and a, and '2 are those roots of the characteristic equation

a, + (D3D5 - D7 1Di + DID7 = 0 (4-12e)

which are defined by

I II

ai, 2 = -- (D3D5 - D- Di) ± -[(D3 D5 -D7- Di) 4DID71/2

(4.12f)

17.

Furthermore,

a l2 -Dl D a.i

Hli a D Di g

* 7

'3, a 2,1(41h

For the specific material properties considered subsequently,

C1 is real; moreover a1,2 > 0 and ai+ 1 c a" In general, the nature of a.

is dependent only upon the elastic properties of the shell material. For

example, if the shell were isotropic, equal roots for the characteristic

equation would be found (i.e., a1,2 = - I). It should also be noteo that

the roots of the characteristic equation do not change with n. The exponents

i were selected to modify Gn in order to avoid numbers of excessive magnitudeI li

in the computations. The arbitrary constants Gn are obtained by satisfyingI i

the boundary condition Eqs. (4.9) which, when combined with Eqs. (4.12)

yields the matrix equation

(C? )[G ,, . 1H i,j = 1,2,3,4 C4.13a)

where

-xCn = [e n- n (4.13b)ii 1 ai3H11 Ei

C n. = e °cr.H 1 - I)2in i Eli (14.13c)

18. i-x

3 = )'( (14. 3d)

-XCn [e (a1 -a1 )- a HiEn (4i. 13.)

also

E n=E n=E n E n I (4.13f)11 12 23 2J4

EraE, :(gi: g c2k (x-xo)(41)

En Enn 210

EH4 = e (14..13h)

Freloading ofdi in ees ahmre etaie disuso on4th and aman-to3nes

Hiat laee shll is ofee in th nA=pend3

nn

Once the constants G nhave been determined from Eq. (4 .13a), the

stresses may be obtained by substituting Eqs. (4.12) and (4.10) into Eqs.

(4.8); this yields

Cr Co 4 -xr= le' (a -a F )-a H IGn e i cos k y Z S cosk y (14.14a)

pc n1i 12 n 13 Iili n nl rn n

c n~ i*

19.

n

E [-e X(a 12a-a b )-a H, IG 1 e cosk y= S6 cosk y (14. 14b)= l [ 22 n 231li n SenPc n=l i=l n=l n

z 0 4 n P i- = Z E [-e x(a +a23 ) + a33H IG.e 'cosky E S cosknY (4 .14c)Pc n=i i=l 23'1 3 i Ii n n=l n

rz n i =-=D )Gne i y sinkny (4.14d)PC n=1 l1 i lI n=l rz n

If more accuracy is desired, the first perturbation can be applied

[i = 2 in Eqs. (4.10) and (4.7)]. This yields

9n2,t - DIgn2 + D3fn2,g = (2DignlD 3 fni,9)9 - D4fnl (4 .15a)

-Dgn2,9 + f n2, Df = (D 9I, + 2D7fl)" D69n (4.15b)

An iteration technique may also be used. The first iteration con-

sists if Eqs. (4.12), which corresponds to an = O. Any further iteration

(gni' fni ) may be obtained by successively solving

2 n n+D n antgni,tt-DIgni+D3fni,t = [Di(e 'n)+D2] 9 ni- 3(1e )ni-1,t-e 'n4fni-!

(4. 16a)

6ng 8 28 g

D5gni,g +fni,9 7Dfni=D5(e .i)gni-i~t5nD6e n gni.l+D7 (e 2n9i)fni.!

(4.16b)

I

20.

where the right hand sides are obtained by inserting results of prior iterations.

In either the perturbation or iteration technique the stress components

corresponding to perturbations or iterations beyond the zeroth must vanish

on the boundaries. In either approach, a correction term is only necessary

in the early terms of the trigonometric series given in Eqs. (4.4), (i.e.,

when n is relatively small), since 8 becomes smaller as n increases and,n

therefore, the approximate solution represented by Eqs. (4.12) becomes more

accurate.n

The selection of P i to modify the constants is indeed appropriate,

for as n grows large the coefficients may be closely approximated by simple

formulae which reveal the dependent of the trigonometric series convergence

on location through the shell thickness. Hence, as n becomes large, kn be-

comes large and [see Eqs. (4.13g) and (4.13h)]

E13 , 22, E14, E2 1 0 (4.1.7a)

Therefore,

C13 , CI1, C23 , C24, C 3, C32, C41, C 2 -. 0 (4.17b)

and Eqs. (4.13) may be replaced by

nnC11 C1,2 0CC n 0n ( 08a)

21 22 12 48an n n

o 0 c c G 0

0 0C n Cn G n H143144 14n

21.

Thus, for large n (n > N say)

)n n

11 =2 (Ii..18b)

nn ~C 34Hn

G

n~~ " 3 C94 - -C) 411 3 "044

gnI= [ "C314e+

0- H

Ge] cnnCn Cn ) (4.18d)3344" 4334

and

- n -aX ('td n ) a](-y )

fn~3 44 43 343H 2

+33H) I ] M ' -n cn' (4.19b)3 C -C33 4 3 34

S = I n -x+ "0- g" I)

Sr [C34 (-aC2e +a13H 2)e

+ e3(-a e-X +a 1H,) )e n c cj. .C4 (4. 19c)

n -xSen = [-C 34 (a,2Ca2e +a 23H12)e

Cn3 a 3 rj~ 3 Hje H n1. e

(a2l-~ ,)e ](n rn 6.rn) (4.-19d)+ 33 (a10i'~ 23H ]n Cm I _J' ^-

33l 44-3 34

n -"(2 (g= )SZ n = -C 34(a 13 2e -x"a33 H12 )e

n -H - ](,- ] ( Hn+ 3 (a13p le -a3 Hl)e ]0 C .r3c ) (4.19e)

33 33 4- C31

53C4_043 3

22.

IH

_n -x- -2( -1 ) n e-X-le'~ (I ) In

rzn = ['C (a2H 12e -J)33 1 (C3C -CWC

(4. l9f)

For the problems under investigation in this report the stresses

converge as

".(2 ( I" l)e H coskn y

The convergence is fastest on the shell outer surface (0 = o > 0) and

slowest on the shell inner surface ( < = l - 0), where i. matches the

convergence of the applied load [Eq. (2.2)].

V'

23.

5. SECOND (GENERALIZED PLANE STRAIN) PROBLEM

The second problem to be considered is that of an infinite, un-

supported shell subjected to internal and external pressure and an applied

external axial force. If the radial displacement is assumed to be only a

function of r, and if the axial strain is assumed to be constant, the strain

displacement equations are given by Eqs. (3.1) in which now

e.= u = const (5.1a)

Yzr = 'zO = YOr = 0 (5.1b)

Hooke's law [Eq. (3.2a)] reduces to

er = B lar - B12Oe - '13ez (5.2a)

e0 = B2 1a r + B22 aO - ' 2 3 ez (5.2b)

where

B -1 (l- r) , -B z zI E- rzzr 12 B21 = E (VrO + vrzVz)r r

B =Vzr , B =1 (1'vz'ze) = 'zO (5.2c)

and the equilibrium equations reduce to

$?

24.

(r) r "e =0 (5.3)r r

The governing differential equation is obtained from Eqs. (3.1)

and (5.1) to (5.3); thus,

r[r(ra),r ],r - ( 1 2 rez (5.4)r 2g2 (d 22

Also, the displacement equation may be obtained by using the plane strain

assumptions in Eqs. (3.5); hence,

r wrr + rw r - D2w D - ezr (5.5)

It should be noted that no matter which of Eqs. (5.4) or (5.5)

is utilized, the previously mentioned singularity, which occurs for either

a transversely isotropic or an isotropic material, persists in the generalized

plane strain equations (i.e., when D4 = 0 and D2 = I or B23 = B13 and

B = B ). This singularity, which determines whether or not the differen-II 22

tial equation is homogeneous, is reflected in the solution. The solution

of the present problem is

a x-l + -X-l -*ea = Ar ' + Ar '; - Oe (5.6 )

r 1 2 o z

X-l -x-lce X (Ar -A2r )-o z(

X1 *X-

r() - +(B 3-.B ) +(E -a (B 3+B )Ie (5.6c)z 13+XB23)A 13 23+2 z 0 1 23ez

25.

D *(XB22 I)A 1r " (XB22+B2)A2 r- -, (-2) re (5.6d)

W= 22B211 22 212 - -D 2 z 5.d

u = ze (5.6e)

In which

X 8 2 / 2 D2 (5.6f)

* (B13-B23 )/(]-X 2)B22 , when X I 1

0 0, when X = I (transversely isotropic or isotropic)

(5. 6 g)

D f D4 /i-D2). when X. vt I

(I )*2 =

0, when X = I (5.6h)

The three conditions to determine A1, A2 and ez are the boundary

conditions

ar = - PO at r = r0 ( outer surface), (5.7a)

( .r = " P = ( p APc) at r = r (inner surface) (5.7b)

and the prescribed axial force condition

ror a rdr = 2 - ro 2 po ) (5.7c)

z 0

Ir

26.

Substitution of Eqs. (5.6) into (5.7) results in three linear algebraic

equations which may be solved to yield

A1 = J1/J , A2 = J2/J , Ez = J/J (5.8a,b,c)A1 12 2 z 3

where

,,m J, { 7 D op" -" - --- x. 2,-xLC- yI 2 -1)+J3 (Y 2 -I)yx ]+Po[~ (Y I+B (72 1)]

rl (5.8d)

M~ ... . 2. 2_1+ I+X

J2 =(' 2 (1-7J ")o+PL[aOp oy1 (7 X+ - i (7i)-po[ o( 'V73 1r, (5.8e)

, -_ X - 1 7- - X - ( 7 1 .l)- x _ o -Xrp "7 - )+P1[13 1 (7Y +1 1) -P"7 (Y -" p (T + 1"I) ,B* (7 1 ) ]

r 1 (5.8f)

j -' a I - h- ! * 1 X) 7--.X (5.8g)

M = (r,2 P - r 2p) (5.8h)

B 13 + XB23= + (5.8j)

4 ~~~ ~lxI

27.

(B13-%B 23 )/(1-X) , when X I

(.8k)

0 , when =l

P 3 = !Ez - ao (B 13 + '23)(

The stresses and displacements of a pressurized cylinder subjected

to periodically spaced band loads may now be obtained by superposing the

individually developed solutions [see Eqs. (4.4), (4.8), and (5.6)]; this re-

sults in

nr0" r-, D4 aPc 4 G .

w = (XB2 2-B2 1)Ai r(B 22+ 2 1)A2 (---) rez+ Z 8 ni=G liG e cosk ny2 11 n1l i=l

(5.9a)

napco 54 Gn Aiu = e zay + Al-- n= b ni=Z H li Gli e sink ny (5.9b)

11 n-l i-l

r -1 +A r --a * e + E [ex( )-a HiG e cosk y (5.9c)r zc i=l 12 n 13 i i n

4 3nX-1 rk-' CO 4 x Pi~ae = X(AI r -Ar2 r')-a0ez+Pc E Z [-e'X(a 12C i-a22n)-a23H li ]G ie cosk y

n=l i=l

(5.9d)

z = (BI3+XB23)ArI + (B13-kB23)A2 rxl + 23e z

4O 4+cF E~ [-e-x(a,3cx.+a2 5 + a H IG n e' cosk y (5.9e)cn=l i=l 23 3l i n

4

T (e.XQ.H I)G'ne 'sink y (5.9f)n=l i=l

28.

6. YIELD CRITERION

Any elasticity solution is valid only up to the pressure at

which yielding begins at some point on the shell. It is therefore of

great interest to attempt to find this initial yeild pressure.

R. Hill, Ref. 9 , developed a plastic potential function f

for the determination of that combination of stresses for which an ortho-

tropic material would begin to yield. This function of the stresses is

defined to be

I (L + 2) 1 1 22 2e 7 2 + * - 7i) (zz'G)

2 2 21 1 2 TeZ Tzr TrQ" ( y + 2 2 (6.1)

x 2 e z 2 re R 2 S 2 T

in which X) e, Z are the radial, circumferential and axial yield stresses,

respectively, and R, S, T are the appropriate shear yield stresses. The

criterion for yielding to begin is that f = I.

The stresses in Eqs. (5.9) could be substituted to leave an expres-

sion for the pressure poy(rz) for which yielding would begin at the position

(r,z)

pa +i-Iz2 a r=(rz) _. 2() z)2+(_ cPc e2 2 Z 2 Z2 2 -2 _pc c

2(i.. Q2 22 °rr rc -1/2+(L + _ I (r.)2 + szr (6.2)

22 2 p 2-2)X 0) Z c PC p

/c

?I

II

28.

6. YIELD CRITERION

Any elasticity solution is valid only up to the pressure at

which yie'ding begins at some point on the shell. It is therefore of

great interest to attempt to find this initial yeild pressure.

R. Hill, Ref. 9 , developed a plastic potential function f

for the determination of that combination of stresses for which an ortho-

tropic material would begin to yield. This function of the stresses is

defined to be

I 1Ik 1 1 2 .a)+. i )a~r2

f = t + - 72(ae +2 + X-2 t ()

2 2 2+ 1 1 2 T ez + T zr rr (6.

X E Z R S T

in which X, G, Z are the radial, circumferential and axial yield stresses,

respectively, and R, S, T are the appropriate shear yield stresses. The

criterion for yielding to begin is that f = I.

The stresses in Eqs. (5.9) could be substituted to leave an expres-

sion for the pressure poy (rz) for which yielding would begin at the position

(r,z)

1o 1 1' z z2 + I I1zP(r,z ) Po/y.v21(l+ 2 +) e 2z2-_I+, l)az O_)

o Pc 02 Z X Pc c P X 8 c c

I' 2

+ 1 1 r - 2 zr -1/2 (6.2)X+ e cc p S2 2 2p 2-2

X P

29.

In the filament wound composite, the radial yield stress X is

essentially that of the resin and is small compared with the axial and

circumferential yield stresses. This enables the approximation

22 (/2) << /2)(16<(1/ / 2 )2) < < )0/<(iX 2 (6. 3a, b)"

a a XTa a ay I zr2 r e z 1/2

(6.3c)

From either of Eqs. (6.2) or (6.3) it can be clearly seen that the

transverse shear term can serve to decrease the yield pressure.

iA

V

- -. F

42

30.

7. NUMERICAL COMPUTATIONS AND DISCUSSION

The approximate solution to the three-dimensional Navier equa-

tions of an orthotropic, infinite circular cylinder requires numerical

verification before it can be used. For this reason the first set of

numerical computations (which were all performed on the IBM 30150 com-

puter located at the Polytechnic Institute of Brooklyn) were devoted to

comparing the approximate results developed here with the exact solution

to a transversely isotropic cylinder subjected only to periodically spaced

band loads. The exact results were given to the author by H. Levine, who

used the exact analysis described in Ref. 4. The cylinder constants

are

EEr E E = 2Ez G =G r/5'28 Vzr =.16 ,' r =0.30

D L rISA = 0.2 -= 0.2 1 0.8 a oL r r0 0

(7. 1)

The comparison is shown in Figures 2 and 3. The results indicate

that although a correction term [.of the type obtained from the solution of

either of Eqs. (4.15) and (4.16)] is desirable, the zeroth-approximation

solution gives results which would be satisfactory to the designer. This

solution gives a fairly accurate representation of the stress distributions

through the thickness in that it demonstrates the nonlinear variation of

the normal stresse. . The tendency of the shear stress to "peak" in the

vicinity of the load discontinuity is predicted.

31.

The second set of computations utilizes the approximate three-

dimensional elasticity solution of an orthotropic shell, subjected to ex-

ternal pressure and an axial force in addition to prescribed periodically

spaced band loads. The results offered by the shell theory developed in Ref. I

are compared to the more exact results obtained here.

In Ref. 1, the band loads were due to periodically spaced elastic

ring supports. The radial displacement of a ring was found as d function

of the magnitude of the band load (p c) in a separate analysis (either ring

theory or an orthotropic Lame analysis). The ratio po/Pc was then obtained

by matching the radial displacements of the ring and the shell at the ring-

shell interface. In the present analysis, the ratio po /c is assumed to

have the same numerical values as those determined in Ref. I.'I

When Flugge-type shell theory was applied to a corresponding

ring-supported orthotropic shell it wa: found that po/Pc = 1.11. The addi-

tion of transverse shear defornarion to this analysis resulted in po/Pc = 1.03.

Obviously, the results of the present three-dimensional elasticity solution do not

reflect the restraining effects of the ring. In order to correct this it

would be necessary to include an analysis of the ring and then match dis-

placements at the ring-shell interface.

The cylinder constants are

O6ps 16 6 1 1~~1 sE = 2.49 x 06psi E =6.14 x i0 psi E6 4.74 x 106psi

rz0.]36 0.176

Vzr 0.5

Letter dated March 24, 1964 from Mr. W.P. Couch, David Taylor Model Basin

32.

-= 0.1755 0.2417 , r = 3.388 in. r, 3 in.

= 0.0769

X =0.22 x 0 i , i = 1.7 x 10p , Z = 1.0 x 0pi

S = 0.09 x 10 psi (7.2)

The transverse constants Er) Grz and vzr were not available, as they are

extremely difficult to obtain (Ref. 6) and were simply given values that

were felt to be representative of this type of material (e.g., see Ref. 7).

The results are presented in Figures 4 to 10. Shell theory gives

excellent results at midbay, but can only approximate the actual state of

stress at the frame. The yield pressures predicted by shell theory [using

the criterion in Eqs. (6.1) with f = 1] are fairly accurate. Although

the present results given more accurate stress distributions for the band

load problem, it does not necessarily follow that the band load problem

exactly reflects a ring-shell interaction problem.

Klosner and Levine (Ref. 4) found that transverse-shear-deformation

shell theory did not lead to improvement over classical shell theory. Such

observations can also be made from the present calculations (see Figures

4 and 5). Furthermore, these results indicate that, at a sufficient dis-

tance away from the band load, the shear stress can be approximated with a

parabolic function. However, the results displayed in Figs. 4 and 8 show

that the axial displacement and axial stress vary cubically (at least)

through the shell thickness. Hence, in the vicinity of the band load, where

the axial displacement reaches its largest value, plane sections do not re-

main plane and the transverse shear stress can not be assumed to vary para-

33.

bolicaily through the thickness. The cubic and higher order thickness terms

combine to result in the "peaking" shown in Figure 6.

The Hill yield criterion Eq. (6.2) predicts that yielding begins

at a fairly low pressure (3500 psi) near the inner surface at the load dis-

continuity. Actual tests (Ref. 2) indicate that the shells do fail at the

frame near the inner surface; but at a significantly higher pressure (12200 psi).

The low theoretical yield pressure results from the low value of the resin

yield stress. When this yield stress was increased (i.e., X was set equal

to 1.0 X 105 psi) the lowest value of the theoretical yield pressure was

found to be Poy(rl, L/2) = 16323 psi.

SOne might speculate that the resin yields at a very low pressure

after which the load is resisted by the glass fibers. If this is the case,

* I the work done here might be extended to a layer analysis (see Appendix),

which could be used to approximate the stresses after yielding has begun.

It appears from these results that the transverse shear stress

distribution has very little effect on the low pressure at which the resin

begins to yield (which was also predicted by the shell theory). However,

as is pointed out in Ref. 1, the actual stresses in the individual con-

stituents of the nonhomogeneous filament wound composite shells must be

obtained by multiplying the stresses obtained here by suitable stress-

concentration factors. This procedure gives rise to substantial changes

in the magnitude of the initial yield pressure (e.g., see Ref. I).

It is interesting to note that the shell wall gets thicker

k,

under pressure (see Fig. 9). This can be traced to the Poisson effect

of the rather large compressive axial and circumferential stresses.

!

34.

In general, the numerical results obtained here indicate that the

approximate elasticity solution developed here to satisfy the orthotropic

Navier equations gives a satisfactory description of the stress distribution

through the shell thickness. However, before this technique can be success-

fully applied, the transverse elastic constants must be experimentally deter-

mined. The low yield pressures obtained here may correspond to the "initial

yielding" state described by Tsai (Ref. 8) in which the load deformation

curve of a similar shell subjected to internal pressure was observed to

have a sudden change in slope at a low pressure. If this is the case, the

layer analysis suggested in the Appendix of this work would certainly be

appropriate.

35.

A.EFERENCES

I. Kempner, Joseph; Misovec, A.P. and Herzner, F.C.: Ring-StiffenedOrthotropic Circular Cylindrical Shell under Hydrostatic Pressure.PIBAL Rep. No. 68-10, May 1968.

2. Hom, K.; Buhl, J.E. and Couch, W.P.: Hydrostatic Pressure Tests ofUnstiffened and Ring Stiffened Cylindrical Shells Fabricated of GlassFilament Reinforced Plastics, David Taylor Model Basin Report 1745,Sept. 1963.

3. Klosner, J.M. and Levine, H.S.: Further Comparisons of Elasticity andShell Theory Solutions, Polytechnic Institute of Brooklyn, PIBAL ReportNo. 689, July 1964.

4. Levine, H.S. and Klosner, J.M.: Transversally Isotropic Cylinders underBand Loads, Jour. Eng. Mech. Div., Proceedings of the ASCE, June 1967,p. 157.

5. Lekhnitskii, S.G.: Theory of Elasticity of an Anisotropic ElasticBody, Holden Day, Inc., San Francisco, 1963.

6. Hom, K.; Couch, W.P. and Willner, A.R.: Elastic Macerial Constantsof Filament Wound Cylinders Fabricated from E-HTS/E787 and S-HTS/E787Prepreg Rovings, David Taylor Model Basin Report 1823, Feb. 1966.

7. Myers, N.C.; Lee, G.D.; Wright, F.C. and Daines, J.V.: Investigationof Structural Problems with Filament Wound Deep Submersibles, FinalReport, H.I. Thompson Fiber Glass Co., January 1964.

8. Tsai, S.W.; Adams, D.F. and Doner, D.R.: Analyses of CompositeStructures, NASA CR 620, November 1966.

9. Hill, R.: A Theory of the Yielding and Plastic Flow of Anisotropic

Metals, Proceedings Royal Society, A193, pp. 281-286, 1948.

I

¢-

36.

APPENDIX

it may be desirable to prescribe a variety of combinations of

radial stress, shear stress, axial displacement and radial displacement as

periodic functions of z on either of the shell surfaces (for example, the

ring supported shell may be analysed by matching ring and shell displacements).

Therefore a more general discussion of possible boundary conditions is offered.

On each shell surface (outer and inner), any two of the following

four quantities must be prescribed:

E le o a) a3H i]Gni = Hn (radial stress) (Ala)

2 4

-x4 0E (e a. H -)Gn E Hn (transverse shear stress)(eIi i = Ii 12i=l 2 3 (Alb)4 (a

G =(radial displacement)

(Alc)4F GnH = Fn (axial displacement) (Aid)

i=l

n n nwhere Hj, G and F are Fourier series coefficients obtained by expanding

the prescribed functions of z.

It is possible to perform a layer analysis by matching the four

quantities listed in Eqs. (Al) across each interface. If a shell consists

of m layers, 4m arbitrary constants would have to be determined The inter-

action equations, which would be n4m-4 in number, could be obtained from

zI

37.44

j

n) = (H?) (A2b)

2Ab

G . = Gn (A2c)i -I j

Fn _ = Fn. (A2d).i-I .I

where j = 2, " m. The subscripts I and m correspond to the inner and

outer shell layers, respectively. The remaining 4 equations are found by

applying two of Eqs. (Al) on layers I and m. The corresponding generalized

plane strain problem must also be solved layer by layer with the radial

stress and both displacements being matched at each interface.

If the layers were permitted an axial motion relative to each

other (as might occur after initial yielding) Eqs. (A2b) and (A2d) could be

replaced by

(H2) =Tn (A2'b)j-I

(Hn). = Tn (A2'd)r3

where Tn would be a constant corresponding to a maximum shear stress at

resin yield.

I

ACKNOWLEDGEMENT

The authors are grateful to Mr. Eugene Golub, who was responsible

for the success of the numerical computations.

I:

SPo fr,-

PP

D 2 D

FIG. ISHELL GEOMETRY AND LOADING

A

o i z

< U)

-JL II Ca:

z LI

La: cc - LU

~~00~ z-z :Oui ~UJ U)

0 _ L'0j.

OD' 0 <~U

-J 0 -

U9 00 0 <-~ tLi i U) LI

4: 000

tU) ew4 V( < w LU<

oo U) <

dd0_ 0_ 0 _ _ ___ _ _

oo (N 00 0 0

-0

0 ~00 I

INIIN

i I

Ja 0

-oo-_0-o

00

SIC,

A A/

r/ a .r A

1.00

FI

0 .94 F I --7 -6 -5 -4 -6 -5 -4 -6 -5 -4

az/p Oz/pc az/pcz=o Z/a -0.109 /a *0.199

M I DBAY

1.08-

r/aT -A 4 A- -A A:

, 00

0f .94 / 1 /0-8 - 7 -6 5 -4 -6 -5 -4 -6 5 -4

Oz/pc az/pcZ/a-O.1993 Z/a -0.2 Z/-0.2-417

EDGE OF CENTER BAND LOADBAND LOAD

F-FLOGGE SHELL THEORYF.- FLUGGE SHELL THEORY WITH TRANSVERSE SHEARA- APPROXIMATE ELASTICITY SOLUTION

c(/P -- -- - o/c" -t.0 34

FIG. 4AXIAL STRESSES IN ORTHOTROPIC CYLINDER

r/O a

ios 1 I - -

tk/1.00-

0.94M1-9 -a -7 -9 -8 -9 -8 -7

I a/ PC a#1 PC 4e PCz/L-o z/a-o.109 z/a -o.199

1.06---

A. F AV_ A J1

,o0. 1993 Z/I=0.2 /a = .2417

FIG. 5 CIRCUMFERENTIAL STRESSESIN ORTHOTROPIC CYLINDER

/ i / 1

1.08 1.086

r/ a tAr/ a A

1.00- 1.00-

0.94 -- 0.94-A.0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4

Z/I -0-109 Z/a -0199

1.06 1.06 -

r/a r/ a

1.01.00

0.94 0. 940 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4

Tr z /Pc Trz/Pcz11a -0.1993 -/A - 0. 2

EDGE OF BAND LOAD

FIG 6 TRANSVERSE SHEAR STRESSESIN ORTHOTROPIC CYLINDER

1.06 -

r/ a A A A A A

1.00-

0.94 --I 0 -1 0 -I 0

ar/ PC Or/ PC ar/Pc-o z/a -0.109 Z/a 0.199

MIDBAY

1.06

r/a AVAI A," A A- "

1.00 L

0.94--1.0-0.5 -1.0-0.5 -I.I - .0

ar/Pc Or/Pc Or/PcZ/a=0.1993 Z/a -0.2 Z/a -0.2417

EDGE OF CENTER BAND LOADBAND LOAD

FIG. 7

RADIAL STRESSES IN ORTHOTROPIC CYLINDER

1 1.08I.r/a A A A A ,A A A

t. 0I I

1.00II

I

0.94-0.3 -0.2-0.5 -0.4-0.5 -0.4-0.5 -0.4

ANUU A11U A11UaP cC Pc IPc apc

Z/a=o.1o9 ./&=o.1g z/a=o.1993 Z/a-0.2

FIG. 8 AXIAL DISPLACEMENTS OFORTHOTROPIC CYLINDER

1.06- 1.r/A AAYI

1.00T

0.94-3.6 -3.2 -3.8 -3.2 -3.4 -3.0

A,W AW AWApC :Pc JPc

Za -o Z/a -o.Io9 z/A -0.19

1.06

Ar/ F A A A Aj

1.00-JI

I: I0.94 --3.4 -3.0 -3.4 -3.0 -3.4 -3.0

A , A AiW V

./A= 0.1993 /a = 0.2 Z/ - 0.2417

FIG. 9 RADIAL DISPLACEMENTS OFORTHOTROPIC CYLINDER

a FLUGGE SHELL THEORY0 FLUGGE SHELL THEORY WITH TRANSVERSE SHEAR

1.00

,0 -/A A A A A

0.94 0- -

3 4 5 3 4 5 6 3 4 5 6pI x|0" p x 3 P -x 3

1-0 Z/a- 0.199 L/ - 0.1993MIDSAY EDGE OF BAND LOAD

1.06"A A A/

r/a

1.00 -

0.94 1,3 4 5 6 3 4 5 6

py x I01 py x I0-

z/a-o.2 Z/a -o.2417CENTER BAND LOAD

FIG. 10 YIELD PRESSURES

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Dept. Aerospace Engrg. & Applied Mechanics 2 ~u333 Jay Street, Brooklyn, N.Y. 11201

APPROXIMATE ELASTICITY SOLUTION FOR ORTIJOTROPIC CYLINDER UNIDER HYDROSTATIC PRESSURI

Andrew P. Misovec and Joseph Kempner

6.111 RPO FT OA T9 a 7a. TOTAL #40. OF PAG9ST 7b NOof0 Raev

ti. CONTRACT ON GRANT NO. to ORIGtNATOW1, REPORT NUhASER(S)

Nonr 839(39)bPR0J9CT NO. PIBAL Rep. No. 68-11

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It. SUPPLEMENTARY NOTES 12 SPONSORING MILITARY ACTIVITY

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13 ABSTRACT An approximate solution to the Navier equations of the three-dimensionaltheory of elasticity for an axisyninetric orthottmpic circular cylinder subjectedto internal and external pressure, axial loads, and clo~sely spaced periodic radialloads is introduced. Numerical comparison with the exact solution for a transver-sely isotropic cylinder subjected to periodic band loads shows that very goodaccuracy is obtainable.

When the results of thils approximate solution are compared with pre--viously obtained results of a Flugge-type shell solution of a ring-reinforcedorthotropic cylinder, it is found that the shell theory gives a fairly accurateArepresentation of the deformation except in the neighborhood of discontinuousloads. The addition of transverse shear deformations does not improve the accuracyof the shell solution.

When Hill's orthotropic yield criterion is applied, it is found thatyielding could begin rather early at the inner surface of the shell adjacent tothe frame. It Is noted that the transverse shearing stress has no greateffect on tne initial yield pressure.

DD 'JA'1!64 1473 LtNCLASS IF111

UNCLASSIFIEDSeCutty Classification_____ _____ ____

I.LINK A LINKS9 LINKCKEY WORDS NL i nt ? N L S

Forthotropic cylinder

band loaded cylinder

composite cylinder

orthotropic elasticity solution

perturbation solution

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UNCLASSIFIEDSecurity Classification

-4

POLYTECHNIC INSTITUTE OF BROOKLYN

EfRR A TA

APPROXIMATE ELASTICITY SOLUTION FOR ORTHOTROPIC CYLINDER INDER

HYDROSTATIC PRESSURE AND BAND LOADS

by

A. P. Mlsovec and Joseph Kempnet

May 1968

Replace Figs. 4 to 10 by the enclosed

*p. 3.1 The first two sentences in the third paragraph should read as follows:

When Flugge-type shell theory was applied to a corresponding ring-supported orthotropic shell it was found that po/pC = 0.960. The additionof transverse shear deformation to this analysis resulte6 in po/PC = 0.972.

PIBAL REPORT NO. 68-11

A

r/a A= . ./-I A4I A

1.00 ---

F/ AA

F I

0 .94 ------

-6 -5 -4 -3 -6 -5 -4 -3 -8 -5 -4 -3

az/Pc az/Pc zz/Pco /a.1990 Z/a/ o..199

MIDBAY EDGE OF BAND LOAD

1.06

r/la

A AA A1.00 - T

-6 -5 - - -5 -4 -3

cz/pc ar/p,Z/a - 0.2 Z/a =0.2417

CENTER BAND LOADF -FLUGGE SHELL THEORYFT - FLOGGE SHELL THEORY WITH TRANSVERSE SHEARA- APPROXIMATE ELASTICITY SOLUTION

Po/Pc =0.960 Po/P, - 0. 972

FIG. 4AXIAL STRESSES IN ORTHOTROPIC CYLINDER

1.0-

M1 A I A A

I

0.94-8 -7 -8 -7 -8-8 -7 -8

a./Pe eO/PC ao/pCZ-O Z/ " 0.g1990 Z/a - O. 1993

1.08 - ,

r/a /I Ii_

A _A A

0.94L--_ //-7 -7/--

./a =0.2 Z/a = 0.24 17

FIG. 5CIRCUMFERENTIAL STRESSES IN ORTHOTROPIC CYLINDER

-= ...I ..

1.06 -}1.06:

r/a NA r/a -A

1.00 00--

0.94V 1 0.94L-i 00 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4

-rz/pc 'rz/ PC

1.08 1.06

r/a r/ a

A F A1.00

094L" ,Ij 0.94 ---

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4Tr z/ *rrz/Pc

z/a -0.1993 I/A -0.2EDGE OF BAND LOAD

FIG. 6 TRANSVERSE SHEAR STRESSESIN ORTHOTROPIC CYLINDER

1.06

r/aA AA

1-001-_A

-lr c0 0 -I0d 0 P r/ Pc a/pC

Z-0a -0.1990 Z/a -. 1,993

1.06 -

A

r/a A A

1.00 7Z

0. 94 [: - -

1I 0 -1.1 -1.0 -0.9Cyr/PC Ulr/P C

Z/A =0.2 /a = 0.2417

FIG. 7'RADIAL STRESSES IN ORTHOTROPIC CYLINDER

1.06

r/a

AA A A -

1.00

0.941-AX-0.45 -0.35 -0.45 -0.35 -0.45 -0.35

A,,U All U A11U

i./a = 0. 199 -/a0.1993 Z/8 - .2

FI1G. 8AXIAL DISPLACEMENTS OF ORTHOTROPIC CYLINDER

r/a

// AA1.00 / ,

At

0. 94 4-3.2-3.0-2.8-.2.6 -2.8 -2.4 -2.8 -2.4

AW AW AUWaPC apc apC

Z= 0 Z/a -0.1990 Z/a= 0.1993

1.08

r/a

[.00

0.94M I-2.8 AW-2.4 -2.8 AW- 2.4

Z/a -0.2 i/a - 0. 2417

FIG. 9RADIAL DISPLACEMENTS OF ORTHOTROPIC CYLINDER

.cc -----

r/a

1-0 - A -- 4 -

0.00,L IL

3 4 5 3 4 5 a 3 4 5 6p x 1-(p- ) p x Iop(ps,) p x ,o- s

zwO z/a -0.1990 Z/a - 0.1993

0 FLUGGE SHELL THEORYa FLUGGE SHELL THEORY WITH TRANSVERSE SHEAR

1.08-

r/a

A A-A A

1.00 - -

_____ /

0.94-3 4 5 6 3 4 5 6 7p X 163 (pS1) P X1- iO(PSI)y/ = 0.2 Z/a -0.2417

FIG. 10 YIELD PRESSURES