THE EFFECT OF IMPERFECTIONS

ON THE

BUCKLING OF THIN WALLED

CYLINDRICAL SHELLS

Thesis by

Robert L. Ditchey

In Partial Fulfillment of the Requirements

For the Degree of

Aeronautical Engineer

California Institute of Technology

Pasadena, California

1973

(Submitted May 25, 1973)

-ii-

ACKNOWLEDGMENT

The author feels very sincerely that this work is but an

incremental step in comparison to the accomplishments of many

engin~ers and theoreticians who have studied this topic. The

intellect and the dedication of so many inspires both awe and

humility.

The author is deeply indebted to Dr. C. D. Babcock, Jr. for

his guidance, patience, and encouragement during the course of

this work, and to Dr. J. Arbocz for his generous assistance and

understanding. In particular, a great debt is owed to Dr. E. E.

Sechler, without whose assistance none of this would have been

possible.

Gratitude is due to Elizabeth Fox for her great skill and

the patience shown in typing this thesis, and to Betty Wood for

preparing the figures.

Finally, a very special word of gratitude is expressed

to the author's wife and children for their encouragement, patience,

and the many sunny weekends spent at horne.

-iii-

ABSTRACT

A theoretical study of the effect of imperfections on the buck­

ling load of a circular cylindrical shell under axial compression was

carried out.

The nonlinear shell equations of Donnell are solved in an

approximate manner using the Potential Energy Theorem. The

method of derivation and the resulting algebraic equations are com­

pared to previous derivations which use a different method of solu­

tion. It was found that the resulting equations are identical to the

ones derived by Galerkin's procedure, lending credence to that method.

A parametric study of an imperfection model was carried

out numerically. The imperfection model is similar to that origi­

nally used by Donnell, having amplitude decay with increasing wave

number. The results show the effect of various rates of decay of

amplitude with wave number, as well as the effect of various

assumed amplitudes. Using these parameters, it is shown when

shell buckling is dominated by long wave imperfections as opposed

to classical short wc:.ve imperfections.

It was found that if the decay of asymmetric imperfections

with axial wave number is sufficiently high, that long wave imper­

fections can dominate. In the range of values chosen for these

parameters, changes in the axisymmetric imperfection model had

little effect on buckling behavior as compared to the changes in

sensitivity to long waves which occur as the asymmetric model

is varied.

A}, A 2 , A 3 , etc.

B}, B 2 , B 3 , etc.

a}, a 2 , a 3 , etc.

bI

, b2

, b3

, etc.

c

E

G}, G2

i, k

I.

L

Ml

, M 2 , M3

,

M 6, M9

, M9

N l ,NS,N 6,NS'

N9

Nx ' Ny' Nxy

p}' P 3' P 4

q

r

R

s

t

u

U 1,U2,U3

-iv-

LIST OF SYMBOLS

coefficients given on page 29

coefficients given on page 29

coefficients given on page 30

coefficients given on page 30

= -V3(1-V2

)

Young's modulus

operators defined on pages Sand 6

axial half wave numbers

circumferential wave numbers

length of shell

coefficients of N y

coefficients of N x

Stress Resultants, defined on page S

coefficients of N xy

decay constant operating on i

decay constant operating on k

shell radius

decay constant operating on I.

shell thickness

Potential Energy

integrals representing contributions to the total

potential energy

u,v

w

w

x,y

XA,X

o.i , o.k , o.i+k' o.i _k ,

{3, 'Yk , Yk , 'Yi+k'

'Yi_k

€x' €y' 'Y xy

v

A

a

s· 1

~. 1

-v-

LIST OF SYMBOLS (Cont'd)

displacements

radial displacement, po sitive outward

radial imperfection (displacement) from perfect

circular cylinder

axial and circumferential coordinates on the

middle surface of the shell, respectively

axisymmetric and asytnmetric imperfection

amplitude used in the imperfection models,

respectively

mode shape parameters defined on page 34

strain, defined on page 8

Poisson's Ratio

nondimensional loading parameter

axial stress

Rca =-:Et

nondimensional radial displacement amplitude

nondimensional initial imperfection amplitude

-viii-

TABLE OF CONTENTS

Part Page

I GENERAL INTRODUCTION 1

II THEORETICAL FORMULATION 4

l. Donnell Shell Equations 4

2. Mode Selection 5

3. Boundary Conditions 7

4. Stress Resultants 8

5. Potential Energy 11

III NUMERICAL ANALYSIS 15

1. Imperfection Model 15

2. Shell Geometry and Material Properties 16

3. Wave Numbers and Mode Shape Parameters 16

4. Numerical Procedure 17

IV DISCUSSION OF NUMERICAL RESULTS AND

RECOMMENDATIONS 20

1. The Effect of Varying the Axisymmetric

Model Parameters (X A and q) 20

2. The Asymmetric Model - The Effect of

Varying X 22

3. The Asymmetric Model - The Effect of

Varying s 22

4. The Asymmetric Model - The Effect of

Varying r 24

5. Items Which Induce Short Wave Imperfection

Sensitivity 24

-I-

I. GENERAL INTRODUCTION

The buckling load of isotropic thin-walled cylindrical shells

under axial compression cannot be accurately predicted by linearized

small deflection theory. Clas sical theory of this general category

predicts a collapse load which is never reached in experiznental work.

Moreover, the results of numerous experiments indicate that a very

complex phenomenon must be involved.

The general problem of cylinders in axial compression has

been investigated and discussed by a very great number of authors.

The reader is referred to Fung and Sechler (Ref. 1) for a detailed

discussion and a listing of some of the many writings on this topic.

For a discussion of the role of imperfections in shell buckling, the

reader is referred to Arbocz and Babcock (Ref. 2).

The work presented herein represents an amplification of

the theory presented by Arbocz and Babcock. In that paper the im­

perfect shell equations are solved in an approximate manner using

a two mode deflection assumption and a Galerkin approach. The

resulting algebraic equations are simplified by neglecting cubic

term.s in com.parison to the quadratic terms (Ref. 3). The circum­

ferential periodicity condition (Ref. 4) is not considered in the

fo rmulation.

A study of the influence of these assumptions on the result­

ing algebraic equations is one objective of this work. Figure 1 is

a graphical display of the rationale involved.

Starting with the as sumptions made in the derivation of the

nonlinear shell equations of Donnell (Ref. 5) one may choose to use

-2-

a Galerkin type of approxim.ation procedure, or to follow a potential

energy m.ethod of form.ulation. Having chosen either of these paths,

two branches then exist for each - whether to seek a particular solu­

tion to the com.patibility equation (a stress function form.ulation) or to

seek a particular solution to the inplane equilibriUITI. equations (a dis­

placem.ent form.ulation). Variations in m.ethod then follow depending

upon the specific way by which periodicity of the circum.ferential

displacem.ent is satisfied. Considering just the Galerkin procedure,

different m.ethods can be obtained depending upon variations in the

weight~g function. Considering the m.ethod of solving the inplane

equilibrium. equations, different m.ethods m.ay be obtained depending

upon whether or not one allows for a constant term. in the radial

displacem.ent. This term. would perm.it a uniform. expansion.

The Galerkin procedure used by Arbocz and Babcock (and

others) left open the m.athem.atical validity of the specific weighting

function to be used - an essential and all-im.portant factor in the

analysis. Since it was recognized that the Galerkin procedure was

approxim.ate, it was desirable to derive a solution to the Donnell

equations using a potential energy m.ethod and com.pare the results.

It should also be understood that an essential factor which

will introduce variations in all m.ethods is the m.athem.atic m.odel for

radial displacem.ent and the im.perfections (m.odal decom.position).

For exam.ple, the m.ethod followed herein is a two m.ode representa­

tion using cosine axial representation for a general axisyrnm.etric

term. and sine axial representation for a general asym.m.etric term..

One could, of course, introduce com.plexity by using m.ore than two

-3-

modes .and one could use a different trigonometric representation.

It is clear from earlier work that a two mode solution does lead to

numerical inaccuracy, as compared to using more than two modes,

and that experimental measurements show significant prebuckling

growth of more than two model components (Ref. 2). On the other

hand, since it was an objective of this work to compare the end

result of the Galerkin method used by Arbocz and Babcock with a

potential energy derived solution, it was considered appropriate to

use the simplest model for radial displacement which captures the

essential nature of the physics, i. e., the two mode case. As will

be described subsequently, the parametric study of sensitivity to

imperfections also dictated simplicity from a computational time

standpoint.

A second objective of this work was to carry out a numeri­

cal study of imperfection sensitivity using as a basis the data from

a recent investigation conducted by Arbocz and Babcock (Ref. 6).

This work, which uses an imperfection model introduced by Irnbert

(Ref. 7), clearly showed that proper modal decomposition of imper­

fections should include asymmetric components, and should include

a decay of imperfection amplitudes with increasing wave numbers.

The numerical work presented in this paper is an expansion

on this topic. Presented herein is a parametric study which shows

the effect that various decay rates have on the buckling load, when

buckling can be dominated by lower order modes, and how the

various parameters of the imperfection models proposed in refer­

ence 6 are interrelated.

-4-

II. THEORETICAL FORMULATION

This section is devoted to the derivation of the appropriate

algebraic equations which represent an approximate solution to the

nonlinear Donnell equations. The approach taken is to solve the

inplane equilibrium for the displacements u and v after an assumption

is made on the radial displacement wand the initial imperfection Vi.

The third equilibrium equation is satisfied in an approximate manner

using the Potential Energy Theorem.

1. Donnell Shell Equations

Using the nonlinear Donnell equations, the two inplane equi-

librium equations for an isotropic cylindrical shell may be written

(in terms of displacement) as:

2 2 2 a u+(l-v) au +(l+V ) a v - G -~ 2 ~ 2 axay - - l(w,w) ax ay 222 a v + (1 - v) a v + ( 1 +v ) a u -

-2 -2- -Z -2- axa = -G2(w,w) ay ax y

where the operators G1

(w, Vi) and G2

(w, w) are given by:

and

G (w, Vi) = a2w. a(w+Vi) + (1 +v ) Ow. a

2 (w+w)

1 ax2 ax 2 ay axay

2 + (l-v) aw 2 ay2

a(w+w) + ~ Ow + ax R ax

2 - 2 + a w. Ow + (1 +v) Ow. ~ ax2 ax 2 ay axay

2-+(l-v)aw. Ow 2 ay2 ax

(1 )

(2)

(3)

-5-

2 - 2 -- 0 w o(w+w) + (I+v) Ow 0 (w+w) + G2 (w, w) = -2 • oy -Z ax· axoy oy

2 - 2-+ (I-v) 0 w • o(w+w) + .!.. Ow + 0 w. Ow +

2 ax2 oy R oy oy2 oy

+ (1 +v) Ow • 2 2-

o w + (I-v) 0 w Ow ( 4) 2 ax oxoy -2- ax2· oy

2. Mode Selection

Assuming the following representation for the displacement w

and the imperfections w:

w = £1 t cos i7r ~ + £2t sin k7r l cos.£ f + £3t (5)

w = ~1 t cos i7r ~ + ~2t sin k7r I. cosl f ( 6)

and performing the operations given by G I and G2 , the right hand

side of equations (1) and (2) are given by: .

G I (w, w) = Al sin 2i7r ~ + A2 sin i7r l +

+ A3 sin 2k1T I. + A4 cos k7r l cos.£ f +

+ A5 sin 2k7r l cos 2.£ f +

+ A6 cos(itk)7r l cosl f +

+ A7 cos(i .. k)7r l cosl f (7)

-6-

G 2 (w, Vi) = BI sin21 i + B2 cos2k7T ~ sin21 ~ +

+ B3 sin(i+k)7T L sinl f +

+ B 4 sin(i-k)7T ~ sinl f +

+B . k x . Y S sm 7T L sml R

where the coefficients A. and B. are given in Appendix I. 1 1

(8)

A particu-

lar solution to the two inplane equilibrium equations is then obtained.

It is assumed that the form of the displacements u and v are given by:

and

up = a l sin 2i7T ~ + a 2 sini7T ~ +

+ a 3 sin 2k7T I. + a 4 COSk7T ~ cosl ~ +

+ as sin 2k7T ~ cos 21 ~ +

+ a 6 cos(i+k)7T ~ cosl i +

+ a 7 cos(i-k)7T L cosl ~

vp = b l sin21 i + b 2 cos2k7T ~ sin21 i

+ b 3 sin(i+k)7T ~ sinl ~ +

+ b 4 sin(i-k)7T L sinl f +

b 'kx 'nY + S sm 7T L smA: R

(9 )

(I 0)

-7-

Note that periodicity of the circumferential displacement is satisfied

without the need of introducing a homogeneous solution as is necessary

in a stress function approach (Ref. 4).

Differentiating u and v as given by the left hand side of p p

equations (1) and (2), and equating coefficients of like trigonometric

terms, algebraic equations which determine the coefficients of u p

and v are obtained, i. e., the coefficients a. and b. are determined p 1 1

in terms of the coefficients A. and B.. These coefficients are given 1 1

in Appendix II.

3. Boundary Conditions

In order to satisfy the boundary conditions on the displace-

ment u, one must find an appropriate homogeneous solution

u(x, y) = u (x, y) + u + ulx P 0 (11 )

The boundary condition at x = 0 is

u(O, y) = 0 (12)

or

u = -u (0, y) o p

now

up(O, y) = a 4 cosl i + a 6 cosl i + a 7 cosl i

The satisfaction of this boundary condition with the simplified homo-

geneous solution is not possible. However, an "average" type

boundary condition, i. e.

27TR f [up(O,y) + U o ] dy = 0 o

can be satisfied and gives

u = 0 o

-8-

The stress resultant Nx

at x = L will be used to determine u 1 0

4. Stress Resultants

Using the constitutive relations

N x _ Et ( - ~ € +v€ )

I-v x Y

Et N =--2(€ +v€)

y I-v y x

N = xy Et

y 2(1+v) xy

and the strain-displacement relations

8u 1 Ow Ow Ow € =-+-(-+2-)-

x Ox 20x 0x0x

8v w 1 Ow Ow Ow € =-+-+-(-+2-)-

Y 8y R 2 8y 8y 8y

8u 8v Ow Ow Ow Ow Ow Ow Y =_+_+_0_+_0_+-0-xy 8y Ox Ox 8y 8y Ox Ox 8y

the stres s resultants are given in terms of displacement as:

(13 )

( 14)

-9-

N -2 x [au 1 Ow Ow Ow] ( 1- v ) - = - + - (- + 2 -) - +

Et Ox 2 Ox ax Ox

+ v [OV + w + ..!. ( oW + 2 ow) Ow] oy R 2 oy oy oy

N -( 1 - v 2) Ef = v [~ + ~ ( : + 2 ~;) :] +

+ [OV + w + 1.. (Ow + 2 Ow) Ow] oy R 2 oy oy oy

N a a - -2(1+v) 2Y = ~ + 2 + [Ow. o(w+w)] + Ow. Ow (15)

Et oy Ox Ox oy oy ox

Since the displacements are now fully determined, one has that:

_ Et { Y... x Y.. Nx - --2 Nl + NS cos2l R + N6 smk7T L cosl R I-v

+NS sin(i+k)7T i cosl f +

+N9 sin(i-k)w ~ cosl i} (16)

Et { . x x N = --2 M1 + M2 COSl7T L + M3 cos 2k7T L y I-v

+M6 sin k7T l cosl f +

+MS sin(i+k)7T l·COSl i +

+M9 sin(i-k)w ~ cosl i} (17 )

-IO-

N = Et {p cosk17' x sin£ y + xy 2(I+v) 1 L R

+P 3 cos(i+k)17' r. sin£ f +

+p 4cos(;-k)" L smi it}

The coefficients N., M., and P. are given in Appendix III. 1 1 1

In order to determine u1

, one must enforce that

217'R J [Nx(L, y) - at] dy = 0 o

Et Et v Nx(L, y) = -.:-2 Nl + --2 N5 cos 2£ R

I-v I-v

therefore

2 a N = (I-v) E 1

(18 )

(19 )

(20)

Since the coefficient Nl in equation (16) contains the as yet undeter­

mined coefficient u1

, one has that:

vt t 2 i17' 2 1 2 u 1 = - It ~3 - 2 (1:) (2 £1 + ~1 ~1)

t 2 k17' 2 £ 2 1 2 - 4" [(r) + v (R)] ('2 £2 + ~2~2)

2 a + (I-v) E

Note that this means that N is given by x

(21)

-11-

Nx = at + { Et 2 N 5 cos 21 f + I-v

Et N . k x Y --2 6 sIn 1T L cosl R + I-v

+ Et2 NS Sin,(itk)1T l cosl i + I-v

+ ~ N9 sin (i-k)1T x cosl Y..} I-v L R

(22)

5. Potential Energy

It can be shown that the total potential energy can be written as

(Ref. 4):

U=U l +U2 +U3 (23)

where:

Ul

= strain energy due to the bending moments (M , M , M ) x y xy

U 2 = strain energy due to the straining of the middle surface

by the membrane forces (N , N , N ) x y xy

U 3 = wo rk done by the load.

These terms are given by the integrals:

Et 3 21T R L {2 2 } U I = 2 J J (K +K ) -2(l-v)(K K -K ) dxdy

24(l-v ) 0 0 xx yy xx yy xy

1 21fR L{ 2 2 } U 2 = 2Et J J (Nx +Ny ) -2(l+v)(Nx N y -Nxy ) dxdy

o 0

21TR

U 3 = - J o

L a Nx(L, y) J ~ dx dy

o (24)

-12-

But it is also true that

K = w, xx xx

K = w . yy , yy C=~3(1_v2)

K =w xy 'xy

thus the integral U I is given by

U I

3 217R L { 2 = Et 2 f f (w, xx + w, yy)

Bc 0 0

-2(I-v)(w, • w, -w~ >}dXdy (25) xx yy xy

also

211'R{ } U 3 = ta f u(L, y)-u(O, y) dy o

yielding:

U 3 = 211' R ta u I L (2 6)

The integrals UI

and U2

are readily integrable. It should

be noted that many of the trigonometric forms which result from

the various products have no contribution to the total potential

energy. The result is as follows:

-13-

The total potential energy is given by

5 . 4 2 22} U - Et 11'RL {(~) t: 2 + 1 [(k11') + (1)] s2

- 2 L '='1 "2" L R 2 8c

{ 2 a vt t

2 i 11' 2 1 t: 2 t: "E )

+ 211'RLta (I-v )E -1f£3 - Z("L) (2'='1+'='1 5 1

t 2 k11' 2 1. 2 1 2 } - "'4 [( T) + v (R) ] (2' s 2 + S 2 ~ 2 )

+ 11'RLEt {N2 + 1.. N 2 + 1.. N 2 + 1.. N 2 + 1.. N 2 (I_v2)2 1 2 5 4 6 4 8 4 9

+ M2 + .!. M2 + 1.. M2 + 1.. M2 + 1.. M2 + .!. M2 } 1 222 3 4 648 4 9

+ 11'RLEt 8(l+v) {

2 2 2} PI + P 3 + P 4

-2v11'RLEt { 1 II} ---;:;:2-:2~ Nl Ml +"4 N 6M 6 + 4 N8M 8 + 4 N9M 9

(I-v )

k11' 2 1. 22 11'RLEt 11'RLEt [(T) +(R) ]

+ M M + (l_v2 )2 2 3 (1_v2 )2 2(1)4

R N6N 9 (27)

As an expression of minimization of total potential energy. it

is enforced that:

aU _ aU _ aU - 0 ~-arz-~-

(28 )

Taking these partial derivatives and performing algebraic \

simplification yields the following three equations:

-14-

VA c 2 1 Z ""E £3 =7 -z i3 (z£z + £Z s 2) (29 )

(A c. - A) £ 1 + C 8 [ £ 1 ( £ Z + ~ 2) + ~ 1 £ Z] (£ Z + ~ Z) + 1

1 Z -C1 (Z£2+£2~Z) - C ZO £Z(£Z+~2) = AS1

(30)

1 Z c (ACk-A)£Z + (C 9+C10 )(Z £2+£2S 2)(£2+ S Z) +

+Cll [£1(£2+;2) + II £2] (£l+I l) +

-C4 £1 (£2+~Z) - C 3 [2£Z(£1 +%1) + £1£2] = AIz (31)

The coefficients in these equations are given in Appendix IV. Cozn-

parison of these equations to those of Arbocz and Babcock (Ref. 2)

will reveal that they are identical if one reverses the sense of the

radial displaceznent. (Ref. 2 uses positive inward vice outward. )

-15-

Ill. NUMERICAL ANALYSIS

1. Imperfection Model

As given in equation 6, the shell im.perfections were asswned

to be fully des cribed by two modal components, one generalized axi-

symmetric mode and one asymmetric mode, i. e. ,

Vi = sIt co s i7r ~ + I2 t sin k'1l' ~ co s £ i

Following the method of modeling given in reference 7, the ampli-

tudes II and ~2 are considered to have wave number dependence.

Decay constants are introduced which decrease the amplitude with

increasing wave nwnber:

I = 1

I2 =

XA .q 1

X

kr £8

(32)

(33)

Nwnerical values for the five parameters (XA

, X, q, r, s)

were obtained experimentally by Arbocz and Babcock (Ref. 6) by

fitting data for 31 different shells. In order to conduct a parametric

study which would include values representative of that experimental

work and be reasonably conservative of computational time, the

following numerical values were used in the computations:

XA

= O. OS, O. 10, 0.20

X = -0.50, -1. 00, -2. 00

q = 1. 0, 2. 0, 3. 0

r = O. 5, 1. 0, 1. 5

s = O. 5, 1. 0, 1. 5

-16-

Hence, 243 different imperfection amplitude models were used in

computation. Eight different wave number combinations were used.

As will be described subsequently, wave numbers were chosen such

that quadratic terms (as well as cubic) were included in the analysis,

and all asymmetric modes fell along the Koiter circle.

2. Shell Geometry and Material Properties

Throughout the numerical analysis, the geometry and mater­

ial properties of the shell were held constant. The following numer­

ical values were used:

R = 4.00

L = 7.00

-3 t = 4. 64 x 10

v = O. 30

3. Wave Numbers and Mode Shape Parameters

The quadratic terms in the buckling equations will vanish

identically unless the restriction i = 2k is enforced. Without the

quadratic terms, these equations describe a structure insensitive

to imperfections. Accordingly, the relationship i = 2k was main­

tained in all calculations.

The wave numbers of the asyrn.rn.etric mode term were

chosen so that this mode corresponded to one of the so-called classi­

cal modes of the Koiter circle. This insures minimization of the

asyrrunetric buckling load. For a discussion of this topic, the

reader is referred to Irnbert (Ref. 7). The Koiter circle is de-

fined by the relationship

-17-

2 2 a k - a k + 13 = 0

or 132

= ak(l-ak ) (34)

with the restriction that for positive, nonzero i.

ak < 1

In practice, the computational scheme was coded to perform

the following operations:

(I) The wave number k was entered (a whole number)

along with shell geometry and properties.

(2) The mode shape parameter ak

was computed.

(3) 13 was computed based upon this computed value

of ak

using equation 34.

(4) The wave number i. was computed from the value

of 13, then rounded to the nearest integer value.

( 5) 13 was then recomputed based upon the rounded

value of i.. This value was then used to compute

all coefficients of the buckling equations.

Accordingly, the values of a k and 13 were such that they were

very close to but not exactly on the Koiter circle.

4. Numerical Procedure

Equation 30 can be solved for £1 as follows:

£1 = (1);3+XI);4)

(ljJ6-X ) ( 35)

-lB-

The expressions for the terms lfJ· are given below. Using this 1

expression for the displacement amplitude Sl' the second buckling

equation may be written as a cubic polynomial in A:

3 2 A + El A + E2 A + E3 = 0

The coefficients E. are given by: 1

El = (l(J1-2l(J6) + l(J4( lfJ 2l(J4-lfJ S)

E2 = lfJ6(l(J6 - 2lfJ l +l(J4l(JS)+lfJ3(2lfJ2l(J4-l(JS)

2 E3 = lfJ6(lfJllfJ6+lfJ3lfJS)+lfJ2lfJ3

The expres sions for l(J. are as follows: 1

_ AckS2 l(JI- - - (C +C )(1 c

2 c (S +I ) 9 10 Z':>2+ S2 S 2)

2 2

2 Cll II s2

(s2+I 2)

4J2 = -Cll

2C3~1 s2 + ----

(s2+~2)

1 2 4J3 = C l (Z S2+ S2I 2) + C 20S2 (S2+I 2)

-CBSI S2(S2+S 2)

lfJ4 = II

( 36)

ljJs C ll SI (2£2+S2)

(£2+S2)

ljJ6 = Xc. + C 8(£2+S2)2 1

+

-19-

C 3(2£2+S2)

(£2+S2) + C 4

Since the im.perfection am.plitudes were given by equations 32

and 33, and the m.ode shape param.eters determ.ined as in paragraph

3 above, one then has a polynom.ial of the form.:

3 2 X + EI (£2)X + E 2(£2)X + E 3(£2) = 0

The com.putational schem.e was coded to solve for the buckling load

as follows:

( 1) The displacem.ent am.plitude £2 was increm.ented

from. the origin, using very sm.all steps.

(2) The polynom.ial was solved to obtain the lowest

real root.

( 3) The slope of the X vs £2 trace was then com.puted.

(4) The value of X at which one first attains a zero

slope was specified as the buckling load.

-20-

IV. DISCUSSION OF NUMERICAL RESULTS

AND RECOMMENDATIONS

1. The Effect of Varying the Axisym.m.etric Model Pararn.eters

(XA

and q)

As shown by figures 3 through 8, the traces of buckling load

versus wave nurn.ber for rn.odels which have q = 2. 0 and 3. 0 coalesce,

regardless of the value of XA

• The reader is referred to Table III

for a corn.parison of the nurn.erical values listed for rn.odels 49, 76,

130, 157, 211, and 238 - all of which have identical asyrn.rn.etric

rn.odel pararn.eters. Loads for these rn.odels are plotted on figure

5. For these six rn.odels, the spread between the highest and

lowest calculated buckling load is O. 01300 for k = 1. 0, decreasing

by a factor of roughly one half as wave nurn.ber increases by two,

to a spread of O. 00194 at k = 14. O. This behavior is repeated for

other cases (a total of 27 plots were possible of the type

shown in figures 3 through 8). In rn.ore understandable terrn.s, it

is recalled that the decay constant q operates on the wave nurn.ber i,

and that i = 2k. It is also true that the axisyrn.rn.etric arn.plitudes

are assurn.ed to be srn.aller nurn.erically than the asym.m.etric arn.pli­

tudes, and that the decay rate q is higher (2.0 and 3.0) than assurn.ed

values for rand s (1. 5 is the highest). For the cases where q is

2. 0 or 3. 0 therefore, the axisyrn.rn.etric irn.perfections are not only

srn.all in corn.parison to the as yrn.rn.etric rn.odels, but the corn.puted

loads show that changes in the pararn.eters X A and q produce only

very srn.all changes in the algebraic equations, i. e., in this case

the coupling is weak.

-21-

Using numerical values fo r comparison, in figure 5, models

76 and 211 are plotted. Imperfection amplitudes for these cases are:

for k = 1. a

model 76 model 211

r = O. as = o. 00 625 1 ( 2)3

c: 0.20 s 1 = --2 = O. 05

( 2)

I = -2.00 = -0. 63246 2 (l 0)0.5 I2 = - O. 63246

A = 0.74368 A = O. 73068

for k = 14.0

model 76 model 211

I = O. 05 = O. 00000 1 ( 28)3

I = O. 20 = O. 00026 1 ( 28)2

g2 = -2.00 ( 14)1. O( 27) 0.5 = -0.02749 £2 = -0. 02749

A = 0.80224 A = O. 80030

Of the six models shown on figure 5 with q = 2.0 and 3.0, model 76

has the smallest II and model 211 the largest.

For a given asym.m.etric model, changes in the axis ym.m.etric

model do not change the sensitivity of the shell to long versus short

wave imperfections. It is seen on these plots that as the axisym.m.et-

ric imperfection amplitudes increase, the traces shift downward,

but do not appreciably change in curvature.

-22-

Because of the observed behavior of the data for q = 2. 0 and

3. 0, it was po s sible to neglect the data for 135 of the 243 models as

being repetitive. Hence it is possible to show all significant data on

twelve figures, each having nine common traces.

2. The Asymmetric Model - the Effect of Varying X

As can be seen in figures 9 through 20, as X increases, a

translation downward of the buckling load trace occurs, without

appreciably altering the curvature of the trace. The behavior is

very similar to the case for varying X A' although somewhat more

pronounced.

In the range of values chosen for X, changes in this parameter

do reduce the buckling load, but do not influence the sensitivity of

the shell to long versus short wave imperfections. It was also

noticeable that as X becomes large, there is some tendency for

short wave sensitive structures to shift the wave number for mini­

mum buckling load from k = 14 (or greater) to k = 11. This can also

be seen in Table III where underlined loads are in the column for

k = 12. One should realize, of course, that while the lowest buck­

ling loads listed in Table III or shown on the figures are at k = 14,

it is certainly true that in some cases buckling may be dominated by a

higher wave number. However, no data are presented for these

cases.

3. The Asymmetric Model - the Effect of Varying s

As can be seen by comparing figures 21 through 32, as s is

increased, all other parameters held constant, the traces tend to

collapse and shift upward, with curvature changes occurring which

-23-

shift the m.inim.um. buckling load to short wave sensitivity. Only in

the cases where the axis yrn.m.etric im.perfections are sm.all and decay

rapidly (XA

= 0. OS; q = 2.0,3.0) is low wave num.ber (long wave)

buckling observed when s gets large (1. 5). In other cases, since s

is the decay rate operating on the circurn.ferential wave number i.,

and since 1 varies from. 1 ° to 27, a high value of s tends to m.ake

the asyrn.m.etric im.perfection am.plitude nearly equal to the axisyrn.-

m.etric im.perfection am.plitude. In that case, buckling is dom.inated

by classical short wave im.perfections.

It would perhaps be clearer to review the m.eaning of the

various decay rates and the numerical range of wave numbers on

which these decay rates operate. For the asym.m.etric m.ode1, the

decay rate r operates on the axial wave nurn.ber, k, which ranges

from. 1. ° to 14. 0. The decay rate s operates on the circumferential

wave nurn.ber, 1, which in this case varies from. 10. ° to 27.0. Both

rand s were varied from. 0.5 to 1.5. If both decay rates have the

value 0. 5, then at k = I, the asyrn.m.etric im.perfection am.plitude

is given by:

r2 = X ° 5 -(1) • (10)°·5

At k = 14, the sam.e term. is given by:

X ~ = --2 (14)0.5 (27)0.5

That is, the axial decay rate r decreases the am.plitude by a factor

of approxim.ate1y 4, while the circurn.ferential decay rate s does the

sam.e by a factor of less than 2 in going from. low wave numbers to

-24-

high wave numbers.

For rand s equal to 1. 0, the asymmetric imperfection ampli­

tude is decreased by a factor of 14 for r, while for s this factor is

less than 3. At rand s equal to 1. 5, the effect is even greater, as

the decay with circumferential wave number is much greater than

that for axial wave number.

4. The Asymmetric Model - the Effect of Varying r

Of all the parameters in the imperfection models, the parame­

ter r was found to have the greatest influence in shifting the sensitivity

of the shell from classical short wave imperfections to long wave

imperfections. As can be seen in figures 9 through 20, if the decay

of asymmetric imperfections with axial wave number is relatively

low (r = 0.5), then buckling is always dominated by short wave im­

perfections, regardless of variations of the other four parameters.

As r is increased to 1. 5, many models show a shift to long wave

sensitivity, the greatest sensitivity occurring when X and rare

both large (2. 0 and 1. 5 respectively).

5. Items Which Induce Short Wave I:mperfection Sensitivity

a. If the decay of asymmetric imperfections with increasing

axial wave number is relatively slow (r = 0.5), then

buckling is dominated by short wave imperfections,

regardless of other factors.

b. If the axisymmetric imperfections are large and decay

slowly in comparison to the asymmetric imperfections,

then short wave imperfections dominate (compare fig­

ures 19 and 20 to figures 13 and 14).

-25-

c. If the asymmetric imperfections are nearly equal in

magnitude to the axisym.m.etric imperfections, then

buckling is dominated by short wave imperfec­

tions. The important parameter in causing the

axisym.m.etric imperfection to have a significant

value is the decay rate q.

6. Items Which Induce Long Wave Imperfection Sensitivity

a. If the decay of axisymmetric imperfections with increas­

ing axial wave number of this mode is relatively high

(q = 2.0,3.0) and the decay of the asym.m.etric imper­

fections is high ( r = 1. 0, 1. 5), then buckling is

dominated by long wave imperfections. If

however, the decay of asym.m.etric imperfections with

increasing circumferential wave number gets suffi­

ciently large (s = 1. 5, X = 0.5) then this effect is lost.

This is, however, not an interesting case, for in

realistic terms, this means that the imperfections

are virtually non-existent - the decay rates are simply

too high.

b. With large axisymmetric imperfections decaying

slowly with wave number, it is necessary to have

large asym.m.etric imperfections decaying rapidly

with axial wave number and slowly with circumfer­

ential wave number to have long wave dominance.

The most striking case of long wave dominance

occurred when X and r were both large.

-26-

7. Sununa ry

Experimental evidence shows that imperfection amplitudes

tend to decrease with wave number. The rate of decrease varies

from shell to shell and covers a fairly wide range. If the decay

constant of the axisyrnrnetric imperfection is greater than one, then

the buckling load becomes insensitive to variations of the decay

constant in this imperfection.

The decay of the asymmetric imperfection amplitude with

wave number is governed by two parameters. The axial wave

parameter appears to be the more important parameter in govern­

ing the critical buckling modes. If this parameter is between zero

and somewhat greater than one half, the shell behavior is dominated

by short waves. This results since the size of the imperfection

going from k = 1. 0 to k = 14.0 is decreased by only a factor of ap­

proximately 4. Due to the greater imperfection sensitivity of

modes with a k and j3 close to O. 5, this behavior is to be expected.

On the other end of the scale, if r is one to one and one half, the

longer wave imperfections tend to be more important. This is not

true if the decay with circumferential wave number gets sufficiently

large. The behavior between these two extremes is less one sided,

being sensitive to the influence of the decay of the imperfection

with circumferential wave number, the size of the axisymmetric

imperfection, and the axisymmetric imperfection decay rate. The

lack of a clear cut dominance of short or long wave imperfections

in this range of r is unfortunate since much of the available experi­

mental data fall in this range. From an experimental point of view. the

-27-

range of r is not too great. However, in this range of r the behavior

is sensitive to all of the other parameters in the model and the effects

must be sorted out with care.

8. Recommendations

The numerical data generated for this study indicates that

further investigation of the decay rate parameters would be enlight­

ening. It is clear that one could interpolate between the curves

obtained to gain information regarding behavior of shells with

different values of the parameters X A and X within the range of

values used herein, and that the parameter q is relatively uninter­

esting (if greater than one). However, more attention could be

devoted to the decay rates rand s, particularly in view of the fact

that experimentally derived values for these parameters are scat­

tered widely between the values 1. 0 and 1. 5. It is apparent that

the nature of shell sensitivity to imperfections changes fairly

rapidly within this range of those parameters.

-28-

REFERENCES

1. Fung, Y. C. and Sechler, E. E.: Instability of Thin Elastic

Shells, Structural Mechanics, Pergalllon Press, 1960.

2. Arbocz, J. and Babcock, C. D.: The Effect of General

Imperfections on the Buckling of Cylindrical Shells, J. of

Appl. Mech., March 1969.

3. Hutchinson, T.: Axial Buckling of Pressurized b:nperfect

Cylindrical Shells, AIAA J., Vol. 3, Aug. 1965.

4. K~rlll~n, T. von and T sien, H. S.: The Buckling of Thin

Cylindrical Shells under Axial COlllpression, J. Aeronautical

Sci., Vol. 8, No.8, p. 302, 1941.

5. Donnell, L. H.: A New Theory for the Buckling of Thin

Cylinders under Axial COlllpression and Bending. Trans.

of A. S. M. E., Vol. 56, p. 795, Nov. 1934.

6. Arbocz, J. and Babcock, C. D.: On the Role of Irnperfec­

tions in Shell Buckling, Paper presented at the XIII IUT AM

Congress of Theoretical and Applied Mechanics, Moscow,

Aug. 1972.

7. Illlbert, J.: The Effect of Imperfections on the Buckling of

Cylindrical Shells. Aeronautical Engineer Thesis, Cali­

fornia Institute of Technology, June 1971.

-29-

APPENDIX I

Operator Coefficients

2 i1l' 3 1 2 A 1 =t(r) (2£1+£1;1)

v t i1l' ~ A2 = - R (r) ';,1

t 2 k1l' k1l' 2 J. 2 1 2 A3 = - 2" (T) [(T) - v (R) ] (2' ~2+~2~2)

A - ~ (k1l') £ 4-R L 2

t 2 k1l' k1l' 2 J. 2 1 2 A5 = - 2" (T) [(17) + (R)] (2;2 + ~2;2)

_ t2

i1l' [k1l'. 11' (I-v) J. 2] [ c c ] A6 - - 2" (r) (T) (l+k) L + 2 (R) • ~1 (;2+52)+51 ~2

t2

i1l' [k1l'. 11' (I-v) J. 2 [ c c ] A7 = - 2" (1:) (T)(l-k>i - -2- (R)]· ~1(£2+';,2)+sl£2

t 2 £. k1l' 2 J. 2 1 2 Bl = - 2" (R) [v(T) - (R)] (2;2 + ~2;2)

t 2 I. k1l' 2 £. 2 1 2 B2 = - 2" (It) [(T) + (R) ] (2 £2 + £2;2)

t 2 . B3 = 4"" (l{) (~) [(i+k) ~ - v (i -k) D . [;1 (~2+!2)+!1 ~2]

t 2 . B4 = 4"" (l{)( ~) [v (i+k) ~ - (i-k)y) • [~1 (~2+;2)+;1 ~2]

t £. B5 = -R (R)~2

-30-

APPENDIX II

Disp1acem.ent Coefficients

t 2 hI' 1 2

a l =""4 (L)(2 £1 + £1;1)

vt a = - . £

2 R(l7T) 1 L

[ k7T 2 1. l 2 (-) -v (-) t L R

a = -- 1 2 ~ (2 £2 + £2 5 2) 3 8 (k7T)

L

[ k7T 2 1. 2.. t k7T v (17) -(R) J

a = - (-) £ 4 R L [(k7T)2+(!)2]2 2

L R

t2

k7T I 2 ~ as = - 8 (L) ('2£2 + £2 5 2)

2 . ~(~) 2 L

a 6 = {[(i+k)y)2+ (~)2 (li V)}

~ 7T 2 1. 2J L[ (i+k) y) -v (W • [[(i+k)~]2+(ft>2J2

• {(i7T)(!) [(i+k).!(!) (l+v )] L R L R 2

- [(Hk);) + (Ii v) (:/l}. [t1 (tZ+;Z)+;1 tzl

21 (i7T)(k7T) (i7T)2(!)2 a =_~ L L + L R

7 2 [( i-k).!] [( i-k).!] L L

IV[(i-k)L]2_(!)2] I [(i_k)~]2+(!)212 •

• [£1 (£2+;2) + ~1 £2]

-31-

Z [v (~)Z -(~1 (!. gZ + sz";Z) t 2 2 b = - 8 (!)

1 R

[ £. 2 k11' 2 t £. (R) +(T) (v +2)]

b = - - (-) £ 5 R R [ ( k7r ) 2 + (!) 2,2 2

L R

t 2 £. 1 2

b 2 = - 8" (R) (Z g2 + g2~2)

2 (i11')2(!)r[(i+k)~]2 -v (i)2] b _~ L R~ L R

3 - 2 [[ (i+k) ~]2 + (~)2J2 • [gl(g2+;2)+~lg2]

~[' 11'] 2 £. 2] t 2 i11' 2 £. ~ (l-k) L -v ('R) b = - (-) (-)

4 2 L R r[ (i-k) 11'] 2 + (!)212 ~ L R

• [gl (g2+~2)+rl g2]

-32-

APPENDIX III

Stres s Resultant Coefficients

v t t 2 i1T 2 1 2 N 1 = u l + It" £3 +"2 (1:") (2 £1 + £1 II)

t 2 k1T 2 1. 2 1 2 + "4 [(L"") +v (~ ] (2 £2 + £2I 2)

2 a N -(l-v )E 1 - (when u i is sUbstituted)

t 2 2 k1T 2 1 2 NS ="4 (I-v ) (L"") (2£2 + £2£2)

t N6=R

2 2 (I_v 2 )(k1T) (!)

L R [(~)2~ (~)2]2 - £2

i1T2 1. 4 2 (1:) (:R) (I-v ) t 2

NS =2 [[ (i+k)-t] 2 + (~)2J~

t2

N =-2 9

i1T 2 £. 4 2 (1:) (R) (I-v)

~(i-k>i] 2+(~)2J2

[ £ 1 ( £ 2 + I 2) + II £ 2]

[~1 (£2+;2) + ;1 £2]

2 t 2 a t 2 1 2 2 1 2 c

Ml = (l-v )R £3 +v(I-v ) E +"'4 (~ (I-v )(2 £2+£2 5 2)

2 t t. M2 = (1- v ) R ':> 1

2 t2

1. 2 1 2 'F M3 = -(I-v)"4 (R) (2"£2 + £2 5 2)

-33-

(1_v2)!.. (k7T)4 M = R L ~

6 [t7T)2t(i)2)2 2 L R

t2

MS = 2 (l_V2 )(i7T)2[(Hk) E]2(i)2

L L R

[[ (Hk)i) 2 + (fv2J2

(1 2 i7T 2 2

[51 (52t~2)t~1 52]

t 2 M =-2 9

-v )(L) [(i-k)i) (~2 [[(i-k)DZ + (~)zy [~l(~Z+SZ) + sl~Zl

t P = - R 1

k7T 3 £ (L) (RJ 2(1 + v)

[ (k7T)2 £ 2, 2 ~2 L + (RJ J

2 i7T 2 P _ t (L) (Hk) E (i)3

3 - -(ltv) L R [[(i+k) D Z +(~)J Z [~l (gz +SZ) + Sj gzl

p - t 2 4 -

(ltv)(-\:)2(f>3[(i-k) i) [W-k) D Z +(iJjZ [ g 1 (gz +gz )+g 1 ~Zl

-34-

APPENDIX IV

Mode Shape Param.eter Coefficients

121 A = -2 (Cl!. + -.", ) c. 1 C.

1 Cl!. 1

1 rCl!; + (32

)2 AC = '2 2

k Cl!k

a~ 1 + --2 ~-2 (Cl!k +(3 )

C = 1

2 ~ for i = 2k

2 ' (if i =I' 2k, C1

= 0) 4Cl!.

1

2 2 CCl!.(3

1

C 3 = 2[ Yk

] , for i = 2k (if i=l' 2k, C

3 = 0)

2 C 4 = ~, for i = 2k (if i"* 2k, C

4 = 0)

2Cl!k

c2

2 4 { 1 1} C = - Cl!. (.). -- + --8 4 11-" - -

Yi+k Yi - k

C - c2

(3z4 9 -

4 Cl!k

2 C - c 2 10 --Cl! 4 k

2 c

Cll = T

c C20 = 4

Cl!~(34 1

--z Cl!k

2(.).2 Cl!kl-"

Yk

2 .2 Rt (17')2 Cl!i = 1 2c I.

2 _ k2 Rt (17')2 Cl!k - 2c L

{_1 + _1 }

Yi +k Y i-k

, for i = 2k (if i =I' 2k, C 20 = 0)

-37-

TABLE II

Irn.perfection Model Surn.rn.ary

Model XA

X Nwnber q r s

1 0.05 1.0 O. 5 0.5 0.5

2 0.05 1.0 0.5 O. r. 1.0

3 0.05 1.0 0.5 O. 5 1.5

4 0.05 1.0 0.5 1.0 O. 5

5 O. Ot) 1.0 0.5 1.0 1.0

6 O. 05 1.0 O. 5 1.0 1.5

7 0.05 1.0 0.5 1.5 0.5

8 0.05 1.0 0.5 1.5 1.0

9 0.05 1.0 O. 5 1.5 1.5

10 0.05 1.0 1.0 O. 5 0.5

11 0.05 1. 0 1.0 0.5 1.0

12 0.05 1.0 1.0 O. 5 1.5

13 0.05 1.0 1..0 1.0 0.5

14 0.05 1.0 1.0 1.0 1.0

15 0.05 1.0 1.0 1.0 1.5

16 O. 05 1.0 1.0 1.5 0.5

17 0.05 1.0 1.0 1.5 1.0

18 0.05 1.0 1.0 1.5 1.5

19 0.05 1.0 2.0 O. 5 O. 5

20 0.05 1.0 2. 0 O. 5 1. 0

21 0.05 1.0 2.0 O. 5 1.5

22 0.05 1.0 2. 0 1.0 O. 5

-38-

TABLE II (Cont'd)

Im.perfection Model Sum.m.ary

Model XA q X r 8 Num.ber

23 0.05 1.0 2.0 1.0 1.0

24 0.05 1.0 2.0 1.0 1.5

25 0.05 1.0 2.0 1.5 0.5

26 0.05 1.0 2. 0 1.5 1.0

27 0.05 1.0 2. 0 1.5 1.5

28 0.05 2. 0 O. 5 O. 5 0.5

29 O. 05 2.0 0.5 0.5 1.0

30 0.05 2.0 0.5 O. 5 1.5

31 0.05 2. 0 0.5 1.0 O. 5

32 0.05 2. 0 O. 5 1.0 1.0

33 0.05 2.0 0.5 1.0 1.5

34 0.05 2. 0 0.5 1. 5 O. 5

35 0.05 2.0 0.5 1.5 1.0

36 0.05 2. 0 0.5 1.5 1.5

37 0.05 2.0 1.0 O. 5 O. 5

38 0.05 2.0 1.0 0.5 1.0

39 0.05 2.0 1.0 0.5 1.5

40 0.05 2. 0 1.0 1. 0 0.5

41 0.05 2.0 1.0 1.0 -1. 0

42 0.05 2.0 1.0 1.0 1.5

43 0.05 2. 0 1.0 1.5 O. 5

44 0.05 2.0 1.0 1.5 1.0

-39-

T ABLE II (Cont'd)

Imperfection Model Summary

Model Number X

A q X r s

45 0.05 2.0 1.0 1.5 1.5

46 0.05 2.0 2.0 O. 5 O. 5

47 0.05 2. 0 2. 0 0.5 1.0

48 0.05 2.0 2. 0 0.5 1.5

49 0.05 2.0 2.0 1.0 O. 5

50 0.05 2.0 2. 0 1.0 1.0

51 0.05 2. 0 2. 0 1.0 1.5

52 0.05 2. 0 2. 0 1.5 0.5

53 0.05 2.0 2. 0 1.5 1.0

54 0.05 2.0 2.0 1.5 1.5

55 0.05 3. 0 O. 5 O. 5 0.5

56 0.05 3.0 O. 5 0.5 1.0

57 0.05 3. 0 O. 5 0.5 1.5

58 0.05 3.0 O. 5 1.0 0.5

59 0.05 3. 0 0.5 1.0 1.0

60 0.05 3. 0 O. 5 1.0 1.5

61 0.05 3.0 O. 5 1.5 O. 5

62 0.05 3.0 0.5 1.5 1.0

63 0.05 3. 0 0.5 1.5 1.5

64 0.05 3.0 1.0 0.5 0.5

65 0.05 3.0 1.0 0.5 1.0

66 0.05 3.0 1.0 0.5 1.5

67 0.05 3.0 1. 0 1.0 O. 5

-40-

T ABLE II (Cont'd)

Im.perfection Model Summary

Model XA q X r s Number

68 0.05 3.0 1.0 1.0 1.0

69 0.05 3. 0 1.0 1.0 1.5

70 O. 05 3. 0 1.0 1.5 O. 5

71 0.05 3. 0 1.0 1.5 1.0

72 0.05 3.0 1.0 1.5 1.5

73 0.05 3. 0 2.0 O. 5 0.5

74 0.05 3. 0 2. 0 O. 5 1.0

75 0.05 3. 0 2. 0 0.5 1.5

76 0.05 3.0 2.0 1.0 0.5

77 0.05 3.0 2. 0 1.0 1.0

78 0.05 3.0 2. 0 1.0 1.5

79 0.05 3. 0 2.0 1.5 0.5

80 0.05 3. 0 2. 0 1.5 1.0

81 0.05 3.0 2.0 1.5 1.5

Notes:

1. For models 82 through 162 inclusive, XA

becomes 0.10, all other parameters sequence as in models 1 through 81.

2. For models 163 through 243 inclusive, X A becomes 0.20, all other parameters sequence as in models 1 through 81.

3. In all cases, the sign of the axisymmetric imperfection is posi­tive (outward) while the asymmetric imperfection is always negative (inward).

TA

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M

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at

part

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lar

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del

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2.

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r to

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le I

fo

r w

av

e n

um

.bers

i a

nd

I

co

rresp

on

din

g

to t

he l

iste

d k

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3.

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r to

Tab

le I

I fo

r d

esc

rip

tio

n o

f th

e i

m.p

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on

ln

od

el

co

rresp

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