Creep buckling of spherical shells.

PAUL CHRISTOS XIROUCHAKIS

Diploma Mechanical and Electrical Engineering

National Technical University of Athens

July 1971

Diplome d'Ingenieur Grande Ecole du Genie Maritime

Ecole Nationale Superieure de Techniques Avancees

July 1973

S.M. Naval Architecture and Harine Engineering

Massachusetts Institute of Technology

August 1975

Submitted in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

at the

Massachusetts Institute of Technology

February:> 1978

Signature of Author

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Signature redacted

Certified by

Accepted by

Depantment of Ocean Engineering

Signature redacted

February

, 1978

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Signature redacted

Thesis Supervisor

···~~~~:~~~~~~:i·~~~~~~;;·~~·~;:~::~;·;~:~;~~;

0

Paul Christos Xirouchakis,

197 8

1\RCHIVES

MASSACHUSETIS INSTITUTE

OF TECHNOLOGY

AUG 161978

CREEP BUCKLING OF SPHERICAL SHELLS

by

PAUL CHRISTOS XIROUCHAKIS

Submitted to the Department of Ocean Engineering on February in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

The creep buckling behavior of a complete spherical shell subjected to uniform external pressure is investigated for a material obeying

Norton's generalized secondary creep law including elastic contributions. The shell is considered to be initially imperfect with an imperfection in the shape of the axisymmetric classical buckling mode. The axisymmetric problem is formulated incrementally using Sander's nonlinear shell

equations. Computations are carried out by the finite difference method. The critical times when axisymmetric snap--through or nonsymmetric

bifurcation occurs are calculated.

The results of the finite difference model are compared with the predictions of the creep buckling of a sandwich model of a shallow section

of a spherical shell.

The boss-loaded spherical shell is the numerical results are compared with

Thesis Supervisor: Title:

investigated subsequently, and available experimental data.

Norman Jones

The aurhor wishes to thank his thesis supervisor Professor N. Jones for his continuous advice and encouragement during the preparation of this thesis. The author is grateful to Professors H. Evans, T.H. Pian, and M. Cleary for their invaluable suggestions and constructive

discussions.

The author is indebted to Professor C. Chryssostomidis for his major support during the earlier period at M.I.T., and to the Department of Ocean Engineering of NI.I.T. and its Head Professor I. Dyer.

The author is grateful for the encouragement and moral support of his early educators at the Faculty of N.T.U.A., especially Professor N. Dimopoulos.

The author is indebted to Fran Forman for her infinite friendship and continuous understanding, and to Ms. Linda Sayegh for her excellent typing of the final draft.

Last but not least, this work is dedicated to Elli and Mihalis, the author's parents.

Page

1. INTRODUCTION 21

2. SHALLOW SHELL EQUATIONS 27

2.1. Linear Viscous Material 30

2.1.1. Snap-through Buckling-An Asymptotic Matching 30

Technique

2.1.2. Bifurcation Buckling and Postbuckling 38

2.2. Sandwich Model 47

2.2.1. Creep Buckling - Creep Exponent Equal to three 48 2.2.1.1. First Solution Method - An 56

Approximate Analysis

2.2.1.2. Second Solution Method - A 67

Perturbation Approach

2.2.1.3. Numerical Solution 79

2.2.1.4. Influence of Elastic Effects 80 2.2.1.5. Bifurcation Buckling and Postbuckling 93 2.2.1.5.1. Approximate Solution 93

2.2.1.5.2. Numerical Solution 95

2.2.1.6. Interaction of X and Y Dependent 98

Deformations

2.2.2. Creep Buckling - Creep Exponent Equal to Five 113 2.2.3. Creep Buckling - Creep Exponent Equal to Seven 122

3. FINITE DIFFERENCE MODEL

3.1. Axisymmetric Creep Buckling

3.2. Bifurcation

3.3. Computer Program Description

3.4. Boss Loaded Spherical Shell - A Comparison with

Experiments

4. CONCLUSION

APPENDICES

A. Computer program for initial postbuckling behavior of a linear viscous material ( 2.1.2.)

B. Computer program for creep buckling of sandwich model -creep exponent equal to three ( *2.2.1.3.)

C. Computer program for creep buckling of sandwich model including elastic effects - bifurcation buckling and

postbuckling creep exponent equal to three ( 2.2.1.4,4 2.2.1.5 D. Numerical scheme for creep buckling including

inter-action of X and Y dependent deformations (2.2.1.5.)

E. Computer program for creep buckling including inter-action of X and Y dependent deformations (2.2.1.5.)

F. Numerical scheme for creep buckling - Creep exponent

equal to five (A2.2.2.)

G. Computer program for creep buckling - Creep exponent

equal to five ( 2.2.2.) Page 128 137 149 157 160 167 172 177 180 ) 187 204 213 224

P age H. Numerical scheme for creep buckling - Creep exponent 231

equal to seven ( 2.2.3.)

I. Computer program for creep buckling - Creep exponent 245

equal to seven ( 2.2.3.)

K. Reduction of the field equations for asymmetrical 253

bifurcation to a system of four second order differential equations ( 3.2.)

L. Reduction of the field equations for axisymmetrical 262

deformation to a system of six first order differential equations ( 3.1.)

M. Finite differenced equations for axisymmetrical 268

deformations ( 3.1.)

N. Numerical integration across the thickness of the 274

shell ( 3.1.)

0. Creep buckling of a long .ylindrical shell under uniform 276

external pressure (ring problem)

P. Axially symmetric creep buckling of circular cylindrical 284

shells in axial compression

Q.

Axially symmetric creep buckling of circular cylindrical 294

Figure Page

1 Shallow section of a sphere 27

2 Sandwich model 47

3 Spherical coordinates 128

4 Discretization by central differences 140

5 Boss loaded spherical shell 160

6 Critical times for a linear viscous material 299 7 Postbuckling behavior for a linear viscous material 300 8 Postbuckling parameter as a function of applied 301

pressure for a linear viscous material

9 Comparison of critical times of sandwich model with 302

results from approximate and perturbation analysis

-Creep exponent equal to three

10 Critical time variation with respect to wavelength 303

parameter for sandwich model. Creep exponent equal to three

11 Displacement component W1 as a function of time- 304 sandwich model - Creep exponent equal to three

12 Displacement rate component W1 as a function of 305

time - Sandwich model - Creep exponent equal to three

13 Stress component C0 as a function of time - Sandwich 306

model - Creep exponent equal to three

14 Stress component D0 as a function of time - Sandwich 307

Figure 15 16 17 18 19

and the elastic Creep exponent Critical timesI effects - Creep Critical timesJ effects - Creep Critical timesJ effects - Creep Critical times Page 308 309 310 311 312 *

Stress component C as a function of time - Sandwich

model - Creep exponent equal to three

*

Stress component D as a function of time - Sandwich model - Creep exponent equal to three

Stress component C1 as a function of time - Sandwich model - Creep exponent equal to three

Stress component D as a function of time - Sandwich model - Creep exponent equal to three

Stress component D as a function of time - Sandwich model - Creep exponent equal to three

Critical times as a function of the applied pressure buckling pressure - Sandwich model

-equal to three for sandwich model

exponent equal to for sandwich model exponent equal to for sandwich model

exponent equal to

for sandwich model

effects - Creep exponent equal to

Critical times for sandwich model effects - Creep exponent equal to

314 including elastic three including elastic three including elastic three including elastic three including elastic three 315 316 317 318 313 20 21a 21b 21c 21d 21e

Figure 22 23 24 25 26 27 28 29

Displacement component W1 as a function of time for sandwich model including elastic effects - Creep exponent equal to three

Displacement rate component W as a function of time for sandwich model including elastic effects - Creep exponent equal to three

Stress component C as a function of time for sandwi model including elastic effects - Creep exponent equal to three

Stress component D0 as a function of time for sandwi4 model including elastic effects - Creep exponent equal to three

*

Stress component C as a function of time for sandw: model including elastic effects - Creep exponent equal to three

Stress component D as a function of time for sandw: model including elastic effects - Creep exponent equal to three

Stress component C as a function of time for sandwi model including elastic effects - Creep exponent equal to three

Stress component D1 as a function of time for sandwi model including elastic effects - Creep exponent equal to three ich ich ch ch ch Page 319 320 321 322 323 324 325 326 ch

Figure Page

*

30 Stress component D as a function of time for sandwich 327

model including elastic effects - Creep exponent equal to three

31 Correction factor for bifurcation of sandwich model - 328

Creep exponent equal to three

32 Critical times for interaction of x and y dependent 329

deformations - Creep exponent equal to three

33a Displacement components to₁₁, L_{2 0} as a function of 330

time - Creep exponent equal to three

33b Displacement components o11, o2 0 as a function of 331 time - Creep exponent equal to three

33c Displacement components wo1 1, t_{2 0} as a function of 332

time - Creep exponent equal to three

34 Critical time variation with respect to wavelength 333

parameter for sandwich model - Creep exponent equal to five

35 Displacement component W as a function of time - 334 Sandwich model - Creep exponent equal to five

36 Displacement rate component W as a function of time - 335

Sandwich model - Creep exponent equal to five

37 Stress component C0 as a function of time - Sandwich 336

model - Creep exponent equal to five

38 Stress component D0 as a function of time - Sandwich 337 model - Creep exponent equal to five

Figure Page

*

39 Stress component C0 as a function of time - Sandwich 338

model - Creep exponent equal to five

*

40 Stress component D as a function of time - Sandwich 339

model - Creep exponent equal to five

41 Stress component C1 as a function of time - Sandwich 340 model - Creep exponent equal to five

42 Stress component D as a function of time - Sandwich 341 model - Creep exponent equal to five

*

43 Stress component D as a function of time - Sandwich 342 model - Creep exponent equal to five

44 Critical time variation with respect to wavelength 343 parameter for sandwich model - Creep exponent equal

to seven

45 Displacement component WI as a function of time - 344 Sandwich model - Creep exponent equal to seven

46 Displacement rate component W as a function of time - 345 Sandwich model - Creep exponent equal to seven

47 Stress component C0 as a function of time - Sandwich 346 model - Creep exponent equal to seven

48 Stress component D0 as a function of time - Sandwich 347 model - Creep exponent equal to seven

*

49 Stress Component C as a function of time - Sandwich 348 model - Creep exponent equal to seven

Figure Page

*

50 Stress component D as a function of time - Sandwich 349 model - Creep exponent equal to seven

51 Stress component C1 as a function of time - Sandwich 350

model - Creep exponent equal to seven

52 Stress component D1 as a function of time - Sandwich 351

model - Creep exponent equal to seven

53 Stress component D as a function of time - Sandwich 352

model - Creep exponent equal to seven

54 Comparison of critical times between numerical and 353

analytical results - Sandwich model - Creep exponents three, five and seven

55 Elastic buckling pressure for a complete spherical 354 shell under uniform external pressure as a function

of imperfection amplitude

56 Rotation l as a function of the meridional angle 60355 and time

57 Tangential displacement U as a function of the 356

meridional angle 0 and time

58 Radial displacement W as a function of the meridional 357

angle 0 and time

59 Membrane force N 1 as a function of meridional angle 358 0 and time

60 Shear force

Q

as a function of meridional angle 0 359

Bending moment M1 as a function of the Page 360 meridional 62a 62b 62c 62d 62e 63a 63b 63c 63d 63e 64a 64b 64c 65a 65b 65c 65d 65e

angle 6 and time

Stress component _{T 11} Stress component _{T 11} Stress component _{T 11} Stress component _{T 11} Stress component _{T 11} Stress component _T22 Stress component _T22 Stress component _T22 Stress componentT2 22 Stress component _T22 Effective stress a a e Effective stress a a e a a a a a a a a function of function of function of function of function of * function of function of t fun:tion of1 function of z z and and and and and and and and and time time time time time time time time time

as a function of z and time

Is is a a function of function of z z and and time time

Effective stress a as a function of z and time e

Critical times for a complete uniform external pressure

Critical times for a complete uniform external pressure Critical times for a complete uniform external pressure Critical times for a complete uniform external pressure Critical times for a complete uniform external pressure

61 as as as as as as as as as a 2 z z z z z

spherical shell under

spherical shell under

spherical shell under

spherical shell under

spherical shell under

375 376 377 378 361 362 363 364 365 366 367 368 369 370 371 372 373 374

Figure _Page

66 _{Critical timas as a function of applied pressure 379}

and the elastic buckling pressure for a complete spherical shell

67 _{Displacement component W as a function of meridional 380}

angle 0 for boss loaded spherical shell - Initial elastic response

68 Membrane force component N as a function of 381

meridional angle 0 for boss loaded spherical shell

-Initial elastic response

69 _{Displacement of boss loaded spherical shell - A 382}

comparison with experiments

70 _{Critical times of a long cylindrical shell under 383}

uniform external pressure (Ring problem)

71 _{Comparison of growth of initial imperfection as a 384}

function of time with Ref. [31] results for a long cylindrical shell under uniform external pressure

72 _{Comparison of critical times between a cylinder 385}

and a spherical section of a shell under uniform external pressure

73 _{Critical times for axially symmetric buckling of 386}

E v K n S.. ii a h c t p CL (0) LIST x, Y OF SYMBOLS

z

EX)N,EXy

K,

Kj,

KXY

u,

V,9

W

Young's modulus Poisson's ratio creep constant stress deviator sphere radius wall thickness = [3(1-v2)]1/2 time

applied external pressure

classical elastic buckling pressure ( -j

(

) c

-/CL

at

ON CHAPTER TWO

cartesian coordinates in the base plane of the shallow section

coordinate normal to the base plane middle surface strains

bending strains

displacement components along the X, Y and Z coordinate

axes respectively

Ti _{Tx 1TIy} ,i ,fl_Ixy

NX, NyN Mx _MYMXY

initial normal displacement component strains

stress resultants

(

)

()

e LIST OF SYMBOLS a, x y w w u v k m flT -' 3Y 1/2 = (3 S.. S.

./2)

effective stress ii 3 ON SECTION 2.1 = 4h _{viscous constant} 3K1 - a viscous constant 1/2 2ca 1/2 h 0 X a Y a = W/h = W/h = U q0/2ch = Vq0/2ch = E a/2ch = K a/2c = N a/Eh = M a a/Eh3 aS 4Eht 3a

amplitude of initial impefection

U1 f w 2

1

LIST OF SYMBOLS d T _,Ty * * cx ,cx a7, y x y xY y C9 , C * * C9 ,CDi D ,D o R U3 . p P tE

coefficients of series expansion of W (equation (2.1.1.13) coefficient of series expansion of u (Eq. 2.1.1.14)

stress function (eq. 2.1.2.4)

postbuckling parameter (eq. 2.1.2.24)

ON SECTION 2.2

distance of sandwich layers (eq. 2.2.1 and 2.2.2) stresses

stresses at z=d/2 (outer face) stresses at z=-d/2 (inner face)

coefficients of series expansion of ax (eq. 2.2.1.9)

f.f

coefficients of series expansion of a (eq. 2.2.1.9)

coefficients of series expansion of Gx (eq. 2.2.1.9)

coefficients of series expansion of a Y (eq. 2.2.1.9)

coefficients of series expansion of W (eq. 2.2.1.9) coefficient. of series expansion of u (eq. 2.2.1.9) half-wavelength of x-dependent deformations

2 =rr ad 2

=4

0.6 _hEuler's time KPn a n

6 6 CR LIST OF SYMBOLS d =2

bifurcation correction factor eq. (2.2.1.5.4)

ON SECTION 2.2.1.6 C0 0,C_{1 1,C} 2 0 * * * C Q0 1 1, 2 0 * * * v 1 1 w00 ,wifw20 p , py

r

coefficients coefficients coefficients coefficients coefficients of of of of of series series series series series expansion expansion expansion expansion expansion of of of of of a (eq. * a _x (eq. a (eq. * a (eq. y U (eq . 2.2.1.6.6) 2.2.1.6.6) 2.2.1.6.6) 2.2.1.6.6) 2.2.1.6.6)

coefficients of series expansion of V (eq. 2.2.1.6.6)

coefficients of series expansion of W (eq. 2.2.1.6.6)

half-wavelengths in the x and y directions respectively

= (p /p )2

x/y

EL r) i rl iU. 1J Ul,U₂ ,W

(

'l

2 Ell,E2, E 12 K_{1 1},K_{2 2 ,K} 12 Nll,N _{22,N 12} QlQ2 M ₁ M22 ,N_{1 2} m polar angle longitude elastic strains creep strains

components of displacement in the meridional,

circumferential and outward normal directions respectively. initial imperfection

components of rotation in the directions of tangents and normal to the middle surface

membrane strains bending strains

membrane force components transverse shear components moment resultants

second stress invariant

y (n-1)

t U. 13 k.. 13 n.. 13 M.. 13 q p m 2 1_ 2 n-1 K En n U. h

w

h a ri.. =-p ₁₃ -a. = z/h time scale =a K. . 13 = h/a = N.. /Eh 13 = M.. /Eh2 13 = Q./Eh Eh a _E. h ij T. ESI e. . 13 ij. ij3

1. INTRODUCTION

The design of some ocean engineering structures (e.g., submersibles, habitats, etc.) has necessitated, due to material and weight limitations,

the development of accurate stress analysis into the inelastic range of deformation. ONR's vehicle ALVIN which consists of a pressure spherical hull submerged to a large depth is a typical example of such ocean

exploration design. New materials used in ocean engineering applications (e.g., plastics, high strength alloys, etc.) may exhibit creep deformations in the ocean environment. Creep buckling analyses are also of importance for the design of structural elements in the presence of high temperature environments (e.g., chemical or nuclear reactor plants).

The creep buckling behavior of cylindrical shells subjected to various loading conditions has been studied by a number of authors. N.J. Hoff [1]

has reviewed most of this work. However relatively few studies have been published on the creep buckling of spherical shells.

Creep refers to the phenomenon of a gradual increase in strain under constant stress. The time dependent stress-strain relations for metals undergoing creep deformations is highly nonlinear.

Creep buckling refers to the phenomenon in which a structural element undergoing creep and subjected to compressive stresses collapses. The time when the deformations or deformation rates tend to infinity or exceed prescribed limits is called the critical creep buckling time. Bifurcation into a different mode of deformation may also result which may or may not lead to loss of load carrying capacity. The present study is concerned

shells. Following this approach [1] the structure is assumed initially imperfect and the growth of these imperfections with time will ultimately lead to creep buckling. The elastic buckling behavior of a structure is related to the creep buckling deformation. Thus the study of the elastic buckling properties of a complete spherical shell under uniform external pressure [2] has revealed strong imperfection-sensitivity of the buckling load. Similar conclusions are drawn in the present study concerning the critical time dependence on the magnitude of the imperfection level.

N.C. Huang [3] investigated the axisymmetric creep buckling behavior of

clamped shallow spherical shells under uniform external pressure for a

viscoelastic material. _{The critical buckling time when the rate of increase} of the average vertical deflection becomes very high (initiation of

snapping) was computed by a finite difference scheme as function of the geometry and the applied pressure. The Galerkin technique was used by P. Varpasuo [4] to study the axisymmetric and asymmetric buckling of a clamped shallow spherical shell for a viscoelastic material. The unknown quantities of the axisymmetric problem were expanded in a series of Bessel functions of the first kind of the first order. Expansions in the fourth power of the sine of the axisymmetric coordinate and the cosine of the asymmetric coordinate were used for the study of the asymmetric bifurcation. Only the first term in the series expansions was retained. The transition values between the two forms of buckling of a geometric parameter characterizing

the shallowness of the shell were computed.

buckling behavior of simply supported thin shallow spherical shells under uniform external pressure using the variational theorem for creep [6, 7]. The above method had already been used to the study of creep buckling of

curved beams under lateral loading by T.H.H. Pian [8]. The mode shapes have been assumed and the requirement of the stationarity of the creep

functional led to a system six linear first-order differential equations with time as the independent variable which were integrated by the Runge-Kutta method. The creep exponent has been taken equal to one. Experi-ments were also performed for type 6/6 Nylon. The theoretical creep

deflections were consistently smaller than the experimental values,whereas the theoretical collapse times were consistently larger than the experi-mental collapse times. The assumption of rotationally symmetric

deforma-tions was verified by the experiments.

N. Miyazaki, G. Yagawa and Y. Ando [9] performed a parametric analysis

of the creep buckling of clamped shallow spherical shells using the finite element displacement method. The critical time when the stiffness matrix becomes non-positive definite for axisymmetric snap-through or asymmetric bifurcation was computed as function of the geometry and the applied

pressure. Threshold pressures, below which creep buckling of either type may not occur were found.

N.C. Huang and G. Funk [10] investigated the inelastic axisymmetrical

and bifurcation buckling of a deep spherical shell elastically supported around a circular boundary and subjected to a uniformly distributed external pressure. The Ramberg-Osgood constitutive equations together

range of parameters considered.

R.K. Penny and D.L. Marriott [11] have examined the creep buckling behavior of boss loaded spherical shells using a simple approximate pro-cedure based on the so-called reference-stress method. Good agreement was found with some experimental tests on aluminum alloy shells.

N. Jones [12] has investigated the creep buckling of a complete

spherical shell using a perturbation procedure. The growth with time of initial imperfections was computed for a material obeying the generalized Norton's law.

The above mentioned investigations except the last one, considered the creep buckling behavior of "perfect" spherical shells. Creep buckling was triggered by the nonuniformity of deformations due to the boundary or loading conditions.

The present investigation is concerned with the creep buckling behavior of a complete imperfect spherical shell subjected to uniform external

pressure. The load is applied "instantaneously" but quasi-statically and is subsequently held constant. The critical times at which axisymmetric snap-through or nonsymmetric bifurcation occurs are calculated. The shell is considered to be initially imperfect and the imperfection is taken in the same shape of the classical axisymmetric buckling mode [2].

An asymptotic matching technique is applied to study the snap-through buckling of a section of a shallow spherical shell under uniform external pressure. The bifurcation buckling and initial postbuckling behavior is subsequently investigated.

Extrapolation, perturbation and numerical techniques are used. The

influence on the critical buckling time of the wavelength of deformation, the elastic deformations, creep exponent variations, magnitude of initial imperfections, magnitude of applied pressure and the interaction of x and y deformations is calculated. The times when bifurcation occurs are computed. The initial postbuckling behavior is determined to decide whether or not bifurcation is followed by loss of load carrying capacity. Similar to the above studies have been made for a cylindrical shell and are presented in the appendix.

The assumptions of the shallow section sandwich model are relaxed by the subsequent analysis of a finite difference model of the complete spherical shell. The finite difference technique has been used by H. Obrecht [15] to study the creep buckling and postbuckling of circular cylindrical shells under axial compression. Using Sander's nonlinear shell equations [14] modified to take into account the effect ofinitial imperfections the axisymmetric prebuckling problem is formulated

incremer ally for a material obeying Norton's generalized creep law

including elastic contributions. The critical times when the deflections increase very rapidly are computed for the axisymmetric problem. Upper bounds on the time step [15] are prescribed so that the numerical

stability of the time integration scheme is assured. The effect of the number of finite difference stations used is studied. The Gauss-Legendre quadrature is used for the integration across the thickness which is more accurate than equal spacing techniques.

The bifurcation problem is investigated by examining the possibility of loss of uniqueness of the axisymmetric into a nonsymmetric form of deformation. Comparisons are made between the results of the finite difference model and the shallow section sandwich model.

The boss-loaded spherical shell is examined subsequently, in order to compare the results of the present analysis with the experiments reported in [11].

2. SHALLOW SHELL EQUATIONS

We consider a shallow section S0 of a sphere [16] as shown in Fig. 1. The x and y cartesian coordinates are in the base plane of the shallow section and the z axis is normal to this plane.

Figure 1

Nonlinear shallow shell equations will be used which are adequate when the characteristic buckle wavelengths are small compared to the shell

radius [16].

In regard with the required continuity of the deflection pattern over the entire surface of the sphere W.T. Koiter [2] remarked that the

ignoration of the above requirements makes it possible to write simple expressions for the buckling modes and the results of the approximate analysis [16] are quite satisfactory from the physical point of view. We use Sander's shallow shell equations [14] including initial x dependent imperfections [17]. Thus we get the following kinematic rela-tions:

U. +

x2

-a

E~ ..

+T4 W

+WW

CL

2.

K

KX ='-'YwY (2.1)

The assumptions underlying Sander's equations are the following:

1. middle surface strains are assumed small 2. rotations are assumed moderately small

3. transverse shear and normal strains are neglected 4. Kirchhoff hypotheses is made

5. the theory admits a principle of virtual work

Using assumption #4 (initially plane cross sections remain plane through-out deformation) we get:

I=A

a

(2.3

Using the principle of virtual work we get the following equilibrium equations:

Nx~y + VN

i

0

M2) M

-I-

+

+

N,

(-vs7TW),,,

+NYN-Y

J

4<(

2 Nxy~y

jV

(2.4)

Writing above equations in polar coordinates we get the same equations as other authors used [18, 19] for the study of elastic buckling of thin shallow spherical shells.

Constitutive Relations Odqvist's

takes the form

generalization of Norton-Bailey's law for uniaxial creep with the aid of the Prandtl-Reuss incremental relations

[12, eq(12), p. 451]

4n-2

deL

S

q K" _2.5)

412

2.1.1. Snap-through Buckling an Asymptotic Matching Technique In this chapter the linear viscous y-independent response of a shallow section of a spherical shell under uniform external pressure will be evaluated. The calculation of the critical time when the deflection rates tend to infinity will be performed. It will be shown that the critical time is finite for an initially imperfect shell. In the present derivation the geometric nonlinearities are retained and this fact gives rise to a finite critical time. If linear kinematic relations are used then the critical time of a linear viscous material tends to infinity. The initial imperfection is assumed to be y-independent.

The strain-displacement and change of curvature relations for a shallow section of a spherical shell for a y-independent deformation modified to take into account the initial y-independent imperfection

are:

Vsg :-\l~gg(2.1.1.1)

Ex

=O - \

where

\J

is the initial imperfection.

The following equilibrium equations are obtained by applying the

x,

A.

-.

(Nn

4Nx ) +(W+\i),xxNx-p

0

ax.xv

Y

(2.1.1.2)

The constitutive relations for a linear viscous material become [12]

Nx

cxv(

,

A+

2.

(2cl.1.3)

where J=io ,his the thickness of the shell and

Or

is a viscous 12

constant.

Taking the rates of equations (2.1.1.1) we get:

Ex

I

+

J

+

Cv+AVx

.

0

Ex

=-ixx

(2.1.1.4)

Substituting equations (2.1.1.3) (2.1.1.4) into the equilibrium equations (2.1.1.2) we get:

jw q

" %A

CL

+

.1

12X Nla

(c2.+0 L.5) CX

2.

~~

4~~~(W~n~~v~

(2.1.1.6)

The elastic response of the material will be only considered for the

case of instantaneous response.

2.

No-..m...nTh- 'P/t

Nondimens lonalizat ion

(2.1.1.7)

Equation (2.1.1.5), (2.1.1.6), (2.1.1.7) will be nondimensionalized

by defining the following dimensionless quantities:

3oK *h C 2 we get: ++w s ,X

\kz1x

+

(W+W,, hj +

v32.W=

04 0 =X (2.1.1.10) Initial Condition At - W. 0 (2.1.1.11)

The initial imperfection is taken to have the same form as the y

independent elastic buckling mode. Thus:

w

=

6

oosx.

(2.1.1.8)

higher harmonics is neglected. Thus,

V..Wo.

+

*.Vcosr

(2.1.1.13)

U5" .=U,. co Sx(2.1.1.14)

Substituting equations (2.1.1.12), (2.1.1.13), (2.1.1.14) into (2.1.1.9), (2.1.1.10) we get:

&

00

TJ ,

V

~A..

4;4

₀

45CA.4&).4.

4

AZ.

=

5

6ti

+

5\46.

+ 3 CV4+lL2.V4

4C+CE)

=0

(2.1.1.15) (2.1.1.16) (2.1.1.17) Initial Condition (2.1.1.18)

From equations (2.1.1.15), (2.1.1.16), (2.1.1.17), we get:

(2.1.1.19)

kv

C

.)

From equation (2.1.1.19) we see that W, tends to infinity when the denominator tends to zero, i.e., when:

(2.1.1.20)

t((

:&1.A5

i.e., for "small" imperfections equation

(2.1.1.19) can be written as:

...

,

.. Wl

3

-- g

(2.1.1.21)

Assuming

(0) SAM

12

_I

.)

approximations to the solution over two overlapping subintervals. Region I

The solution is assumed to be of order E i.e.

\4

4

=

Q

)

(2.1.1.22)

1.0 3 0

Using equation (2.1.1.22) the terms - \V4

\4

,-j

eA\A4Vi become much smaller than W4 and therefore they are neglected in Region I. We get

=

\4+

(2.1.1.23)

The solution of equation (2.1.1.23) satisfying the initial condition

equation (2.1.1.18) is:

0(

QC

(2.1.1.24)

The region of validity of equation (2.1.1.24) is obtained from the

following inequalities: 2. &

iV4

\AkW

(2.1.1.25) (2.1.1.26)

The inequalities (2.1.1.25) (2.1.1.26) are satisfied if

(2.1.1.27)

The solution is assumed to be of order 1 i.e.:

Vt4 =0(l)

(2.1.1.28)

Using equation (2.1.1.28) the term *6 in equation (2.1.1.21) becomes much smaller than the term)\/4 and therefore is neglected in Region II.

We get:

42

₂

(2.1.1.29)

The solution of equation (2.1.1.29) is:

4

ccSt.

exp

\4

.

-c)

(2.1.1.30)

where const is a constant to be determined from the matching condition.

The region of validity of equation (2.1.1.30) is obtained from the

following inequality:

or

_-t

_{<<4(2.1ol.31)}

Existence of Overlapping Region

WhenW1 is in the following range:

412. 414

fC

-<sVA4KL

(2.1.1.32)

then the inequalities (2.1.1.27), (2.1.1.31) which define II respectively are satisfied. Hence the existence of an region. It is obvious that the matching region chosen only possible choice. For example, the following choice

the Regions I and overlapping

is not the

Matching

The solution in Region I could be written as

(2.1.1.33)

but /J '>4e according to (2.1.1.32) therefore the solution in Region I

becomes in the matching region:

W-eat

(2.1.1.34)

The solution in Region II could be written as

at

+a

(2.1.1. 35)

IL

but Wix4«i in the matching region according to (2.1.1.32) therefore

VS

becomes in the matching region

W4 XCamst (2.1.1.36)

From equations (2.1.1.34), (2.1.1.36) it is seen that\41,Wj-x have the

same functional form in the matching region, therefore the matching

condition becomes the equality of the constants in the above equations, i.e.

coonst

-..

(2.1.1.37)

Critical Time

From equations (2.1.1.30), (2.1.1.37), we get:

(2. .l.38)

c

=2

A. W-

-ex? (I

0

Hence the Wd= tends to infinity when

'q4:I

.(2.1.I

Substituting equation (2.1.1.40) into equation (2.1.1.38) we get the

following expression for the critical time

tCR= t(2.1.

1.40)

[.41)

From equation (2.1.1.41) it is seen that for an initially perfect shell i.e., for &=O the critical time tends to infinity.

We also see that as

2

approaches zero the critical buckling time approaches infinity. When tCaR=O we get the critical elastic buckling load as a function of the imperfection level for Poisson's ratio yt

2

-- 4,A28&(21.142

In this chapter the possibility of loss of uniqueness of the previously studied y-independent response will be investigated. It has been shown

[20, 21] that the viscous components do not explicitly enter into the bifurcation problem. They enter into the bifurcation condition by

determining the current state of stress and deformation. As mentioned in the previous section the elastic properties would only be considered when considering an instantaneous response, and they will be neglected through the viscous deformation history.

The following dimensionless quantities are defined:

a.

E -2ch

e,,

2

2cct

S=J

2cbhLq

_E.C

99

(222C (2.1.2.1)

The equilibrium and compatibility equations for a shallow section of a spherical shell are written in dimensionless form as

TO~c~x+Tf*,n.

Z~hxri

eVJLL+

CWJ+

tb)1:z1h+ W,.-Tw

4.+2V44l

W ₁

e

M

LOX+

e,

₂

-e~

₂

~

W),.

-

C4

8₇

aP

rV4,e-)

In the bifurcation analysis the stress and deformation components are written as follows

ex e. +ex \A/

w+w-e, T% *+e.

m

_{+ A}

(2.1.2.3)

where the barred quantities represent the dominant y-independent response and the unbarred quantities represent the additional components.

represents the additional load increment so that the initial slope of the statIc load-deflection path in the postbuckling regime could be determined. Thus the stability of the bifurcation mode would be

investigated to determine whether or not snap-through occurs following bif urcation.

Introducing equations (2.1.2.3) into equations (2.1.2.2) and

adopting the following stress-function for the perturbed stress components

(2.1.2.4)

we get the following equations:

2.c ~W C1

(2.1.2.6)

where c 3cit

The following expansions are performed [131:

CO C41 2. C2)

V4=W +8EW 4tW 4.\

a=t) 4 - + '" (2.1.2.7)

where WW , are the increments of the fundamental y-independent mode of deformation due to the application of the load increment A2 .

W ,

4C

, are the bifurcation modes and W , - are second order

terms. t is the amplitude of the buckling mode.

The following Taylor series expansion about A)=O is performed [13]

(0) CO 2.

C

A(2.1.2.8)

where primes denote differentiation with respect to

2N2

Using elastic stress-strain relations for the perturbed quantities and introducing equations (2.1.2.7), (2.1.2.8) into (2.1.2.5), we get:

Ca/ I Cc)

\, -,.w,.. x.- A co s-x-c -:n- WXX - w w),

(2.1.2.9)

\qw +? c'enrw

<A

C ( 7 + .W(C2.n.2.10

2cC

a)i^.#C) c) C4

17 +c n ws x c ;; + w 4rz .2 c Y13W,) 41 c

C2) %t 2) Cal 01)0 1) A)2

74 =.W

-cC7

o4)Xcwlo-2

qx\pt

cc l

(2.1.2.11)

where the following incremental solution has been used

S

₂₅

_c~t)Azs

2

+

A cos

(2.1.2.12)

Equation (2.1.2.9) relates the increments of the fundamental y-independent mode of deformation. Equation (2.1.2.10) represents the bifurcation

problem. Equation (2.1.2.11) is needed in order to study the postbuckling behavior.

It is assumed that:

( W+$x),Z=

K

cos x (2.1.2.13)

where Wc would be found from the y-independent response of the

funda-mental mode. Assuming further that bifurcation occurs in Region I we get:

jj__w , (2.1.2.14)

Solution of equation (2.1.2.9) gives: I

_K

A=

_..

_k

The following buckling pattern is assumed [16]

C41 i

W = SiN~tSN

(2.1.2.15)

(2.1.2.10) we get the following exact solution CO C4\) _CO .S (2 _.)

(0~IT

)-\s~

(2.1.2.17)

where (2.1.2.18) (2.1.2.19) 0) C_

D.1

c(

4)xcX

Substituting equation (2.1.2.16), (2.1.2.17) into the equilibrium equation in (2.1.2.10) we get the following liUfurcation condition

[c+0

2

+KcrO+

1)

+.

4K

Q-

c2-*~h

(2.1.2.20)

where

In the elastic case the above expression reduces to that previously give in Ref.[16],

Taking for (= the critical value for the elastic case

and neglecting second order terms in the imperfection parameter we get the following expression for the time to reach bifurcation

2a

a&

+ (2.1.2.21)

When the argument of the logarithmic function equals one we clearly get the elastic bifurcation condition, i.e.,

(i-W)

t:

.

C^

+2.)

(2.1.2.22)

06

(40)

J. R. Fitch [22] generalized Koiter's theory of initial postbuckling

behavior using the Budiansky-Hutchinson approach, to account for non-linear prebuckling deformation.

The load increment A is expanded in terms of the imperfection parameter E

(2.1.2.23)

Since the postbuckling behavior must be independent >f the sign of the buckling mode it follows that

2.=

. Hence the sign of 'I will determine the stability of bifurcation. If 2 is negative snap-buckling follows bifurcation.

The general expression for is given by [22]:

_

3

<K0(1 3 W'i Wci4ici

i)W")>

K~vf~

~

4C 4

A c

~

a(-Y'

o W).%% (2.1 .2.24)

where A is given by equation (2.1.2.15)

and < > denotes averaging over the shell.

In order to evaluate '2j from equation (2.1.2.24) the boundary value problem equation (2.1.2.11) need to be solved. Galerkin's procedure will be used. The inhumogeneous terms in equations (2.1.2.11) suggest the

following expansions:

C2 C C) C2( .

C2) C2)

D a cosz + D csZ.x. .z cosZ.Xt C D.4- D, cosr + D, cos2x)

(2.1.2.26)

Introducing equations (2.1.2.25), (2.1.2.26) into the :ompatibility equation (2.1.2.11) gives:

4cosz

+446

₊

coszx + cos2[ie )D I C2)

0

+

CA Ae

2

_Oj

tosc

16

C4+f) Pa cos2x

+

C

2,

cosc

+

A Ca cs2x

+

cosZ

114

tC?' +

C4+4

1

C4

cosX

+ 4C++

A4

Ce

COS2f-4

2g<c

52

(C,

+(C2,+2C )cosz

+Cc

1

s::

=

c

(Os

+ cos

28')

(2.1.2.27)

Equation coefficients we get

DAO

+

CA

=

o

AAD0L2 + _{COL a} CM C2) c A)

?

Do

+C

-cK

C,

=S.

16 2C4)) c 6 C 2

(AA ) a+ C + )C4 -otf

~i+C

.

2. C+X ) + COD - (2)-C)

'iC+?)R

c,\+)C

1

-c

t\.C., ~

(2.1.2.28) (2.1.2.29) (2.1.2.30) (2.1.2.31) (2.1.2.32)

Using the following expressions from the analysis of the y-independent response:

c (2.1.2.33)

now ..

2

sr.(2.1.2.34)

becomes:-7W + 1

+

2W,

X -2cKc4, cos

+

iiC22+3Qc-osz)w .) C-4 C") C4) Ci C) c )

2C(W~x

i,

_{+W>,dt ipxv2wjxj , )(2.1.2.35)} Introducing equations (2.1.2.25), (2.1.2.26) into equation (2.1.2.35) we get:

C,

cosz

+ A6 C

cos2c

+ cos 2

a46

OAC,

+ C4+4

Cicosx +

16 C W)Cicos2xj

C2) _{D +}

X)

-D2

cosx - 4 D2%cos2Co - c.s2V6Ctj D4,cosx

+ 4

Dacos2x)

- 4

Ao:>s28.CD"

+ D4cos'

+

U

cos2>x)- 4c

C2, cosz

-

2j

C&

₂

cos2x

...

6c

coszg CC2

Cos'

+ 4C

₂

cos2x') +

44

C.

)+o+?

C2)c

T+C

+

SC4 ci s C +RC2cos2xcosh C=

4c4

₊

_{cosx- 00sK\ co}

)co-c

-4

40,cos2x

V)

D.

4

AVCosx

-2 cos

4

2.1 cos 2

(2.1.2.36)

Equating coefficients we get:

4-)

C21 -

D-

(2.1.2.37)

- )-(2.1.2.38)

1CoC, +CNIi C k-..o+ 4

)(2.1.2.39)

++ )a) + .C 41 + 2 )

Bci

('\

C

4

+ Mc

Co

+

' Ua):

-.

a

(2.1.2.40)

4 C+CYCC)" - CIC+. - C+ ) I

(2.1.2.41)

Equations (2.1.2.28) to (2.1.2.32) and (2.1.2.37) to (2.1.2.41) constitute

(ihs2 (2)

equation (2.1.2.24) becomes: CO CI) C-0 ) CO CO

<

C W,4+

C4

+ c A cocs')

CVW,0

c-n

oo-,

ICA)

C) - A t(2.1.2.42) We also get: 2(2..24).4(0 2

)

Is

D(2.1.2.43)

KQ(;COC4C)

M~c~C)

)D

0)+c4+AC

(Q C

, w",omm) 2 C m+ Ca T -C2 C. + Z,

C,-6

C)

c2

)

I2.} (2.1.2.44)

Using equations (2.1.2.42) to (2.1.2.44) we get the following expression

for

2

cCA (2.1.2.45)

(+

-C

A B

CID'.,- D4)I

The system of equations (2.1.2.28) to (2.1.2.32) and (2.1.2.37) to (2.1.2.41) is solved by the Gaussian elimination technique to determine

from equation (2.1.2.45). Appendix A contains a listing of the

In the following we use the sandwich approximation or double membrane model [23, 24]. The actual solid wall is replaced by an equivalent

sandwich wall with two faces, each of thickness h/2, separated by a distance d by a core which is rigid in shear but unable to carry normal stresses, see Fig.2. C

q7 I Figure 2

Rabotnov [25] suggested the following relation between the distance d of the faces of the equivalent sandwich wall and the wall thickness h:

I -n/n4 \a(V(2.2.1)

Using the value n=3 for the creep exponent we get [23]

d = 0.528 h

The above relation is obtained by requiring that the deformations of the solid and sandwich walls be the same in simple compression and in simple bending. L.A. Samuelson [26] suggested the following relation

&= 6

=

0.5

_(2.2.2)

which he obtained by equating the elastic bending stiffness of the double-membrane model to that of the actual shell. He found that his estimate of

the one obtained by several multi-membrane models.

2.2.1. Creep Buckling-Creep Exponent Equal to Three

The creep buckling of y-independent deformations of a sandwich model of a shallow section of a spherical shell under uniform external pressure will be investigated for a material that creeps under the cubic

creep law.

From equations (2.3) we get the following kinematic relations for

y-independent deformations:

CL =(2.2.1.1)

Taking the rates of the above equations we get:

OL

+ i

O:I

From equations (2.5) with r-=3 we get:

a.

c r " (2.2.1.3)

Combining equations (2.2.1.2) and (2.2.1.3) we get

0 0 r u3 3

' V4

4Q 4WX~~V

$

!%Ks

C.

!

-

Ct+"y)T x.

The equilibrium equations (2.4) become:

(2.2.1.4)

MxX

ttNx%)

4.7+W),,,Nx-T.-o

We define Cx,

C

stresses at z=d/2 and CS, 6- at z = -d/2.

Performing the integration across the thickness we get:

Nx

-.

(Cx

4

'

b

t/.;

MVA

(t=bh

A{

Finally we get the following equations for the sandwich model idealization:

6 9

,

M

+4kV~x=

2 ... ( ,xK =

N-

c

-

tV+(5x

K2

x (5y*c5

-x

3 c ( c yexf+ ' I

c

7

6 _x,x

₄

_SxxO

₀

(Cx 2 xx XXWrl- ww)

_CL.

c6xt+

+5x %CY) 4

CV7+V),XX C5 + (')-5h

(2.2.1.7) Neglecting the geometric nonlinearities we get:

)X *V

_am.

+ 4v

IXY

a

a0

_am4 4

~f~3,J+C

'

(2.2.1.5) (2.2.1.6) c 4 * 19 $1

(5v

4* 4 1

4(OA,nXx

X I -A)...A.CCKGY+

6AtC+

t+T

+ W)<'%AC X-+x

2 OL

h

(2.2.1.8)

We now expand the above functions into cosine series for the space variable x and we keep only the constant and the first cosine term.

N.J. Hoff [23] used the same assumption in studying the axially symmetric

creep buckling of circular cylindrical shells in axial compression. He demonstrated [27] that for the case of creep buckling of columns with a cubic creep law retention of higher harmonies does not contribute to greater accuracy in practical calculations. S. Patel and J. Kempner [23]

reached similar conclusions. Finally we have the following expansions:

x& -CO - C4 CCS 716.

P

P

V

V

cd

Vcos~

(2.2.1.9) ** * *

where C ,C ,D D ,C ,Cl, Do, D15, ,W and U are functions of time only .

Cy ₀

&t0cos

S

P is the half-wavelength of the x-dependent deformations. We take the

initial imperfection to be in the shape Of the elastic buckling y-independent mode [16]

where E is the amplitude of the initial imperfection. The critical elastic half-wavelength is [16]

We define

For becomes RcgR 3

We get RCR=1.5 8 4 using Rabotnov's criterion and RCR=1.7 3 2 using Samuelson's criterion.

Introducing the above expansions into the governing equations and using the following approximations for the second and third powers of the cosine function [23] we get:

W3tCC,o4

)

+3S C Co*CCI+

)+QC

i

_(J + 4-.it - C

...

a CC.*5CA+o

~

A) 4342

c,+

~.gC,

CL+D 'I d + c t o) CC4+ D54 D ' C

<W

-c

c* cC.

-J+

XJ

CL

42

C L * M *

a\e

r4_

+

)CV

)

3$G.c

t?)

2T'+

5C:*

4)

4

c +'

3j

vat

2L

C+c,

=0 A.

-hc.*+0

O.

CeC CD . CID, 14 +4)W+ IT C+ C:) =

(2.2.1.10)

Combining the above equations and using the definition of R we get:

M.1.c.*oJ

+4CAS

C1+t.7oCC

4+4

4C

+ 5C

cc

3

2

2.

\ - x L -V D2c+ O + + C+\ + e C ... C C -+ e + )

-C, C (Ce4+4d--etC

2COz +

0

' + ~5 CCottDO)CC

+OG) 4 3ADcCC0

3 S

C Io + b + ) + . C C + + .. t =

'2

C co+ b+ t>4)C t4Do' CCl+D e4+{

Cc*0*$cj _{)+ f*+ DA)co% (}

K

0

4r

0

A)

C4

i-3

c,+

YLc

C

D) n

AJCC

+c

+c 4 +W

0

=

-t

C 4C(f

>A

-C

4

+C

4

=o

C + Cc - tDa - t>o

2:

2

CL

h*

R

C4)

c

t

+-t ) +

Wl+'C

(2.2.1.11)

We nondimensionalize the above equations using the following relations:

whr Oth )Z'nttesar onimnso= l Tefisttw qutin

. * Cd+ 4-A

OV

'5 CL(2.2.1.12)

where the() quantities are nondimensional. The first two equations determine the relation of W40and IJwith the remaining quantities.

Restricting our attention to the remaining equations and dropping the we get the following dimensionless equations (we redefine

4

=V4 )

new old =

VP

o+

i+004-

Q, 4 4~CC.,+D4

C

Q

C

C6ct%)&cc>t-

+

t)

$$i~C

4

tLc'

ti ).A .

(ca+ l+ )c co

+

to)

ccq

+

}

c+

C 3 3 +C 0 2

cco+

t:+

cl +

o+

.3

c

r+Q:cc,

+

D

+'S

c

coc

+,

t)",

CO

c cD ei + ecc,+ t 4 {) Do Dj da

CQ*+,

+C

*

+*

22-2 *A

+

₎

OEM~

Cc+cc-+tg-

+-o%

+A-

CciAr

*oCCJ +t:*

+%t:

A+A

C

C*+

(R-)CQC+

c

e

)

+lRb&

4

+4J&

2(Q+D

4'e +$

-5

+

0

=

c

c'+

c~o c

C c *+oocc+

+

c.cc+*,

c< -

C

*o D

-

D*o-

-R

C-Co

4

+D4

V4(QA+CO

(2.2.1.13)

We have nine equations with the nine unknowns _,

_{CA) C}

+ tI=

We follow here the same method as that employed in [23] for the

solution of the axially symmetric creep buckling of a circular cylindrical shell under axial compression. The above method consists of obtaining the solution when the deformations are very small and when they are very large. By interpolating, the solution for the intermediate range of

deformations is obtained.

Solution for Small Deformations

The assumption is made that at the beginning of the creep buckling process the following inequalities are valid:

co) \cD :) <\

These inequalities express the fact that at the beginning of the creep-buckling process the membrane state of stress is the dominant one. Using

the above inequalities the governing equations are simplified as follows:

z 0 ) '26* '2 * *

gW,

-..

CCo+D) CC 4A+

) 4Ct C

₄

-CCO+ 0

₀

) CC4+

C

-(2.2.1.1.1) (CO+ t>033 + CO = CG*+ t$ )+ CO (2.2.1.1.2) 3 3 *

2

CC4t:')

+ CO

+D=O

(2.2.1.1.3)

(c4%+

C

+ D

₀

C

+

-.

+

CCObO)C~tZ)4@*

2

c2-Kcc*

4

* *(2.2.1.1.4)

(Co+ -6 Cq To S CCc+

(

4t

4%

t-

' (2. 2.1.1.5)

(2.2.1.1. 6)

Cj -=- C4 (2.2.1.1.7)

Co

+rc(6

- bc-% 4

(2.2.1.1.8)

-R C

₄

-Cb

RW

4

CQ+C41

)Q

(2 .2.1.1.9)

From equations (2.2.1.1.2) and (2.2.1.1.4) we get:

53

*'S *

c3 -D-O = Co - DC)

(2.2.1.1.10)

From equation (2.2.1.1.3) by defining C0/D = we get:

3 3

2 (AAr

c)4

0(

*4=O

or (ct+4)QX( 40(4 )=0

2

For O( real oC4+4d. is always positive, therefore

Hence D = -C (2.2.1.1.11)

o o

Using equations (2.2.1.1.2) and (2.2.1.1.11) we get:

3

t

*3 *3

(2.2.1.1.12)

Using equations (2.2.1.1.4) and (2.2.1.1.11) we get:

3 * *( *..3

CO

C C + )0

4D