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TitleCorrelating the Resisting Mechanisms and the Strength of Thin Shells Un

der Follower Loads

Author(s) George, Thomas; Okada, Hiroo

Editor(s)

CitationBulletin of Osaka Prefecture University. Series. A, Engineering and natur

al sciences. 1997, 45(2), p.87-97

Issue Date 1997-03-31

URL http://hdl.handle.net/10466/7751

Rights

Bulletin of Osaka Prefecture UniversitySeries A, Vol. 45, No. 2, 1996, pp. 87-97 87

Correlating

Strength of

the Resisting Mechanisms and the

Thin Shells under Follower Loads

Thomas GEoRGE' and Hiroo OKADA'

(Received November 29, 1996)

This paper deals with the internal mechanisms of resisting an external follower load and the elastic

strength limits of thin shells. Owing to their form resistance, shells are inherently advantaged to

function under very high loading requirements, where the strength and stability are the primary

concerns. A set of governing equations for thin shells defined in a system of monoclinically convectedcoordinate axes are used for this analysis. The resisting mechanisms of a shell are derived from

different components of terms associated either with the extensional or the bending stiffness parame-

ters. The physical nature of equilibrating the governing equation is studied here by numerically

analyzing the share of contributions from these different components. Also, an analysis of the stress

resultants at selected points of the shell geometry during large deformations brings out the develop-

ment of severely stressed zories during the deformation process, and thus provides a realistic picture

of possible local elastic failure points.

Finally, the numerical results for the resisting mechanisms and strength characteristics are brought

together and their underlying features are analyzed with the comparative measures of shell stability.

As a result, some general characteristics of both singly and doubly curved partial shells are estab-

lished. These results are promising to the extend of explaining some of the instability mechanisms

exhibited by thin shells during finite deformations.

1. Introduction

The allowable tolerance for structural systems are

statistically estimated during the design stage within

the scope of known or predictable loading conditions.

It becomes one of primary importance to make the

closet possible estimate of the strength of any struc-

ture before placing it to an operation field with unpre-

dictable conditions. Shells are very prone to have

some uncertain design criteria due to the mostly

unavailable details of their stability or strength charac-

teristics, as compared to other components of a struc-

ture.

The theory of shells by itself is a well researched

subject and numerous studies have been conducted on

its advanced aspectsi-5). However, the elastic stabil-

ity analyses of shells are not necessarily accompanied

by a study of the strength aspects, which in fact is

taken for granted in most situations6-8>. This paper

deals exclusively with a comparative study of the

resisting mechanisms and the elastic strength of thin

shells subjected to follower loads.

'DepartmentEngineering

of Marine System Engineering, College of

The governing equations for thin shells were derived

here in a system of monoclinically convected coordi-

nate axes defined over the middle surface. The sys-

tematic presentation of tensor derivations from the

first principles of continuum mechanics, and other

details of that derivation have been the subject of study

in many of the previous papers5'6). Those analytic

equations were derived with proper consideration of

the shell geometry after deformation, which was

expected to make the numerical results to be more

exact than a formulation based on the pre-deformation

metric and curvature values. The ability of this for-

mulation to deal with the large deformations and the

stability mechanisms of several types of thin shells

have also been verified in some of the earlier studies6･').

The internal resisting mechanism of thin shells can

be considered as derived either from the extensional or

the bending stiffness. These resisting factors are

qualitatively divided into different groups of terms,

each of which contributes in varying proportions to the

total resisting mechanism of the shell at its deformed

states. Also, the elastic stresses developed in the shell

vary across the shell thickness at different points,

some of which would attain the limit states even before

the static failure points. Here, the stress resultants

88 Thomas GEoRGE and Hiroo OKADA

were calculated at selected points of the shell geometry

using the physical components of the normal and tan-

gential stresses due to the membrane forces and the

moments.

The detailed aspects of the resisting mechanism and

the elastic strength were independently analyzed in

different studies, which would be correlated in this

paper to bring out the comparative characteristics.

Partial cylindrical and spherical shells with curvatures

ranging from very shallow to very deep were consid-

ered here. These analyses have brought out a pattern

of resistance mechanism and strength characteristics,

and established some critical curvature ranges of shells

for which the elastic strength might be lower than the

plate strength, though mostly a local effect for shells.

The Galerkin method was used for numerical formu-

lations, for a specific simply supported boundary condi-

tion only. Attempts for generalizations were there-

fore restricted to the partial cylindrical and spherical

shells, which might be extended safely to other types of

shells and different loading and boundary conditions,

gererally for their qualitative aspects.

2. Theoretical Formulations

The general governing equations for a thin shell are

formulated here in the monoclinically convected coordi-

nate exes (e', e2, e3) traced over the shell middle

surface, where the e3 direction is adopted normal to

that surface, as shown in Fig. 1.

The Kirchhczff:Love assumptions are assumed true

throughout the finite deformation process of a thin

shell. The Range Convention adopted here for indices

IFa

R

t H -- The partial The partialR spherical shell cylindrical shell

TTrickness ot the sheil : t/a = O.Of

Poisson's ratio :V =O.3

e3(z)

'- e?

-x -×

s 'l ;i'C

Nt

i,

/t12t12

upper surfacemiddle sur acelower surface

Fig.1 The definition of monoclinically convected coordinate

axes and the geometry of two particular shell types

is such that all Latin indices (i, i le, ･･･) take values 1, 2,

3 and all Greele indices (a, P, 7,･･･) have the range 1, 2.

Various other notations and symbols used in this paper

will be explained when and where they appear first.

2.1 FundamentalEquations

The general surface strain of the shell is expressed

here using the 2-dimensional form of the Green-

Lagrange strain tensor, and its middle surface counter-

part e.p may be defined in terms of the corresponding

metric tensors before deformation (a.p) and after defor-

mation (Adi) of the middle surface.

Eap= -lt (Aap-aafi) ) (1) Adi= aap+Ualp+upla+uil. uilfi 1

The metric tensors can be derived directly from the

first principles using the position vectors before and

after the deformation of any point on the middle sur-

face and the corresponding deformation vector. The

symbol ` 1 ' denotes 3-dimensional (spacial) covariant

differentiation with respect the subscript that follows,

as different from its 2-dimensional (sutuce) counter-

part ` ll ' which will be used later.

Also, the curvature tensor of the middle surface

after deformation B.p may be expressed as the sum of

the curvature tensor before deformation b.p and the

change of curvature tensor x.p.

Bap = bap + Xap ) x.p :u3Idi-U317 UVIaly } (2) +-IF (u71,ztO1e- uV1eu"1,- a7"ui1,ui1e)b.p J

The Cauclly stress tensor d`'fi for isotropic materials

is used here in its 2-dimensional form to express the

stress-strain relationship on the general surface, where

E is the Young's Modulus of elasticity and v is the

Poisson's Ratio.

d4' == (1-E .,) GaP'" op,, (3)

The quantity G"P7e is a fourth order tensor known as

the `Elasticity Tensor' for the general surface, which

may be expressed in terms of its middle surface coun-

terpart A"iY7" using appropriate shijifer parameters for

contravariant quantities after deformation.

The membrane force tensor Nop and the moment

tensor Mdi may be derived through the direct integra-

tion of stress components across the shell thickness t.

After proper substitutions and reductions, the mem-

brane force and moment tensors can be expressed

completely in terms of the middle surface quantities.

Correlating the Resisting Mechanisms and the Strength of

N"P = Da"P6A E ex + KaSfin4( b$ cSf (S: (S}

- bf (s:l (s}- b} (sg (sS)K,, (4)

Mafi =KaapaXx,x+KaeP6X(b$(s::-bg)6"x (5)

In the above equations, D and K are respectively the

extensional stz;fuess and the bending stWizess parame-

ters, (Sg is the Kronecker delta and a`rP7" is the elasticity

tensor of the middle surface before deformation, as

given below.

Et, Et3 D= (1-v2) 'K= 12am.2) (6)

a`es,lts= ( li V ) [aartfe+ al6 tf7] + vfpa7" (7)

The above expressions for membrane force and

moment tensors are simplified for small deformations.

However, the middle surface curvature before defor-

mation (bfi") may be replaced by the corresponding

value after deformation (BS) for the general case of

finite deformations.

Now, the transverse shear force can be eliminated by

substitutions from the equilibrium equations of a thin

shell, whereby the following governing equations can

be formulated for finite deformations.

AI"Pll.+M"711.Bfi,=-PP-m"BP. (8) (a) (a)

N"pt'Bap-M"P(.11)ap=-P3+M"(.Hf (9)

Here the quantity ma is the sum of the surface trac-

tions and body forces contributed moment load, PP and

P3 are respectively the tangential and normal compo-

nents of the applied load. The symbol (a) below the

2-dimensional (sudece) covariant differentiation sign

`Il' denotes differentiations performed on quantitie$.

after deformation.

For analytical simplifications, the middle surface

strain and the change of curvature tensors may be

expressed in their partial differential forms, as given

below.

E`43 = HIF{ Ua,p+ Up,a - 2(b.fi ue + r2p ui) + b2 bxp( ag)2

+ ze3,. ze3,p + b2 [(b7 u, + u3,fi) ux

+ (IiXp ZEp - UA,p) U3 ] + bG(Ux U3,a - Ux,a U3) }

x.p == u3,.p - b2 bpx u3 m r2p ze3,x + (bG uti,. + b2 tig,p) zt3,a

+ -ll-[bi ag(wf2 - b,X b£(ztti)2] b,,7

r2p are the Christqffel symbols

represent the covariant and contravariant

(10)

and uk or uh (le= 1, 2, 3)

deflection

7;hin Shells under Follower Loads 89

components in the three coordinate directions. The

deflection components can be considered here in the

form of an algebraic summation series, as follows:

mn uk=211 IUle",i)dik(ei, e2) (11)

Here, Uk represent the deflection coefficient and

dik(e', e2) may be considered as a double trigonometric

function, where e' and e2 are the angular coordinates

of the middle surface in the principal directions.

2.2 Equations for the Resisting Mechanism

This paper considers the resisting mechanism during

the normal deformations of a thin shell. Eq. (9) which

is the governing equation in the normal direction is to

be analyzed here in detail to bring out the contributions

to the total resisting mechanism by each component

term.

The equilibrium in the normal direction may be

represented in the following form where the S term

denotes the group of terms that club together with the

lixtensional stij7izess Pavaneeter D and the B term

denotes the group of terms that club together with the

Bending stWiiess Parameter K.

s+B :-p3 (12)Now, the terms S and B may be subdivided into the

following components, where the E terms and the B

terms denote a division based on the characteristics of

each group of terms.

SI.:l::i::g::2 } (i3)

The significance and meaning of this arbitrary subdivi-

sion is visible when each group of terms is studied from

the following detailed representation of their compo-

nent structures.

El=D(a")2euBn

E2=Dva"a22[£n&2+e22Bn]

For

E3=D ( 15V ) a"a22 [(Ei2+ E2i)(Bi2+&i)]

E4=D(a22)2E22&2

Bl=K(a")2[xiiBn(b22-bl)-xu,ii-Enaib22]

B2=-Kvaiia22[xn,22+x22,ir+en,22bl+E22,nb22]

B3= -K ( 1i V ) aiia22 [2(xi 2,i2 + x2 i,i2)

+(ei2,i2+E2,,,2)(bl+b,2)]

B4 = K(a22)2 [ x,,Bz2(bl - b22) - x22,22 - E22,22 bl]

the sake of convenience and easy

(14)

(15)

visualization

90 Thomas GooRGE and Hiroo OKADA

effect, the above equations are given in terms of strain

components, curvatures and the change of curvature.

First all, the terms El and Bl contain terms grouped

together with the metric tensor in the first principal

direction ei (the term `primary direction' will be used

in further discourse) and the terms E4 and B4 contain

terms grouped together with the metric tensor in the

second principal direction e2 (the term `secondary

direction' will be used in further discourse). The

terms El and E4 represent the stretch in the respective

principal directions, whereas the terms Bl and B4

represent the bending in the respective principal direc-

tions.

Next, the terms E2, E3, B2 and B3 are groups of

terms accompanying the product of metric tensors in

the principal directions and characterized by the pres-

ence of the Poisson effect. They are basically the

cross components of stretching and bending in the

principal directions. The difference lies in the fact

that the terms E2 and B2 represent the effects due to

strains and curvatures or its changes in the principal

directions, whereas the terms E3 and B3 are associ-

ated with the shear and twist components.

The terms 8 and B will be denoted in further

discourse as the Extensional part and the Bending

part, respectively. Also, their subdivisions will be

denoted as the Component terms. The above nota-

tions are to be followed for all the evaluations made on

the resisting mechanisms of thin shells during the

numerical analyses conducted in this paper.

2.3 Equations for Strength Calculations

The tensor components for stress resultants as calcu-

lated above, may be physically interpreted using the

following conversion scheme2'5), where the quantities

with bracketed subscripts denote the physical compo-

nents. It should be specially noted that the Einstein 's

Summation Convention is not applicable for the identi-

cal repeated indices in these equations.

.?v ,di, = iv"p 'vl-iA2if s,ny -,

M(zes)= MaPVAApaaa(16)

Also, the physical (actual) stress components in the

normal and tangential (shear) directions may be calcu-

lated using either one of the following equations'･2>.

... m- .`n6 vCgkil

(17)

afapJ=LNIfz£!-:IEEtt!grd!E2M,2 (ls)

The quantities G.. and Gpt' are the metric tensors of

the general surface after deformation, which may be

calculated from the corresponding values of the middle

surface using appropriate shofirer parameters for trans-

formations between the two surfaces5).

In this paper, the direct stress and the bending stress

are calculated correspondingly by the N(as) and M(.p)

terms in Eq. (18). There would arise a small differ-

ence, which is negligible for all practical purposes,

between the stresses calculated using the above two

equations. This is mainly due to the differences in the

extent of simplifications during the stages of reduc-

tions leading to these equations, and to some extent

due to the assumption of linear stress distributions

across the shell thicknessi) adopted in Eq.(18).

3. Numerical Analyses of the Resisting Mechanism

and Strength

The most important problems accompanying shell

designs are the many unknown characteristics that the

shell might display at the large deformation stage. In

the numerical analyses conducted in this research, two

of such characteristics, the resisting mechanisms and

the strength, are studied thoroughly. The shell theory

was described briefly in ch. 2 for the general thin shell

continuum and two of the major shell types, partial

cylindrical and partial spherical shells, are chosen to

represent the singly curved (zero Gaussian curvature)

and doubly curved (positive Gaussian curvature) shells.

The shell boundaries are considered simply support-

ed, with all the inplane deflections arrested along the

edges. A uniformly distributed pressure load is con-

sidered to be acting in the anti-radial direction, and the

letter p(== lp31) is used in the graphical representations

of results. The shell geometry is considered to be that

which produces a projected square base of unit area, by

which the principal chord lengths [a == li = 4] are unity.

In order to satisfy the fundamental assumptions, the

shell thickness t, which is considered uniform, is

assumed to be sufficiently small [t/a==O.Ol] in

comparison with the radius R. The numerical value

Correlating the Resisting Mechanisms and the Strength of

of E=1.96×10'i N/m2 for the Ybungls modulus andthe Poissonls ratio of v==O.3 are used in the numerical

7;hin Shells under jFbllower Loads 91

calculations.

The differential geometry of a toroidal shell is used

to express the basic values on the middle surface.

The corresponding equations for the metric and curva-

ture tensors and the Christoffel symbols can easily be

formulatedfromfirstprinciples5). The`primarydirec-

tion' (see sec. 2.2) of partial cylindrical shells in chosen

in its longitudinal direction.

The equilibrium equations [Eqs. (8), (9)] ･under static

considerations are solved using the Galerkin method,

with 16 modal combinations. [m, n=4, in Eq. (11)] .

Curvature ranges of shells are selected from the plate

(R/a=oo) to the subtended angle of rr radinns at the

center (R/a=O.5) for the deep shell. Generally, all

graphical representations of numerical results are

nondimensionalized appropriately. Some details of

the numerical results for the finite deformations and

instability characteristics of different shells6t'), the

results of which are partly incorporated in this paper,

are provided in an accompanying paper.

3.1 The Resisting Mechanism

In the numerical analysis to determine the resisting

mechanism of shells using Eq. (12), instead of consider-

ing only a single monitoring point, for example the

center of the shell, the average value of several sym-

metric mesh points is calculated. The term avenge

defZection denotes the analytically integrated average

of the deflection values u3 over the whole surface.

The allowable error in satisfying Eq. (12) is restricted

well within 1.0% by adjusting the number of mesh

points. About 40 different curvature values each for

the partial cylindrical and spherical shells are numeri-

cally analyzed. The term injluence Parameter used in

the graphical results indicates the fraction of influence

shared by a component term in the total resisting

mechanism of the equilibrium equation for deforma-

tions in the normal direction.

3.1.1 Initial Resisting Mechanism

The initial shares of extensional and bending stiff-

ness parts are shown in Fig. 2. Shallow curvatures

are shown towards the right where the bending part

(B) has complete domination, and at the left extreme

where deep shells are shown, the extensional part (£)

dominates the resisting mechanism.

The curves for partial spherical shells are more

shifted towards the shallow shell end than the partial

6-oENas

aooco==c

1.0

O,9

O,8

O,7

O,6

O.5

O.4

o.d

O.2

O,1

o,o

-ttt:::--- .-s:--- SNs --- SstLSsss

X'x,

Extensjonal part'xx `'x

/Sik cylindricalshells , X x,

X Bending part

x spherical shellsilli/ -illVk StStt

NS ts IS NS tl ss SsstSss

----- . "L '.t".-"

1 10 tOO Radius (R/a) .-!2gtg2-s!]g!!s-epshells -S!t!a!!gw-sbg!!s-.allowshells

Fig.2 The share of extensional and bending stiffness parts

in the initial resisting mechanisms of different shells

cylindrical shell, which is very much in accordance

with the logical arguement that for doubly curved

shells the effect of curvature appears at more shallow

curvatures than singly curved shells for which the

effect of bending in the primary direction lasts until

deeper shells. More detailed analyses of the compo-

nent parts can also show, among various other interest-

ing details, that for deep cylindrical shells the exten-

sional component in the radial direction E4 is the sole

contributor to its resisting mechanism.

3.1.2 Resisting Mechanism in Deformed States

The results for initial influences of various compo-

nents are roughly representative of the deformed equi-

libriums also. However, the mechanism of deforma-

tion is accompanied by geometrical changes which

initiate some considerable changes in the resisting

mechanism of the shell. The total resisting mecha-

nism is composed of several smaller terms that consti-

tute each component term of Eq. (12). For a deeper

understanding of the resisting mechanism, it might be

worthwhile to study some of those terms in detail and

determine their shares in the total resisting mecha-

nism.

Fig. 3 and Fig. 4 show the variations in the influence

of different terms in the total resisting mechanism

during deformed states. The top row of 3 figures each

shows the aggregate influence and the bottom row

shows the influence due to component terms of the

bending and the extensional parts. Figures (a-1) and

(b-1) can be found to be of the same type for both shells

except for the effects of unsymmetry of principal

92 Thomas GEoRGE and Hiroo OKADA

1.0

O.9ts.¢. o.e

E e,7

9cu o.e

a o,s

88 o,-

= e,3

=-C e.2o,t

e.o

' E--i-g-p'irt 'N,.

Exta,,kualpM

x, N,B x,

s

x

N

N

N

'

N'

1-

x

E

'N. B

N.Ns 's

s.

1.e

ests o.e

6E o,79 o.6

E8 os

C O,4o= a3Et o.t

O.1

e.o

o.el o.te 1,co laooLoad (p/E) (xio-.)

(R/a) = 128,O

B"vdir-part.-..t

Exbfitior-lp

1

B 1"'x

.1 ' t

s

(a-1) Radius

N

x

'

x

x

N

s

x..B tNl

NN

o,ol e.lo t.oo to.oo Load (p/E) cxio-`)

Radius (R/a) =16.0

1.0

e,g

ts o,s

6E o.79 e.e

8s es

8 o,,

= e.3¢S 02

O.1

o,e

(a-2)

Extensenalpart

s

BSendiNpm t-ti

(a-3)

o.et

Radius

e.lo 1.ooLoad (p/E)

(R/a) =

10.ooC*lo-e)

2.0

(a) Aggregate influence

ts o.3

-oE9 o.2

888 e.i

lE o.e

--4Pl---

1

e?

..e± El,E2,

El ..- it-L X "s. E4.-' Nt N' N! NN ! ea.-- t "" t. Ytr.'KN l/ /F X .N'U'･.u 1,' N `'t:' ee xee ,.4 -' ""I"k.. "s.

e k.. - -- t '`-v.c:-t"-h ----Sa. 'tu NJ af

O.Ol O.10 t.oo Load (p/eRadius (R/a) =

10,oo

CX10--)

O.7

o,ets6 e,s: O,4

a"to3

E o2

2t o.t

c o,o

-o.t

.... SL.-.s

N

x

,lN, 1

et,es-.".--' .y or.."..m.."...---"--

El,esIEIH----･･･..

El

.1･E

"-'-"a･.1.s/ '-･..y

"Nge

× xx N".s2 NN pa. . .. N. ' ;"';'t::"'>(

t- )t"vr--1..;ror'C.tl'

e.ot o,lo 1,oo Load (p/E)

Radius (R/a) =

lo.co

(x10")

1.0

o,odi o.e

6 o.7

Eas o.etsa e.s8 o.4

8 o.s

S O.2

S o,t

e.o

-O,1

E4

ezes,or

El,E3 e"･･

(b-1)

(b)

128.0 (b-2)

lnfluence16,O (b-3)

of component

e,el o,lo t.oo to.co Load (p/E) {xie-')

Radius (R/a) =2.0

termsFig.3 Variations in the share of different components in the resisting mechanisms

of some partial cylindrical shells

1,e

o,o6 o.s

2 o,T

g ee

a! :･l

¢= e,3¢S e.2

e.t

e.o

+""-"--' Nevdop pvt X ' BN,

Extntiorripm

s

s

x

1'

N,

N,

'

1

N'

N,

s

xxB SS.LXs N s

o.ot o.lo 1.co Load (p/E>

Radius (R/a) =

ze

ts t,s

-¢E i.o

sas

a es88 e.o

c=

E -e.5

-1.0

11 X, B

,N )t ,N +t ,NExior,ndpvts /i, X'x 8

'E--ifp'vt B

s

(a-1)

KK

s.B-

e.oT o.lo 1.co Load (p/E)

Radius (R/a) =

t.o

o.g6 o,e

6E o,7g o,s

88 O,s

C o.4¢= O.3St O.2

o,t

o.e

te.co

(x10-')

128.0

(a)

te.oo

(x10'`)

liontlput

?!njvvpth

s

B

(a-2)

./'

Aggregate16.0 (a-3)

e.ol o.io 1.ee lo.oo Load (p/E) (xio-t)

Radius (R/a) =2.0

influence

ts O.3

-cx)Eg ,.2os

a¢oC e.tm2E

o,o

--eqJ-'t-+s el

ee

El,E

1 E2,E4

Ea

N N x x N N N- 4"-- .. x ." ...k< :N :N..

.. .. ,, X,p3 'N ;r:. ..-.k. NN r' '... ee N

t --L.t -. ...-.Ea- Bl.B-

o,et o,to t.oo Load (p/E)

Radius (R/a) =

to.oo

CX10-`)

56Egcu

amoco==E

e,s

O.7

O,6

O.5

O.4

O.3

O.2

O.1

o.o

-O,1

-O.2

-o.s

-O.4

ElE4

x X EI,E4 x x ･. E2." .. s.-' / ' ..'N .. .lt....epr/.x

.EG h

l-Xee

N

lxlXNI

BI,S-

u-'-B'1""si""'-"'-..B.t......."......."

-'FiE2,,

Et,E4

(b-1)

oot o.ie 1.co lo,co Load (p/E) {xio-`)

e.4

ts-O O.3

E9astr e.2oo=o-;1 o.,

e.e

128.0

1

E4

E?

ee･as

uuuouo" goo""o"el,or

(b-2)

e"==o"E )

Radius (R/a) =16.0 (b-3)

o.ol e.to Load

Radius

1.oo le.oo(P/E) (xio-e)

(R/a) = 2,O

(b) lnfluence of component termsFig.4 Variations in the share of different cornponents in the resisting mechanisms

of some partial spherical shells

Correlating the Resisting Mechanisms and

directions in the case of the cylindrical shell. The

domination of the bending components disappear at

higher loading stages of shallow shells, which shows

the importance of extensional terms for shallow shells

also when considering large deformations.

In Figures (a-2) and (b-2), the resisting mechanisms

do not match between the two shells. For cylindrical

shells in Fig. 3 the initial domination is by the bending

part for the presence of its primary direction, which is

the other way in Fig. 4 for the spherical shells.

However, at higher loading stages the dominant parts

reverse for the cylindrical shell while the initially

dominant part of the spherical shell regains the posi-

tion after an intermediate reversal of roles. The

details of this picture given in (b-2) for both shells in

terms of the corresponding components show the

extremely complicated influence changes occuring

around the static instability point of each shell.

Figures (a-3) and (b-3) are much less complicated,

where the extensional parts clearly dominate and their

components follow the natural conclusions. It may be

seen from the above illustrations that for both types of

shells the curvatures may be broadly divided into an

`extension zone', a `bending zone' and a `combined

zone' based on the resisting mechanisms. Deep shells

in the `extension zone' follow the evidently straight-

forward pattern, and for shells in the `combined zone'

extensional and bending parts demonstrate a very

complicated role playing.

3.1.3 Characteristic Curves of the Resisting

Mechanism The resisting mechanisms of shells from the shallow

to deep curvature ranges are calculated and the results

are shown in Fig. 5 and Fig. 6. Here, the boundaries

of influence zones for each component term is plotted

as continuous lines, where the influence zones are

measured as the range of influence parameters greater

than O.Ol (1.0% share of the total influence or more).

The `Threshold of static instability' is determined

based on the combined estimation using the results of

an earlier study6) on the natural frequencies of vibra-

tion of the deflected shell and the ultimate loading

values from the present study. The resisting mecha-

nism is calculated for loading stages within this thresh-

old load only, since the mechanism undergoes many

changes beyond this point.

As for the method of reading these graphs, the

boundaries of influence zones for each component

the Strength of TZtin Shells under libllower Loacls 93

plotted here represents the appearance of that compo-

nent when moving upwards through the direction of

increasing load for each radius. A curve due to a

component that does not appear at any loading stage

can be considered to have its influence zones for all

Ioading levels at that radius. Thus, at any radius the

portion of an ordinate below the point of an intersec-

tion with a curve is the `no influence zone' of load

levels for that component.

In both figures, shallow shells towards the right end

are dominated by bending terms B only, until a higher

loading stage where the influence of the extensional

part S appears. This can be understood easily by

(x1o-6)

100.00

A-N..avvtuoJ

10.00

1.00

O.10

O.Ol

i

:

i" st's "

sSE ""

Sl ts s

>zoc,9

2oYw

Partial cylindrical shell

N,'

itr･N/Tu?trie.S.h,O.idL.OgdStai,8icinstabiiity/

N, XN

XI!l :, k.

t' l "' ll"`X.11,,,, 'X.k.. I-..E;L -

l V!s, / ,A--Eh2-...--L-- s '-' t lB2 Ns / El '--'- BilP3 11i S"-------･" E,b,/

l li, Extension and Bending/ Beonndl/yng

1 10 100 Radius (R/a) (.!D!g912-sng!!E-hlls -Shallowshells

Fig.5 Influence zones of different components in the resisting mechanisms of partial cylindrical shells

(xlo-6)

1000.00

100.00

A!!!e lo.oo

vasoJ

1.00

O.10

O.Ol

-ssi l"

11 -"

.

Partiat spherical shell

Threshold of static instability// Ultimate loading

X kx Xtx..

1lXN " X<k.. ..E.3-

Extension

only

il 1: 11B21B3tt2

1 ':,

x N XI xx-Y

Extension andBending

-'1E2 i ----L- ' ' El,E4

Bending only

Fig.6

1 10 100 Radius (R/a) <-!D2ggP-sng!!s-hll -Shallowshells

Influence zones of different components in the

resisting mechanisms of partial spherical shells

94 Thomas GEoRGE and Hiroo OKADA

considering that the deflected shell attains curvatures

which bring in the extensional effect. Now, for deep

shells towards the left end the picture is considerably

different for cylindrical and spherical shells.

For the panial cylindrical shells shown in Fig. 5 the

influence zones due to extensional components El, E2

and E3 appear only at very high loading stages at or

near the static instability. The initial influence zone

at the deep shell end is completely dominated by the

principal stretch component E4 in the radial direction.

However, for the partial spherical shells shown in Fig.

6 the only extensional component that does not appear

during the lower loading stages is the inplane twist E3

for the complete range of curvatures, and the principal

stretch components El and E4 are symmetric for all

curvatures.

The appearance of influence zones due to the bend-

ing components Bl to B4 in the case of both the shells

can be seen starting after the `extension only' zone and

remains so for the whole range of shallower shells

towards the right of the figure. In the case of cylindri-

cal shells in Fig. 5 the principal bending component Bl

due to the primary direction appears first and the other

components follow in suit at shallower curvatures as is

expected.

The influence zone that appears in between the

extremes is shared by both the extensional and the

bending parts. The overall width of this zone is slight-

ly larger for cylindrical shells due to the early appear-

ance of the effect of bending at the deep shell side and

the late disappearance of the effect of extension in the

radial direction at the shallow shell end.

3.2 StrengthAnalysis

In the numerical analysis using Eq. (18) to determine

the stress states of shells, instead of considering only a

single point for monitoring the stress variations, for

example the center of the shell, four different points on

a quarter shell area are considered simultaneously.

Fig. 7 shows the points chosen for monitoring, where

point A is the center of a quarter shell (e' :O.25ni,

e2=O.25 02), points B and C being the center points on

the principal curvature lines intersecting at point A

(ei=O.25n,, e2=O.509, and ei=O.50n,, e2=O.25fl,

repectively), and point D is the center of the complete

shell (0'=O.50 fli, e2=O.50 st2). Here, 9i and fl2 are

the subtended angles at the center of curvature in each

principal direction.

About 25 different curvature values each for the

tls----------

Rl a･....e2

: l--.

Partial cylindricat shelt

Fig.7

partial cylindrical and spherical shells are used for this

analysis. For an estimation of the elastic limit stress

state, the yield point stress for the material (oj) is

chosen as 300.0MPa, and the value (a(.p)/qy)=1.0

becomes the limit state of stress at the given monitor-

ing point in the direction indicated.

3.2.1 Calculation of Stresses

As the shell undergoes deformations at subsequent

loading states, the state of stress at each point under-

goes variations, which could be associated either with

the direct stresses due to the membrane force distribu-

tions or the bending stresses due to the moment distri-

butions. The contributions from both sources could

further be divided into normal stresses and tangential

shear stresses.

In Eq. (18) direct stresses are given by the first term

and bending stresses by the second. Further, normal

stresses are calculated when a =P and tangential shear

stresses are obtained when crtP in the subscripts

within brackets. The assumption of linear distribu-

tion of all stresses across a shell cross section is consid-

ered valid here.

Fig. 8'(a) shows the variations of normal stresses at

the monitoring point A of a typical shell (partial

spherical shell, R/a= 2.0). The vertical chain line in

Fig. 8 (a) indicates the limit stress state on the upper

surface A(U) in both e' and e2 directions at the load

level as given by the horizontal chain line. Fig. 8 (b)

elaborates upon the stress levels at other monitoring

points when the stress level A(U) reaches the limit

stress state. This particular case is evidently that of

a fully compression shell since all the monitoring

points give compressive stresses only.

Generally, at the limit stress state for any one of the

monitoring points, some shells have fully compressive

stresses at all points whereas, some other shells have a

compression and tension sharing situation or a tension

dominated situation. Only normal stresses are shown

eD r; eB eD ',. ec ec 'X: e2 'A A･., .,. e' ...,･:::'i'l: t ,.. :. .".".." aJ:" 1 el .-･""'ny "`,, /1,'`' ./'R

- -it -J 1- itJ-- -d l-pt',

Partial sphericat shell

Monitoring points for stress calculations

Correlating the Resisting Mechanisms and the Strength of

-6

1;hin Shetls under ]Fbllower Loatis 95

(xlO }

20,O

A 15.0-xav 10,O

vtoo"

5,O

t2A(u) e,e

t

E'

1,

11

ll

U:UpporsurfaceL:Lowersurface

'/ 12,t A(L} e,e:

:t

:

:

t --

: --

t tt z : i- -t : -- : IL ,

1.0

AOx× o.s

98O o,obco-to

E "e.s

6z

-1.0

1--"----'-'---'--" t-----'

UML UML UML UML

U;VppersurfaceM : Mlddla stttfAoe

L:Lovversurfeoe

UML UML UML UML

A" 'B

l 1 1 1 l I-..-..--.-.-L- -.-..-.-

CDIAB CD

(xlo-6)

A-x.avvtuo"

O'O-3,o -2.s -2.o -1,s -1.o -o.s o.o o･s Direction & Position Normal stress (o/ o, )

(a) Variation of Normal Stresses (b) Stress states forthe Limit state

Fig.8 An example for the variation of normal stresses with load and the stress states at the monitoring points for the limit state (Partial spherical shell, Rla==2.0)

10,O9,O8,O

7.0

6,O

5,O

4.0

3,O

2.0

1.0O,9O,8

O,7

O,6

O.5

ltil-is-

Regions of double curvature advantage

e',1,･ /.li

tss

l- ts Ss ss Ss ss s--- - -li

[.,,,

..

,. .E,egl･g."dO.`.Cn"tarvgae'YIR,,.･･ ･:･

Region of spherical

disadvantage1-:･: ･･････ "-,i

･,:.:.･･･････････ Sub-plate region･･･････ .',.

(sp.shells)

ts -- ' -t x, -t -t -t ttt st ; sl : t't Sub--plate region (cy,sheUs} '

Plate strength'= '- i' in't"

Partial cylindrical shells

7 Partial spherical she"s

,i,.,･1

Fig.9

in Fig.

are found to be much smaller to pose any serious threat

of contributing to an initial limit stress state.

3.2.2 Strength Characteristics

Similar to the typical result shown in Fig. 8, the limit

stresses for different curvatures are calculated and the

results are summarized in Fig. 9 for the two types of

shells. From these elastic strength characteristics a

comparative study is made possible between the two

shell types. The spherical shells are found to be more

highly prone to reach an earlier strength limit for a

particular range of curvatures than their cylindrical

counterparts.

It is interesting to note that shells in a particular

curvature range have their strength limits reached,

10.0 1oo,O Radius (R/a)

Comparison of the elastic strength characteristics

between the two types of shells and the identifica-

tion of different regions

(8) since the order of tangential shear stresses

although at a local monitoring point and not a global

state, even before the corresponding value for a simple

plate. This phenomenon at the first glance runs con-

trary to the popular idea of shells being stronger due to

their' form resistance. However, this analysis con-

siders only the local stress limit and does not include

any other considerations like the geometrical strength,

stability, ultimate strength or global elastic/plastic

limits. Also, there is a region where the addition of a

curvature to the cylindrical shell could enhance its

strength in par with that of spherical shells.

It might be concluded from Fig. 9 that for shallow

shells, the simple plate is the upper limit in a local

strength perspective, and the real manifestations of

shell strength are visible mainly in the deep curvature

ranges for both types of shells.

3.2.3 A Comparative Study of Shell Characteristics

The characteristics of resisting mechanisms and

strength that were demonstrated separately are com-

bined into one in Fig. 10 and Fig. 11, where the curves

of strength limits for each shell are drawn along with

the characteristic curves of resisting mechanisms.

The `extension only' zone of resisting mechanism can

be found to be the same as the region of `shell strength'

determined by the strength analysis. Similarly, the

`bending only' zone is roughly the same as the `equi-

plate strength' zone. The most intricate among these

zones, the `extension and bending' zone, can be seen to

indicate the `sub-plate strength' zone, which explains

the fact that any amount of lowering of the influence of

bending stiffness also renders the shell strength to be

96

(x1o-6}

100.oo

a lo.oo).

1.00

vdioJ o,10

O.Ol

Thomas GEoRGE and Hiroo OKADA

;,. Partial cylindrical shetl '.. Shell l Sub-plate IEqui-plate,, '.N Strength: strength : strength

Xs }

,, ,Se.. / Static tnstability

tt t-- - - N-,

),i"" - K , - - - L4. !r!lfate strength

"i',, /i ... X- Strength limit

IVSisss --i-" it- .-- .-E3- -' --

II'I'l' Bil 'ttsh`s, -/ /-'-E?---.J:.:-

,! B3P41 'SN....--.." /g 1 71,ExtensionandBendingEl//Beonnd/9ng

E .9 = ra s o o E m = ･.-. ca '6 o =Position

Direction

Position

Direction

E .9

Partial cyrindrical shellsVDe'gpi Deep i as'dS"M i sMhead/1"o[ lshatiowigXary"ew/

lt ' t lo e toot iPlatelnitial tnfluenoe Range et Extensienal Stiffness i

Exig:tsyion:/NtMbmedlntluenceRange i Bending only

lnitial lnfluenoe Range ot Bending StMness

th'6

>.N=(e9.o..

co

= E 8 E o = ･.-. m '6 o =

Fig.12

1

2 3

10 1ooRadius (R/a)4 5 678910 20 30

'"''"''-fi"u-e"'i"'1'"'ivsvu''}b'i-'"･`"' iej::"'"' 1'i"'s'd''1,u'.""'Ei'','.'''si',''1'"'i"I'::*i:"'ig'"H'i"i,'";p"E'i";var.'';a'i

A(U) eCompressionshells

:

:

:

D(U) , 21 eiCompression artd Tension i

sharingshells i

D(L) elTensiondominated shells

A(U)

d8

: (e-) D(U)

i ee(o')!

l 1

D(L)

ee

1 10 100 Radius (R/a) -P9t9!2-Et!9!!E-ePShellS pttellowshells

Fig.10 Comparison between the resisting mechanisms and

elastic strength characteristics of partial cylindri-

cal shells

(x 1o -6)

1000.00

100.oo

A!!!9 io.oo

8 i.oo

e O.10

O.Ol

56789 1e 20 Radius (Rla)

1 10 1oo30

lnitial 1nfiuenoe Range of Extensional Stiffriess ;

:Extension only ,

:

Combined lrvfiuence Range,

, Bending only

, Partial spherical shell'. Shell I Sub-plate l Equi-plate '... strength L strength , strength

-s / Static lnstability

N

1..---L--. ]u1N M- i lx l l X 1 1, X< 1･i,

Extension B21331iB2

only 1 'l,

,

/ Plate strength

"'-- Strength Limit

. - -. -" X 1 .-x X - t s-- JExtension andBending

E3

E2

El,E4

Bending only

,lnitial lnfluence Range ot Bending Stiffness

1 10 100 Radius (Rla)

twtepshells wttallowshells

Fig.11 Comparison between the resisting mechanisms and elastic strength characteristics of partial spherical

shells

`sub-plate' until the extensional stiffness regains the

dominating status near to the `extension only' zone,

where the `shell strength' is attained.

Based on the numerical evidence obtained in this

research, a broad classification of shell curvatures

become possible. In Fig.12 the conclusions drawn

from the analyses of resisting mechanisms are demon-

strated at the top and bottom third of the figure along

with the results of strength analyses shown in the

middle portion. The top and bottom halves of the

figure represent the cases of partial cylindrical and

partial spherical shells respectively.

1 10 100 Vety l IMediuml Medium l :Very Deep i Deep e Deep t Shaleovv ,Shallowe Shallow/

t t i i tPlate Partial spherical shells

Deep shells ShaUow shells

A depiction of qualitatively equi-strength curvature

regions and their limiting strength specifications

against the different regions of resisting mecha- nisms of the two types of shells

The terminology used here for the subdivision of

curvatures and different ranges of mechanisms are

arbitrary and self-explanatory. The upper and lower

horizontal scales are staggered in the middle figure,

where the strength analysis is shown, to facilitate

direct comparisons between the equi-strength curva-

tures of the two types of shells. Also indicated in the

middle figure are the monitoring positions and the

principal directions of the initial limit stress state.

The dominant stress states at the two ends of the

medium curvature range [(6'): tensile stress, (a-):

compressive stress] are also shown.

This subdivision is a qualitative depiction comparing

the resisting mechanisms and strength characteristics

of shells in a picto-verbal sense, which is considered

suitable as a quick reference into the different shell

mechanisms.

4. Conclusions

Some characteristics of thin shells with respect to

their resisting mechanisms and strength have been

brought out and the correlations between the two were

graphically demonstrated through the numerical

results presented in this paper. The resisting mecha-

Correlating the Resistiirg Mechanisms and the Strength of T7iin Shells under li;ollower Loads 97

nism was analyzed within the statically stable region

and the strength was evaluated based on certain local

elastic criteria. Both these aspects have shown that

the medium range of curvatures, which has wide prac-

tical applications, is the most intricate area that act as

a transition range between shallow and deep shells,

where the problem has to be approached with extreme

cautlon.

0bviously, any practical design criteria for shell

structures have to take care of all these different

aspects simultaneously. The present approach could

be not applicable to the accompanying dynamic phe-

nomena, if any, where some small nonlinearities at an

initial stage could be a detrimental factor to the equili-

brating mechanism. The results presented here for

the particular shell types could serve as a useful guide-

line for other types of shell problems too.

References

1) Flligge, W.: Stresses in Shells, Springer-Verlag, Berlin

(1960)

2) Green, A. E, Zerna, W.: Theoretical Elasticity, Oxford

University Press, Oxford (1960)

3) Flttgge, W.: Tensor Analysis and Continuum Mechanics,

Springer-Verlag, New York (1972)4) Pietraszkiewicz,W.:GeometricallyNonlinearTheoriesof Thin Elastic Shells, Advances in Mechanics, 12 (1989),

pp. 51N1305) Shinoda, T., George, T., Fukuchi, N.: A Detailed Analysis

of the Theory of Thin Shells and some Particular Applica-

tions, Trans. of the West Japan Society of Naval Archi-

tects, No. 80 (1990), pp. 171t--193

6) Fukuchi, N., George, T., Shinoda, T.: Dynamic Instability

Analysis of Thin Shell Structures subjected to Follower

Forces, (2nd Report) Numerical Solutions and some Theoretical Concepts, Journal of the Society of Naval

Architects of Japan, No. 171 (1992), pp. 597A-609

7) George, T., Fukuchi, N.: Dynamic Instability Analysis of

Thin Shell Structures subjected to Follower Forces, (6 th

Report) Dynamic Threshold Characteristics and Post- critical Stability, Journal of the Society of Naval Archi-

tects of Japan, No. 176 (1994), pp. 319"-330

8) George, T., Okada, H., Fukuchi, N.: The Resisting Mecha-

nism during the Large Normal Deflection'of Thin Shells

subjected to Follower Forces, Journal of the Society of

Naval Architects of Japan, No. 179 (1996), pp. 281-v292