ELASTIC/PLASTIC BUCKLING OFCYLINDRICAL SHELLS WITH ELASTIC CORE

UNDER AXIAL COMPRESSION

byJin Zhang

McGill UniversityDepartment of Civil Engineering and Applied Mechanics

Montreal, Quebec, Canada

June 2009

A thesis submitted to the Graduate and Postdoctoral Studies Officein partial fulfillment of the requirements of the degree of

Master of Engineering

© Jin Zhang 2009

- i -

Abstract

Elastic as well as plastic buckling of circular cylindrical shells filled with a core material

is analyzed under axial compressive loading. A practical example of this situation is the

buckling of concrete filled steel tubular (CFT) columns used widely in high-rise

buildings. The theoretical problem is modeled as the bifurcation buckling of a perfect

"infinitely" long circular cylindrical shell under uniform compression, constrained by a

one-way (tension-less) foundation. An important and useful novelty is that the shell

material is allowed to undergo strain-hardening plasticity before buckling. For simplicity,

the core material is assumed to remain elastic. The approach is analytical. The governing

equations are solved exactly to obtain buckling loads, and wavelengths in contact and no-

contact regions. The theoretical results, when applied to CFT columns, are found to be in

very good agreement with the experimental buckling loads of other researchers.

- ii -

Résumé

Le flambage élastique ainsi que plastique de coquilles cylindriques remplis d'un matériel

sont analysé sous le chargement de la compression axiale. Un exemple pratique de cette

situation est le flambage de colonnes tubulaires en acier remplis de béton (CFT) qui sont

largement utilisées dans les immeubles de grande hauteur. Le problème théorique est

modélisé comme le flambage par bifurcation d'une coquille parfaite cylindrique de

longueur "infinie" sous la compression uniforme, en présence d'une contrainte à sens

unique. Une nouveauté importante et utile est que le matériel de coquille est sollicitée

dans le domaine post-élastique avant de flambement. Pour simplifier, le matériel remplis

est supposé rester élastique. L'approche est analytique. Les équations régissants sont

résolus exactement à obtenir les charges de flambage, et les longueurs d'onde en contact

et sans contact régions. Les résultats théoriques, aux applications de CFT colonnes, se

trouvent en très bon accord avec des charges de flambage d'autres chercheurs.

- iii -

Acknowledgments

The author would like to express sincere thanks to his research supervisor, Professor S.

C. Shrivastava of the Department of Civil Engineering and Applied Mechanics, McGill

University, for his help, suggestions, and encouragement throughout the entire process of

his research and writing of this thesis. He also takes this occasion to express special

thanks to his wife, Bei Ni, for supporting him with love and patience.

- iv -

Table of Contents

Abstract .................................................................................................................. iÞ

Resume (French) ..................................................................................................... iiÞ

Acknowledgments ................................................................................................... iiiÞ

Table of Contents .................................................................................................... ivÞ

List of Major Symbols ............................................................................................ viÞ

List of Figures ......................................................................................................... viiiÞ

List of Tables .......................................................................................................... ixÞ

Chapter 1: Introduction ........................................................................................ 1

1.1 Definition of the problem ................................................................................... 11.2 Literature review ............................................................................................... 71.3 Statement of the objectives ................................................................................ 9

Chapter 2: Elastic Buckling of Infinitely Long Shells with Elastic Core 10

2.1 Governing equations derived by the method of virtual work ........................... 10ÞÞ

2.2 Elastic bifurcation buckling of hollow shell ........................................ 16Ð5 œ !Ñ

2.3 .................... 17Elastic bifurcation buckling of shell with rigid core Ð5 œ _Ñ ÞÞÞÞÞ

2.4 ............... 18Elastic bifurcation buckling of shell with elastic core ( )! 5 _ Þ

2.4.1 Solution for buckling deflections in contact regions ............................... 19ÞÞ

2.4.2 Solution for buckling deflections in no-contact regions ......................... 21ÞÞÞ

2.5 Matching conditions between contact and no-contact regions 23 ........................ÞÞÞ2.6 Equations for buckling load and wave lengths ................................................. 24ÞÞ

2.7 Solutions for buckling load and wave lengths ÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞ................................. 25ÞÞ

Chapter 3: Plastic Buckling of Infinitely Long Shells with Elastic Core 29

3.1 Constitutive relations of the plasticity theories .................................................. 293.2 Constitutive relations for a plastic bifurcation analysis ..................................... 323.3 ............... 33Governing equations for plastic axisymmetric bifurcation buckling ÞÞÞ

3.4 ........................................ 36Plastic bifurcation buckling of hollow shell (5 œ !Ñ Þ

- v -

3.5 Plastic bifurcation buckling of shell with rigid core ( 385 œ _Ñ ..........................3.6 Plastic bifurcation buckling of shell with elastic core ( )! 5 _ .................. 393.7 Equations for buckling load and wave lengths .................................... 42ÞÞÞÞÞÞÞÞÞÞÞÞÞ

Chapter 4: Application to Concrete filled Steel Tubular Columns, andVerificaion with Experiments ............................................................ 43

4.1 Application to concrete filled steel tubular columns ........................................ 43Þ

4.2 Verification with experiments ............................................................................ 454.2.1 Theoretical failure loads versus experimental loads of [30] 46Sakino et al. 4.2.2 Theoretical failure loads versus experimental loads of O'Shea

and Bridge [31] ....................................................................... 48ÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞ

4.2.3 Theoretical failure loads versus experimental loads ofLam and Gardner [32] ...................................................... 49ÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞ

4.3 The scatter of the three sets of results ............................................................... 514.4 Buckling load using simple equation ............................................... 53Ð5 œ _Ñ ÞÞÞ

Chapter 5: Summary and Conclusions ............................................................... 54ÞÞ

5.1 Summary ............................................................................................................ 545.2 Conclusions ....................................................................................................... 565.3 Suggestions for future work .............................................................................. 57Þ

References ............................................................................................................ 58Þ

- vi -

List of Major Symbols

E E-ß = area of concrete core, area of steel shellF ßG ß Hw w w elastic-plastic moduli for plastic bifurcation buckling- ß ." " related to roots of characteristic equation in contact regions- ß .2 # related to roots of characteristic equation in no-contact regionsH shell elastic rigidity, also outside diameter of shell/ß / œ IÎI "ß œ IÎI- - parameters in elastic-plastic moduli, = >

I Young's modulus of elasticityI> tangent modulusI= secant modulus0 w- crushing strength of concrete cylinder test

5 foundation modulusRBB axial force per unit length due to bucklingR R œ >

-< -< buckling load of shell per unit length, 5-<

R-? ultimate strength of concrete coreRI ultimate strength calculated by Eurocode 4 equationsR- ultimate strength calculated by CSA S16 equationsRX theoretical strength predicted by the present methodR? ultimate strength obtained from experiments in literatureQBB BB buckling moment resultant per unit length due to 5UB BD buckling shear force resultant per unit length due to 5V shell middle surface radius> shell thicknessA A_ _

radial buckling displacement, œ AÐBÑ

? ? œ ?ÐBÑ .A

.B

_ _ axial buckling displacement, (

A ÐBÑß A ÐBÑ" # buckling deflections in the contact, and no-contact regions% %))ß BB buckling strains in shell> >, coefficients of differential equations in contact, no-contact regionsw

in elastic range> >: :

wß coefficients of differential equations in contact, no-contact regions in plastic range ( thickness coordinate/ Poisson's ratio

- vii -

5-< buckling stress5 5BBß )) shell stresses# ß #' 0 buckling lengths in contact, no-contact regions

- viii -

List of Figures

Fig. 1-1 Buckling mode of a short shell on one-way elastic foundation .................. 3Þ

Fig. 1-2 Buckling mode of an infinitely long shell on one-way elastic foundationwith regions of contact and no-contact ....................................................... 3

Fig. 1-3 Concrete filled steel tubular (CFT) column ................................................ 4

Fig. 1-4 Typical failure mode of CFT columns [1] .................................................. 6

Fig. 1-5 Axisymmetric buckling mode of cross-section .......................................... 6Þ

Fig. 2-1 Coordinate system of shell .......................................................................... 11Fig. 2-2 Buckling mode of a strip taken from an infinitely long shell resting on

one-way elastic foundation ........................................................................ 18Þ

Fig. 2-3 Buckling load parameter ......................................................................... 27!

Fig. 2-4 No-contact length parameter .................................................................... 28"

Fig. 2-5 Contact length parameter ......................................................................... 28#

Fig. 4-1 Ramberg-Osgood curve of steel (I œ #"!ß !!! œMPa, 853 MPa)5C

used in the experiments conducted by Sakino ............... 46 et al. [30] ÞÞÞÞÞÞÞÞÞÞÞ

Fig. 4-2 Ramberg-Osgood curve for steel used in the experiment conductedby ........................................ 48O'Shea and Bridge [31] ÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞ

Fig. 4-3 Ramberg-Osgood curve for steel used in the experiment conductedby Lam and Gardner [32] ................................................... 50ÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞ

Fig. 4-4 Scatter for the present theoretical predictions vs. experiments in [30] ...... 51Fig. 4-5 Scatter for the present theoretical predictions vs. experiments in [31] ...... 52Fig. 4-6 Scatter for the present theoretical predictions vs. experiments in [32] ...... 52

- ix -

List of Tables

Table 2.1 Buckling parameters for infinitely long shells with differentfoundation moduli .................................................................................... 26

Table 4.1 Comparison of the present theoretical predictions with experiments [30] 47RX Table 4.2 Comparison of the present theoretical predictions with experiments [31] 49RX Table 4.3 Comparison of the present theoretical predictions with experiments [32] 51RX

- 1 -

Chapter 1

Introduction

1.1 Definition of the problem

Since late 1960s, buckling of plates and shells with surrounding solid or fluid constraintshas driven interest of many researchers. As a result, a number of researches have beenconducted with the intention of clarifying the mechanism of such buckling problems.Generally, these problems are treated as stability of plates or shells resting on elasticfoundations and subjected to compressive loads. The buckling loads and buckling modesare usually derived under the assumption that the plates or shells are bonded to the elasticfoundation, and the latter is able to provide both compressive and tensile reactions. Thebuckling analysis is therefore simplified. This kind of problem is termed as buckling of astructure against a two-way foundation. Many studies have been carried out based on thisassumption, despite the fact that this assumption may not be a realistic one. In manysituations, the bond is not positively ensured during construction, and therefore ought tobe ignored for a safe design.

In contrast to this two-way assumption, there are problems in which the plates and shellsare not bonded to the foundation but simply rest on it. In such (non-bonded) cases thefoundation can only provide resistance when compressed. These problems are termed asone-way buckling problems, which can be illustrated by several examples. One exampleis the phenomenon of "lift-off" of welded railway tracks due to their constrained thermalexpansion. In this problem the foundation is unable to restrain the upward buckling of therailway track, and therefore no tensile reaction forces are developed. Other examples aresurface delamination in composites, and temperature induced "pop-up" of a pipeline inthe vertical plane. This category of buckling problems (i.e., one-way buckling) is moredifficult to analyze. The present work falls into this latter category.

Problems regarding elastic buckling of shells have been well studied. On the other hand,there are very few studies on the plastic buckling problems. Since plastic buckling is amore desirable failure mode (for harnessing full strength of a metal column) than theelastic buckling, the former type of buckling is very important from a design point ofview. To the best of the author's knowledge, however, there are no solutions (at least

- 2 -

analytical ones) available on the plastic buckling of shells constrained by one-way(tension-less) elastic foundation. The present thesis carries out exact analyses for bothelastic and plastic ranges of buckling behaviour.

The theoretical problem is that of bifurcation buckling of a thin infinitely long perfectlycircular cylindrical shell (i.e., a tube) resting on a tension-less elastic foundation (i.e., thecore) and subjected to a uniform compressive stress in the longitudinal direction.

When inward movement of the shell wall is restricted by the compressive reaction forcesprovided by the core, the buckling mode can be of one sign (outward with a single wave,Fig. 1-1) or of multi-signs having multiple waves, Fig. 1-2. The former mode is possiblefor short shells, not considered here. The latter mode is possible for long shells, in whichregions of contact and no-contact in the buckling mode would emerge simultaneouslyunder certain compressive forces. In the no-contact regions, the shell wall buckles awayfrom the core, and no reaction forces exist between the shell wall and the foundationcore. Conversely, in the contact regions, the inward buckling displacements of the shellwall induce reaction forces of the foundation. The analysis required to find the solutionfor such buckling problem is quite difficult because, in a sense, a nonlinear problem is tobe solved. The analysis is further complicated by virtue of the fact that strain-hardeningplastic behaviour of the shell material is allowed to occur. The results from the presentanalysis are valid for both elastic and plastic ranges of shell buckling behaviour. For thelatter case, usually treated in an ad hoc fashion, the present analysis furnishes a rationalbasis for computing the column capacity against buckling in plastic range.

- 3 -

Ncr

Short CylindricalShell

Ncr

ShellSurface

One-wayFoundation

Fig. 1-1 Buckling mode of a short shell on one-way elastic foundation

N cr

Wavelength 2ξ+2ζ

N cr

N cr

N cr

One-way Foundation

Core

Shell

Fig. 1-2 Buckling mode of an infinitely long shell on one-way elastic foundation withregions of contact and no-contact

- 4 -

Design of concrete filled steel tubular (CFT) columns is the most relevant practicalapplication of the present research on the above stated one-way buckling problem. CFTcolumns, Fig. 1-3, have become an attractive solution in civil engineering practice inrecent decades. CFT has excellent structural performance characteristics. Steel membershave the advantages of high tensile strength and ductility, while concrete members havethe advantages of high compressive strength and stiffness. CFT as a composite member,combining steel and concrete, has the beneficial qualities of both of these materials [1].In addition to the structural benefits, CFT members have lower cost and lowerconstruction time compared to reinforced concrete columns or purely steel columns. As aresult of these distinct advantages, CFT columns are used extensively in earthquake-resistant structures, high-rise buildings, and bridges. However, the use of CFT columns isstill limited due to a lack of understanding about their true strength, especially theirbuckling strength.

Concrete

Steel

t

R

Fig. 1-3 Concrete filled steel tubular (CFT) column

- 5 -

In the early stages of loading, of a CFT column, Poisson's ratio for concrete is to !Þ"& !Þ#

which is lower than that for steel being around Therefore, in the early stage, the steel!Þ$Þ

tube expands more than the concrete core, and one might say that a gap might developbetween the steel tube and the concrete core. However, as the longitudinal strainincreases with the loading, Poisson's ratio of concrete begins to increase, and approachesclose to in the fully plastic state [2]. In other words, the lateral expansion of the!Þ&

concrete core gradually becomes greater than that of the steel tube, which overcomes anygap and provides a snug fit with the steel tube. Hence, effectively, the tube developstensile hoop stresses due to the radial pressure developed at the steel-concrete interface.For buckling analysis, this possibly small pre-buckling hoop tension in the tube isneglected.

Bond between concrete core and steel tube was studied by Virdi and Dowling [3]. Theystated that the interface bond exists as a result of the interlocking of the concrete core dueto two types of imperfections, namely, the surface roughness of the steel tube, and thevariation in its cross-sectional shape along the length. Nevertheless, Furlong [4] foundthat practically no bond exists between concrete core and steel tube. On the other hand,Shakir-Khalil and Mouli [5] reported, from their experiments, a bond strength of to!Þ$*

!Þ&" N/mm . Thus, it is clear that the existence of bond cannot be relied upon for#

analyzing structural strength. Therefore, in the present buckling analysis, no interfacebond is assumed to exist.

Based on the discussion above, the buckling of the steel tube of CFT columns can beclassified in the category of one-way buckling problem. Typical local buckling of thesteel member is shown in Fig. 1-4 and Fig. 1-5.

- 6 -

Fig. 1-4 Typical failure mode of CFT columns [1]

Steel tube

Concrete Core

wDisplacementBuckling

Fig. 1-5 Axisymmetric buckling mode of cross-section

- 7 -

1.2 Literature review

As mentioned before, although there are many studies on elastic buckling of shells, thereare only a few dealing with plastic buckling, and none on plastic buckling of shells ontension-less foundations.

Euler's investigation [6] of the elastic stability of slender columns is considered as theorigin of the classical theory of elastic buckling of structures. Buckling of a cylindricalshell under uniform compression is a very important problem with numerousapplications. The work of the various researchers since the time of Euler can be found inthe celebrated book by Timoshenko and Gere [7]. For a long hollow shell, assuming thebuckling to be axisymmetric, the critical (i.e., the lowest) bifurcation buckling stress 5-<

is given by Timoshenko and Gere [7] as the classical expression

5/

-<#

œ Ð Ñ Ð"Þ"ÑI >

$Ð" Ñ VÈ .

But experiments since 1930s revealed that experimental buckling loads were invariablymuch lower (and with a wide scatter) than the above prediction of the classical theory.The reason for the wide discrepancy was explained by von Karman and Tsien [8]. Theyshowed that the post-bifurcation path for the cylindrical shell falls steeply to a load muchlower than the bifurcation load. Koiter [9] showed that this post-bifurcation minimumload may be calculated by performing a nonlinear imperfection growth analysis of smallinitial geometric imperfections in the shell. The concept of "imperfection-sensitivity" hasbeen well accepted in estimating the buckling strength of hollow shells undercompressive loading.

The plastic buckling of hollow shells was addressed by Bijlaard [10], Gerard [11] andBatterman [12]. They came to essentially the same conclusion that the load predicted by adeformation theory of plasticity agrees well with the experiments. On the other hand,results of Lee [13] showed that an incremental theory dramatically overestimates thebuckling strength.

Seide [14] studied the elastic buckling of a shell with a soft elastic core, by solvingdifferential equations for equilibrium of the shell. He noted that for a filled shell theaxisymmetric buckling mode gives the lowest buckling load. Karam and Gibson [15]

- 8 -

investigated the same problem, by assuming a sinusoidal radial displacement duringbuckling and the core as an elastic half-space. They concluded that shells with acompliant core result in reduced sensitivity to imperfections.

Ghorbanpour Arani [16] derived stability equations for a simply supported/> +6Þ

cylindrical shell with an elastic core by the energy method. The elastic core wasrepresented by elastic springs with modulus . Two-way foundation assumption was5/

used in all of these research works [14, 15, 16].

Bradford [17] provided a theoretical study of the local and post-local buckling of/> +6Þ

thin-walled circular steel tubes that contain an elastic core. The problem was treated aselastic buckling of shells on one-way foundation. Energy method and Ritz approach wereapplied. They found that the elastic local buckling stress for a circular tube with rigidcore is (or ) times that of its unfilled (hollow) counterpart.È$ "Þ($

However, since late 1990s, the interest of researchers in shell analyses has been orientedtoward experimental and computer solutions. Literature review by O'Shea and Bridge[18] indicates that the concrete filled steel tubes have received extensive attention by anumber of authors. A great number of experiments, with wide range of and HÎ> PÎH

ratios, have been conducted. Review of experimental works can also be found in thestate of the art report by Shanmugam and Lakshmi [19]. By comparison, only limitedattention has been paid to derive analytical solutions for the buckling load.

As mentioned before, at the present time, there are only a few researches dealing withshell buckling against one-way foundation, and none in the plastic range.

- 9 -

1.3 Statement of the objectives

In consideration of the foregoing discussion, the objectives for the present investigationwere defined to be:

(i) Derivation of the exact solutions for the elastic bifurcation buckling of infinitely longshells supported by tension-less elastic foundations, and subjected to uniformcompressive stress in the longitudinal direction.

(ii) Derivation of the exact solutions for the plastic bifurcation buckling of the same typeof shells stated above in (i) and shown in Fig. 1-2, by employing the constitutive relationsof the deformation and incremental theories of plasticityN N Þ# #

(iii) Application of the analytical results to concrete filled tubular steel (CFT) columns,and their validation by comparison with the experimental results of other researchers andalso with the loads calculated by using design codes.

- 10 -

Chapter 2

Elastic Buckling of Infinitely Long Shellswith Elastic Core

The analysis of buckling of infinitely long circular cylindrical shells containing an elasticcore in this thesis is based on the following assumptions:

(i) The shell is considered geometrically perfect. For analysis in this chapter, it isconsidered to be made of an isotropic elastic material. It is assumed to be thin withrespect to length and diameter. Its self weight is considered negligible. It is loaded byuniformly compressive stress in the longitudinal direction only. The boundary conditionsare assumed to allow the shell to remain perfectly circular under slowly increasingmagnitude of this stress, until a critical value is reached, at which point the shell buckleswith wavy displacements. Thus, the buckling considered is of the bifurcation type.

(ii) The shell is filled with concrete or a foam like material, forming a core. It ispresumed that no effort is made to provide a bond between the core and the surroundingshell wall. Thus, the core, is assumed incapable of providing any tensile or frictionalresistance to the shell wall; it only provides compressive resistance against the inwardmovement of the shell wall. The core support is therefore termed as a tension-lessfoundation. The buckling of the shell under such a constraint is called one-way buckling.

(iii) The core is represented by a Winkler type foundation, meaning that the compressivereaction forces are linearly proportional to the radially inward deflections.

2.1 Governing equations derived by the method of virtual work

Figure 2-1 shows the coordinate system with respect to the shell geometry. The pre-bifurcation state of stress is purely uniaxial compression. Let be the compressive5-<

stress at which bifurcation takes place, and let the reference geometric configuration bethat just before the bifurcation. Then, the displacements considered below are thosearising due to bifurcation alone. The corresponding strains and stresses are thereforeincremental ones, being additional to those just prior to bifurcation.

- 11 -

η

θ

Fig. 2-1 Coordinate system of shell

The critical bifurcation (corresponding to the minimum buckling stress), is considered tobe axisymmetric, as demonstrated by Batterman [12] for hollow cylindrical shells.Accordingly, the bifurcation displacements, strains, and stresses are all independent ofthe circumferential coordinate. The circumferential displacement is taken to be

_@

identically zero, and hence the non-zero displacements are axial and radial (positive_ _? A

outward). As usual, for thin shells considered here, the radial displacement is_A

considered independent of the thickness coordinate, and thus it is a function only of theaxial coordinate:

A_

œ AÐBÑÞ Ð#Þ"Ñ

Invoking the Love-Kirchhoff kinematic hypothesis, according to which normals to themiddle surface always remain normal (before as well as after deformation), the axialdisplacement is taken as?

_

- 12 -

? œ ?ÐBÑ Ð#Þ#Ñ.A

.B

_( .

where is the thickness coordinate, and is the axial displacement at the middle( ?ÐBÑ

surface, like is positive in radially outward direction.( (œ !Þ AÐBÑ

The non-zero strains arising due to bifurcation, in compliance with the aboveassumptions are

% % ()) œ œ Ð#Þ$ÑA .? . A

V .B .B, .BB

#

#

For plane stress behaviour (appropriate for thin shells), the linear elastic stress-strainrelations are

5 % /% 5 /% %/ /

BB BB BB# #œ Ð Ñ œ Ð Ñ Ð#Þ%Ñ

I I

" " )) )) )),

where is Young's modulus and is Poisson's ratio of the shell material. SubstitutingI /

the expression for strains in terms of displacements, one obtains buckling stresses interms of buckling displacements as

5 ( //

BB # #

#

œ Ð ÑI .? . A A

" .B .B VÐ#Þ&Ñ

5 / /(/

)) œ Ð ÑI .? . A A

" .B .B V# #

#

.

The virtual displacements of the shell are in the longitudinal direction and in the$ $? A

out-of-plane direction respectively. Thereby, the virtual strains are

$% ( $%$ $ $

BB

#

#œ ß œ Ð#Þ'Ñ

.Ð ?Ñ . Ð AÑ A

.B .B V)) .

The internal virtual work expression can be written as the sum of two parts:

( (Z Z

# #

#

5 $% / /( ) (/

$)) )) .Z œ Ð Ñ .B V. .

I .? . A A A

Ð" Ñ .B .B V V

- 13 -

œ Ð Ñ A .B Ð#Þ(Ñ# I> .? A

Ð" Ñ .B V

1

// $

#!

P(in the circumferential direction; and the second part, related to the longitudinal stress À

( (Z Z

BB BB

# #

# # #5 $% ( ( / ) (

$ $

/.Z œ Ð Ñ Ð Ñ.BV. .

.Ð ?Ñ . Ð AÑ I .? . A A

.B .B Ð" Ñ .B .B V

œ Ð Ñ ?.B# VI> .Ð ?Ñ . ? .A

Ð" Ñ .B .B V .B

1 $ /

/$

# #!

P #(

A.B Ð Ñ ?# VI> . A # VI> .? A

"#Ð" Ñ .B Ð" Ñ .B V

1 1

/ /$ / $

$ %

# % #!

P P

!( ’ “

. A Ð#Þ)Ñ# VI> . A .Ð AÑ # VI> . A

"#Ð" Ñ .B .B "#Ð" Ñ .B

1 $ 1

/ /$

$ # $ $

# # # $

P P

! !’ “ ’ “

The external forces acting on the shell are (i) the longitudinal compressive force

T œ # V > Ð#Þ*Ñ1 5-<

and (ii) the resisting force from the foundation

J œ # V 5 A.B Ð#Þ"!Ñ1 (!

P

where is the foundation modulus, which in the present case of tension-less foundation5

is required to be zero in the no-contact region. Now, the axial shortening of the shell dueto buckling is

J œ Ð .BÑ Ð.BÑ .B ¸ Ð Ñ .B Ð#Þ""Ñ.A " .A

.B # .B( (š ›Ê! !

P P# # # .

Thus the virtual work of external forces is

T J A œ # V > .B # 5V A A.B.Ð AÑ .A

.B .B$? $ 1 5 1 $

$-<

! !

P P( (

- 14 -

œ # V Ð > 5 AÑ A.B # V > A Ð#Þ"#Ñ. A .A

.B .B1 5 $ 1 5 $( ’ “

!

P

-< -<

#

#

P

!.

The principle of virtual work requires equality of the virtual work of the internal stressesto the virtual work of the external forces for arbitrary virtual displacements and :$ $? A

( š ›!

P $ % #

# % # #-<1 1

/ // 1 5 1 $

VI> . A # I> .? A . A

'Ð" Ñ .B Ð" Ñ .B V .B Ð Ñ # V > # V5 A A.B

?.B Ð Ñ ?# VI> . ? .A # VI> .? A

Ð" Ñ .B V .B Ð" Ñ .B V

1 / 1

/ /$ / $

# # #!

P # P

!( š › ’ “

# V > A œ ! Ð#Þ"$Ñ# VI> . A .Ð AÑ .A # VI> . A

"#Ð" Ñ .B .B .B "#Ð" Ñ .B

1 $ 1

/ /1 5 $

$ # $ $

# # # $

P P

! !-<’ “ ’š › “ .

For the above to be true, for arbitrary and in the domain, satisfaction of the$ $? A

following two differential equations of equilibrium is necessary:

I> . ? .A

Ð" Ñ .B V .BÐ Ñ œ ! Ð#Þ"%Ñ

/

/# #

#

VI> . A I> .? A . A

"#Ð" Ñ .B Ð" Ñ .B V .B Ð Ñ V > V5A œ ! Ð#Þ"&Ñ

$ % #

# % # #-</ /

/ 5 .

and also the vanishing of each of the boundary terms:

I> .? A I> . A .Ð AÑ

Ð" Ñ .B V "#Ð" Ñ .B .BÐ Ñ ? œ !ß œ !ß

/ // $

$# # #

P P

! !

$ #’ “ ’ “

> A œ ! Ð#Þ"'Ñ.A I> . A

.B "#Ð" Ñ .B’š › “5 $

/-<

$ $

# $

P

!.

The interpretation of the vanishing of the boundary terms at the ends , isB œ ! P

(i) either or is prescribedR œ Ð Ñ œ !ß ? Ð#Þ"(ÑI> .? A

Ð" Ñ .B VBB #/

/

- 15 -

(ii) either or is prescribed Q œ œ !ß Ð#Þ")ÑI> . A .A

"#Ð" Ñ .B .BBB

$ #

# #/

(iii) either or is prescribed.U œ > œ !ß A Ð#Þ"*Ñ.A I> . A

.B "#Ð" Ñ .BB -<

$ $

# $5

/

For simply supported shells, considered here, at the ends, and atA œ œ ! ? œ !. A

.B

#

#

B œ ! œ ! B œ P.? A

.B V, the supported end, and at , the loaded end./

The integration of the first differential equation gives

I> .? A

Ð" Ñ .B VÐ Ñ œ - Ð#Þ#!Ñ

//

#

where is a constant. However, in view of the boundary condition, at the loaded end-

where is not prescribed, the left hand side is required to be zero. Hence Thus? - œ !Þ

.? A

.B V œ ! Ð#Þ#"Ñ/

in the domain as well. Substitution of this result in the second differential equation gives

I> . A . A A

"#Ð" Ñ .B .B V > I> 5A œ ! Ð#Þ##Ñ

$ % #

# % # #-</

5 .

Introducing the standard notations

H œ R œ > Ð#Þ#$ÑI>

"#Ð" Ñ

$

# -</

5, -<

the equation governing axisymmetric bifurcation buckling of the shell is:

H R A 5A œ ! Ð#Þ#%Ñ. A . A I>

.B .B V

% #

% # #-< .

- 16 -

Let and represent the buckling deflections in the contact regionsA ÐB Ñ A ÐB Ñ Ð5 Á !Ñ" # #"

and no-contact regions respectively. Then the applicable differential equationsÐ5 œ !Ñ

are

H R A 5A œ ! Ð#Þ#&Ñ. A . A I>

.B .B V

% #" "

"% -< " "

"# #

in contact regions, and

H R A œ ! Ð#Þ#'Ñ. A . A I>

.B .B V

% ## #

#% -< #

## #

in no-contact regions.

2.2 Elastic bifurcation buckling of hollow shell Ð5 œ !Ñ

When no core is present, one has the case of classical shell buckling analysis. Assumingthe shell to be simply supported, a mode shape satisfying the boundary conditions is:

A œ - Ð#Þ#(ÑB

6" sin

1

where is an arbitrary constant, and is the unknown buckling wave length. Satisfaction- 6

of the governing differential equation requires

H R œ ! Ð#Þ#)Ñ6 6 V

I>1 1% #

% # #-<

which gives

R œ Ð Ñ Ð#Þ#*ÑH 6 I>

6 V-<

# #

# # #

1

1.

For a minimum critical load, should be such that This condition requires6 œ !Þ. R

.6-<

#H #6 œ !ß 6 œ Ð#Þ$!Ñ6 V

I> V>

"#Ð" Ñ

1 1

1 /

#

$ # # #or .

ÈÈ%

- 17 -

Accordingly, the minimum critical load and stress are

R œ œ Ð#Þ$"ÑI> ÎV I>ÎV

$Ð" Ñ $Ð" Ñ-< -<

#

# #È È/ /5, .

The above expression for the critical buckling load and buckling length are identical tothose given by Timoshenko and Gere [7].

2.3 Elastic bifurcation buckling of shell with rigid core Ð5 œ _Ñ

With rigid core, the shell, in axisymmetric buckling, has only a line contact with thefoundation, around the circumference at , where the deflection and slope ofB œ „ 6Î#

the longitudinal mode shape are zero.

The buckled shape defined by (omitting the arbitrary multiplicative constant)

AÐBÑ œ #7 8#7 B 8 B #7 B 8 B

6 6 6 6cos cos sin sin

1 1 1 1Ð#Þ$#Ñ

in which and are positive odd integers, satisfies the boundary conditions that7 8

AÐB œ 6Î#Ñ œ AÐB œ 6Î#Ñ œ ! A ÐB œ 6Î#Ñ œ A ÐB œ 6Î#Ñ œ !and . w w Ð#Þ$$Ñ

The function is periodic and the buckling mode is symmetric about AÐBÑ B œ !Þ

Substituting the assumed mode shape into the differential equation gives, for all :B

#7 Ð%7 8 Ñ Ð%7 $8 Ñ 6 R 6 #7 B 8 B

H H 6 6š ’ “ ›# # # # # #

# %-<

1 11 1I>

V#cos cos +

8 Ð%7 8 Ñ Ð"#7 8 Ñ œ !6 R 6 #7 B 8 B

H H 6 6š ’ “ ›# # # # # #

# %-<

1 11 1I>

V#sin sin

Ð#Þ$%Ñ

This requires:

Ð%7 8 Ñ Ð%7 $8 Ñ 6 R 6

H H# # # # # #

# %-<

1 1’ “ I>

Vœ !

#,

Ð#Þ$&Ñ

Ð%7 8 Ñ Ð"#7 8 Ñ 6 R 6

H H# # # # # #

# %-<

1 1’ “ I>

Vœ !

#.

- 18 -

Solving these two equations for R 6ß-< and one obtains

R œÐ%7 8 Ñ

Ð%7 8 Ñ-<

# #

# #

I>

V $Ð" Ñß

#

#È /

Ð#Þ$'Ñ

6 œV>

"#Ð" ÑÈ È

È%7 8# #1

/% #

The buckling load must be a minimum at the bifurcation, which means and .7 œ " 8 œ "

This gives the critical buckling load and wave length as

R œ $&

$-<(min)

I> V>

V $Ð" Ñ "#Ð" Ñ6 œ Ð#Þ$(Ñ

#

# #È ÈÈ È/ /

1, .

%

We observe that, for a shell with rigid core , the buckling load is increased byÐ5 œ _Ñ

&Î$ times that of a hollow shell . This new and exact result is lower than theÐ5 œ !Ñ

approximate one derived by Bradford [17] which gives a factor of /> +6Þ È$. Thecorresponding buckling length here is times that for a hollow shellÈ$ Þ

2.4 Elastic bifurcation buckling of shell with elastic core ( )! 5 _

When the foundation modulus is between zero and infinity, there are regions of contactand no-contact of the shell with the foundation. Buckles and coordinate system are shownin Fig. 2-2.

N crN cr

Infinitely Long Shell-ζ ζ -ξ ξ

One-way Foundation

Fig. 2-2 Buckling mode of a strip taken from an infinitely long shell resting on one-wayelastic foundation

- 19 -

2.4.1 Solution for buckling deflections in contact regions

In a contact region, the applicable differential equation is

H R Ð 5ÑA œ ! Ð#Þ$)Ñ. A . A I>

.B .B V

%" "

"% -< "

#

"# #

, or

. A R . A

.B A œ ! Ð#Þ$*Ñ

H .B

% #" -< "

"%

"# ">

where

R œ > Ð#Þ%!ÑI> 5

HV H-< -< #

5 , and .> œ

To solve this ordinary differential equation with constant coefficients, let

A ÐB Ñ œ + / Ð#Þ%"Ñ" " ":B"

where is an arbitrary constant. Then, satisfaction of the differential equations requires + :"

to be a root of

: : œ ! Ð#Þ%#ÑR

H% #-<

>

Solving the quadratic equation, one obtains

: œ „ Ñ Ð#Þ%$Ñ" R

# H# -<( È?

where

? >œ Ð Ñ % œ Ð Ñ %Ð Ñ Ð#Þ%%ÑR R I> 5

H H HV H-< -<# #

#

? can be negative, positive, or zero. One must consider all these three possibilities. Inthe first case of , the four roots are the following complex conjugate pairs:? !

- 20 -

: œ - 3. : œ - 3. : œ - 3. ß : œ - 3." " " # " " $ " " % " ", , , Ð#Þ%&Ñ

where

- œ # . œ # " "

# #" "Ê ÊÈ È> >

R R

H HÐ#Þ%'Ñ

-< -< and,

The solution can therefore be written as

A ÐB Ñ œ E Ð- B Ñ Ð. B Ñ E Ð- B Ñ Ð. B Ñ " " " " " " " " "w w" #"cosh cos cosh sin

E Ð- B Ñ Ð. B Ñ E Ð- B Ñ Ð. B Ñw w$ %" " " " " " " "sinh cos sinh sin Ð#Þ%(Ñ

in which , , , and are constants depending on the matching conditions withE E E Ew w w w" # $ %

the solution for the no-contact part. Taking the origin for this solution at the center of thebuckle then the solution must be symmetric about . This symmetry conditionß B œ !"

yields Hence the solution becomesE œ E œ !Þw w# $

A Ð Ñ œ E Ð- Ñ Ð. Ñ E Ð- Ñ Ð. Ñ" " " " " " " " "w w" %B B B B Bcosh cos sinh sin" Ð#Þ%)Ñ

At the ends, of this buckle , which then requiresB" "œ „ ß A Ð „ Ñ œ !' '

E E . -w" %

w" "œ Ð Ñ Ð Ñtan tanh' ' . Ð#Þ%*Ñ

Then, finally the solution of roots for this case ( is expressed as? !)

A Ð Ñ œ E - . - . -" " " " " " " "w%B Ö Ð B Ñ Ð B Ñ Ð B Ñ Ð. B Ñ×sinh sin cosh cos" " " "tanh tanÐ Ñ Ñ Ð#Þ&!Ñ' '(

Since E E œE

-w% %

w

"is arbitrary, it is permissible to replace it as and write

A Ð Ñ œ - . - . -" " " " " " " "B Ö Ð B Ñ Ð B Ñ Ð B Ñ Ð. B Ñ×E

-"" " " "sinh sin cosh costanh tanÐ Ñ Ñ Ð#Þ&"Ñ' '(

By this transformation, the solution has a general form which is valid for all values of - ß"

real, zero, or purely imaginary.

- 21 -

When then purely imaginary and the general solution becomes, with? !ß - œ 3 - œ ß" "w

- œ 3 -" "w :

A Ð Ñ" "B œ Ö Ð3 - B Ñ Ð. B Ñ 3 - . Ð3 - B Ñ Ð. B Ñ×E

3 -"w " " "

w w w" " " " " " "sin sin cosh tan( coshtanh Ð Ñ Ñ' '

Ð#Þ&#Ñ

By using the connection between hyperbolic and trigonometric functions, the above canbe written in the same form as when ? !

A Ð Ñ" "B œ Ö Ð- B Ñ Ð. B Ñ - . Ð- B Ñ Ð. B Ñ×E

-"w " " " "

w w w" " " " " "sin sin costan Ð Ñ Ñ Ð#Þ&$Ñ' 'tan( cos

When it is found that and the above solution becomes its limit as? œ !ß - œ !ß"

- Ä !" :

A Ð Ñ E . . ." " " " "B œ ÖB Ð B Ñ Ð B Ñ×" " "sin cos' 'tan( Ñ Ð#Þ&%Ñ

2.4.2 Solution for buckling deflections in no-contact regions

When there is no contact with the foundation, the modulus and the governing5 œ !ß

equation reads

. A R . A

.B A œ ! Ð#Þ&&Ñ

H .B

% ## -< #

#%

## #>w

where .R œ >ßI>

HV-< -< #

5 >w œ

Analogous to the analysis for the contact case, the applicable solution for the no-contactregions have been found. Assuming

A ÐB Ñ œ + /# # #;B# Ð#Þ&'Ñ

where is some constant For a solution of the differential equation, must be a root of+ Þ ;#

; œ !% # wR

H; Ð#Þ&(Ñ

-<>

- 22 -

which gives

; œ Ð „ Ñ"

## w

R

HÐ#Þ&)Ñ

-< È? ,

where

? >w wœ Ð Ð R R %I>

H H HVÑ % œ Ñ Ð#Þ&*Ñ

-< -<# ##

Again three cases of roots are considered. When , the four roots are the following?w !

complex conjugate pairs:

8 œ - 3. 8 œ - 3. 8 œ - 3. ß 8 œ - 3." # # # # # $ # # % # #, , , ; Ð#Þ'!Ñ

where

- œ # . œ # " "

# ## #

w wÊ ÊÈ È> >R R

H HÐ#Þ'"Ñ

-< -< ,

The solution can therefore be written as

A ÐB Ñ œ F Ð- B Ñ Ð. B Ñ F Ð- B Ñ Ð. B Ñ# # # # # # # # # #w w" #cosh cos cosh sin

F Ð- B Ñ Ð. B Ñ F Ð- B Ñ Ð. B Ñw w$ %# # # # # # # #sinh cos sinh sin Ð#Þ'#Ñ

where , , , and are constants depending on the matching conditions with theF F F Fw w w w" # $ %

solution for the contact part. Now, again, if the origin for this solution is taken at thecenter of the buckle, then the solution must be symmetric about . This symmetryB# œ !

condition yields Hence the solution becomesF œ F œ !Þw w# $

A ÐB Ñ œ F Ð- B Ñ Ð. B Ñ F Ð- B Ñ Ð. B Ñ# # # # # # # # # #w w" %cosh cos sinh sin Ð#Þ'$Ñ

At the ends, of this buckle , which then requiresB2 œ „ ß A Ð „ Ñ œ !0 0#

F F . -" " "w%œ Ð Ñ Ð Ñtan tanh0 0 . Ð#Þ'%Ñ

- 23 -

The solution may be written as

A ÐB Ñ œ Ö Ð- B Ñ Ð. B Ñ - . Ð- B Ñ Ð. B Ñ×F

-# # # # # # # # # # # #

#sin sin cosh tan( coshtanh Ð Ñ Ñ Ð#Þ'&Ñ0 0

where is a constant. As explained before, this form of the solution is applicable to allF

three cases of ? ? ?w w w !ß œ !ß !Þand

2 5 Matching conditions between contact and no-contact regionsÞ

In the previous section, solutions were obtained for contact and no-contact regions of thebuckled shell respectively. Since these two kinds of regions belong to the same shell,they must be compatible along their common circles. As shown in Fig. 2-2, commoncircles occur at for the contact region, and at for the no-contact region.B œ B œ " #' 0

At common circles the two separate solutions must have the same deflection, same slope,same bending moment, and same shear force. The equality of deflections, has alreadybeen satisfied by requiring them to be zero at the common circles. The remainingmatching conditions are:

.A .A

.B .Bœ

" " # #ÐB Ñ ÐB ÑÐ#Þ''Ñ

" #B œ B œ¹ ¹

" #' 0

Q œ Q Ð#Þ'(ÑBB BBB œ B œ¹ ¹

" #' 0

U œ U Ð#Þ')ÑB BB œ B œ¹ ¹

" #' 0

For the second condition, i.e., the equality of bending moment, since

Q œ Ð#Þ'*ÑI> . A

"#Ð" Ñ .BBB

$ #

# #/

it is equivalent to

. ÐB Ñ . ÐB Ñ

.B .BÐ#Þ(!Ñ

# #" " # #

" ## #B B

A Aœ Þ¹ ¹

" #œ œ' 0

- 24 -

For the third condition, the expression for the shear force per unit length isUB

U œB

5/

-<

$ $

# $> Ð#Þ("Ñ.A I> . A

.B "#Ð" Ñ .B

In view of the required equality of the slopes, the shear force matching condition at thecommon circles becomes

. ÐB Ñ . ÐB Ñ

.B .BÐ#Þ(#Ñ

$ $" " # #

" #$ $B B

A Aœ¹ ¹

" #œ œ' 0

These three matching conditions are used to determine theÐ#Þ''Ñ Ð#Þ(!Ñ Ð#Þ(#Ñ, , and buckling equations in the next section.

2.6 Equations for buckling load and wave lengths

Adopting the approach proposed by Yang [20], let the first derivatives at the commoncircle of the two regions be expressed as

.A ÐB Ñ .A ÐB Ñ

.B .Bœ E+ œ E+ Ð Ñ œ F+ œ F+ Ð Ñ #Þ($

" " # #

" #B œ B œ"" "" "# "#¹ ¹ a b

" #' 0' 0, and

where is a function of , is a function of . The first matching condition Eq. + +"" "#' 0 Ð#Þ''Ñ

requires that

.A ÐB Ñ .A ÐB Ñ

.B .B œ + E + F œ ! #Þ(%

" " # #

" #B œ B œ"" "#¹ ¹ a b

" #' 0

Similarly, matching conditions may be expressed asÐ#Þ(!Ñ Ð#Þ(#Ñ and

+ E + F œ ! #Þ(&#" ## a b+ E + F œ ! #Þ('$" $# a bwhere

E+ Ð Ñ œ F+ Ð Ñ œ #Þ((. A ÐB Ñ . A ÐB Ñ

.B .B#" ##

# #" " # #

# #" #B œ B œ

' 0¹ ¹ a b" #' 0

and

- 25 -

E+ Ð Ñ œ F+ Ð Ñ œ #Þ(). A ÐB Ñ . A ÐB Ñ

.B .B$" $#

$ $" " # #

$ $" #B œ B œ

' 0¹ ¹ a b" #' 0

and

are evaluated as indicated.

There are thus three equations with two unknown constants and . However, theE F

buckling lengths , and the buckling load , which are in terms of etc., are# # R + ß +' 0 -< "" "#

also unknowns of the problem. For obtaining equations only contain , , and , the' 0 R-<

constants and are eliminated from Eqs. and , and from Eqs. andE F #Þ(% #Þ(& #Þ(&a b a b a ba b#Þ(' . This gives the following two equations:

+ +

+ + œ ! #Þ(*

"" #"

"# ##a b

+ +

+ + œ ! #Þ)!

$" $#

#" ##a b

The third relation obtained from Eqs. and , is a lineara b a b#Þ(% #Þ(' œ !ß+ +

+ +"" "#

$" $#

combination of the previous two, and is therefore an identity. This relation is not used inthe following, but it might be used to check the accuracy of the numerical results. Thetwo equations, which may be called the buckling equations, are expressed as

Eq1sin + sinh sin + sinh

cos cosh cos coshœ œ !

" - Ð#. Ñ . Ð#- Ñ " - Ð#. Ñ . Ð#- Ñ

- . Ð#. Ñ Ð#- Ñ - . Ð#. Ñ Ð#- Ñ" " " " # # # #

" " " " # # # #' ' 0 0

' ' 0 0Ð#Þ)"Ñ

Eq2 sin sinhsin sinh

œ- Ð#. Ñ . Ð#- Ñ

- Ð- $ . Ñ Ð#. Ñ . Ð$- . Ñ Ð#- Ñ" " " "

" " " "" " " "# # # #

' '' '

œ !- Ð#. Ñ . #- Ñ

- Ð- $ . Ñ Ð#. Ñ . Ð$- . Ñ Ð#- Ñ

# # # #

# # # ## # # ## # # #

sin sinh(sin sinh

0 0

0 0Ð#Þ)#Ñ

2. Solutions for buckling load and buckling wave lengths(

The two buckling equations, ( . ) and ( . ), are used to calculate the critical buckling# )" # )#

load and the corresponding buckling wave lengths and for a shell of givenR # # ß-< ' 0

dimensions, and material properties and and a given foundation modulus .H >ß ß I ß 5/

These equations, however, are nonlinear. In these two equations, there are threevariables, namely buckling load , contact buckling length and no-contact bucklingR # ß-< '

- 26 -

length . One cannot find a solution for three unknown variables from two equations. A#0

third condition is needed. This condition is provided by the fact that the critical bucklingload must be a minimum. A trial and error procedure is then used to solve theR-<

buckling equations. One assumes a value of , the contact buckling length, and then#'

finds the other two unknowns from the two buckling equations. Different values of are#'

used for obtaining corresponding and . Among these obtained values of , theR # R-< -<0

minimum one is deemed as the critical buckling load; and the corresponding , the#0

buckling wave length of the no-contact region, is also then obtained. A Mathematica [21]program can be written for doing these calculations.

The buckling load obtained from the trial and error procedure is expected to fallR-<

between the lower bound, i.e. the buckling load value of a hollow shell, and the upperß

bound, i.e., the buckling load value of a shell with rigid core. A nondimensionalparameter , the ratio of buckling load to that of a hollow shell, is introduced!

! œ Ð#Þ)$ÑR

R-<

-<!

where buckling load of a hollow shell. In the same way,R œ-<!

I> ÎV

$Ð" Ñ

#

#É / is the

" #0 '

œ ß œ Ð#Þ)%Ñ6 6! !

where buckling wave length of a hollow shell.6 œ!1

/

ÈÈ V>

"#Ð" Ñ% # is the

A set of numerical results were obtained from the [21] program and areMathematicalisted in Table 2.1 below.

Table 2.1 Buckling parameters for infinitely long shells with different foundation moduli5 œ ! 5 œ "! 5 œ "!! 5 œ "Þ! ‚ "! 5 œ _" "Þ$#! "Þ'#* "Þ''( "Þ''(!Þ& !Þ'*" !Þ)&* !Þ)'' !Þ)''!Þ& !Þ%%" !Þ!"! !Þ!!$ !

'

!"#

- 27 -

As shown in Table 2.1, the buckling load and no-contact wave length increase with theincreasing of foundation modulus , to a limit of and respectively. On the5 "Þ''( !Þ)''

contrary, the contact wave length decreases, to a limit of zero.

The influence of on the buckling load and buckling wave lengths is shown graphically5

in Figs. 2-3, 2-4, and 2-5. These graphs supplement the information of Table 2.1. It isapparent that the buckling parameters are quite sensitive for low values of say5ß

! 5 #&Þ 5 œ #&ß "Þ&#ß For the buckling load parameter is already compared to its!

ultimate value of for Similarly, at the contact length parameter "Þ''( 5 œ _Þ 5 œ #&ß #

is compared to its ultimate value of zero for the rigid core. As well, the no-¸ !Þ!#ß

contact length parameter at is , compared to the ultimate value of 5 œ #& ¸ !Þ)& !Þ)''Þ"

As may be expected, the graphs in these figures are not smooth in the range ofsensitivity, , but they do correctly indicate the trend in the influence of the! 5 #&

foundation modulus on the buckling parameters.

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0 25 50 75 100 125 150 175 200 225k

α

Fig. 2-3 Buckling load parameter !

- 28 -

0.500

0.550

0.600

0.650

0.700

0.750

0.800

0.850

0.900

0.950

1.000

0 25 50 75 100 125 150 175 200 225k

β

Fig. 2-4 No-contact length parameter "

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0.450

0.500

0 25 50 75 100 125 150 175 200 225k

γ

Fig. 2-5 Contact length parameter #

- 29 -

Chapter 3

Plastic Buckling of Infinitely Long Shellswith Elastic Core

In this chapter, plastic buckling behaviour of infinitely long shells on elastic one-wayfoundations is analyzed. The main difference with the preceding chapter is that now thematerial moduli are those belonging to an elastic-plastic, strain-hardening solid. Thereare some types of steel which do not exhibit a yield plateau, stainless steel for instance,but have a rising strain-hardening behaviour. The kinematic assumptions made here areidentical to those in the previous chapter.

3.1 Constitutive relations of the plasticity theories

A brief and elementary introduction of the theory of plasticity is presented for thepurpose of this thesis in the following. For further discussion, reader may consultstandard books on the theory of plasticity, such as by Hill [22] or Chen [23]. The formeris more theoretical, while the latter is more practical.

Upon exceeding a critical stress value, called yield stress, materials begin to undergoplastic, or irreversible, deformation. For multi-axial state of stress, the yield limit isdetermined by a yield criterion. The most often used criterion is that of Tresca or that ofvon Mises. In this thesis, the latter one is adopted, also called the yield criterion. TheN#

von Mises criterion says that the yielding begins when the second invariant of thedeviatoric stress tensor N# reaches a critical value:

N œ Î$ Ð$Þ"Ñ##C5

where is the current yield stress of the material. This criterion is also expressed as5C

5 5/ Cœ Ð$Þ#Ñ

where effective stress.5/ #œ $N œÈ

- 30 -

This criterion is used to predict yielding of materials (generally metals) under anymultiaxial loading condition from the results of a simple uniaxial tensile test. defines aN#

yield surface in stress space. Plastic deformation is possible as long as the stress point ison the current yield surface. This yield surface, however, is not fixed and depends on thetotal plastic strain accumulated during previous history of plastic straining. After&T

initial yielding, a new yield surface is established based on the present state ofdeformation, with stresses and strains . If the state of the stress is changed such that5 %9 9

3434

the stress point moves inward of the new yield surface, there is unloading. The behaviourof the material is then again elastic, and small enough increments of stresses will onlyproduce elastic strains. If, on the other hand, the stress point moves outside the new yieldsurface, there is loading, i.e. there is further plastic deformation. Neutral loading willß

happen if the stress point is still on the yield surface. In this case, no further plasticdeformation takes place but only the elastic ones.

For their simplicity, two approaches have been used in developing the constitutiverelations of plasticity. They are the so called incremental and deformation theoriesN N# #

of plasticity. In addition to using the von Mises criterion, they both incorporate theobserved behaviour of metals that the plastic strain involves no change in volume, inrecognition of the fact that plasticity arises due to slip displacements of metal crystalsover slip planes.

The deformation theory assumes that the state of the stress determines the state of thestain, and is therefore like a nonlinear elasticity theory with stress dependent moduli.Relations between the deviatoric stress components and the plastic as= œ

$34 34 34

555 $

5

well as elastic strain components, , , are postulated as% %:34 34

/

% : % $5

34:

34 3434/ 34 55

œ = ß œ Ð$Þ$Ñ=

#K *O

where is a scalar determined by experiments. The total strain components are the sum:

of the elastic and the plastic parts:

% % %34 34:

34/

œ Ð$Þ%Ñ

The deformation theory cannot adequately describe the observed phenomenon of loading,unloading, and neutral loading. The constitutive relations of deformation theory are only

- 31 -

valid in the case of proportional loading [23]. In fact, for proportional loading, thedeformation theory becomes identical to the incremental theory. However, bifurcationbuckling involves non-proportional loading, and in such a case there is divergencebetween the two theories.

The incremental theory relates the increment of plastic strain to the state of stress, and istherefore load path dependent. Relations between the deviatoric stress components= œ . .34 34 34$

:34

/345 $ % %555 and the plastic as well as elastic strain increments, , , are

postulated as

. œ 1 .N = ß . œ Ð$Þ&Ñ.

*O #K

.=% % $

534:

# 34 34/ 55

3434

where is a hardening parameter related to the position of the stress point on the uniaxial1

stress-strain curve of the material. The total strain increment is assumed to be the sum ofplastic and elastic strain increments:

. œ . . Þ Ð$Þ'Ñ% % %34 34:

34/

The incremental theory is considered a correct theory of plasticity based on theoreticalconsiderations. However, its application is more difficult. Therefore, despite its weakfoundations the deformation theory continues to be used because of its comparativesimplicity. Moreover, for bifurcation problems, the bifurcation loads predicted by theincremental theory are sometimes absurdly higher than the experimental values.Although by performing imperfection growth analyses, the maximum loads computedusing the incremental theory may be made to agree with the experimental loads, therequired growth analysis is time consuming and gives uncertain answers depending uponthe amplitudes of the imperfections. Therefore, the incremental theory of plasticity isusually not recommended. Paradoxically, the bifurcation loads predicted by thedeformation theory for shells are in good and conservative agreement with the testresults, and also invariably lower than the bifurcation loads from the incremental theory[10-12]. Therefore from a practical point of view it is the deformation theory which isused in the present work. There also exists a theoretical justification for the use of thedeformation theory according to Hutchinson [24]. He states [24] that "... for a restrictedrange of deformations, deformation theory coincides with a physically acceptableN#

incremental theory which develops a corner on its yield surface ... most of the results

- 32 -

which have been obtained using deformation theory are (his italics) validN# rigorouslybifurcation predictions ..."

3.2 Constitutive relations for a plastic bifurcation analysis

In axisymmetric bifurcation buckling of axially stressed shells, the state of stress changessuddenly from uniaxial to multiaxial, with increments in stress and strain occurring inother directions due to buckling. Therefore, for a bifurcation analysis, one needsincremental relations, even for the deformation theory. Without further going into details,it can be said that the applicable (plane stress and axisymmetric) constitutive relations forthe deformation theory can be expressed as [10, 25]N#

. œ F . G . . œ G . H . $Þ(5 % % 5 % %BB BB BBw w w w

)) )) )), a bwhere it can be shown that [25]

F œIÐ $/ $Ñ

Ð& $ / % Ñ Ð" # Ñw

#

-

- / /

G œ $Þ)#I Ð " # Ñ

Ð& $ / % Ñ Ð" # Ñw

#

- /

- / /a b

H œ%I

Ð& $ / % Ñ Ð" # Ñw

#

-

- / /

and where in the above

/ œ "ß œ $Þ*I I

I I= >- a b

In the expressions for the two parameters and , is the secant modulus, and/ I œ Î- 5 %=

I œ . Î.> 5 % is the tangent modulus, which are found from the uniaxial stress-straincurve of the material at the stress level equal to the applied axial stress.

The above relations can be used for the incremental theory by putting in the/ œ !

expressions for the moduli. Furthermore, putting 1, and results in the elastic- œ / œ !

relations.

- 33 -

According to Shanley's concept [26], plastic bifurcation occurs under increasing load.Thus, there is no elastic unloading of any fibers of the shell from the state of yield atbifurcation. This is a standard assumption, known as the tangent modulus assumption, inanalyzing plastic bifurcation buckling of columns, plates and shells.

It is convenient to change the notation slightly now. From now on, the increments instrain and stress due to bifurcation buckling will be simply denoted as and% %BBß ))

5 5BBß Þ Ð$Þ(Ñ)) Thus, the constitutive relations will read as

5 % % 5 % %BB BB BBw w w wœ F G œ G H $Þ"!)) )) )), a b

with exactly the same meaning for the moduli as above.

3.3 Governing equations for plastic axisymmetric bifurcation buckling

Governing equations are now derived for the shell buckling in the plastic range. Thekinematic assumptions, unaffected by material behaviour, remain the same as for theelastic case. The bifurcation strains are therefore, as before,

% % ()) œ œ V .B .B

.? .A A, BB

#

#a b$Þ""

The stresses are however determined from the elastic-plastic constitutive relations are

5 % % (BB BB

#

#œ œ

.? . A A

.B .B VF G F F Gw w w w w

))

a b$Þ"#

5 % % ()) ))œ œ .? . A A

.B .B VG H G G Hw w w w w

BB

#

#

The governing equations and possible boundary conditions are derived from the principleof virtual work by following the same procedure as for the elastic case. The virtualdisplacements of the shell are in the longitudinal direction and in the out-of-plane$ $? A

direction respectively. Thereby, the change in the virtual strains are

- 34 -

$% ( $%$ $ $

BB

#

#œ ß œ

.Ð ?Ñ . Ð AÑ A

.B .B V)) a b$Þ"$

The virtual work of the circumferential stress is

( (Z Z

#

#5 $% ( ( )

$)) )) .Z œ Ð Ñ . V. .B

.? . A A A

.B .B V VG G Hw w w

œ # > Ð Ñ A.B.? A

.B V1 $(

!

P

G H $Þ"%w w a band that of the longitudinal stress is

( (Z Z

BB BB

# #

# #5 $% ( ( ( )

$ $.Z œ Ð ÑÐ Ñ . V. .B

.? . A A .Ð ?Ñ . Ð AÑ

.B .B V .B .BF F Gw w w

œ # V> Ð Ñ .B .B.? A .Ð ?Ñ V> . A . Ð AÑ

.B V .B ' .B .B1 (

$ 1 $( (! !

P P$ # #

# #F G Fw w w

œ # V> Ð Ñ ? Ð Ñ ? .B.? A . ? .A

.B V .B V .B1 $ $š’ “ ›(F G F Gw w w w

P

! !

P #

#

A A.BV> . A .Ð AÑ . A . A

' .B .B .B .B

1 $$ $

$ # $ %

# $ %

P P

! ! !

Pš’ “ ’ “ ›(F F F $Þ"&w w w a bThe external forces in the present work are longitudinal compressive force T

T œ # V >1 5-< a b$Þ"'

and the resistant force of the foundation

J œ # V 5 A.B1 (!

P a b$Þ"(

where is the foundation modulus, which in the present case of tensionless foundation is5

allowed to be zero in the no-contact regions. The shortening displacement of the shellunder compression is

- 35 -

J œ Ð Ñ .B" .A

# .B(!

P# a b$Þ")

Thus the virtual work of external forces is

T J A œ # V > .B # V 5 A A.B.Ð AÑ .A

.B .B$? $ 1 5 1 $

$-<

! !

P P( (

œ # V > A # V > A.B # V 5 A A.B.A . A

.B .B1 5 $ 1 5 $ 1 $-< -<

P

! ! !

P P#

#’ “ ( ( a b$Þ"*

Equating the virtual work of internal stresses to that of external forces gives

# V> Ð Ñ ? .B # > Ð Ñ A.B. ? .A .? A

.B V .B .B V1 $ 1 $( (

! !

P P#

#F G G Hw w w w

A.B # V > A.B # V 5 A A.BV> . A . A

' .B .B

1$ 1 5 $ 1 $

$ % #

! ! !

P P P

% #-<( ( (Fw

# V> Ð Ñ ? A.? A V> . A .Ð AÑ V> . A

.B V ' .B .B ' .B1 $ $

1 $ 1’ “ ’ “ ’ “F G F Fw w w wP P P

! ! !

$ # $ $

# $

# V > A œ !.A

.B1 5 $-<

P

!’ “ a b$Þ#!

Since and are arbitrary and independent functions of , their collected coefficients$ $A ? B

must be zero. In the domain, this requires satisfaction of the differential! B Pß

equations:

F $Þ#"Gw

w. ? .A

.B V .B œ !

#

#a b

11 1 5 1

V> . A .? A . A

' .B .B V .B # >Ð Ñ # V > # V5 A œ !

$ % #

% #-<F G H $Þ##w w w a bThe boundary terms provide the conjugate boundary conditions at the ends B œ !ß P

(i) either or is specifiedR œ >Ð Ñ œ !ß ?.? A

.B VBB F G $Þ#$w w a b

- 36 -

(ii) either , or is specifiedQ œ œ !> . A .A

.B .BBB

$ #

#

F

"#$Þ#%

w a b(iii) either or is specifiedU œ > œ !ß A

> . A .A

.B .BB -<

$ $

$

F

"#$Þ#&

w

5 a bThe first equation can be integrated to yield:

R œ >Ð Ñ œ.? A

.B VBB F G $Þ#'w w constant a b

Assuming that end is supported ( specified and the end is allowed toB œ ! ? œ ! Ñ B œ P

move ( is unspecified , the latter condition requires ? Ñ R œ >Ð Ñ œ !Þ.? A

.B VBB F Gw w

Hence the constant is zero, and one can substitute

.? A

.B Vœ

G

F$Þ#(

w

wa b

in the second equation to give the following governing equation on :A

F F H G

F$Þ#)

w w w w

w

> . A . A >

"# .B .B VR Ð 5ÑA œ !

3 % #

% # #-<

# a b

3.4 Plastic bifurcation buckling of hollow shell Ð5 œ !Ñ

As in the elastic case, consider now the plastic buckling of a long hollow circularcylindrical shell with simply supported ends. A suitable buckling shape satisfying theseboundary conditions is

AÐBÑ œ -B

6 sin

1 a b$Þ#*

where is an arbitrary constant, and is the unknown wave length of the ensuing buckles.- 6

The satisfaction of the governing equation requires

F F H G

F$Þ$!

w w w w

w

> >

"# 6 6 V R Ð Ñ œ !ß

3 1 1% #

% # #-<

#

from which a b

R œ Ð Ñ > 6 >

V "# 6-< # # #

# # #F H G F

F$Þ$"

w w w w

w 1

13 a b

- 37 -

The condition for to be a minimum requires which isR œ !ß.R

.6-<

-<

#Ð Ñ # œ !> 6 >

V "# 6# # $

# #F H G F

F$Þ$#

w w w w

w 1

13 a bwhich gives the wave length and the corresponding critical bifurcation buckling load

6 œ V> ß"#Ð

1È Ë% F

F H G Ñ$Þ$$

w#

w w w#a b

R œ œ> >

V $ V $-< -<

# # #Ë ËF H G F H G$Þ$%

w w w w w w

, 5 a bThese results were obtained by Batterman [12]. Further simplification can be obtained bynoting that

ÈF H G $Þ$&#I

Ð& $ / % Ñ Ð" # Ñw w w

# œ# È a b

- / /

which can be used to express more transparently

6 œ V> ßÎ%)

(def) 1È Ë% Ð $/ $Ñ

Ð& $ / % Ñ Ð" # Ñ$Þ$'

-

- / /

#

#a b

5-<(def) , œ# >ÎV

$È I

Ð& $ / % Ñ Ð" # Ñ$Þ$(È a b

- / / #

where it may be recalled that and obtained from a uniaxial- œ IÎI / œ IÎI "> =

compression test. The above formulas are for the deformation theory of strain-N#

hardening plasticity.

For incremental theory, one must put . Then, the formulas for the incremental/ œ ! N#

theory of strain-hardening plasticity, are:

6 œ V> ßÎ%)

(inc) 1È Ë% Ð $Ñ

Ð& % Ñ Ð" # Ñ$Þ$)

-

- / /

#

#a b

- 38 -

5-<(inc) , œ# >ÎV

$È I

Ð& % Ñ Ð" # Ñ$Þ$*È a b

- / / #

The formulas for the elastic range are recovered by putting and . One gets/ œ ! œ "-

6 œ V> ß"

(elas) 1È Ê%"#Ð" Ñ

$Þ%!/#

a b

5-<(elas) œ" >ÎV

$È I

Ð" ÑÞ $Þ%"È a b

/#

3.5 Plastic bifurcation buckling of shell with rigid core (5 œ _Ñ

With rigid core, the shell only has circumferential line contact with the foundation atB œ „ 6Î#, where the deflection and slope of the buckling mode shape are zero.

As for the elastic case, the buckled shape is taken as

AÐBÑ œ #7 8#7 B 8 B #7 B 8 B

6 6 6 6cos cos sin sin

1 1 1 1 a b$Þ%#

in which and are positive odd integers. It satisfies the boundary conditions:7 8

AÐB œ 6Î#Ñ œ AÐB œ 6Î#Ñ œ ! A ÐB œ 6Î#Ñ œ A ÐB œ 6Î#Ñ œ !and .w w a b$Þ%$

Satisfaction of the governing equation for all requires B À

’ š › “#7 Ð%7 8 Ñ Ð%7 $8 Ñ 6 R #7 B 8 B

F 6 61 1

1 1# # # # # ##

-<

w

"#

>3"#

> V

6%

# #

#F H G

F

w w w

w#cos cos

8 Ð%7 8 Ñ Ð"#7 8 Ñ œ !6 R #7 B 8 B

F 6 6’ š › “# # # # # #

#-<

w1 1

1 1"#

>3"#

> V

6%

# #

#F H G

F

w w w

w#sin sin

a b$Þ%%

This then means that

#7 Ð%7 8 Ñ Ð%7 $8 Ñ 6 R 6

1 1# # # # # ## %

-<’ “"# "#

> > Vœ !

F

F H G

F$Þ%&

w

w w w

w3 # #

#

# a b

- 39 -

and

Ð%7 8 Ñ Ð"#7 8 Ñ 6 R 6# # # # # ## %

-<1 1’ “"# "#

> > Vœ !

F

F H G

F$Þ%'

w

w w w

w3 # #

#

# a b

Solving these two equations for R 6-< and , one obtains:.

R œÐ%7 8 Ñ

Ð%7 8 Ñ-<

# #

# #

> Ð

V $

# #Ë F H G Ñ$Þ%(

w w w a b

6 œV>

"#Ð ÑÎ

È ÈÉ%7 8# #

1

% F H G F$Þ%)

w w w w##a b

The critical buckling load is the minimum bifurcation load, which means and7 œ "

8 œ ". Therefore the corresponding critical buckling stress and buckling wave length aregiven as

6 œ V>"#Ð

> Ð

V $È È Ë Ë$ R œ

&

$1

% F F H G Ñ

F H G Ñ$Þ%*

w#

w w w

w w w

#

# #

, .-< a b

These results for the plastic buckling of a cylindrical shell with rigid core are new, aswere those for the elastic case. A comparison this with the hollow case, reveals the factthe rigid core increases the buckling stress by a factor of , and the buckle length by a&Î$

factor of È$.

3.6 Plastic bifurcation buckling of shell with elastic core ( )! 5 _

The analytical procedure for the plastic case follows closely that for the elastic case. Thegoverning equation for plastic buckling in contact region is

F > F H G

F$Þ&!

w w w w

w

> . A . A

"# V.B R Ð 5ÑA œ !

.B

3 % #" "

"% -< "

"# #

# a bor

- 40 -

. A "#R . A

.B A œ !

> .B

% #" -< "

"%

"# "

F$Þ&"

w 3 >: a b

whereR œ >ß œ Ð 5Ñ Þ Ð$Þ&#Ñ> V

-< -< #

#

5 >:"# > F H G

F Fw w

w w w

3

Let , where is an arbitrary constant. Then, satisfaction of theA ÐB Ñ œ + / +" " " "7B"

differential equations requires that be a root of7

7 7 œ !"#R

>% #-<

F$Þ&$

w 3 >: a bSolving the quadratic equation, one obtains

7 œ Ð „ Ñß" "#R

# ># -<

:F

$Þ&%w 3

È? where a b

?:-< -<# #

#

#

œ Ð Ñ % Ð Ñ Ð 5Ñ"#R "#R

> > > VF F F F

%) > F H G$Þ&&

w w w w

w w w

3 3 3>: œ a bAs in the elastic case, can be negative, positive, or zero, and all three possibilities?:

must be considered. Considering first the case , one finds that the four roots are?: !

the following complex conjugate pairs.

7 œ - 3. 7 œ - 3. 7 œ - 3. ß7 œ - 3." " " # " " $ " " % " ", , , ; a b$Þ&'

where , ß - œ # . œ # " "

# #" : " :Ê ÊÈ È> >

"#R "#R

> >-< -<

F F$Þ&(

w w3 3 a bThe solution form is exactly similar to the elastic case, namely

A Ð Ñ œ - . - . -" " " " " " " "B Ö Ð B Ñ Ð B Ñ Ð B Ñ Ð. B Ñ×E

-"" " " "sinh sin cosh costanh tanÐ Ñ Ñ' '(

a b$Þ&)

As remarked earlier this solution form is good for all the three cases of ?:Þ

When there is no contact with the foundation, the modulus and the governing5 œ !ß

equation reads

- 41 -

. A "#R . A

.B A œ !

> .B

% ## -< #

#%

## #

F$Þ&*

w 3 >:w a b

where >:w œ

"#Ð Ñ

V >

F H G

F$Þ'!

w w w

w#

#

# #a b

Analogous to the analysis for the contact cases, one must now find the applicablesolutions. Assuming where is some constant, it is found that for it toA ÐB Ñ œ + / ß +2 # # #

8B#

be a solution of the differential equation, must be a root of8

8 8 œ !% # w:

"#R

>-<

F$Þ'"

w 3 > a bSolving the quadratic equation, one obtains

8 œ Ð „ Ñ"

##

:w

"#R

>-<

F$Þ'#

w 3 É? a b

where ? >: :w wœ Ð % œ Ð

"#R "#R %)Ð Ñ

> >Ñ Ñ

V >

-< -<# ##

# #F F

F H G

F$Þ'$

w w

w w w

w#3 3 a bAgain three cases of roots might arise. However, by using the device introduced earlier,all three cases can be considered by one formula. For , the four roots are the?:

w !

following complex conjugate pairs:

8 œ - 3. 8 œ - 3. 8 œ - 3. ß 8 œ - 3." # # # # # $ # # % # #, , , ; a b$Þ'%

where , - œ # . œ # " "

# ## #: :

w wÊ ÊÉ É> >"#R "#R

> >-< -<

F F$Þ'&

w w3 3 a bThe solution form, exactly of the same form, as the elastic one is

A ÐB Ñ œ Ö Ð- B Ñ Ð. B Ñ - . Ð- B Ñ Ð. B Ñ×F

-# # # # # # # # # # # #

#sin sin cosh tan( coshtanh Ð Ñ Ñ0 0

a b$Þ''

- 42 -

3.7 Equations for buckling load and wave lengths

As for the elastic case, the matching conditions at the common circles of the contact andno-contact regions require equality of the functions to zero, and equality of the first,second, and third derivatives. The two buckling equations are of the same form as for theelastic case, repeated here for convenience

Eq1 ,sin + sinh sin + sinh

cos cosh cos coshœ œ !

" - Ð#. Ñ . Ð#- Ñ " - Ð#. Ñ . Ð#- Ñ

- . Ð#. Ñ Ð#- Ñ - . Ð#. Ñ Ð#- Ñ" " " " # # # #

" " " " # # # #' ' 0 0

' ' 0 0 a b$Þ'(

Eq2 sin sinhsin sinh

œ- Ð#. Ñ . Ð#- Ñ

- Ð- $ . Ñ Ð#. Ñ . Ð$- . Ñ Ð#- Ñ" " " "

" " " "" " " "# # # #

' '' '

œ !Þ $Þ')- Ð#. Ñ . # Ñ

- Ð $ . Ñ Ð#. Ñ . Ð$- . Ñ Ð#- Ñ

# # # #

# # # ## # # ## # # #

sin sinh( cc sin sinh

0 0

0 0a b

Therefore, the method of solving Eqs. and explained previously in ChapterÐ#Þ)"Ñ Ð#Þ)#Ñ

2 can also be applied to the plastic buckling equations. However, in solving theseequations, one must remember that the moduli to be used here are stress dependent (andnot constants as was the case with elastic buckling).

The influence of the foundation modulus on the buckling parameters, in the manner ofFigs. 2-2, 2-3, and 2-4 for the elastic case, was not computed for the present plastic case.Since the moduli in the plastic buckling case are dependent on the particular stress-straincurve of the shell material, such graphs will not have general validity, being relevant onlyfor the chosen shell material. However, the general trend of the influence of thefoundation modulus on the buckling parameters can be expected to be similar to the5

elastic case. In other words, the buckling parameters will be sensitive for low values of5ß 5 and insensitive for high values of .

- 43 -

Chapter 4

Application to Concrete filled Steel Tubular Columns, andVerification with Experiments

This chapter presents application and verification of the theoretical buckling analyses ofthe preceding chapters. The present theory is employed to calculate the ultimate failureloads of concrete filled steel tubular (CFT) columns. The theoretical results are comparedwith three sets of available experimental results of other researchers. Also, theoreticalresults are compared with the values calculated by empirical equations of the currentdesign codes of different countries.

4.1 Application to concrete filled steel tubular columns

As described in Chapter 1, concrete filled steel tubular (CFT) columns have widepractical application in high-rise buildings and bridges, especially in high seismicregions. For a given cross-sectional area, CFT columns have an advantage over steel orconcrete sections alone when used as compression members. They have increased loadbearing capacity and improved ductility. CFT columns are constructed by pouringconcrete into steel tubes after the tubes have been positioned on site. The nature of thisconstruction technique gives CFT columns an additional advantage as the tubes serve asforms during pouring of concrete, thus reducing the construction time and providingmore stability during the construction phase.

Tests to investigate the axial strength of CFT columns have been performed on a varietyof and ratios by a number of researchers. These tests reveal that the typicalHÎ> PÎ>

failure modes of CFT columns are associated with the buckling of the steel tube and thecrushing of the concrete core. Based on the present theory, the strength of the steel tube isidentified with the buckling strength of the steel shell, taking into consideration theunilateral constraint of the in-filled concrete. Accordingly, the ultimate strength or thefailure load of a CFT circular tube is calculated by assuming simultaneous local bucklingof the shell and the crushing of concrete, and is expressed as

R œ R R œ E 0 E œ E Ð" Ñ Ð%Þ"Ñ0 E

EX = -? -< = - -< =

w-

w-

-< =

-5 5

5

- 44 -

in which, , is the buckling load of the steel tube alone, being the criticalR œ E= -< = -<5 5

buckling stress of the shell constrained unilaterally by the in-filled concrete. R œ 0 E- -w-

is the crushing strength of concrete, being the cylinder test strength of the concrete and0 w-

E H- denotes the area of the concrete section. Denoting by the outside diameter of thetube and by its thickness, one has for the steel and concrete areasß >

E œ ÐH #>Ñ E œ ÖH ÐH #>Ñ × œ ¸ Ð%Þ#Ñ% % E H ÐH #>Ñ %>

E ÐH #>Ñ H- =

# # # -

=

#

# #

1 1, ,

Equation shows that the concrete contribution to column strength varies directlyÐ%Þ#Ñ

with ratio. In other words concrete contribution is higher for thinner tubes.HÎ>

The usual reduction of concrete strength by the performance factor of is omitted!Þ)&

since the development of concrete strength is better achieved due to protection againstthe environment, and against splitting of concrete [27]. Therefore, implicitly, in theformula , the effects of concrete confinement on the steel shells as well as the steelÐ%Þ"Ñ

shell confinement of the concrete are both taken into consideration. The foundationmodulus of concrete is generally taken as GPa [28].5 #$

The most crucial step in calculating the ultimate strength of concrete filled steel tubecolumns is to determine the value of . For the purpose of determining , the5 5-< -<

properties of the steel used to fabricate the tube should be ascertained, especially thestress-strain covering both elastic and plastic ranges. This stress-strain curve is generallyacquired from tension tests on coupons cut from the steel tubes used in the experiments.As usual, it is assumed that the compression behaviour of steel is the same as the tensionone (i.e., no Bauschinger effect is present). The curve is then modeled approximately byan analytical expression, in which the strain is expressed as a function of the stress,% % 5œ Ð ÑÞ In the present work, the stress-strain curves are modeled by using the well-accepted Ramberg-Osgood formula [29].

In the experiments of other researchers used below, if the plasticity effects are neglectedand the buckling is considered purely elastic, it is found that the critical stresses are5-<

too high, much beyond the initial yield stress of the material. Hence, it appears that forrealistic predictions, plastic buckling must be considered. Although the theoretical resultscan be obtained for both the incremental and the deformation theories of plasticity,N N# #

it was proved earlier in Chapter 3 that the bifurcation stresses predicted by the N#

- 45 -

incremental theory are always higher compared to the deformation theory results. Now,the bifurcation stresses calculated by the deformation theory are found to be quite closeto the experimental results, in fact with predictions on the safe side. Therefore, for thepractical purposes, only the theoretical results of the deformation theory will be usedN#

for comparison with the experimental loads. This is quite acceptable as the deformationtheory has been widely used in engineering practice.

CFT columns used in reality generally possess a much greater height compared to theirdiameter and thickness. Therefore, the steel tubes of the experiment can be considered asthe "infinitely" long shells to fulfill the conditions of the present theory. Thus, the theorydeveloped for infinitely long shells can be adopted for the practical cases of long shells offinite length.

4.2 Verification with experiments

As mentioned in Chapter 1, there are numerous experiments that have been reported inthe literature on the strength of concrete filled steel tube columns. Both normal strengthand high strength steels have been used in these tests. Because of their comparativelymore careful reporting, the results of (1) Sakino et al. [30], (2) O'Shea and Bridge [31],and (3) Lam and Gardner [32] have been selected for the verification purposes herein.These experiments cover a variety of steel and concrete types, and tube sizes in terms ofthe ratios. HÎ> The Poisson's ratio of steel was not given in the above three references.This ratio is needed in the present work of calculating the buckling stress of the shell.5-<

Therefore the standard value of this ratio was used in the calculation of the/ œ !Þ#*

buckling stress of the steel tube for the first two references [30,31]. The tubes used in thetest of the third reference [32] were made of stainless steel, for which the Poisson's ratiowas taken as in the calculation of the buckling load Recall that, the Poisson's!Þ$!& R Þ-<

ratio for concrete (not needed in the calculations here) is close to in the plastic range!Þ&

of its compression behaviour. The ratios of the steel tubes used in all three sets ofPÎH

experiments were kept low (in the range to 4) in order to ensure local bucklingPÎH œ #

and avoid overall buckling. The lengths of the buckles are quite small compared to thetube length, thus allowing development of several half waves ( 10) over the length.

The theory developed in the present work (assuming infinite tube length) is thereforeconsidered applicable to the columns of the experiments.

- 46 -

4.2.1 Theoretical failure loads versus experimental loads of [30]Sakino et al.

According to the test data reported by [30], the stress-stain curve of the steelSakino et al. used by them was approximated by the author as a Ramberg-Osgood curve by thefollowing equation [29]:

% 55 5

5œ !Þ!!# Ð Ñ ß I œ #"!ß !!! ß œ )&$ Ð%Þ#Ñ

I C

&#CMPa MPa

Not all specimens had the above yield stress. However, it seems they all belonged to thesame batch of steel as the difference in the yield stresses of the specimens is not large.The above stress-strain curve is used for all specimens, by keeping the exponent the samebut changing the yield stress to that for the specimen under consideration. The graphicalrepresentation of this equation is shown in Fig. 4-1.

0

100

200

300

400

500

600

700

800

900

1000

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

ε

σ(M

Pa)

Fig. 4-1 Ramberg-Osgood curve of steel (I œ #"!ß !!! œMPa, 853 MPa) used in the5C

experiments conducted by Sakino et al. [30]

- 47 -

The Ramberg-Osgood representation is now used to determine the secant and tangentmoduli, needed in the buckling equations of Chapter 3, as functions of the axial stress.Using these moduli in the constitutive relations of the deformation theory of plasticity,the critical buckling stress can be determined. Once the steel buckling stress isdetermined, formula can then be used to obtain the theoretically predicted failureÐ%Þ"Ñ

load for the individual specimen of the above mentioned experimental set.

As is evident from Table 4.1, twelve specimens were tested in study [30] with ratiosHÎ>

of and . High strength steel, and normal as well as high strength concrete"'Þ(ß $%Þ$ß &#Þ"

were used. It is found that the results from the present theory are invariably lower thanthe experimental results, being in the range of % to % thus giving safe)*Þ* *'Þ% ß

predictions. On average, the predictions are lower by % with a standard deviation of'Þ(

"Þ& ))Þ$ *%Þ!%. In comparison, the strengths calculated by Eurocode 4 range from % to %of the experimental loads. These loads are lower on the average by % with a standard*Þ!

deviation of %. The strengths calculated by CSA S16 range from % to % of"Þ( ((Þ' "!$Þ)

the experimental loads. CSA S16 strengths are lower on the average by % with a(Þ(

standard deviation of %. Therefore, the strengths calculated by the present theory are(Þ!

in better agreement with the test results than those from the empirical equations ofEurocode 4 [33] and CSA S16 [34]. In other words, the predictions of the present theoryare conservative, but not overly. Note that the variability in the predictions is quiteindependent of the ratio.HÎ>

Table 4.1 Comparison of the present theoretical predictions with experiments [30]RX † Specimen CC8-A-2 5CC8-A-4-1

H PÎH HÎ> 0 R R R R R ÎR R ÎR R Î R

"!) # "'Þ( )&$ #&Þ% )) #"#$ #!&$ ##)% ##(& !Þ*$$ !Þ*!# "Þ!!%

"!* # "'Þ) )&$ %

5 5C -< X I G ? X ? I ? G ?w-

!Þ& )) ##&% #")$ #$'% #%%' !Þ*## !Þ)*$ !Þ*'(

"!) # "'Þ( )&$ %!Þ& )) ##$! #"'! #$$* #%!# !Þ*#) !Þ)** !Þ*(%

"!) # "'Þ( )&$ (( )) #%)* #%"* #%

5CC8-A-4-2 5CC8-A-8 5 (" #("$ !Þ*") !Þ)*# !Þ*""

### # $%Þ$ )%$ #&Þ" ) %()( %''' &"&% %*'% !Þ*'% !Þ*%! "Þ!$)

### # $%Þ$ )%$ %!Þ& ) & "' &"*% &%#% &'$* !Þ*%$ !Þ*

CC8-C-2 70CC8-C-4-1 70 3 #" !Þ*'#

### # $%Þ$ )%$ %!Þ& ) & "' &"*% &%#% &("% !Þ*$! !Þ*!* !Þ*%*

### # $%Þ$ )%$ (( ) '&'* '%%( '!'" ($!$ !Þ)** !Þ))$ !Þ)$!

$$

CC8-C-4-2 70 3CC8-C-8 70CC8-D-2 ( # &#Þ" )#$ #&Þ% ) (*"" (($# )")( )%(& !Þ*$$ !Þ*"# !Þ*''

$$( # &#Þ" )#$ %"Þ" ) *#!& *!#( ))%' *'') !Þ*&# !Þ*$% !Þ*"&

$$( # &#Þ" )#$ %"Þ

49CC8-D-4-1 49CC8-D-4-2 " ) *#!& *!#( ))%' *)$& !Þ*$' !Þ*") !Þ)**

$$( # &#Þ" )#$ )&Þ" ) "#)$% "#'&' "!'*% "$((' !Þ*$# !Þ*"* !Þ(('

49CC8-D-8 49

†Note that the symbols in the above table have the following meaning: diameter of steel tube in mm,H œ> œ œ 0 œ thickness of steel tube in mm, yielding stress of steel in MPa, ultimate strength of5C

w-

concrete cylinder test in MPa, buckling stress of steel shell in MPa, ultimate testing load of5-< ?œ R œspecimens in kN, ultimate load predicted by the present theory in kN, ultimate load predictedR œ R œX I

by Eurocode 4 in kN, ultimate load predicted by CSA S16 in kN.R- œ

- 48 -

4.2.2 Theoretical failure loads versus experimental loads of O'Shea and Bridge [31]

The Ramberg-Osgood curves were derived for the experiments conducted by O'Shea andBridge [31]. These curves for five types of steel used in this series of 15 tests are found as

% 55 5

5œ !Þ!!# Ð Ñ ß I œ #!!ß '!! ß œ $'$ Ð%Þ$Ñ

I C

%(C MPa MPa

% 55 5

5œ !Þ!!# Ð Ñ ß I œ #!( %!! ß œ $!' Ð%Þ%Ñ

I C

%&C, MPa MPa

% 55 5

5œ !Þ!!# Ð Ñ ß I œ #!% (!! ß œ #&' Ð%Þ&Ñ

I C

$)C, MPa MPa

% 55 5

5œ !Þ!!# Ð Ñ ß I œ "(( !!! ß œ #"" Ð%Þ'Ñ

I C

$!C, MPa MPa

% 55 5

5œ !Þ!!# Ð Ñ ß I œ "() %!! ß œ ")' Ð%Þ(Ñ

I CC

25 , MPa MPa

The graphs of these equations are shown in Fig. 4-2Þ

0

50

100

150

200

250

300

350

400

450

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

ε

σ(M

Pa)

Steel Type 1 Steel Type 2Steel Type 3 Steel Type 4Steel Type 5

Fig. 4-2 Ramberg-Osgood curve for steel used in the experiment conductedby O'Shea and Bridge [31]

- 49 -

For the experiments reported in [31], the specimen ranged from to .HÎ> &)Þ& ##"

Therefore, these tubes are thinner than those of the first set. The predicted failure loadsrange from % to % of the experimental ones. For 5 out of the total 15)&Þ! "!#Þ$

specimens, the predictions were slightly on the unsafe side. The average of the differenceof the predicted and the experimental loads is % lower, with and a standard deviation%Þ%

of %. In comparison, the strengths calculated by Eurocode 4 vary from % to&Þ" *"Þ(

"!*Þ*% with unsafe predictions for 10 out of 15 specimens. Thus, clearly the Eurocode 4does not satisfyingly predict the CFT column strength for this set of experiments,especially for specimens with high ratios. It is also worth noting that the unsafeHÎ>

theoretical results for five specimens were with those possessing very-high-strengthconcrete cores. This suggests that for very-high-strength concrete, the confinement effectis probably less, and hence in such cases, the concrete part of the strength might bereduced by introducing a performance factor. The strengths calculated by CSA S16 varyfrom % to % with an average of % of the experimental ones. Only two(*Þ" "!"Þ$ *!Þ#

predictions are slightly higher than the experiments results. Therefore CSA S16 equationsare safe but conservative in this set of experiments compared to theoretical results.

Table 4.2 Comparison of the present theoretical predictions with experiments [31]RX † Specimen S30CS50B 3S20CS50A

H PÎH HÎ> I 0 R R R R R ÎR R ÎR R ÎR

"'& $Þ% &)Þ& #!!'!! $'$ %)Þ$ )' "&#) "()( "')$ "''# !Þ*"* "Þ!(& "Þ!"$

"*! $

5 5C -< X I - ? X ? I ? G ?w-

Þ& *(Þ* #!%(!! #&' %" #') "%#' "&)$ "%$) "'() !Þ)&! !Þ*%$ !Þ)&(

"*! $Þ& "#& #!(%!! $!' %)Þ$ $$( "'$# "('' "&)' "'*& !Þ*'$ "Þ!%# !Þ*$'

"*! $Þ

S16CS50BS12CS50A & "') "()%!! ")' %" ")) "#'# "$#% ""(# "$(( !Þ*"' !Þ*'# !Þ)&"

"*! $Þ& "*) "((!!! #"" %" ##! "#'& "$"" ""') "$&! !Þ*$( !Þ*(" !Þ)'&

"'& $Þ% &)Þ

S10CS50AS30CS80A & #!!'!! $'$ )!Þ# $)' #"'% #$*" ###" ##*& !Þ*%$ "Þ!%# !Þ*')

"*! $Þ& *(Þ* #!%(!! #&' (%Þ( #') #$%$ #%*# ##"$ #&*# !Þ*!% !Þ*'# !Þ)&%

"*! $Þ& "

S20CS80BS16CS80A #& #!(%!! $!' )!Þ# $$( #&!( #'#) #$#& #'!# !Þ*'% "Þ!"! !Þ)*%

"*! $Þ& "') "()%!! ")' )!Þ# ")) #$%( #%"' #!)( ##*& "Þ!#$ "Þ!&$ !Þ*!*

"*! $Þ& #

S12CS80AS10CS80B #" "((!!! #"" (%Þ( ##! #"*$ ##%* "*$* #%&" !Þ)*& !Þ*"( !Þ(*"

"'& $Þ% &)Þ& #!!'!! $'$ "!) $)' #(") #*$( #')* #'($ "Þ!"( "Þ!** "Þ!!'

"*! $Þ& *

S30CS10AS20CS10A (Þ* #!%(!! #&' "!) #') $#%* $$'! #*(( $$'! !Þ*'( "Þ!!! !Þ))'

"*! $Þ& "#& #!(%!! $!' "!) $$( $#(" $$'" #*'( $#'! "Þ!!$ "Þ!$" !Þ*"!

"*! $Þ& "'

S16CS10AS12CS10A ) "()%!! ")' "!) ")) $""( $")& #($$ $!&) "Þ!"* "Þ!%# !Þ)*%

"*! $Þ& ##" "((!!! #"" "!) ##! $"#! $"'& #("( $!(! "Þ!"' "Þ!$" !Þ))&S10CS10A†Note: that the elastic modulus of steel in MPa is variable here. The other symbols have the same meaningas in the previous table.

4.2.3 Theoretical failure loads versus experimental loads of Lam and Gardner [32]

Materials used in experiments [32], Table 4.3, were stainless steel and concrete ofstrengths , and MPa. In this series of experiments, tensile coupons were$! '! "!!

machined from the stainless steel tubes and were tested to determine the basic material

- 50 -

stress-strain curves. A total of six specimens were tested. The Ramberg-Osgood curvesfor the two steels are represented as

% 55 5

5œ !Þ!!# Ð Ñ ß I œ "*" *!! ß œ %"# Ð%Þ)Ñ

I CC

4.25 , MPa MPa

% 55 5

5œ !Þ!!# Ð Ñ ß I œ ")$ '!! ß œ #'' Ð%Þ*Ñ

I CC

6 , MPa MPa

The resulting graphs are shown in Fig. 4-3.

0

100

200

300

400

500

600

700

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

ε

σ(M

Pa)

CHS 104X2CHS 114X2

Fig. 4-3 Ramberg-Osgood curve for steel used in the experiment conductedby Lam and Gardner [32]

It is found that the present theory and existing design guide for carbon steel can also besafely applied to concrete filled stainless steel tubes. he specimen were andT HÎ> "*

&#ßmeaning that the tubes were not very thin. The predicted results from the theory rangefrom % to % of the experimental results. One exact value is obtained for(*Þ& "!!Þ!

specimen CHS104 2 - C30. In comparison, the strengths calculated by Eurocode 4 vary‚

from % to % with no unsafe prediction; and predictions of CSA S16 vary from)#Þ' *)Þ!

((Þ( *$Þ'% to %. Therefore, the present theoretical prediction and empirical equations ofEurocode and CSA S16 can be safely used for stainless steel.

- 51 -

Table 4. Comparison of the present theoretical predictions with experiments [32]$ RX Specimen CHS104 2 - C30 36CHS104 2 - C60

H PÎH HÎ> I 0 R R R R R ÎR R ÎR R ÎR

‚ "!% #Þ* &# "*"*!! %"# $" '*( '** ')& '&% '** "Þ!!! !Þ*)! !Þ*‚

5 5C -< X I G ? X ? I ? G ?w-

"!% #Þ* &# "*"*!! %"# %* '*( )%! )"* ((% *!" !Þ*$$ !Þ*!* !Þ)&*‚ "!% #Þ* &# "*"*!! %"# '& '*( *'' *$' ))" ""$$ !Þ)&$ !Þ)#' !Þ(((‚

CHS104 2 - C100CHS114 6 - C30 ""% #Þ' "* ")$'!! #'' $" $(# "!&$ ""!' "#&% !Þ)$* !Þ*!" !Þ))#

‚ ""% #Þ' "* ")$'!! #'' %* $(# "#!! "#$" "$%! !Þ)*& !Þ*&# !Þ*"*‚

1130CHS114 6 - C60 1276CHS114 2 - C100 1407""% #Þ' "* ")$'!! #'' '& $(# "$$" "$%# "'(% !Þ(*& !Þ)%! !Þ)!#

4.3 The scatter of the three sets of results

The scatter of the present theoretical predictions and those from the Eurocode 4 and CSAS16 empirical equations are shown graphically for each set of experiments in Fig. 4-4, 4-5, and 4-6. The conclusion already discussed and now clearly evident from these figuresis that the present theoretical predictions are almost always below the experimentalvalues but not overly conservative (barring a few which are slightly above theexperimental values).

Fig. 4-4 Scatter for the present theoretical predictions vs. experiments in [30]

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 10 20 30 40 50 60D/t

N/N

u

Present Theory NT/NuEurocode 4 NE/NuCSA S16 Nc/Nu

- 52 -

Fig. 4-5 Scatter for the present theoretical predictions vs. experiments in [31]

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 50 100 150 200 250D/t

N/N

u

Present Theory NT/Nu

Eurocode 4 NE/Nu

CSA S16 Nc/Nu

Fig. 4-6 Scatter for the present theoretical predictions vs. experiments in [32]

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 10 20 30 40 50 60D/t

N/N

u

Present Theory NT/NuEurocode 4 NE/NuCSA S16 Nc/Nu

- 53 -

4.4 Buckling load using simple equation Ð5 œ _Ñ

The foundation modulus of the concrete was assumed to be GPa. This is a high#$

enough value to be considered as infinity. Therefor the simple equation

R œ&

$-<(min)

> Ð

V $

# #Ë F H G Ñw w w

,

for the rigid core, can be used to calculate buckling load or as initial values in solving thebuckling equations in Chapter 3. It was found the buckling load obtained from simpleequation is quite close to that obtained from buckling equations.

- 54 -

Chapter 5

Summary and Conclusions

5.1 Summary

Elastic as well as plastic buckling of cylindrical shells with elastic core under axialcompression has been studied. The shell is considered to be thin, and usual kinematicassumptions are assumed to hold. The shell geometry is considered to be perfect withoutany imperfections. Hence the problem is solved as a bifurcation buckling type. The shellmaterial is considered to be elastic as well as plastic. The elastic behaviour is based onthe standard linear relations for shell structures. The plastic behaviour is modeled as thenonlinear strain-hardening type, employing deformation as well as incrementalN N# #

theories of plasticity. The core is considered to remain elastic. Any bond between theshell wall and the core is neglected. The core is considered as a one-way tension-lessfoundation of the Winkler type, providing only compressive resistance lineallyproportional to the inward displacement of shell, with a constant foundation modulus.The bifurcation problem is solved by exact analysis. Equations have been derived forobtaining buckling loads and wave lengths. Closed form exact solutions are obtained forelastic as well as plastic buckling in the presence of a rigid core. One of the novelties ofthe present work lies in including the plasticity effects, and applying the theoreticalresults to the practical problem of determining failure loads of concrete filled steeltubular (CFT) columns.

Chapter 1 gives some background on the general topic of shell buckling. Although thereis considerable literature on the general subject of this thesis, most of the previous workshave been experimental. Theoretical investigations are limited by the assumption of abonded foundation. The most important work assuming unbonded or tension-lessfoundation is that by Bradford [17]. However, this work deals only with the elastic/> +6Þ

buckling and is based on an assumed approximation of the critical mode shape. Thus, thepresent work is the only one which deals with both elastic and plastic buckling of suchunbonded shells by exact analysis.

Chapter 2 investigates theoretically the elastic buckling of infinitely long shells withelastic core. The theory is developed by employing the method of virtual work, which is a

- 55 -

general method applicable to elastic as well as plastic behaviours. Bifurcation equationsand boundary conditions are derived for the shell constrained by a tension-less core. Thesolutions obtained for the contact and no-contact regions are subjected to the matchingconditions at the common circles of the two regions. These conditions yield two bucklingequations, which in conjunction the condition that the critical load be the minimum,determine the critical loads and the wave lengths of the contact and no-contact regions.Apart from the foregoing general procedure, a closed form solution is derived for the caseof buckling with a rigid core, which is a new and exact result.

Also investigated in Chapter 2, is the influence of the foundation modulus on thebuckling loads and the wave lengths. It is found that the buckling parameters are quitesensitive to the foundation modulus for low values, but quite insensitive for high values.5

They begin to reach their ultimate values for the rigid core quite early on, implying thateven a moderately stiff core (much less than a concrete core) will increase the bucklingload capacity close to that for a rigid core.

Chapter 3 deals with plastic buckling of shells made of stain-hardening plastic materialand containing Winkler type elastic core. The theoretical development essentially followsthat of Chapter 2, except that the constitutive relations of strain-hardening plasticitytheories, incremental and deformation, are employed. The solution procedure isN N# #

similar except that the moduli are now stress-dependent. To the best of the author'sknowledge, the theoretical analysis in this chapter on the plastic bifurcation buckling ofshells with one-way elastic core has not been investigated by other researchers.

In Chapter 4, the theoretical results based on deformation theory of plasticity areN#

compared with experiments on CFT columns from three different literature sources, aswell as with the empirical equations of Eurocode 4 and CSA S16. These experimentswere conducted on tubes of different types of steel and concrete core, and cover a widerange of the ratio of the tubes. It is found that the present theoretical results are inHÎ>

very good agreement with the experimental results, being less conservative but mostly onthe safe side.

- 56 -

5.2 Conclusions

The research presented in this work has wide applications, in civil, mechanical, andaerospace engineering. Listed below are some of the significant conclusions arising fromthis work.

(i) Exact analyses of elastic as well as plastic bifurcation buckling of shells with one-way(tension-less) elastic core and subjected to uniform compressive force are performed. Theanalyses and the associated results are new even for the case of elastic buckling.

(ii) Closed form exact expressions for the buckling load and wave length are obtained forboth elastic and plastic cases, when the core can be assumed rigid.

(iii) The buckling parameters are found to be sensitive only for low values of thefoundation modulus (soft cores). This means that even moderately stiff cores can increasethe buckling load close to the maximum attainable for the rigid core.

(iv) The failure loads of CFT columns, computed simply as the sum of the individualbuckling strength of the steel tube and the crushing strength of the concrete core, wherethe former is determined by the present deformation theory bifurcation load, are foundN#

to be in very good agreement with different test results. The agreement on the whole isbetter than that obtained by using the empirical equations of Eurocode 4 and CSA S16.Thus, in contrast to the empirical equations of the code bodies, the present research offersa rational method, grounded in fundamental engineering principles, for determining thestrength of CFT columns. This research may therefore help the code formulatingauthorities to reassess and possibly modify their present recommendations.

(v) For easy calculations one may use the simple and the exact formula of the N#

deformation theory derived for the case of a rigid core, especially for the CFT columns inview of concrete providing a very stiff core.

(vi) The exact results given or obtainable from the analytical procedure given in thisthesis may be used as bench marks for validating the results of FE models of the similarbuckling problems by using a constructed or a commercial FE code.

- 57 -

5.3 Suggestions for future work

The work presented in this thesis may be rendered more convenient and useful by

(i) Constructing charts and tables for determining the critical loads.

(ii) Suitably modifying the theoretical results to obtain equations for safe and economicaldesign of CFT columns.

- 58 -

References

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[4] Furlong, R. W., , Journal of theStrength of steel-encased concrete beam columnsStructural Division, ASCE, 1976, vol. 93, no. 5, pp.113-24.

[5] Shakir-Khalil, H., Mouli, M., Further tests on concrete-filled rectangular hollow-section columns, Structural Engineer, vol. 68, no. 20, 1990, pp.405-412.

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[9] Koiter, W. T., The effect of axisymmetric imperfections on the buckling of cylindricalshells under axial compression, Proceedings of Konninklijke Nederlandse Akademie vanwetenschappen 66, 1963, pp. 265-279.

[10] Bijlaard, P. P., , Journal ofTheory and Tests on Plastic Stability of Plates and Shellsthe Aeronautical Sciences, vol. 16, 1949, pp.529-541.

[11] Gerard, G., Compressive and Torsional Buckling of Thin-wall Cylinders in YieldRegion, NACA technical note no. 3726, 1965.

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[13] Lee, L. H. N., Inelastic Buckling of Initially Imperfect Cylindrical Shells Subject toAxial Compression, Journal of Aerospace Science, Vol. 29, 1962, pp.87-95.

[14] Seide, P., The stability under axial compression and lateral pressure of circular-cylindrical shells with a soft elastic core, Journal of Aerospace Science, vol. 29, 1962,pp. 851-862.

- 59 -

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[26] Shanley, F. R., , Journal of Aerospace Science, vol.14,Inelastic Column Theory1947, pp. 261-267.

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- 60 -

[29] Ramberg, W. and Osgood, W. R., Description of Stress-Strain Curves By ThreeParameters, NACA, Tech. Note, no.902, 1946, pp.1-13.

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[33] CEN, Eurocode4: Design of Composite Steel and Concrete Structures - Part 1-1:General Rules and Rules for Buildings, EN1994-1-1:2004, European Committee forStandardization, Brussels, Belgium, 2004.

[34] CSA, , CAN/CSA-S16-01,Steel Structures for Buildings (Limit States Design)Canadian Standards Association, Toronto, Ontario, Canada, 2001.