ELASTIC/PLASTIC BUCKLING OF CYLINDRICAL SHELLS WITH ELASTIC CORE UNDER AXIAL COMPRESSION

Transcription

1 ELASTIC/PLASTIC BUCKLING OF CYLINDRICAL SHELLS WITH ELASTIC CORE UNDER AXIAL COMPRESSION y Jin Zhang McGill University Department of Civil Engineering and Applied Mechanics Montreal, Queec, Canada June 2009 A thesis sumitted to the Graduate and Postdoctoral Studies Office in partial fulfillment of the requirements of the degree of Master of Engineering Jin Zhang 2009

2 Astract Elastic as ell as plastic uckling of circular cylindrical shells filled ith a core material is analyzed under axial compressive loading. A practical example of this situation is the uckling of concrete filled steel tuular (CFT) columns used idely in high-rise uildings. The theoretical prolem is modeled as the ifurcation uckling of a perfect "infinitely" long circular cylindrical shell under uniform compression, constrained y a one-ay (tension-less) foundation. An important and useful novelty is that the shell material is alloed to undergo strain-hardening plasticity efore uckling. For simplicity, the core material is assumed to remain elastic. The approach is analytical. The governing equations are solved exactly to otain uckling loads, and avelengths in contact and nocontact regions. The theoretical results, hen applied to CFT columns, are found to e in very good agreement ith the experimental uckling loads of other researchers. - i -

3 Résumé Le flamage 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 flamage de colonnes tuulaires en acier remplis de éton (CFT) qui sont largement utilisées dans les immeules de grande hauteur. Le prolème théorique est modélisé comme le flamage par ifurcation 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 flamement. Pour simplifier, le matériel remplis est supposé rester élastique. L'approche est analytique. Les équations régissants sont résolus exactement à otenir les charges de flamage, et les longueurs d'onde en contact et sans contact régions. Les résultats théoriques, aux applications de CFT colonnes, se trouvent en très on accord avec des charges de flamage d'autres chercheurs. - ii -

4 Acknoledgments The author ould 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 riting of this thesis. He also takes this occasion to express special thanks to his ife, Bei Ni, for supporting him ith love and patience. - iii -

5 Tale of Contents Astract... Þ Resume (French)... Þ Acknoledgments... Þ Tale of Contents... Þ List of Major Symols... Þ List of Figures... Þ List of Tales... Þ i ii iii iv vi viii ix Chapter 1: Introduction Definition of the prolem Literature revie Statement of the ojectives... 9 Chapter 2: Elastic Buckling of Infinitely Long Shells ith Elastic Core Governing equations derived y the method of virtual ork... ÞÞ Elastic ifurcation uckling of hollo shell Ð5 œ!ñ Elastic ifurcation uckling of shell ith rigid core Ð5 œ _Ñ... ÞÞÞÞÞ Elastic ifurcation uckling of shell ith elastic core (! 5 _)... Þ Solution for uckling deflections in contact regions... ÞÞ Solution for uckling deflections in no-contact regions... ÞÞÞ Matching conditions eteen contact and no-contact regions...þþþ Equations for uckling load and ave lengths... ÞÞ Solutions for uckling load and ave lengths ÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞ... ÞÞ 25 Chapter 3: Plastic Buckling of Infinitely Long Shells ith Elastic Core Constitutive relations of the plasticity theories Constitutive relations for a plastic ifurcation analysis Governing equations for plastic axisymmetric ifurcation uckling... ÞÞÞ Plastic ifurcation uckling of hollo shell (5 œ!ñ... Þ 36 - iv -

6 3.5 Plastic ifurcation uckling of shell ith rigid core ( 5œ_Ñ Plastic ifurcation uckling of shell ith elastic core (! 5 _... ) Equations for uckling load and ave lengths ÞÞÞÞÞÞÞÞÞÞÞÞÞ Chapter 4: Application to Concrete filled Steel Tuular Columns, and Verificaion ith Experiments Application to concrete filled steel tuular columns... Þ Verification ith experiments Theoretical failure loads versus experimental loads of Sakino et al. [30] Theoretical failure loads versus experimental loads of O'Shea and Bridge [31]... ÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞ Theoretical failure loads versus experimental loads of Lam and Gardner [32]... ÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞ The scatter of the three sets of results Buckling load using simple equation Ð5 œ _Ñ... ÞÞÞ 53 Chapter 5: Summary and Conclusions... ÞÞ Summary Conclusions Suggestions for future ork... Þ 57 References... Þ 58 - v -

7 List of Major Symols E -ß E = area of concrete core, area of steel shell FßGßH elastic-plastic moduli for plastic ifurcation uckling -ß. " " related to roots of characteristic equation in contact regions -ß. 2 related to roots of characteristic equation in no-contact regions H shell elastic rigidity, also outside diameter of shell /ß - parameters in elastic-plastic moduli, / œ IÎI= "ß - œ IÎI> I Young's modulus of elasticity I > tangent modulus I = secant modulus 0- crushing strength of concrete cylinder test 5 foundation modulus R BB axial force per unit length due to uckling R uckling load of shell per unit length, R -< -< œ 5 -< > R -? ultimate strength of concrete core R I ultimate strength calculated y Eurocode 4 equations R- ultimate strength calculated y CSA S16 equations R X theoretical strength predicted y the present method ultimate strength otained from experiments in literature R? Q BB U B uckling moment resultant per unit length due to 5 BB uckling shear force resultant per unit length due to 5 BD shell middle surface radius V > shell thickness A radial uckling displacement, A œaðbñ? axial uckling displacement,? œ?ðbñ (.A.B AÐBÑßAÐBÑ " uckling deflections in the contact, and no-contact regions % % uckling strains in shell ))ß BB >>, coefficients of differential equations in contact, no-contact regions in elastic range > : ß > : coefficients of differential equations in contact, no-contact regions in plastic range ( thickness coordinate / Poisson's ratio - vi -

8 5 -< uckling stress 5BBß 5)) shell stresses ' ß 0 uckling lengths in contact, no-contact regions - vii -

9 List of Figures Fig. 1-1 Buckling mode of a short shell on one-ay elastic foundation... Þ 3 Fig. 1-2 Buckling mode of an infinitely long shell on one-ay elastic foundation ith regions of contact and no-contact... 3 Fig. 1-3 Concrete filled steel tuular (CFT) column... 4 Fig. 1-4 Typical failure mode of CFT columns [1]... 6 Fig. 1-5 Axisymmetric uckling mode of cross-section... Þ 6 Fig. 2-1 Coordinate system of shell Fig. 2-2 Buckling mode of a strip taken from an infinitely long shell resting on one-ay elastic foundation... Þ 18 Fig. 2-3 Buckling load parameter! Fig. 2-4 No-contact length parameter " Fig. 2-5 Contact length parameter Fig. 4-1 Ramerg-Osgood curve of steel (I œ "!ß!!! MPa, 5 C œ 853 MPa) used in the experiments conducted y Sakino et al. [30]... ÞÞÞÞÞÞÞÞÞÞÞ 46 Fig. 4-2 Ramerg-Osgood curve for steel used in the experiment conducted y O'Shea and Bridge [31]... ÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞ 48 Fig. 4-3 Ramerg-Osgood curve for steel used in the experiment conducted y Lam and Gardner [32]... ÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞ 50 Fig. 4-4 Scatter for the present theoretical predictions vs. experiments in [30] Fig. 4-5 Scatter for the present theoretical predictions vs. experiments in [31] Fig. 4-6 Scatter for the present theoretical predictions vs. experiments in [32] viii -

10 List of Tales Tale 2.1 Buckling parameters for infinitely long shells ith different foundation moduli Tale 4.1 Comparison of the present theoretical predictions R X ith experiments [30] 47 Tale 4.2 Comparison of the present theoretical predictions R X ith experiments [31] 49 Tale 4.3 Comparison of the present theoretical predictions R X ith experiments [32] 51 - ix -

11 Chapter 1 Introduction 1.1 Definition of the prolem Since late 1960s, uckling of plates and shells ith surrounding solid or fluid constraints has driven interest of many researchers. As a result, a numer of researches have een conducted ith the intention of clarifying the mechanism of such uckling prolems. Generally, these prolems are treated as staility of plates or shells resting on elastic foundations and sujected to compressive loads. The uckling loads and uckling modes are usually derived under the assumption that the plates or shells are onded to the elastic foundation, and the latter is ale to provide oth compressive and tensile reactions. The uckling analysis is therefore simplified. This kind of prolem is termed as uckling of a structure against a to-ay foundation. Many studies have een carried out ased on this assumption, despite the fact that this assumption may not e a realistic one. In many situations, the ond is not positively ensured during construction, and therefore ought to e ignored for a safe design. In contrast to this to-ay assumption, there are prolems in hich the plates and shells are not onded to the foundation ut simply rest on it. In such (non-onded) cases the foundation can only provide resistance hen compressed. These prolems are termed as one-ay uckling prolems, hich can e illustrated y several examples. One example is the phenomenon of "lift-off" of elded railay tracks due to their constrained thermal expansion. In this prolem the foundation is unale to restrain the upard uckling of the railay track, and therefore no tensile reaction forces are developed. Other examples are surface delamination in composites, and temperature induced "pop-up" of a pipeline in the vertical plane. This category of uckling prolems (i.e., one-ay uckling) is more difficult to analyze. The present ork falls into this latter category. Prolems regarding elastic uckling of shells have een ell studied. On the other hand, there are very fe studies on the plastic uckling prolems. Since plastic uckling is a more desirale failure mode (for harnessing full strength of a metal column) than the elastic uckling, the former type of uckling is very important from a design point of vie. To the est of the author's knoledge, hoever, there are no solutions (at least - 1 -

12 analytical ones) availale on the plastic uckling of shells constrained y one-ay (tension-less) elastic foundation. The present thesis carries out exact analyses for oth elastic and plastic ranges of uckling ehaviour. The theoretical prolem is that of ifurcation uckling of a thin infinitely long perfectly circular cylindrical shell (i.e., a tue) resting on a tension-less elastic foundation (i.e., the core) and sujected to a uniform compressive stress in the longitudinal direction. When inard movement of the shell all is restricted y the compressive reaction forces provided y the core, the uckling mode can e of one sign (outard ith a single ave, Fig. 1-1) or of multi-signs having multiple aves, Fig The former mode is possile for short shells, not considered here. The latter mode is possile for long shells, in hich regions of contact and no-contact in the uckling mode ould emerge simultaneously under certain compressive forces. In the no-contact regions, the shell all uckles aay from the core, and no reaction forces exist eteen the shell all and the foundation core. Conversely, in the contact regions, the inard uckling displacements of the shell all induce reaction forces of the foundation. The analysis required to find the solution for such uckling prolem is quite difficult ecause, in a sense, a nonlinear prolem is to e solved. The analysis is further complicated y virtue of the fact that strain-hardening plastic ehaviour of the shell material is alloed to occur. The results from the present analysis are valid for oth elastic and plastic ranges of shell uckling ehaviour. For the latter case, usually treated in an ad hoc fashion, the present analysis furnishes a rational asis for computing the column capacity against uckling in plastic range

13 Ncr Ncr Short Cylindrical Shell Shell Surface One-ay Foundation Fig. 1-1 Buckling mode of a short shell on one-ay elastic foundation N cr N cr One-ay Foundation Wavelength 2ξ+2ζ Core Shell N cr N cr Fig. 1-2 Buckling mode of an infinitely long shell on one-ay elastic foundation ith regions of contact and no-contact - 3 -

14 Design of concrete filled steel tuular (CFT) columns is the most relevant practical application of the present research on the aove stated one-ay uckling prolem. CFT columns, Fig. 1-3, have ecome an attractive solution in civil engineering practice in recent decades. CFT has excellent structural performance characteristics. Steel memers have the advantages of high tensile strength and ductility, hile concrete memers have the advantages of high compressive strength and stiffness. CFT as a composite memer, comining steel and concrete, has the eneficial qualities of oth of these materials [1]. In addition to the structural enefits, CFT memers have loer cost and loer construction time compared to reinforced concrete columns or purely steel columns. As a result of these distinct advantages, CFT columns are used extensively in earthquakeresistant structures, high-rise uildings, and ridges. Hoever, the use of CFT columns is still limited due to a lack of understanding aout their true strength, especially their uckling strength. Concrete R Steel t Fig. 1-3 Concrete filled steel tuular (CFT) column - 4 -

15 In the early stages of loading, of a CFT column, Poisson's ratio for concrete is!þ"& to!þ hich is loer than that for steel eing around!þ$þ Therefore, in the early stage, the steel tue expands more than the concrete core, and one might say that a gap might develop eteen the steel tue and the concrete core. Hoever, as the longitudinal strain increases ith the loading, Poisson's ratio of concrete egins to increase, and approaches close to!þ& in the fully plastic state [2]. In other ords, the lateral expansion of the concrete core gradually ecomes greater than that of the steel tue, hich overcomes any gap and provides a snug fit ith the steel tue. Hence, effectively, the tue develops tensile hoop stresses due to the radial pressure developed at the steel-concrete interface. For uckling analysis, this possily small pre-uckling hoop tension in the tue is neglected. Bond eteen concrete core and steel tue as studied y Virdi and Doling [3]. They stated that the interface ond exists as a result of the interlocking of the concrete core due to to types of imperfections, namely, the surface roughness of the steel tue, and the variation in its cross-sectional shape along the length. Nevertheless, Furlong [4] found that practically no ond exists eteen concrete core and steel tue. On the other hand, Shakir-Khalil and Mouli [5] reported, from their experiments, a ond strength of!þ$* to!þ&" N/mm. Thus, it is clear that the existence of ond cannot e relied upon for analyzing structural strength. Therefore, in the present uckling analysis, no interface ond is assumed to exist. Based on the discussion aove, the uckling of the steel tue of CFT columns can e classified in the category of one-ay uckling prolem. Typical local uckling of the steel memer is shon in Fig. 1-4 and Fig

16 Fig. 1-4 Typical failure mode of CFT columns [1] Concrete Core Buckling Displacement Steel tue Fig. 1-5 Axisymmetric uckling mode of cross-section - 6 -

17 1.2 Literature revie As mentioned efore, although there are many studies on elastic uckling of shells, there are only a fe dealing ith plastic uckling, and none on plastic uckling of shells on tension-less foundations. Euler's investigation [6] of the elastic staility of slender columns is considered as the origin of the classical theory of elastic uckling of structures. Buckling of a cylindrical shell under uniform compression is a very important prolem ith numerous applications. The ork of the various researchers since the time of Euler can e found in the celerated ook y Timoshenko and Gere [7]. For a long hollo shell, assuming the uckling to e axisymmetric, the critical (i.e., the loest) ifurcation uckling stress 5 -< is given y Timoshenko and Gere [7] as the classical expression 5 -< I > œ Ð Ñ. Ð"Þ"Ñ È$Ð" / Ñ V But experiments since 1930s revealed that experimental uckling loads ere invarialy much loer (and ith a ide scatter) than the aove prediction of the classical theory. The reason for the ide discrepancy as explained y von Karman and Tsien [8]. They shoed that the post-ifurcation path for the cylindrical shell falls steeply to a load much loer than the ifurcation load. Koiter [9] shoed that this post-ifurcation minimum load may e calculated y performing a nonlinear imperfection groth analysis of small initial geometric imperfections in the shell. The concept of "imperfection-sensitivity" has een ell accepted in estimating the uckling strength of hollo shells under compressive loading. The plastic uckling of hollo shells as addressed y Bijlaard [10], Gerard [11] and Batterman [12]. They came to essentially the same conclusion that the load predicted y a deformation theory of plasticity agrees ell ith the experiments. On the other hand, results of Lee [13] shoed that an incremental theory dramatically overestimates the uckling strength. Seide [14] studied the elastic uckling of a shell ith a soft elastic core, y solving differential equations for equilirium of the shell. He noted that for a filled shell the axisymmetric uckling mode gives the loest uckling load. Karam and Gison [15] - 7 -

18 investigated the same prolem, y assuming a sinusoidal radial displacement during uckling and the core as an elastic half-space. They concluded that shells ith a compliant core result in reduced sensitivity to imperfections. Ghoranpour Arani /> +6Þ [16] derived staility equations for a simply supported cylindrical shell ith an elastic core y the energy method. The elastic core as represented y elastic springs ith modulus 5 /. To-ay foundation assumption as used in all of these research orks [14, 15, 16]. Bradford /> +6Þ [17] provided a theoretical study of the local and post-local uckling of thin-alled circular steel tues that contain an elastic core. The prolem as treated as elastic uckling of shells on one-ay foundation. Energy method and Ritz approach ere applied. They found that the elastic local uckling stress for a circular tue ith rigid core is È $ (or "Þ($ ) times that of its unfilled (hollo) counterpart. Hoever, since late 1990s, the interest of researchers in shell analyses has een oriented toard experimental and computer solutions. Literature revie y O'Shea and Bridge [18] indicates that the concrete filled steel tues have received extensive attention y a numer of authors. A great numer of experiments, ith ide range of HÎ> and PÎH ratios, have een conducted. Revie of experimental orks can also e found in the state of the art report y Shanmugam and Lakshmi [19]. By comparison, only limited attention has een paid to derive analytical solutions for the uckling load. As mentioned efore, at the present time, there are only a fe researches dealing ith shell uckling against one-ay foundation, and none in the plastic range

19 1.3 Statement of the ojectives In consideration of the foregoing discussion, the ojectives for the present investigation ere defined to e: (i) Derivation of the exact solutions for the elastic ifurcation uckling of infinitely long shells supported y tension-less elastic foundations, and sujected to uniform compressive stress in the longitudinal direction. (ii) Derivation of the exact solutions for the plastic ifurcation uckling of the same type of shells stated aove in (i) and shon in Fig. 1-2, y employing the constitutive relations of the N deformation and N incremental theories of plasticityþ (iii) Application of the analytical results to concrete filled tuular steel (CFT) columns, and their validation y comparison ith the experimental results of other researchers and also ith the loads calculated y using design codes

20 Chapter 2 Elastic Buckling of Infinitely Long Shells ith Elastic Core The analysis of uckling of infinitely long circular cylindrical shells containing an elastic core in this thesis is ased on the folloing assumptions: (i) The shell is considered geometrically perfect. For analysis in this chapter, it is considered to e made of an isotropic elastic material. It is assumed to e thin ith respect to length and diameter. Its self eight is considered negligile. It is loaded y uniformly compressive stress in the longitudinal direction only. The oundary conditions are assumed to allo the shell to remain perfectly circular under sloly increasing magnitude of this stress, until a critical value is reached, at hich point the shell uckles ith avy displacements. Thus, the uckling considered is of the ifurcation type. (ii) The shell is filled ith concrete or a foam like material, forming a core. It is presumed that no effort is made to provide a ond eteen the core and the surrounding shell all. Thus, the core, is assumed incapale of providing any tensile or frictional resistance to the shell all; it only provides compressive resistance against the inard movement of the shell all. The core support is therefore termed as a tension-less foundation. The uckling of the shell under such a constraint is called one-ay uckling. (iii) The core is represented y a Winkler type foundation, meaning that the compressive reaction forces are linearly proportional to the radially inard deflections. 2.1 Governing equations derived y the method of virtual ork Figure 2-1 shos the coordinate system ith respect to the shell geometry. The preifurcation state of stress is purely uniaxial compression. Let 5 -< e the compressive stress at hich ifurcation takes place, and let the reference geometric configuration e that just efore the ifurcation. Then, the displacements considered elo are those arising due to ifurcation alone. The corresponding strains and stresses are therefore incremental ones, eing additional to those just prior to ifurcation

21 η θ Fig. 2-1 Coordinate system of shell The critical ifurcation (corresponding to the minimum uckling stress), is considered to e axisymmetric, as demonstrated y Batterman [12] for hollo cylindrical shells. Accordingly, the ifurcation displacements, strains, and stresses are all independent of _ the circumferential coordinate. The circumferential is taken to e identically zero, and hence the non-zero displacements are axial? and radial A(positive _ outard). As usual, for thin shells considered here, the radial displacement A is considered independent of the thickness coordinate, and thus it is a function only of the axial coordinate: A _ œ AÐBÑÞ ÐÞ"Ñ Invoking the Love-Kirchhoff kinematic hypothesis, according to hich normals to the middle surface alays remain normal (efore as ell as after deformation), the axial displacement? _ is taken as

22 _.A? œ?ðbñ (. ÐÞÑ.B here ( is the thickness coordinate, and?ðbñ is the axial displacement at the middle surface, ( œ!þ( like AÐBÑis positive in radially outard direction. The non-zero strains arising due to ifurcation, in compliance ith the aove assumptions are %)) œ A, % œ.? (. A BB. ÐÞ$Ñ V.B.B For plane stress ehaviour (appropriate for thin shells), the linear elastic stress-strain relations are I I 5 BB œ Ð% BB /% Ñ 5 œ Ð/% BB % Ñ ÐÞ%Ñ / )), )) " " / )) here I is Young's modulus and / is Poisson's ratio of the shell material. Sustituting the expression for strains in terms of displacements, one otains uckling stresses in terms of uckling displacements as I.?. A A 5 BB œ Ð ( / Ñ " /.B.B V 5 )) œ I / /( / Ð.?. A A ".B.B V Ñ. ÐÞ&Ñ The virtual displacements of the shell are $? in the longitudinal direction and $ Ain the out-of-plane direction respectively. Therey, the virtual strains are.ð $?Ñ. Ð$ AÑ $ A $% BB œ ( ß $% œ ÐÞ'Ñ.B.B )). V The internal virtual ork expression can e ritten as the sum of to parts: I.?. A A $ A ( 5 )) $% )).Z œ ( Ð/ /( Ñ.B V. ).( Ð" / Ñ.B.B V V Z Z

23 1I>.? A œ Ð/ Ñ$ A.B ÐÞ(Ñ Ð" / ( Ñ.B V! P in the circumferential direction; and the second part, related to the longitudinal stress À ( 5 BB$% BB.Z œ ( Ð.Ð $?Ñ. Ð ( $ AÑ ( / ) ( Ñ I / Ð.?. A A.B.B Ð" Ñ.B.B V Ñ.B V.. Z Z 1 VI>.Ð $?Ñ.? /.A œ Ð Ñ $?.B Ð" / ( Ñ.B.B V.B! P $ P % 1VI>. A 1VI>.? A A.B Ð Ñ? "Ð" / ( Ñ.B $ % Ð" / Ñ.B / V $! $ $ $ 1 VI>. A.Ð$ AÑ P 1VI>. A P $ A. ÐÞ)Ñ "Ð" / Ñ.B.B! "Ð" / Ñ.B$! P! The external forces acting on the shell are (i) the longitudinal compressive force T œ 1V 5 -< > ÐÞ*Ñ and (ii) the resisting force from the foundation J œ 1 V( 5 A.B ÐÞ"!Ñ! P here 5 is the foundation modulus, hich in the present case of tension-less foundation is required to e zero in the no-contact region. No, the axial shortening of the shell due to uckling is P.A ".A J œ ( š ÊÐ.BÑ Ð.BÑ.B ( Ð Ñ.B. ÐÞ""Ñ.B.B!! P Thus the virtual ork of external forces is P T J A œ V >.Ð $ $? $ 1 5 AÑ.A -< (.B 15V( A $ A.B.B.B!! P

24 œ V Ð >.A 5 AÑ A.B V >.A P 1 ( 5 -< $ 1 5 -< $ A ÐÞ"Ñ.B.B.!! P The principle of virtual ork requires equality of the virtual ork of the internal stresses to the virtual ork of the external forces for aritrary virtual displacements $? and $ A: 1VI>. A 1I>.? A. A ( š Ð/ Ñ 1V 5 % -< > 1 V5A $ A.B 'Ð" / Ñ.B Ð" / Ñ.B V.B! P $ % 1 VI>.? /.A 1VI>.? A $?.B Ð / Ñ $? Ð" / ( š Ñ.B V.B Ð" / Ñ.B V! P P $ $ $ 1 VI>. A.Ð$ AÑ P.A 1VI>. A V > A œ! ÐÞ"$Ñ "Ð" / Ñ.B.B 1 5.B "Ð" / Ñ.B$ $ P š -<.!!! For the aove to e true, for aritrary $? and $ A in the domain, satisfaction of the folloing to differential equations of equilirium is necessary: I>.? /.A Ð Ñ œ! ÐÞ"%Ñ Ð" / Ñ.B V.B $ % VI>. A I>.? A. A "Ð" Ñ.B Ð" Ñ Ð /.B V Ñ V 5 % -< / / >.B V5A œ!. ÐÞ"&Ñ and also the vanishing of each of the oundary terms: $ I>.? A P I>. A.Ð$ AÑ P Ð / Ñ $? œ!ß œ!ß Ð" / Ñ.B V! "Ð" / Ñ.B.B! $ $ š.a I>. A 5 -< > A œ! ÐÞ"'Ñ.B "Ð" Ñ.B $ / $ P.! The interpretation of the vanishing of the oundary terms at the ends Bœ!, Pis I>.? A (i) either RBB œ Ð / Ñ œ!ß or? is prescried ÐÞ"(Ñ Ð" / Ñ.B V

25 $ I>. A.A (ii) either QBB œ œ!ß or is prescried ÐÞ")Ñ "Ð" Ñ.B.B / $ $.A I>. A (iii) either UB œ 5-< > œ!ß or A is prescried. ÐÞ"*Ñ.B "Ð" / Ñ.B$.A For simply supported shells, considered here, Aœ œ! at the ends, and?œ! at.b.? A Bœ!, the supported end, and / œ! at BœP, the loaded end..b V The integration of the first differential equation gives I>.? A Ð / Ñ œ - ÐÞ!Ñ Ð" / Ñ.B V here - is a constant. Hoever, in vie of the oundary condition, at the loaded end here? is not prescried, the left hand side is required to e zero. Hence - œ!þthus.? / A œ! ÐÞ"Ñ.B V in the domain as ell. Sustitution of this result in the second differential equation gives $ % I>.A.A A "Ð" Ñ.B 5 % -< / >.B I> V 5A œ!. ÐÞÑ Introducing the standard notations $ I> H œ R œ > ÐÞ$Ñ "Ð" /, 5 -< -< Ñ the equation governing axisymmetric ifurcation uckling of the shell is: %.A.A I> H R A 5A œ! ÐÞ%Ñ.B% -<..B V

26 Let AÐBÑ " and AÐBÑ represent the uckling deflections in the contact regions Ð5Á!Ñ " and no-contact regions Ð5 œ!ñ respectively. Then the applicale differential equations are %.A".A" I> H R A 5A œ! ÐÞ&Ñ.B% -<.B " " V " in contact regions, and % ".A.A I> H R A œ! ÐÞ'Ñ.B% -<.B V in no-contact regions. 2.2 Elastic ifurcation uckling of hollo shell Ð5 œ!ñ When no core is present, one has the case of classical shell uckling analysis. Assuming the shell to e simply supported, a mode shape satisfying the oundary conditions is: B A" œ - sin 1 6 ÐÞ(Ñ here -is an aritrary constant, and 6is the unknon uckling ave length. Satisfaction of the governing differential equation requires % 1 1 I> H R œ! ÐÞ)Ñ 6% -< 6 V hich gives 1 H 6 I> R-< œ Ð Ñ. ÐÞ*Ñ 6 1 V.R-< For a minimum critical load, 6should e such that œ!þthis condition requires.6 1 I> 1ÈV> H 6 œ!ß or 6 œ. ÐÞ$!Ñ 6$ 1 V È % "Ð" / Ñ

27 Accordingly, the minimum critical load and stress are I> ÎV I>ÎV R -< œ, 5 œ ÐÞ$"Ñ $Ð" Ñ -<. È / È$Ð" / Ñ The aove expression for the critical uckling load and uckling length are identical to those given y Timoshenko and Gere [7]. 2.3 Elastic ifurcation uckling of shell ith rigid core Ð5 œ _Ñ With rigid core, the shell, in axisymmetric uckling, has only a line contact ith the foundation, around the circumference at Bœ 6Î, here the deflection and slope of the longitudinal mode shape are zero. The uckled shape defined y (omitting the aritrary multiplicative constant) AÐBÑ œ B 8 1 B B 8 1 cos cos sin sin B ÐÞ$Ñ in hich 7and 8 are positive odd integers, satisfies the oundary conditions that AÐB œ 6ÎÑ œ AÐB œ 6ÎÑ œ! and A ÐB œ 6ÎÑ œ A ÐB œ 6ÎÑ œ!. ÐÞ$$Ñ The function AÐBÑ is periodic and the uckling mode is symmetric aout B œ!þ Sustituting the assumed mode shape into the differential equation gives, for all B: % 6R-< 6 I> 71B 81B 7 š Ð%7 8 Ñ 1 Ð%7 $8 Ñ 1 H HV cos cos % 6R-< 6 I> 71B 81B 8š Ð%7 8 Ñ1 Ð"7 8 Ñ1 œ! H HV sin sin 6 6 ÐÞ$%Ñ This requires: % 6R-< 6I> Ð%7 8 Ñ 1 Ð%7 $8 Ñ 1 H HV œ! %, 6R-< 6I> Ð%7 8 Ñ1 Ð"7 8 Ñ1 H HV œ!. ÐÞ$&Ñ

28 Solving these to equations for R-< and 6ß one otains R -< œ Ð%7 8 Ñ Ð%7 8 Ñ 6œÈ%7 8 1 I> V È $Ð" / Ñ ß ÈV> È% "Ð" / Ñ ÐÞ$'Ñ The uckling load must e a minimum at the ifurcation, hich means 7œ" and 8œ". This gives the critical uckling load and ave length as & I> V> R-<(min) œ $ $ V $Ð" Ñ 6 œ È 1È,. È / "Ð" Ñ ÐÞ$(Ñ È% / We oserve that, for a shell ith rigid core Ð5 œ _Ñ, the uckling load is increased y &Î$ times that of a hollo shell Ð5 œ!ñ. This ne and exact result is loer than the approximate one derived y Bradford /> +6Þ [17] hich gives a factor of È $. The corresponding uckling length here is È $ times that for a hollo shell Þ 2.4 Elastic ifurcation uckling of shell ith elastic core (! 5 _) When the foundation modulus is eteen zero and infinity, there are regions of contact and no-contact of the shell ith the foundation. Buckles and coordinate system are shon in Fig ζ ζ -ξ ξ Infinitely Long Shell N cr N cr One-ay Foundation Fig. 2-2 Buckling mode of a strip taken from an infinitely long shell resting on one-ay elastic foundation

29 2.4.1 Solution for uckling deflections in contact regions In a contact region, the applicale differential equation is %.A".A" I> H R Ð 5ÑA œ! ÐÞ$)Ñ.B% -<.B ", or V " % ".A" R.A.B -< " % H.B > A " œ! ÐÞ$*Ñ " " here I> 5 R-< œ 5 -< >, and > œ. ÐÞ%!Ñ HV H To solve this ordinary differential equation ith constant coefficients, let A ÐB Ñ œ + / " " " :B " ÐÞ%"Ñ here + " is an aritrary constant. Then, satisfaction of the differential equations requires : to e a root of % R-< : : > œ! ÐÞ%Ñ H Solving the quadratic equation, one otains " R-< : œ ( È? Ñ ÐÞ%$Ñ H here R-< R-< I> 5? œ Ð Ñ % > œ Ð Ñ %Ð Ñ ÐÞ%%Ñ H H HV H? can e negative, positive, or zero. One must consider all these three possiilities. In the first case of?!, the four roots are the folloing complex conjugate pairs:

30 :" œ -" 3. ", : œ -" 3. ", : $ œ -" 3. ", ß: % œ -" 3. " ÐÞ%&Ñ here " " -" œ Ê È R." œ Ê È R > and, > H H -< -< ÐÞ%'Ñ The solution can therefore e ritten as AÐBÑœE " " " coshð-bñ " " cosð.bñ E " " coshð-bñ " " sinð.bñ " " E sinhð- B Ñ cosð. B Ñ E sinhð- B Ñ sinð. B Ñ ÐÞ%(Ñ $ " " " " % " " " " in hich E", E, E$, and E% are constants depending on the matching conditions ith the solution for the no-contact part. Taking the origin for this solution at the center of the uckleß then the solution must e symmetric aout B œ!. This symmetry condition $ yields E œ E œ!þ Hence the solution ecomes " % A" ÐB" Ñ œ E coshð-" B" ÑcosÐ. " B" Ñ E sinhð-" B" ÑsinÐ. " B" Ñ ÐÞ%)Ñ At the ends, œ ' of this uckle ß A Ð ' Ñ œ!, hich then requires B " " " " % " " E œ E tanð. ' ÑtanhÐ- ' Ñ. ÐÞ%*Ñ Then, finally the solution of roots for this case (?!) is expressed as " " % " " " " " " " " " " AÐBÑœE ÖsinhÐ- BÑsinÐ. BÑ tanhð- ' Ñtan(.' ÑcoshÐ- BÑcosÐ.BÑ ÐÞ&!Ñ % % " Since E is aritrary, it is permissile to replace it as E œ E - and rite E AÐBÑœ ÖsinhÐ- BÑsinÐ. BÑ tanhð- ' Ñtan(.' ÑcoshÐ- BÑcosÐ.BÑ ÐÞ&"Ñ " " " " " " " " " " " " - " By this transformation, the solution has a general form hich is valid for all values of - ß " real, zero, or purely imaginary

31 When?!ßthen -" œ3-" œpurely imaginaryß and the general solution ecomes, ith -" œ 3-" : E AÐB Ñœ ÖsinhÐ3 - B Ñ sinð. B Ñ tanh Ð3 - ' Ñtan(. ' Ñ coshð3 - B Ñ cosð. B Ñ " " 3- " " " " " " " " " " " ÐÞ&Ñ By using the connection eteen hyperolic and trigonometric functions, the aove can e ritten in the same form as hen?! E AÐB Ñœ ÖsinÐ- B Ñ sinð. B Ñ tan Ð- ' Ñ tan(. ' Ñ cosð- B Ñ cosð. B Ñ ÐÞ&$Ñ " " - " " " " " " " " " " " When? œ!ß it is found that -" œ!ßand the aove solution ecomes its limit as -" Ä!: AÐB Ñœ E ÖB sinð. B Ñ ' tan(.' Ñ cosð. B Ñ ÐÞ&%Ñ " " " " " " " " Solution for uckling deflections in no-contact regions When there is no contact ith the foundation, the modulus equation reads 5œ!ß and the governing %.A R.A.B -< % H.B > A œ! ÐÞ&&Ñ I> here R-< œ 5 -< >ß > œ. HV Analogous to the analysis for the contact case, the applicale solution for the no-contact regions have een found. Assuming AÐBÑœ+/ ;B ÐÞ&'Ñ here + is some constantþ For a solution of the differential equation, ; must e a root of % R-< ; œ! H ; > ÐÞ&(Ñ

32 hich gives " R-< ; œ Ð È? Ñ, H ÐÞ&)Ñ here R-< R -< %I>? œð Ñ % > œ Ð Ñ ÐÞ&*Ñ H H HV Again three cases of roots are considered. When?!, the four roots are the folloing complex conjugate pairs: 8" œ - 3., 8 œ - 3., 8$ œ - 3., ß8% œ - 3. ; ÐÞ'!Ñ here " " - œ Ê È R. œ Ê È R >, > H H -< -< ÐÞ'"Ñ The solution can therefore e ritten as AÐBÑœF " coshð-bñ cosð.bñ F coshð-bñ sinð.bñ F sinhð- B Ñ cosð. B Ñ F sinhð- B Ñ sinð. B Ñ ÐÞ'Ñ $ % here F", F, F$, and F% are constants depending on the matching conditions ith the solution for the contact part. No, again, if the origin for this solution is taken at the center of the uckle, then the solution must e symmetric aout B œ!. This symmetry condition yields F œ F œ!þhence the solution ecomes $ " % AÐBÑœFcoshÐ-BÑcosÐ.BÑ FsinhÐ-BÑsinÐ.BÑ ÐÞ'$Ñ At the ends, B 2 œ 0 of this uckle ß A Ð 0Ñ œ!, hich then requires " % " " F œ F tanð. 0ÑtanhÐ- 0 Ñ. ÐÞ'%Ñ

33 The solution may e ritten as F AÐBÑœ ÖsinhÐ-BÑ sinð.bñ tanh Ð- 0Ñ tan(. 0Ñ coshð-bñ cosð.bñ ÐÞ'&Ñ - here F is a constant. As explained efore, this form of the solution is applicale to all three cases of?!ß? œ!ßand?!þ 2Þ5 Matching conditions eteen contact and no-contact regions In the previous section, solutions ere otained for contact and no-contact regions of the uckled shell respectively. Since these to kinds of regions elong to the same shell, they must e compatile along their common circles. As shon in Fig. 2-2, common circles occur at B œ ' for the contact region, and at B œ 0 for the no-contact region. " At common circles the to separate solutions must have the same deflection, same slope, same ending moment, and same shear force. The equality of deflections, has already een satisfied y requiring them to e zero at the common circles. The remaining matching conditions are:.a" ÐB" Ñ.A ÐB Ñ ¹ œ ¹.B Bœ.B " " ' Bœ 0 ÐÞ''Ñ Q ¹ œ Q ¹ ÐÞ'(Ñ BB BB Bœ' Bœ 0 " U ¹ œ U ¹ ÐÞ')Ñ B B Bœ' Bœ 0 " For the second condition, i.e., the equality of ending moment, since $ I>. A QBB œ ÐÞ'*Ñ "Ð" Ñ.B / it is equivalent to. A" ÐB" Ñ. A ÐB Ñ ¹ œ ¹.B B œ.b " " ' B œ 0 Þ ÐÞ(!Ñ

34 For the third condition, the expression for the shear force U B per unit length is U œ B 5 -< $ $.A I>. A > ÐÞ("Ñ.B "Ð" / Ñ.B$ In vie of the required equality of the slopes, the shear force matching condition at the common circles ecomes $ $. A" ÐB" Ñ. A ÐB Ñ ¹ œ ¹.B $ B œ.b $ " " ' B œ 0 ÐÞ(Ñ These three matching conditions ÐÞ''Ñ, ÐÞ(!Ñ, and uckling equations in the next section. ÐÞ(Ñ are used to determine the 2.6 Equations for uckling load and ave lengths Adopting the approach proposed y Yang [20], let the first derivatives at the common circle of the to regions e expressed as.a" ÐB" Ñ.A ÐB Ñ ¹ œe+ "" œe+ "" Ð' Ñ, and ¹ œf+ " œf+ " Ð0Ñ aþ($.b Bœ " '.B Bœ 0 " here + is a function of ', + is a function of 0. The first matching condition Eq. ÐÞ''Ñ requires that "" ".A" ÐB" Ñ.A ÐB Ñ ¹ ¹ œ + "" E + " F œ! aþ(%.b Bœ " '.B Bœ 0 " Similarly, matching conditions ÐÞ(!Ñ and ÐÞ(Ñ may e expressed as + " E + F œ! aþ(& + $" E + $ F œ! aþ(' here.aðbñ " ".AÐBÑ E+ " Ð' Ñ œ F+ Ð Ñ œ Þ((.B ¹ and 0 Bœ.B ¹ a " ' Bœ 0 "

35 $ $.AÐBÑ " ".AÐBÑ E+ $" Ð' Ñ œ F+ $ Ð Ñ œ Þ() $ ¹ and 0 $ ¹ a.b Bœ " '.B Bœ 0 are evaluated as indicated. " There are thus three equations ith to unknon constants E and F. Hoever, the uckling lengths ', 0and the uckling load R -<, hich are in terms of + "" ß + " etc., are also unknons of the prolem. For otaining equations only contain ', 0, and R -<, the constants E and F are eliminated from Eqs. aþ(% and aþ(&, and from Eqs. aþ(& and aþ('. This gives the folloing to equations: + "" + " œ! aþ(* + + " + $" + $ œ! aþ)! + + " + "" + " The third relation otained from Eqs. aþ(% and aþ(', œ!ß is a linear + $" + $ comination of the previous to, and is therefore an identity. This relation is not used in the folloing, ut it might e used to check the accuracy of the numerical results. The to equations, hich may e called the uckling equations, are expressed as sin + sinh sin + sinh Eq1 œ " -" Ð. "' Ñ." Ð-" ' Ñ cos cosh " - Ð. 0Ñ. Ð- 0Ñ -. Ð. ' Ñ Ð- ' Ñ -. cosð. 0Ñ coshð- 0Ñ œ! " " " " ÐÞ)"Ñ Eq2 œ -" sinð. "' Ñ." sinhð-" ' Ñ - Ð- $. ÑsinÐ. ' Ñ. Ð$-. ÑsinhÐ- ' Ñ " " " " " " " " - sinð. 0Ñ. sinh( - 0Ñ œ! - Ð- $. ÑsinÐ. 0Ñ. Ð$-. ÑsinhÐ- 0Ñ ÐÞ)Ñ 2. ( Solutions for uckling load and uckling ave lengths The to uckling equations, ()". ) and (). ), are used to calculate the critical uckling load R -< and the corresponding uckling ave lengths ' and 0ß for a shell of given dimensions, H and >ß material properties ß I and / ß and a given foundation modulus 5. These equations, hoever, are nonlinear. In these to equations, there are three variales, namely uckling load R, contact uckling length ß and no-contact uckling -< '

36 length 0. One cannot find a solution for three unknon variales from to equations. A third condition is needed. This condition is provided y the fact that the critical uckling load must e a minimum. A trial and error procedure is then used to solve the R -< uckling equations. One assumes a value of ', the contact uckling length, and then finds the other to unknons from the to uckling equations. Different values of ' are used for otaining corresponding R -< and 0. Among these otained values of R-<, the minimum one is deemed as the critical uckling load; and the corresponding 0, the uckling ave length of the no-contact region, is also then otained. A Mathematica [21] program can e ritten for doing these calculations. The uckling load R -< otained from the trial and error procedure is expected to fall eteen the loer ound, i.e. ß the uckling load value of a hollo shell, and the upper ound, i.e., the uckling load value of a shell ith rigid core. A nondimensional parameter!, the ratio of uckling load to that of a hollo shell, is introduced R-<! œ ÐÞ)$Ñ R -<! here R -<! œ I> ÎV É $Ð" / Ñ is the uckling load of a hollo shell. In the same ay, 0 ' " œ ß œ ÐÞ)%Ñ 6 6!! here 6 œ ÈV> È % "Ð" / Ñ! 1 is the uckling ave length of a hollo shell. A set of numerical results ere otained from the listed in Tale 2.1 elo. Mathematica [21] program and are Tale 2.1 Buckling parameters for infinitely long shells ith different foundation moduli! " 5 œ! 5 œ "! 5 œ "!! ' 5 œ "Þ! "! 5 œ _ " "Þ$! "Þ'* "Þ''( "Þ''(!Þ&!Þ'*"!Þ)&*!Þ)''!Þ)''!Þ&!Þ%%"!Þ!"!!Þ!!$!

37 As shon in Tale 2.1, the uckling load and no-contact ave length increase ith the increasing of foundation modulus 5, to a limit of "Þ''( and!þ)'' respectively. On the contrary, the contact ave length decreases, to a limit of zero. The influence of 5 on the uckling load and uckling ave lengths is shon graphically in Figs. 2-3, 2-4, and 2-5. These graphs supplement the information of Tale 2.1. It is apparent that the uckling parameters are quite sensitive for lo values of 5ß say! 5 &Þ For 5 œ &ß the uckling load parameter! is already "Þ&ß compared to its ultimate value of "Þ''( for 5 œ _Þ Similarly, at 5 œ &ß the contact length parameter is!þ!ß compared to its ultimate value of zero for the rigid core. As ell, the nocontact length parameter at 5 œ & is "!Þ)&, compared to the ultimate value of!þ)''þ As may e expected, the graphs in these figures are not smooth in the range of sensitivity,! 5 &, ut they do correctly indicate the trend in the influence of the foundation modulus on the uckling parameters α k Fig. 2-3 Buckling load parameter!

38 β k Fig. 2-4 No-contact length parameter " γ k Fig. 2-5 Contact length parameter

39 Chapter 3 Plastic Buckling of Infinitely Long Shells ith Elastic Core In this chapter, plastic uckling ehaviour of infinitely long shells on elastic one-ay foundations is analyzed. The main difference ith the preceding chapter is that no the material moduli are those elonging to an elastic-plastic, strain-hardening solid. There are some types of steel hich do not exhiit a yield plateau, stainless steel for instance, ut have a rising strain-hardening ehaviour. The kinematic assumptions made here are identical to those in the previous chapter. 3.1 Constitutive relations of the plasticity theories A rief and elementary introduction of the theory of plasticity is presented for the purpose of this thesis in the folloing. For further discussion, reader may consult standard ooks on the theory of plasticity, such as y Hill [22] or Chen [23]. The former is more theoretical, hile the latter is more practical. Upon exceeding a critical stress value, called yield stress, materials egin to undergo plastic, or irreversile, deformation. For multi-axial state of stress, the yield limit is determined y a yield criterion. The most often used criterion is that of Tresca or that of von Mises. In this thesis, the latter one is adopted, also called the N yield criterion. The von Mises criterion says that the yielding egins hen the second invariant of the deviatoric stress tensor N reaches a critical value: N œ Î$ Ð$Þ"Ñ 5 C here 5 C is the current yield stress of the material. This criterion is also expressed as 5/ œ 5C Ð$ÞÑ here œ È$N œ effective stress. 5 /

40 This criterion is used to predict yielding of materials (generally metals) under any multiaxial loading condition from the results of a simple uniaxial tensile test. defines a yield surface in stress space. Plastic deformation is possile as long as the stress point is on the current yield surface. This yield surface, hoever, is not fixed and depends on the total plastic strain accumulated during previous history of plastic straining. After & T initial yielding, a ne yield surface is estalished ased on the present state of 9 9 deformation, ith stresses 5 and strains % 34. If the state of the stress is changed such that 34 the stress point moves inard of the ne yield surface, there is unloading. The ehaviour of the material is then again elastic, and small enough increments of stresses ill only produce elastic strains. If, on the other hand, the stress point moves outside the ne yield surface, there is loading, i.e. ß there is further plastic deformation. Neutral loading ill happen if the stress point is still on the yield surface. In this case, no further plastic deformation takes place ut only the elastic ones. N For their simplicity, to approaches have een used in developing the constitutive relations of plasticity. They are the so called N incremental and N deformation theories of plasticity. In addition to using the von Mises criterion, they oth incorporate the oserved ehaviour of metals that the plastic strain involves no change in volume, in recognition of the fact that plasticity arises due to slip displacements of metal crystals over slip planes. The deformation theory assumes that the state of the stress determines the state of the stain, and is therefore like a nonlinear elasticity theory ith stress dependent moduli. 555 Relations eteen the deviatoric stress components = 34 œ 534 $ 34 and the plastic as $ : ell as elastic strain components, %, % /, are postulated as : 5 % 34 : 34 % 34 / œ = ß œ = $ 34 Ð$Þ$Ñ K *O here : is a scalar determined y experiments. The total strain components are the sum of the elastic and the plastic parts: : / % œ % % Ð$Þ%Ñ The deformation theory cannot adequately descrie the oserved phenomenon of loading, unloading, and neutral loading. The constitutive relations of deformation theory are only

41 valid in the case of proportional loading [23]. In fact, for proportional loading, the deformation theory ecomes identical to the incremental theory. Hoever, ifurcation uckling involves non-proportional loading, and in such a case there is divergence eteen the to theories. The incremental theory relates the increment of plastic strain to the state of stress, and is therefore load path dependent. Relations eteen the deviatoric stress components 5 : / = 34 œ $ $ 34 and the plastic as ell as elastic strain increments,.% 34,.% 34, are postulated as :. 5.=.% 34 œ 1.N = 34ß.% 34 / œ $ 34 Ð$Þ&Ñ *O K here 1 is a hardening parameter related to the position of the stress point on the uniaxial stress-strain curve of the material. The total strain increment is assumed to e the sum of plastic and elastic strain increments: : /.% œ.%. % Þ Ð$Þ'Ñ The incremental theory is considered a correct theory of plasticity ased on theoretical considerations. Hoever, its application is more difficult. Therefore, despite its eak foundations the deformation theory continues to e used ecause of its comparative simplicity. Moreover, for ifurcation prolems, the ifurcation loads predicted y the incremental theory are sometimes asurdly higher than the experimental values. Although y performing imperfection groth analyses, the maximum loads computed using the incremental theory may e made to agree ith the experimental loads, the required groth analysis is time consuming and gives uncertain ansers depending upon the amplitudes of the imperfections. Therefore, the incremental theory of plasticity is usually not recommended. Paradoxically, the ifurcation loads predicted y the deformation theory for shells are in good and conservative agreement ith the test results, and also invarialy loer than the ifurcation loads from the incremental theory [10-12]. Therefore from a practical point of vie it is the deformation theory hich is used in the present ork. There also exists a theoretical justification for the use of the deformation theory according to Hutchinson [24]. He states [24] that "... for a restricted range of deformations, N deformation theory coincides ith a physically acceptale incremental theory hich develops a corner on its yield surface... most of the results

42 hich have een otained using N deformation theory are rigorously (his italics) valid ifurcation predictions..." 3.2 Constitutive relations for a plastic ifurcation analysis In axisymmetric ifurcation uckling of axially stressed shells, the state of stress changes suddenly from uniaxial to multiaxial, ith increments in stress and strain occurring in other directions due to uckling. Therefore, for a ifurcation analysis, one needs incremental relations, even for the deformation theory. Without further going into details, it can e said that the applicale (plane stress and axisymmetric) constitutive relations for the N deformation theory can e expressed as [10, 25] BB BB )) )) BB )). 5 œ F.% G.%,. 5 œ G.% H.% a$þ( here it can e shon that [25] F œ G œ H œ IÐ- $/ $Ñ - Ð& $/ % / Ñ Ð" / Ñ I Ð- " / Ñ - Ð& $/ % / Ñ Ð" / Ñ %I- - Ð& $/ % / Ñ Ð" / Ñ a$þ) and here in the aove I I / œ "ß œ $Þ* I - I a = > In the expressions for the to parameters / and -, I œ 5 Î% is the secant modulus, and I œ. Î. > 5 %is the tangent modulus, hich are found from the uniaxial stress-strain curve of the material at the stress level equal to the applied axial stress. = The aove relations can e used for the incremental theory y putting /œ! in the expressions for the moduli. Furthermore, putting - œ 1, and / œ! results in the elastic relations

43 According to Shanley's concept [26], plastic ifurcation occurs under increasing load. Thus, there is no elastic unloading of any fiers of the shell from the state of yield at ifurcation. This is a standard assumption, knon as the tangent modulus assumption, in analyzing plastic ifurcation uckling of columns, plates and shells. It is convenient to change the notation slightly no. From no on, the increments in strain and stress due to ifurcation uckling ill e simply denoted as % BBß %)) and 5BBß5)) ÞThus, the constitutive relations Ð$Þ(Ñ ill read as BB BB )) )) BB )) 5 œf% G%, 5 œg% H % a$þ"! ith exactly the same meaning for the moduli as aove. 3.3 Governing equations for plastic axisymmetric ifurcation uckling Governing equations are no derived for the shell uckling in the plastic range. The kinematic assumptions, unaffected y material ehaviour, remain the same as for the elastic case. The ifurcation strains are therefore, as efore, A.?. A %)) œ % œ ( a$þ"" V.B.B, BB The stresses are hoever determined from the elastic-plastic constitutive relations are 5 BB œ % BB % œ.? (. A F G F F G A )).B.B V 5 )) œ % %)) œ.? (. A G H G G H A BB.B.B V a$þ" The governing equations and possile oundary conditions are derived from the principle of virtual ork y folloing the same procedure as for the elastic case. The virtual displacements of the shell are $? in the longitudinal direction and $ A in the out-of-plane direction respectively. Therey, the change in the virtual strains are