BEHAVIOUR OF THIN-WALLED CYLINDRICAL AND CONICAL SHELLS-CARBON vs . STAINLESS STEEL

Tanke cilindrične ljuske su konstruktivni elementi koji su našli široku primenu u praksi. Bilo da je reč o limenci piva ili o delu za rakete, neophodno je temeljno poznavanje svih njihovih karakteristika. Posebno treba naglasiti analizu stabilnosti i raspored (formu) izbočine koja se formira u postkritičnoj oblasti − dijamantsku formu, ili u literaturi još poznatu kao Yoshimura šablon [19]. Stvarne ljuske su elementi koji imaju određene imperfekcije. Moguće je klasifikovati imperfekcije na tri osnovna tipa: geometrijske, strukturne i imperfekcije u opterećenju. U analizi, najčešće se koriste geometrijske imperfekcije zbog jednostavnosti njihovog definisanja, a pokazale su se i kao adekvatne za opisivanje bilo koje vrste imperfekcija. Dobro je poznato da najveći uticaj na stabilnost cilindra imaju geometrijske imperfekcije zadate u obliku koji odgovara sopstvenim oblicima izbočavanja (jednom imperfekcijom ili kombinacijom više njih). Ovakva pretpostavka značajno odstupa od realne slike, ali je u građevinarstvu našla široku primenu. Jedan od razloga za to jeste mali broj dostupnih merenja imperfekcija na


INTRODUCTION
Thin-walled cylindrical shells are structural elements that are widely used in practice.No matter if it is about a beer can or a rocket section, it is necessary to thoroughly know all their characteristics.The analysis of stability and buckling pattern formed in the post-buckling area should be especially emphasized.Buckling pattern can be a diamante pattern or the Yoshimura pattern, as it alternative known in literature [19].
The actual shell structures are elements which have certain imperfections.It is possible to classify these imperfections to three basic types: geometrical, structural and loading imperfections.The most often used are geometrical imperfections due to the simplicity in their definition.Moreover, it is proved that geometrical imperfections are adequate for describing any kind of imperfections.It is well known that geometrical imperfections have the highest influence on the cylinder stability.The geometrical imperfections are set in the form which corresponds to the eigenmodes (one or several of them combined).Such assumption considerably deviates from izvedenim konstrukcijama.Pored velikih ekonomskih izdataka, koji su neophodni da bi se jedno ovakvo merenje izvršilo, treba napomenuti i to da su imperfekcije funkcija s velikim brojem promenljivih.Znatan uticaj na pojavu imperfekcija imaju različite faze gradnje, način eksploatacije, klimatski uslovi (temperatura), kao i mnogi drugi uslovi.
In civil engineering, buckling of cylindrical shell is a relevant criterion for design of chimneys, wind turbine towers, silos and similar structures.It is necessary to find a balance between safety and cost-effectiveness, but to remain in the financially justified limits at the same time, using the less conservative design procedures that will retain safety in terms of design resistance and stability.This is the aim of the paper, i.e. the influence of different values of amplitudes of initial geometrical imperfections, set in the form of the first eigenmode, on shell behaviour with the analysis of the choice of materials from which the shells are made.
The results of the numerical analysis of the stability of cylindrical and conical shells using software package Abaqus are presented in the paper.The influence of material and geometric nonlinearity on critical buckling stress and buckling resistance is analysed.Material nonlinearity has been taken into calculation using the experimental stress-strain curves of carbon and stainless steel, while the geometric nonlinearity has been included through various levels of initial imperfections.The results of the conducted numerical analysis have been compared with the recommendations given in EN 1993-1-6 [9] and the values of the reduction factor, calculated as ratio between critical buckling stress and buckling resistance have been defined.The values of the reduction factor are compared with the data available in the reference literature.

AXIALLY COMPRESSED CYLINDRICAL AND CONICAL SHELLS
During the recent decades, a large number of researches have been focused on the analysis of shell stability problem.The case of axially loaded cylindrical shells exposed to compression stands out as characteristic for this type of structural elements, and it is analysed in this paper.The reason lies in considerably different behaviour of cylindrical shells under axial load than of the plates and columns [3].This is graphically presented in figure 1 for different structural elements exposed to axial compression.The force-strain curves obtained by the analysis for the initial phase, ultimate phasewhere the loss of stability occurs and postbuckling phase are given.In figure 1, for perfect plates the load (P) can actually increase above the Euler buckling load with increasing axial deformation (∆).For perfect bars the load (P) neither increases nor decreases, while for perfect shells with further increase of deformation (Δ) the load (P) decreases beyond the Euler buckling load.When considering the influence of initial imperfections, the Euler load for bars is the maximum load for all axially compressed elements, with and without initial imperfections.The plate shows an increase in deformation (Δ) in the case of an element koje je veće od Eulerovog opterećenja.Cilindrična ljuska s početnim imperfekcijama gubi stabilnost pri opterećenju znatno manjem od Eulerovog opterećenja, čime je ponašanje ovog elementa okarakterisano kao izuzetno katastrofalno [10].
with initial imperfections, but in both cases they can accept the load that is greater than Euler's.A cylindrical shell with initial imperfections loses stability at a load significantly smaller than Euler's which has been characterized as extremely catastrophic behaviour [10].Slika 1. Krive sila-deformacija za različite pritisnute konstruktivne elemente [10] Figure 1.The force-strain curves for different axially compressed structural elements [10]

Linear-elastic theorycritical load
The behaviour of cylindrical shells can be described by a single set of equations, but due to its complexity they are impracticable.Therefore, certain simplifications are made by ignoring the parameters that have a small influence on the considered phenomenon.One such set of equations for description of shell behaviour was proposed by Donnell [6].In the case of axially loaded shells, the set of equations can be reduced to one equation of the eighth order, which is known as Donnell's equation and which can be used for determining the critical load, both under the axial compression and to the torsion and internal pressure.The critical load value can be obtained by solving the Donnell's equation and such a solution was provided by Batdorf [2] for a cylinder pinned on both ends (pinned).The obtained solution, alternatively called the classical solution in the literature, is presented by the eq.( 1):  1) is applicable for cylinders of medium length which are the most common in practice, and therefore very important.

Failure and post-buckling behaviour
Experimental research showed that the critical buckling stress at which failure occurs is often considerably lower than the theoretically calculated one obtained using the linear-elastic theory.Therefore, it can be concluded that the linear-elastic theory is inadequate for description of behaviour of axially compressed cylinders, and that describing actual behaviour can be accomplished using the nonlinear large deformation theory.
Postoje različite analitičke formulacije redukcionog faktora za izotropne cilindrične ljuske, aksijalno opterećene.Neke od prvih preporuka potiču s početka 20.veka, kao što su NASA SP-8007 [14].Prema njima, kritičan napon dobija se iz sledeće formule: further research proved that the obtained results are less accurate, the following conclusions were drawn:  even the slightest imperfection causes a considerable lower critical stress value;  the minimal value is not significantly impacted by the size of the imperfection, so adopting this value as a critical one would lead to a conservative solution for the designers.
The solution which greatly contributed to solving the problems in practice was discovered by Koiter [11] using a simplified large deformation theory on the example of axisymmetric initial imperfections.The obtained result has a good agreement with the lower values obtained by experimental research.In figure 2b the results of the experimental investigation of cylindrical shells published by various authors are presented.The ratio of the diameter (R) and the shell thickness (t) is indicated on the abscissa, while the normalized critical load (λ) is defined on the ordinate, which represents the ratio of the critical load obtained by the experiments (Pexp) and the load obtained using the classical solution (Pcl).The full line on the graphic is a solution obtained by multiplying the load obtained by the classical theory with the correction (knockdown) factor defined in NASA's design recommendations [14].An explanation for a large dispersion of the results of experimental tests, in comparison with the theory was established and initial imperfections were allocated as the main reason.

Design recommendations
Different standards and manuals used in engineering practice give design recommendations for cylindrical and conical shells.All of them basically implement a classical solution obtained using the linear theory, eq. ( 1), and then multiply it by the corresponding knockdown factor to obtain the load which can be transferred by the cylinder.This factor is determined based on the full curvature marking the lower boundary of test results in figure 2b.
Oba rešenja i preporuke u EN 1993-1-6 [9], kao i preporuke koje je dala NASA-a u SP8007 [14], u osnovi su potekle iz rada Weingarten-a [17].Osnovna razlika When the ratio of the largest amplitude w relative to the corresponding lr is less than 0,01, eq. ( 5) and ( 6) should be applied to γ; when equal 0,02 values of γ should be taken in the amount of 50%, and between the interpolations should be done.When this ratio is greater than 0,02, there are no recommendations, suggesting that such elements should be excluded from the application.
In the case of conical shells, the value of the knockdown factor for the range 10°  α  75° amounts to 0,33, while for the range α ≥ 75° it must be verified experimentally, as recommended in [15].The mentioned knockdown factor, as in the case of cylindrical shells, is multiplied with the classical solution provided in the eq.( 2).

NUMERICAL ANALYSIS
In order to quantify the influence of main parameters on the resistance of cylindrical and conical shells, a numerical parametric analysis was performed using the finite element method.A detail comparative analysis was conducted in order to quantify the compliance of numerical analysis results with design recommendations given in European standard EN 1993-1-6 [9].Numerical analysis of cylindrical and conical thin-walled shells was performed using the Abaqus software, version 6.12-3 [1].
In the numerical analysis of the stability problem, two methods were used:  analysis of the linear bifurcation eigenvalue (LBA -Linear Buckling Analysis);  analysis of response after stability loss or nonlinear analysis (Postbuckling analysis) with GMNIA analysis (GMNIA -Geometrically and materially nonlinear analysis with imperfections included).
A linear elastic analysis was used in the initial phase of the calculation which provides prediction of eigenmodes.The element stability analysis with GMNIA is based on solving the nonlinear equilibrium equation using an appropriate numerical method.An arc length method, also known as the Riks method [12] is implemented in the paper.
The numerical analysis presented in this paper included the cylindrical shell 10 m long with 2,5 m radius, with the wall thickness ranging between 6,0 mm and 30,0 mm, which belongs to the medium long cylindrical shells as defined in EN 1993-1-6, Annex D [9].In addition, the numerical analysis included the conical shell of variable radius ranging between 1,25 m and 2,5 m, with 10,0 mm wall thickness.The models of cylindrical and conical shells pinned on both ends are defined in Abaqus [1] using shell S4R finite elements.The mesh for all analysed models has been formed using S4R finite elements of the approximate size of 200 x 200 mm for which the results start to converge more considerably (the critical buckling stress which corresponds to the first eigenmode is lower than 0,20%).

NUMERICAL ANALYSIS RESULTS AND COMPARISON WITH RECOMMENDATIONS GIVEN IN EN1993-1-6
The values of the elastic critical buckling stress for cylindrical shells with thickness from 6,0 mm to 30,0 mm and conical shell with 10,0 mm thickness are defined using the LBA analysis in Abaqus [1] and according to design recommendations given in EN 1993-1-6 [9] which is at the same time the classical solution presented in the eq.( 1).Material nonlinearity is introduced in the numerical models through the real stress-strain relation for two analysed materials, and the geometrical imperfection is set as displacement of a certain amplitude, which corresponds to the first eigenmode (LBA analysis in Abaqus [1]).
The impact of different values of imperfections on the critical buckling stress is analysed on the example of cylindrical and conical shells 10,0 mm thick, with the geometrically and materially nonlinear analysis with imperfections included (GMNIA).

Results of numerical analysis -cylindrical shells with thickness of 10 mm
Buckling resistance of the cylindrical shell of S275 carbon steel and 1.4301 stainless steel is given in figure 5.
In the case of the cylindrical shells of carbon steel, the imperfections of 1,0 and 2,0 mm cause axially symmetrical buckling in the zone immediately next to supports, i.e. immediately below the zone where the load is applied, as given in figures 6a and 6b.The response of the stainless steel cylindrical shell at lower values of imperfections has the form of dimple buckling in the zone where the load is applied, as displayed in figures 6c and 6d, which is a characteristic response of the structure retained even when the imperfections are increased up to 10,0 mm.
In addition, in the case of stainless steel, in the stress range between the proportionality limit stress fp and the 0.2% proof stress f02 there is a progressive decline of tangent elastic modulus Et, which causes the considerable decrease of stiffness as observable in figure 5b, in cylindrical shells of stainless steel.

Results of numerical analysis -conical shells with thickness of 10mm
The impact of different values of initial geometrical imperfections on the buckling resistance of the conical shell made of S275 carbon steel is given in figure 7.
Critical buckling stress obtained from LBA analysis in Abaqus [1] is 504,3 MPa for conical shell made of carbon steel with 10,0 mm thickness.The response of a conical shell made of carbon steel is in a form of dimple buckling in the zone where the load is applied which is a characteristic response of the structure retained even when the imperfections are increased up to 10,0 mm, as displayed in figure 8.
Comparison of the results of the numerical analysis for cylindrical shells from carbon and stainless steel and the design recommendations given in EN 1993-1-6 [9] for the buckling curve defined for class A of fabrication tolerances is given in figure 9b.The analysis included cylindrical shells with thicknesses in the range from 6,0 mm to 30,0 mm made from carbon steel S275 and shell made from stainless steel 1.4301 with 10,0 mm thickness.According to EN 1993-1-6 [9] initial geometrical imperfection wk should be determined as a function of shell radius, thickness and dimple tolerance parameter U0,max=0,006, as given in eq. ( 8).The values of initial geometrical imperfections are presented in Table (1).The suitable prediction of results is achieved for analysed cylindrical shells and design recommendations according to class A of fabrication tolerances.Besides, the results of numerical analysis for cylindrical shells with 10,0 mm thickness made form carbon and stainless steel with initial geometrical imperfections in the range from 1,0 mm to 10,0 mm are presented in figure 9b.
The results of numerical analysis satisfy empirically defined recommendations in EN 1993-1-6 [9] as shown in figure 9b in the case of cylindrical shell with 10,0 mm thickness made of S275 carbon steel with geometrical imperfections up to 4,0 mm.The results of numerical analysis fail to satisfy the design recommendations in the case of imperfections larger than 4,0 mm which is at the same time the value of imperfection recommended by the Eurocode for the analysed cylindrical shell and class A of fabrication tolerances (Table 1).Even though the design recommendations for the stainless steel shells have not been defined yet, the analysis of behaviour of shells made from this material was particularly significant due to the specific mechanical properties of this material.When the mechanical properties of stainless steel are implemented in the design recommendations for shells provided in EN 1993-1-6 [9], one may also observe that the best agreement of the numerical analysis results with such defined design recommendations are obtained for imperfections of up to 4,0 mm (figure 9b Komparativna analiza rezultata numeričke analize, metodom konačnih elemenata i proračunskih preporuka datih u EN 1993-1-6 [9] za cilindrične ljuske od ugljeničnog čelika S275, debljine od 6,0 do 30,0 mm, prikazana je u Tabeli (1).Geometrijske imperfekcije zadate su prema preporuci za definisanje geometrijske imperfekcije wk date u [9] za klasu A proizvodnih tolerancija.Iste proračunske preporuke u pogledu zadavanja početnih geometrijskih imperfekcija wk primenjene su i na konusnu ljusku od ugljeničnog čelika S275, s debljinom zida ljuske od 10,0 mm, kako je prikazano u Tabeli (2).
Comparative analysis of results obtained from numerical analysis with design recommendations given in EN 1993-1-6 [9] for cylindrical shells made of S275 carbon steel 6,0 to 30,0 mm thick, is given in Table (1).Comparison of the results is performed in order to define the compliance of numerical analysis with recommendations given in European standard.Geometrical imperfections are defined according to the recommendations for geometrical imperfection wk provided in EN 1993-1-6, Annex D [9] for class A of fabrication tolerances.The same design recommendations considering definition of geometrical imperfections wk are also implemented on the conical shell made of S275 carbon steel, with the shell wall thickness of 10,0 mm, as given in Table (2).
Figure 10 shows comparisons of numerical analysis results for a cylindrical shell of 10,0 mm thick of carbon and stainless steel in Figure 5 (with geometric imperfections from 1,0 mm to 10,0 mm), with the currently developed empirical recommendations for the knockdown factor (KDF).The defined recommendations are provided in the form of the curves showing dependence of the knockdown factor and ratio of the shell radius and thickness (R/t).Figure 10 shows recommendations for the reduction factor according to NASA SP-8007 [14], modified curves depending on the ratio of the shell length and radius L/R according to Wagner [18], experimental results by Flügge and the results of the presented numerical analysis.The modified curve TH -L/R=5 according to Wagner [18] includes the mutual impact of imperfection in load and geometric imperfection (SBPA -single boundary perturbation approach), and for the ratio of length and radius of shell L/R=5 defines significantly lower value of the knockdown factor (KDF) from the recommendations given in NASA SP-8007 [14].Cilindrične ljuske od ugljeničnog čelika S275 pokazuju relativno dobro slaganje sa modifikovanom krivom prema Wagner-u [18] za geometrijske imperfekcije definisane u skladu s preporukama datim u EN 1993-1-6 [9] za analiziranu dužinu i prečnik ljuske.Za ljusku debljine 10,0 mm od ugljeničnog čelika S275 i geometrijske imperfekcije od 1,0 do 3,0 mm rezultati numeričke analize zadovoljavaju empirijski definisane preporuke prema Wagner-u [18].Za imperfekcije od 4,0 mm, što su ujedno i vrednosti imperfekcije koju preporučuje Evrokod za debljinu zida ljuske od 10,0 mm i klasu A proizvodne tolerancije, rezultati su nešto konzervativniji.Isto ponašanje uočeno je za veće vrednosti početnih imperfekcija.Rezlutati numeričke The cylindrical shells made of S275 carbon steel exhibit relatively good agreement with modified curves by Wagner [18] for geometrical imperfections defined in accordance with the recommendations provided in EN 1993-1-6 [9] for the analysed shell length and radius.In the case of the shell 10,0 mm thick made of S275 carbon steel and the geometrical imperfections from 1,0 to 3,0 mm, the results of the numerical analysis satisfy empirically defined recommendations according to Wagner [18].For imperfections of 4,0 mm, which are simultaneously the imperfection values recommended by the Eurocode for the shell wall thickness of 10,0 mm and class A of fabrication tolerance, the results are slightly more conservative.The same behaviour is observed for analize za cilindričnu ljusku od nerđajućeg čelika takođe su konzervativni u poređenju s preporukama prikazanim na slici 10b, čak i za najmanje vrednosti imperfekcija od 1,0 i 2,0 mm.Ovakav rezultat je očekivan, imajući u vidu to što su empirijske krive za redukcioni faktor, prikazane na slici 10, razvijene na osnovu rezultata eksperimentalnih ispitivanja cilindričnih ljuski od ugljeničnog čelika.

ZAHVALNOST
Ovo istraživanje je finansijski podržalo Ministarstvo prosvete, nauke i tehnološkog razvoja Republike Srbije, u okviru naučnih projekata TR-36048 i III 42012.the higher values of initial imperfections.The results of the numerical analysis for cylindrical shell of stainless steel are also conservative in respect to the recommendations displayed in figure 10b, even for the lowest values of imperfections of 1,0 and 2,0 mm.Such a result is expected regarding that the empirical curves for the knockdown factor given in figure 10 are developed based on the results of experimental tests of cylindrical shells made of carbon steel.

CONCLUSION
Based on the results of numerical analysis presented in this paper, the following conclusions can be drawn:  Initial geometrical imperfections have a great impact on the buckling resistance of thin-walled cylindrical and conical shells.An imperfection of 1,0 mm causes the buckling resistance reduction of 12%, i.e. 17%, for carbon steel and stainless steel cylindrical shells, respectively.By increasing initial geometric imperfections up to 5.0 mm, the reduction of buckling resistance reaches up to 45% for carbon steel and 50% for stainless steel.This reduction in the case of conical, carbon steel shells reaches 30% for imperfection of 5mm.
 Cylindrical shells of stainless steel shows a progressive decline of tangent elastic modulus Et, in the stress range between the proportionality limit stress fp and the 0.2% proof stress f02 in comparison with carbon steel cylindrical shells, which causes the considerable decrease of stiffness.
 With the increase of the cylindrical shell thickness, the impact of initial geometrical imperfections on the buckling resistance is decreasing, and the impact of material nonlinearity is increasing.
 The value of KDF factor for geometrical imperfections determined according to the recommendations given in EN 1993-1-6 [9] for cylindrical shells made of carbon steel represent a good agreement with recommendations given by Wagner [18].
As the reduction of the theoretical value of critical stress, prescribed by the standards is high, the future improvement of design recommendations is based on the optimization of the knockdown factor.One way is forming and implementation of a data base on imperfections of cylindrical shells based on the conducted experimental tests (Imperfection Data Bank).This facilitates optimization of calculation and design of a specific type of cylinder, by analysing the result obtained on the similar types of cylinders.In addition to the previous, there is also a question whether the initial imperfections effect, resulting in reduced bearing capacity, can be avoided using appropriate design, i.e. production quality.There are many papers on this topic, and all of them are striving to attain the shells insensitive to initial deformations.Therefore, it can be concluded that the field of analysis of cylindrical and conical shells behaviour is still largely open to research.

ACKNOWLEDGEMENT
This research is financially supported by the Ministry of Education, Science and Technological development of the Republic of Serbia, in the framework of the scientific projects TR-36048 and III 42012.
t debljina ljuske; Imperfekcije se u [9] uzimaju u obziru skladu sa slikom 3. Dužina poremećaja definisana je izrazom: where: L length of shell; R radius of curvature; t shell thickness.Imperfections are taken into account in [9] in accordance with figure 3. The length of the disorder is defined by the expression: Figure 4. a) stress-strain curves for carbon steel S275 and stainless steel 1.4301 b) model of conical and cylindrical shell and boundary conditions

8 .Figure 8 .
Figure 8. Buckling of cylindrical shell 10,0 mm thick made of S275 carbon steel for different values of imperfections: a) 1 mm, b) 2 mm, c) 3 mm, d) 4 mm a) redukcija napona izbočavanja za ljuske debljine 10 mm a) reduction of bucking stress of 10 mm thick shells b) rezultati numeričke analize b) comparison of the results of the numerical analysis with design recommendations Slika 9. Poređenje rezultata numeričke analize s preporukama za proračun Figure 9. Impact of geometrical imperfections