Non-linear Buckling and Post-buckling Analysis of Cylindrical Shells Subjected to Axial Compressive...

Nonlinear Engineering, Vol. 2 (2013), pp. 83–95 Copyright © 2013 De Gruyter. DOI 10.1515/nleng-2013-0009

Non-linear Buckling and Post-buckling Analysis of CylindricalShells Subjected to Axial Compressive Loads:A Study on Imperfection Sensitivity

Y. Venkata Narayana,1 Jagadish Babu Gunda,2;�P. Ravinder Reddy3 and R. Markandeya4

1 Department of Mechanical Engineering, SreeNidhi In-stitute of Science and Technology, Ghatkesahar, An-drapradesh, India

2 Advanced Systems Laboratory, Kanchan Bagh, Hyder-abad, India

3 Department of Mechanical Engineering,ChaitanyaBharathi Institute of Science and Technology, Gandipet,Andrapradesh, India

4 Department of Metallurgical Engineering JNTU Kukat-pally, Andraprdesh, India

Abstract. Imperfection sensitivity of cylindrical shellssubjected to axial compressive load is investigated bymeans of non-linear buckling analysis and post-bucklinganalysis. Non-linear buckling analysis involves the deter-mination of the equilibrium path (or load-deflection curve)upto the limit point load by using the Newton-Raphson ap-proach, whereas post-buckling analysis involves the deter-mination of the equilibrium path beyond the limit pointload and up to the collapse load by using the arc-lengthapproach. Limit point loads evaluated from these two ap-proaches for various imperfection magnitudes show an ex-cellent agreement which clearly confirms the numerical re-sults obtained.

Keywords. Non-linear buckling, post-buckling, cylindricalshells, imperfection sensitivity, axial compressive loads.

1 Introduction

The problem of cylindrical shell buckling subjected to axialcompressive loads has been solved by many researchers [1-10] who have used approximate analytical methods as wellas finite element methods. Theoretically evaluated classicalbuckling load is generally much higher than the actual buck-ling load of the cylindrical shell and a knock-down factor is

Corresponding author: Jagadish BabuGunda, E-mail: [email protected].

Received: 25 June 2013. Accepted: 7 August 2013.

introduced to furnish a better approximation of actual buck-ling load based on experimental tests. Fischer [11], Yamakiand Kodama [12], investigated the effect of bending stressesand pre-buckling deformations and emphasized that the ef-fect of pre-buckling deformations was not a primary reasonfor the difference between the classical prediction and theexperimental results. The pioneering contributions of vonKarman and Tsien [13], Donnell and Wan [14], Koiter et al.[15], Budiansky and Hutchinson [16] on cylindrical shellbuckling shows that initial geometric imperfections is thesingle dominant factor for the discrepancy between theoryand experiments.

Arboczand Hol [17] demonstrated that the form andamplitude of imperfections are dependent on the fabrica-tion process and quality. Buckling of imperfect cylindri-cal shells thus remains a subject of active area of researchwith special emphasis on modeling of the real imperfectionsas well as of boundary conditions and load eccentricity, ifany. Shen and Li [18], and Schneider [19] investigated thebuckling of shell structures by taking dimples as a geomet-ric imperfection pattern. Frano and Forasassi [20], Prabuet al. [21] investigated the buckling behavior of imperfectthin cylindrical shells under lateral pressure by taking oval-ity as imperfection sensitivity parameter and observed thatthe buckling load decreases with an increase in imperfec-tion magnitude. Recently, Kobayashi et al. [22] employed astabilization technique by using artificial damping to inves-tigate the post-buckling behavior of perfect Yamaki cylin-der subjected to axial compression. These researchers em-phasized the difficulty of using the conventional arc-lengthmethod when applied for the post-buckling analysis of im-perfect cylindrical shells. Spagnoli et al. [23] investigatedthe buckling behavior of laminated composite cylinders anda correlation study on theoretical versus experimental endshortening behavior is discussed and a summary of knock-down factors as well as FE (finite element) reduction fac-tors are reported. To the best of the authors’ knowledge adetailed and generalized approach of a qualitative study onnon-linear buckling and post-buckling behavior of cylindri-cal shells including the influence of geometric imperfectionhas not been well reported in the literature.

This study makes an attempt to accurately evaluate thelimit point load of an isotropic and composite, imperfectcylindrical shells by means of non-linear buckling analysisas well as post-buckling analysis by using a general purpose

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84 Y. Venkata Narayana, Jagadish Babu Gunda, P. Ravinder Reddy and R. Markandeya

finite element software (ANSYS) [24] (shell281 element).A generalized procedure is employed here which can be ex-tended to investigate the behavior of any other modes ofpractical interest (for example ovality mode for long cylin-ders) for any given choice of imperfection shapes.

2 Finite element formulation

The finite element discretization process for geometricallynon-linear analysis yields a set of simultaneous equations:

ŒKT �ı¹uº D ı¹f ºa (1)

where ŒKT � is the tangent stiffness matrix, ı¹uº is the in-cremental nodal displacement vector and ı¹f ºa is the in-cremental nodal force vector. For determining the bucklingload, the equation can be greatly simplified by taking smalldeformation and we can omit the nonlinear terms whichare functions of nodal displacements in the tangent stiff-ness matrix. The following expression [24] gives the tan-gent stiffness matrix after linearization

ŒKT � D ŒKL�C ŒK� � (2)

where ŒKL� is a linear stiffness matrix and ŒK� �, a stressstiffness matrix. If a stress stiffness matrix ŒK� �nr is gen-erated according to a reference load F nr , for another loadlevel F a with �, a scalar multiplier, we have

¹F ºa D �¹f ºnr ; ŒK� � D �ŒK� �nr : (3)

When buckling occurs, the external loads do not change,i.e., ı¹f ºa D 0. The resulting bifurcation solution for thelinearized buckling problem may be determined from thefollowing eigenvalue equation�

ŒKL C �cr ŒK� �nr

�ı¹uº D 0 (4)

where �cr is an eigenvalue and ¹uº becomes the eigenvec-tor defining the buckling mode. The critical load Fcr can beobtained from ¹f ºcr D �¹f ºnr . In ANSYS, a subspace it-eration technique is employed to extract the eigenvalues andthe corresponding eigenvectors. Geometric non-linearity isconsidered by using the von-Karman strain–displacementrelations, where the moderately large rotations and dis-placements of the order of characteristic dimension of theproblem are allowed. For the non-linear problem, the stiff-ness matrix [K] itself is function of the unknown degreesof freedom which leads to system of non-linear equations.An iterative process of solving the non-linear equations isrequired and these can be written as (ANSYS version 13.0)

ŒKTi �¹uiº D ¹f ºa� ¹fiº

nr (5)

¹uiC1º D ¹uiº � ¹�uiº (6)

where ŒKTi � is the tangent stiffness matrix, i represent-ing the current equilibrium iteration and ¹fiºnr vector ofrestoring loads corresponding to the element internal loads.Equation (5) represents a generalized system of simultane-ous non-linear equations which needs to be solved for de-termining the equilibrium path of the thin cylindrical shellsubjected to an axial compressive load. Figure 1 shows atypical equilibrium paths of a geometrically perfect and im-perfect shells, where points A and B denotes the limit pointloads obtained from the equilibrium path approach whichindicates the maximum load carrying capability of an eithera perfect or an imperfect cylindrical shell and is generallyreferred as non-linear buckling load. In present work loadededges of the cylindrical shell are assumed to be simply sup-ported while the edge displacement in the direction of theapplied load is freely allowed. Generally non-linear buck-ling load can be evaluated by performing either the non-linear buckling or the post-buckling analysis approaches.The following summary explains a detailed procedure in-volved in these analysis.

Figure 1. General buckling phenomenon of cylindricalshell.

3 Non-linear Buckling & Post-buckling Analysis

Non-linear buckling analysis for a geometrically perfect orimperfect cylindrical shell generally involves the determi-nation of the equilibrium path up to the limit point loadand beyond which the slope of the load-deflection curve (orequilibrium path) ceases to be positive. Post-buckling anal-ysis generally involves the determination of the full equilib-rium path which also includes tracing of the unstable solu-tion of the equilibrium path. Salient steps involved in theseapproaches are briefly summarized.



Non-linear Buckling and Post-buckling Analysis of Cylindrical Shells Subjected to Axial Compressive Loads 85

3.1 Steps for non-linear buckling and post-buckling analysisusing load–deflection curve

Step 1. For the given structural configuration, a linearbuckling analysis has been performed and for the sakeof simplicity the linear buckled modeshape itself hasbeen chosen as the shape of initial imperfection. Themagnitude of imperfection is referred with referenceto the thickness parameter of the cylindrical shell. Itmust be noted that the shape of imperfection can begiven in the form of a linear combination of buck-led modeshapes or random imperfection or experi-mentally measured imperfection shape can also be im-parted.

Step 2. After applying an initial geometric imperfection,a non-linear analysis (non-linear buckling or post-buckling) has been performed to trace the equilibriumpath of interest.

Step 3. Non-linear buckling analysis involves the appli-cation of Newton-Raphson method to solve Eq. (1)whereas the post-buckling analysis involves the appli-cation of arc-length method.

Step 4. Load–deflection curve obtained using the Newton-Raphson method represents the primary path whereasthe load-deflection curve traced in arc-length methodincludes the primary (stable) as well as secondary (un-stable) and tertiary equilibrium paths.

Step 5. Accuracy of these two approaches for any givenimperfection magnitude is established in evaluatingthe limit point load of isotropic and composite cylin-drical shells.

For the sake of conciseness, details of the Newton-Raphson and the arc-length methods are not discussedhere in detail but the information may be obtained fromRefs. [22 – 25].

4 Results and Discussion

Consider an isotropic cylindrical shell subjected to an axialcompressive load (N/. The adopted geometrical and ma-terial properties of the shell for this study [22] are: radius(R) of the cylindrical shell is 100 mm, thickness (t) of thecylindrical shell is 0.247 mm, length (L) of the cylindricalshell is 113.9 mm, modulus of elasticity (E) is 5560 MPaand Poisson’s ratio � is 0.3

The critical buckling load .Ncr / evaluated for the abovementioned problem on the basis of classical formula [22,24] indicates that it is 1290 N against a predicted value of1282 N which is obtained from the linear buckling (eigen-value) analysis. Figure 2 shows the mesh convergence studyobtained from the linear buckling analysis and an elementsize of 10 mm is chosen for all (for non-linear buckling and

post-buckling) the numerical results discussed herein. Fig-ures 3 and 4 show the first ten linear buckled modeshapesof an isotropic cylinder subjected to axial compressive load.Steps outlined in the earlier section are followed for per-forming the non-linear buckling and post-buckling analy-sis of imperfect isotropic cylindrical shell. Linear bucklingmodeshape has been considered as the basis of initial imper-fection. Figure 5 shows the post-buckling behavior of cylin-drical shell for various imperfection magnitudes, whereasFig. 6 shows the results obtained from the non-linear buck-ling as well as post-buckling analysis for various imperfec-tion magnitudes (� D w�=t , where w� is the maximumimperfection amplitude and t is the thickness of the cylin-drical shell). Note that the primary equilibrium paths ob-tained from these two approaches show an excellent agree-ment and it indicates the confidence gained on the numeri-cal results discussed. Table 1 shows the comparison of limitpoint loads obtained for various eigen-imperfection ampli-tudes which demonstrate the accuracy and exactness of thetwo approaches discussed. In general, it is observed that thelimit point load (or non-linear buckling load) reduces withincreasing magnitude of imperfection. However, for post-buckling analysis the secondary equilibrium path (equilib-rium path beyond limit point load) clearly provides the com-plete load-deflection behavior (primary, secondary as wellas tertiary) of an isotropic cylinder subjected to axial com-pressive load. Figure 7 shows the comparison of the buck-led modeshapes obtained from the linear buckling (or eigen-value) and the non-linear buckling analysis approaches (for� D1.0, corresponds to limit point load). Figure 8 shows thebuckled modeshapes obtained from the post-buckling anal-ysis which corresponds to the limit point load and the otherreference points along the equilibrium path for the givenmaximum imperfection amplitude (� D1.0). From Figs. 7and 8, the number of lobes (or circumferential waves) ob-tained from all the analysis are observed to be equal for theisotropic cylindrical shell throughout the equilibrium path.

Figure 2. Mesh convergence study on cylindrical shell sub-jected to axial compressive load.




Mode 1 (n = 20) Ncr = 1281.25 N

Mode 2 (n = 20) Ncr = 1281.25 N

Mode 3 (n = 1) Ncr = 1315.10 N

Mode 4 (n = 2) Ncr = 1315.60 N

Mode 5 (n = 2) Ncr = 1315.60 N

Figure 3. First five linear buckled modeshapes of an isotropic cylinder subjected to axial compressive load.




Mode 6 (n = 4) Ncr= 1316.90 N

Mode 7 (n = 4) Ncr= 1316.90 N

Mode 8 (n = 6) Ncr= 1318.10 N

Mode 9 (n = 6) Ncr= 1318.10 N

Mode 10 (n = 8) Ncr= 1319.4 N

Figure 4. Linear buckled modeshapes (modes 6 to 10) of an isotropic cylinder subjected to axial compressive load.




Table 1. Imperfection sensitivity study on isotropic cylindrical shell.

Linear Non linear buckling loadBuckling load (1282 N)

Imperfection � D0.0 � D 0:1 � D 0:3 � D 0:5 � D 1:0

SensitivityLimit point load (N) 1277 1061 911 807 574(Nonlinear buckling)Limit point load (N) 1271 1045 905 784 557

(Post buckling)

Figure 5. Post buckling analysis of cylindrical shell showsinfluence of various geometric imperfection amplitudes.

Figure 6. Non-linear buckling and post-buckling behaviorof cylindrical shell show influence of various geometricimperfection amplitudes.

Table 2. Mechanical properties [23] of E-glass wo-ven roving-isophthalic polyester matrix composite(DF1400) [23].

Material Property Direction Value

Elastic modulus (GPa) 1-Weft (E1) 16.42-Warp (E2) 12.7

Shear modulus (GPa) G12 3.1Poisson’s ratio �12 0.20

The confidence gained on the analysis of thin, isotropiccylindrical shells subjected to axial compressive load is fur-ther utilized here briefly for investigating the non-linearbuckling and post-buckling behavior of laminated compos-ite cylindrical shell (geometric dimensions considered aresame as isotropic cylinder). Mechanical properties of thelaminated ([02] or 0°/0°) (ply thickness 0.1235 mm) GlassFiber Reinforced plastic (GFRP) [23] material are shown inTable 2. Figures 9 and 10 show the first ten linear buck-led modeshapes of composite cylindrical shell. Figure 13shows the non-linear buckling and the post-buckling be-haviour of the laminated composite cylindrical shell. Itis clearly observed that these two approaches again showgood agreement in predicting the primary equilibrium pathas well as in predicting the limit point load of the laminatedcomposite cylindrical shell. Table 3 shows a brief compari-son of the limit point loads of GFRP cylindrical shell whichare obtained from the non-linear buckling and the post-buckling analysis approaches. Figure 11 shows the buck-led modeshapes obtained from the linear and the non-linearbuckling analysis which corresponds to the limit point loadfor the given maximum imperfection amplitude (� D 0.3).Figure 12 shows the buckled modeshapes obtained from thepost-buckling analysis which corresponds to the limit pointload and the collapse load (that corresponds to the lowestload after limit point) (� D 0.3). From Figs. 11 and 12, thenumber of lobes (or circumferential waves) obtained fromall the analysis is not equal (and it can be due to anisotropicnature of composites) for the composite cylindrical shell




Table 3. Imperfection sensitivity study on composite (GFRP) cylindrical shell.

Imperfection magnitude (�)Imperfection Sensitivity � D 0:0 � D 0:01 � D 0:1 � D 0:3

(Linear buckling loadD 2737.70)Limit point load (N) 2735 2282 1802 1374(Nonlinear buckling)Limit point load (N) 2740 2280 1800 1372

(Post buckling)

Mode 1 (n = 20) Linear

(a)

Mode 1 (n = 20) Non-linear

(b)

Figure 7. Buckled modeshapes obtained from (a) linear eigenvalue analysis and (b) non-linear bucklinganalysis(� D 1.0) (scaled) for isotropic cylindrical shell.




(a)Mode 1 (n = 20) Post-buckling

analysis

(b)Mode 1 (n = 20) Post-buckling

analysis

(c)Mode 1 (n = 20) Post-buckling

analysis

Figure 8. Modeshapes obtained from post-buckling analysis (� D 1.0) of isotropic cylinder subjected to axial compres-sive load a) Primary limit point load b) Secondary reference point (Lowest point of secondary equilibrium path afterlimit point load) c) Third reference point (traced end point of the tertiary equilibrium path).




Mode 1 (n = 1) Ncr = 2558.2 N

Mode 2 (n = 2) Ncr = 2563.1 N

Mode 3 (n = 2) Ncr = 2563.1 N

Mode 4 (n = 4) Ncr = 2578.5 N

Mode 5 (n = 4) Ncr = 2578.5 N

Figure 9. First five linear buckled modeshapes of composite cylinder subjected to axial compressive load.




Mode 6 (n = 6) Ncr = 2605.6 N

Mode 7 (n = 6) Ncr = 2605.6 N

Mode 8 (n = 8) Ncr = 2643.3 N

Mode 9 (n = 8) Ncr = 2643.3 N

Mode 10 (n = 10) Ncr = 2687.8 N

Figure 10. Linear buckled modeshapes (mode 6 to 10) of composite cylinder subjected to axial compressive load.




Linear (n = 12)(a)

Non-linear (n = 24)(b)

Figure 11. Buckled modeshapes obtained from (a) linear eigenvalue analysis (mode 13) and ( b) non-linear bucklinganalysis (� D 0.3)(scaled) (for composite cylindrical shell)

Post-buckling analysis (Limitpoint load) (n = 24)(a)

Post-buckling analysis (Collapse load) (n = 6)(b)

Figure 12. Modeshapes obtained from the post-buckling analysis (� D 0.3) (a) Primary limit point load (b) Collapse load(end point of the secondary equilibrium path) for composite cylindrical shell (scaled).




which is in contrast to the behavior of the isotropic cylin-drical shell (Figs. 7 and 8).

Confidence gained on the analysis presented on isotropicand composite cylindrical shells can be directly utilizedfor the analysis of advanced grid-stiffened structures withthe various forms of imperfection with generalized loadingconditions such axial, bending and torsion loads.

Figure 13. Non-linear buckling and Post-buckling analysisof composite cylindrical shells for various eigen imperfec-tion amplitudes (�).

5 Conclusions

Non-linear buckling and post-buckling analysis of thincylindrical shells subjected to axial compressive load isbriefly investigated for various eigen imperfection ampli-tudes. For the sake of simplicity the shape of initial im-perfection is chosen as the shape of the linear buckledmodeshape. Non-linear buckling analysis uses the Newton-Raphson approach to predict the primary equilibrium pathwhereas the post-buckling analysis uses an arc-length ap-proach to predict the primary as well as secondary equi-librium paths. Limit point loads obtained from these twoapproaches are in excellent agreement which provides in-dependent check on the numerical results presented. Thesetwo approaches are applied for solving both isotropic andcomposite cylindrical shells. In general, it is observed thatthe magnitude and shape of initial imperfection plays a pre-dominant role in evaluating the variation of the limit pointloads of imperfect cylindrical shells.

References[1] Z. Lorenz, AchsensymmetrischeVerzerrungen in dunwandi-

gen Hohlzylinder. Z. Ver. Deut. Ingr. 52, 1908, 1766 – 1793

[2] S.P. Timoshenko, Einige Stabilitatsrobleme der Elastizats-theorie. Z. Math. Phys. 58, 1910, 337 – 357

[3] R.V. Southwell, On the general theory of elastic stability.Phil. Trans. Royal Soc., London, Series A, 213, 1914, 187 –202

[4] A. Robertson, The Strength of tubular struts. ARC Reportand Memorandum No. 1185, 1929

[5] W. Flugge, Die Stabilitat der Kreiszylinderschalen.Ingenieur-Archiv. 3, 1932, 463 – 506

[6] E.E. Lundquist, Strength test of thin-walled duralumin cylin-ders in compression. NACA Tech. Note, No 473, 1933

[7] W.M. Wilson and N.M. Newmark, The strength of thin cylin-drical shells as columns. Bulletin No 255, Eng. ExperimentalStation, University of Illinois, USA, 1933

[8] M. Stein, The effect on the buckling of perfect cylinders ofpre-buckling deformations and stresses induced by edge sup-port, in Collected Papers on Instability of Shell Structures,NASA Tech. Note, No. D-1510, 1962, 217 – 226

[9] N.J. Hoff and W. Rehfield, Buckling of axially compressedcylindrical shells at stresses smaller than the classical criticalvalue. J. Appl. Mech., ASME 32, 1965, 542 – 546

[10] B.O. Almroth, Influence of edge conditions on the stabil-ity of axially compressed cylindrical shells. AIAA J., II (II),1966, 134 – 140

[11] G. Fischer, Influence of boundary conditions on stability ofthin-walled cylindrical shells under axial load and internalPressure. AIAA J. 3, 1965, 736 – 738

[12] N. Yamaki and S. Kodama, Buckling of circular cylindricalshells under compression-report 3: solutions based on theDonnell type equations considering pre-buckling edge rota-tions. Report of the lnst.of High Speed Mech., No. 25, To-hoku Univ., 1972, 99 – 141

[13] T. Von Karman and H.S. Tsien, The buckling of thin cylin-drical shells under axial compression. J. Aerospace Sci. 8,1941, 303 – 312

[14] L.H. Donnell and C.C. Wan, Effect of imperfections onbuckling of thin cylinders and columns under axial compres-sion. J. Appl. Mech., ASME 17(1), 1950, 73 – 83

[15] W.T. Koiter, I. Elishakoff, Y.W. Li, and JH. Starnes Jr., Buck-ling of an axially compressed cylindrical shell of variablethickness, lnt. J. Solids Struct. 31(6), 1994, 797 – 805

[16] B. Budiansky, and J.W. Hutchinson, A survey of some buck-ling problems. AIAA J. 4(9), 1966, 15051510

[17] J. Arbocz, and J.M.A.M. Hol, Collapse of axially com-pressed cylindrical shells with random imperfections. AIAAJ. 29(12), 1991, 2247 – 2256

[18] H.S. Shen and Q.S. Li, Thermo mechanical post bucklingof shear deformable laminated cylindrical shells with localgeometric imperfections, Int. J.Solids and Struct. 39, 2002,4525 – 4542

[19] W. Schiender, Stimulating equivalent geometric imperfec-tions for the numerical buckling strength verification of axi-ally compressed cylindrical steel shells, Comput. Mech. 37,2006, 530 – 536

[20] R. Lo Frano, and G. Forasassi, Buckling of imperfect thincylindrical shell under lateral pressure, Science and Tech-nology of Nuclear Installations, 2008, Article ID 685805

[21] B. Prabu, N. Rathinam, R. Srinivasan, and K.A.S. Naarayen,Finite Element Analysis of Buckling of Thin CylindricalShell Subjected to Uniform External Pressure, J. Solid Mech.1(2), 2009, 148 – 158

[22] T. Kobayashi, Y. Mihara, and F. Fuji, Path-tracing analysisfor post-buckling process of elastic cylindrical shells underaxial compression, Thin Walled Struct. 61, 2012, 180 – 187




[23] A. Spagnoli, A.Y. Elghazouli, and M.K. Chryssanthopoulos,Numerical simulation of glass-reinforced plastic cylindersunder axial compression, Marine Struct. 14, 2001, 353 – 374

[24] ANSYS, Inc theory of reference

[25] R.D. Cook, D.S. Malkus, and M.E. Plesha, Concepts andApplications of Finite Element Analysis. [Chapter 18], 4thEd. New York, Wiley, 2002



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