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Failure of Engineering Materials & Structures

Code 48

UET TAXILA

MECHNICAL ENGINEERING DEPARTMENT

Buckling Stability of Thin Walled Cylindrical Shells Under

Axial Compression

Himayat Ullah 1 and Sagheer Ahmad 2

1 NESCOM, Islamabad and 2 MED UET Taxila ABSTRACT Light weight thin walled cylindrical shells subjected to external loads are prone to buckling rather than strength failure. In this paper, buckling investigation of thin walled cylindrical shells under axial compression is presented. Buckling failure is studied using analytical, numerical and semi empirical models. Analytical model is developed using Classical Shell small deflection theory. A Semi empirical model is obtained by employing experimental correction factors based on the available test data to the theoretical model. A finite elements model is built using ANSYS FEA Code for the same shell. Finally, the different results obtained using the three analysis methods are compared. The comparison reveals that analytical and numerical linear model results match closely with each other but are higher than the empirical values. To investigate this discrepancy, non linear buckling analysis with large deflection effect, is carried out. The effect of geometric imperfection is also studied through a nonlinear model. These nonlinear analyses show that the effects of nonlinearity and geometric imperfections are responsible for the difference between theoretical and experimental results. NOTATION A Extensional stiffness of shell per unit length Et/1-ν 2

a Radius of cylinder L Length of cylinder t Wall thickness of cylinder amn, bmn, cmn Buckling amplitudes D Flexural stiffness of shell per unit length Et3/12(1- 2ν ) Pcr Critical Buckling Load m Number of waves in longitudinal direction within length of cylinder n Number of waves in circumferential direction

θNN x , Normal in plane forces per unit length (load/unit length) on x and θ planes

xx NN θθ , In plane shear forces per unit length (load/unit length)

θMM x , Bending moments per unit length on x and θ planes

θxM Twisting moment per unit length on axial of plane of cylindrical shell Qx, Qθ Shear force per unit distance on x and θ planes

Himayat Ullah and Sagheer Ahmad FEMS (2007) 48

206

u,v,w Displacements in x, y and z directions; axial, tangential and radial displacements in shell mid surface

Z Batdorf Parameter (L/r)2 r/t 21 ν− ) zxyzxy γγγ ,, Shear strains in xy, yz and zx planes

zyx εεε ,, Normal strains in x, y and z directions

rεεθ , Tangential and radial normal strains χ Change of curvature in shell λ Numerical factor

crσ Compressive stresses at critical loads INTRODUCTION Cylindrical shells have been extensively used in many types of structures. They are subjected to various combinations of loading. The most critical load which challenges the stability of thin shells is axial compression. The usual failure mode associated with thin shell structures is buckling. Many investigations have focused on the axial compression problem for more than 60 years. Love [1] was the first investigator to present a successful linear shell theory based on classical elasticity. Flugge and Byrne [2] presented the second order approximation theory. Donnell [3] developed an eighth order differential equation for determination of critical strength of cylinders with simply supported edges under torsion. Donnell and Fllugge[3,4] highlighted that initial imperfections and the deviation of the actual edge supports from the theoretical support conditions were responsible for observed discrepancy between experimental and theoretical buckling stress values. Batdorf [4] presented a simplified method of elastic stability analysis for thin cylindrical shells. Batdorf, Schildcrout, and Stein [5] employed linear theory as a guide and constructed empirical curves using the data of several of the early investigators. Their Experimentation revealed reduction in critical stress as compared to theoretical values. They highlighted that the observed buckle pattern is different from that predicted on the basis of theory. Von Karman and Tsien [6] introduced a large deflection theory to account for the buckling behavior of long cy1nders. They showed that a long cylinder can be in equilibrium in a buckled state at a stress smaller than the critical stress of linear theory and also succeeded in accounting for the buckle pattern observed in the early stages of buckling. Harris, Seurer, Skeene and Benjamin [7] conducted a series of tests on cylinders ranging from short to long subjected to axial compression. They developed design curves for buckling coefficients versus Batdorf Parameter for r/t ratios from 100 to 2000 and above 2000. Koiter [11] gave theoretical explanation for the influence of initial geometric imperfections on the shell buckling loads. NASA [8] developed monograph which covers the design and analysis criteria of both stiffened and unstiffened thin walled circular shells subjected to various loading conditions. Empirical formulae and design charts based on experimentation are also provided.

MED UET Taxila (2007)

Himayat Ullah and Sagheer Ahmad FEMS (2007) 48 207

S-E Kim and C-S Kim [9] developed practical design equations and charts estimating the buckling strength of the cylindrical shell and tank both perfect and imperfect, subjected to axially compressive loads based on parametric study using ABACUS. Chen & Li [10] investigated nonlinear buckling in thin-walled members with the effect of initial imperfections due to geometry and residual stress. K. Athiannan and R.Palaninathan [11] conducted tests on imperfect shells under transverse load and modeled the imperfection in nonlinear FE model using ABACUS. They developed two models. In the 1st, the real imperfections are imposed at all nodes, and in the 2nd FE model, the imperfections are imposed by renormalizing the eigen mode, using the maximum measured imperfection. G. Catellani, F. Pellicano, D. Dall_Asta, M. Amabili [12] analyzed compressed circular cylindrical shell having geometric imperfections for static and dynamic loading environment. They used Donnell’s nonlinear shallow-shell theory and Sanders shell theory for the analysis. In this paper, shell buckling problem is investigated using classical shell theory, Semi empirical shell model and FE model in ANSYS. Eigen Value Buckling analysis is carried out for critical buckling load and buckling mode shapes. To investigate the discrepancy between theory and experiment, Non Linear large displacement buckling analysis is carried out. The effect of geometric imperfection on the buckling strength is also studied. The imperfections are imposed on the FE models in two ways: (i) Buckling mode imperfect geometry i.e. by renormalizing the eigen mode with an imperfection factor (ii) geometry with real imperfection. Geometric imperfection factors of 0.5,5 and 50 % are used. The numerical buckling loads obtained from these FE models are in agreement with the experimental values. ANALYTICAL SHELL MODEL The analytical model is based on the Kirchoff–Love hypothesis. The problem is analyzed by the Method of Equilibrium. Equilibrium equations, kinematic relations and constitutive equations are obtained. The three sets of field equations: kinematic, constitutive and equilibrium, along with appropriate boundary conditions comprise the governing equations of the mathematical model. Differential Equations of Equilibrium (Statics) Pressure Vessels exemplify the axisymmetrically loaded cylindrical shell. Owing to symmetry an element cut from a cylinder of radius ‘a’ will have acting on it the internal pressures Px , Py, Pz , surface force resultants Nx , Nθ , Nθx , Qx , Qθ and moment resultants Mx, Mθ , Mxθ .

Eliminating the shear forces Qx and Qθ from the above equations and assuming Px = Py = Pz (Pr) = 0, the final equilibrium equations are, a ∂Nx/∂x + ∂Nxθ/∂θ = 0 ∂Nθ/dθ + a ∂Nxθ/∂x + a Nx∂2v/∂x2 –∂Mxθ/∂θ – 1/a ∂Mθ/∂θ= 0 (1) aNx∂2w/∂x2 + Nθ + a∂2Mx/∂x2 + 2∂2Mxθ/∂x∂θ + 1/a ∂2Mθ/∂θ2 = 0



208

Fig1. Cylindrical shell element (a) with internal force resultants and surface loads (b) with internal moment resultants. Kinematic Relationships The strain components at any point through the thickness of the shell, may be written as

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

θ

θ

γεε

x

x

- z ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

xyo

xo

γεε

θ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

θ

θ

χχχ

x

x

Kinematic expressions relating the mid surface strains to the displacement are,

xoε = ∂u/∂x

oθε = 1/a (∂v/∂θ)-w/a (2)

oxθγ = 1/a (∂u/∂θ) + ∂v/∂x Similarly the Changes in curvature at any shell point, xχ and θχ and twist θχ x are expressed by

xχ = ∂2w/∂x2

θχ = 1/a2(∂x/∂θ + ∂2w/∂θ2) (2)

θχ x = 1/a(∂v/∂x + ∂2w/∂x∂θ) Constitutive relations For an isotropic cylindrical shell, the constitutive relations are



{ }{ }

[ ] [ ][ ] [ ]⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

DA

MN

00 { }

{ }⎭⎬⎫

⎩⎨⎧χε

(3)

Where { = }N { }θθ NxNNx ,, T and { } }{ θθ MxMMxM ,,= T being the resultant membrane forces and bending moments. The elastic matrices are given by

[ ]A = Et/1-ν 2

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−2

1000101

νν

ν and [ ]D = Et3/12(1- 2ν )

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−νν

ν

1000101

The governing equations for deflection

The expressions governing the deformation of cylindrical shells subjected to direct and bending forces can now be developed. This is accomplished by introducing the compatibility relations (3) into constitutive relations (4) and then subsequently into the equilibrium equations (2). After differentiation and simplification, the set of differential equations of the buckling problem is as follows: ∂2u/∂x2 + (1+ν)/2a ∂2v/∂x∂θ - ν/a ∂w/∂x+ (1-ν)/2a2∂2u/∂θ2 = 0 (1+ν)/2a ∂2u/∂x∂θ + (1-ν)/2 ∂2v/∂x2+1/a2 ∂2v/∂θ2-1/a2 ∂w/∂θ + α [1/a2∂2v/∂θ2+1/a2∂3w/∂θ3 ∂3w/∂x2∂θ + (1-ν) ∂2v/∂x2] + q (4) 0/ 22 =∂∂ xν a q ∂2w/∂x2 + ν ∂u/∂x +1 /a ∂v/∂θ - w/a – α [1/a ∂3v/∂θ3 +a (2-ν)∂3v/∂x2∂θ + a3∂4w/∂x4 + 1/a ∂4w/∂θ4 + 2a ∂4w/∂x2∂θ2] = 0 ,where Nx = N, α = t2/12a2 and q = (1-ν2) N/Et Solution The general solution of equations (5), if the origin of coordinates is placed at one end of the shell, can be expressed by the series,

u = (c1/va) x + c2 + ∑ a ∞

m∑∞

nm n sin (nθ) cos (mπ x/L)

v = b∑∞

m∑∞

nmn cos (nθ) sin (mπ x/L) (5)

w = c1 + ∑ c∞

m∑∞

nmn sin (nθ) sin (mπ x/L)

Substituting the solution (6) into (5), the trigonometric functions drop out entirely, and equating the determinant of coefficients equal to zero, neglecting the smaller and higher order terms, after simplification we get, Ncr = D [m2 π 2/L2 + E t L2/D m2 π 2a2] (6)



210

Equation (7) is for symmetrical buckling under uniform axial pressure. Neglecting the 2nd term on the RHS of equation (7), as it becomes smaller as compared to the 1st term, we get,

crσ = m2 E t2π 2 /12(1- 2ν ) L2,

For m = 1, crσ = E t2 /12(1-2π 2ν )L2 (7)

By incorporation of buckling coefficient Kc by Batdorf [14], the equation (7) for the compressive buckling stress is given by,

crσ = Kc E /12(1- 2π 2ν ) * (t/ L) 2 (8)

This is the theoretical buckling stress equation. SEMI EMPIRICAL MODEL

The classical small deflection theory has not proved adequate for determining the buckling strength of thin walled cylinders or curved sheet panels. Since there is a large discrepancy between theory and test data of cylindrical shells, thus correction factors (Buckling Coefficients) are employed to the small deflection theory models. These models are then called semi empirical models. Correction factors are based on experimental tests of various researchers and aerospace companies [8, 14]. Fig.2 shows the plot of extensive test data and a 90 percent probability curve derived by the author of [14] by a statistical approach. It shows a set of design curves of Buckling Coefficient ‘Kc’ versus Batdorf Parameter ‘Z’ for various r/t values and for 90 percent probability. In semi empirical models, equation (8) is used, except the buckling coefficient Kc is read from fig.2 test data. The theoretical results are far above the test values as shown by the fig 2.

Fig 2.Compressive Buckling Coefficient for Cylinders (90 % Probability) (Ref.14)



These results show that theoretical buckling strength is 5.3 times greater than experimental An example problem is taken to investigate the difference between theoretical and experimental critical buckling load. A circular cylindrical shell of diameter 2540 mm, Length 1905 mm. and thickness 1.27 mm is subjected to axial compression load. The cylinder is made of Aluminum Alloy Al 2024-T3. The theoretical and experimental buckling loads are determined based on equation (9) and fig.2, and are presented in table1.

Table 1.Determination of Theoretical and Experimental Buckling Loads

(R/t) (L/R) Z Kc (theory)

Kc (Expt)

σcr (Theory) Equ (9)

σcr (Expt) Bruhn[14]

Fig2

Pcr(Theory)

(kN)

Pcr (Expt) (kN)

1000 1.5 2048 1500 280 44.4 MPa

8.3 MPA

450

84.2

test value. To avoid buckling in compression, the axial applied load must be less than P cr (Exp) to provide safety against buckling failure. NUMERICAL BUCKLING ANALYSIS Buckling analysis is a technique used to determine buckling loads - critical loads at which a structure becomes unstable, and buckled mode shapes - the characteristic shape associated with a structure's buckled response. There are two methods to predict buckling load and buckling mode of a structure using FEM.

Linear Buckling Analysis

Linear (Eigenvalue) buckling analysis predicts the theoretical buckling strength (the bifurcation point) of an ideal linear elastic structure. However, imperfections and nonlinearities prevent most real-world structures from achieving their theoretical elastic buckling strength. Thus linear buckling analysis often yields unconservative results, and is not generally used in design of real life structures.

A 3-D FE model using SHELL 93 quadratic element is built for the cylindrical shell of the dimensions given in the above example, using ANSYS 9.0 finite elements software. Linear Isotropic material model is defined for Alum 2024-T3. Mesh convergence check is performed on the FE model by varying the mesh density to validate the results of the buckling analysis. For this purpose, the element size was kept 125 mm on both circumferential and longitudinal shell sides. The size was reduced until mesh convergence is achieved at 50 mm for both buckling load and buckling mode as shown in figs 4 and 5 respectively. Simply supported boundary conditions are imposed on the cylinder under axial compressive load.

Linear buckling analysis in ANSYS finite-elements software is performed in two steps. In the first step, Pre buckling stress of the structure is calculated in a static solution with prestress effects. The second step involves solving the eigenvalue problem given in the form of equation



212

(10). This equation takes into consideration the prebuckling stress effect matrix [S] calculated in the first step. The eigenvalues calculated by the buckling analysis represent buckling load factors. Therefore, if a unit load is specified, the load factors represent the buckling loads.

436.00

438.00

440.00

442.00

444.00

446.00

448.00

450.00

452.00

125X125 100X100 75X75 50X50 25X25

Mesh Size ( mm )

Buc

klin

g Lo

ad (k

N

0

2

4

6

8

10

12

14

125X125 100X100 75X75 50X50 25X25

Mesh Size (mm)N

o of

Axi

al M

odes

Fig 4. Effect of Mesh Density on Fig 5. Effect of Mesh Density on

Buckling Load Buckling Mode

}0{}]){[]([ =+ ii SK ψλ (10) Where = Stiffness matrix, = Stress stiffness matrix ][K ][S

iλ = ith eigenvalue determining buckling load (or Load Factor)

i}{ψ = ith Eigen vector of displacement determining buckling mode

Fig 6.1st Mode Shape, m =12, Axisymmetric, Fig 7. 10th Mode Shape, m = 12, n= 10 Eigenvalue Buckling Analysis yields results in the form of buckling loads and deformed structure shapes called mode shapes. The critical load is the lowest load factor, which is 450 kN. Mode shapes are extracted for each. load sub step. Eigenvalue buckling analysis is generally used to observe the possible failure modes. Various buckling modes obtained by numerical analysis for axial compressive load case are shown in Figures 6 and 7. These mode shapes show



axial (Longitudinal) as well as Circumferential Waves. Nonlinear Buckling Analysis Nonlinear buckling analysis is usually the more accurate approach and is therefore recommended for design or evaluation of actual structures. This technique employs a nonlinear static analysis with gradually increasing loads to seek the load level at which a structure becomes unstable. Using the nonlinear technique, we can include features such as initial imperfections, plastic behavior, gaps, and large-deflection response in FE models. In addition, using deflection-controlled loading, you can even track the post-buckled performance of your structure. To investigate the discrepancy between theoretical and experimental results using numerical technique, the buckling of cylindrical shells is modeled in three ways: (i) Non Linear perfect model (ii) Buckling mode imperfect geometry (iii) geometry with real imperfection Non Linear Perfect Model The Linear model is now solved with nonlinear geometric option. The cylinder is simply supported and the applied load is 10 % higher than critical load predicted by the eigenvalue buckling analysis. As the critical stress is 45 Mpa well below the material yield limit, so linear material model is used. The load is applied in 30 sub steps to seek the limit point of the shell. The limit load is given in table 2 and the load deflection curve is shown in fig.10. Buckling Mode Imperfect Geometry Geometric Imperfection is employed in the non linear FE model. The imperfection shape (an idealized one) imposed is in the form of critical buckling mode and hence the name ‘buckling mode imperfect geometry’. The critical mode shape is renormalized using an imperfection factor. The FE model is scaled by this amount. If the critical mode deflection is ‘δ’, it is normalized by an imperfection factor ‘k’. Thus the FE mesh is scaled by a factor ‘kδ’. Three values of imperfection factors of 0.5%, 5% and 50 % are employed to determine the limit load comparable with the experimental value. The results are shown in table 2 and figure 10.The result of 50 % factor model is in agreement with experimental result. The results show that as the imperfection factor is increased, the value of limit load is reduced and the solution is unconverged at lower sub step. Geometry with Real Imperfection The shell is modeled in five segments along the length. The top and bottom rings have the nominal thickness; the 2nd and 4th ring thickness is reduced by 0.0127 mm and the center ring by 0.025 mm. The boundary conditions are the same as nonlinear model. Load deflection analysis is carried out using geometric non linear option. The solution became unconverged at a lower sub step than the other two models. The limit load is 81.55 kN.



214

DISCUSSION OF RESULTS Application of theory to the design of actual cylindrical shells is complicated by apparent discrepancies between theory and experiment. This behavior can be seen in comparison of the cylindrical shell results in table 2. The table shows that the analytical and linear numerical model results match very well in case of axial compression.

Fig 8 .Non Linear Perfect Model Fig 9 .Buckling Mode Imperfect Geometry (m =13, n=0)(Axisymmetric) with 50 % Factor (m =13, n = 0)

0

50

100

150

200

250

300

350

400

450

500

0.00E+00 2.50E-01 5.00E-01 7.50E-01 1.00E+00 1.25E+00

Deflection (mm)

Forc

e (k

N)

Non Linear without Geometric Imperfection

Geometric Imperfection Factor 0.005



Linear Model

Geometry with Real Imperfection

Fig 10. Effect of Geometric Non Linearity and Geometric Imperfection on Buckling Strength of

Cylindrical Shells.



The results show that limit load is reduced by 67 % of the theoretical load by using large deflection theory. It is evident from the table that the linear classical theory is responsible for the discrepancy between theoretical and empirical results. The deflection plot of fig 8 and 9, show that the axisymmetric critical mode shape is the same as that of the eigenvalue buckling analysis. The table shows that the shell is sensitive to initial geometric imperfections. As the imperfection factor is increased, the limit load is reduced and the shell becomes unstable at a lower load step as shown in fig 10. At factor of 50 %, the limit load is near to the empirical value. The result of shell with real imperfection is the same as that of empirical.

Analytical Model

Numerical Models

Semi-Empirical Bruhn [14] 90% Probability

Linear Model

Nonlinear (Perfect)

Buckling Mode Imperfect Geometry

Geometry with Real Imperfect ion

GIF 0.5 %

GIF 5 %

GIF 50 %

Buckling Load (kN)

450

450.1

146.8 114

98

81.6

81.6

84

Table2: Comparison of buckling loads of the case studied using Analytical, Numerical and

Empirical approach CONCLUSIONS The following conclusions are based on the conducted research work:

1. There is large discrepancy between theoretical and experimental (design) results for buckling under axial compressive loads. Theoretical buckling stress of the example problem is 5.3 times greater than experimental.

2. The investigation augmented that the linear classical theory and initial geometric imperfections are responsible for the discrepancy between theoretical and empirical results.

3. Both the nonlinear models of ‘Buckling Mode Imperfect Geometry’ and ‘Geometry with Real Imperfection’ estimate the limit load closer to the experimental value.

4. Buckling mode shape is the same for linear, perfect and imperfect nonlinear models. 5. It is revealed that the design curves based on test data include both shell nonlinearity and

imperfection effects. 6. It is concluded that the classical buckling formulations should be used for the design of

shells subjected to compressive buckling. Linear theory can only be used for determination of mode shapes. Therefore it is recommended to use nonlinear theory with



216

geometric imperfections and Semi empirical models based on test data for buckling under axial compression.

REFERENCES

1. Love, A.E.H, “A Treatise on the Mathematical Theory of Elasticity” 4 the Ed, Dover

Publications, Inc., New York 1944. 2. Baker E. H., Kovalevsky L. and Rish F. L., “Structural Analysis of Shells”, McGraw-Hill

Book Company, New York, 1972. 3. Donnel L.H, “Stability of Thin Walled Tubes under Torsion” NACA Report No 479. 4. Batdorf S.B, “A simplified Method of Elastic Stability Analysis for Thin Cylindrical

Shells” NACA Report 874. 5. S.B.Batdrof, Murry Schildcrout, and Manuel Stein, NACA Tech Repot No: 1343,

“Critical Stress of Thin Walled Cylinder in Axial Compression” Washington 1947. 6. Von Karma and Tsian, “The Buckling of Thin Cylindrical Shells Under Axial

Compression” Jour. Aero Sciences. Vol 8, June, 1941. 7. Harris, Seurer, Skeene and Benjamin, “The Stability of Thin Walled Unstiffened Circular

Cylinders under Axial Compression” Jour. Aero Sciences. Vol 24, August, 1957. 8. NASA SP-8007, NASA, Space vehicle design criteria. “Buckling of thin walled circular

cylinder” 1968. 9. Seung-Eock Kim and Chang-Sung Kim “Buckling strength of the cylindrical shell and

tank subjected to axially compressive loads” Journal of Thin-Walled Structures 2002; 40,329-353.

10. S.-L. Chen and S.-F. Li “Study on the Nonlinear Buckling in Thin-Walled Members with Arbitrary Initial Imperfection” Journal of Thin- Walled Structures. 1994; 19,253-268

11. K. Athiannan and R. Palaninathan “Buckling of cylindrical shells under transverse shear” Journal of Thin-Walled Structures. 2004; 42, 1307–1328.

12. G. Catellani , F. Pellicano , D. Dall_Asta and M. Amabili ”Parametric instability of a circular cylindrical shell with geometric imperfections Journal of Computers and Structures 2004; 82, 2635–2645.

13. Ugural C Ahsel, “Stresses in Plates and Shells”, 2nd ed., McGraw-Hill Book Company, New York, 1999

14. E.F Bruhn, “Analysis and Design of Flight Vehicle Structures” Jacobs Publishing, USA. 15. Ansys, 2004, “ANSYS Structural Analysis Guide, Buckling Analysis, Rlease 10.0”

ANSYS, Inc. 16. Gerard and Becker, “Handbook of Structural Stability, Part III, Buckling of Curved

Plates and Shells” NACA T.N. 3783, Aug 1957. 17. Rodney Pinna, Beverley F. Ronalds, “Hydrostatic Buckling of Shells with various

boundary conditions” Journal of Constructional Steel Research, May 1999. 18. Philip Walker “Buckling of Cylinders with local loads normal to the shell” The

University of Edinburgh, Edinburgh EH9 3JN, 2001 ABAQUS UK Users Group Conference.

19. Eduardo Manuel Sosa, “Computational Buckling Analysis of Cylindrical Thin Walled Above Ground Tanks” PhD, Thesis University of Puerto Rico.

20. Samuel Kidane, “Buckling Analysis of Grid Stiffened Composite Structures” MS Thesis, Louisiana State University, 2002.


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