A Study on Effects of Initial Deflection on Ultimate Strength of Ring-stiffened Cylindrical...

Thin-Walled Structures 31 (1998) 187–202

Effects of openings of the buckling ofcylindrical shells subjected to axial compression

J.F. Jullien*, A. LimamResearch Unit of Civil Engineering, Institut National des Sciences Applique´es, Lyon, France

Abstract

Experimental and numerical methods are used to study the stability problem of cylindricalshells with cut-outs. The paper presents parametric research of the shape (square, rectangular,circular), the dimensions (axial and circumferential sizes, diameter) of the hole. The effect ofthe location and the number of the holes are also studied. The analysis indicates that thecritical load is sensitive to the opening angle or circumferential size of the hole. The function(critical load-opening angle) is linear for large openings and independent of the geometricalimperfections of the shell. However for small openings, it is necessary to take into accountthe coupling between the initial geometrical imperfections and the openings. The linearapproach does not fit because of the importance of the evolution of the displacements nearthe openings. These results will be used for the development of European rules. 1998Elsevier Science Ltd. All rights reserved.

Keywords:Opening; Buckling; Cylindrical shell; Axial compression

1. Introduction

In many industrial applications, shells are equipped with openings of variousshapes, sizes and locations within the lateral surface. The objective of the presentpaper is to improve the understanding of the effect of cut-outs on the critical bucklingload of thin cylindrical shells.

Little work has been done in the literature on the subject. Early work dates backto Tennyson [1], who performed an experimental study on the effect of small circular

* Corresponding author

0263-8231/98/$—see front matter 1998 Elsevier Science Ltd. All rights reserved.PII: S0263 -8231(98)00003-2

188 J.F. Jullien, A. Limam/Thin-Walled Structures 31 (1998) 187–202

openings on shells which buckle in the elastic range. This work was followed byStarnes [2] and Toda [3], who tested shells with a larger range of opening diameters.

Knodel and Schulz [4] were the first to perform tests on steel cylinders with a largespectrum of imperfections (geometry, loading, material). This gave the possibility fora statistical evaluation of the buckling strength of shells. At the same time no physicalunderstanding of the phenomenon was gained.

The first numerical calculations on cylindrical shells with cut-outs were reportedby Almroth and Holmes [5]. Based on the available results, Samuelson and Eggwertz[6] proposed a simple analytical description of the effect of the diameter of the holeon the buckling strength of shells.

None of the above-mentioned researches have provided sufficient information todevelop a reliable physical model of the considered problem, or to deduce a generalunderstanding. Many tests had been carried out with specimens in Mylar, giving apurely elastic buckling, and the studied parameters did not take into account themeridional or circumferential size of rectangular openings, or the position of theopening in relation to the boundaries. Furthermore, the analysis of coupling geometri-cal imperfections with openings has not been carried out.

In order to achieve this goal, a new study has been performed on steel shellssubjected to uniaxial compressive load. The non-uniform axial stress induced by theopening induces an “internal” bending, and is included in this work.

A series of tests have been undertaken with different shapes (square, rectangular,circular), sizes and positions of openings with respect to the boundaries of the shell.All tests were of high quality with procedures reproducing all the parameters, likethe initial geometrical imperfections. In fact, it is important to know to what extentthe effect persists in the case of shells with openings. The present paper gives aclear answer to that question.

In parallel to the physical testing, an extensive parametric numerical study wasperformed, in order to develop the “know-how” for such a complex case, taking intoaccount the relative influences of all the parameters.

Furthermore, this study gives a guide on developing the design rules.

2. Experimental methods

2.1. Definitions of shells

The notation used is defined in Fig. 1. The upper and lower cross-sections (A andB) were assumed to remain plane and refrained from warping and/or ovalisation.

The upper cross-section was free to translate and rotate around the axes of thesection, while the lower cross-section was fixed. The loading was applied at thecentre of the upper cross-section by means of a displacementuz.

In the circumferential direction, the opening was defined by the central angleu,or the curvilinear length 2c 5 ru or by the dimensionless lengthr

r 5c

√rt

189J.F. Jullien, A. Limam/Thin-Walled Structures 31 (1998) 187–202

Fig. 1. Section of a cylindrical shell and notation.

In the axial direction, the height of square and rectangular cut-outs was definedby h0. The distance from the lower edge of the opening to the lower cross-sectionB of the shell was denoted byhb.

hb 5 f·l 2 h0

2

The experimental study was limited to relatively short shells with approximately,/r 5 2, with an aspect ratio ofr/t 5 280, diameter 2r 5 99 mm, thicknesst 50.175 mm and length, 5 104 mm.

Different types of square, rectangular and circular openings were made, as shownin Table 1. Characteristic dimensions of openings and their positions are defined forthe different cases studied (cases 2 to 7), and for comparison the case of a shellwithout a cut-out (case 1).


Table 1Characteristic dimensions of openings and their positions

Case 1 Without openingCase 2 Dimensions of centrally positioned square opening

Fixed parameterh0 = 2c Variable parameter 2c2c 8.64 17.28 25.92 38.88 51.84

(mm)r 1.47 2.94 4.40 6.61 8.81u 10 20 30 45 60

(degree)Case 3 Distance of the lower edge of the square opening from cross-section B.

Fixed parameterr = 4.40f 1 0.75 0.50 0.25 0

Case 4 Number of equidistanced square openingsnb: number of openingsr: 1 openingr 1.47 4.40 4.40 8.80nb 2 2 3 2

Case 5 Height of centrally positioned rectangular openingFixed openingr = 2.94

h0/, 0.17 0.25 0.37 0.50 0.75Case 6 Dimensionless curvilinear width of centrally positioned rectangular opening

Fixed parameterh0/, = 0.17r 2.94 4.40 6.61 8.81 13.23

Case 7 Dimensionless diameter of centrally positioned circular openingr 0.15 0.28 0.73 1.46 2.88 4.34 5.75

2.2. Material and manufacturing

Over 100 specimens were manufactured. Cylindrical shells were made from sheetmetal by means of rolling and electric welding along a generator. These operationswere automated to ensure that the resulting geometric imperfections were the samefor all specimens, which is a necessary condition for the present comparative study.

The opening was always situated on the opposite side of the weld. Great care wastaken so that the process of cutting the opening did not introduce any additionalimperfections.

The tests were carried out on a special device, under displacement control accord-ing to the geometrical and boundary conditions defined above (Fig. 1). The boundaryconditions were introduced in the specimens to be tested by means of two rigidplates, which were connected to cylinders by a sobering technique.

2.3. Initial geometrical imperfections

For each specimen tested the geometrical imperfections were measured by meansof a contactless sensor at 25,000 points. For that purpose a special automated scan-ning system was mounted on the testing machine.

The geometric imperfections were located mainly in the vicinity of the weldedgenerator. The imperfections were characterised in the circumferential direction bythe wavelength corresponding ton 5 7 or 8, with a maximum amplitude of


2w(m) 5 2 3 157mm. The corresponding dimensionless amplitude iswt

(n) 5 0.9.

There were no imperfections in the meridional direction (m 5 0) except along thewelded generator which shows two half waves (m 5 2) with an amplitude

2w(m) 5 2 3 98 mm Swt

(m) 5 0.56D (Fig. 2).

2.4. Reference shell

The results for different cases of openings are compared with those correspondingto shells without cut-outs. This resulted in a large experimental and numerical studyof shells without cut-outs and with typical geometrical imperfections.

The reproductibility of the present test methodology is verified by comparingresults for eight specimens without an opening. The mean value of buckling loadwas P0 5 11.78 KN, with a standard deviation of 58 N, which corresponds to 5%of P0. The corresponding stress isscr 5 216 MPa representing 47% of the theoreticalbuckling stressscr.

3. Effect of shape of cut-out

3.1. Square openings

For given dimensions of the square and centred opening under consideration, theforce-displacement characteristic of the shell remains linear up to the first local buck-ling (LB), which occurs in the vicinity of a contour of the opening. After this pointthe behaviour is linear and stable with a reduced slope until a global buckling(collapse buckling, CB) takes place which involves the entire shell, Fig. 3.

In the pre-buckling range, before the first local buckling, large radial displacementsdevelop near the edges of the openings, in particular around the rim of the hole.

Fig. 2. Measured profiles of geometrical imperfections for one case of the opening.


Fig. 3. Compression of load-deflection curves for cases 1 and 2 showing the effect of hole size.

These observations are visualised in Fig. 4, which shows radial displacements fortwo types of square openings at 50 and 80% of the local buckling, at the first localbuckling and finally at the collapse buckling.

A square cut-out in a cylindrical shell subjected to axial compression reduces thefirst local buckling load and the global collapse buckling load as compared to thereference shell without a hole. The drop of critical loads with the dimensionlessgeometrical parameter of the openingr is described by a linear function in the studiedrange 1.47, r , 8.81. The ratio between these two critical loads is a function ofr, with a mean value of approximately 1.1 (Fig. 5). Beyond the point of approxi-mately r 5 10, the first local buckling does not develop.

For chosen dimensions of the square cut-outr 5 4.40 the position of the openingalong the meridional direction (axis of a cylinder) with respect to cross-sections Aor B does not change the critical load. This interesting result indicates that the criticalload is not linked to the direct distribution of stresses above or below the hole. Theabove conclusion is valid as long as cross-sections A and B remain plane but couldrotate. In practice, this boundary condition could be satisfied by placing reinforcingrings of sufficient rigidity at these cross-sections, Fig. 5.

Multiple and equidistanced openings over the circumferential direction produce asimilar type of instability (first local buckling followed by collapse buckling) witha corresponding loss of rigidity. In the case of multiple openings the reduction inthe critical load is smaller when the comparison is made taking the sum of theopening angle, as shown in Fig. 6. Because of symmetry, the appearance of criticalloads is not accompanied by the development of “internal” bending. The abovedependence can be converted to a different co-ordinate system in which the abscissa


Fig. 4. Evolution of buckling forms (prebuckling, local buckling (LB), global buckling (CB). Whiterepresents peaks and blue represents valleys as viewed from inside.

is not the sum of openings but the width of a single opening. It is interesting to notethat in the above co-ordinate system, the first local buckling load is the same for ashell with single and multiple openings.


Fig. 5. Reduction of critical loads with an opening angle, showing weak effect of the height and positionof a hole in the meridional direction.

Fig. 6. Reduction of the first local buckling load with the size and number of openings (cases 2 and 6).

3.2. Rectangular openings

A cylindrical shell with a rectangular opening carries the same critical bucklingload as an identical shell with a square opening, provided the cut-outs are of anidentical length in the circumferential direction, Fig. 5. The height of the openingdoes not intervene in the studied range of parameters. At the same time this con-clusion cannot be extended to cut-outs in the form of a slot, without a separate study.


3.3. Circular openings

A cylindrical shell with a circular opening of a diameter equal to the width of thesquare opening develops the same collapse buckling load (Fig. 7). At the same timethe first load buckling is not observed in this case. The above result indicates thatthe occurrence of the first local buckling is preceeded by a redistribution of stressesdue to stress concentration at the corners of square or rectangular cut-outs.

For this type of opening the effect of a small hole dimensionr 5 0.15; 0.28; hasalso been analysed. It was found that very small holes do not cause any appreciablechange in the critical load.

4. Effect of geometrical parameters

4.1. Opening angle

The present experimental study has determined the effect of the shape of the open-ing on the critical load of cylinders subjected to axial compression. It was shown thatthe key parameter controlling the buckling strength of the shell was the dimension ofa cut-out in the circumferential direction. In the case of multiple openings positionedequidistantly around the circumference, the control parameter was the sum of holedimensions of all openings.

Fig. 7. Dimensionless critical load versus and opening angle for square and circular holes. Note that thereis no local buckling for a circular opening and a generalised buckling load is the same for both shapes.


The above experimental results are confirmed by finite element calculations. Thecalculations were run using the three-dimensional code CASTEM 2000, developedby the French Atomic Energy Commission. The code is capable of dealing withvarious types of singularities. Half of the shell was modelled by 3500 triangularelements DKT (three displacements and three rotations per node), as shown in Fig.8. Contrary to previous analysis on buckling of shells, linear analysis (LA) was foundto give a smaller critical load than geometrically non-linear analysis (GNA), asshown in Fig. 9. This can be explained by an inability of the geometrically linearmodel to account for large radial displacements around the opening.

In fact, already at 20% of the critical load, radial displacements exceed by approxi-mately two times the displacements predicted by linear theory. Further refinementof the model by introducing the non-linear material behaviour (GMNA) brings thelevel of stresses even lower.

The result of the linear (LA) and non-linear (GMNA) analyses mentioned abovecorresponds to loading case (a) when the cross-sections A and B remain parallel.This condition does not fully reflect reality because a single opening produces anasymmetry and so-called internal bending. At the same time, the results of calcu-lations with geometrical and material non-linearities referred to in Fig. 9 as case (b)correspond to the boundary condition in which cross-section A is free to displaceand rotate in all three directions.

It should be noted that for design purposes one has to consider rather conserva-tively only the first critical buckling load, even though there could be a stable post-critical range until collapse buckling occurs. Remember that for circular holes thereis no first local buckling. Therefore the design should be made on the basis of globalcollapse buckling.

Fig. 8. Finite element mesh of a cylinder with an opening (3500 elements for one half of the shell).


Fig. 9. Finite element results for two types of boundary and loading conditions (a) and (b) and linear(LA) and non-linear (GNA and GMNA) formulations.

4.2. Length to radius ratio

As was mentioned in Section 2.1 all the results presented so far are valid onlyfor shells with,/r 5 2.

The amount of bending deformation introduced by the presence of non-symmetri-cally positioned openings will depend on the distance between two cross-sections,taking into account the second order effect and greater susceptibility for shape distor-tion in the critical cross-section in the case of longer cylinders. Fig. 10 indicates thatthere is an asymptotic limit for the length dependence. A more detailed analysis isnecessary to quantify the above dependence. In the mean time, the existing resultscan be used for shells reinforced by two heavy rings at cross-sections A and Bdistanced, 5 2r apart. This will ensure that cross-sections remain plane withoutwarping and ovalisation.

4.3. Geometrical imperfections

The experimental results for different forms of openings correspond to metal cylin-ders characterised by reproducible geometric imperfections encountered in industrialpractice. These results were compared with earlier results reported by Starnes [2], seeFig. 11. While the shell material is different (Mylar versus steel) and the geometricalimperfections in Starnes’s experiments are much smaller compared with the presentones, the overall behaviour of both types of shells is identical in a certain range of


Fig. 10. Effect of dimensionless length,/r on critical buckling load (numerical study GMNA (b)).

Fig. 11. Comparison of INSA tests with earlier results by Starnes [2].

the parameterr. The range of interest corresponds to angular openings larger thana limiting value denoted byr,.

In the first approximation, the geometrical imperfections are seen not to have anysignificant effect compared with the effect of opening beyond the critical limitr,.

It is possible to identify a reduction function for a given opening over the ranger > r,. From Fig. 11 it transpires that for small openings up to this limiting value,


the effect of imperfections must be coupled with that of the cut-out. At the sametime, for very small openings, their size is not important compared with a significantimperfection sensitivity of cylinders subjected to compressive loads.

As shown in Fig. 12 a comparison between calculations and experimental resultsis in full agreement.

The obtained results are limited to the aperture sizer , 5 6.61 (i.e.u 545°) and 212, r/t , 400. These limits could be extended in the continuation of

this research.In Fig. 12, the limits of two distinct zones are clearly identified. In the case of

small openings, it is necessary to include in the analysis a coupling between thegeometrical imperfection and presence of an opening. In the range of moderatelylarge openings, the existence of geometrical imperfections of any size can be neglect-ed.

A careful examination of post-critical geometry for all cases considered indicatesthat the modal geometry is not effected by the presence of openings. This statementis valid for different dimensions and shapes of shells tested. One can conclude thatthe mode numbersn andm are “invariants”, because they depend on the geometryof a shell rather than that of an opening.

It should be recalled that the critical load has an asymptote with increasing ampli-tude of imperfections. It is important to note that the dependence of the critical loadon the hole size has a similar form as the known imperfection sensitivity. There

Fig. 12. Comparison between experimental results and numerical calculations for perfect (GMNA) andimperfect (GMNIA) shells. Note the existence of two ranges. For smallr there is a coupling betweengeometrical imperfection and hole size. For medium opening, there is no coupling.


the reduction factor reaches a maximum (asymptotic) value where the opening sizecorresponds to the half-length of the buckling wave [7,8]. It is shown in Fig. 12 thatgeometrical imperfections no longer have any effect onscr for r > 2.94, i.e. for anopening equal or larger than the half-wave of the Yoshimura pattern.

4.4. Principle of the proposed rule

The proposed rule uses a reduction (knock-down) factor to the critical Donnellstressscr in the same way as is done for a cylindrical shell without an opening. Thisrule is developed for the most unfavourable case, i.e. for cut-outs with sharp corners(squares, rectangles). For the purpose of evolving a European rule [9], the samenotation is used for the reduction factora, as in the classical buckling analysis ofimperfect shells. The reduction factora is a linear piece-wise function of the geo-metrical parameter of the openingsr, (Fig. 13). In the design rule it is proposed thatthree distinct regimes be distinguished, denoted respectively by (1), (2) and (3).Regime (1) corresponds to very small openings in which the size of the hole haspractically no effect on the critical load. The shell response is dominated by thegeometrical imperfections. The reduction factor is constant with respect to the open-ing parameterr, and its magnitude is equal to the reduction factor of an imperfectcylinder without an opening subjected to axial compression.

Responses of shells with relatively large openings are dominated by the presenceof cut-outs rather than geometrical imperfections. This function is described by alinear dependency given by segment (3) in Fig. 13. For intermediate hole sizes thereis a strong coupling effect between the opening parameterr and geometrical imper-

Fig. 13. Conceptual sketch of the “opening reduction factor” function.


fections. The reduction factor in this range is approximated by a straight line, desig-nated in Fig. 13 as segment (2).

In order to define the position of the above three lines on thea 2 r plane it isnecessary to identify six parameters.

5. Conclusion

The above experimental and numerical work leads to the following conclusions,while enhancing the European rules relating to shell buckling.

O The opening parameterr characterizes the effect of a hole on the critical load ofa cylindrical shell with an opening under axial compression;

O The behaviour shows two large zones:O for small openings in which there is a coupling between the geometrical imperfec-

tion and the presence of an opening;O for a moderately larger opening the existence of geometrical imperfections of any

size can be neglected;

O The limit between these two zones is defined by an opening angle equivalent tothe length of the half-wave of the shell without any opening;

O The critical mode is independent of an opening;O For a moderately larger opening the reduction of critical load is linear with the

angle of the opening;O The height and the position of the hole have little influence;O The linear bifurcation theory underestimates the carrying capacity because it pre-

dicts buckling of the vertical edge of the cut-out.

Acknowledgements

The authors wish to thank the CECA-ECSC, which has supported this research(contract no. 7210-SA/208, part E “Enhancement of ECCS design recommendationand development of Eurocode 3, parts related to shell buckling”).

References

[1] Tennyson RC. The effect of unreinforced circular cutouts on the buckling of circular cylindrical shellsunder axial compression. Journal of Engineering for Industry 1968;November:541–6.

[2] Starnes JH. The effects of cutouts on the buckling of thin shells. In: Fung YC, Sechler EE, editors.Thin shell structures. Englewood Cliffs, New York: Prentice Hall, Inc., 1974:289–304.

[3] Toda S. Buckling of cylinders with cutouts under axial compression. Journal of Experimental Mech-anics 1983;3:414–7.

[4] Knodel P, Schulz U. Stabilite´ de chemine´es d’acier a` ouvertures dans les tuyaux. Stahlbau (Der)1988;57(1):13–21.


[5] Almroth BO, Holmes AM. Buckling of shells with cutouts. Experimental and analysis. Journal ofSolids Structures 1972;8:1057–566.

[6] Samuelson AL, Eggwertz S. Shell stability handbook. London: Elsevier Applied Science, 1992:278pp.

[7] Jullien JF, Limam A. Effect of openings on the buckling of cylindrical shells subjected to axialcompression. International Conference on Advances in Steel Structures, Hong Kong, 11–14December 1996.

[8] Al Sarraj M. Effets des ouvertures sur la stabilite´ des coques cylindriques minces soumises a` com-pression axiale. The´se de Doctorat, INSA de Lyon (France), 5 Octobre 1995:260 pp.

[9] ECCS/TWG8.4 Buckling of steel shells. European recommendations, 4th edn, no. 56, 1988.

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