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Experimental studying of buckling of stringer cylindrical shells under axial compression
ARTICLE IN PRESS
Thin-Walled Structures 45 (2007) 877–882 www.elsevier.com/locate/tws
Experimental studying of buckling of stringer cylindrical shells under axial compression V.L. Krasovskya,, V.V. Kostyrkob a
Department of Building Mechanics and Strength Materials, Prydneprovsk State Academy of Civil Engineering and Architecture, Dniepropetrovsk, Ukraine b Department of Theoretical Mechanics, Dniepropetrovsk National University, Dniepropetrovsk, Ukraine Available online 4 October 2007
Abstract Results of tests on axial compression of small-sized quality steel cylinder shells strengthened by 24 and 36 longitudinal thin-walled stiffeners are presented. The shell length was varied. Shells both with inside and outside stiffening were tested at simply supported and clamped edges. The shell carrying capacity that was governed in the tests by overall buckling in the elastic range was compared with the estimated critical loads based on structural-orthotropic theory. The satisfactory quantitative correlation has been received only for the long simply supported shells with 36 inner stiffeners, which demonstrated insignificant effect of local undulation that preceded overall deflections. The experimental and the theoretical results differed significantly (twice as much) when the actual mechanism of lateral deflection caused by the intensive local undulation differed from the adopted model. r 2007 Elsevier Ltd. All rights reserved. Keywords: Buckling; Cylindrical shells; Stiffened shells; Compression; Experiment
1. Introduction In the class of longitudinally stiffened shells (stringer shells) subjected to axial compression, deformation and buckling of cylinders with multiple stiffeners and with relatively weak bending stiffness has been the subject of careful researches. Behavior of such structures can be rather satisfactorily described by the linear theory of structurally orthotropic shells [1–3]. However, correlation of theoretical and experiment results obtained within the broad range of structural parameter changes is found to be unsatisfactory if spacing between the stiffeners is not enough small (stringer shells almost close to rational ones) [4]. This work presents data of the experimental investigation carried out with the aim to study systematically the influence of the following structural factors on the mechanism of the stringer shell carrying capacity exhaustion: (1) number of stiffeners; (2) stiffeners eccentricity
Corresponding author. Tel.: +38 0 56 247 03 79.
E-mail address: [email protected] (V.L. Krasovsky). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.08.028
sign; (3) shell length; (4) edge fixing requirements (simple support or clamping). 2. Specimens and the tests procedure The experiments were carried out on small-sized specimens with diameter 2R ¼ 143 mm and the wall thickness h ¼ 0.19 mm (R/h ¼ 376). All shells were stiffened with identical equidistant longitudinal thin-walled stiffeners of angular profile with dimensions: 4.0 4.3 0.34 mm. Depending on the number of stiffeners (k), the specimens were subdivided into two series: series 1, k ¼ 24 and series 2, k ¼ 36. Length of the specimens in each series varied in a wide range. The shells of each dimension had either external or internal stiffeners. The shells were made of the cold-rolled stainless steel grade X18H9H (elastic modulus E ¼ 191 GPa, 0.2% proof stress ¼ 800 MPa, Poisson’s factor n ¼ 0.3) by applying the contact spot welding with one longitudinal lap seam whose width was equal to 0.02 of the shell perimeter. The stiffeners were made of metal strips on a special bending appliance and spot welded abreast to the shell’s outer or
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V.L. Krasovsky, V.V. Kostyrko / Thin-Walled Structures 45 (2007) 877–882
inner surface along the narrow side of the angle bar. Prior to fitting the shells to the test machine, special edge devices were attached to them. The compressive force was developed by the UME-10tm mechanical universal testing machine. All shells were subjected to the kinematical loading at constant speed of 0.05 mm/min. Pairs of specimens with inner and outer arrangement of stringers, with either simply supported or clamped edges, were tested. In the case of hinged edges, the load to the specimens was transmitted through the shell only. In the case of clamping, loading was uniformly distributed along the whole cross-sectional area of the specimen. 3. Test results 3.1. Series 1 A total of 35 specimens with L/R ¼ 0.28–1.80 were tested. We are going to describe the buckling process in the hinged shells. The buckling was preceded by the local undulation between the stiffeners. The first dimples of the local buckling appeared, as a rule, in the form of a ‘‘light’’ clap during the process of initial bending deflection (Fig. 1a). The process of intensive local undulation started at a certain level of the load. These loads were assumed as local buckling loads Nm (between the stiffeners). Formation of the local dents was accompanied with claps. As a result of the shell’s ‘‘rapid’’ undulation process, the local post-critical configuration of dimples in ‘‘chess’’ order was
formed (Fig. 1b), with twisted stiffeners. The local dimples developed as the load increased, as well as the stiffeners’ twist angles. The tests performed at low rates of loading have allowed observing the process of overall buckling, different in the shells with inside and outside stiffeners. At loading rate V ¼ 0.005 mm/min, the following sequence of carrying capacity exhaustion was observed for outside stiffened specimens, with L ¼ 80–130 mm. Local folding of stringers along the diagonals that linked local dimples on adjacent panels was accompanied by the acute clap and the walls distortions (out of the plane). As a result, the shells surface was covered with rather small, but clearly distinctive diamond-shaped dents (Fig. 1c). When the motor of the test machine was turned off, the described form could not keep for too long time (1–3 s). The next clap followed with formation of a new shape: large elongated shells dents (typical for stringer stiffening), covering several adjacent stiffeners (Fig. 1d). At reduced length L, such a clap, as a rule, was not observed, and the initial dents developed and spread on the whole shell length. Within small areas, the diamond-shaped dimples were arranged in a single row, and after the stiffeners folding the shells used to crumple. Buckling of shells with inner stiffeners was developing with not so sharp changes in the form. The overall postcritical shape dents were originated on the boundary of two local adjacent dents separated by a stiffener. At first, its development was accompanied with bending of only this element. Then, the dents started growing and covered several adjacent stringers. Usually, the process was rather
Fig. 1. Local and overall buckling shapes for shells of series 1.
ARTICLE IN PRESS V.L. Krasovsky, V.V. Kostyrko / Thin-Walled Structures 45 (2007) 877–882
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Fig. 2. Critical loads of local and overall buckling for shells of series 1.
rapid and was accompanied with a dull clap in the case of long shells (Fig. 1e and f). Clamping of the edge did not result in any significant changes in the carrying capacity exhaustion mechanism. Critical loads of global buckling (Ncr) for all shells turned out to be ultimate too. Fig. 2 shows values of Nm (circles) and Ncr (triangles) versus the shell length, of the specimens with outside (light marks) and inside (dark marks) stiffening, in the case of simple support (Fig. 2a) and clamping (Fig. 2b) of the edges. The theoretical solutions with account of the prebuckling deformation are presented by the curves ‘‘1’’ and ‘‘2’’ (for inside and outside stiffening, correspondingly). The curves for zero-torque prebuckling theory are marked ‘‘10 ’’ and ‘‘20 ’’ (this relates also to Fig. 4). The analysis of these data demonstrates that the level Nm practically does not depend on the shell length. Only in the case of short shells, Nm depends on the length. At inner stiffening, local buckling of the shells occurred at slightly higher loads in comparison with the outer stiffening. In the m case of hinged edges, the ratio N m þ =N (‘‘+’’ relates to inner stiffening, ‘‘’’ to outer one) ranges, in average, from 1.3 to 1.5. The boundary conditions practically do not affect the value of Nm, and the difference between values of Nm for shells with inside and outside stiffeners remains. Unlike Nm, values of Ncr essentially depend on the shell length. Reduction of L, in all cases, leads to monotonous growth of Ncr. The experimentally obtained increase in critical loads due to outside stiffening (comparing with the inside stiffeners) in the case of simple support is insignificant and varies from 5% to 20%. At clamping of the
edges, the stiffeners eccentricity sign does not practically affect the carrying capacity. At inside stiffening, the increase in critical loads caused by the edge clamping, in comparison with the simple support, ranges from 10% to 20%. In the case of outside stiffening, no increase of Ncr has been observed. Advantages of outside stiffening were clearly visible only in the case of simple supported edges and for moderate length shells, and attained 25%. 3.2. Series 2 This series covers the shells with k ¼ 36. The total of 28 specimens, the length of which varied from 20 to 200 mm (L/R ¼ 0.28–2.80), were tested. The local buckling of the shells between the stiffeners also preceded the overall buckling of specimens of this series. The overall buckling developed following the same mechanism described above for the shells of series 1 (Fig. 3a). At the same time, no short wave modes with folding stiffeners, preceding formation of typical large dimples of overall buckling, were observed. Overall buckling of the long shells (L ¼ 130–200 mm) was accompanied with formation of large dimples with smooth outlines (Fig. 3b). When the stiffeners were fitted from outside, they had folds in the middle cross-section. The shells with minimal length buckled in mode close axisymmetric one (Fig. 3c and d). The dependence of critical loads Nm and Ncr on the shell length is shown in Fig. 4a and b (here we used the same notation as in Fig. 2a and b for shells of series 1).
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Fig. 3. Local and overall buckling shapes for shells of series 2.
Fig. 4. Critical loads of local and overall buckling for shells of series 2.
It is seen that the difference between the levels of Nm for the shells with outside and inside stiffening at both simply supported and clamped edges has practically disappeared. Independently of the boundary conditions, the advantage of the outside stiffening is more vivid. In addition, in the case of the specimens with L ¼ 120 mm, this increase reached 45% at hinged edges, and 31% at clamping. When the shell length was reduced, the cr difference between N cr and N þ also reduced, and at buckling in the axisymmetric mode (L ¼ 120 mm), the critical load turned out to be higher at inside arrangement of the stiffeners.
The comparison of the test results for both types of boundary conditions reveals that critical loads for inside stiffening of the shells with clamped edges significantly exceed Ncr for hinged edges practically in the whole range of lengths tested. The effect of boundary conditions on Ncr in the case of shells with outside stiffening was not noted. 4. Comparison of the experimental and theoretical results The experimental values of Ncr were compared with theoretical predictions based on the structural-orthotrophic model of stiffened shells. Computations were
ARTICLE IN PRESS V.L. Krasovsky, V.V. Kostyrko / Thin-Walled Structures 45 (2007) 877–882
carried out both for initial momentless state of the shells and with taking into account non-linear axisymmetric prebuckling deformation. Analysis of the theoretical dependences showed that the main factor influencing Ncr (when we take into account the prebuckling deformation) is partially caused by the presence of transverse deformations in the shell, and also partially due to the asymmetrical structure of the shell. Note that we took into account the real character of applying the load to the shell for hinged edges (external compressive force was applied only to the shell skin) and for clamped edges (the load was applied to the whole cross-section). The theoretical dependencies of Ncr on the shell length for series 1 and 2 are given in Figs. 2a, b and 4a, b, respectively. The solid line represents the solution with account of the prebuckling deformation (inside stiffening— curve 1, outside stiffening—curve 2). Computed curves for Ncr at zero-torque initial condition are represented by the dotted line (inner stiffening—curve 10 , outer stiffening— curve 20 ). It is seen in the presented graphs that in all cases the experimental values of critical loads lie below the theoretical values. Let us consider, in more details, the case of the simply supported edges. At inside stiffening of shells of series 1, the best correlation between the theory and the experiment was attained for the shells with the large length (L ¼ 130 mm), however, even in this case the difference between the estimated and experimental values made about 30%. At reduction of L the discrepancy increased. The larger the number of the stiffeners the better was the correlation of the estimated and experimental data. In the case of specimens of series 2 strengthened by the inside stiffeners, in a wide range of L, the estimated values of Ncr differed from the experimental values by not more than 17%. In the case of outside arrangement of the stiffeners, the difference between the experimental and estimated values increased. In case of specimens of series 1 with L ¼ 130 mm, the difference was 50%, and it grew with reduction of length. The discrepancy between the theoretical results and the experimental data for shells of series 2 in the range L ¼ 200–120 mm did not exceed 20%. However, further reductions in length increased it significantly. The discrepancy between the theoretical and experimental results significantly increased in the case of clamped edges. It was not less than 60–80% in the shells with inside stiffeners and more than 100% for outside stiffened shells. In addition, unlike the case of hinged edges, the correlation did not improve as the stiffeners number increased. The theoretically predicted significant advantage of the outside stiffening comparing with the inside stiffening was not confirmed by the experiment for this case. The cr estimated level of ratio N cr =N þ was attained only for the long shells of series 2. The effect of the clamping edges was exhibited mostly for the shells with the inside stiffeners. However, in this case cr the obtained ratios N cr z =N f (‘‘z’’ relates to clamping, ‘‘f’’ to
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simple support) were significantly below the estimated one. Their maximum value did not exceed 1.3 for shells of series 1 and 1.25 for series 2. 5. Discussion and conclusion It is evident that the main reason for discrepancies between the experimental and theoretical values of loads of overall buckling Ncr is the prior local buckling of the shell between the stiffeners. Here, we revealed two mechanisms of its influence on the process of overall buckling and value of Ncr. First, the local buckling of a shell reduces its rigidity to tension–compression and shear, this leads to lowering rigidity of the shell as a whole, as well as to additional compression and bending of the stiffeners. Second, the shell chess-type dimples developed due to the local buckling contributed to the stiffeners twisting. This causes lowering of longitudinal bending rigidity of the shell, in particular in the areas with large tangential shift of the stiffener’s upper ends. The degree of influence of these factors on the shell carrying capacity is determined by the magnitude of the local mode to the moment when the overall buckling occurs. In the case of long shells with simply supported edges and average length shells of series 2, the intensity of local mode was comparatively low. The level of carrying capacity is mainly influenced only by the factor of the structure rigidity reduction due to tension–compression. The shells with inside stiffening have critical loads that were rather close to the estimated ones in the wide range of changes of their length. In the case of outside stiffening, due to higher critical loads, the discrepancy between the theory and the experiment was somewhat higher. In this case, the lowering of shell rigidity may be taken into account within the applied structural-orthotrophic model by reducing the shell skin with the better resulting correlation between the theory and the experiment. The intensive local undulation that is typical for the shells of series 1, as well as for the short shells of series 2, has an effect not only on the Ncr value, but also on the mechanism of overall buckling. That is clearly demonstrated by the shells with outside stiffening in the form of the short-wave mode of buckling with folding of the stiffener walls. There was no local folding of the stringer’s walls in the shells with inside stiffeners; however, the presence of the developed local mode had a substantial effect on the process of the overall undulation. A typical feature for both types of stiffening is the considerable reduction of shell rigidity both to tension–compression and to bending (due to deformation of the stiffener wall out of the plane initiated by the local undulation of the shell). That is the main reason of the significant difference between the experimental and the estimated critical loads and buckling modes. Estimation of carrying capacity for such shells is possible only when all above-mentioned effects are taken into account, especially the ones that have a clearly expressed non-linear nature. It should be noted
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here that local folding of the stiffeners walls was also observed at testing the shells, buckling of which was the result of interaction between the local modes of the stiffeners buckling and the overall modes. A non-linear theory of the ‘‘coupled’’ lateral buckling of stiffened shells developed mainly by Manevich [5], has allowed, at present, to improve significantly the agreement between the experimental and the theoretical data for a rather wide class of shells stiffened by thin-walled stiffeners. But, being an asymptotic theory, this theory does not take into account certain features of the shell prebuckling deformation prior to the overall buckling. Certainly, in the case of the shells with intensive local undulation, the experimentally obtained increase in Ncr at more rigid boundary conditions turns out to be less than the one predicted by the structural-orthotropic theory, in particular for the outside stiffened shells with short wave buckling. This result may be explained by difference between the actual mechanism of buckling and the theoretical model. At the same time, even for the specimens, the buckling mechanism of which agreed with the model, the theoretical values of Ncr at clamped edges remained significantly higher than the experimental ones (long shells, series 2). It means that assessment of boundary conditions effects based on the structural-orthotrophic
model for the considered class of shells may lead to wrong results. The local buckling of the shell and related effects are also the reason for the significant reduction of the actual effect of the stiffeners arrangement eccentricity sign on Ncr comparing with the theoretical prediction. References [1] Singer J. The influence of stiffener geometry and spacing on the buckling of axially compressed cylindrical and conical shells. In: Theory of thin shells—Proceedings of Second IUTAM symposium, Copenhagen, 1967. Berlin: Springer; 1969. p. 234–63. [2] Singer J. Buckling of integrally stiffened cylindrical shells—a review of experiment and theory. In: Contribution theory of aircraft structuring, Delft, 1972. p. 325–57. [3] Weller T, Singer J. Further experimental studies on buckling of integrally ring-stiffened cylindrical shells under axial compression. Exp Mech 1974;14(7):267–73. [4] Manevich AI, Demeshko MF, Krasovsky VL, Kucherenko VM. An experimental investigation of stability of longitudinally stiffened cylindrical shells under axial compression. In: Raschiot prostranstennykh konstrukciy. Moscow: Stroyizdat; 1971. no. 14, p. 87–102 [in Russian]. [5] Manevich AI. Coupled instability of cylindrical shells stiffened with thin stiffeners. In: Proceedings of third international conference on thin-walled structures: advances and developments. Cracow: Elsevier Science Ltd.; 2001. p. 683–91.
Thin-Walled Structures 45 (2007) 877–882 www.elsevier.com/locate/tws
Experimental studying of buckling of stringer cylindrical shells under axial compression V.L. Krasovskya,, V.V. Kostyrkob a
Department of Building Mechanics and Strength Materials, Prydneprovsk State Academy of Civil Engineering and Architecture, Dniepropetrovsk, Ukraine b Department of Theoretical Mechanics, Dniepropetrovsk National University, Dniepropetrovsk, Ukraine Available online 4 October 2007
Abstract Results of tests on axial compression of small-sized quality steel cylinder shells strengthened by 24 and 36 longitudinal thin-walled stiffeners are presented. The shell length was varied. Shells both with inside and outside stiffening were tested at simply supported and clamped edges. The shell carrying capacity that was governed in the tests by overall buckling in the elastic range was compared with the estimated critical loads based on structural-orthotropic theory. The satisfactory quantitative correlation has been received only for the long simply supported shells with 36 inner stiffeners, which demonstrated insignificant effect of local undulation that preceded overall deflections. The experimental and the theoretical results differed significantly (twice as much) when the actual mechanism of lateral deflection caused by the intensive local undulation differed from the adopted model. r 2007 Elsevier Ltd. All rights reserved. Keywords: Buckling; Cylindrical shells; Stiffened shells; Compression; Experiment
1. Introduction In the class of longitudinally stiffened shells (stringer shells) subjected to axial compression, deformation and buckling of cylinders with multiple stiffeners and with relatively weak bending stiffness has been the subject of careful researches. Behavior of such structures can be rather satisfactorily described by the linear theory of structurally orthotropic shells [1–3]. However, correlation of theoretical and experiment results obtained within the broad range of structural parameter changes is found to be unsatisfactory if spacing between the stiffeners is not enough small (stringer shells almost close to rational ones) [4]. This work presents data of the experimental investigation carried out with the aim to study systematically the influence of the following structural factors on the mechanism of the stringer shell carrying capacity exhaustion: (1) number of stiffeners; (2) stiffeners eccentricity
Corresponding author. Tel.: +38 0 56 247 03 79.
E-mail address: [email protected] (V.L. Krasovsky). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.08.028
sign; (3) shell length; (4) edge fixing requirements (simple support or clamping). 2. Specimens and the tests procedure The experiments were carried out on small-sized specimens with diameter 2R ¼ 143 mm and the wall thickness h ¼ 0.19 mm (R/h ¼ 376). All shells were stiffened with identical equidistant longitudinal thin-walled stiffeners of angular profile with dimensions: 4.0 4.3 0.34 mm. Depending on the number of stiffeners (k), the specimens were subdivided into two series: series 1, k ¼ 24 and series 2, k ¼ 36. Length of the specimens in each series varied in a wide range. The shells of each dimension had either external or internal stiffeners. The shells were made of the cold-rolled stainless steel grade X18H9H (elastic modulus E ¼ 191 GPa, 0.2% proof stress ¼ 800 MPa, Poisson’s factor n ¼ 0.3) by applying the contact spot welding with one longitudinal lap seam whose width was equal to 0.02 of the shell perimeter. The stiffeners were made of metal strips on a special bending appliance and spot welded abreast to the shell’s outer or
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V.L. Krasovsky, V.V. Kostyrko / Thin-Walled Structures 45 (2007) 877–882
inner surface along the narrow side of the angle bar. Prior to fitting the shells to the test machine, special edge devices were attached to them. The compressive force was developed by the UME-10tm mechanical universal testing machine. All shells were subjected to the kinematical loading at constant speed of 0.05 mm/min. Pairs of specimens with inner and outer arrangement of stringers, with either simply supported or clamped edges, were tested. In the case of hinged edges, the load to the specimens was transmitted through the shell only. In the case of clamping, loading was uniformly distributed along the whole cross-sectional area of the specimen. 3. Test results 3.1. Series 1 A total of 35 specimens with L/R ¼ 0.28–1.80 were tested. We are going to describe the buckling process in the hinged shells. The buckling was preceded by the local undulation between the stiffeners. The first dimples of the local buckling appeared, as a rule, in the form of a ‘‘light’’ clap during the process of initial bending deflection (Fig. 1a). The process of intensive local undulation started at a certain level of the load. These loads were assumed as local buckling loads Nm (between the stiffeners). Formation of the local dents was accompanied with claps. As a result of the shell’s ‘‘rapid’’ undulation process, the local post-critical configuration of dimples in ‘‘chess’’ order was
formed (Fig. 1b), with twisted stiffeners. The local dimples developed as the load increased, as well as the stiffeners’ twist angles. The tests performed at low rates of loading have allowed observing the process of overall buckling, different in the shells with inside and outside stiffeners. At loading rate V ¼ 0.005 mm/min, the following sequence of carrying capacity exhaustion was observed for outside stiffened specimens, with L ¼ 80–130 mm. Local folding of stringers along the diagonals that linked local dimples on adjacent panels was accompanied by the acute clap and the walls distortions (out of the plane). As a result, the shells surface was covered with rather small, but clearly distinctive diamond-shaped dents (Fig. 1c). When the motor of the test machine was turned off, the described form could not keep for too long time (1–3 s). The next clap followed with formation of a new shape: large elongated shells dents (typical for stringer stiffening), covering several adjacent stiffeners (Fig. 1d). At reduced length L, such a clap, as a rule, was not observed, and the initial dents developed and spread on the whole shell length. Within small areas, the diamond-shaped dimples were arranged in a single row, and after the stiffeners folding the shells used to crumple. Buckling of shells with inner stiffeners was developing with not so sharp changes in the form. The overall postcritical shape dents were originated on the boundary of two local adjacent dents separated by a stiffener. At first, its development was accompanied with bending of only this element. Then, the dents started growing and covered several adjacent stringers. Usually, the process was rather
Fig. 1. Local and overall buckling shapes for shells of series 1.
ARTICLE IN PRESS V.L. Krasovsky, V.V. Kostyrko / Thin-Walled Structures 45 (2007) 877–882
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Fig. 2. Critical loads of local and overall buckling for shells of series 1.
rapid and was accompanied with a dull clap in the case of long shells (Fig. 1e and f). Clamping of the edge did not result in any significant changes in the carrying capacity exhaustion mechanism. Critical loads of global buckling (Ncr) for all shells turned out to be ultimate too. Fig. 2 shows values of Nm (circles) and Ncr (triangles) versus the shell length, of the specimens with outside (light marks) and inside (dark marks) stiffening, in the case of simple support (Fig. 2a) and clamping (Fig. 2b) of the edges. The theoretical solutions with account of the prebuckling deformation are presented by the curves ‘‘1’’ and ‘‘2’’ (for inside and outside stiffening, correspondingly). The curves for zero-torque prebuckling theory are marked ‘‘10 ’’ and ‘‘20 ’’ (this relates also to Fig. 4). The analysis of these data demonstrates that the level Nm practically does not depend on the shell length. Only in the case of short shells, Nm depends on the length. At inner stiffening, local buckling of the shells occurred at slightly higher loads in comparison with the outer stiffening. In the m case of hinged edges, the ratio N m þ =N (‘‘+’’ relates to inner stiffening, ‘‘’’ to outer one) ranges, in average, from 1.3 to 1.5. The boundary conditions practically do not affect the value of Nm, and the difference between values of Nm for shells with inside and outside stiffeners remains. Unlike Nm, values of Ncr essentially depend on the shell length. Reduction of L, in all cases, leads to monotonous growth of Ncr. The experimentally obtained increase in critical loads due to outside stiffening (comparing with the inside stiffeners) in the case of simple support is insignificant and varies from 5% to 20%. At clamping of the
edges, the stiffeners eccentricity sign does not practically affect the carrying capacity. At inside stiffening, the increase in critical loads caused by the edge clamping, in comparison with the simple support, ranges from 10% to 20%. In the case of outside stiffening, no increase of Ncr has been observed. Advantages of outside stiffening were clearly visible only in the case of simple supported edges and for moderate length shells, and attained 25%. 3.2. Series 2 This series covers the shells with k ¼ 36. The total of 28 specimens, the length of which varied from 20 to 200 mm (L/R ¼ 0.28–2.80), were tested. The local buckling of the shells between the stiffeners also preceded the overall buckling of specimens of this series. The overall buckling developed following the same mechanism described above for the shells of series 1 (Fig. 3a). At the same time, no short wave modes with folding stiffeners, preceding formation of typical large dimples of overall buckling, were observed. Overall buckling of the long shells (L ¼ 130–200 mm) was accompanied with formation of large dimples with smooth outlines (Fig. 3b). When the stiffeners were fitted from outside, they had folds in the middle cross-section. The shells with minimal length buckled in mode close axisymmetric one (Fig. 3c and d). The dependence of critical loads Nm and Ncr on the shell length is shown in Fig. 4a and b (here we used the same notation as in Fig. 2a and b for shells of series 1).
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Fig. 3. Local and overall buckling shapes for shells of series 2.
Fig. 4. Critical loads of local and overall buckling for shells of series 2.
It is seen that the difference between the levels of Nm for the shells with outside and inside stiffening at both simply supported and clamped edges has practically disappeared. Independently of the boundary conditions, the advantage of the outside stiffening is more vivid. In addition, in the case of the specimens with L ¼ 120 mm, this increase reached 45% at hinged edges, and 31% at clamping. When the shell length was reduced, the cr difference between N cr and N þ also reduced, and at buckling in the axisymmetric mode (L ¼ 120 mm), the critical load turned out to be higher at inside arrangement of the stiffeners.
The comparison of the test results for both types of boundary conditions reveals that critical loads for inside stiffening of the shells with clamped edges significantly exceed Ncr for hinged edges practically in the whole range of lengths tested. The effect of boundary conditions on Ncr in the case of shells with outside stiffening was not noted. 4. Comparison of the experimental and theoretical results The experimental values of Ncr were compared with theoretical predictions based on the structural-orthotrophic model of stiffened shells. Computations were
ARTICLE IN PRESS V.L. Krasovsky, V.V. Kostyrko / Thin-Walled Structures 45 (2007) 877–882
carried out both for initial momentless state of the shells and with taking into account non-linear axisymmetric prebuckling deformation. Analysis of the theoretical dependences showed that the main factor influencing Ncr (when we take into account the prebuckling deformation) is partially caused by the presence of transverse deformations in the shell, and also partially due to the asymmetrical structure of the shell. Note that we took into account the real character of applying the load to the shell for hinged edges (external compressive force was applied only to the shell skin) and for clamped edges (the load was applied to the whole cross-section). The theoretical dependencies of Ncr on the shell length for series 1 and 2 are given in Figs. 2a, b and 4a, b, respectively. The solid line represents the solution with account of the prebuckling deformation (inside stiffening— curve 1, outside stiffening—curve 2). Computed curves for Ncr at zero-torque initial condition are represented by the dotted line (inner stiffening—curve 10 , outer stiffening— curve 20 ). It is seen in the presented graphs that in all cases the experimental values of critical loads lie below the theoretical values. Let us consider, in more details, the case of the simply supported edges. At inside stiffening of shells of series 1, the best correlation between the theory and the experiment was attained for the shells with the large length (L ¼ 130 mm), however, even in this case the difference between the estimated and experimental values made about 30%. At reduction of L the discrepancy increased. The larger the number of the stiffeners the better was the correlation of the estimated and experimental data. In the case of specimens of series 2 strengthened by the inside stiffeners, in a wide range of L, the estimated values of Ncr differed from the experimental values by not more than 17%. In the case of outside arrangement of the stiffeners, the difference between the experimental and estimated values increased. In case of specimens of series 1 with L ¼ 130 mm, the difference was 50%, and it grew with reduction of length. The discrepancy between the theoretical results and the experimental data for shells of series 2 in the range L ¼ 200–120 mm did not exceed 20%. However, further reductions in length increased it significantly. The discrepancy between the theoretical and experimental results significantly increased in the case of clamped edges. It was not less than 60–80% in the shells with inside stiffeners and more than 100% for outside stiffened shells. In addition, unlike the case of hinged edges, the correlation did not improve as the stiffeners number increased. The theoretically predicted significant advantage of the outside stiffening comparing with the inside stiffening was not confirmed by the experiment for this case. The cr estimated level of ratio N cr =N þ was attained only for the long shells of series 2. The effect of the clamping edges was exhibited mostly for the shells with the inside stiffeners. However, in this case cr the obtained ratios N cr z =N f (‘‘z’’ relates to clamping, ‘‘f’’ to
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simple support) were significantly below the estimated one. Their maximum value did not exceed 1.3 for shells of series 1 and 1.25 for series 2. 5. Discussion and conclusion It is evident that the main reason for discrepancies between the experimental and theoretical values of loads of overall buckling Ncr is the prior local buckling of the shell between the stiffeners. Here, we revealed two mechanisms of its influence on the process of overall buckling and value of Ncr. First, the local buckling of a shell reduces its rigidity to tension–compression and shear, this leads to lowering rigidity of the shell as a whole, as well as to additional compression and bending of the stiffeners. Second, the shell chess-type dimples developed due to the local buckling contributed to the stiffeners twisting. This causes lowering of longitudinal bending rigidity of the shell, in particular in the areas with large tangential shift of the stiffener’s upper ends. The degree of influence of these factors on the shell carrying capacity is determined by the magnitude of the local mode to the moment when the overall buckling occurs. In the case of long shells with simply supported edges and average length shells of series 2, the intensity of local mode was comparatively low. The level of carrying capacity is mainly influenced only by the factor of the structure rigidity reduction due to tension–compression. The shells with inside stiffening have critical loads that were rather close to the estimated ones in the wide range of changes of their length. In the case of outside stiffening, due to higher critical loads, the discrepancy between the theory and the experiment was somewhat higher. In this case, the lowering of shell rigidity may be taken into account within the applied structural-orthotrophic model by reducing the shell skin with the better resulting correlation between the theory and the experiment. The intensive local undulation that is typical for the shells of series 1, as well as for the short shells of series 2, has an effect not only on the Ncr value, but also on the mechanism of overall buckling. That is clearly demonstrated by the shells with outside stiffening in the form of the short-wave mode of buckling with folding of the stiffener walls. There was no local folding of the stringer’s walls in the shells with inside stiffeners; however, the presence of the developed local mode had a substantial effect on the process of the overall undulation. A typical feature for both types of stiffening is the considerable reduction of shell rigidity both to tension–compression and to bending (due to deformation of the stiffener wall out of the plane initiated by the local undulation of the shell). That is the main reason of the significant difference between the experimental and the estimated critical loads and buckling modes. Estimation of carrying capacity for such shells is possible only when all above-mentioned effects are taken into account, especially the ones that have a clearly expressed non-linear nature. It should be noted
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here that local folding of the stiffeners walls was also observed at testing the shells, buckling of which was the result of interaction between the local modes of the stiffeners buckling and the overall modes. A non-linear theory of the ‘‘coupled’’ lateral buckling of stiffened shells developed mainly by Manevich [5], has allowed, at present, to improve significantly the agreement between the experimental and the theoretical data for a rather wide class of shells stiffened by thin-walled stiffeners. But, being an asymptotic theory, this theory does not take into account certain features of the shell prebuckling deformation prior to the overall buckling. Certainly, in the case of the shells with intensive local undulation, the experimentally obtained increase in Ncr at more rigid boundary conditions turns out to be less than the one predicted by the structural-orthotropic theory, in particular for the outside stiffened shells with short wave buckling. This result may be explained by difference between the actual mechanism of buckling and the theoretical model. At the same time, even for the specimens, the buckling mechanism of which agreed with the model, the theoretical values of Ncr at clamped edges remained significantly higher than the experimental ones (long shells, series 2). It means that assessment of boundary conditions effects based on the structural-orthotrophic
model for the considered class of shells may lead to wrong results. The local buckling of the shell and related effects are also the reason for the significant reduction of the actual effect of the stiffeners arrangement eccentricity sign on Ncr comparing with the theoretical prediction. References [1] Singer J. The influence of stiffener geometry and spacing on the buckling of axially compressed cylindrical and conical shells. In: Theory of thin shells—Proceedings of Second IUTAM symposium, Copenhagen, 1967. Berlin: Springer; 1969. p. 234–63. [2] Singer J. Buckling of integrally stiffened cylindrical shells—a review of experiment and theory. In: Contribution theory of aircraft structuring, Delft, 1972. p. 325–57. [3] Weller T, Singer J. Further experimental studies on buckling of integrally ring-stiffened cylindrical shells under axial compression. Exp Mech 1974;14(7):267–73. [4] Manevich AI, Demeshko MF, Krasovsky VL, Kucherenko VM. An experimental investigation of stability of longitudinally stiffened cylindrical shells under axial compression. In: Raschiot prostranstennykh konstrukciy. Moscow: Stroyizdat; 1971. no. 14, p. 87–102 [in Russian]. [5] Manevich AI. Coupled instability of cylindrical shells stiffened with thin stiffeners. In: Proceedings of third international conference on thin-walled structures: advances and developments. Cracow: Elsevier Science Ltd.; 2001. p. 683–91.