Buckling of thin-walled conical shells under uniform external pressure

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acting until the bifurcation point at the apex of two or three yield lines caused resurgence of buckling lobes in the opposite side of the frusta and eventually it led to complete failure of one side of the frusta which took place in the supporting point with luxating the lower edge. 7.2. SCC specimens SC1 In Fig. 12 a contrast is carried out between initial and ultimate geometry of test specimen SCC1 in which the utmost deformation is positioned at the height of 1 4L of the specimen. ARTICLE IN PRESS -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 SC4 FEA Expev FEA Exper. B.S. Golzan, H. Showkati / Thin-Walled Structures 46 (2008) 516–529 527 Radial displacement (mm) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Radial displacement (mm) Fig. 13. Radial displacement vs. external pressure for specimens, SC1 and SC4. 0 0 The load–displacement curves for the net displacements are plotted for quite a few points in Fig. 5(a). In some of the models the displacements were similar in the initial stage of loading, while in the other ones the displacements started to differ early in the loading stage of SCC1. These two genuses of models were thus selected to contrast the two types of behavior. These curves show similar divergence as observed from the load–strain curves, Fig. 5(a). The first set, Fig. 5(a) which had similar displacements in the initial stage of loading, experienced rapid increases in displacements at increased loads. It is hence recommended that for the determination of the buckling load of a model conical shell in its corresponding 0 70 65 60 55 50 45 40 35 30 25 20 15 10 5 65 60 55 50 45 40 35 30 25 20 15 10 5 0 Pressure (KPa) Pressure (KPa)
528 perfect state, a suitable load–deflection curve which shows an obvious slope change be identified. This is likely to be for a point in a relatively more perfect region of the shell. The intersection point between the initial slope of this curve and a tangent to the postbuckling part of the curve can be taken as a good rough figure to the buckling load of a corresponding perfect model. To identify such a suitable point, load–displacement curves of all points at or near wave crests and troughs can be contrived. This method emerges to offer a rational approach for the determination of buckling loads for conical caps. 8. Concluding remarks This paper has described a recently developed experimental facility for buckling experiments on conical shells. The facility consists of a loading system, a simple measurement strategy for rather accurate geometric imperfection and deformation surveys, and compulsory equipment for the fabrication of quality test models. Distinctive results of sample tests have been presented to illustrate the competence of this facility. Procedures for processing the test results to verify both the buckling load and the modes of buckling have also been presented. The deliberate data and obtained domino effects are reported for six frusta and four SCC specimens with simply supported ends subjected to uniform peripheral pressure. The salient concluding tips are as follows [9]: Fabrication and testing of small-scale models have been undertaken to examine the buckling behavior of unstiffened shallow conical shells. In all models, smallscale manufacturing has produced relatively high imperfection values. However, since full imperfection scans have been recorded, the test results can be used to validate numerical or other models. Since the predictions are influenced by imperfection amplitudes, a reasonable assumption must be made in the absence of test models. In all specimens, the initial buckling occurred when one or more buckling lobes were detected. The applied pressure increased until an overall buckling mode was formed. The yield lines in the lower part of the frusta in all specimens are in the form of ‘‘V’’ shape, which were recognized in the range of postbuckling. In all specimens, the inward deformations are so larger than the outward deformations. In all specimens, difference between initial buckling and overall buckling loads was substantial. Postbuckling potency exists apparently in all specimens under the effect of external pressure. In all specimens, it is experimentally corroborated that the longer shells are more flexible in radial direction and therefore, they are weaker than shorter shells. The buckling loads obtained from experiments are lower than the ones derived from FEA and Jawad equation ARTICLE IN PRESS B.S. Golzan, H. Showkati / Thin-Walled Structures 46 (2008) 516–529 while the two latter ones are different in some aspects related to the tapering ratio of specimens (R/r) and initial imperfections present in FE models but absent in the arithmetical come up, which is worth noticing as mentioned in the context. Clearly, for more slender shells the collapse mode will change and the kinematical assumptions of the mechanism approach would be inappropriate. Acknowledgments The work depicted herein outlined part of a scheme on ‘‘Stability and Strength of Conical Cones’’ subsidized by the Ministry of Science, Technology and Research in the I.R. Iran and carried out in collaboration with the Structural Research Center at Urmia University. We would like to put across gratitude to the technicians in the Structures Laboratory of Urmia University, in particular Mr. Jafar Azim Zadeh and our best friend Mr. Emad Jahangiri for their enthusiasm and professionalism in conducting the probes. The authors are so appreciative to Prof. J.G. Teng, for his great favors in providing commentaries and presenting his constitutive remarks regarding this research. References [1] Teng JG, Zhao Y, Lam L. Techniques for buckling experiments on steel silo transition junction. Thin-Walled Struct 2001;39:685–707. [2] Ross CTF, Little APF, Adeniyi KA. Plastic buckling of ring-stiffened conical shells under external hydrostatic pressure. Ocean Eng 2005;32:21–36. [3] Wintersetter ThA, Schmit H. Stability of circular cylindrical steel shells under combined loading. Thin-Walled Struct 2001;40:893–909. [4] Popov AA. Parametric resonance in cylindrical shells: a case study in nonlinear vibration of structural shell. Eng Struct 2003;25:789–99. [5] Ansourian P. On the buckling analysis and design of silos and tanks. J Construct Steel Res 1992;23(1–3):273–94. [6] Holst FG, Rotter JM, Calladine ChR. Imperfection in cylindrical shells resulting from fabrication misfits. J Eng Mech 1999;125(4): 410–8. [7] Shen HS, Chen TY. Buckling and post buckling behavior of cylindrical shells under combined external pressure and axial compression. Thin-Walled Struct 1991;12:321–34. [8] Yamaki N. Elastic stability of circular cylindrical shells. Amsterdam: North-Holland; 1984. [9] Golzan SB. Investigation of the buckling and postbuckling behavior of conical and truncated conical shells under uniform external pressure. MEng thesis, Faculty of Civil Engineering, Urmia University, Urmia, Iran; 2006. [10] Babcock CD. Experiments in shell buckling. In: Fung YC, Sechler EE, editors. Thin-shell structures: theory, experiment and design. Englewood Cliffs, NJ: Prentice-Hall; 1974. p. 345–69. [11] Singer J. Buckling experiments on shells—a review of recent developments. Solid Mech Arch 1982;7:213–313. [12] Dowling PJ, Harding JE. Experimental behavior of ring and stringer stiffened shells. In: Harding JE, Dowling PJ, Agelidis N, editors. Buckling of shells in offshore structures. London: Granada; 1982. p. 73–107. [13] Walker AC, Andronicou A, Shridharan S. Experimental investigation of the buckling of stiffened shells using small scale models. In:
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