icccbe 2010 Nottingham University Press Proceedings of the International Conference on Computing in Civil and Building Engineering W Tizani (Editor) Plastic non-symmetric bifurcation buckling of circular cylinders under uniform external hydrostatic pressure Carl T.F. Ross, Chris Bull & Andrew P.F. Little University of Portsmouth, UK Abstract The paper presents a theoretical and an experimental analysis of the buckling of 4 thin-walled circular cylinders, which were tested to destruction under uniform external hydrostatic pressure. The mode of failure of the vessels was lobar buckling or shell instability, where the vessel imploded inwards with evenly spaced waves spaced around its circumference. The theoretical analysis adopted the finite element method, where the commercial computer package, namely ANSYS was used. The theoretical work involved the inclusion of both material and geometrical non-linearity, using an incremental stepby-step method. About 50 steps were used per model. Comparison between theory and experiment was good and this appeared to indicate that the method could be applied to full-scale vessels. Keywords: submarines, pressure hulls, buckling, ANSYS 1 Introduction A typical submarine pressure hull consists of a combination of thin-walled circular cylinders, cones and domes. In this paper, we will only consider the cylinder. Now under uniform external pressure, such a vessel can implode through shell instability or lobar buckling, (Ross, 1965; Ross, 2001; Windenburg and Trilling, 1934) at a fraction of the pressure to cause the same vessel to explode under uniform internal pressure. This mode of failure is shown in figures 1 and 2, and it is an undesirable mode owing to its poor resistance to withstand uniform external hydrostatic pressure. One method of improving its poor resistance to withstanding uniform external hydrostatic pressure is to ring stiffen it, with ring stiffeners spaced suitably apart. In this case, the vessel can still fail due to shell instability, if adjacent ring stiffeners are spaced too far apart. It is for this reason that the present study was carried out, where both geometrical and material non-linearity was considered. In this case, the initial out-of-circularity plays an important role, because the initial out-of-roundness increases non-linearly with increasing pressure, until parts of the shell become plastic. When this occurs, the tangent modulus of some parts of the shell rapidly decreases and this considerably worsens the situation, until catastrophic failure takes place. A description of the experimental and the theoretical analyses will now be carried out.
Figure 1, Shell instability of cylinders 1 to 3 Figure 2, Shell instability for cylinder 7 2 Experimental analysis The dimensions of the four cylinders that will be modelled, using the commercial computer package, namely ANSYS, are shown below in table1. The picture, in figure 3, shows a schematic diagram of the circular cylinders. The, value of N in the table, represents the number of ring stiffeners present. Models 1, 2 & 3 were made from HE9 WP aluminium alloy and Model 7 was made from good quality mild steel. The test tank and the hand-driven hydraulic pump are shown in figure 4.
Figure 3. Geometrical details of the Aluminium Alloy Circular cylinders. In figure 3, h = wall thickness = 2mm; a = mean radius = 13.1 cm. Table 1. Geometrical details of the circular cylinders Cylinder No. L 1 L L b b F b f d 1 88.90 114.30 676.28 8.26 8.26 15.75 5.00 2 95.25 95.25 616.59 10.16 8.26 15.75 5.00 3 63.50 57.15 400.69 10.16 8.26 15.75 5.00 7-254.00 254.00 - - - - N Figure 4, The test tank and hydraulic pump The inside surface of each circular cylinder was fitted out with a number of electrical resistance strain gauges, including 10 circumferential strain gauges at mid-bay, in the largest bay of each vessel. The purpose of these strain gauges was to study the circumferential deformation pattern, just prior to buckling. The buckled forms of the vessels are shown in figures 1 & 2, where it can be seen that all 4 vessels collapsed by shell instability. The experimentally obtained collapse pressures are shown
below in bar; the figures represent the number of circumferential waves or lobes (n), that each vessel buckles into. Model 1 = 23.10(8); Model 2 = 24.14(9); Model 3 = 27.72(12); Model 7 = 39.17(5). 3 Theoretical analysis The theoretical analysis was based on the finite element method, using the commercial computer package, namely ANSYS. The method used was a step-by-step incremental method; which allowed for both material and geometrical non-linearity. The method is described with the aid of figure 5. The element used to model the vessels was Shell93; which was an 8-node isoparametric element. The wall of the shell was meshed and the structures were assumed to be fixed at the left ends of the computer images. Attached to each right end of the computer image, was a circular disc of thickness about 10 times the wall thickness of each vessel. It was considered that this best represented the actual experimental case. A typical mesh is shown in figure 6. Figure 5, Table showing the incremental method used Figure 6, Mesh for a cylinder The theoretical buckled pattern for cylinders 7 & 1 to 3 are shown in figures 7 to 10.
Figure 7, ANSYS screen dump for cylinder 7 Figure 8, ANSYS screen dump for cylinder 1 Figure 9, ANSYS screen dump for cylinder 2 The results for the non-linear and Eigen buckling analyses are shown in Table 3; where the Eigen buckling results use only linear elastic theory.
Figure 10, ANSYS screen dump for cylinder 3 Table 2. Experimental and the ANSYS non-linear & elastic results Buckling pressures for models 1 to 3 & 7 Cylinder Von mises Ansys Eigen buckling Ansys non-linear number (bar) (bar) (bar) Experimental (bar) 1 25.0 (8) 32.1 (8) 26.5 23.1 (8) 1.1842 2 30.6 (8) 39.9 (9) 26.2 24.1 (9) 1.0810 3 48.9 (10) 76.0 (10) 27.5 27.7 (12) 0.8820 7 42.7 (6) 37.5 (6) 48.5 39.1 (5) 1.4240 The numbers in parentheses represents the number of lobes(n) In Table 2, the constant λ was called the thinness ratio, where λ= {(L/2a) 2 /(h/2a) 3 }/(σ yp /E) 0.5 } (1) σ yp = Yield stress & E = Young s modulus. λ 4 Conclusions The results show that the ANSYS non-linear analysis gave good results for Cylinders 1 to 3, but were not too good for Cylinder 7. The reason why the non-linear theory gave poor results for Cylinder 7 was probably because Cylinder 7 collapsed elastically. Moreover, the ANSYS screen dump for Cylinder 7 was not exactly the same as the experimental mode shown in figure 2. For the cylinders, which collapsed inelastically, namely Cylinders 1 to 3, the ANSYS non-linear analysis was very good, whereas the linear Eigen buckling analysis, grossly overestimated the collapse loads; this behaviour was expected. It was interesting to note that the ANSYS screen dumps showed a partial collapse mode; the same as that occurred in the experiments. To apply the ANSYS non-linear theory to full-scale vessels, more work needs to be done. References ROSS, C.T.F., 1965. The collapse of ring-reinforced cylinders under uniform external pressure. Trans., RINA, 107, 375-394. ROSS, C.T.F., 1965. The instability of ring-stiffened circular cylindrical shells under uniform external pressure. Trans., RINA, 107, 156-161. ROSS, C,T.F., 2001. Pressure Vessels: External Pressure Technology, Chichester, UK, Horwood. WINDENBURG, D.F. and TRILLING, C., 1934. Collapse by instability of thin cylindrical shells under external pressure. Trans. ASME, 11, 819-825.