1. Introduction

Cylindrical shells are often used in ship & land-based platform structures at deck plating with a camber, side shell plating at fore and aft parts, and bilge circle part. It has been believed that such cylindrical shell can be modelled fundamentally as a part of a cylinder under axial compression. In general, it is known that curvature increases the buckling strength of a curved plate subjected to axial compression, and that curvature is also expected to increase the ultimate strength. We conduct series of elastic large deflection analysis in order to clarify the fundamentals in buckling and post-buckling behaviour. Shell buckling is probably the most complex instability phenomenon. Similar to plate buckling, it involves the cross-sectional shape rather than displacement of the section as a whole. Compressive stresses can arise from compressive axial load, bending and locally applied concentrated loads. In addition, however, shells can also be subject to internal or external pressures (pipes, silos, tanks, off-shore jacket legs, etc.). The load-shortening behaviour shows drastic loss of carrying resistance as soon as the critical buckling load is reached, and the shell tends to be a buckled shape, which is in equilibrium with an external compression load and it is much lower than the critical load. This is general characteristics of shells, whatever the type of shell and the type of loading. Unlike plates, they are not able to exhibit any post-buckling reserve of resistance. A brief review is made on previous research works related to buckling behavior of circular cylinder.

Wang et al. (2018) developed finite element (FA) numerical procedure for predicting the buckling load. Results indicate that the buckling load predicted by the FE analysis is very close to that from the experiments. Knockdown factor (KDF) is discussed with reference to the NASA design document.

Jiao et al. (2018) conducted experimental and numerical studies to investigate the effects of ringed stiffener on the buckling behavior of perforated cylindrical shells under axial compression. Three test specimens with without ringed stiffener were manufactured and tested. The finite element method using static analysis with artificial damping is used to simulate the displacement controlled compression tests. Good agreement was found between the numerical and experimental results. It is found that the global buckling loads of perforated shells are improved by ringed stiffener.

Wagner and Hühne (2018) studied the affects that led to new improved knockdown factors for the design of cylindrical shells under axial compression. The corresponding design methods were derived by means of three step procedures involving deterministic methods for the buckling load prediction, modern experimental results and an extensive probabilistic analysis. The new design procedure was demonstrated by means of a full-scale primary launch-vehicle shell and the results show an estimated weight reduction of about 20 %.

2. Analysis and FEM (Finite Element Method) method

2.1. Analysis element and arc-length method

A linear material model is satisfactory when only small quantities of the material are exposed to the yield stress or greater. The type of constitutive law is adopted to model the behaviour of steel; bilinear isotropic hardening. The Bilinear Isotropic Hardening options uses von Mises yield criteria (which includes most ductile metals) coupled with an isotropic work hardening assumption. This option is often preferred for large strain analyses. In practical designs, some engineers want to know the factor of safety of the structure for a nonlinear buckling load. The numerical simulations were carried out using the general commercial finite element code ANSYS (ANSYS Inc, 2015). The shell-181element was used to model the shells. This element is extremely suitable for analyzing from thin to moderately thick shell structures. The element is a four-node Reissener-Mindln shell element with six degrees of freedom at each node and is well suited for large strain nonlinear applications. Shell 181 uses a penalty method to relate the independent rotational degrees of freedom about the normal (to the shell surface) with the in-plane components of displacements. The arc-length method is suitable for nonlinear static equilibrium solutions of unstable problems. Applications of the arc-length method involves the tracing of a complex path in the load-displacement response into the buckling/post buckling regimes. The arc-length method uses the explicit spherical iterations to maintain the orthogonality between the arc-length radius and orthogonal directions. It is assumed that all load magnitudes are controlled by a single scalar parameter (i.e., the total load factor). Unsmooth or discontinuous load-displacement response in the cases often seen in contact analyses and elastic-perfectly plastic analyses cannot be traced effectively by the arc-length solution method.

2.2. Classical theory and convergency test

Thin-walled circular cylinders are widely used in numerous applications such as tanks, silos, space launchers. A correct design has to be paid attention to buckling phenomena, which may occur under specific loading conditions, and might cause global collapse of the structure. The circular cylinder can also be subject to axial compression (pipes, silos, tanks, off-shore jacket legs, etc) as shown in figure 1. The simplest case to analysis is the axially compressed circular cylinder, and the elastic critical buckling stress is calculated as follows equation (1).

where, R : radius, t : thickness, ν : poisson’s ratio, E : Young’s elastic modulus, σ_cr : critical buckling strength

This buckling load is derived on two assumptions; (1) the pre-buckling increase of the radius due to the Poisson effect is unrestrained, and (2) two edges are held against translational movement in the radial and circumferential directions during buckling. However, they are able to rotate about the local circumferential axis. These edge restraints are usually called "classical boundary conditions" as follows equation 2.

According to classical theory of shell buckling, the perfect cylindrical shell can be buckled either axisymmetrically - with a succession of ring buckles - or in the form of a chessboard (Figure 2-a, b); depending on the buckling circumstances, the shell may achieve a diamond-like post buckling configuration.

The preparatory calculations were conducted to determine the level of mesh refinement required modeling to ensure convergence for a cylinder both L/R=1.0 and L/R=3.0, plate thickness t=5 mm and radius of the cylinder fixed 1,000 mm and cylinder height (L) adopted 1,000 and 3,000 mm separately. Figure 3 shows relationships of elastic buckling stress and division number of elements for the circumferential direction of the circular cylinder varying the change the mesh sizes.

In general, the possible collapse modes of the circular cylinder under axial loading are known as Euler buckling shape, Diamond buckling shape, Chess board buckling shape and Concertina buckling shape (Ring buckling shape). On the basis of the calculated results, three kinds of buckling patterns (Concertina buckling shape, Chess board buckling shape and Diamond buckling shape) can be confirmed for the circular cylinder. When the number of meshing of the length to radius ratio 1.0 is not enough, i.e. less than 80, a different first buckling mode with different buckling stress is obtained. However, the first buckling modes converged to Diamond buckling shape when the number of subdivision is increased as shown in Figure 3. In case of taking place concertina buckling shape at the less than 80 of the division number for circumferential edge, deflection mode for circumferential direction does not happen, but the abrupt increase of mesh size changes to diamond buckling shape accompanied with deflection mode for circumferential direction. When the number of meshing of the length to radius ratio (L/R) 3.0 is not enough, i.e. less than 120, a different first buckling mode is appeared (so called Diamond buckling shape) as shown in Figure 3. Results from classical formula show very good correlation from 280 of the mesh numbers and the eigen buckling of diamond shape is also preserved. Thus, the element number has to be carefully considered since it sensitively affects the buckling shape and buckling stress. Circular cylinder cases have been calculated with L/R 1.0 and 3.0 optimum element, number 210 and 280, and it convergence ratios 0.067 % and 0.3193 %, respectively.

In order to verify the calculation of this study, the load carrying capacity calculations for the curved plates subjected to compressive loading were compared using two programs (ANSYS and ULSAS (Yao et al., 1998)). The thickness of the plate is taken as 15 mm. The maximum magnitude of the initial deflection is 1 % of the thickness.

Average stress and average strain curves of the cylindrically curved plate are compared with various flank angles under axial compression. In case of the thin plate (t=15 mm) with a flank angle 5 degrees the secondary buckling takes place accompanied by a snap-through. Except the case of 5 degree flank angle, the ultimate strength also increases with the increase in the flank angle. The collapse behaviour simulated using different FEM codes shows good agreement with evaluating the ultimate strength. Almost the same local yielding near the loading edge part of the curved plate are observed commonly among all the cases with different flank angles as shown in Figure 4 and 5.

3. Elastic Large Deflection Analysis and results

The pre-buckling behavior of the circular cylinder is also approximately linear behaviour as shown in Figure 6. The circular cylinder essentially retains its original shape until it buckles at point B. The circular cylinder not only evades any adding load increase by buckling, but the load also drops down abruptly to a fraction of the buckling load at point C with a rounded buckle shape. When the axial load increases, the path moves to point D accompanied by a decrease an in-plane rigidity.

The cylinder wall deformations at point B are characterized by one or several localized diamond shaped buckling patterns as shown in Figure 7.

Figure 8 indicates the relationship between average stress and strain of circular cylinder with different lengths under axial compression. The elastic buckling stress behaviors are almost the same among all the cases with various lengths. However, post-buckling behaviors are very different. Otherwise, the effect of aspect ratio can be ignored if focus is placed on the estimation of buckling stress of the circular cylinder.

Figure 9 shows a single radius to thickness ratio of R/t=100 and a single height to radius ratio of L/R=1.0 is used. The primary buckling takes place at point A and then the in-plane rigidity is abruptly reduced due to re-distribution of the in-plane stress. When the path reaches at point C, a buckling mode changes patterns from one-tier elliptical mode to two-tier diamond shape due to the secondary buckling phenomenon on the side wall as shown in Figure 8. From the results, the cylinder structure subjected to the compressive load has made post-buckling behavior complicated and it is necessary to reflect these characteristics for the safe structure design.10

Figure 11 shows the comparison of average stress and ratio of radius to thickness with various magnitudes of the imperfection under compression. As the magnitude of initial imperfection increases, the elastic buckling strength decreases, and the larger the radius to thickness ratio, the greater the difference in the elastic buckling strength.

5. Conclusions

The present paper focuses on the following area which has been studied numerically as appropriate:

To examine and clarify the buckling and post buckling behaviour of the cylindrical shell under axial compression, it is considered of effects of several parameters.