Buckling of imperfect cone-cylinder transition subjected to external pressure

Highlights

• First Numerical prediction into buckling behaviour of imperfect cone-cylinder transition subjected to external pressure.

• Comparison of results show that single load indentation imperfection is more detrimental compared to Eigenmode imperfection.

• Proposed knockdown factor for both Eigenmode imperfection and single load indentation imperfection for design purposes.

Abstract

The paper presents a result of the numerical investigation into the buckling behaviour of geometrically imperfect cone-cylinder transition subjected to external pressure. The models are assumed to be made from the Hiduminium alloy (HE-15). Various initial geometric imperfections such as eigenmode imperfection and Single Load Indentation (SLI) imperfections were superimposed on the perfect cone-cylinder shell. The reduction of the buckling strength was then quantified numerically. As expected, the buckling strength of cone-cylinder shells were strongly affected by initial geometric imperfection and the reduction in buckling strength was seen to be strongly dependent on the choice and location of imperfection. Overall, the lowest SLI imperfection curve produces the worst sensitivity as compared to the eigenmode imperfection curve. Conservative knockdown factors that can be implemented for the design of cone-cylinder transition have been proposed for both eigenmode imperfection and SLI imperfection. Finally, a simple empirical solution based on several case studies has been proposed that could reasonably be used to estimate the buckling of externally pressurized cone-cylinder shell transition under Geometrical and Materially Non-linear Analysis (GMNA) and Geometrical and Materially Non-linear Imperfection Analysis (GMNIA) cases. A good correlation with the ultimate strength is observed with the application of the proposed empirical formula.

Introduction

Shell combinations such as cylinder with conical end closure are commonly used in many engineering industries. They find application in pressure vessels as transition elements between two cylinders of different diameters, in offshore as offshore platform legs and in nuclear and chemical industries as flue gas desulphurization (FGD) vessel assembly. More often than none, when they are used as pressure vessels component, they are subjected to external pressure which can lead to structural instability of the shell structures. Research into the buckling behaviour of externally pressurized cone-cylinder transition can be found in Ref. [[1], [2], [3], [4], [5], [6]]. Wenk Jr. and Taylor [1] derived equilibrium equations that are practically useful in solving conical and reinforced cone-cylinder intersection buckling problems. In 1969, Manning [2] widen the study by analyzing similar segmented structural assembly numerically. Experimental and numerical investigation into the buckling behaviour of cone-cylinder transition subjected to uniform external pressure is presented in Ref. [3,5,7,8].

Initial geometric imperfection is considered to be a challenging scenario the engineer/designer may expect, as it possibly will reduce the buckling strength of the shell structures. Questions on how it has been defined, positioned, maximum amplitude, worst shape, etc. allow the designer to plan for the worst possible scenario such as unexpected catastrophic structural failure [9]. As most cone-cylinder transition assembly in practice is susceptible to initial geometric imperfection, the buckling behaviour of imperfect cone-cylinder transition would be beneficial to the industries. A widely used form of initial geometric imperfection is the Eigenmode imperfection. However, in practice, most imperfections found in structures do not have this buckling mode. Hence, the need for a more realistic form of imperfection such as the Single Load Indentation (SLI) imperfection approach which has been adjudged to be more conservative, Hühne et al. [10]. Reference [11] presents a comprehensive review of the buckling behaviour of imperfection sensitivity of cone or cylinder under various loading conditions. Whilst, there have been several investigations on the initial geometric imperfection sensitivity of cones (Refs [4,[12], [13], [14], [15], [16], [17], [18]]), cylinders (Refs [[19], [20], [21], [22], [23], [24], [25], [26]]), etc. Nonetheless, to date, there has been no information in the open literature on the imperfection sensitivity of cone-cylinder transition subjected to external pressure.

In Refs [4,[12], [13], [14], [15],18], the initial geometric imperfection of the conical shell is considered in the form of Eigenmode imperfection. Ifayefunmi and Błachut [15,18] studied the influence of initial Eigenmode geometric imperfections on the buckling strength of truncated cones subjected to axial compression, lateral pressure, and combined axial compression and external pressure. The cones were assumed to be relatively thick, thereby falling within the elastic-plastic domain. The ratio of the amplitude of imperfection, w_o, to the cone wall thickness, t (i.e., w_o/t), was modelled numerically. Ifayefunmi and Błachut [15], also, examined initial geometric imperfection in the form of the axisymmetric outward bulge (for axial compression only), a “single wave” extracted from Eigenmode, and a localized smooth dimple (for lateral pressure only), combined axisymmetric outward bulge, and a single wave extracted from Eigenmode (for combined loading). Numerical investigation into initial geometric imperfection sensitivity cone subjected to combined axial load and lateral pressure, within the elastic-plastic regime was presented in Refs. [16,17]. Spagnoli [27,28] investigated the local and global buckling modes of axially compressed stiffened cones. Linear eigenvalue finite element analysis was employed to examine the modes of conical shells under axial compression. The changes of buckling modes were captured with the range of stiffener slenderness, several stiffeners and a tapering angle that acts as design parameters. To conclude, the paper presented the existence of simultaneous buckling modes, a starting point from which the imperfection sensitivity of cones, particularly in terms of cones to cylinders, can then be investigated (e.g. through non-linear analysis of FE models with eigenmode-affine imperfections).

Furthermore, imperfection sensitivity of cylindrical shell via Eigenmode shape has also attracted some attention, Refs [[19], [20], [21], [22], [23]]. Several researchers argued that the Eigenmode imperfection technique may not serve as the worst-case geometrical imperfection [19,24]. The single load indentation commonly referred to as single perturbation load approach (SPLA) proposed by Hühne et al. [10], is a relatively new method for designing an imperfection of shell structures. This method uses the lateral load imposed on the outer surface of the model to simulate the geometrical imperfection on the structural models such as cylindrical shells. Refs [20,25] presented studies where the cylinder knockdown factor was estimated using SLI. In Refs [26,29], multiple loads indentation along the cylinder circumference were introduced to demonstrate the worst kind of imperfection level. The test was carried out to describe and initiate the range of worst multiple load indentation approach. As a result, the multiple load indentation gives more conservative results than the single load indentation imperfection approach and in some extreme cases leads the knockdown factor closest to NASA SP 8007 guideline [30]. In several studies, a correlation between Eigenmode imperfection and SLI imperfection techniques were reported, Refs [19,27]. Castro et al. [20] explained the consistency of localized dent due to the perturbation load and its relationship with the global cylinder buckling load. Throughout its development, the SLI imperfection received many positive reviews towards its direct and robust approach but limited to axially compressed composite shells. In 2015, Błachut [31], extend this approach to externally pressurized steel domes, where alternative imperfection shape (i.e., dimple) utilizing (i) Legendre polynomials, (ii) increased-radius patch, and, (iii) localized inward dimple were investigated. Interestingly the localized inward dimple approach is found to be identical with the SLI method, in which a concentrated radial force is used to produce a dimple as an imperfection shape. More recently, the single load indentation technique was employed for the case of the axially compressed steel cylinder-cone-cylinder shell [32] and steel cones in Ref. [33]. In Ref. [32], it was reported that the location of the dimple tends to have a significant effect on the sensitivity of the structures. Whilst, in Ref. [33] the comparison between experimental and numerical prediction was seen to be good. This tends to provide an explanation to the appropriateness and relevancy of this approach to other structures and materials.

The literature shows that there is still limited knowledge of the imperfection sensitivity of cone-cylinder transition in the open literature. This paper aims to provide relevant insight into the influence of imperfection amplitude on the buckling behaviour of cone-cylinder transition. Two types of imperfection techniques are considered. They are; (a) Eigenmode imperfection and (b) Single Load Indentation (SLI) imperfection. It must be mentioned here, that the Single Load Indentation (SLI) imperfection is similar to the Single Perturbation Load Analysis (SPLA) employed for axially compressed composite shells. For the SLI imperfection, the lateral perturbation load is applied at (i) cylinder mid-section, (ii) cone mid-section and (iii) both cone & cylinder mid-sections. The present work is entirely numerical using ABAQUS finite element (FE) code and it complements the experimental results reported in Refs. [3,6].