Lagae, Guy

Van Impe, Rudy

Vanlaere, Wesley

Undetermined

Ghent University Academic Bibliography

COMPARISON OF NUMERICALLY DETERMINED FAILURE LOADS OF LIQUID-FILLED CONICAL SHELLS WITH EXPERIMENTAL TEST RESULTS   Wesley Vanlaere, Guy Lagae, Rudy Van Impe  Laboratory for Research on Structural Models, Department of Structural Engineering, Ghent University, Technologiepark-Zwijnaarde 904, B9052 Zwijnaarde, Belgium, Wesley.Vanlaere@UGent.be, Guy.Lagae@UGent.be, Rudy.VanImpe@UGent.be  ABSTRACT- The ECCS design rules for liquid-filled conical shell are verified in this paper by performing numerical simulations. There is a discrepancy between these two that is caused by the chosen imperfection shape. With this axisymmetric shape, the conical shells fail by plastic buckling. However, the realism of this imperfection shape is questioned.  INTRODUCTION:  The main body of a typical steel water tower is the conical vessel that acts as a water reservoir. The conical shell structure has a vertical axis of revolution and contains a liquid with a specific weight γ’ up to a height h’ above the base circle of the cone. In this paper, the structural behavior of such liquid filled conical shells is studied. We assume the cone to have a constant wall thickness and to be simply supported at the lower rim of the cone. An example of such a cone is shown in Fig. 1. The presence of the liquid in the shell leads to meridional compressive membrane stresses which increase rapidly towards the lower rim. If these compressive stresses surpass a critical value, the shell buckles in spite of the stabilizing effect of the circumferential tensile stresses. For these liquid-filled conical shells, design rules can be found in the Fourth Edition of the ECCS Recommendations of Buckling of Shells [1988]. These rules are based on the results of numerous experiments that were performed on scale models. In this paper, these design rules are compared with the results of numerical simulations and the discrepancy between these two is explained.   PROCEDURES, RESULTS AND DISCUSSION:  The aim of the paper is to compare the ECCS procedure for the liquid-filled cones with the results of numerical simulations with the finite element package ABAQUS. For this purpose, the water level Rdh′  corresponding with the meridional buckling design stress σxRd was determined with the ECCS procedure for seven cone geometries and for the quality classes C (see Table 1). This is the worst quality class permitted by Eurocode 3 [2006]. With ABAQUS, these seven geometries were also modelled and the GMNIA (Geometrically and Materially Nonlinear Analysis of the Imperfect structure) analyses for the quality class C were performed.  For the nonlinear material properties, perfect elastic-plastic material was assumed. The geometric imperfections were given the shape of the first eigenmode of the perfect cone, which is an axisymmetric mode. The amplitude of the geometric imperfection was 310interpreted in a manner consistent with the gauge length method of the Eurocode 3 [2006] (see Fig. 2). For each geometry, both GMNIA analyses with the first half wave oriented outward and with the first half wave oriented inward were investigated. In the analyses, the density of the liquid was increased until bifurcation or snap-trough occurred. The formula      βββ′+′γ′=σcostr2tg)tg3hr(h112x         (1) was used to determine the meridional compressive stress at the lower rim for the critical specific weight γ’ and the liquid level h’. With this meridional stress, an estimation of the critical water level could be obtained if the specific weight of water was inserted in Eqn. (1). For this new liquid level, a new GMNIA analysis was performed and a new critical liquid density was obtained. This procedure was repeated until the critical density was (almost) equal to the density of water. The results of this iterative process are given in Table 1 for the seven cone geometries and quality class C.    Fig. 1 The cone geometry.   Fig. 2 Measuring the size of the geometric imperfection. r -N -N Δw 0x ϕ ϕ  Table 1: The Cone Geometries: Results With ECCS Procedure And With ABAQUS Number 1 2 3 4 5 6 7 r1 (mm) 90 3000 350 579 200 3794 3794 t (mm) 0.3239 10 0.30 0.310 0.1229 8 15 β (°) 49.93 45 40 40 39.98 51 51 E (MPa) 195420 210000 210000 200000 5220 196200 196200fyk (MPa) 240 240 240 240 elastic 240 240 ECCS Procedure for the liquid-filled conical shell Class C σxRd (MPa) 43.96 78.30 23.13 17.05 0.7112 50.73 79.41 Rdh′  (mm) 641 7780 872 836 118 5193 8113 ABAQUS GMNIA Class C σxRd (MPa) 58.62 49.27 23.91 16.98 0.81 31.98 48.61 Rdh′  (mm) 712 6436 885 835 126 4263 6637  The results for quality Class C can be compared with the results obtained with the procedure as described in the ECCS Recommendations (Table 1). As can be seen, the critical water 311levels predicted by the ECCS are not always conservative, which is of course a problem. For geometry 2, 6 and 7, the procedure overestimates the critical water level with almost 20%. In Fig. 3, the failure pattern for geometry 2, quality class C is given. The figure also shows a contour plot of the von Mises stresses. The red zones indicate the places where the von Mises stresses are equal to the yield stress. It is clear that the failure pattern for the cone is Elephant foot and thus a plastic failure phenomenon. This plastic failure phenomenon is at first sight totally unexpected. The ultimate meridional compressive stresses are all well below the yield stress and therefore an elastic failure was expected. However, due to the presence of the large axisymmetric shape imperfections with the first half wave oriented outward, the circumferential tensile stresses obtain large values and the combination of these circumferential and meridional stresses lead to the premature yielding, causing the Elephant foot failure pattern. This effect is not covered in the Fourth Edition of the ECCS Recommendations. However, it is clear that this failure pattern is only to be feared when large axisymmtric shape imperfections are included in the simulations. Therefore it is useful to question the realistic nature of this imperfection shape. This study shows that there is a need for discussion on which imperfection shapes should be taken into account in the design process and which imperfection shapes are too deleterious.   Fig. 3. Contour plot of a failed cone. The colours indicate the von Mises stresses.   CONCLUSIONS:  Numerical simulations of imperfect liquid-filled conical shells show that Elephant foot is a failure pattern that is to be feared. With this failure pattern, the European design rules prove to be unsafe. However, this failure pattern only occurs if large axisymmetric imperfections are included in the simulations. With this paper, the authors hope to start a discussion on the imperfection shapes that should be taken into account in the design process.  ACKNOWLEDGEMENTS: The first author is a Postdoctoral Fellow of the Research Foundation – Flanders (FWO). Therefore, the authors would like to express their gratitude for the financial support of the FWO.  REFERENCES:   ECCS, Technical working group on the stability of shells, (1988), Buckling of Steel Shells, European Recommendations, Fourth Edition, No 56,  ECCS, Brussels. EN 1993-1-6, 2006, Eurocode 3: Design of steel structures-part 1-6: Strength and Stability of Shell Structures, CEN, European Committee for Standardization, Brussels. 312

Comparison of Numerically Determined Failure Loads of Liquid-Filled Conical Shells with Experimental Test Results

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Comparison of Numerically Determined Failure Loads of Liquid-Filled Conical Shells with Experimental Test Results

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