Proceedings of the ASME 2017 Pressure Vessels and Piping Conference PVP2017 July 16-20, 2017, Waikoloa, Hawaii, USA PVP2017-65724 NONLINEAR BUCKLING ANALYSIS OF CYLINDRICAL SHELL WITH NORMAL NOZZLE SUBJECTED TO AXIAL LOADS Qianyu Shi Harbin Boiler Co., Ltd. Harbin, China, 150046 Email: [email protected] Zhijian Wang Harbin Boiler Co., Ltd. Harbin, China, 150046 Email: [email protected] Hui Tang Harbin Boiler Co., Ltd. Harbin, China, 150046 Email: [email protected] structures [2]. Whereas most pressure vessel codes provide methods of stability design by rules, and only for basic geometrical structures(e.g. cylindrical shell, conical shell, spherical shell) and simple load conditions (e.g. axial uniform compression, uniform external pressure and bending moment and their combinations) [3]. ABSTRACT Design of Large-scale and light-weight pressure vessels is an inexorable trend of industrial development. These large thinwalled vessels are prone to buckling failure when subjected to compression loads and other destabilizing loads. Thus, buckling analysis is a primary and even the most important part of design for these pressure vessels. Local buckling failure will probably occur when cylindrical shells with nozzle subjected to axial loads. In this paper, a FE model of cylindrical shell with a normal nozzle is established in ANSYS Workbench. The bifurcation buckling analysis is performed by using an elasticplastic stress analysis with the effect of nonlinear geometry, and a collapse analysis is performed with an initial imperfection. The axial buckling loads are obtained by these two types of method. Some issues about nonlinear buckling analysis are discussed through this study case. Since 2007, ASMEⅧ-2 provides three types of method to evaluate buckling failure using numerical solution (e.g. FEA) as follows: Type 1 is a bifurcation buckling analysis that is performed using an elastic stress analysis without geometric nonlinearities in the solution to determine the pre-stress in the component. Type 2 is a bifurcation buckling analysis that is performed using an elastic-plastic stress analysis with the effects of nonlinear geometry in the solution to determine the pre-stress in the component. ABBREVIATIONS ASMEⅧ-2 ASME B&PV Code Section VIII Division 2 [1] Fx Nozzle force in radical direction Fy Nozzle force in circumferential direction Fz Nozzle force in axial direction node_max Maximum deformation of the lowest mode e Allowable maximum tolerance of shell Type 3 is a collapse analysis in which imperfections are explicitly considered in the analysis model geometry. It should be noted that a collapse analysis can be performed using elastic or plastic material behavior. It depends on the stress state in the component. The former two types are bifurcation buckling analysis for perfect structures, which can be performed through eigenvalue buckling analysis in some general FEA software (e.g. ANSYS, ABAQUS) [4], another way for Type 2 is a nonlinear structural analysis for all process, in which a perturbation can be used. For example, deformation perturbation method can be used for solving a global buckling, load perturbation method is more suitable for solving a local buckling [5]. Collapse analysis is a limit point buckling analysis for imperfect structures. It is same as the elastic-plastic stress analysis method for protecting against plastic collapse failure. INTRODUCTION Cylindrical shells are commonly used in engineering structures such as aircraft, missiles, silos, pipelines, tanks, and some submarine structures. During their service life, these components are often subjected to axial loads. In addition, these structures often have geometric discontinuities, such as stiffeners and nozzles, which can lead to substantial stress concentrations and subsequently influence the stability of the 1 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use The effects of cylindrical shell with a cutout subjected to axial loads were studied by some previous work [2, 6-7]. Reference [8] addresses the issue of nozzle loads, which were applied to a circular opening in a shell, reducing the buckling capacity. However, the nozzle itself was not modeled. The influence of nozzle loads to shell with nozzle modeled subjected to external pressure was investigated in reference [9]. In this paper ， a FE model of cylindrical shell with a normal nozzle is established, type 2 and type 3 are implemented to evaluate the axial buckling loads in the case. Different directions of nozzle force are used as the perturbation loads in type 2. The lowest mode shape of eigenvalue buckling analysis is used as the initial imperfection in type 3. FINITE ELEMENT ANALYSIS Elastic-plastic buckling analysis are carried out in ANSYS Workbench, with a material curve derived using ASME Ⅷ -2 Annex 3D. The stress-strain curve of material is shown in Fig.1. The elastic-plastic buckling analysis of shells subjected to axial loads has been verified reliable in ANSYS [2]. Fig.2 The geometric model GEOMETRY AND MESHING A cylindrical shell is modeled with a normal nozzle at the half height of shell in Fig.2. The parameters of the geometry are shown in table 1. Solid 186 element with 20-node is used to mesh the structure in Fig.3. This element is suitable for analyzing nonlinear behaviors and stress concentrations. Fig.3 The meshing model BOUNDARY CONDITION With reference to the coordinate system shown in Fig.2, the boundary condition of the FE model in ANSYS Workbench is shown in Fig.3. The uniform axial force is applied to the top end of the shell, and the top end is restricted in translational circumferential and radial directions using a local cylindrical coordinate system, whereas the bottom end is fixed by setting the translational DOF in x-, y- and z- directions to be zero. A nozzle force is applied to the end of the nozzle. The different directions of nozzle force are applied respectively in three directions: Fx, Fy, Fz. Fig.1 The stress-strain curve of material STABILIZATION METHOD Nonlinear stabilization method and arc-length method are generally used for nonlinear buckling analysis. Nonlinear stabilization method cannot detect the negative-slope portion of a load-deformation curve, which is not suitable for simulating the post-buckling behavior of structures, but it can be used for global and local buckling. Arc-length method can detect the negative-slope portion of a load-deformation curve, which is suitable for simulating the post-buckling behavior of structures, but it cannot solve problems with local buckling [4]. In this Table 1 Parameters of the geometry (mm) Shell diameter 2000 Nozzle diameter 500 Shell thickness 10 Nozzle thickness 20 Shell length 8000 Nozzle length 1500 2 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use paper, the local buckling may occur for a shell with a nozzle, and the post-buckling is not considered, so the nonlinear stabilization method is used. When this method is used, the buckling load can be obtained in load-deformation curve as shown in Fig.5. The buckling point is indicated by the inability to keep structure stable for a small increase in load. Fig.6 Bifurcation axial buckling loads of shell under different directions of nozzle loads COLLAPSE ANALYSIS Collapse analysis requires that initial imperfections are considered in the model. In generally, it is difficult to measure the imperfections of structures in the design phase. Reference [11] has showed that the lowest buckling mode can be used to determine the initial imperfections, and a good correlation was observed between the buckling analysis and the experiments. One of the popular methods is consistent mode imperfection method. In this case, firstly, an eigenvalue buckling analysis is carried out. The imperfection shape of structure is the lowest buckling mode shape obtained by using command UPGEOM in ANSYS Workbench. The magnitude of imperfection is defined by the lowest buckling mode deformation multiplied a factor, ∆ (∆ = e / node_max) [12]. e is the allowable maximum shell tolerance derived from requirement of ASMEⅧ-2, (e = 1% × Shell Diameter = 1%×2000 = 20 mm) . Fig.4 Displacement and force boundary condition of the FE model The axial buckling load of collapse analysis is 7874KN in this case. The lowest eigenvalue buckling mode shape of shell is shown in Fig.10 and the collapse shape is shown in Fig.11. For this analysis model, the second eigenvalue buckling mode shape is used, as the first mode shape is affected by the nozzle deformation. Fig.5 The force – deformation cur CONCLUSIONS The nonlinear buckling analysis shows that the nozzle loads and initial imperfections have an impact on the buckling capacity of shells with a nozzle. BIFURCATION BUCKLING ANALYSIS Bifurcation buckling analysis is carried out using nonlinear structural analysis, in which initial imperfections are not considered. The nozzle force is regarded as the lateral perturbation load according to reference [10]. The nozzle load Fz has the most influence on the buckling capacity of the shells with a nozzle subjected to axial loads, whereas the nozzle load Fx has the least influence. In actual engineering, the nozzle loads can be controlled in a safety limit through nonlinear buckling analysis. The bifurcation buckling loads under different directions of nozzle loads are shown in Fig.6. The buckling mode shapes under 40KN nozzle loads in three directions are shown in Fig.7~Fig.9. The nozzle load Fz has the most influence on the buckling capacity of the shells with a nozzle subjected to axial loads, whereas the nozzle load Fx has the least influence. Local buckling will occur around the nozzle when shells with a nozzle subjected to axial loads, the local reinforcement 3 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use can be used to improve the buckling capacity (e.g. the pad reinforced nozzles and the local stiffeners). REFERENCES [1] 2015 ASME Boiler & Pressure Vessel CodeⅧ Division 2, American Society of Mechanical Engineering, New York. [2] Han, H. P., Chen, J. Q., Taheri, F., et al, 2006, “Numerical and Experimental Investigations of the Response of Aluminum Cylinders with a Cutout Subject to Axial Compression”. Thin-Walled Structures, Vol. 44, pp. 254270. [3] Shen, J., Tang, Y. F., Liu, Y. H., 2015, “Buckling Analysis of Pressure Vessel Based on Finite Element Method,” 14th International Conference on Pressure Vessel Technology, Shanghai, China, Vol. 130, pp. 355-363. [4] ANSYS,Inc., 2015, ANSYS Mechanical User’s Guide for Release15.0, ANSYS,Inc.. [5] Zhou, Y. Z., 2008, “Nonlinear Stability Analysis for Single Layer Lattice Shells with Complex Surface,” Master Thesis, Zhejiang University, China. [6] Zhang, Z. Z., Zhao, G. W., Huang, H., et al, 2012, “Buckling Analysis and Experiment of Open Cylindrical Thin Shells under Axial Load,” Journal of Beijing University of Aeronautics and Astronautics, China, Vol. 38, No. 4, pp. 557-562. [7] Yan, G., 2013, “The Design and Experimental Study of Composite Cylindrical Shell under Axial Compressive Load,” Ph.D. Thesis, Jilin University, China. [8] Maharaj, Ashveer. 2003, “A Comparative Study on the Effects of Internal Vs External Pressure for a Pressure Vessel Subjected to Piping Loads at the Shell-to-nozzle Junction.” Ph.D. Thesis, University of Natal Durban. [9] Clarke, E., Frith, R., 2015, “The Effect of Nozzles and Nozzle Loadings on Shell Buckling,” ASME 2015 Pressure Vessels and Piping Conference. American Society of Mechanical Engineering, Boston, USA. [10] Khakimova, R., Zimmermann, R., Wilckens, D., et al, 2016, “Buckling of Axially Compressed CFRP Truncated Cones with Additional Lateral Load ： Experimental and Numerical Investigation,” Composite Structures, Vol. 157, pp. 436-447. [11] See, T., Mcconnel, R. E., 1986, “Large Displacement Elastic Buckling of Space Structures,” Journal of Structural Engineering, Vol. 112, pp. 1052-1069. [12] Wei, X. Y., Chen, B. B., Zheng, H. Q., et al, 2015, “Discussion of Regulation of Initial Geometric Deviations on Cylindrical Shells under External Pressure in Pressure Vessel Design Standard,” Pressure Vessel Technology, Vol. 32, No. 4, pp. 20-28. One issue is worthy of being taken into consideration when the eigenvalue buckling analysis of shells with a nozzle is carried out for a collapse analysis. That is, a long nozzle can affect the lower mode shape of the shells, the most portion of large deformation would occur on the nozzle in the first or second mode shape because of the unrestricted end of the nozzle. These lower buckling modes cannot be used as the imperfections for shells. A third or fourth mode may be useful in this situation. FUTURE WORK The effect that eccentric loads have on the buckling capacity of a cylindrical shell with normal nozzle will be investigated. The nonlinear-based eigenvalue buckling analysis will be used to calculate the bifurcation buckling Load. The presence of multiple nozzles and their relative positions could have an impact on the buckling capacity. Different shell and nozzle diameters can be considered. 4 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use FIGURE Fig.7 The buckling mode shape of shell subjected to axial load with nozzle load Fx Fig.10 The lowest eigenvalue buckling mode shape of shell (second mode) Fig.8 The buckling mode shape of shell subjected to axial load with nozzle load Fy Fig.11 The collapse shape of shell subjected to axial load Fig.9 The buckling mode shape of shell subjected to axial load with nozzle load Fz 5 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

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