NONLINEAR BUCKLING ANALYSIS OF A CYLINDRICAL SHELL STRUCTURES

Pressure hulls are designed to stand the hydro static and dynamic bearing when they operate under water. These structures are also applied to naval submarines, remotely operated vehicle (ROV) and autonomous underwater vehicles (AUV). The underwater vehicle plays an important role in the petroleum industry, ocean engineering, and fishing, etc. To do so, ROV and AUV are studied for design and fabrication. This paper focuses on studying linear and nonlinear buckling analysis of stiffened cylindrical pressure hull under bearing load in Vietnam sea environment. The modeling of this pressure hull structure has been carried out by finite element analysis software packages.


INTRODUCTION
AT The buckling strength of ship and offshore structures are always an important part in design and analysis procedures.Thin-walled cylindrical members with specific geometrical and structural characteristics are also applied to various fields of industries, particularly in underwater engineering.When structures under compressive load, they meet various buckling modes with the change of dimensional ratios.There are two groups of dimensional ratios relate length to diameter and radius to thickness, the buckling behaviors are different from each group.
The classical buckling strength of shell under axial compression is based on Timoshenko and continues to be studied in numerical and experimental model reference loads for the cylindrical shell [1].In addition, the buckling strength of shell under hydrostatic pressure is also studied, for convenience, Warren C.Young and Richard G.Budynas provide a handbook to rapidly calculate for the various problem of solid structures [2].This method helps designers can easily estimate the linear buckling behavior without needing numerical analysis.However, in case of a complicated object such as stiffened model and nonlinear buckling strength, the finite element method is a predominant assessment.Paik numerically investigated the ultimate strength of plate under compressive load, Mingcai Xu and C [3]. Guedes Soares make an assessment of five narrow stiffened panels tested with two stiffeners under axial compression until collapse and beyond is determined by finite element analysis, and is compared with experimental results [4].
A study the ultimate strength of notched cylinders subjected to axial compression in experimental and numerical testing [5].M.R. Khedmati and M. Nazari using a numerical and experiment investigation into the structural behavior of preloaded tubular members under lateral impact loads using nonlinear finite element method [6].Minjie Cai, J. Mark F.G. Holst and J. Michael Rotter regarding two different buckling phenomena of a typical thin cylindrical silo shell under localized axial compression, and the influence of geometric imperfections on the buckling strength of the shell [7].Seung-Eock Kim and Chang-Sung Kim develop practical design equations and charts estimating the buckling strength of the cylindrical shell and tank subjected to axially compressive loads with geometrically perfect and imperfect shells and tanks [8].
The aim of this paper is to investigate the linear and nonlinear buckling strength of the cylindrical with radius to thickness ratio of less than 400.The numerical model of HRC -AUV with a length of 9.46 m and a radius of 0.4 m is investigated [9].The maximum operating depth of 120 m, it complies with continental shelf environment of Vietnam sea [10][11][12].

Structural geometry model
The HRC-AUV's geometrical and inertial characteristics are taken into account and shown in Figure 1.
For short tube, of length l, ends held circular, but not otherwise constrained, or long tube held circular at intervals l, the critical external pressure pcr is calculated by approximate formula as follows, For thin-walled circular tube under uniform longitudinal compression (radius of tube = r) shown in Figure 4, For steel material, the obtained results are shown in Table 1.A numerical model of the cylindrical shell including a necessary number of elements and the appropriate element type in numerical analysis is presented.In the present paper, element SHELL181 with 4 edges and 4 nodes is a candidate, it is suitable for thin-walled plate and shell.The full model with a length of 6460 mm is analyzed.However, for calculating rapidly, a half model with length of 3230 mm is used as shown in Figure 5.For nonlinear buckling analysis, the coupling rigid body condition is applied to this model with two master nodes.The first master is at the center of the circle with UX, UY, UZ, ROTY, ROTZ = 0 and the second master is used with UX, UY, ROTY, ROTZ = 0 in Figure 6.

Figure 6: Mesh and boundary condition
The number elements in the axial direction increased from 50 to 200, the error reduced from 1.119% to 0.034%, when 100 elements to 200 elements in the axial direction, the difference of buckling strength in two circumstances is 0.034%.Thus 100 elements in the axial direction are used for analysis model.The load is applied to the model in two cases: Load case 1 is uniform longitudinal compression without hydrostatic load and load case 2 is uniform longitudinal compression with the hydrostatic load when cylindrical shell under 120 m depth water, the appropriate external pressure is p = 1.1772MPa.

Cylindrical shell under uniform longitudinal compression
When the cylindrical shell under uniform longitudinal compression without hydrostatic pressure, the obtained results of buckling strength for thickness model t of 1, 5, 10, 15, and 20 mm are 308.434MPa, 7293.93 MPa, 27085 MPa, 57504.3MPa, and 93160.5 MPa, respectively.The finite element analysis obtained results have a value higher than the buckling strength in Table 1.In order to determine the converge of good results for the finite element analysis, the model is divided with three strategies of meshing, the Lmesh size of 50mm, 75 and 100mm, the ultimate strength of the cylindrical under axial compressed load with hydrostatic pressure p = 1.1772MPa, the obtained results are: 284.688MPa, 284.984MPa and 285.2429Mpa in Figure 12 and 13, respectively.It is clear that the results from the model with Lmesh size of 50mm and 75mm is a very small deviation, thus for nonlinear analysis model, the Lmesh size =50 is the best choice.
The ultimate strength is also obtained from the model with Lmeshsize of 50mm, and the wall thickness of 5mm, 10mm, 15mm and 20mm, these values are shown in Figure 14-16.According to the yielding of the material of 355MPa, the ratio of ultimate to yielding strength in the range of 0.8019-0.8214,this is very good for buckling and ultimate strength in compressed load.Thus, the cylindrical hull structure can be operated under the 120m deepth sea as the wall thick ness above 5mm.

Figure 1 :
Figure1: HRC-AUV mechanical design[9] In the present paper, the geometrically cylindrical shell with length is l of 6.46 m (Command/ Power), and the radius is r of 0.4m, the thickness of present model is t of 1 to 20 mm, the radius to thickness ratio of 20 to 400 are illustrated in Figure2

Figure 2 :
Figure 2: Cylindrical shell model 2.2 Linear buckling of the cylindrical shell

Figure 4 :
Figure 4: Thin-walled circular tube under uniform longitudinal compression And the critical uniform longitudinal compression cr is calculated by approximate formula as follows,

Figure 5 :
Figure 5: Numerical model of cylindrical shell: (a) half model; (b) full model

Table 1 :
Critical uniform longitudinal compression cr for thin-walled circular tube