LINEAR AND NONLINEAR BUCKLING ANALYSIS OF STIFFENED CYLINDRICAL SUBMARINE HULL

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1 LINEAR AND NONLINEAR BUCKLING ANALYSIS OF STIFFENED CYLINDRICAL SUBMARINE HULL SREELATHA P.R * M.Tech. Student, Computer Aided Structural Engineering, M A College of Engineering, Kothamangalam , sreelathathayil@gmail.com ALICE MATHAI Associate Professor, Civil Engineering Department, M A College of Engineering, Kothamangalam Abstract : Submarine is a watercraft capable of independent operation under water. Use of submarines includes marine science, offshore industry underwater exploration etc. The pressure hull of submarine is constructed as combination of cylinders and domes. The shell is subjected to very high hydrostatic pressure, which creates large compressive stress resultants. Due to this the structure is susceptible to buckling. The introduction of stiffeners in both directions considerably increases the buckling strength of the shell. Since the stiffened cylindrical shell is susceptible to initial imperfections, nonlinear analysis is essential. The objective of this work is the linear and nonlinear analysis of the stiffened cylindrical shell subjected to very high hydrostatic pressure. Finite element method is a powerful tool for analysis of complex structures. Finite element package ANSYS is used for modeling and analysis the submarine hull. Keywords: Buckling analysis; Nonlinear buckling analysis; Stiffened Cylindrical shell. 1. Introduction Submarines are designed to use at great depths. The Hull structure, which is a very important part of the submarine become more and more important since its strength is the main concern. When submerged, the water pressure on the submarine hull increases while the pressure inside stays the same i.e., one atmospheric pressure (McDaniel, 2011). The hull surrounding them must be able to withstand high water pressure at the desired depth, usually around 300 m. A thin walled cylindrical shell is used for the submarine. Stiffeners in circumferential and longitudinal directions considerably increase the resistance of the shell. 2. Types of analysis 2.1. Linear buckling analysis Buckling phenomenon is the major failure mode associated with thin walled cylindrical shell subjected to external pressure. Hydrostatic pressure is a polygenetic force, because the pressure has to rotate while buckling. The bifurcation buckling pressure can be determined using linear stability analysis. Determination of the bifurcation buckling pressure by linear stability analysis is carried out by constructing linear elastic stiffness matrix and the geometric stiffness matrix. Since the buckling stage is defined as that in which the proximate displacement starts growing indefinitely, the total stiffness must be singular at this stage or determinant of total stiffness must vanish. [[ K] +λ[kg]] [δ] = 0. (1) Buckling pressure can be determined by the condition, * SREELATHA P.R. Padinjarethayil,kudayathoor.P.O Idukki,Kerala, sreelathathayil@gmail.com. ISSN : Vol. 4 No.06 June

2 [K] + λ[kg] =0 (2) Where λ is the nondimensional buckling pressure. It is an Eigen value problem. Since the cylindrical shell is comparatively short shell it buckles with n number of waves in circumferential direction and one half wave in the longitudinal direction. The cylinder buckles with n = 0 corresponds to axisymmytric form and n=1corresponds to rigid body motion. These values of n are not used. The first buckling mode occurs when n=2. Minimum of all these pressure defines the buckling pressure at corresponding wave number (Rajagopalan, 1993) Nonlinear analysis Nonlinearity arises when the load displacement graph is nonlinear. The cause of nonlinearity may be material or geometric. Material nonlinearity may be due to the nonlinear stress strain relation and geometric nonlinearity due to nonlinear strain displacement relation. The critical load could not be determined with sufficient accuracy if prebuckling nonlinearity is neglected. Normally the loss of stability occurs at the limit point, rather than at the bifurcation point. In such cases the critical load must be determined through the solution of non linear system of equations. Geometric nonlinearity may be due to the follower force effect of hydrostatic force. In nonlinear analysis the stiffness should be taken in tangential sense i.e. the displacement should be calculated in incremental load step till convergence is achieved. Newton Raphson method is sued for nonlinear analysis. 3. Design concept Stiffened cylindrical shell can buckle elastically under external pressure in different ways.fig.1 has many longitudinal lobes and many circumferential lobes and is characteristics of strongly framed cylinders. This buckling lobe is referred to as interstiffener buckling or shell instability. Fig 2 has only one longitudinal lobe and is characteristics of more weakly framed cylinders. This can be referred as general instability. In this combined shell and framing might buckle in an overall manner before the frame buckle locally (Kendrick S). Fig. 2. General instability Fig. 1. Shell instability 4. Analytical Investigation D Faulkner (Professor of Naval architecture and Ocean Engineering) presented a paper on The collapse strength and design of submarines in The model for this analysis is the model suggested by Prof. D Faulkner. The model is of steel with T shaped ring stiffeners and the geometry and material properties are given in the table 1. Middle portion of the model given in Fig.3 for a length of 10 m is taken for the present study Table 1. Geometry and material properties of the model Type of submarine Large research submarine Length of the shell between compartments mm Radius of the shell 4000 mm Thickness of the shell 34 mm Length of the shell between the stiffeners 664 mm Depth of stiffener 336 mm Cross sectional area of the stiffener mm 2 Thickness of web and flange of stiffener 18 mm, 28mm Material steel Poisson s ratio 0.3 Permissible stress 540 N/mm 2 ISSN : Vol. 4 No.06 June

ANSYS is general purpose software used for different types of structural analysis mainly for marine structures.

3 Fig. 3. Geometry of Faulkner s model 5. Finite element modeling Finite element is an essential and powerful tool for solving structural problem. FEM can be used for a variety of linear, nonlinear and structural stability problems. Finite element package ANSYS is used for modeling and analysis of the structure. ANSYS is general purpose software used for different types of structural analysis mainly for marine structures.it provides a powerful pre and post processing tool for mesh generation from only geometry source to produce almost any element type. Stiffeners are modeled by beam element and cylindrical shell is modeled by shell elements (Paleti Srinivas et al, 2010). Element used is 8 nodded SHELL93 and is particularly well suited to model curved shells (Andrew et al, 2008). The element has six degrees of freedom at each node, translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z-axes. The deformation shapes are quadratic in both in-plane directions. The element has plasticity, large deflection, and large strain capabilities. 3 nodded BEAM189 is an element suitable for analyzing slender to moderately thick beam structures. This element is based on Timoshenko beam theory. Shear deformation effects are also included.beam189 is a quadratic (3-node) beam element, has six degrees of freedom at each node. These include translations in the x, y, and z directions and rotations about the x, y, and z directions. This element is well-suited for linear, large rotation, large strain nonlinear applications. These elements work best with the full Newton-Raphson solution scheme. The stiffened cylindrical sell consists of ring stiffeners. For modeling such stiffeners 3 nodded beam elements are used to obtain the exact properties of the curved beam. The mesh size influences the accuracy of convergence and speed of the solution. After meshing the stiffener and the area, merging of nodes, elements, key points, lines are done to establish connectivity between shell elements and beam element so that the all structure act integrally as a single unit. For performing linear and nonlinear analysis submarine is modeled for a length of 10m. In Fig.4, One quarter of the submarine is modeled for a length of 664 mm (c/c distance between the stiffeners). Shell between the stiffeners is then completed as in Fig.5. Fig.4. One quarter of the model between stiffeners Fig.5. Shell between the stiffeners ISSN : Vol. 4 No.06 June

6. Interstiffener analysis The hydrostatic pressure acting on the submarine is 3.015 N/mm 2 for a design depth of 300 m.

Linear buckling analysis (Eigen buckling analysis) The ANSYS Eigen results obtained for the interstiffener shell are 113.69 N/mm 2 and 28.28 N /mm 2 for fixed and simply supported boundary conditions.

69N/mm 2 and is corresponding to wave number n=26. Fig. 6. Shell with fixed boundary condition and loading. Fig.7. Buckling mode. Fig. 8. Circumferential wave pattern of buckling mode. Fig. 9 shows the shell with simply supported boundary condition with loading.

4 6. Interstiffener analysis The hydrostatic pressure acting on the submarine is N/mm 2 for a design depth of 300 m. The type of analysis done is static and is carried out by incorporating two different boundary conditions. Fixed-fixed and simply supported simply supported Linear buckling analysis (Eigen buckling analysis) The ANSYS Eigen results obtained for the interstiffener shell are N/mm 2 and N /mm 2 for fixed and simply supported boundary conditions. In Fig.6, Shell with fixed boundary condition and loading is given. Fig. 7 and Fig. 8 Shows buckling mode with 26 lobs in circumferential direction. It indicates that the buckling pressure is N/mm 2 and is corresponding to wave number n=26. Fig. 6. Shell with fixed boundary condition and loading. Fig.7. Buckling mode. Fig. 8. Circumferential wave pattern of buckling mode. Fig. 9 shows the shell with simply supported boundary condition with loading. Fig.10. and Fig.11 shows buckling mode with 21 numbers of circumferential lobs. The buckling pressure is 28N/mm 2 corresponding to wave number n=21. Fig. 9. Shell with simply supported boundary condition and loading. Fig. 10 Buckling mode. Fig 11. Circumferential wave pattern of buckling mode. ISSN : Vol. 4 No.06 June

6.2 Nonlinear analysis ANSYS employs Newton Raphson approach to solve nonlinear problem (Prabu et al, 2009).

The iterative procedure continues until the problem converges. In addition to pressure loading, an out of plane load is also applied to cause buckling.

5N/mm 2 for fixed and simply supported boundary conditions. 7. Interbulkhead analysis Full submarine model for a length of 10 m for the present study is shown in Fig.

14 shows the deflected shape and buckling mode for fixed boundary condition. It indicates 3 lobs in the circumferential direction. Fig 13. Deflected shape-fixed fixed.

5 6.2 Nonlinear analysis ANSYS employs Newton Raphson approach to solve nonlinear problem (Prabu et al, 2009). In this problem the loads are subdivided into a series of load increments. The load increments are applied over several load steps. The iterative procedure continues until the problem converges. In addition to pressure loading, an out of plane load is also applied to cause buckling. The point where nonlinearity starts is the nonlinear buckling load. The nonlinear buckling loads obtained are 97.5 N/mm 2 and 22.5N/mm 2 for fixed and simply supported boundary conditions. 7. Interbulkhead analysis Full submarine model for a length of 10 m for the present study is shown in Fig.12. Fig 12. Full model ANSYS Eigen results obtained for full model are 90 N/mm 2 and 81 N/mm 2 for fixed and simply supported boundary condition respectively. Fig. 13 and fig. 14 shows the deflected shape and buckling mode for fixed boundary condition. It indicates 3 lobs in the circumferential direction. Fig 13. Deflected shape-fixed fixed. Fig 14. Buckling mode. In the case of simply supported end condition, no much reduction in the buckling pressure on release of end restranits. Fig.15 and Fig.16 gives the deflected shape and the buckling mode. It indicate 3 lobs in the circumferential direction. ISSN : Vol. 4 No.06 June

Fig.15. Deflected shape Simply supported. Fig.16. Buckling mode. The results obtained for various boundary conditions for interstiffener and interbulkhead analysis are summarized in the table 2.

6 Fig.15. Deflected shape Simply supported. Fig.16. Buckling mode. The results obtained for various boundary conditions for interstiffener and interbulkhead analysis are summarized in the table 2. Table 2. Comparison of analysis results Interstiffener Interbulkhead Boundary condition Fixed Simply supported Fixed Simply supported Collapse pressure (N/mm 2 ) No of circumferential lobs Failure criteria Maximum yield stress for this model by design of D Faulkner is 540 N/mm 2. Fig. 17 and Fig. 18 shows the von Mises stress and principal stress for the shell just before buckling and the value of maximum stress is N/mm 2. Since the stress is very less compared to yield stress there is no possibility of material failure and the structure may fail by form failure. The stress is maximum at the ends indicates the constraint of degrees of freedom at the ends. Fig 17. von Mises stress before buckling. Fig 18.Principal stress before buckling. 9. Results and discussions Comparative study of linear buckling and nonlinear buckling has been done for two configuration, interstiffener and interbulkhead. The scope of investigation has been extended to realize the influence of possible boundary conditions, which reflects the effect of end restraint. The ends have been treated as fixed-fixed and simply supported- simply supported. For interstiffener configuration it is concluded that interstiffener buckling pressure is higher for fixed boundary condition compared to simply supported boundary condition. For simply supported the ends will have more flexibility and hence will have more effective length and failure occurs with less number of circumferential waves. Fixity reduces the effective length and shell fails with more number of circumferential waves. For shorter shell the nonlinear buckling pressure is susceptible to rotation restraint at the ends indicated by higher buckling load for fixed boundary condition. ISSN : Vol. 4 No.06 June

7 Between the bulkheads the shell is comparatively longer and shell buckles with less number of circumferential waves. The lower value of n refers to the general instability. From the result it is clear that the influence of rotation restraint is nominal. There is no such reduction in buckling pressure on release of rotation restraint. There is a reduction of % in buckling pressure. This is because in long shell the shell buckles with 3 waves in circumferential direction. From the linear and nonlinear analysis results it is found that failure will occur prior to buckling load. The percentage reduction in the buckling pressure in nonlinear analysis are 14.2 and 20.4 for fixed and simply supported boundary condition respectively. Considering these effects nonlinear analysis becomes even more important in the damage prediction of submarine shell. Therefore a definite need is felt for the nonlinear analysis while considering the design criteria of the submarine hull. References [1] Andrew P.F,Little,Ross C T F,Daniel,Graham (2008);Inelastic buckling of geometrically imperfect tubes under external hydrostatic pressure. [2] Kendrick S ;The buckling under external pressure of ring stiffened circular cylinders. [3] McDaniel J T (2011) ;J T McDaniel Official Website. [4] Prabu N,Rathinam N,Srinivasan R,Narayanan K A S ( 2009);Finite elemnt analysis of buckling of thin cylindrical shell subjected to uniform external pressure. [5] Paleti srinivas,krishnachaithanya Sambana,Rajeshkumar (2010);Finite element analysis using ANSYS [6] Rajagopalan, K.;. (1993); Finite Element Buckling Analysis of Stiffened Cylindrical Shells, Oxford &IBH publishing Co.Pvt Ltd. ISSN : Vol. 4 No.06 June

CHAPTER 1 INTRODUCTION

CHAPTER 1 INTRODUCTION 1.1 GENERAL More than two third of the earth's surface is covered with water. Submersibles are primarily employed to observe and explore the subsea environment. Submarine is a submersible