Abstract

In this study, the optimum structure of submarine pressure hulls under hydrostatic pressure was investigated by finite element method. The weight/volume ratio, which is considered as the buoyancy factor, and the maximum strength were taken as the basis for structural efficiency in the submarine pressure hull design. The collapse diving depth pressure was used for both the scantling of structural components and the calculation of critical buckling pressures. General instability buckling shape was observed during the analysis and the results were verified by DNV-GL Classification Society rules. Finally, the results of the different structures optimized to give the maximum strength at the same weight were presented in tables.

1. Introduction

The structural stability of submarines is very important for them to carry out their missions without any problems. Accidents at submarines from the past to the present have made these studies valuable for both the safety of the crews and the survival of the structure. Finally, the accident on the Titan submarine, which was planned to dive to a depth of 3800 meters to see the wreckage of the Titanic and resulted in the death of five people, showed the importance of these studies [1].

Submarines generally have two hulls. The first of these hulls is the pressure hull, which contains the living spaces, weapon control systems, weapons communication and control room, batteries, main and auxiliary machinery and provides the strength of the submarine under hydrostatic pressure. The function of the outer shell covering the submarine pressure hull is to add a hydrodynamic feature to the structure [2, 3].

Experimental and numerical studies on the strength of cylindrical shells have been studied in the past. The first theoretical solutions for cylindrical shells of uniform thickness were presented by Von Mises, Windenburg and Trilling and Von Sanden and Gunther in 1929, 1934 and 1952 respectively. These theories are still used because they are relatively simple [4].

For the calculation of symmetric buckling of a cylindrical shell between stiffeners, Lunchick [5] presented the report numbered 1393 called “plastic axial symmetric buckling of stiffened cylindrical shell made of strain hardening materials and subjected to external hydrostatic pressure”. In this study, Lunchick [5] took the hardening behaviour of the material into the calculation and considered the Poisson ratio as a variable ranging from its elastic value to 1/2, which is the upper limit for an incompressible material. He presented a shell length criterion for long and short cylinders. He said that below this value the shell buckles in only one
half-wave [5].

For the asymmetrical buckling calculations of the stiffened cylindrical shell, Reynolds [6] presented the report numbered 1392, named inelastic buckling of the cylindrical shell under hydrostatic pressure. In this report, a solution of Gerard’s differential equations for plastic buckling of cylindrical shells was found by Reynolds [6] for asymmetric buckling under hydrostatic pressure. According to this study, the critical buckling pressure was found as a function of the cylindrical shell geometry and the secant and tangent moduli for the shell material according to the stress-strain diagrams [6].

For the calculation of the general instability buckling shape of stiffened cylindrical shells, Bryant [7] presented his report number 306, named buckling of a stiffened cylindrical shell under hydrostatic pressure. In his paper, Bryant [7] solved the general instability problem by using the elastic potential energies of the shell and girders.

Aileni et al. [8] calculated the critical buckling pressure of stiffened cylindrical shells under external pressure using finite element method (FEM). Linear and nonlinear buckling analysis results were compared with experimental results and it was observed that the nonlinear analysis results were closer to the experimental results. The variation of critical buckling pressures was investigated by using reinforcing elements in Z, square, rectangular, C, I and T sections. At the end of the analysis, it was observed that the critical buckling pressure decreased as the distance between the reinforcements increased. In addition, it was found that higher critical buckling pressures occurred in Z and square section reinforcements [8].

Oh and Koo [9] analysed 7 case studies with analytical solutions for the optimum design of submarine pressure hull and validated the study with finite element analysis. As a result of the study, they proposed initial scantling formulas for weight optimization in relation to radius (R), yield strength and design pressure for shell thickness, flange width, flange thickness, web height and web thickness. Using the proposed initial sizing formulas, it was found that the pressure hull weight was reduced between 6% and 19% [9].

Kine [10] aimed to optimize the buckling performance of a pressure hull under hydrostatic pressure. His study was based on the critical buckling pressure and the weight corresponding to this pressure. The optimization includes four design parameters including cylindrical shell thickness, unsupported length between stiffeners, stiffener height and stiffener thickness. The results obtained from this study indicated that the critical buckling pressure increased by 7.09%. It was found that the unsupported length between reinforcements and cylinder thickness have a significant effect on the buckling performance of the pressure hull [10].

Wei et al. [11] investigated the optimization of a trapezoidal reinforced composite cylindrical pressure hull under hydrostatic pressure. The composite cylindrical shell was manufactured with carbon fiber reinforced epoxy, while the stiffeners were made of aluminium alloy. An analytical buckling model was derived for the stiffened composite cylindrical shell under hydrostatic pressure. Then, the FEM was used to verify the accuracy of the analytical solution. After verification, the analytical solution was combined with the genetic algorithm to obtain the maximum buckling pressure and optimize the cross-sectional shape of the stiffeners. It was found that there is a linear relationship between the cross-sectional moments of inertia of the stiffeners and the critical buckling pressure [11].

Rathinam et al. [12] generated finite element models with the help of the Ansys package to predict the shell and general instability damage shapes in stiffened cylindrical shells. Finite element analysis results were compared with analytical and experimental results. As a result, it was found that the minimum critical pressure value determines in which buckling mode the pressure hull will be damaged [12].

Şenol [13] presented a structural optimization based on the FEM for the optimum structural design of a submarine pressure hull. He compared the finite element results with the DNV-GL Classification Society results and observed that they converged. In his studies to determine the optimum geometry, a single ring stiffened geometry with effective shell length was subjected to external hydrostatic pressure using non-linear material properties and large deformations. As a result, an optimum design is defined in a way to obtain the minimum weight and maximum internal volume targets [13].

Fu et al. [14] proposed a pressure hull design method within the limits of outer envelope size and inner space utilization using variable section ribs. The pressure hull bearing mechanism was theoretically and numerically calculated and experimentally verified. They used the energy method to calculate the critical buckling pressure. As a result, it was found that variable cross-section ribs increase the buckling load by about 26.7% by maximizing the space utilization [14].

Shinoka and Netto [15] conducted three different optimization studies to minimize the weight of submarine pressure hulls. The optimization tools used were: Differential Evolution, Particle Swarm and Simulated Annealing. As a result, they found that the Differential Evolution algorithm is the most reliable and consistent in minimizing the objective functions [15].

In this study, the optimum structure was investigated for submarine pressure hulls where different reinforcements are used. The FEM was used to calculate the critical buckling pressure and the Screening optimization method was used to calculate the optimum structure. The results of the analysis were also verified with DNV classification society rules. When these results were analyzed in terms of structural efficiency, it was determined that the structures with Transverse Internal Reinforced Systems were the most effective.

2. Submarine Pressure Hull Design

2.1. Calculation of Safety Factor and Diving Depth Pressures

The pressure hull is the basic structure of a submarine and it constitutes half or more of the total weight of the submarine. A typical pressure hull is produced by welding T-section girders to high-strength axisymmetric shells produced by cold rolling. Since the late 1970s, the typical form of a submarine pressure hull has been a ring stiffened circular cylindrical shell, closed by a parabolic bow and an elliptical stern hatch, as shown in Figure 1 [16, 17].

The biggest problems faced by the submarine designer are to reduce the weight of the pressure hull and the cost of submarine construction, to increase the payload carrying capacity and the ship’s speed.

There are some diving depths that form the basis for submarine pressure hull design. These depths are mainly: Nominal diving depth (NDD) is the diving depth at which the submarine can operate unrestrictedly. Collapse diving depth (CDD) is the theoretical maximum depth at which the submarine can dive. Collapse diving pressure (CDP) is the pressure value at which the pressurized hull can collapse under a 1-minute load. This depth value is used for the scantling of pressure hull structural components [19, 20].

The conventional submarine considered in this study was assumed to operate at a NDD of 400 meters. The DNV-GL Classification Society gives the calculation of the hydrostatic pressure depending on the NDD in Equation 1 as follows [19].

​​​Nominal Diving Pressure = 
Nominal Diving Depth . 0.101 ​(​​bar​)​​​

(1)

According to the calculation, it was found that the submarine would be exposed to 40.4 bar a hydrostatic pressure at the NDD, which would be 4.04 MPa because 1 bar pressure corresponds to 0.1 MPa pressure.

Considering the constraints given in Table 1, the safety factors S₁ and S₂ corresponding to the nominal diving pressure of 40.4 bar were found to be 1.20 and 1.7984 respectively. The CDP was found 7.26554 MPa and the Test diving pressure applied to test the tightness and function of the pressure hull and equipment was found 4,848 MPa.

2.2. Structure Material

Pressure hull materials must be able to withstand high external pressures and the adverse effects of the environment. The commonly used material for a submarine pressure hull is steel with high yield strength, usually obtained by alloying or heat treatment. HY grade steels are metallurgically quenched and tempered martensitic steels. This martensitic lattice structure is formed as a result of heat treatment with alloying elements such as nickel, chromium, molybdenum and vanadium. Table 2 shows the general physical properties of the materials used in the construction of submarine pressure hulls. In this study, HY100 steel was preferred [21, 22].

2.3. Scantling of Structural Components

The strength and stiffness of any structure vary according to the properties of the material used and the geometry of the structure. Without changing the geometry of a structure, the strength and stiffness of the structure can be increased by adding small weight stiffeners to the structure. It has been observed that the buckling strength of the hull is greatly improved by using stiffeners. In pressure hulls, T-section stiffeners are most commonly used and are usually placed inside the hull. In this study, different stiffeners with the same cross-sectional moment of inertia were used for transverse, longitudinal and both transverse and longitudinal (combined) systems [8]. Figure 2 shows the effective length shell with T-section stiffeners.

Oh and Koo [9] proposed initial sizing formulas for shell thickness, flange width, flange thickness, web height and web thickness based on R, yield strength and design pressure in their study on the optimum design of submarine pressure hull. These formulas are given in Table 3. In this study, these formulas were used for the initial sizing.

In this study, the R was defined as 3123 mm, and the length of the pressure hull was specified as 9369 mm.

The structure lengths obtained depending on the input parameters were given in Table 4.

For the scantling of the frame space, the effective length formula in Equation 2 in the fourth chapter of the DNV-GL Classification Society Rulebook was used [19].

The structural component lengths chosen as a result of the scantling were given in Table 5.

The moment of inertia of the singular ring stiffener with reference to the neutral axis was calculated as 7.70536.108 mm⁴ (Parallel Axis Theorem).

3. Calculation of Critical Buckling Pressure of Submarine Pressure Hull

When a submarine pressure hull is exposed to a pressure equal to or exceeding the CDD during operation, the hull may experience the following types of damage:

1. Asymmetric Interstiffener Buckling

2. Symmetric Interstiffener Buckling

3. General Instability

4. Tripping of Frames

5. Fore and Aft Buckling [23].

Some of the factors affecting the buckling shape of ring stiffened cylindrical shells are the bending stiffness of the frames and the cylindrical shell, the type of stiffeners such as longitudinal, ring or combination of these two [10].

Buckling shapes for stiffened cylindrical shells can be classified into two main groups. These are interstiffener (local) buckling and general instability of the structure. In local buckling, if the bending stiffness of the frames is such that they will not buckle when subjected to a critical load, the shell will buckle between the frames. This buckling is classified as asymmetric and symmetric interstiffener buckling [10].

Asymmetric buckling is also called lobe buckling and occurs when the sizes of the frames are small and they are located far apart from each other. As shown in Figure 3, the shell buckles in a wave along the circumference between the frames [18].

Symmetrical buckling occurs when the ring stiffeners have large dimensions and are placed very close to each other. As shown in Figure 4, the cylindrical shell buckles in the shape of an accordion [18].

General instability of the structure is described as buckling of the pressure hull between bulkheads, web frames or dished ends. In this buckling mode in order to resist displacements in the shell, there isn’t enough bending stiffness in the frames [18].

In our design, general instability buckling mode was found in our finite element analysis. This failure mode is caused by a lack of strength of the material or the submarine diving deeper than the collapse depth [18]. Figure 5 shows the general instability buckling shape.

DNV-GL Classification Society equations for the calculation of the general instability critical buckling pressure are as follows [19]:

The membrane pressure:

The pressure in the web frames:

The pressure in the frames:

The pressure in the bulkheads:

The general instability pressure:

Table 6 shows the general instability critical buckling pressure and its components.

4. Finite Element Analysis and Optimization Method

In this part of the study, the finite element analyses were used to validate the critical buckling pressures found in the previous section and to optimize the structure.

The criteria defined for the optimum design:

CDP≤P_{cr_asym}

CDP≤P_{cr_sym}

CDP≤P_g

Minimum Buoyancy Factor

Eigenvalue buckling analysis is used to determine the theoretical critical buckling load of an ideal linear elastic structure. It is assumed that the initial imperfections of the structure are neglected. It is used to predict the bifurcation point using a linearized model of an elastic structure. A full 360 degree model is required for the analysis. Because buckling occurs, the deformation of the structure is no longer axisymmetric [10].

To determine the buckling load factor λ for the structure under pressure P, a linear static analysis is first performed. In the eigenvalue problem given in Equation 8, Κ is the stiffness and S is the stress matrix of the structure. These are calculated by static analysis with prestressing effects established. The solution of the eigenvalue problem gives the i. eigenvalue λ_i (buckling load factor), where ψi is the i. eigenvector of the displacement corresponding to the eigenvalue. Not all eigenvalues are necessary and the critical buckling load is calculated with the lowest eigenvalue [25].

Depending on the buckling load multiplier obtained at the end of the linear buckling analysis, the critical buckling pressure is calculated as given in Equation 9 [26].

The critical load multiplier was calculated here:

If λ_c < 1, the structure buckles.

If λ_c > 1, the structure is safe.

4.1. The Geometry Design

The structure was designed as a shell element. Within the scope of the design, 21 frames with a distance of 453 mm between them were used. Figure 6 shows the design process of the structure.

4.2. Mesh Generation

When the convergence analysis was examined in Table 7, the result obtained using a mesh element size of 30 mm was taken as the reference value. According to this value, the deviation amounts from this value for other element sizes were given as percentages. When the average mesh quality, minimum mesh quality and percentage change values generated depending on the mesh element size were examined, it was seen that the most ideal mesh size was 50 mm. Analyses were continued using a mesh size of 50 mm.

The cross-sectional view of the structure as a result of the meshing process was shown in Figure 7.

4.3. Load and Boundary Conditions

The linear buckling analysis of our pressure hull model was analyzed under a collapse buckling pressure of 7.26554 MPa. It was accepted that the pressure hull was constrained by bulkheads and the two ends of the hull are supported by fixed supports.

4.4. Buckling Analysis

After the mesh model of the pressure hull model was designed and the load and boundary conditions were entered into the system, the static analysis was performed first. Following this static analysis, the results obtained were linked to the buckling analysis and the first linear buckling analysis of the structure was done. Figure 8 shows the buckling mode of the structure as a result of buckling analysis.

5. Investigation of the Optimum Submarine Pressure Hull

In this section, the Screening optimization method was used to determine the ideal design in the model analyzed and to realize the objective functions mentioned in the previous sections. Within the scope of the optimization study, shell, web and flange thicknesses were considered as input parameters, while the pressure hull weight, internal volume and buckling load factor were considered as output parameters. Depending on the changing input parameters in the cylindrical pressure hull geometry, it was desired to resize the structure and recalculate the buckling calculations.

The Screening method can be used for both Response Surface Optimization and Direct Optimization systems. This method enables you to create a new set of samples and order the samples according to the objectives and constraints. This non-iterative method can be used for any type of input parameter. Screening is typically used to find initial candidate points for preliminary design. These candidate points can subsequently serve as initial points for gradient-based methods to refine the solution. Three candidate designs were selected from 9000 samples. The shell, web and flange thicknesses used in the design scheme created for the optimization study are shown as graphs in Figure 9, respectively [26].

The 9000 sample design points generated within the scope of the optimization and the related output parameters are shown in Figure 10. Here, the points shown in gray are the samples that are out of the targets.

The influence of the input parameters on the output parameters is shown in Figure 11. As can be seen from the graph, the flange thickness has a high sensitivity with the buckling load factor and a much lower sensitivity with the mass of the structure compared to the shell and web thickness.

From the 9000 samples analyzed, 3 design points were identified by Ansys. These 3 candidate design points are shown in Figure 12.

A comparison of the optimized geometry and the initial geometry is given in Table 8.

5.1. Designs Using T-Section Stiffeners

Different designs using T-section stiffeners in the optimization stage are shown in Figures 13, 14, 15, and 16, respectively.

The optimization results for the T-sections stiffened systems are given in Table 9.

5.2. Designs Using Flat Bar Stiffeners   

Different from the T-sections stiffened systems, the change of buckling behavior was investigated by using flat bar stiffeners in the structure. The flat bar stiffeners to be used should have an equal moment of inertia with the T-section reinforcing element. The flat bar reinforced systems with different layouts are shown in Figures 17, 18 and 19, respectively.

The optimization results for the flat bars stiffened combined system are given in Table 10.

6. Conclusions

Different geometries were investigated for the optimum design of the submarine pressure hull. The objective functions were the lightest structure and the highest internal volume corresponding to the maximum critical buckling pressure. The thicknesses of the structural elements of the pressure hull were the input parameters and the sensitivity of the output parameters on weight and critical buckling pressure were investigated. Different frame profiles were used to investigate the effect of frame geometry on the design. In addition, transverse, longitudinal and combined systems using different frame profiles were investigated and appropriate frames layout was examined. As a result of the optimizations, the minimum weights of the systems with the same critical buckling pressure are given in Table 9. Table 10 shows the flat bar stiffened systems with the same critical buckling pressure. When the results were compared, it is seen that T-sections stiffened systems are more suitable in terms of buckling. When examined in terms of critical buckling pressures at the same weight, transverse internally reinforced system > transverse externally reinforced system > combined internally reinforced system > combined externally reinforced system > longitudinal internally reinforced system.

As can be seen from Table 9, the most ideal design concept in terms of buckling strength is the transverse internal T-sections stiffened system, which is widely used in the construction of many submarine pressure hulls. This system provides the same critical buckling pressure with a lighter weight compared to other systems. Although the Transverse External Reinforced System was advantageous in terms of internal volume, it was disadvantageous compared to the Transverse Internal Reinforced System in terms of weight. The results also indicated that longitudinal reinforcements were insufficient for supporting the submarine pressure hull in terms of buckling strength.