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Abstract

The design of deep submersible pressure hull's structural is one of the core technologies of submersible development of human history. Submersible pressure hulls with fiber-reinforced multilayer constructions have been developed in the recent years as substitutes for classical metallic ring-stiffened pressure hulls; strength and stability are its top priority. This paper investigates the optimum design of a composite elliptical deep-submerged pressure hull under hydrostatic pressure to minimize the buoyancy factor of the submersible pressure hull under constraints on the failure criteria and the buckling strength of the hulls to reach the maximum operating depth. The thickness and the fiber orientation angles in each layer, the radii of the ellipse, and stringers dimensions were taken as design variables and determined in the design process. The optimization procedures are performed using commercial finite element analysis software ANSYS. Additionally, a sensitivity analysis is performed to study the influence of the design variables on the structural optimum design. Results of this study provide a valuable reference for designers of underwater vehicles.

1. Introduction

Many structural elements made of fiber-reinforced composites are increasingly utilized in aerospace, marine, and civil engineering applications [1]. The most important benefits from using such advanced materials are the attainment of high stiffness-to-weight ratio and long fatigue life. An important element of any submersible is the pressure hull. The pressure hull is the main watertight structural component of a submarine that houses personnel, propulsion machinery, weapons and sensor systems, and other sensitive equipment. Pressure hulls are designed to withstand the hydrostatic load associated with diving to depths usually measured in the hundreds of meters [2]. One of the major problems confronted by the designer of deep submersibles is to minimize the weight of the pressure hull and to increase the payload, propulsion, and velocity of submersible vessels, which can be achieved through the advanced hull concept and efficient material systems [3, 4].

Buoyancy requirement for the weight-sensitive structures can be easily met by applying a composite material, which has an excellent specific strength and stiffness to reduce the weight of the structure. The danger of being detected by sonar can be reduced and the resistance to corrosion can be increased because the composite sandwich has excellent sound absorption properties and is stable against the chemical reaction [5]. A submarine designer pointed out that the safety factor of 1.5–2.0 is considered acceptable in most engineering practice [6]. Liang et al. [7, 8] optimized the design of filament-wound multilayer sandwich submersible pressure hulls. The facings become thicker and the core becomes thinner as the operational depth increases. Alvarez et al. [9] optimized the hull geometry to reduce the total resistance. The hull shape resulting from the optimization process reduces (up to 25%) the estimated total resistance. Cui et al. [10] performed the structural optimization of a bulk carrier with two conflicting objectives (weight and fatigue) as a case study. Zu et al. [11] presented the optimal design problem of a half-cell dome profile for filament wound articulated pressure vessels which aims to maximize the structural performance of pressure vessels based on the continuum theory and the Tsai-Wu failure criterion. Adali et al. [12] presented an approach for the optimization of symmetrically laminate cylindrical pressure vessels subjected to strength constraint. The optimization is carried out with respect to the fiber orientations and thickness distributions subject to the Tsai-Wu failure criterion.

Kim et al. [13] designed an automotive composite lower arm using carbon-epoxy composite materials and optimized the stacking sequence of the composite layer to maximize the buckling load capability. The composite lower arm had two times higher stiffness and buckling strength compared to the conventional steel lower arm while having 50% less weight. Lee et al. [14] optimized composite sandwich cylinders under external hydrostatic pressure. Both buckling and material failure were considered. The optimization results suggest that when the core thickness increases, the material failure determines the design maximum pressure instead of the buckling. Tauchert [15] studied the optimum design of a reinforced cylindrical pressure vessel. Messager et al. [16] presented the optimal design of deep submarine by determining the laminate stacking sequences that maximize the critical external buckling pressure, using a genetic algorithm procedure coupled with an analytical model. Bruggi and Taliercio [17] presented a numerical approach for the optimal design of any unidirectional fiber-reinforcement to improve the structural performance of existing structural elements. The Tsai-Wu failure criterion is implemented to detect highly tensile-stressed regions in the existing structural components.

Ca et al. [19] presented a reliability-based load and resistance factor design procedure for subsea composite pressure vessel subjected to external hydrostatic pressure. The longitudinal modulus, inside radius of composite layers, unsupported length, and external pressure significantly affect the design results, whereas transversal modulus, Poisson's ratio, shear modulus, and winding angle have little effects. Maalawi [20] developed a practical approach for enhancing the buckling stability limits of thin-walled anisotropic rings-long cylinders. Bakshi and Chakravorty [21] studied the first-ply failure of thin composite conoidal shells subjected to uniformly distributed load. To achieve high values of the failure loads, the bottommost lamina of the cross-ply laminates should have the fibers oriented along the arch direction.

Chang [24] investigated analytical and experimental first-ply failure loads of laminate pressure vessels subjected to internal pressure loads. Chang and Chiang [25] reported experimental and theoretical first-ply failure loads of antisymmetrically laminate composite plates subjected to central point loads. Prusty et al. [26] reported first-ply failure of laminate cylindrical and spherical shells with and without stiffeners, respectively. Failure due to structural buckling is also a major risk factor for thin laminate cylindrical shells. Anastasiadis and Simitses [27] studied the buckling of long laminate cylindrical shells under external radial pressure using higher order deformation theory. The results provide valuable information concerning the applicability of the various shell theories.

Gohari et al. [28] studied a circular cylindrical thin-walled shell failure made of GRP composite subjected to static internal and external pressure. Deformation, delamination, shear deformation, and microbuckling failure were investigated. Stacking sequence and fiber angle orientations were mainly effective on failure strength. Xiong et al. [29] studied the failure behavior of sandwich-walled cylindrical shells with metallic lattice truss cores and fiber-reinforced composite face sheets under uniaxial compression. The local buckling and face crushing modes can exist together and are the most important modes of failure of the fabricated structure. Graham [30, 31] studied composite pressure hulls for deep ocean submersibles and evaluated some of the popular failure criteria. Hsu et al. [32] studied the curvature effect on stress concentrations around a circular hole in the curved shell and the failure modes of the Guppy type pressure hull under external pressure loading using the finite element method.

Kolakowski [33] presented a modified Tsai-Wu criterion, which can be used in the multicriterion optimal design of composite structures. Jen [34] presented the optimal design of a small-scale midget submersible vehicle (MSV) pressure hull using a powerful optimization procedure combining the extended interior penalty function method with the Davidon-Fletcher-Powell method. Liang et al. [35] investigated the optimum design of multiple intersecting spheres deep-submerged pressure hull subjected to hydrostatic pressure. Ross [36 –38] introduced an alternative design for submarine pressure hull without ring-stiffeners and predicted the four-lobe buckling mode of a corrugated carbon fiber vessel under external pressure. In this study, a composite submarine pressure hull in the form of elliptical cylinder subjected to hydrostatic pressure as shown in Figure 1 is optimized using nonlinear finite element analysis software ANSYS. The objective function is to minimize the buoyancy factor by simultaneously considering the buckling and material failure loads.

Figure 1:

Geometry of elliptical submersible pressure hulls.

2. Submarine Pressure Hull

The submersible pressure hull structure form is different according to the structure of submersible pressure shell and material selection principle. The usual shape of a submarine pressure hull is in the form of a ring-stiffened circular cylinder, blocked by end caps, as shown in Figure 2. This is a good structure to resist the effects of external hydrostatic pressure [39]. Figure 3 shows various wall architectures used for pressure hulls which include pure monocoque constructions, gas-storage toroidal tube-stiffener hull wall construction, traditional ring-stiffened hull wall constructions, trapezoid corrugated hull wall construction, ring-stiffened corrugated pressure hull wall construction, and sandwich hull wall construction [7, 18].

Figure 2:

The usual shape of a submarine pressure hull.

Figure 3:

Shape of various wall architectures used for pressure hulls [7, 18].

In this study, an elliptical pressure hull as shown in Figure 1 is proposed for a better space utilization, good hydrodynamic form, and stability; this is better than a spherical form of the same volume and is easy to manufacture. The stacking sequence of the shell cross section is shown in Figure 4. Plies are numbered from the bottom of the laminate upwards. The stacking sequences ${{[(α_{1}^{o} / α_{2}^{o} / α_{3}^{o} / α_{4}^{o})}_{4}]}_{s}$ and (α₃^o = − α₂^o) are considered with thicknesses t₁,t₂,t₃,t₄, respectively.

Figure 4:

The stacking sequence of the cross section of the shell.

3. Material Property Requirements

The materials for underwater pressure vessels must not only be capable of withstanding very high external pressures, but must also have other suitable properties that can withstand the environment [39]. The main materials for the design of submarine pressure hulls are high strength steels, aluminum alloys, titanium alloys, and composites [39]. Substituting composites for the metallic structures has many advantages because of the higher specific stiffness and strength of the composite materials. In this study, composite materials are used due to their superior properties, like strength-to-weight ratio. Also, composite structures offer great flexibility and stiffness in design, allowing change of the material system in many ways like number of fibers and different lay-up conFigurations. The model is assumed to be constructed from T700/epoxy composites to model the laminates which have high stiffness and strength. The material property and the strength parameters of T700/3234 epoxy are given in Table 1 [22, 23].

Table 1:

Material properties of the T700/epoxy composites [22, 23].

Material	Material and strength properties

T700/epoxy composites	E₁₁ = 132 GPa, E₂₂ = 10.3 GPa, E₃₃ = 10.3 GPa, G₁₂ = 6.5 GPa, G₁₃ = 6.5 GPa, G₂₃ = 3.91 GPa, υ₁₂ = 0.25, υ₁₃ = 0.25, υ₂₃ = 0.38, X_t = 2150 MPa, X_c = 2150 MPa, Y_t = 298 MPa, Y_c = 298 Mpa, S = 778 MPa, ρ = 1570 kg/m³

4. First-Ply Failure of Angle-Ply Laminate

It is important to note that laminate composites fail in a progressive manner which initiates from the weakest lamina of the laminate and the corresponding load is termed as the first-ply failure load. Failure modes of a composite lamina are dependent on the applied load. If a lamina is subjected to tensile stresses, then its failure modes could be fiber breakage, transverse matrix cracking in the plane of the lamina, or interfiber shear failure of the matrix. In case of compressive stresses, the failure modes could be fiber buckling which dominates failure of the lamina along the fiber direction and matrix crushing which leads to the failure of the composite matrix. All these failure modes can be identified through maximum stress and maximum strain failure theories. In this study, the first-ply failure criterion of angle-ply laminate is considered to be the first strength constraint on the present optimization design problem. The Tsai-Wu failure criterion is employed to assess the capacity of angle-ply laminate of multilayer elliptical pressure hull to withstand overstressing failure. Additionally, a first-ply failure strength load factor (FI) is introduced to identify the characteristics of the first-ply failure of the angle-ply laminate.

The Tsai-Wu failure criterion is the most generalized criterion since it distinguishes between the compressive and tensile strength of a lamina and can be written as the following expanded form [40]:

FI = F_{11} σ_{1}^{2} + F_{22} σ_{2}^{2} + F_{66} σ_{6}^{2} + F_{1} σ_{1} + F_{2} σ_{2} + 2 F_{12} σ_{1} σ_{2},

(1)

where the symbols σ₁ and σ₂ are the stresses in the longitudinal and transverse fiber directions, respectively, and σ₆ is the in-plane shear stress. The expressions for the coefficients F₁₁,F₂₂,F₆₆,F₁,F₂, and F₁₂ are given by

\begin{matrix} F_{11} = \frac{1}{X_{t} X_{c}}, F_{22} = \frac{1}{Y_{t} Y_{c}}, F_{66} = \frac{1}{S^{2}}, \\ F_{1} = \frac{1}{X_{t}} - \frac{1}{X_{c}}, F_{2} = \frac{1}{Y_{t}} - \frac{1}{Y_{c}}, F_{12} = - \frac{1}{2} \sqrt{F_{11} F_{22}}, \end{matrix}

(2)

where X_t and X_c are the longitudinal tensile and compressive strengths, respectively, Y_t and Y_c are those in the transverse direction, and S is the in-plane shear strength. The Tsai-Wu failure criterion is used for the failure prediction in a ply, and the Tsai-Wu failure function must be lower than 1.0 for the whole model. Accordingly, failure occurs when the above function reaches or exceeds the value 1. Moreover, the following condition must be satisfied to ensure that the angle-ply laminate does not exhibit first-ply failure:

FI \leq 1 .

(3)

5. Optimization Methodology of Composite Elliptical Pressure Hull

The optimal design problem of composite elliptical submersible pressure hull under hydrostatic pressure is to minimize the buoyancy factor of the submersible pressure hull. Under constraints on both the failure and buckling strength of the hull shell, according to the design requirements. Buckling and failure loads have to be considered in the optimal structural design of the composite elliptical submersible pressure hull under hydrostatic pressure because the buckling load can change according to geometric conditions and stacking sequence while local failure can occur even before buckling. Consequently, the structure's resistance to failure and buckling can be improved by changing the effective modulus or strength by choosing the appropriate materials and lay-up sequence. Therefore, an optimized stacking sequence for the composite elliptical submersible pressure hull is needed. The failure strength is based on the Tsai-Wu failure criterion. The hull buckling failure strength is based on the buckling strength factor (λ) which is the ratio between critical buckling load (P_cr) and actual operating load (P) of the hull shell; it must be greater than one. There are many algorithms and software for implementing the structural optimization [41].

Figure 5 shows the optimization procedure for elliptical submersible pressure hull. The optimization model of the design includes design variables, objective function, and design constraints. The optimization design is described as follows.

Figure 5:

Flow chart of optimal design procedure.

5.1. Design Variables

The radii of the ellipse of the elliptical submersible pressure hull (the major diameter (D_major) and the minor diameter (D_minor)), the stringers dimensions (W₁₁,W₂₁,W₁₂, and W₂₂), the orientation angle of the fiber direction for every layer (α₁,α₂,α₃, and α₄), the thickness of each layer (t₁,t₂,t₃, and t₄), and the operating depth (H) as a state variable are considered for the optimal design problem.

5.2. Objective Function

The optimal design problem involves determining the best geometrical configuration for minimizing the buoyancy factor. Buoyancy factor is used to compensate loss of weight due to immersion in water. Buoyancy is the upward force that keeps things afloat. The net upward buoyancy force equals the amount of the weight of fluid displaced by the body volume. This force will make objects lighter when it immerses in fluid. The objective in the design of the elliptical submersible pressure hull is to minimize the buoyancy factor, which is the common basis for evaluating the structural efficiencies of the various hulls. The objective function is stated as follows:

F (X) : Minimize B . F = \frac{Total hull weight}{The weight of fluid displaced by the body volume} .

(4)

5.3. Design Constraints

The minimum buoyancy factor design problem of the composite elliptical submersible pressure hull under hydrostatic pressure treated here has two behavior constraints that control the failure modes and side constraints which specify the permissible range of design variables to specify spatial and human requirements. Such constraints are presented as follows.

5.3.1. Behavior Constraints

In order to ensure the optimum structural design on the elliptical submersible pressure hull to reach the maximum operating depth, the material and buckling strength constraints must be satisfied and are defined as follows.

Material Strength Constraint. In order to avoid materials failure, the Tsai-Wu failure criterion is used for the failure prediction in a ply, and the Tsai-Wu failure function (FI) must be lower than 1.0 for the whole model. Hence, the strength inequality constraint can be expressed as

g_{1} : FI (i) - 1 \leq 0, i = 1,2, 3, \dots, n,

(5)

where FI is first-ply failure of the angle-ply laminate for each ith layer and n is the number of layers.

Instability Constraint (Buckling Constraint). The minimum critical buckling load (P_cr) must exceed the maximum actual operating loading (P) to ensure the stability of the pressure hull. Therefore, a general instability of elliptical submersible pressure hull is presented as

g_{2} : \frac{P}{P_{cr}} - 1 \leq 0 .

(6)

Operating Depth. In order to reach the maximum operating depth withstanding the applied hydrostatic pressure,

H \leq H^{U},

(7)

where H and H^U represent the operating depth and its upper limit, respectively.

5.3.2. Side Constraints

In addition, corresponding to the choice of the design variables, the following conditions of the dimensioning belong to the group of side constraints in these investigations:

D_{i}^{L} \leq D_{i} \leq D_{i}^{U}, i = \max, \min,

(8)

where D_i, D_i^L, and D_i^U represent the ith elliptical submersible pressure hull diameters and their upper and lower limits, respectively.

For the ring beams and stringers with rectangular cross-section.

W_{i 1}^{L} \leq W_{i 1} \leq W_{i 1}^{U}, W_{i 2}^{L} \leq W_{i 2} \leq W_{i 2}^{U}, i = 1,2,

(9)

where W_i1, W_i2, W_i1^U, W_i1^L, and W_i2^L represent the ith rings beam and stringers cross-section dimensions and their upper and lower limits, respectively. Consider

t^{L} \leq t_{i} \leq t^{U}, i = 1,2, 3, \dots, n,

(10)

where t_i, t^L, and t^U represent the ith thickness and the lower and upper bounds imposed on the individual ply thicknesses. Consider

α^{L} \leq α_{i} \leq α^{U}, i = 1,2, 3, \dots, n,

(11)

where α_i, α^L, and α^U represent the ith orientation angle of each layer and their upper and lower bounds, respectively. The angles are given in degrees and varied from 1° to 90°.

6. Finite Element Model Simulation

The main requirement of composite elliptical submarine submersible pressure hulls is to resist hydrostatic pressure exerted by water at maximum operating depth to protect personnel and equipment inboard when submerged. When hydrostatic pressure exceeds a critical value, shell buckling may occur to the pressure hull and the angle-ply lamina failure stress exceeds the strength. To understand the mechanical behaviors of composite elliptical submarine submersible pressure hulls and to avoid these failures, the buckling strength, the failure strength, and the maximum deflection are analyzed using FEA software ANSYS. The structure is modeled using SHELL99 (linear layered structural shell element with 8-node, 3-D shell element, with six degrees of freedom at each node). The BEAM189 (six degrees of freedom occur at each node) is used for rings beam and stringers. BEAM189 is a quadratic three-node beam element in 3D [42, 43].

The shell consists of thirty-two layers with stacking sequence ${[{(0^{°} / 45^{°} / - 45^{°} / 90^{°})}_{4}]}_{s}$ before the optimization. The displacement boundary conditions applied in the FE model were as follows: (1) Symmetry boundary condition in all nodes at y = 0, and x = 0 but this boundary condition was not sufficient since the submarine would tend to move in the z-direction due to the fluid flow. Therefore, a 2nd boundary condition was employed; (2) one single node close to center of gravity of the hull was constrained in the z-direction (longitudinal); that is, Uz = 0. No restriction was imposed in the y-direction movement. Submarine motion in the vertical direction (y) was balanced by gravity and buoyancy. These boundary conditions allowed us to capture heave and pitch motion of the submarine in the simulation [44].

7. Numerical Results and Discussions

Results of the optimization procedures for composite elliptical deep submersible pressure hull are presented and summarized in Table 2 and Figures 6 –11. Table 2 indicates the optimal design point and shows that the optimal objective function (buoyancy factor (B.F)) is 0.27023 and the maximum Tsai-Wu failure is 0.98381 which occurs in the bottommost fiber. Also, it is noted that the optimal angle-ply orientations for thirty-two layers are $[{{({85.5}^{°} / {1.27}^{°} / - {1.27}^{°} / {89.4}^{°})}_{4}]}_{s}$ . These results can overcome all structural failures until the operating depth (H) of 4211.1 m in the optimum configuration of the composite elliptical submersible pressure hull. Furthermore, a sensitivity analysis is presented to understand thoroughly the effects of the design variables on the optimal structural strength design of the elliptical submersible pressure hull.

Table 2:

The results of the optimal design.

FI ₁	0.9838	FI ₁₁	0.635	FI ₂₁	0.7788	FI ₃₁	0.59774	α₄	89.408°
FI ₂	0.8150	FI ₁₂	0.825	FI ₂₂	0.5606	FI ₃₂	0.77501	W ₁₁	5.02 (cm)
FI ₃	0.8059	FI ₁₃	0.8075	FI ₂₃	0.564	D _major	1.9884 (m)	W ₂₁	5.06 (cm)
FI ₄	0.9461	FI ₁₄	0.5580	FI ₂₄	0.7739	D _minor	1.9732 (m)	W ₁₂	0.13087 (m)
FI ₅	0.921	FI ₁₅	0.5502	FI ₂₅	0.7786	t ₁	1.79 (mm)	W ₂₂	0.15768 (m)
FI ₆	0.7285	FI ₁₆	0.7895	FI ₂₆	0.5751	t ₂	1.66 (mm)	Operating depth (H)	4211.1 (m)
FI ₇	0.7200	FI ₁₇	0.785	FI ₂₇	0.5797	t ₃	0.52 (mm)	Buoyancy factor (B.F)	0.27023
FI ₈	0.8853	FI ₁₈	0.5487	FI ₂₈	0.7744	t ₄	1.08 (mm)	Maximum deflection (δ_MAX)	1.97 (cm)
FI ₉	0.861	FI ₁₉	0.5512	FI ₂₉	0.7796	α₁	85.501°	Buckling strength factor (λ)	105.1
FI ₁₀	0.6436	FI ₂₀	0.774	FI ₃₀	0.5920	α₂ = − α₃	1.2723°	Buckling strength factor (λ)	105.1

Figure 6:

Effect of design variables on buoyancy factor.

Figure 7:

Effect of design variables on buckling strength.

Figure 8:

Displacement distribution for elliptical submersible pressure hull.

Figure 9:

Effect of design variables on maximum deflection.

Figure 10:

Tsai-Wu failure distribution for elliptical composite submersible pressure hull.

Figure 11:

Effect of design variables on Tsai-Wu failure index.

7.1. Effect of Design Variables on Buoyancy Factor

The buoyancy factor is a parameter expressing the efficiency of the structure in terms of its ability to provide for an excess of displacement over that required to support its own weight. 1.0 is seen to be the critical value of the buoyancy factor, where the empty pressure hull is in a condition of neutral buoyancy with zero efficiency with respect to its ability to support other weights. Figure 6 shows that the B.F is directly proportional to each shell thickness (t). The B.F is inversely proportional to the major diameter (D_major) and minor diameter (D_minor). In addition, larger t₁,t₂,t₃, and t₄ all imply a higher B.F. Also, the B.F will be decrease as D_major and D_minor increase.

7.2. Effect of Design Variables on Buckling Strength Factor

Figure 7(a) shows the effect of α upon the buckling strength factor; in case of α₁, as it increases the buckling strength factor decreases with small rate until α₁ equals 45°. After that the buckling strength factor decreases rapidly until α₁ equals 70°. With increasing α₁ the buckling strength factor increases again. The α₄ has the same trend as α₁. In case of α₂, as α₂ increases the buckling strength factor increases until α₂ equals 20°. After that the buckling strength factor decreases as α₂ increases. Figure 7(b) shows the effect of D_major upon the buckling strength factor. This figure shows that as D_major increases the buckling strength factor decreases rapidly. The maximum variation occurs at D_major (1.9–2.1 m). After that, as D_major increases buckling strength factor decreases with minimum variation. In order to avert the failure, the D_major can be appropriately chosen to be less than 2.2 m.

Figure 7(c) shows the effect of D_minor upon the buckling strength factor, which shows that as D_minor increases the buckling strength factor increases and reaches to the maximum value when D_minor is nearly equal to D_minor. In order to avert the failure, the D_minor can be appropriately chosen to be more than 1.7 m. Figure 7(d) shows the effect of H upon the buckling strength factor. This curve shows that as H increases the buckling strength factor decreases with relatively high slope until H equals 4000 m; after that the slope is decreasing.

Figure 7(e) shows the effect of layer thickness upon the buckling strength factor. It can be seen that as t₁ and t₄ increase the buckling strength factor decreases but t₁ has more effect on buckling strength factor than t₄ which indicates that the geometric effects have a great influence upon the stability. As t₂ and t₃ increase, the buckling strength factor increases and the maximum variation occurs when the range of layer thickness is between 0.0005 and 0.002 m. Also, the figure illustrates that t₂ and t₃ heavily influence the buckling strength factor.

7.3. Effect of Design Variables on Maximum Deflection Value

The displacement distributions in both X and Y directions for elliptical submersible pressure hull are shown in Figure 8. The maximum magnitude of the displacement occurs at centroid regions of the submarine in Y direction as shown in Figure 8(a). In case of X direction, the maximum magnitude of the displacement occurs at the right of main part of the submarine as shown in Figure 8(b).

Figure 9(a) shows the effect of α on maximum deflection value; for α₁, the maximum deflection occurs when α₁ equals 0°. With increasing α₁, the maximum deflection value decreases and reaches the minimum value when α₁ equals 90°. The orientation of fiber direction in layer 4 (α₄) has the same trend as α₁ but its effect upon the maximum deflection value is less than α₁. For the orientation of fiber direction in layer 2 (α₂) and layer 3 (− α₂) as α₂ increases the maximum deflection value is nearly constant until α₂ equals 38°. After that, as α₂ increases the maximum deflection value increases and reaches the maximum value when α₂ equals 90°.

Figures 9(b) and 9(d) show the effect of D_major and H upon the maximum deflection value. The curve illustrates that D_major and H are directly proportional to maximum deflection value. The influence of D_major upon deflection value is greater than the operating depth. Figures 9(c) and 9(e) show that the maximum deflection value is inversely proportional to each shell thickness (t) and minor diameter (D_minor). Also, t₁ has the largest effect on the maximum deflection value and t₃ has the minimum effect on the maximum deflection value.

7.4. Effect of Design Variables on Maximum Tsai-Wu Failure

The Tsai-Wu failure distributions for elliptical composite submersible pressure hull are shown in Figure 10. The maximum Tsai-Wu failure occurs at layer (1) and layer (4), which reveals that the failure will be initiated first in the bottommost lamina at the middle of the sides of the plates near the stringers. Figure 11(a) shows the effect of α₁ upon the Tsai-Wu failure; the maximum Tsai-Wu failure occurs when α₁ equals 0° for all layers. The maximum value of Tsai-Wu failure occurs at layer (4). As α₁ increases the maximum Tsai-Wu failure decreases until α₁ equals 40°. The maximum Tsai-Wu failure for layer (1) decreases and reaches the minimum value but the failure still exists in layers 2, 3, and 4. With increasing α₁, the maximum Tsai-Wu failure decreases until α₁ equals 85.5°; at this value there is no failure and the maximum Tsai-Wu failure for all layers will be less than one.

Figure 11(b) shows the effects of α₂ upon Tsai-Wu failure and it is observed that all the layers nearly have the same trend. The minimum Tsai-Wu failure occurs when α₂ equals 0°. As α₂ increases, the changes in the maximum Tsai-Wu failure are too small until α₂ equals 40°. After that, as α₂ increases, the maximum Tsai-Wu failure increases and reaches the maximum value when α₂ equals 90°. Figure 11(c) shows the effects of α₄ upon Tsai-Wu failure; it shows that α₄ has the same effect on the maximum Tsai-Wu failure as α₁. Figure 11(d) shows the effect of D_major upon the maximum Tsai-Wu failure, which indicates that as D_major increases the Tsai-Wu failure increases. To avert the failure, the D_major can be appropriately chosen to be less than 2.05 m.

Figure 11(e) shows the effect of D_minor upon Tsai-Wu failure which indicates that as D_minor increases the Tsai-Wu failure decreases. In order to avert the failure, the D_minor can be appropriately chosen to be not less than 1.93 m. Figure 11(f) presents the effect of H upon the maximum Tsai-Wu failure of the elliptical composite submersible pressure hull. The curve shows that the maximum Tsai-Wu failure is directly proportional to operating depth. In order to avert the failure, the maximum operating depth must be less than 4200 m. Figure 11(g) shows the effect of t₁ upon the maximum Tsai-Wu failure. The curve shows that the maximum Tsai-Wu failure is inversely proportional to t₁.

Figure 11(j) has the same trend as Figure 11(g) but t₁ has a greater influence on the value of the maximum Tsai-Wu failure than t₄. In order to avert the failure, t₁ and t₄ must be more than 1.79e–3m and 1.08E–03m, respectively. Figure 11(h) presents the effect of t₂ upon the maximum Tsai-Wu failure. It can be observed that the maximum Tsai-Wu failure is inversely proportional to t₂. Figure 11(i) has the same trend as Figure 11(h) and shows that t₂ has a greater influence on the value of the maximum Tsai-Wu failure than t₃. In order to avert the failure, t₂ and t₃ must be more than 1.66e–3m and 0.52E–03m, respectively.

8. Conclusions

This paper presents the optimum design of elliptical composite submersible pressure hull under hydrostatic pressure to minimize the buoyancy factor of the pressure hull. Sensitivity analyses were performed to study the effects of the design variables on the optimal structural strength design. The minimum buoyancy factor provides a valuable reference for the submarine pressure hull design so as to increase the payload and the performance with respect to controlling the submarine. The Tsai-Wu failure criteria predict efficiently the ply failure of elliptical composite submersible pressure hull under hydrostatic pressure. Resistance to in-plane forces rather than shell buckling is the critical design consideration at extreme operational depths in this case. It implies that a high material strength is important for deep-diving pressure hulls. To achieve high values of the material failure loads, the bottommost lamina of the elliptical composite submersible pressure hull should have the fibers oriented nearly in the circumferential direction. The design variables (D_major,D_minor,t₂, and t₃) heavily influence buoyancy factor. The design variables (α,H,D_major,D_minor,t₂, and t₃) heavily influence the buckling strength factor. The design variables (D_major and D_minor) heavily influence the maximum deflection value; the design variables (H,α₁,α₄, and t) have a moderate influence upon maximum deflection value. The design variables (α,D_major,D_minor, and t) heavily influence the Tsai-Wu failure. The results can serve as a valuable reference for designers of underwater vehicles.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Authors’ Contribution

Professor Tong Lili and Dr. Mahmoud Helal have the same contribution.

References

Vinson

J. R.

, The Behavior of Shells Composed of Isotropic and Composite Materials, Springer, New York, NY, USA, 1993.

MacKay

J. R.

and Keulen

, “Partial safety factor approach to the design of submarine pressure hulls using nonlinear finite element analysis,” Finite Elements in Analysis and Design, vol. 65, pp. 1–16, 2013.

Lian

C. C.

Liao

C. C.

, and Lu

C. M.

, The Study of Minimum-Weight Optimal Design for Submarine Pressure Hull, Institute of Technology, Chung Cheng, Singapore, 1989.

Reynolds

Lomacky

, and Krenzke

, “Design and analysis of small submersible pressure hulls,” Computers & Structures, vol. 3, no. 5, pp. 1125–1143, 1973.

Ross

C. T. F.

, “A conceptual design of an underwater vehicle,” Ocean Engineering, vol. 33, pp. 2087–2104, 2006.

Mills

, Submarine Design and Development, Conway Maritime, London, UK, 1984.

Liang

C. C.

Chen

H. W.

, and Jen

C. Y.

, “Optimum design of filament-wound multilayer sandwich submersible pressure hulls,” Ocean Engineering, vol. 30, pp. 1941–1967, 2003.

Liang

C. C.

Hsu

C. Y.

, and Tssi

H. R.

, “Minimum weight design of submersible pressure hull under hydrostatic pressure,” Compurers and Strucrtures, vol. 63, pp. 187–201, 1997.

Alvarez

Bertram

, and Gualdesi

, “Hull hydrodynamic optimization of autonomous underwater vehicles operating at snorkeling depth,” Ocean Engineering, vol. 36, pp. 105–112, 2009.

10.

Cui

Turan

, and Sayer

, “Learning-based ship design optimization approach,” Computer-Aided Design, vol. 44, pp. 186–195, 2012.

11.

Koussios

, and Beukers

, “Shape optimization of filament wound articulated pressure vessels based on non-geodesic trajectories,” Composite Structures, vol. 92, pp. 339–346, 2010.

12.

Adali

Summers

E. B.

, and Verijenko

V. E.

, “Optimisation of laminated cylindrical pressure vessels under strength criterion,” Composite Structures, vol. 25, pp. 305–312, 1993.

13.

Kim

D. H.

Choi

D. H.

, and Kim

H. S.

, “Design optimization of a carbon fiber reinforced composite automotive lower arm,” Composites Part B: Engineering, vol. 58, pp. 400–407, 2014.

14.

Lee

G. C.

Kweon

J. H.

, and Choi

J. H.

, “Optimization of composite sandwich cylinders for underwater vehicle application,” Composite Structures, vol. 96, pp. 691–697, 2013.

15.

Tauchert

T. R.

, “Optimum design of a reinforced cylindrical pressure vessel,” Journal of Composite Materials, vol. 15, no. 5, pp. 390–402, 1981.

16.

Messager

Pyrz

Gineste

, and Chauchot

, “Optimal laminations of thin underwater composite cylindrical vessels,” Composite Structures, vol. 58, pp. 529–537, 2002.

17.

Bruggi

and Taliercio

, “Topology optimization of the fiber-reinforcement retrofitting existing structures,” International Journal of Solids and Structures, vol. 50, pp. 121–136, 2013.

18.

Ross

C. T. F.

, Pressure Vessels External Pressure Technology, British Library, London, UK, 2nd edition2011.

19.

Liu

Tian

, and Li

, “Reliability-based load and resistance factor design of composite pressure vessel under external hydrostatic pressure,” Composite Structures, vol. 93, pp. 2844–2852, 2011.

20.

Maalawi

K. Y.

, “Optimal buckling design of anisotropic rings/long cylinders under external pressure,” Journal of Mechanics of Materials and Structures, vol. 3, no. 4, pp. 775–793, 2008.

21.

Bakshi

and Chakravorty

, “First ply failure study of thin composite conoidal shells subjected to uniformly distributed load,” Thin-Walled Structures, vol. 76, pp. 1–7, 2014.

22.

Zheng

J. Y.

and Liu

P. F.

, “Elasto-plastic stress analysis and burst strength evaluation of Al-carbon fiber/epoxy composite cylindrical laminates,” Computational Materials Science, vol. 42, no. 3, pp. 453–461, 2008.

23.

Panahi

Ghavanloo

, and Daneshmand

, “Transient response of a submerged cylindrical foam core sandwich panel subjected to shock loading,” Materials and Design, vol. 32, no. 5, pp. 2611–2620, 2011.

24.

Chang

R. R.

, “Experimental and theoretical analyses of first-ply failure of laminated composite pressure vessels,” Composite Structures, vol. 49, no. 2, pp. 237–243, 2000.

25.

Chang

R. R.

and Chiang

T. H.

, “Theoretical and experimental predictions of first-ply failure of a laminated composite elevated floor plate,” Proceedings of the Institution of Mechanical Engineers, PartE: Journal of Process Mechanical Engineering, vol. 224, no. 4, pp. 233–245, 2010.

26.

Prusty

B. G.

Ray

, and Satsangi

S. K.

, “First Ply failure analysis of stiffened panels—finite element approach,” Composite Structures, vol. 52, pp. 73–81, 2001.

27.

Anastasiadis

J. S.

and Simitses

G. J.

, “Buckling of pressure-loaded, long, shear deformable, cylindrical laminated shells,” Composite Structures, vol. 23, no. 3, pp. 221–231, 1993.

28.

Gohari

Golshan

Mostakhdemin

Mozafari

, and Momenzadeh

, “Failure strength of thin-walled cylindrical GFRP composite shell against static internal and external pressure for various volumetric fiber fraction,” International Journal of Applied Physics and Mathematics, vol. 2, pp. 111–116, 2012.

29.

Xiong

Ghosh

Vaziri

Wang

, and Wu

, “Sandwich-walled cylindrical shells with lightweight metallic lattice truss cores and carbon fiber-reinforced composite face sheets,” Composites A, vol. 56, pp. 226–238, 2014.

30.

Graham

, “Composite pressure hulls for deep ocean sub-mersibles,” Composite Structures, vol. 32, no. 1–4, pp. 331–343, 1995.

31.

Graham

, “Buckling of thick-section composite pressure hulls,” Composite Structures, vol. 35, no. 1, pp. 5–20, 1996.

32.

Hsu

Liang

Shiah

, and Jen

, “A study of stress concentration effect around penetrations on curved shell and failure modes for deep-diving submersible vehicle,” Ocean Engineering, vol. 32, no. 8–9, pp. 1098–1121, 2005.

33.

Kolakowski

, “On some aspects of the modified TSAI-WU criterion in thin-walled composite structures,” Thin-Walled Structures, vol. 41, pp. 357–374, 2003.

34.

Jen

C. Y.

, “Coupled acoustic structural response of optimized ring stiffened hull for scaled down submerged vehicle subject to underwater explosion,” Theoretical and Applied Fracture Mechanics, vol. 52, pp. 96–110, 2009.

35.

Liang

C. C.

Shiah

S. W.

Jen

C. Y.

, and Chen

H. W.

, “Optimum design of multiple intersecting spheres deep-submerged pressure hull,” Ocean Engineering, vol. 31, pp. 177–199, 2004.

36.

Ross

C. T. F.

, “A novel submarine pressure hull design,” Journal of Ship Research, vol. 31, pp. 186–188, 1987.

37.

Ross

C. T. F.

and Little

A. P. F.

, “The buckling of a corrugated carbon fibre cylinder under external hydrostatic pressure,” Ocean Engineering, vol. 28, pp. 1247–1264, 2001.

38.

Ross

C. T. F.

and Popken

, “Buckling of tube-stiffened prolate domes under external water pressure,” Thin-Walled Structures, vol. 22, no. 3, pp. 159–179, 1995.

39.

Ross

C. T. F.

, “A conceptual design of an underwater missile launcher,” Ocean Engineering, vol. 32, no. 1, pp. 85–99, 2005.

40.

Lopez

R. H.

Miguel

L. F. F.

Belo

I. M.

, and Cursi

J. E. S.

, “Advantages of employing a full characterization method over form in the reliability analysis of laminated composite plates,” Composite Structures, vol. 107, pp. 635–642, 2014.

41.

Awad

Z. K.

Aravinthan

Zhuge

, and Gonzalez

, “A review of optimization techniques used in the design of fibre composite structures for civil engineering applications,” Materials and Design, vol. 33, no. 1, pp. 534–544, 2012.

42.

Inc

, “ANSYS Theory Reference Release 14.5,” October 2012.

43.

Erdogan

G. I. M.

, The Finite Element Method and Applications in Engineering Using ANSYS, Springer, New York, NY, USA, 2006.

44.

and Mahfuz

, “Finite element simulation of composite ship structures with fluid structure interaction,” Ocean Engineering, vol. 52, pp. 52–59, 2012.

Design Optimization of Composite Elliptical Deep-Submersible Pressure Hull for Minimizing the Buoyancy Factor

Abstract

1. Introduction

2. Submarine Pressure Hull

3. Material Property Requirements

4. First-Ply Failure of Angle-Ply Laminate

5. Optimization Methodology of Composite Elliptical Pressure Hull

5.1. Design Variables

5.2. Objective Function

5.3. Design Constraints

5.3.1. Behavior Constraints

5.3.2. Side Constraints

6. Finite Element Model Simulation

7. Numerical Results and Discussions

7.1. Effect of Design Variables on Buoyancy Factor

7.2. Effect of Design Variables on Buckling Strength Factor

7.3. Effect of Design Variables on Maximum Deflection Value

7.4. Effect of Design Variables on Maximum Tsai-Wu Failure

8. Conclusions

Conflict of Interests

Footnotes

Authors’ Contribution

References