Analysis of plated-hull structure strength against hydrostatic and hydrodynamic loads: A case study of 600 TEU container ships

2.1 Finite element approach

One of the most important factors to consider during the shipbuilding process is the hull structural resistance to working loads. Testing of the ship's structural design is required to determine how strong the hull is. Mathai [2] tested a container ship's structure and compared the stress (von Mises) obtained by the simulation with the material's allowable stress and the deformation of the simulation results with the container ship's permissible deformation.

A simulation based on the finite element approach is a beneficial technique in design [17] and as an alternative to reduce the costs incurred throughout the design phase [18]. Testing of the ship structures using Finite Element Analysis has been conducted by many previous researchers [19,20,21]. The test results by Paik [20] concluded that the computational estimates with finite element analysis compared with the calculation results have a difference of less than 6 percent.

Shell elements are used in this simulation, and many earlier researchers [22,23,24] have used shell elements on FEA to analyze the hull structure. Shell elements can be used in Finite Element Analysis (FEA) to obtain good results. As they allow for modeling of thin features with fewer mesh elements, they can save a lot of time in the computation. Shell elements are suitable for the geometry tested in this simulation. They are also less prone to negative Jacobian errors and are easier to mesh. The materials used in the ship structure affect the strength of the ship structure. In his research, Raja [25] concluded that the value (von Mises), strain, and deformation of each tested material are different. Material selection is also essential to obtaining a suitable material for use in shipbuilding.

In the ship structure analysis as a thin-walled component, analyzing this engineering problem using plane stress conditions is more relevant. Plane stress is defined as a stress state in which the normal and shear stress perpendicular to the plane is assumed to be zero, specifically, the normal stress σz and shear stresses τ_xz and τ_yz are assumed to be zero. Generally, a thin member (which has a small z dimension compared with the dimensions in the x and y planes) in which the loads act only in the x–y plane can be considered plane stresses. The stress–strain relationship in the in-plane stress condition is as follows:

where σ_x is the normal stress in the x-direction, σ_y is the normal stress in the y-direction, τ_xy is the shear stress in the xy-direction, ɛ_x is the normal strain in the x-direction, ɛ_y is the normal strain in the y-direction, γ_xy is the shear strain xy-direction, E is Young's modulus, and v is the Poisson ratio.

The thickness of the plate on the ship structure has a significant impact; the thickness of the structure must be suitable and must meet the allowable stress and deformation criteria to prevent the ship structure from failing. The ship structure becomes more robust as the plate thickness increases, but it is also heavier [26] and requires more fuel. The weight of the cargo carried by the container ship also impacts the ship draft; according to a test conducted by Putra [27], the heavier the cargo, the deeper the ship draft. The results of this test also concluded that, the larger the ship draft, the greater the fuel consumption on the ship. The depth of the ship draft also affects the hydrostatic pressure and hydrodynamic pressure generated by ocean waves following the route upon which the ship travel. Tests of the ship structure against hydrostatic pressure and hydrodynamic pressure have been carried out by several previous researchers [2, 25]. However, in a study conducted by Raja [25], the value of hydrodynamic pressure was calculated using the Heller and Jasper theory rather than the Common Structural Rules. Parunov [28] concluded that the failure of the ship structure due to sagging decreased five times with the use of the Common Structural Rules compared with the previous method.

2.2 Container ship structure

A container ship is a ship built specifically to transport standard-size containers. Container placement is mobile, with vertical frames. The capacity of the container ship is measured in Twenty-Foot Equivalent Units (TEUs). A standard 20-foot container size is 20 × 8.0 × 8.5 feet (6.1 × 2.4 × 2.6 meters). Currently, container ships carry about 90% of the world's non-bulk cargo. These containers are standard in size so that they can be easily transferred to various modes of transport. Ships range in size from about 500 TEU to about 22000 TEU, which can fit containers that are 20 ft and 40 ft in length. Each ship generally lists its maximum carrying capacity for each container size. The larger the ship, the larger the container that can be transported by ship and the greater the burden on the ship. However, larger vessels are cheaper to build per ton and operating costs per ton also decrease as the size of the ship increases [29]. Since its use more than 50 years ago, the size of container ships has been growing, becoming increasingly sophisticated, and becoming more efficient.

A midship section is the part that shows where the components attach to the ship. In the midship structure of container ship construction, the most significant part is a very large deck opening [30]. These ships are also referred to as open deck vessels. This feature is required for easy and fast loading and unloading of containers. However, this results in structural challenges when providing adequate longitudinal and torsional strength to the hull as large deck openings cause a loss of longitudinal and torsional strengths. The longitudinal structural strength of the midship section is the most important factor in ensuring a safe structure of the ship [31, 32]. All types of loads must be considered to calculate the structural strength, such as ship and cargo weights, and ocean wave loads [33, 34]. The structural layout of a midship section of a container ship is depicted in Figure 1.

Figure 1

Midship of a container ship [35].

2.3 Longitudinal strength

The longitudinal strength of a ship is critical because too much load on the ship's hull might cause a structural failure. A ship receives a longitudinal bending moment due to a variation in weight and the buoyancy distribution throughout, with its maximum being around the midship region [36].

Of the many accidents that occur on ships, structural failure in the ship's midship section is the most frequent. This is because the midship section of the ship is the most critical part of the ship structure. The structural failure of a ship can be seen in Figure 2. We chose the midship section for an analysis of the hull structure and to run simulations on the midship area of the ship to test the structure because, if the midship does not fail, the fore and stern structures of the ship do not fail against the load received by the ship.

Figure 2

Failure of a ship's structure.

2.4 Environmental loads

Environmental loads are loads acting on structures resulting from environmental conditions [37] such as air, waves, currents, ice, etc. [38]. This analysis focuses on hydrostatic loads (hydrostatic and deadweight pressures) and hydrodynamic loads (hydrodynamic pressures) produced by a ship's operations and acting on the vessel. The load scenario designed for strength assessment consists of a static plus dynamic load case [25], where static and dynamic loads depend on the loading conditions considered. Hydrostatic pressure is the pressure that seawater exerts in all directions on the hull submerged in water and caused by gravitational forces. This pressure depresses the structure inwards and holds and stiffens the hull layer. The distribution of pressure caused by hydrostatic pressure at the center of the ship can be seen in Figure 3.

Figure 3

Distribution of hydrostatic pressure at the center of the ship [35].

To find the hydrostatic pressure on the midship side, the amount of pressure can be calculated based on CSR [39]. The Common Structural Rules (CSR) is an empirical approach to predicting the ship's load. A significant value can be expressed according to Eq. (2).

where P_s is the hydrostatic pressure, ρ is the seawater density (t/m³), g is the gravity acceleration (m/s²), T_lci is the draught in the considered cross section (m), and z is the coordinates along the vertical axis (m). The value of hydrostatic pressures on the lower side of the midship can be expressed based on Eq. (3).

The pressure generated from the water flowing to and around the ship is called hydrodynamic pressure. Hydrodynamic pressures in this study are assumed to include the effects of damaged and unruptured waves that attack the hull. The wave conditions in which the ship operates affect the amount of hydrostatic pressure on the hull. The distribution of hydrodynamic pressure at the center of the ship is depicted in Figure 4.

Figure 4

Distribution of hydrodynamic pressure at the center of the ship [39].

To determine the hull's hydrodynamic strength and surface pressure, guidelines and rules were used by various classification bureaus [40]. Hydrodynamic pressure is calculated based on Common Structural Rules [39]. Common Structural Rules (CSR) is used to predict hydrodynamic press loads on ships as in Eq. (4).

where P_hf is the hydrodynamic pressure; f_p is the coefficient corresponding to the probability; f_nl is the coefficient considering the nonlinear effect; C is the wave coefficient; L_csr−b is rule length (m); λ is the wavelength (m); x, y, and z are the X, Y, and Z coordinates (m); T_Lci is the draught in the considered cross section (m); and Bi is the molded breadth at the waterline (m). The C value at hydrodynamic pressure for a vessel with 90 m ≤ L_csr−b ≤ 300 m can be expressed using Eq. (5).

The value of λ on hydrodynamic pressure can be expressed as in Eq. (6).

2.5 Allowable stress and deformation

Allowable stress is the maximum stress allowed for the materials used in the ship's structure. Allowable stress can be expressed using Eq. (7) [25].

where σ_a is the allowable stress, σ_f is the tensile strength, and S is the safety factor.

Moreover, allowable deformation is the maximum allowed deformation value for specific materials used in the ship's structure. When the dimensions of the original design are reduced, the value of deformation increases. To create a design that can withstand the load, an allowable deformation value must be determined. The permissible deformation value for container ships is allowed based on a Common Structural Rule of 168.5 mm [2].

2.6 Material selection for ship structure

Many types of materials are applied in the shipbuilding industry. The type of material is chosen according to the location and uses of the ship, including an assumption of the type of loads to be experienced by the structural member. Various steel types are widely used because of their properties being able to withstand extreme conditions compared with other materials.

Low carbon steel has a carbon content of ≤ 0.25% C [41]. This low carbon steel is relatively soft and weak but has outstanding ductility and toughness. This type of steel is not responsive to heat treatment, and steel reinforcement is conducted with cold work. Moreover, medium carbon steel has a carbon content of 0.25% – 0.60% C [41]. This type of steel can be heat treated to improve the mechanical properties of medium carbon steel. This type of steel has stronger mechanical properties and a higher degree of hardness than low carbon steel. Moreover, high carbon steel has a carbon content of 0.60% – 1.40% C [41]. This type of steel is the hardest, strongest, yet least ductile of carbon steels. It has a high enough martensite that makes the results less optimal when the surface hardens. High-strength, low-alloy steel (HSLA) is included in the low-carbon alloy category [41]. This type of steel contains several alloy elements such as copper, vanadium, nickel, and molybdenum, with a combination of concentrations of up to 10%. It provides higher strength compared with low-carbon steel. This type of steel has ductile, formable, machinable, and more corrosion-resistant properties than carbon steel.

3.1 Design and validation test

This research aims to determine the strength of the 120 m length midship section of a hull container ship against environmental loads applying hydrostatic and hydrodynamic loads using finite element analysis. The midship structure of the ship was designed using SolidWorks Software. All of the primary dimension data that were determined were then inserted into the SolidWorks software. In the early stages of the formation of the midship structure, a 2D sketch was made according to the dimensions of the midship of a container ship designed from the previous work [25]. The design of the midship geometry was finalized in Solidworks, with the dimensions in Figure 5a. The design of the 3D Mid-ship is shown in Figure 5b.

Figure 5

(a) Geometry of the midship section and (b) 3D model of the midship.

A convergence study is critical to obtaining an optimal element within optimal computational time. In this case, the validation process was conducted by comparing the structural response with a reference work [25]. Validation was performed by creating the structure, material, and construction in accordance with previous work so that the test results can be compared with the benchmark results. Before performing a simulation, setting and configuration tests must be conducted. Nowadays, titanium type materials have been used in the shipbuilding industry [42]. This material is used in benchmark studies, and its convergence results were further validated in [25]. In this case, the Element-length-to-thickness (ELT) ratio varied in elemental size with the geometry thickness, ranging from ELT 5 to 11. In this case, the loading scenario on the ship structure was divided into static load and dynamic load. The static load was divided into the deadweight load at 0.0535 MPa, and the hydrodynamic pressure was 0.01 MPa at the bottom shell and 0.0075 MPa on the side shell. For dynamic loads, a hydrodynamic pressure of 0.005786 MPa was used. The loading scenario and boundary condition in this simulation can be seen in Figure 6.

Figure 6

Applied load and boundary conditions.

The results of Reference [25] provide the values for deformation, strain, and stress (von Mises). The numerical results obtained from the current work are compared with the results from the reference work, which is summarized in Table 4. The result shows that an ELT ratio of 10 has the closest value to the reference work based on the validation tests performed. Compared with the benchmark study, the result was obtained at 26.94 mm deformation and a stress of 5.579 × 10⁻⁴ at an ELT ratio of 10.

3.2 Scenario of simulation parameters

In this case, the influences of material types, plate thickness, and draft height on the structural responses were evaluated. The materials were chosen based on the type of steel that has been standardized by the American Bureau of Shipping (ABS) and the type of material commonly used on ships. In this study, four carbon steel materials were simulated. ASTM A131 Grade AH36 and AH32 are included as low carbon steel. AISI 1020 has a carbon content of 0.17%–0.23%, where the carbon content <0.25% makes this type of steel AISI 1020, also included as a low carbon steel. The mechanical properties of AISI 1020 can be seen in Table 12. AISI 1035 is included as medium carbon steel [41]. The different mechanical properties of each material are given in Table 2. Regarding the failure criterion, we assume in the analysis that the material is still in their elastic conditions, so it is still below the yield point when we applied the load. Therefore, the model is not subjected to plastic deformation or geometrical failure. The plate thickness of the hull structure varied by 100, 150, 200, and 250 mm. Besides thickness varying, the structural response varied with the draft height, in the range of 5 – 9 m.

The mesh size used in this simulation is based on the ELT (Element-Length-to-Thickness) ratio that we already obtained from the validation test. The Element-length-to-thickness (ELT) ratio value is obtained by the element size divided by the thickness of the geometry. A welded joint can be modeled on a finite element analysis, as an earlier researcher did [43, 44], but a welded joint is not modeled and defined in this simulation because it is already represented by the designed model used on this simulation without the welded joint modeled, as used in a work by pioneer researchers [2, 24] in a simulation without defined the welded joint. Modeling welded joints will require a more complicated design and will take a long time when running simulations but will be considered in further research.

4.1 Structural response due to material type variations

The four types of materials used in this simulation are ASTM A131 Grade AH36, ASTM A131 Grade AH32, AISI 1020, and AISI 1035. In this simulation, the water draft was at 9 m and the structure thickness was 100 mm with an ELT ratio of 10. In this case, the load applied to the structure is divided into two types: static load and dynamic load. Static loads are divided into deadweight at 0.0535 MPa [25] and hydrostatic pressure (bottom shell of 0.0905 MPa; side shell of 0.08366 MPa), and the dynamic loads consist of hydrodynamic pressures of 0.0356 MPa. Deformation of the hull's structure is caused by the hydrostatic and hydrodynamic loads on the hull, in which the deformation value of the structure must be smaller than the material's allowable deformation. The deformation value and status for each material tested are shown in Table 6 and the contour plots in Figure 7.

Figure 7

Deformation contour plots for different material types: (a) Grade AH36; (b) Grade AH32; (c) AISI 1020; and (d) AISI 1035.

Based on the deformation results of the simulation, the ASTM A131 Grade AH36 material has a deformation value of 51.14 mm with a pass status. ASTM A131 Grade AH32 has the same deformation value as Grade AH36, which allows us to conclude that the strengths of both materials are the same. For AISI 1020 and AISI 1035, the values are still below the allowable deformation value so the material is considered safe for use in the design. According to the theory, the test result data were obtained when ASTM A131 had a higher tensile strength than other materials to obtain the smallest deformation in this type of material. The same phenomenon can also be seen in AISI 1020 material properties, where the tensile strength of this material is the smallest compared with other materials tested. The highest deformation value in this test is found in the AISI 1020 material. We can also see from the testing analysis that all materials have deformation values below the allowable deformation value. All materials tested are still safe for use in the hull's structural design.

Strain is the dimensional changes in a material caused by stress applied to the material. The strain contour plots are depicted in Figure 8. Based on the simulation results, ASTM A131 Grade AH36 material has a strain value of 9.123 × 10⁻⁴, and Grade AH32 material obtained a strain value of 9.123 × 10⁻⁴. AISI 1020 material has a strain value of 9.577 × 10⁻⁴, and AISI 1035 has a strain value of 9.349 × 10⁻⁴. The smallest strain values were experienced with the models using the ASTM A131 Grade AH36 and ASTM A131 Grade AH32 materials. This same strain value can also show that the strengths of both materials are the same. In the properties of AISI 1020, the tensile strength of this material is the smallest compared with the other materials tested, so the highest strain value in this test is found in this type of material.

Figure 8

(a) Comparison of a strain contour plot with various material types: (a) Grade AH36, (b) Grade AH32, (c) AISI 1020, and (d) AISI 1035.

The applied load causes stress on the hull's structure. The stress value in the structure must be smaller than the allowable stress criteria for each material to increase the safety factor. The amount of stress value for each material tested is provided in Table 4, and the results of each test stress simulation can be seen in Figure 9. Based on the results, ASTM A131 Grade AH36 material has a stress value of 344.8 MPa, and ASTM A131 Grade AH32 material obtained a deformation value of 344.8 MPa. For the AISI 1020 material, the stress value is 344.8 MPa, and AISI 1035 has a stress value of 344.9 MPa. From the result, the stress value of the entire material is almost the same, ranging from 344.8–344.9 MPa. All of the materials have stress values that are still below the allowable stress value of each material.

Figure 9

Stress contour plots at (a) Grade AH36, (b) Grade AH32, (c) AISI 1020, and (d) AISI 1035.

4.2 Response due to structural plate thickness variations

There are four variations in the thicknesses used in this simulation: 100 mm, 150 mm, 200 mm, and 250 mm. In this simulation, the water draft was 9 m, the material used was ASTM A131 Grade AH36 with an ELT ratio of 10. The load on the structure is similar to that in the previous case. The deformation value and status for each thickness variation tested are provided in Table 5. The comparison of contour deformation plots with different plate thicknesses is depicted in Figure 10.

Figure 10

Deformation contour plots at various plate thicknesses: (a) 100 mm, (b) 150 mm, (c) 200 mm, and (d) 250 mm.

The simulation results produce a deformation value of 51.14 mm at a structure thickness of 100 mm. Then, the thicknesses of 150, 200, and 250 mm obtained deformation values of 39.37 mm, 25.67, and 18.59 mm. The deformation results of all thickness variations tested show that the smallest deformation value is found in the thickness of 250 mm. The largest deformation can be found in the model with 100 mm in thickness. The deformation values of the test become smaller as the thickness of the ship's structure increases. We can also see from the test analysis that all of the thicknesses have deformation values below the allowable deformation value.

The strain results of each contour strain plot can be seen in Figure 11. Based on the simulation results, the strain value in each test with a structure thickness of 100 mm has a strain value of 9.123 × 10⁻⁴. The thickness of 150 mm has a strain value of 7.873 × 10⁻⁴. For thicknesses of 200 mm and 250 mm, the strains are 5.387 × 10⁻⁴ and 4.04 × 10⁻⁴, respectively. The smallest strain is experienced in the model at the thickness of 250 mm, and the largest one can be found at the thickness of 100 mm. We can conclude that the strain value in the test becomes smaller as the thickness of the ship's structure increases.

Figure 11

Strain contour plots at various plate thicknesses: (a) 100 mm, (b) 150 mm, (c) 200 mm, and (d) 250 mm.

The stress value for each thickness tested is provided in Table 6, and the simulation contour plot results of each test stress can be seen in Figure 12. The models with 100 mm and 150 mm thicknesses have stress values of 344.8 MPa and 343.2 MPa. For a thickness of 200 and 250 mm, the models experience stress values of 203.2 MPa and 155.6 MPa. We can see that the smallest stress value is found when the thickness of the structure is 250 mm and that the largest one can be found at a thickness of 100 mm. We can also see that all thickness have a stress value that is still below the value of allowable stress material. All variations of thickness tested are still safe for use in the design of a hull's structure.

Figure 12

Comparison of stress contour plots at different plate thicknesses: (a) 100 mm, (b) 150 mm, (c) 200 mm, and (d) 250 mm.

4.3 Response due to draft variations

Five draft variations in the range of 5 – 9 m are used in this simulation. In this simulation, the model discretization is similar to that in two previous case studies. The load on the structure is divided into static and dynamic loads. Static loads are divided into deadweight pressure at a magnitude if 0.0535 MPa [25], and the hydrostatic pressure changes with each draft variation. Dynamic loads consist of hydrodynamic pressure, in which the value also varies according to the draft. The value is provided under the existing theory, where the deeper the draft height, the greater the pressure on the structure. The magnitude of the hydrostatic pressure and the hydrodynamic pressure in this test can be seen in Table 7.

The comparison of deformation results for each draft variation tested is presented in Table 8. The deformation contour plot for each draft variation can be seen in Figure 13. From the deformation analysis results of all thickness variations tested, the smallest deformation value is found in the draft variation of 5 m, which is 28.73 mm. The largest deformation value is found in the 9 m draft variation of 51.14 mm. It makes the value of deformation in testing even greater as the draft increases. We can also see from the test analysis that all thickness variations have deformation values that are still below the allowable deformation value so that all draft variations tested are still safe for use in the design of the hull's structure.

Figure 13

(a) Deformation contour plot at various drafts: (a) at draft 9 m, (b) at draft 8 m, (c) at draft 7 m, (d) at draft 6 m, and (e) at draft 5 m.

The strain contour plots for each test simulation can be seen in Figure 14. Based on the strain results, the strain value in the test at a draft 9 m has a value of 9.123 × 10⁻⁴. For the drafts of 8 m and 7 m, the strain values are 7.672 × 10⁻⁴ and 7.332 × 10⁻⁴, respectively. Moreover, at drafts of 6 m and 5 m, the structure has strain values of 6.061 × 10⁻⁴ and 5.291 × 10⁻⁴, respectively. We can see that the smallest strain value is found at a draft depth of 5 m and that the largest strain value is found at a draft of 9 m. This makes the strain in the test even greater as the draft increases.

Figure 14

Comparison of strain contour plots at different draft heights: (a) 9 m, (b) 8 m, (c) 7 m, (d) 6 m, and (d) 5 m.

The stress value for each draft variation and the status are presented in Table 9, while the contour stress plots at different draft variations can be seen in Figure 15. In each test obtained with a draft of 9 m, the stress value is 344.8 MPa. For the drafts 8 m, 7 m, 6 m, and 5 m, the structures experience deformation values of 288.6 MPa, 276.5 MPa, 227.3 MPa, and 198 MPa, respectively. All stress values are still below the allowable stress values. The material stress analysis results show the smallest stress values at a draft of 5 m, and the greatest stress value is at a draft of 9 m. It makes the stress value of the test even greater as the draft increases. We can also see that all variations of the draft have a stress value that is still below the allowable value of stress so that all variations in the draft tested are still safe for use in the design of a hull structure.

Figure 15

(a) von Mises stress contour plots for drafts at (a) 9 m, (b) 8 m, (c) 7 m, (d) 6 m, and (d) 5 m.