Research on Uprighting Process of a Capsized Ship in Combined Wind and Wave Parameters

Abstract

At present, most salvage schemes are designed based on the calculation results of ship statics, which is still an empirical method. However, many projects are always affected by the wind and waves, and there is a significant difference between the force situation of ships with wind area and the calculation results of ship statistics. This paper analyzes the methods and tools of righting a capsized ship, establishes the righting force model of a capsized ship in wind and waves in accordance with floatation, stability, righting force, waves and wind, and then derives the method of longitudinal strength calculation. The floating state and stability of a capsized ship in four different sea conditions are calculated by GHS software, and three uprighting schemes are designed based on the number and position of the righting force. According to the size of the wind and waves, this paper simulates the uprighting process of the capsized hull in four cases. According to the results, it found that waves and wind can play a significant role in the uprighting process. Case I could reduce the righting force by 76.2~78.3% more than case III. The wind force moment of case II is about 34% less than that of case I. The size of the waves is negligible to the stability of a capsized ship; however, it also has an impact on the hull loads during the uprighting process. When the winds and waves are large, setting the location of righting force in a proper cross-section can reduce the righting force and hull loads to a certain extent. Through comparative analysis, the paper gives the righting schemes for different cases.

1. Introduction

The attitudes and characteristics of the shipwreck, positions and quantity of goods, hull strength, buoyancy and stability directly impact the design of refloating sunken ships in salvage operation [1]. A perfectly designed salvaging program especially in the cases that need high computational accuracy of lifting force can effectively improve the quality of work [2]. Common methods of salvaging program design are the empirical method, US Navy salvage manual, and virtual simulation method. The design case shall comprehensively consider the impact of the actual situation, for example, weight, buoyancy, lifting force, auxiliary buoyancy, wind force and wave force, etc.

Captain George H. Reid advised that a capsized ship with enough reserve buoyancy can be towed to an appropriate area before uprighting. Some ships with obstacles on deck should be righted before towing [3]. During uprighting, the stress distribution of the hull is directly decided by the salvage method. Michael S. Dean introduced nine methods of righting a capsized ship, and the influence of salvage method on the hull strength is studied [4]. An introduced a new technology of pile rooting to right a capsized ship, which could achieve the goal of reducing cost [5]. Lamberti studied the effect of the parbuckling method on righting COSTA CONCORDIA, conducting a comparative study about the CFD method and providing analytical results. The effect of waves on the uprighting process-based CFD method was summarized [6]. The application of SuSy devices for increasing the buoyancy/gravity of the damaged cabins/ballast tanks is presented, which provides the theoretical foundation for salvaging sunken ships with pontoons [7]. In order to refloat sunken ships in the inland waters of the Republic of Serbia and reduce the difficulty of calculation, the impact of multiple cranes on the ship and precautions are analyzed [8]. Sails and human gravity can assist in the righting of capsized sail-assisted ships; the positions of human and sails should vary with the attitude of the hull [9,10]. The causes of a capsized ship are different. The lifting force of a pontoon and crane can change the stability of a capsized ship dramatically [11]. The displacement of Seaspan 240 is 12,300 tons. When Seaspan 240 capsized in an accident, methods of water injection of the side compartments, transverse loading on hull, and hatch sealing were used in one day and still maintain a high speed at its present age [12].

The force analysis is closely related to the attitude and stability of the ship and evaluation of hull strength and safety. Righting a capsized ship is a complicated problem while refloating shipwrecks. During the process, the calculation and analysis method are based on a refloating method [13]. Drobyshevski deduced the calculation method of righting force and studied the effect of variation of gas parameters of air cushion on the floating condition of a capsized ship [14]. The righting force of a capsized ship with or without air pockets was also solved by a theoretical method. A comparison has been carried out for the ballast water method and lifting force method during righting a capsized ship. The stress calculation method in the sagging and hogging state has a certain significance for salvage engineering [15]. Luo applied GHS software and simulated the uprighting process of a Trans Summer--57000T bulk carrier; the draft, heel angle and longitudinal strength were calculated [16].

The traditional small-scale ship model experiment only considers a wave-induced interference effect. However, further experimental results indicated that wind may have a significant impact on ships [17]. Wind forces account for a proportion of the total load on the offshore platform, of which about 10% is a fixed offshore platform and about 25% is a mobile offshore platform [18]. Rules’ method is the basic method for calculating wind force, which is a summary of the theoretical formula of classification society and wind tunnel test. This method has the advantages of simple formulation and quick calculating speed. However, Rules’ method cannot fully consider the complex flow problems, which leads to conservative results [19]. Calculation results of wind force according to different calculation methods have not fully considered the shielding effect from a multi-structure, which brings an inaccurate result of shape coefficient and lift force [20].

Generally, considering project safety and cost, many salvage projects are carried out under the right circumstances. The influence of wind and waves is impossible to eliminate during continuous work. The process of righting a capsized ship involves the competence of righting force moment and righting moment. The influence of wind moment and wave moment must be considered. The effects of wind and waves on ship stability are directly related to the hull lines, draft, wind and waves, etc. [21]. Nicola Petacco et al. created a large ship set to study the effect of volume distribution on waves [22]. The ship stability drops significantly when the wave excitation frequency reaches a natural frequency of ship motion [23]. Refloating engineering must make a complete analysis according to each site’s differences; waves and wind lead to performance degradation during the angle measurement process [24]. It was found that the solitary wave height from 12 to 24 m may capsize the contour of vessels up to 1000 t and 3000 t, respectively, according to Kitaev and Dorozhko’s research [25].

With the development of shipbuilding technology and the shipping market, the main dimensions of ships become larger and larger. The ship in distress at sea has the characteristics of a large-scale hull, large dead weight and load capacity. The salvage technology of a large tonnage sunken ship has attracted the extensive attention of scholars. During righting Costa Concordia, salvage experts were still not exactly sure whether they would succeed or not after a long preparation; they had heightened fears of abnormal deformation or local damage of the hull structure [26]. Traditional working methods cannot be directly applied to large-scale ship accidents, because much better experience and more complete data are needed to deal with the challenges ahead. At present, Trans Summer is the largest shipwreck, which was refloated in China. For complex working conditions and sea conditions, there needs to be a comprehensive application of various methods to achieve ship salvage. These need a great righting force moment from a wide variety of tools and technologies [27,28].

The evaluation methods for wind and wave loading include model tests, numerical calculation, and industry standards. Model tests have advantages of high reliability and convenience, but they also have high economic costs and are time consuming, which is not suitable for a wide range of uses. The numerical calculation method has high flexibility and reliability in solving complex problems, but it is also time-consuming and occupies more resources. Numerical simulation results and actual results differ considerably when calculating the loading of wind and waves on a complex marine structure. Industry standards include a summary of the theoretical formula and test law by the Classification Society, and they have the advantages of simple calculating and fast velocity, which is one of the mainstream approaches to calculating wind force and wave force. However, they cannot be used in an analysis of complicated flow phenomena. A person who operates such a method must be a fully qualified person with loads of experience.

The heel angle changes largely during righting a capsized ship, and the calculation amount is too large, especially in heavy weather. This paper studies the theory of ship mechanics, and the GHS calculation software application principle in wind and waves is expounded. In view of the advantages of the seakeeping module in designing the salvage scheme in the wind and wave environment, GHS software is applied to simulate the process of righting a capsized ship in the wind and wave environment [29,30,31,32,33]. The research on ship stability, righting force, floatation, sheer force, bending moment, torque in different environments, presence in calm water, wave environment, and joint action of wind and waves is also compared.

2. Theoretical Calculation

2.1. Theoretical Calculation of Uprighting Process in Wave Environment

2.1.1. Wave Force Calculation

In the uprighting process of a capsized ship, the wave effect of advanced sea conditions cannot be ignored, which makes the stress situation of the hull more complicated. The theories for calculating wave force include micro-amplitude wave theory, Stokes fifth-order wave theory, elliptic cosine wave theory, solitary wave theory, etc.

Most methods for calculating wave forces are complex. In this paper, the approximate calculation formula of wave drift force recommended by Liu is adopted [34].

$$\left\{\begin{array}{c}{X}_{wave}=0.5\rho g{\xi }^{2}L{C}_{xw}\cos \left({\theta }_F+{\theta }_s\right)\\ Y_{wave}=0.5\rho g{\xi }^{2}L{C}_{yw}\cos \left({\theta }_F+{\theta }_s\right)\\ N_{wave}=0.25\rho g{L}^2{\xi }^2{C}_{Nw}\cos \left({\theta }_F+{\theta }_s\right)\end{array}\right.$$

(1)

where $\rho$ is seawater density, kg/m³. $g$ is gravitational acceleration, m/s². $\xi$ is mean wave amplitude, m. $L$ is hull length, m. ${\theta }_F$ is wave encounter angle, °. ${\theta }_s$ is heading angle, °. $C_{xw}$, $C_{yw}$, $C_{Nw}$ are wave drift force coefficients, which can be solved from regression expression.

$$\left\{\begin{array}{c}{C}_{xw}=0.05-0.2\left(\lambda /L\right)+0.75{\left(\lambda /L\right)}^2-0.51{\left(\lambda /L\right)}^3\\ C_{yw}=0.46-6.83\left(\lambda /L\right)-15.65{\left(\lambda /L\right)}^2+8.44{\left(\lambda /L\right)}^3\\ C_{Nw}=0.11+0.68\left(\lambda /L\right)-0.79{\left(\lambda /L\right)}^2+0.21{\left(\lambda /L\right)}^3\end{array}\right.$$

(2)

Here, $\lambda$ is mean wavelength, m.

This is a slowly varying wave drift force, which will produce a force that changes slowly and periodically with time on the wreck. Although the magnitude is not large, the natural frequency of the horizontal motion of the wreck is usually very low, and the frequency of the wave slow drift force is close to it, so it is easy to cause long-period and large-scale low-frequency resonant horizontal motion.

For the convenience of calculation, the heading angle can be taken as 0° by human activities.

According to reference [35], the moment of wave force is resolved into three components along the coordinate axes [35]:

$$\left\{\begin{array}{c}{M}_{Xwave}=-z_{wave}{Y}_{wave}\\ M_{Ywave}=z_{wave}{X}_{wave}\\ M_{Zwave}=N_{wave}\end{array}\right.$$

(3)

where $M_{wavex}$, $M_{wavey}$ and $M_{wavez}$ are the moment the wave force is resolved into three components along the coordinate axes, N/m.

2.1.2. Wind Force Calculation

Wind action includes longitudinal wind force, transverse wind force and torsional moment, which is related to ship-shape lines, ship dimension, wind direction and wind scale. It can be obtained from wind tunnel test or towed-model experiments.

Through comparative analysis of wind action, Liu recommends using the following formula to compare in the field of ship salvage, which is easier to calculate, and the stability of the calculation result is good [34].

$$\left\{\begin{array}{c}{X}_{wind}=\frac{1}{2}{\rho }_a{V}_R^2{A}_{TV}{C}_{Ax}\\ Y_{wind}=\frac{1}{2}{\rho }_a{V}_R^2{A}_{LV}{C}_{Ay}\\ N_{wind}=\frac{1}{2}{\rho }_a{V}_R^2{A}_{LV}L{C}_{AN}\end{array}\right.$$

(4)

Here, $V_R$ is wind speed, m/s. ${\rho }_a$ is air density, kg/m3. $L$ is ship length, m. $A_{TV},A_{LV}$ are the projected area of the frontal plane of projection and the side plane of hull portions above water, m². $X_{wind}$ is the longitudinal wind force, N. $Y_{wind}$ is the transverse wind force, N. $N_{wind}$ is the torsional moment, N.m. $C_{Ax},C_{Ay},C_{AN}$ is the wind drag coefficient in the vertical direction, transverse direction and rotation about the z-axis, which can be deduced from the following regression formulas.

$$C_{Ax}=\left\{\begin{array}{cc}0.505427-0.004771\beta -0.000057{\beta }^2& 0^{°}\le \beta \le {60}^{°}\\ 0& {60}^{°}<\beta <{120}^{°}\\ 1.9984-0.022485\beta +0.000048{\beta }^2& {120}^{°}\le \beta \le {180}^{°}\end{array}\right.$$

(5)

$$C_{Ay}=\left\{\begin{array}{cc}-0.011487+0.0244477\beta -0.000162{\beta }^2& 0^{°}\le \beta \le {50}^{°}\\ 0.8& {50}^{°}<\beta <{130}^{°}\\ -0.859721+0.033872\beta -0.000162{\beta }^2& {130}^{°}\le \beta \le {180}^{°}\end{array}\right.$$

(6)

$$C_{AN}=\left\{\begin{array}{cc}0.00075+0.00355\beta -0.000028{\beta }^2& 0^{°}\le \beta \le {45}^{°}\\ -0.002357\beta +0.211071& {45}^{°}<\beta <{135}^{°}\\ 0.26026-0.00645\beta +0.000028{\beta }^2& {135}^{°}\le \beta \le {180}^{°}\end{array}\right.$$

(7)

where $\beta$ is the wind direction angle.

According to reference [35], the moment of wind force is resolved into three components along the coordinate axes:

$$\left\{\begin{array}{c}{M}_{Xwind}=-z_{wind}{Y}_{wind}\\ M_{Ywind}=z_{wind}{X}_{wind}\\ M_{Zwind}=N_{wind}\end{array}\right.$$

(8)

where $M_{wavex}$, $M_{wavey}$ and $M_{wavez}$ are the moment of wind force, which is resolved into three components along the coordinate axes, N/m.

To calculate the wind force for obscured components, the wind force of a single component should be multiplied by the shielding factor. However, it can only be used for some simple cases. For large-scale and complex random structures, there is no proper calculation method, and the wind tunnel experiment is the best choice. In many cases, simplified coefficients or an empirical coefficient can only be obtained because there are limitations in the experimental equipment. So, simulation software should be used to calculate the shielding effect for complex engineering [20].

2.2. Righting Force Calculation

Then, Equation (10) is obtained:

$$\left\{\begin{array}{c}{M}_X=M_{XG}+M_{XB}+M_{XF}+M_{Xwave}+M_{Xwind}\\ =\left(-Y_{G}G+Y_B\mathrm{\Delta }+Y_{F}F\right)\cos \varnothing \cos \theta +\left(-Z_{G}G+Z_B\mathrm{\Delta }+Z_{F}F\right)\sin {\varnothing }_{\cos \theta }-z_{wave}{Y}_{wave}-z_{wind}{Y}_{wind}\\ M_Y=M_{YG}+M_{YB}+M_{YF}+M_{Ywave}+M_{Ywind}\\ =\left(-Z_{G}G+Z_B\mathrm{\Delta }+Z_{F}F\right)\sin \theta +\left(X_{G}G-X_B\mathrm{\Delta }-X_{F}F\right)\cos \varnothing \cos \theta +z_{wave}{X}_{wave}+z_{wind}{X}_{wind}\\ M_Z=M_{ZG}+M_{ZB}+M_{ZF}+M_{Zwave}+M_{Zwind}\\ =\left(X_{G}G-X_B\mathrm{\Delta }-X_{F}F\right)\sin \varnothing \cos \theta -\left(-Y_{G}G+Y_B\mathrm{\Delta }+Y_{F}F\right)\sin \theta +N_{wave}+N_{wind}\end{array}\right.$$

(9)

Here, $M_X$, $M_Y$, and $M_Z$ are the moment of gravity, buoyancy, righting force and wave force, which are resolved into three components along the coordinate axes.

The relationship between $M_X$, $M_Y$, and $M_Z$ is represented by:

$$\begin{array}{l}{M}_X\sin \theta -M_Y\sin \varphi \cos \theta +M_Z\cos \varphi \cos \theta =\\ -(z_{wave}{X}_{wave}+z_{wind}{X}_{wind})\sin \varphi \cos \theta -(z_{wave}{Y}_{wave}+z_{wind}{Y}_{wind})\sin \theta +(N_{wave}+N_{wind})\cos \varphi \cos \theta \end{array}$$

(10)

The static equilibrium equation of gravity, buoyancy force and righting force can be obtained.

$$\mathrm{\Delta }+F-W=0$$

(11)

where $\mathrm{\Delta }$ is buoyancy force, $F$ is righting force, and $W$ is gravity.

Then, the mechanical model of uprighting is established.

$$\left\{\begin{array}{c}\mathrm{\Delta }+F-W=0\\ M_Y=M_{YG}+M_{YB}+M_{XF}+M_{Y\mathit{wave}}+M_{Y\mathit{wind}}\\ =\left(-Z_{G}G+Z_B\mathrm{\Delta }+Z_{F}F\right)\sin \theta +\left(X_{G}G-X_B\mathrm{\Delta }-X_{F}F\right)\cos \varnothing \cos \theta +z_{\mathit{wave}}{X}_{\mathit{wave}}+z_{\mathit{wind}}{X}_{wind}\\ M_Z=M_{ZG}+M_{ZB}+M_{ZF}+M_{Zwave}+M_{Zwind}\\ =\left(X_{G}G-X_B\mathrm{\Delta }-X_{F}F\right)\sin \varnothing \cos \theta -\left(-Y_{G}G+Y_B\mathrm{\Delta }+Y_{F}F\right)\sin \theta +N_{\mathit{wave}}+N_{\mathit{wind}}\end{array}\right.$$

(12)

2.3. Longitudinal Strength Calculation

The uprighting process is a quasi-static process. The sum of the weight, the buoyancy, the perpendicular component of wave force, the perpendicular component of wind force and the righting force is zero. However, the distribution of five forces along the ship length is not uniform. Then, shear force and bending moment are produced. Here, $w(x)$ expresses gravity force per unit length at section x, $b(x)$ expresses buoyancy force per unit length at section x, $X_{wave}(x)$ expresses the perpendicular component of wave force per unit length at section x, $X_{wind}(x)$ expresses the perpendicular component of wind force per unit length at section x, and $f(x)$ expresses the righting force per unit length at section x [36].

$$q(x)=w(x)-b(x)-X_{wave}(x)-X_{wind}(x)-f(x)$$

(13)

Then, the shear force and the bending moment at section x can be expressed as follows [36]:

$$N(x)={\displaystyle {\int }_0^{x}q(x)dx}$$

(14)

$$M(x)={\displaystyle {\int }_0^x{\displaystyle {\int }_0^{x}q(x)dx}}dx$$

(15)

Assuming that the horizontal loading at section x can be expressed as $q(y,z)$, the torque of the section is $T(y,z)$ [36].

$$T(y,z)={\displaystyle {\int }_0^z{\displaystyle {\int }_0^{y}q(y,z)\cdot l}}(y,z)dydz$$

(16)

where $l(y,z)$ is the distance between the load acting point and balance position.

3. Simulation Calculation

3.1. Introduction of Simulation Analysis on Wind and Wave

The seakeeping module (SK) of GHS software can provide solutions for hydrodynamic analysis. Integration in the GHS environment also means that calculations can be performed using existing geometry files, load conditions, and run files. This translates into minimal additional user input, unparalleled flexibility and ease of use in its segment.

SK is a unique and powerful analysis tool. There are a wide range of environmental input options: sine curve, spectrum, data, range, expansion and flexible sampling. The ability to specify any loading conditions at a special level of detail and include all inertial properties enables many cases to be run with little effort and maximum repeatability. This module integrates GHS command language and running files with high flexibility. With minimal syntax, it can quickly and easily set up hydrodynamic analysis, allowing a rapid re-evaluation of complex geometries.

Hydrodynamic analysis using GHS geometry files (GF) does not need to create other models, and it automatically identifies the ship loading conditions specified in GHS, including all cabin weights, increased weights and hull weight distribution. It is not necessary to specify hydrostatic parameters, and the required contents can be accessed directly.

SK supports the calculation of the six degree of freedom motion of the floating body, including displacement, velocity, acceleration, amplitude and phase angle, and absolute and relative motion. The motion can be calculated with the center of gravity of the ship and the designated critical point can be designated as the designated point, including the coupling effect (such as the pitch to heave effect near the bow).

In order to analyze cases with a large amount of calculation, SK provides an overall summary table, which displays the position, speed and acceleration values in the X (pitch), y (yaw) translation directions and X (roll), y (pitch) and Z (yaw) angle directions. These tables are ideal reference points in iterative calculations.

3.1.1. Simulation Method of Wave Force

The virtual wave is built on a fixed horizontal plane, which is defined by the heel angle, trim angle and the vertical distance from the origin to the horizontal plane. The software will automatically calculate the waterline height according to the rise or fall of the wave and the change of the heel angle of the floating body caused by the wave. GHS can set periodic waves and then calculate the stability and strength of the ship under the action of waves. If the wavelength parameter is omitted, the default wavelength of the software is the length between the vertical lines. The wavelength is taken as the projected length of the length between the perpendiculars on the actual waterplane for just this situation.

When calculating the hull strength, the sectional area changes with the wave parameters. Given any state of the hull, the shear force and bending moment at any position along the length of the hull can be obtained. During the calculation process, the software can solve the intersection of multiple waterplanes and the hull section line, and it can also identify the waterplane formed by the waves and the hull. When calculating the strength of the floating body in waves, it is necessary to use the defined wave parameters before calculating the longitudinal strength of the floating body, so the calculation results take into account the influence of the waveform.

3.1.2. Simulation Method of Wind Force

By displaying the wind speed and wind pressure at a series of heights above the water surface, the wind force on the floating body can be obtained. The wind speed is generally in knots. The wind pressure is obtained by linear interpolation between the given points. The wind pressure can also be directly defined in the pressure mode. If the wind pressure defines n height points, when 2 ≤ n ≤ 4, an n − 1 polynomial curve will be used to fit the equation of height and wind pressure. When n > 4 (n max. 20), a third-order polynomial calculated by the least squares method will be used for fitting.

There are two methods to obtain the effective side area to calculate the wind heeling moment. First, traditional lateral projection includes all parts of the hull and superstructure above the waterline. Second, the band method measures from the area of strip distribution wind pressure above the waterline to the highest point of the model.

The lateral projection method ignores the shielding effect of the lateral projection plane. Therefore, when the heeling is not zero, a larger value that does not conform to the reality is often obtained. However, in the state of zero heel, it is reasonable to calculate the wind heeling moment according to the lateral projected area, which is also used in some specifications.

The band method can identify the shielding of the lateral area, and the heeling moment generated with the heel angle is more real and reasonable. However, compared with the traditional calculation method, the calculation time of this method is longer, especially for the righting arm. The smaller the bar width, the slower the calculation. If the bar width is too small, the speed will be very slow, which obviously does not have much significance for the accuracy of the calculation results.

3.2. Simulation Design Scheme

3.2.1. Establishment of the Ship Model

GHS geometry files can be modeled by software interface operation and an editing program. The method of a direct editing program is applied in this paper. The hull model program is edited according to the ship drawings to further establish the cabin and encrypt the cabin at the bow, stern and cabin. Taking a ship as an example, the model is shown in Figure 1, which is used for simulation calculation.

Take Figure 1 for example, where the origin $O$ crosses the area of the base plane, the midship section, and the longitudinal mid-section, which is 8.6 m away from the bow. The axis direction is as follows: $Ox$ is the intersection of the base plane and the longitudinal mid-section; the stern represents the positive direction. $Oy$ is the intersection of the base plane and the midship section; the starboard represents the positive direction. $Oz$ is the intersection of the longitudinal mid-section and the midship section; the positive direction is over the base plane.

Table 1. The principal dimensions of the intact ship.

Length Overall (m)	Breadth (m)	Molded Depth (m)	Draft (m)
139	22	10.556	3744.25 t

shows the ship’s principal dimensions.

According to the principal dimensions, loading condition and ship lines, the floatation and stability can be solved quickly by using GHS. Highly accurate results can be obtained by increasing the offset table values.

3.2.2. Working Condition Simulation

In general, the salvage work needs to be carried out under good sea conditions to avoid adverse environmental forces preventing the salvage work or causing damage to the ship. However, in most projects, it is difficult to avoid the effects of wind and waves. In the actual project, due to the limitation of salvage equipment, the normal salvage work requires that the wind force should not exceed level 6 and the wave height should not exceed 1.5 m. According to the experience [37], this paper simulates the uprighting process of capsized ships in the following four environments.

Case I: The ship listed −163.65 to starboard, had a trim of −1.49°, and had an origin draft of −4.259 m. The wave height is 1.5 m, and the average wind is 6 degrees.

Case II: The ship listed 174.57° to starboard, had a trim of −1.54°, and had an origin draft of −4.757 m. The wave height is 1 m, and the average wind is 5 degrees.

Case III: The ship listed 179.99° to starboard, had a trim of −1.55°, and had an origin draft of −5.287 m. There is a swell of height 1 meters from seaward.

Case IV: The ship listed 180° to starboard, had a trim of −1.55°, and had an origin draft of −4.787 m. The wave and wind have a small influence on the hull, which can be neglected.

Considering the influence of wind and waves on rescue equipment, this paper only simulates the maximum wind and wave level allowed by the working condition.

In each working condition of a capsized ship, the floating state of the ship is the result of the combined action of gravity, buoyancy, wind and waves. It is found that a 1 M wave height has little influence on the inclination angles of the hull, but it has a certain influence on the ship draft of the hull due to the effect of wave peak by comparing the floating states of the hull in case III and case IV. By comparing the floating state of the hull in case II and case III, it is found that the transverse stability of the hull decreases greatly under the combined action of waves and wind. By comparing the floating state of the hull under case I and case II, it is found that the stability of the hull is significantly reduced when the wind wave value is increased.

3.3. Stability Analysis of the Capsized Ship

In order to analyze the stability of the capsized ship under four sea conditions in detail, Figure 2 shows the righting arm curves of the capsized ship under a calm sea environment and wind and wave environments. The horizontal axis shows the heel angle in degrees, and the vertical axis shows the righting arm in m. The ship displaces a large quantity of water, which topped 21,089.07 tons. The stability of the capsized ship is greater, while the effect of waves on the capsized ship is negligible. However, the combined action of wind and waves has a great influence on the stability of the ship hull. The greater the wind and waves, the smaller the stability range and negative-going righting arm. The stability of the capsized ship under case III and case IV is almost equal. This is because the wave direction is perpendicular to the hull, the wavelength is longer, and there are waves cresting at the origin. However, this is just a special case; the floatation, stability and mechanical properties of capsized ships should be further investigated when waves and ships are in different locations. The theory shows that a capsized ship naturally recovers to upright floating attitude when the heel angle restores to a certain degree. It is found from the result of calculation that some capsized ships cannot naturally recover to upright floating attitude due to wind and waves. Late in the course of the uprighting process, a reversal righting force moment is needed to maintain the balance under the first two cases, and case I needs a greater righting force moment. A comparison between case II and case III indicates that the wind force moment is greater than the wave force moment. The wind and waves reduce the negative stability value of the hull and increase the positive stability value of the hull, but they do not change the trend of the hull stability. This is beneficial to reduce the righting moment in the early stage of the uprighting process, but in the later stage of the uprighting process, a large reverse moment may be needed to reduce the speed of the hull’s automatic return.

4. Simulation Calculation and Analysis

4.1. Scheme Design

Considering the effect of heel angle, trim angle, structure, strength and positions of righting force points, righting force points can set in three longitudinal places when designing the uprighting scheme: the longitudinal coordinates are −33.15 m, −70.5 m, and −102.672 m, which are, respectively, located in ①②, ③④, ⑤⑥ in Figure 1. In order to achieve good positions of righting force points designs, three uprighting schemes were carried out.

Scheme A: Positions of righting force points are located in ③④.

Scheme B: Positions of righting force points are located in ①②, ⑤⑥.

Scheme C: Positions of righting force points are located in ①②, ③④, ⑤⑥.

The shear force, bending moment and torque of position of the righting force point for case I scheme A, case III scheme A and case IV scheme A exceed the allowable values, which could deform or damage the hull. The below calculations do not take these three schemes into account. From the simulation results of other schemes, no immediate abnormalities of trim angle and draft are obtained. So, the scheme shows a certain feasibility.

Without question, when the righting force point is under the action of a large external force, that results in the deforming and damaging of the hull. Therefore, monitoring the strength values of righting force points will aid in designing a more perfect uprighting scheme. To distinguish positions of righting force points, based on each working environment and each uprighting scheme, they are respectively marked A③④, B①②, B⑤⑥, C①②, C③④, and C⑤⑥.

In practical engineering, the righting force exerted on the hull is often less than the buoyancy force, so it is a common method for the floating crane to exert a vertical force to overturn the ship. The operation process has the advantages of fast speed and easy control, but it also has the disadvantages of high cost. In some projects, other methods are selected because the crane ship with large lifting capacity cannot be used. In recent years, with the rapid development of marine engineering science and technology, the number of large lifting vessels has increased, but the number of large lifting vessels worldwide is still limited, and small and medium-sized marine engineering companies can use fewer large floating cranes. The program is developed according to actual operation condition, every cross-section of righting force point is acted upon by two forces. First, we set the horizontal forces on the right side of the ship bottom, the angle is 10°, and we direct the pull downward. Second, we set the horizontal forces on the left side of the deck, the angle is 10°, and we direct the pull upward.

4.2. Analysis of the Calculation Results

4.2.1. Wind Force and Righting Force

The wind force, wind force moment, righting force, righting force moment and the maximum righting force of a ship under typical attitude are shown in

Table 2. Wind force and righting force of case I.

Schemes Parameters	Heel Angle /°	Trim Angle /°	Wind Force /MN	Wind Force Moment /MN.m	Righting Force /MN	Righting Force Moment /MN.m	Heel Angle /°	Trim Angle /°	Maximum Righting Force /MN
Case B	−149.16	−1.52	19.1595	186.6941	12	78.3192	6.19	0.54	15
Case C	−149.12	−1.48	19.1768	186.9567	12	78.3294	5.99	0.6	15

,

Table 3. Wind force and righting force of case II.

Schemes Parameters	Heel Angle /°	Trim Angle /°	Wind Force /MN	Wind Force Moment /MN.m	Righting Force /MN	Righting Force Moment /MN.m	Heel Angle /°	Trim Angle /°	Maximum Righting Force /MN
Case A	−149.02	−1.49	12.4754	121.50	22	124.6092	2.92	0.59	24
Case B	−149.06	−1.58	12.6061	122.6640	30	171.7928	2.81	0.5	33
Case C	−149.08	−1.47	12.6039	122.5313	27	167.2199	2.65	0.61	30

,

Table 4. Wind force and righting force of case III.

Schemes Parameters	Heel Angle /°	Trim Angle /°	Wind Force /MN	Wind Force Moment /MN.m	Righting Force /MN	Righting Force Moment /MN.m	Heel Angle /°	Trim Angle /°	Maximum Righting Force /MN
Case B	−149.02	−1.69	-	-	66	352.8452	0.07	0.41	69
Case C	−149.02	−1.43	-	-	60	343.5245	0.17	0.63	63

and

Table 5. Wind force and righting force of case IV.

Schemes Parameters	Heel Angle /°	Trim Angle /°	Wind Force /MN	Wind Force Moment /MN.m	Righting Force /MN	Righting Force Moment /MN.m	Heel Angle /°	Trim Angle /°	Maximum Righting Force /MN
Case B	−149.03	−1.68	-	-	63	347.8286	0.41	0.28	66
Case C	−149.04	−1.44	-	-	57	338.5778	0.05	0.63	60

. The tables are used to analyze the effects of wind force and wave force on the uprighting process and determine the maximum righting force of schemes.

When the righting force moment is greater than the righting moment of a capsized ship, the heel angle decreases gradually to negative 0°. The ship would list to the other side if the righting force moment is too large. In fact, no excessive righting force moment will be applied in practical engineering. The righting force will decrease when the heel angle exceeds the corresponding angle of the maximum righting moment. Take case I as an example; the ship will automatically return to the equilibrium location when the heel angle exceeds 119.79°, because the stability of the capsized ship has a range of 169.79°–119.79°.

A simulation program of a superposition algorithm for righting the capsized ship is programmed, and the righting force has accelerated at a quantity of 3 MN. Take case I scheme C as an example: the heeling angle is −149.12° when the righting force is 12 MN; then, the heeling angle is 5.99° when the righting force further increases to 15 MN. So, the righting force is required for righting a capsized ship is between 12 and 15 MN. It is verified by the simulation results that 12 MN is the marginal value of righting force.

The calculation results revealed that the stability is greatly affected by the joint action of wind and waves. An uprighting scheme should be determined in terms of the actual developing environment. When the stability of a capsized ship is large, a small wave has no obvious effect on stability, but it has a certain influence on the righting force calculation. Comparing the calculated results of case I, case II, and case III, the effect of wind force on the ship is much bigger than that of the wave force, and it is bigger than the righting force in case I.

The calculation results of case II, case III, and case IV were compared, and the righting force of schemes are generally larger, which plays a very important role in longitudinal strength analysis to calculate the load of righting force points. The righting forces of the case I schemes are equal, which further explains that the wind has a remarkable effect on the hull. The righting force of case II is small, with the lowest numbers of righting force point, which might result from the wind and waves. In the engineering practice, according to the equipment condition and engineering requirement, calculation results can offer the support for scheme selection. For instance, in order to minimize the gesture change and deformation extent of a hull, steel beams are located above the ship bottom to lift the Korean ferry SEWOL through numerous calculations.

4.2.2. Shear Force

Figure 3, Figure 4, Figure 5 and Figure 6 represents the shear force of righting force points under every operational condition. There is a large difference in the shear force when the heel angle exceeds the corresponding angle of the maximum righting moment, and the shear force of A③④ changes greatly. This indicates that the shear force of a single righting force point is more sensitive. The shear forces of the same righting force point under the four cases are decidedly different from each other, and the shear force difference of case III and case IV is small.

The shear force of most righting force points in various cases decreases gradually (shown in Figure 3, Figure 4, Figure 5 and Figure 6), but the shear force of case III C③④, case III A③④, case IVC③④, and case IV A③④ increase gradually, as illustrated in Figure 5 and Figure 6. The shear forces of the righting force point of different schemes under various cases are close to each other. The shear force difference of C①② and B①② increases when the heel angle decreases; the maximum differences are, respectively, 1.66%, 4.05%, 9.97%, and 9.233%, and C①② is smaller than B①②. The shear force difference of C⑤⑥ and B⑤⑥ increases when the heel angle decreases, the maximum difference is, respectively, 3.70%, 8.80%, 2.16%, and 2.00%, and C⑤⑥ is smaller than B⑤⑥. The shear force difference of C③④ and A③④ increases when the heel angle decreases, and the maximum difference is 1.72%. By comparing scheme A with scheme C, the shear force did not decrease, and part of the righting force of scheme C may be used to control the attitude of the ship. In scheme B and scheme C, the change of shear force at the righting point in each working condition gradually increases, among which C③④ and B⑤⑥ increase greatly. In the salvage process, attention should be paid to the influence of wind force on the local force of the hull.

4.2.3. Bending Moment

Figure 7, Figure 8, Figure 9 and Figure 10 represent the bending moment of righting force points under every operational condition, and they have the same variation trend. In scheme B and scheme C, the bending moment of the first half of the hull is significantly reduced under windless conditions, which is corresponding to the weight distribution of the hull and can provide a reference for salvage engineering. Through comprehensively analyzing four cases, the bending moment difference of C①② and B①② increases when the heel angle decreases, the maximum difference is, respectively, 0.84%, 1.99%, 4.88% and 4.53%, and C①② is smaller than B①②. The bending moment difference of C⑤⑥ and B⑤⑥ increases when the heel angle decreases, and the maximum difference is, respectively, 1.01%, 2.38%, 5.72% and 5.3%, and C⑤⑥ is greater than B⑤⑥. The bending moment of C③④ is smaller than A③④, and the maximum difference is 4.80%. A comparison and analysis of the three schemes shows that the righting force points of multiple cross-sections do not always reduce the bending moment.

4.2.4. Torque

The heel angle changes greatly during the righting of a capsized ship. The torque varies complicatedly when the hull is affected by the magnitude, position and direction of the righting force. The study of torque variation is of importance for structure safety. Figure 11, Figure 12, Figure 13 and Figure 14 represent the torque of righting force points under every operational condition. The torque of A③④, C①②, B①② and C③④ later in applying the righting force is largely fluctuation, and the torque of A③④ and C③④ changes direction, while the torque direction of B①② and C①② stays in the same direction. Under the condition of no wind, the torque value at each righting force point varies in a larger range, because the uprighting project needs to apply a larger righting force to overcome the larger stability of the hull. When the torque is low, the torque direction changes frequently, which may lead to the fatigue damage of the hull.

Through comparing the four cases, the torque difference of C①② and B①② increases when the heel angle decreases, the maximum difference is, respectively, 0.9%, 1.99%, 3.98% and 3.40%, and C①② is smaller than B①②. The torque difference of C⑤⑥ and B⑤⑥ increases when the heel angle decreases, and the maximum difference is, respectively, 0.77%, 1.54%, 3.10% and 2.94%, and C⑤⑥ is smaller than B⑤⑥. The torque of C③④ is larger than that of A③④, and the maximum difference is 6.02%. Analysis showed that multiple righting force points do not necessarily reduce torque effectively.

4.3. Comparison of Uprighting Schemes

To evaluate the effects of different salvage schemes, effects of floatation, stability, righting force, wave, wind and longitudinal strength should be comprehensively analyzed. Of the three schemes, in the view of strength, scheme C is preferred to scheme B, and the shear force and torque of scheme A are smaller than those of scheme C. Combining with the calculation results of righting force, case II is adapted using scheme A, while the attitude control of the hull is an important factor to be investigated in the actual project. Through comprehensively analyzing four cases, the righting force of scheme B is 0 MN, 3 MN, 6 MN and 6 MN higher than that of scheme C. In order to have an integrative consideration, case I can choose scheme A, while case II, case III and case IV can choose scheme C.

5. Conclusions

Righting a capsized ship is a quite complex engineering task, especially when the hull is interfered with by wind and waves. The main work and results of this paper are as follows.

Wind has obvious effects on the stability of a capsized ship when the hull has a larger windward area. The wave action cannot be calculated directly, but it can be evaluated through comparing cases. The wave forces on a hull are much weaker than the action of wind, and they also have a certain effect on the stability and uprighting process. The action of a favorable wind can greatly reduce the righting force, and it can also reduce the internal force.

The maximum righting force of case II scheme A is 24 MN, which is less than the other two cases by 27.3% and 20%, respectively. The shear force and torque of righting force points are relatively small. In a situation of high winds and rough seas, the righting force of single righting force points can also produce better results. It is important to note that the direction of righting force and direction of the wind and waves should be the same.

Through the simulation analysis of case I, case III, and case IV, the righting force of scheme C is, respectively 0, 8.7%, and 9.1% less than that of scheme B. Most of the maximum internal force values of the righting force points of scheme C are smaller than those of scheme B. Only the bending moment of C⑤⑥ is larger than that of B⑤⑥. So, the scheme is more suitable for these three cases. According to the calculation results, properly increasing the righting force of ①② under case III and case IV can reduce the bending moment of ⑤⑥.