1 Introduction

Over the last few years, the worldwide shipping business and sea-related traffic have gotten busier and busier [1], stimulated by the recovery of trade and strong demand for commodities. This situation speeds up the launch of ships. The scale of the world’s fleet has been increasing, and ship accidents have increased significantly. Many ships have capsized due to the loss of stability.

Serious accidents often occur after the ships lose their stability. The main factors affecting the stability of the ship are the position of the ship’s center of gravity, the marine environment, the positions of the cargo and the operation error [2]. The fuzzy analytic hierarchy process (FAHP) was used to analyze the hydrostatic mechanical properties of ships by Thaddeus C. NWAOHA and Fabian I. IDUBOR. The effects of ship shape, draft and weight on ship stability were analyzed in detail [3]. Cristian-Milică Niţă used Modelmaker software to study the influence of ship shape and ballast tank size on ship stability [4]. Borlase studied the stability of different structures under different loads [5]. Weidle W S studied the influence of geometric parameters of a single ship, catamaran and trimaran on the stability of intact ship and damaged ship and obtained the effect of waterline area on ship stability [6]. Karimirad M and Michailides C studied the static stability of different types of semi-submersible offshore wind turbines and found that the restoring moment can be improved by changing the stability height [7].

Changes in underwater volume affect the stability of the hull. Sun et al. evaluated the stability of the damaged ship by using the loss buoyancy method, volume change method and Newton iteration method. They obtained the effect of the liquid level height of the tank and the floating parameters of the hull when the ship was in any floating states [8]. Degan et al. studied the seaworthiness of large-scale ships through methods of the multi-attribute decision making (MADM) and mathematical design model (MDM) and obtained the calculation method of ship stability through a large number of analyses. The error between the calculation results and the actual results is small [9]. The movement of the floating body load will cause a change in stability. Due to the effect of inertia, the speed of load movement on the floating body should be limited [10]. For a long time, a large number of studies have been carried out on the stability of normal ships, while there are few studies on the stability of complex structures. For example, the underwater shape of the hull changes continuously in the uprighting process of a capsized ship, which increases the difficulty of analyzing the stability of the ship [11, 12]. Varsami used a Trainer 5000 simulator to simulate the salvage process of a stranded ship and obtained the change in hull stability [13].

Under the action of wave force, not only the force distribution and direction of the floating body are changed, but also the stability of the floating body. Both tidal waves and air pressure waves will reduce the ship’s stability, cause the ship to sway or tilt and make the ship deviate from the equilibrium state [14]. Manderbacka et al. discussed the effects of different wave parameters on ship stability based on different cases [11]. Ueng put forward a static method to calculate the stability and buoyancy of ships and further deduced the calculation methods of heave, pitch and roll motions of ships, which can be used for ship motion simulation [3]. Through simulation analysis, Armudi et al. found that both wave velocity and wave height had significant effects on ship drift [15]. Ermina Begovic et al. studied the change of hull stability caused by the change of hull position in waves [16]. Petacco studied the effects of hull volume distribution, hull weight distribution and wave profile on hull stability [17]. For the floating body with a special underwater shape, it is difficult to accurately solve the analytical solutions of wave first-order and second-order potentials and their external forces [18]. Gu et al. studied the motion of paralyzed ships under sea waves through experiments and simulations. Their results show that the probability of inclination of the paralyzed ship with drift motion is higher than that of the paralyzed ship without drift motion [19].

The simulation calculation of wave force is helpful to determine the motion of the floating body. With reference to the Harbour hydrological code (Chian), Li and Shi calculated the wave force of the towing systems. The calculation method has high accuracy when applied to low-speed structures [20]. Yu et al. compared the motion of a jack-up drilling platform in waves through test and simulation calculations. The error rate between the simulation results and the test results is less than 10%, which shows that the numerical simulation method can better simulate the motion response of this platform [21]. When Pekka Ruponen used Archimedes law to calculate the wave force of the hull, the hull surface was divided into several panels, and the buoyancy was obtained by integrating the pressure on the panels. This method is also suitable for analyzing the ship motion in waves and for the stability analysis of offshore structures with complex geometry [22]. In order to calculate the motion of the ship in waves, Kuang Ueng et al. regarded the hull as a rigid body, assuming that the motion amplitude of the hull is small, and the six degrees of freedom are independent and superimposed to describe the motion of the hull [23]. Cien and Tey adopted the statics method to calculate the floating state and stability of the floating platform for marine agricultural microalgae and further studied the influence of scale and wave frequency on the tilt angle of the floating platform [24]. Armudi et al. used numerical analysis methods to study the motion law of ships under the action of linear and nonlinear waves. The results show that waves seriously affect the ship’s motion [25].

Many salvage projects are operated when the waves are small and the impact of the swell is ignored while formulating the uprighting scheme [26, 27]. However, sometimes swell can also have a non-negligible impact on the hull, so it is necessary to study the impact of swell on the uprighting process [28]. Based on previous research, this paper focuses on the calculation of the influence of swell on the uprighting process, establishes the mechanical model of righting a capsized ship considering the swell effect and clarifies the influence of the encounter angle on the hull. Furthermore, GHS software is used to simulate the uprighting process of the capsized ship in seven environments. The influences of a calm sea environment, wave environment and encounter angle on the uprighting process of the capsized ship are analyzed.

In view of the influence of waves on the uprighting process, the calculation method of the righting force and righting force moment is derived. In order to determine the effect of swell on a capsized ship under three-level sea conditions, the righting force and righting force moment of different ship headings are compared, and the encounter angle suitable for righting a capsized ship is obtained. Furthermore, the dangerous positions of the hull and the influence of the wave direction on the longitudinal strength of the hull are determined by analyzing the mechanical distribution of the hull, which provides a theoretical reference for the uprighting process in the swell environment.

This paper consists of five parts. The first part is literature research, including ship floating state research, ship static stability research, wave force research and the effect of waves on uprighting process research. The second part is the mechanical analysis of the capsized ship, including the wave force model, the force analysis of the hull and the solution of the righting force. The third part is the simulation of the uprighting process of the capsized ship. The fourth part is the analysis and discussion of the hull stability, the righting force, the righting force moment and the longitudinal strength of the hull. The fifth part belongs to the conclusion.

2 Theoretical calculation of the uprighting process in wave environments

2.1 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 on the hull more complicated. While designing the uprighting scheme, the influence of wave force on the uprighting process of the capsized ship must be taken into consideration.

At present, the calculation methods of wave force include the Morison equation method, three-dimensional source-sink distribution method, Green function method, slice method, design wave method, nonlinear Stokes fifth-order method and slender body theory calculation method [29].

Most methods for calculating wave forces are complex. It is difficult to meet the needs of the calculation of the uprighting process. In this paper, the approximate calculation formula of wave drift force recommended by Liu is adopted [30].

$$ \left\{ {\begin{array}{*{20}l} {X_{{{\text{wave}}}} = 0.5\rho g\xi ^{2} LC_{{xw}} \cos \left( {\theta _{F} + \theta _{s} } \right)} \hfill \\ {Y_{{{\text{wave}}}} = 0.5\rho g\xi ^{2} LC_{{yw}} \cos \left( {\theta _{F} + \theta _{s} } \right)} \hfill \\ {N_{{{\text{wave}}}} = 0.25\rho gL^{2} \xi ^{2} C_{{Nw}} \cos \left( {\theta _{F} + \theta _{s} } \right)} \hfill \\ \end{array} } \right. $$
(1)

In the formula, \(\rho\) represents the seawater density. \(g\) represents gravitational acceleration. \(\xi\) represents the mean wave amplitude. \(L\) represents the hull length. \(\theta_{F}\) represents the wave encounter angle. \(\theta_{s}\) represents the heading angle. \(C_{xw}\), \(C_{yw}\), \(C_{Nw}\) are wave drift force coefficients, which can be solved from the regression expression.

$$ \left\{ {\begin{array}{*{20}l} {C_{xw} = 0.05 - 0.2\left( {\lambda /L} \right) + 0.75\left( {\lambda /L} \right)^{2} - 0.51\left( {\lambda /L} \right)^{3} } \hfill \\ {C_{yw} = 0.46 - 6.83\left( {\lambda /L} \right) - 15.65\left( {\lambda /L} \right)^{2} + 8.44\left( {\lambda /L} \right)^{3} } \hfill \\ {C_{Nw} = 0.11 + 0.68\left( {\lambda /L} \right) - 0.79\left( {\lambda /L} \right)^{2} + 0.21\left( {\lambda /L} \right)^{3} } \hfill \\ \end{array} } \right. $$
(2)

Here, \(\lambda\) represents the mean wavelength.

For the convenience of calculation, the wave encounter angle can be taken as 0°.

2.2 Righting force calculation

According to reference [26], the moment of wave force is resolved into three components along the coordinate axes:

$$ \left\{ {\begin{array}{*{20}l} {M_{{X{\text{wave}}}} = - z_{{{\text{wave}}}} Y_{{{\text{wave}}}} } \hfill \\ {M_{{Y{\text{wave}}}} = z_{{{\text{wave}}}} X_{{{\text{wave}}}} } \hfill \\ {M_{{Z{\text{wave}}}} = N_{{{\text{wave}}}} } \hfill \\ \end{array} } \right. $$
(3)

In this above formula, \(M_{{X{\text{wave}}}}\), \(M_{{Y{\text{wave}}}}\) and \(M_{{Z{\text{wave}}}}\) are the moments of wave force which is resolved into three components along the coordinate axes.

Then, formula (4) is obtained as follows [26].

$$ \begin{aligned} M_{X} & = M_{XG} + M_{XB} + M_{XF} + M_{{X{\text{wave}}}} \\ & = \left( { - Y_{G} G + Y_{B} \Delta + Y_{F} F} \right){\text{cos}}\phi {\text{cos}}\theta + \left( { - Z_{G} G + Z_{B} \Delta + Z_{F} F} \right){\text{sin}}\phi {\text{cos}}\theta - z_{{{\text{wave}}}} Y_{{{\text{wave}}}} \\ M_{Y} & = M_{YG} + M_{YB} + M_{YF} + M_{{Y{\text{wave}}}} \\ & = \left( { - Z_{G} G + Z_{B} \Delta + Z_{F} F} \right){\text{sin}}\theta + \left( {X_{G} G - X_{B} \Delta - X_{F} F} \right){\text{cos}}\phi {\text{cos}}\theta + z_{{{\text{wave}}}} X_{{{\text{wave}}}} \\ M_{Z} & = M_{ZG} + M_{ZB} + M_{ZF} + M_{{Z{\text{wave}}}} \\ & = \left( {X_{G} G - X_{B} \Delta - X_{F} F} \right){\text{sin}}\phi {\text{cos}}\theta - \left( { - Y_{G} G + Y_{B} \Delta + Y_{F} F} \right){\text{sin}}\theta + N_{{{\text{wave}}}} \\ \end{aligned} $$
(4)

Here, \(M_{X}\), \(M_{Y}\) and \(M_{Z}\) are the moments of gravity, buoyancy, righting force and wave force which is resolved into three components along the coordinate axes.

The relationship between \(M_{X}\), \(M_{Y}\) and \(M_{Z}\) is represented by formula (5).

$$ \begin{aligned} M_{X} \sin \theta - M_{Y} \sin \varphi \cos \theta + M_{Z} \cos \varphi \cos \theta = & - z_{{{\text{wave}}}} X_{{{\text{wave}}}} \sin \varphi \cos \theta \\ & - z_{{{\text{wave}}}} Y_{{{\text{wave}}}} \sin \theta + N_{{{\text{wave}}}} \cos \varphi \cos \theta \\ \end{aligned} $$
(5)

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

$$ \Delta + F - W = 0 $$
(6)

Then, the mechanical model of uprighting is established as follows.

$$ \begin{gathered} \Delta + F - G = 0 \hfill \\ M_{Y} = M_{YG} + M_{YB} + M_{YF} + M_{{Y{\text{wave}}}} \hfill \\ = \left( { - Z_{G} G + Z_{B} \Delta + Z_{F} F} \right){\text{sin}}\theta + \left( {X_{G} G - X_{B} \Delta - X_{F} F} \right){\text{cos}}\phi {\text{cos}}\theta + z_{{{\text{wave}}}} X_{{{\text{wave}}}} \hfill \\ M_{Z} = M_{ZG} + M_{ZB} + M_{ZF} + M_{{Z{\text{wave}}}} \hfill \\ = \left( {X_{G} G - X_{B} \Delta - X_{F} F} \right){\text{sin}}\phi {\text{cos}}\theta - \left( { - Y_{G} G + Y_{B} \Delta + Y_{F} F} \right){\text{sin}}\theta + N_{{{\text{wave}}}} \hfill \\ \end{gathered} $$
(7)

In each working condition of righting a capsized ship, the floating state of the ship is the result of the combined action of gravity, buoyancy, waves and righting force. According to Eq. (7), the state of the hull under any positive force can be obtained.

During the process of designing the scheme, the calculation workload is particularly large. There is no other literature to study the uprighting process of a capsized ship. Based on the study of the force of the capsized ship in still water, this paper considers the effect of waves on the ship. And then the force of the capsized ship in the process of uprighting can be obtained, which provides a theoretical basis for practical engineering.

3 Simulation calculation of the uprighting process of a capsized ship in wave environments

3.1 Calculation method

According to the environmental conditions of the salvage work, the GHS software is used in this paper to simulate the uprighting process of the capsized ship under the action of sea state III.

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. The GHS software 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 is the length between the vertical lines. In this case, the wavelength is the projected length between perpendiculars on the actual waterplane. When calculating the hull strength, the cross-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 can also identify the waterplane formed by waves and the hull.

When calculating the strength of the floating body in waves, it is necessary to use the determined wave parameters before calculating the longitudinal strength of the floating body, so the influence of the waveform is taken into account in the calculation results.

The LS module calculates the longitudinal strength of the floating body and then obtains the integrals of shear force and bending moment. The specific calculation method is to regard the ship as a beam supported by water, load it with a given weight distribution and consider the effects of other external forces when calculating along the longitudinal integral. Similarly, the torque module can calculate torque caused by weight, buoyancy and other external forces. During the calculation, if the moment of inertia of the hull section is input, the bending deflection of the hull can be calculated. The load distribution can be in the form of point loads or piecewise linear load curve, and finally, the various loads are automatically summed to obtain the target curve. During the calculation process, the software will take the influence of changes in parameters such as the inclination angle, draft and flooding quantity of the floating body into account.

3.2 Simulation design scheme

The hull model is shown in Fig. 1, and the main dimensions of the hull are shown in Table 1.

Fig. 1
figure 1

Hull model

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Table 1 The main dimensions of the hull
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Generally, salvage work needs to be carried out in good sea conditions to avoid harsh environmental forces to hinder salvage work or damage to the ship. However, in most projects, it is difficult to avoid the effect of waves [31, 32]. To study the effect of waves, the uprighting process of a capsized ship under different wave conditions is simulated in this paper, that is, the calm sea environment and the wave encounter angles at 0°, 60°, 120°, 180°, 240° and 300° under the action of sea state III. In the uprighting scheme, the positions of cables for righting the capsized ship are shown as a and b in Fig. 1. Under the sea condition of level 3, the wave height is 1 m, the wavelength is 20 m, and the wave crest is at the origin. During the uprighting process, different environments will lead to different working conditions.

Case A (calm sea): The ship listed 180° to starboard and had a trim of 0.88° and an origin draft of − 6.897 m.

Case B (wave encounter angle is at 0°): The ship listed 180° to starboard and had a trim of 0.79° and an origin draft of − 6.927 m.

Case C (wave encounter angle is at 60°): The ship listed 180° to starboard and had a trim of 0.80° and an origin draft of − 6.93 m.

Case D (wave encounter angle is at 120°): The ship listed 180° to starboard and had a trim of 0.88° and an origin draft of − 6.863 m.

Case E (wave encounter angle is at 180°): The ship listed 180° to starboard and had a trim of 0.96° and an origin draft of − 6.793 m.

Case F (wave encounter angle is at 240°): The ship listed 180° to starboard and had a trim of 0.96° and an origin draft of − 6.78 m.

Case G (wave encounter angle is at 300°): The ship listed 180° to starboard and had a trim of 0.88° and an origin draft of − 6.842 m.

Under each working condition, the floating state of the hull is the result of the combined effects of weight, buoyancy, wave force and righting force. Although the difference in wave encounter angles is large, the difference in floating parameters of the ship hull is small.

4 Simulation calculation

4.1 Righting arm

Figure 2 shows the righting arm curves of the capsized ship under a calm sea environment and wave actions at different encounter angles. This curve shows the stability of the ship when the heel angle is 180° and also demonstrates the difficulty of the uprighting process. The convex part of the curve represents that the righting force moment is required to balance the hull, while the concave part represents that no righting force moment is required. However, a reverse righting force moment is required to prevent the hull from returning to the equilibrium position quickly and avoid the inertial effect of the hull due to the faster recovery speed under the action of the righting moment. This may cause the hull to tilt to the left after returning to its equilibrium position.

Fig. 2
figure 2

The static stability curves of capsized ship

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During the uprighting process, the righting force moment must be greater than the righting moment of the capsized ship. The two partial enlargements in Fig. 2 are partial screenshots of the peaks and troughs of the righting arm curve. At the stage of applying the positive righting force moment, when the encounter angle is 240°, the maximum righting arm of the ship is 2.077 m; when the encounter angle is 0°, the maximum righting arm of the ship is 1.94 m; under the two conditions, the difference of the maximum righting arm value is 7.06%. At the stage of applying the reverse righting force moment, when there is no wave, the maximum reverse righting arm of the ship is − 1.609 m; when the encounter angle is 240°, the maximum reverse righting arm of the ship is − 1.593 m; under the two conditions, the difference of the maximum reverse righting arm value is 1%.

At the early stage of the uprighting process, the working condition of a large righting force moment is needed, and at the later stage, a large righting force moment is also needed to stabilize the ship. At the early stage of the uprighting process, the working condition of a small righting force moment is needed, and at the later stage of the uprighting process, a small righting force moment is also needed to stabilize the ship.

4.2 Righting force and righting force moment

Tables 2 and 3 show the righting force and righting force moment of the hull under the calm sea environment and at different angles of wave direction, respectively. During the uprighting process, the righting force is continuously applied to reduce the inclination angle of the hull. It should be noted that at the later stage of uprighting, excessive righting force was applied, resulting in the deviation of the hull from the equilibrium position.

Table 2 Righting force of the hull under different environments (kN)
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Table 3 Righting force moment of the hull under different environments (kN m)
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The calculation results in Table 2 show that the righting force is the smallest under the action of 0° wave direction, which is 400 N smaller than that under the calm sea environment. Under the action of 180° and 240° wave directions, the maximum righting force is 42,766.8 kN. The maximum difference of the maximum righting force in different wave environments is 1 kN. In the forward and aft waves, the longitudinal inclination of the ship varies greatly, which leads to the obvious difference in the floating state and the righting moment. The wave in the aft direction of the ship requires a large righting force.

The calculation results in Table 3 show that in the uprighting process, under the action of 0° wave direction, the righting force moment is the smallest, 549.5 kN m less than that under a calm sea environment. Under the action of 240° wave direction, the maximum righting force moment is 111,002.1 kN m. The maximum difference of the maximum righting moment in different wave environments is 1177.5 kN m. When waves in the direction of 180° and 240°, a large righting force moment is required to balance the hull, and the maximum righting moment in other environments is less than that in a calm sea environment.

The maximum righting force and the maximum righting force moment under the first three conditions are relatively large. Especially in the waveless sea environment, the uprighting process needs a large righting force and righting force moment. When the wave encounter angle is at 60°, the required righting force and the righting force moment are the largest. In practical work, it should be avoided to right a capsized ship in this environment.

The maximum righting force and the maximum righting force moment of the bow wave environment are relatively small, while the maximum righting force and the maximum righting force moment in the calm sea environment are not relatively small.

5 Calculation of shear force, bending moment and torque

While designing the righting scheme of a capsized ship, we should not only pursue reducing the righting force and righting force moment but also pay attention to reducing the damage to the hull. By analyzing the maximum shear force value, maximum bending moment value and maximum torque value of the hull under various working conditions, the stress changes of the hull during the uprighting process can be predicted, which is convenient for arranging the salvage construction workshop in advance. In order to express the force range of all parts of the hull in the uprighting process, the shear force, bending moment and torque of each part of the hull are calculated and compared in detail.

5.1 Shear force calculation

Shear force is the result of ship gravity, buoyancy, wave force and righting force acting directly on the hull. Figure 3a–g shows the maximum and minimum shear force value curves under each working condition, as well as the difference curves between the maximum and minimum shear force value under each working condition (Fig. 3h). In the figures, the horizontal axis is the longitudinal coordinate of the hull, and the vertical axis is the shear force value of the hull. By referring to the following curves, we can judge whether it is suitable to right the capsized ship in the current environment.

Fig. 3
figure 3

The maximum and minimum shear force value curves

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It should be noted that the peaks or troughs in Figure a–g are only relatively large shear force values at a certain position, which is not necessarily a dangerous position.

As can be seen from Fig. 3a–g, in the environment without waves and with an encounter angle at 300°, the shear force value of the capsized ship changes little.

In Figure a–c, the maximum shear force gradually approaches the middle of the ship from the bow position. In Figure d–f, the maximum shear force gradually approaches the stern position. In Figure g, the maximum shear force appears near the bow and stern position.

In Figure a–d, the minimum shear force value is near the stern. In Figures e–f, the minimum shear force value is near the bow. In Figure g, the minimum shear force value appears near the middle of the ship.

The hull position corresponding to the maximum value of Figure h is not the hull position corresponding to the crest of the maximum value curve in Figures a–g. When it is in a calm sea environment and the encounter angle is at 60° and 120°, the hull position corresponding to the maximum value of Figure h is the hull position corresponding to the trough of the minimum value curve in Figure a–g.

In the uprighting process, the change of shear force value at each position of the hull varies under different conditions. Attention should be paid to the change range of shear force value at any positions of the hull so as not to exceed the allowable value. To timely check the righting force, the appropriate wave environment should be selected or the heading angle of the capsized ship should be adjusted before the righting plan is made.

5.2 Bending moment calculation

Figure 4a–g shows the maximum and minimum bending moment value curves under each working condition, as well as the difference curves between the maximum and minimum bending moment value under each working condition (Fig. 4h). In the figures, the horizontal axis is the longitudinal coordinate of the hull, and the vertical axis is the bending moment value of the hull. The data in the following figures can be used to judge whether the bending moment value at each position exceeds the allowable value.

Fig. 4
figure 4

The maximum and minimum bending moment value curves

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It can be seen from Fig. 4 that the bending moment value of the capsized ship is relatively small in the calm sea environment. The bending moment in the environment with an encounter angle at 300° is much smaller than that in other wave environments. The variation trend of the maximum and minimum value curves is inconsistent under various working conditions.

The overall trend of the curves in Fig. 4b and e, c and f, d and g is opposite, and the wave direction is also opposite.

In Fig. 4b and c, the hull is subjected to a large positive bending moment. In Fig. 4 e and f, the hull is subjected to a large negative bending moment. In Fig. 4a, d and g, the hull is subjected to both positive and negative bending moments. From Fig. 4h, the difference between the maximum and the minimum bending moments is concentrated in the middle of the hull.

5.3 Torque calculation

Generally, ships are paid more attention to the calculation of shear force and bending moment in normal operation. However, the role of torque under the action of external forces should be properly considered. When designing the righting scheme of the capsized ship, the hull torque value must be carefully analyzed to avoid accidents.

Figure 5 a–g shows the maximum and minimum torque value curves under each working condition, as well as the difference curves between the maximum and minimum torque value under each working condition (Fig. 5 h). In the figures, the horizontal axis is the longitudinal coordinate of the hull, and the vertical axis is the torque value of the hull.

Fig. 5
figure 5

The maximum and minimum torque value curves

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There are maximum positive torque and minimum negative torque values under all working conditions, and the positive torque values are not large, while the negative torque values under C and D working conditions are relatively large. At the beginning of the uprighting operation, most of the hull positions have positive torque values, and only a few positions in the middle and aft of the hull have negative torque values.

Under A-E conditions, the minimum negative torque value corresponding to the longitudinal coordinates of the ship is 80.3, which also corresponds to the positions of the righting force points and the superstructure. Under F and G conditions, the minimum negative torque value appears at the longitudinal coordinates of 54.267 and 38.647, and the torque value is larger than that under other conditions.

In Figure h, there is a large difference between the maximum torque and the minimum torque under each working condition in the middle positions near the aft positions, which is related to the positions of the righting force points and the position of the superstructure.

In the uprighting process, the hull torque distribution under each working condition is different.

Under working condition A, after the uprighting operation starts, all the other positions of the hull show negative torque values except for the bow part. As the ship’s heel angle decreases, most positions of the ship are subjected to positive torque when the heel angle is close to 75°.

Under working condition B, after the uprighting operation starts, all positions on the hull except the bow show negative torque values. As the heel angle decreases to 135°, the bow position of the hull becomes positive torque. When the heel angle is close to 105°, the front half of the hull is subjected to positive torque. When the heel angle is close to 75°, all positions except the stern positions of the hull are affected by the positive torque. The hull is subjected to positive torque when the hull heel angle is from 75° to 0°.

Under condition C, after the uprighting operation starts, all the other positions of the hull show negative torque values except for the bow part. During the uprighting process, the influence of negative torque on the position of the hull near the bow gradually increases. When the hull heel angle decreases to 140°, the bow positions are subjected to positive torque. The bow part subjected to positive torque increases gradually. When the heel angle is close to 65°, the ship head positions are subjected to positive torque. When the heel angle is close to 55°, all the positions except the stern positions are affected by positive torque.

Under working condition D, after the uprighting work starts, all positions of the hull become negative torque as the ship’s heel angle decreases to 80°. The head position of the hull is subjected to positive torque. When the ship’s heel angle is close to 55°, all the positions except the stern are affected by positive torque.

Under working condition E, after the uprighting work starts, all positions of the hull become negative torque as the ship’s heel angle decreases to 80°. The other positions are subjected to negative torques except for the positive torque at the bow and stern of the ship. As the heel angle decreases, there is more and more positive torque acting on the middle part of the hull at both ends, and finally, only the middle part of the hull suffers the negative torque.

Under working condition F, after the uprighting work starts, the bow is subjected to positive torque. As the heel angle decreases gradually, the ship gradually receives the positive torque from bow to midship. At the heel angle of 80°, the rear part of the ship suffers the negative torque. During the heel angle from 80° to 0°, only the middle of the hull is subjected to negative torque.

Under working condition G, after the uprighting work starts, all positions of the hull except the bow are subjected to negative torque. With the decrease in the ship’s heel angle, the torque from the bow position to the middle position of the ship gradually becomes positive torque. When the heel angle of the hull is close to 80°, all positions of the hull are subjected to positive torque.

During the process of uprighting, the longitudinal distribution of torque is not always in the same direction, resulting from the combined action of hull gravity, buoyancy and righting force. The influence of the wave direction on the torque distribution is obvious, and the part near the stern of the hull is more susceptible to the effect of negative torque. When the heel angle of the hull reaches 80 degrees, the positive moment of the hull increases rapidly, which is consistent with the changing trend of the stability curve.

6 Conclusion

To study the influence of waves on the uprighting process of a capsized ship, a mechanical model of righting force is established based on previous literature. By simulating the uprighting process of the capsized ship in different environments, the conclusions can be drawn as follows.

The shear force of the ship in the calm sea environment and wave environment is quite different. When the wave encounter angle is at 0°, the shear force of some positions of the ship is 2–3 times that of the calm sea environment. Therefore, it is necessary to consider the effect of waves on the ship in the process of righting the capsized ship. And, the shear force at the same position of the ship may vary greatly with different wave angles. When the wave encounter angle is at 0°, the shear force value of some positions of the ship is 3–4 times that of the 300° environment. Therefore, adjusting the angle between the ship and the wave and the direction of righting the capsized ship can effectively reduce the damage of the wave to the ship.

In the uprighting project, the wave encounter angle will significantly affect the bending moment distribution, and the bending moment value of individual locations can be changed by more than 200%. In the environment with encounter angles at 0°, 60°, 180° and 240°, the variation trends of the maximum and the minimum bending moment curves of the hull are consistent, and the variation range of the bending moment value is larger than that in other cases. Therefore, the encounter angle should be avoided from being close to the above angle, or the encounter angle of the hull should be avoided from being close to the above angle for a long time.

When the encounter angles are at 60° and 120°, the variation range of hull torque value is large during the uprighting process. The negative torque generated near the superstructure is large, while the positive righting force moment needed to be applied in the uprighting process is small. It shows that the local position of the ship under these two conditions makes it easy to produce a large torque value, but the total torque value of the ship is relatively small. When the encounter angle is at 300°, the negative torque value of the ship changes little, but it needs to apply a relatively large righting force moment during the uprighting process. Under this condition, the righting force moment is not large, and the position of the righting force point should be changed.