Formulation of Ice Resistance in Level Ice Using Double-Plates Superposition

Abstract

The estimation of ship resistance in ice is a fundamental area of research and poses a substantial challenge for the design and safe use of ships in ice-covered waters. In order to estimate the ice resistance with greater reliability, we develop in this paper an improved Lindqvist formulation for the estimation of bending resistance in level ice based on the superposition of double-plates. In the developed method, an approximate model of an ice sheet is firstly presented by idealizing ice sheeta as the combination of a semi-infinite elastic plate and an infinite one resting on an elastic foundation. The Mohr–Coulomb criterion is then introduced to determine the ice sheet’s failure. Finally, an improved Lindqvist formulation for estimation of ice resistance is proposed. The accuracy of the developed formulation is validated using full-scale test data of the ship KV Svalbard in Norway, testing the model as well as the numerical method. The effect of ice thickness, stem angle and breadth of bow on ship resistance is further investigated by means of the developed formulation.

1. Introduction

Global warming may cause the Arctic ice cover to shrink, i.e., to reduce its extent and its average thickness. This climate trend promotes increasing scientific and commercial activities in the Arctic, which are primarily related to offshore oil and gas exploration and production [1]. The sea transportation from Northern Europe to the Far East will certainly increase with the possibility of year-round navigation in the near future. It is known that the presence of sea ice is the main factor hindering operations in the Arctic, and the determination of ice resistance is more complicated than open water resistance due to the characteristics of ice properties and icebreaking phenomena [2]. The estimation of ship resistance in ice is a fundamental area of research and poses a substantial challenge for the design and safe use of ships in ice-covered waters. As a basic component in all ice interactions, level ice load is often studied. Most of the available methods used to determine the ship resistance in level ice can generally be categorized into three groups: empirical models [3,4,5,6], numerical models [7,8,9,10] and experimental models [11,12,13]. All of these models help optimize the design and operation of ships in Arctic waters. In particular, the empirical models and their variants are popularly used in the early design stage for a ship when choosing a hull form and a propulsion system that give the best possible performance in terms of global resistance, available thrust, maximum speed and fuel consumption. The empirical methods may also be used during the operation of the ship as a guiding tool to help the crew optimize their route.

In the past decades, a great deal of attention has been paid to the development of empirical models for predicting ice resistance of ships based on experience and observation of ships in service or model-scale tests. In the literature, several famous and commonly used models exist, including semi-analytical and purely empirical variants, e.g., Lindqvist [5], Keinonen et al. [14], Riska et al. [15,16] and Spencer and Jones [17]. Lindqvist described the ice-breaking process in a physical way by identifying and simplifying the problem, and then presented a relatively straightforward empirical formula in which the model is a function of the main dimensions of the ship, hull form, ice thickness, ice strength and friction. In [5], Lindqvist did not include all resistance components to describe the ice-breaking process, but chose the components that are generally accepted as dominating; thus ice resistance was finally divided into three components: crushing, bending and submersion, and bending was identified as the main component in the resistance. A modification of the Lindqvist empirical model was proposed by Riska et al., who based the formulation of ice resistance on the assumption that the resistances in open water and ice can be separated and superimposed to obtain the total resistance [13,16]. Their model can be used for calculating resistance in level ice, with the velocity-related empirical coefficients derived from the full-scale data of a number of ships in the Baltic Sea. The concept of energy consideration was also introduced to estimate the ice resistance of a ship. Based on experience from a number of full-scale experiments in the Baltic Sea, Keinonen et al. presented a formulation to estimate ice resistance for a ship at low velocity, in which the resistance is a function of the main dimensions and environment of the ship, the thickness of ice and snow, the ice bending strength, and ice salinity and porosity. Spencer and Jones investigated methods of predicting ice resistance and proposed a component-based ice resistance prediction method [15], which assumed that four different resistance components occur during an ice breaking process. These forces are open-water resistance, buoyancy resistance, clearing resistance and breaking resistance [18]. It is well recognized that a good understanding of the process of ship–ice interaction is essential for developing reliable empirical models [18,19,20], and it is necessary to develop new ice resistance formulations with greater reliability as the knowledge of the physics behind ice-breaking advances and more experience has been gained [21,22,23].

The main contribution of this study is the development of a more reliable formulation for the estimation of the bending component in the Lindqvist formulation for calculating ice resistance. In most of the popular formulations, e.g., in the Lindqvist formulation, the bending resistance is estimated with the assumption that the ice sheet fails once its bending stress reaches its bending strength. However, besides bending failure, the ice plate also exhibits crushing failure and shear failure due to the effect of the bow. Moreover, due to the fact that the compressive strength of ice is far higher than its bending strength, the failure of the ice sheet may change from bending failure to crushing failure when subjected to a combination of moment and horizontal pressure [24,25,26]. Therefore, the ice-breaking resistance, when calculated from the existing formulations, is in fact underestimated and may lead to an inappropriate scheme in an early design stage for a ship [27,28]. In order to estimate the ice breaking resistance, before ice cracks are formed, the ice field is idealized as a semi-infinite plate on an elastic foundation, and the local bending failure of each wedge-shaped ice block in the bow area is considered as the bending failure of the ice plate as a whole. The radial cracks are simulated by the failure of an infinite elastic-based plate, and the circumferential cracks are simulated by the failure of a semi-infinite elastic based-plate [7,10,29,30]. The Mohr–Coulomb criterion is then introduced to determine the failure of the ice sheet by considering the fact that the compressive strength of sea ice is far higher than its bending strength. In the process of ice breaking, the ice plate is essentially a two-directional plate. According to the linear elastic assumption, the ice sheet can be decomposed into a semi-infinite elastic foundation plate along the heading direction and an infinite elastic foundation plate perpendicular to the heading direction. The accuracy of the developed ice resistance formulation is validated by full-scale test data of the ship KV Svalbard in Norway [31]. The effect of ice thickness and stem angle on ships’ ice resistance is finally investigated based on the developed formulation.

The remainder of this paper is organized as follow. The interaction between ship and ice is briefly introduced in Section 2. In Section 3, the establishment of a new formulation for calculating bending resistance is presented, and the procedure for determining total ice-breaking resistance is summarized in Section 4. In Section 5, the validity of the new contribution is discussed by comparing the results of our new formulation with full-scale experiments, further clarifying that the new model has higher accuracy than the classical Lindqvist model; this is followed by a comprehensive discussion of the effect of ice thickness, angle of ship bow and ship breadth on the ice resistance in Section 6.

2. Interaction between Ship and Ice

A brief introduction of the ice-breaking process is given in this section. The interaction between a ship and sea ice is a complex process. It depends strongly on the ice conditions, the hull geometry and the relative velocity between the ship and the ice. Kotras et al. [6] and more recently Valanto [32] presented an overview of ship–ice interaction, where the interaction process was divided into several phases: breaking, rotating, sliding and clearing. When a ship advances in a large ice floe, the ice forces on the hull will increase by increasing the penetration depth up to a limit set by the failure of the ice sheet. The failure of the ice sheet may occur in different modes: pure crushing, bending, buckling, shearing, splitting or mixed-mode, where two or more of the failure modes are active at the same time. The ice properties and thickness, the hull characteristics and the relative drift velocity control the ice sheet’s mode of failure.

The ice-breaking phase begins with a localized crushing of the free ice edge at the contact zone. The ice forces on the hull will increase by increasing the penetration depth up to a limit set by the failure of the ice sheet. That causes the ice sheet to deflect and the bending stresses to build up; when the accumulated force is high enough, bending failure of the ice at a certain distance in front of the contact surface is initiated [33]. Finally, the failure of the ice sheet may occur in different modes: pure crushing, bending or buckling. Upon the formation of broken ice pieces, the pieces start rotating downward until they are parallel to the hull. During the rotation, the broken ice pieces push the previously broken ice further down. In the final phase of interaction, the broken ice floes slide along the hull until they are cleared away to the sides, to the wake or to the propellers, where they are milled.

As a conceptually simple and operationally straightforward empirical formulation, the Lindqvist formula is capable of describing the ice-breaking process and offering a reliable estimation of ice resistance, and it is a widely used tool for the early design of ship structures. The main components of Lindqvist formulation include bending, crushing and submersion resistance, and the explicit Lindqvist formulation is given as follows:

$$R_{\mathrm{ice}}=(R_{\mathrm{c}}+R_{\mathrm{b}})(1+1.4\frac{V}{\sqrt{gh}})+R_{\mathrm{s}}(1+9.4\frac{V}{\sqrt{gL}})$$

(1)

where $R_{\mathrm{ice}}$ is the total ice resistance, $R_{\mathrm{c}}$ is the crushing resistance, $R_{\mathrm{s}}$ is the resistance due to submersion and $R_{\mathrm{b}}$ is the bending resistance, which is an innegligible proportion.

In the Lindqvist formulation, the determination of bending resistance resorts to the assumption that the ice plate is a one-directional plate. Nevertheless, This assumption is only valid for the determination of crushing resistance and resistance due to submersion, while it is unsuitable for determining the bending resistance component. Since the contact area between the ship’s stem and the ice floe cannot be fully considered with this assumption, a poor approximation of the bending component and thereby a poor estimation of ice resistance may result. In addition, only flexural failure has been considered as the failure mode of ice in the determination of bending resistance in the Lindqvist model. However, besides bending failure, the ice sheet also exhibits crushing failure and shear failure due to the effect of the bow. It is urgent to develop a new formulation to estimate the bending component in light of the evident drawbacks that exist in the aforementioned method for the estimation of bending resistance.

3. A New Formulation for Estimation of Bending Resistance

In this section, a modified empirical formulation in the framework of the Lindqvist model for better prediction of ship resistance in ice is developed by idealizing the ice sheet as a two-directional elastic plate. Our approach considers the bending failure of the ice sheet that occurs in all bow regions, and this bending has been approximated by that of an infinite elastic plate under the vertical component of the contact force. In addition, the Mohr–Coulomb criterion is introduced to determine the failure of the ice sheet by considering the fact that the compressive strength of ice is far higher than its bending strength.

3.1. The New Model for Estimating Bending Resistance

It has been mentioned that the bending failure of the ice sheet generally dominates over the other modes of failure [27,34,35,36,37]. Based on this hypothesis, it became feasible to use the theory of an elastic plate resting on an elastic foundation. Furthermore, by virtue of the principle of superposition, the ice sheet is idealized as the combination of two elastic plates with equal thickness resting on an elastic foundation, as shown in Figure 1. One is the semi-infinite elastic plate whose width equals that of the ship, and the other is an infinite elastic plate whose width depends on the breaking length or the characteristic length of the ice sheet. Since the remainder of the ice sheet is outside the area of fracture length, the effect of the remainder of the ice sheet on bending resistance is limited and can be approximated as the boundary constraint. The reaction force of the foundation that supports these two elastic plates is used to approximate the buoyancy force of the ice sheet. We emphasize that radial failure and circumferential failure will be formed in the ice sheet since the bending failure most probably occurs during the ship–ice interaction process. By setting the semi-infinite elastic plate along the direction of the ship advancing, the bending failure of this plate naturally simulates the circumferential failure of the ice sheet.

Similarly, by setting the infinite elastic plate perpendicular to the direction of the ship advancing, the bending failure of this plate can be used to simulate the radial failure of the ice sheet. Note that the compatibility of vertical displacement in the two elastic plates is held in the whole process, as depicted in Figure 1b. The two components H and P have the relation:

$$H=\zeta P$$

(2)

where $\zeta$ denotes the proportionality coefficient (or relation coefficient) of the horizontal component and a vertical component, which can be determined by:

$$\zeta =\frac{sin\varphi +\mu cos\varphi }{cos\varphi -\mu sin\varphi }$$

(3)

where $\varphi$ is the angle of the ship bow, which is shown in Figure 2. It is straightforward that the bending resistance $R_{\mathrm{b}}$ in ice, or equivalently the horizontal resistance of the ice sheet to the ship stem, is actually the reacting force of the horizontal component H shown in Figure 1b. Thus, the estimation of bending resistance $R_{\mathrm{b}}$ in ice is then transformed to the determination of the horizontal component H.

On the other hand, the vertical distributed load P that is distributed on the ice sheet posed by the ship stem can also be decomposed into $P_1$ and $P_2$, as shown in Figure 3. It is clear that $P_1$ is distributed on the edge of the semi-infinite elastic plate, as shown in Figure 4, and that $P_2$ is distributed on the area of the middle of the semi-infinite elastic plate, as shown in Figure 5. Once the vertical component P in Figure 3 could be determined on the basis of the compatibility of vertical displacement in the two elastic plates together with the failure criterion of the ice sheet, the horizontal component H and thereby the bending resistance $R_{\mathrm{b}}$ in ice can be obtained by using the relation given in Equation (2).

3.2. Compatibility between the Semi-Infinite Elastic Plate and Infinite Elastic Plate

3.2.1. Computation of Semi-Infinite Elastic Plate

Figure 4 describes the semi-infinite elastic plate resting on an elastic foundation that subjects to the distributed load $P_1$. The width and thickness of the plate are denoted by B and h, respectively. The vertical load $P_1$ is distributed on the edge, i.e., the origin $x=0$, of the semi-infinite elastic plate. Under the vertical load $P_1$, the general solution [38,39] of the deflection curve of the semi-infinite elastic plate is determined as:

$$z_{\mathrm{A}}=e^{{\beta }_{\mathrm{A}}x}\left(A_{1}cos{\beta }_{\mathrm{A}}x+A_{2}sin{\beta }_{\mathrm{A}}x\right)+e^{-{\beta }_{\mathrm{A}}x}\left(A_{3}cos{\beta }_{\mathrm{A}}x+A_{4}sin{\beta }_{\mathrm{A}}x\right)$$

(4)

where $A_1$, $A_2$, $A_3$ and $A_4$ are integration constants to be determined, and the coefficient ${\beta }_{\mathrm{A}}$ is computed as:

$${\beta }_{\mathrm{A}}=\nu {\left(\frac{{\rho }_{\mathrm{w}}g}{4D}\right)}^{\frac{1}{4}}$$

(5)

It is also known that the moment, M, and deflection of the semi-infinite elastic plate have the relation:

$$M=-D\frac{d^2{z}_A}{d{x}^2}$$

(6)

where the bending stiffness of the plate, D, can be determined as:

$$D=\frac{E{h}^3}{12\left(1-{\nu }^2\right)}$$

(7)

The integration constants $A_1$, $A_2$, $A_3$ and $A_4$ can be determined from the boundary condition that the deflection $z_{\mathrm{A}}$ in Equation (4) and the moment M in Equation (6) tend to be zero with the increase of the distance from the origin; together with the initial state the shear force of the edge of the semi-infinite foundation plate is equal to $P_1$. By substituting the obtained integration constants into Equation (4), the deflection of the semi-infinite elastic plate is then formulated as:

$$z_{\mathrm{A}}=\frac{P_1{e}^{-{\beta }_{\mathrm{A}}x}cos{\beta }_{\mathrm{A}}x}{2{\beta }_{\mathrm{A}}^{3}D}$$

(8)

and the moment of the semi-infinite elastic plate is accordingly derived as:

$$M_{\mathrm{A}}=-\frac{P_1}{{\beta }_{\mathrm{A}}}{e}^{-{\beta }_{\mathrm{A}}x}sin{\beta }_{\mathrm{A}}x$$

(9)

3.2.2. Computation of Infinite Elastic Plate

Figure 5 describes an infinite elastic plate resting on an elastic foundation that is subjected to the distributed load $P_2$. The width of the plate equals the characteristic length of the ice sheet, $l_{\mathrm{c}}$, which is computed as:

$$l_{\mathrm{c}}={\left(\frac{E{h}^3}{12{\rho }_{\mathrm{w}}g\left(1-{\nu }^2\right)}\right)}^{\frac{1}{4}}$$

(10)

and the thickness of the plate is h, as shown in Figure 5. The vertical load $P_2$ is distributed on a half-circle area at the middle of the infinite elastic plate, and B is the diameter of the loaded half-circle area. The vertical deflection of the infinite elastic plate is represented by using the middle-layer section in the o-xy plane, and the deflection of all points in the section is assumed to be the same. Under the micro-load $qdy$, the deflection of the infinite elastic plate, at the origin $x=0$, can be derived as:

$$dz=\frac{qdy}{8{\beta }_B^{3}D}{e}^{-{\beta }_{\mathrm{B}}y}\left(cos{\beta }_{\mathrm{B}}y+sin{\beta }_{\mathrm{B}}y\right)$$

(11)

where $q=\frac{P_2}{B}$, and ${\beta }_{\mathrm{B}}={\beta }_{\mathrm{A}}=\nu {\left(\frac{{\rho }_{\mathrm{w}}g}{4D}\right)}^{\frac{1}{4}}$. By further integrating Equation (11) along the diameter of the loaded area, B, the deflection of the infinite elastic plate under the distributed load $P_2$, at the origin $x=0$, is formulated as:

$$z_{\mathrm{BO}}=2{\int }_0^{B/2}\frac{qdy}{8{\beta }_{\mathrm{B}}^{3}D}{e}^{-{\beta }_{\mathrm{B}}y}\left(cos{\beta }_{\mathrm{B}}y+sin{\beta }_{\mathrm{B}}y\right)=\frac{P_2}{4D{\beta }_{\mathrm{B}}^{4}B}\left(1-e^{-{\beta }_{\mathrm{B}}\frac{B}{2}}cos{\beta }_{\mathrm{B}}\frac{B}{2}\right)$$

(12)

Further, on the basis of the relationship between the moment and deflection of the plate, the moment of the section at $y=0$ is finally determined as:

$$M_{\mathrm{B}}=\frac{P_2}{2{\beta }_{\mathrm{B}}^{2}B}{e}^{-{\beta }_{\mathrm{B}}\frac{B}{2}}sin{\beta }_{\mathrm{B}}\frac{B}{2}$$

(13)

3.2.3. Superposition of Semi-Infinite Elastic Plate and Infinite Elastic Plate

According to the independence principle of motion, the vertical deflections of the two different elastic plates, $z_{\mathrm{A}}$ in Equation (8) and $z_{\mathrm{BO}}$ in Equation (12), are independent under the distributed loads $P_1$ and $P_2$, respectively, while their displacement in the vertical direction should be consistent. The compatibility of vertical displacement in the two elastic plates should be held in the superposition, i.e., the vertical deflection of the semi-infinite elastic plate, $z_{\mathrm{A}}$, should be equal to that of the infinite elastic plate, $z_{\mathrm{BO}}$, at the origin. With this condition, the relation between $P_1$ and $P_2$ is derived as:

$$f=\frac{P_1}{P_2}=\frac{(1-e^{-{\beta }_{\mathrm{B}}\frac{B}{2}}cos{\beta }_{\mathrm{B}}\frac{B}{2})}{2{\beta }_{\mathrm{B}}B}$$

(14)

3.3. Failure of the Ice Sheet

It has been mentioned that ice sheets exhibit complex mechanical properties and various failure modes during the ship–ice interaction process [40,41,42,43]. In this paper, the Mohr–Coulomb failure criterion, instead of the commonly used maximum stress criterion, is adopted to determine the failure of the sea ice, as this criterion describes the pressure characteristics and thus can reveal the difference between tensile strength and compressive strength of the ice sheet [44,45]. From the Mohr–Coulomb criterion, the failure results both from normal stress and shearing stress, and the maximum principal stress ${\sigma }_1$ and minimum principal stress ${\sigma }_3$ have the following relation:

$${\sigma }_3=-f_{\mathrm{c}}+m{\sigma }_1$$

(15)

where $f_{\mathrm{c}}$ represents compressive strength and the coefficient m is the ratio of compressive strength to tensile strength, i.e., $m=f_{\mathrm{c}}/f_{\mathrm{t}}$, where $f_{\mathrm{t}}$ denotes tensile strength.

Radial cracks and circumferential cracks will be formed in the ship–ice interaction process. Because the ice plate is regarded as the superposition of two-directional elastic plates, the generation of circumferential and radial fractures is also regarded as the failure characteristics of each one-directional plate. The radial cracks occur in the middle of an infinite elastic plate, while circumferential cracks occur on a section whose moment is at a maximum over the semi-infinite elastic plate.

On the other hand, since horizontal compression stress to the ice sheet will be induced by the ship stem, it can be concluded that the failure of the ice sheet occurs at the junction of radial cracks and circumferential cracks. The stress on the element body that originates from the lower surface of the damaged ice sheet, as shown in Figure 6, is then calculated as:

$${\sigma }_{\mathrm{A}}=\frac{\left|M_{\mathrm{A}}\right|}{W_{\mathrm{A}}},{\sigma }_{\mathrm{B}}=\frac{\left|M_{\mathrm{B}}\right|}{W_{\mathrm{B}}},{\sigma }_{\mathrm{H}}=\frac{H}{L\mathrm{h}}$$

(16)

where ${\sigma }_A$ and ${\sigma }_B$ represent the bending stress under moments $M_{\mathrm{A}}$ and $M_{\mathrm{B}}$, respectively, ${\sigma }_{\mathrm{H}}$ represents the compression stress under horizontal load H, and $W_{\mathrm{A}}$ and $W_{\mathrm{B}}$ stand for the bending modulus of the section of the semi-infinite elastic plate and the infinite elastic plate, respectively. According to the stress on the micro-elements under spatial stress conditions from Figure 5, the principal stress on the micro-elements that from the ice sheet is finally determined as:

$${\sigma }_1=\frac{-{\sigma }_{\mathrm{A}}-{\sigma }_{\mathrm{H}}+{\sigma }_{\mathrm{B}}}{2}+\frac{1}{2}\sqrt{{\left(-{\sigma }_{\mathrm{A}}-{\sigma }_{\mathrm{H}}-{\sigma }_{\mathrm{B}}\right)}^2}={\sigma }_{\mathrm{B}}$$

(17)

$${\sigma }_3=\frac{-{\sigma }_{\mathrm{A}}-{\sigma }_{\mathrm{H}}+{\sigma }_{\mathrm{B}}}{2}-\frac{1}{2}\sqrt{{\left(-{\sigma }_{\mathrm{A}}-{\sigma }_{\mathrm{H}}-{\sigma }_{\mathrm{B}}\right)}^2}=-{\sigma }_{\mathrm{A}}-{\sigma }_{\mathrm{H}}$$

(18)

3.4. Determination of Bending Resistance

On the basis of compatibility of the semi-infinite elastic plate and infinite elastic plate together with the failure criterion of the ice sheet, the bending resistance can be derived by substituting.

$$-{\sigma }_A-{\sigma }_H=-f_c+m{\sigma }_B$$

(19)

$$f_c=\frac{f}{f+1}\frac{6P{e}^{{\beta }_{A}x}sin({\beta }_{A}x)}{B{h}^2{\beta }_A}+\frac{\zeta P}{Bh}+\frac{m}{f+1}\frac{3P{e}^{-\frac{B}{2}{\beta }_B}sin(\frac{B}{2}{\beta }_B)}{l_c{h}^2{{\beta }_B}^{2}B}$$

(20)

$$R_{\mathrm{b}NEW}=\frac{\zeta f_{\mathrm{c}}}{U_1{e}^{-\frac{\pi }{4}}sin\frac{\pi }{4}+U_2+U_3}$$

(21)

where $U_1=\frac{6f}{(f+1)B{h}^2{\beta }_{\mathrm{A}}},U_2=\frac{\zeta }{Bh}$ and $U_3=\frac{3m{e}^{-{\beta }_B\frac{B}{2}}sin{\beta }_B\frac{B}{2}}{(f+1)l_{\mathrm{c}}{h}^2{\beta }_{\mathrm{B}}^{2}B}$.

Similar to the bending resistance in the Lindqvist model in Equation (1), this new model also includes parameters that are easily accessible, e.g., the main dimensions and hull form of a ship, bending strength and thickness of the ice sheet, friction etc. The Mohr–Coulomb failure criterion is used to judge the failure of ice during the interaction between ship and ice, which reflects the difference between the tensile strength and compressive strength of sea ice. Furthermore, the compressive strength of the ice sheet, which is an additional important parameter, is added to better approximate the failure mode of sea ice. Once the bending resistance is determined by means of the proposed new formulation, the total ice resistance can be readily obtained by substituting the $R_{bNEW}$ in Equation (21) into Equation (1), i.e.,

$$R_{\mathrm{ice}}=(R_{\mathrm{c}}+R_{\mathrm{b}NEW})(1+1.4\frac{V}{\sqrt{gh}})+R_{\mathrm{s}}(1+9.4\frac{V}{\sqrt{gL}})$$

(22)

4. A Procedure for Calculation of Ice Resistance

In this section, the procedure for the estimation of ice-breaking resistance is summarized. We begin with the determination of parameters including the bending strength of the ice sheet ${\sigma }_{\mathrm{b}}$, the thickness of the ice sheet h, the friction coefficient $\mu$, the stem angle $\varphi$, the waterline entrance angle $\alpha$, the angle between the normal of the surface and a vertical vector $\psi$, the seawater density ${\rho }_{\mathrm{w}}$, the density of ice ${\rho }_{\mathrm{i}}$, Young’s modulus E, the Poisson ratio of sea ice $\nu$, the breadth B, the draft T, the length L, and the speed V of the ship. Secondly, in order to obtain the total ice-breaking resistance, the three main components, i.e., the crushing resistance, the resistance due to submersion and the bending resistance have to be determined. Since the ice sheet is regarded as a one-directional plate, the accuracy of the resulting bending resistance is not exact enough. On the other hand, the resulting bending resistance calculated by the proposed new formulation is more accurate in comparison to the real one due to the fact that the two failure modes (bending failure and compression-bending failure) are considered simultaneously instead of only considering a single bending failure. Thus, in our current work the new formulation is chosen to determine the bending resistance. In addition, due to the excellent performance of the Lindqvist method for estimating crushing as well as submersion resistance, the determination of these two components continues to be used in our procedure, i.e.,

$$R_{\mathrm{c}}=0.5{\sigma }_{\mathrm{b}}{h}^2\frac{tan\varphi +\frac{\mu cos\varphi }{cos\psi }}{1-\frac{\mu sin\varphi }{cos\psi }}$$

(23)

$$R_{\mathrm{s}}=({\rho }_{\mathrm{w}}-{\rho }_{\mathrm{c}})ghB(T\frac{B+T}{B+2T}+\mu (0.7L-\frac{T}{tan\varphi }-\frac{B}{4tan\alpha }+Tcos\varphi cos\psi \sqrt{\frac{1}{{sin}^2\varphi }+\frac{1}{{tan}^2\alpha }}))$$

(24)

where the parameters in Equations (23) and (24) are detailed above.

The proposed methodology is capable of offering a reliable and straightforward estimation of ice resistance by using easily accessible parameters without prohibitive cost caused by precision sensors; thus it is quite suitable for practical implementation. Moreover, it is worth mentioning that the proposed method can be used to investigate the height of sea ice pile-ups on the slope structures, and it can be further generalized to the analysis of ice resistance caused by the interaction between ice sheets and slope structures.

5. Validation of the Proposed Model

5.1. Full-Scale Data

In this section, the proposed model for estimating ship resistance in ice is validated usind a set of full-scale experiments on KV Svalbard in the Storfjorden Sea and the Barents Sea. KV Svalbard is a Norwegian coast guard icebreaker that operates in the Arctic [31]. The performance of KV Svalbard in ice was studied, and a set of full-scale experimental data was collected in 2007. These data include, among other things, the speed of the ship, the thickness, density, temperature and salinity of the sea ice, and the ship’s resistance in ice. For the set of tests, KV Svalbard was driven straight (with a constant heading) in a fairly uniform ice sheet.

Table 1. Data on KV Svalbard and sea ice.

Description	Symbol	Value
Ship length between perpendiculars [m]	L	89
Ship breadth [m]	B	19.1
Ship depth [m]	D	10.8
Ship draft [m]	T	6.5
Stem shape angles [deg]	$\varphi$	35
Waterline entrance angle [deg]	$\phi$	34
Hull/ice friction coefficient	$\mu$	0.15
Tensile strength of ice [kPa]	$f_{\mathrm{t}}$	500
Difference in density (water/ice) [kg/m3]	$\rho$	125

gives an insight into the important parameters of the KV Svalbard and properties of the sea ice. The data of ice resistance were collected and calculated by a monitoring system installed on KV Svalbard. The system is a prototype of a planned ice load monitoring system, which includes fiberoptic sensors, electromagnetic equipment and computational software. The value of ice resistance was measured setting fiberoptic sensors on the selected regions in the bow of KV Svalbard.

Figure 7 describes the ice resistance estimated from the new model, the Lindqvist model, and the full-scale record from the KV Svalbard, in which each point represents the obtained ice resistance that corresponds to various test cases. Furthermore,

Table 2. Experiment and calculation of ice resistance of the KV Svalbard icebreaker under different ice thicknesses.

Case	Ice Thickness (m)	Experiment Results (kN)	The New Formula (kN)		Lindqvist Formuala (kN)
			Ice Resistance (kN)	Error	Ice Resistance (kN)	Error
1	0.41	776	560	27%	405	47%
2	0.44	803	600	25%	433	46%
3	0.51	815	720	11%	517	36%
4	0.65	844	949	12%	673	20%
5	0.93	1340	1520	13%	1054	21%
6	1.04	1435	1530	6%	1058	26%
7	1.29	2081	2113	1.5%	1438	30%
8	1.33	2006	2304	14%	1563	22%
9	1.41	2312	2507	8%	1694	26%

quantitatively compares the estimated ice resistance from a different model and full-scale data for different cases. It can be seen that the ice resistance from the developed model is closer to the full-scale data when compared with that from the Lindqvist model; the average errors are 17% and 34%, respectively when the ice thickness is less than 1 m. Furthermore, the average error from the developed model is reduced to 7% when the ice thickness is larger than 1 m, whereas the average error of the Lindqvist model is 27% in this case. The accuracy of the proposed model is improved by about 18% when compared with the Lindqvist model. It can be seen from Figure 7 that the calculation results of the icebreaking resistance evaluation method in this paper are closer to the full-scale data compared with the calculation results of the Lindqvist formula.

5.2. Numerical Method

The numerical method predicting ice resistance was introduced by Su et al. [7]. In this numerical method, the coupling between continuous ice forces and ship motions is considered, and the three rigid body degree of freedom equations of surge, sway and yaw are solved by numerical integration. The numerical results were in good agreement with the ship performance data from the ice trails of AHTS IB Tor Viking II icebreaker [7,18]. Therefore, this numerical method is used to investigate the accuracy of the new formula proposed in this paper.

Table 3. The ice resistance values obtained by model test, numerical methods proposed in [7], the new formula and the Lindqvist formula.

Case No.	Model Test	Numerical Simulation		The New Formula (kN)		Lindqvist Formula (kN)
		Ice Resistance (kN)	Error	Ice Resistance (kN)	Error	Ice Resistance (kN)	Error
1	470	440	6.38%	391	16.81%	320	31.91%
2	560	530	5.36%	482	13.93%	380	32.14%
3	670	630	5.97%	585	12.69%	510	23.88%
4	720	690	4.17%	655	9.03%	560	22.22%

lists the results of the numerical method, the new formula and Lindqvist formula. As shown in Table 3, it can be concluded that the new formula improved the accuracy of the Lindqvist formula by 14%.

6. Discussions

6.1. Effect of Ice Thickness on the Ice Resistance

Ice thickness has a direct impact on the navigation safety and icebreaking performance of ships. Generally, the icebreaking resistance of ships will increase with increasing ice thickness, and the icebreaker will choose different icebreaking methods according to the different ice thicknesses. Figure 8 shows the variation of the icebreaking resistance and bending resistance of the icebreaker with varying ice thickness. The variation range of sea ice thickness h is $0.5\sim 1.5$ m, with a gradient of $0.1$ m. Other parameters of the icebreaker and ice plate are kept constant: ship speed $V=0.5$ m/s, bow angle $\varphi ={30}^{\circ }$, friction coefficient $\mu =0.04$ [46,47], sea ice bending strength ${\sigma }_{\mathrm{b}}=0.5$ MPa, compressive strength $f_{\mathrm{c}}=1.5$ MPa and tensile strength $f_{\mathrm{t}}=0.5$ MPa. As shown in Figure 8, the ice and bending resistances calculated by the new formula and Lindqvist formula increase with the ice thickness, and the resistance calculated by the new formula is always slightly higher than that of the Lindqvist formula. In addition, it can be observed that the difference between the new formula and Lindqvist formula is relatively small when the ice thickness is small, and this difference increases with the ice thickness.

6.2. Effect of Ship Bow Angle on the Ice Resistance

Ships’ icebreaking capacity mainly depends on a sturdy and sharp bow. Ships’ bow can not only break sea ice but also clear crushed ice. The bow angle of ships determines the force applied to the sea ice, which is the component force in the horizontal and vertical directions. The bow’s angle also has a decisive influence on the fracture mode of sea ice under the action of the slope. Figure 9 shows the variation of the ice resistance and bending resistance of icebreakers with the bow angle. The change range of the bow angle $\varphi$ is $20\sim {35}^{\circ }$. Other parameters of the icebreaker and ice plate are kept unchanged: ship speed $V=0.5$ m/s, sea ice thickness $h=1$ m, friction coefficient $\mu =0.04$, sea ice bending strength ${\sigma }_{\mathrm{b}}=0.5$ MPa, compressive strength $f_{\mathrm{c}}=1.5$ MPa and tensile strength $f_{\mathrm{t}}=0.5$ MPa. It can be seen from Figure 10 that the ice resistance and bending resistance increase with increasing bow angle.

6.3. Effect of Ship Breadth on the Ice Resistance

The ice resistance is not only related to the physical performance of the ice sheet, but is also influenced by the parameters of the ship. An appropriate stem shape can effectively reduce the icebreaking force acting on the ship. The stem breadth is an important parameter of the stem shape, and thus its influence is studied in this section. Figure 10 shows the variation of total ice resistance and bending resistance with the ship breadth. The ship breadth varies by $10\sim 30$ m, while the velocity is $V=0.5$ m/s, the ice thickness is $h=1$ m, the friction coefficient is $\mu =0.04$, the bending strength of the ice is ${\sigma }_{\mathrm{b}}=0.5$ MPa and the compression strength is $f_{\mathrm{c}}=1.5$ MPa. As shown in Figure 10, both the proposed formula and the Lindqvist formula provide larger predictions of resistances, and the variation of ice resistance with ship breadth is more notable than that of bending resistance.

It can also be seen from Figure 8, Figure 9 and Figure 10 that the results from the improved method is larger than that of the Lindqvist formula.This is mainly because the failure modes of ice plates in the proposed method consider not only bending failure but also extrusion and shear failure. In addition, the difference between the new formula and the Lindqvist one increases with ice thickness, and is almost irrelevant to the angle and breadth of the ship, as shown in Figure 9 and Figure 10.

7. Conclusions

In this paper, a calculation method of bending resistance based on double-plate superposition and the Mohr–Coulomb failure criterion is put forward, and then the improved Lindqvist formula of ice resistance is proposed. The reliability of the proposed evaluation method was verified by comparing with the icebreaking test data of Norway’s KV Svalbard, a model test and a numerical method. The comparison of calculation results showed that the accuracy of the proposed model in this paper is higher. The developed approximation method for ice sheets can be extended to the interaction between inclined structures and ice sheets. The evaluation method of ships’ icebreaking resistance proposed in this paper has important engineering application value, and it can provide a theoretical basis for the design of ship structures in cold regions.