1. Introduction
To meet the requirements of the maritime transportation and to reduce energy consumption, and also emission of ships and the size of certain types of merchant vessels has kept increasing. For example, 400 kDWT ultra large ore carriers have been put into service and the design of over-20000TEU ultra large container carriers is also under way. Due to the increase in ship size and the extensive use of high-strength steels as a common material in the whole ship to reduce the ship weight, the flexibility of the hull girder increases and the natural frequency of ship decreases [
1,
2], which result in wave encounter frequencies that are very close to the first order natural frequency of the hull, and consequently, the resonance between the hull and waves, i.e., springing, occurs [
3,
4]. Because of the small damping of hull structures, springing decays very slowly and the vibration stresses of high frequency remains in the ship structures, which will lead to severe structural fatigue damage. A number of researchers have studied the springing responses of large vessels theoretically and experimentally.
Based on full scale measurements on board an iron-ore carrier operating in the North Atlantic Ocean, Storhaug et al. [
5] showed that the wave-induced vibrations caused 44% of the fatigue damage. The hydroelasticity model test that was conducted by Storhaug et al. [
6] on a large ore carrier also showed that 56% of the damage was caused by springing and whipping. Drummen et al. [
7] studied, experimentally and also numerically, the fatigue damage that was caused by wave induced vibrations of a containership operating in the North Atlantic Ocean. The measurements showed that the wave-induced vibrations accounted for approximately 40% of the total fatigue and the numerical results were found to overestimate the total fatigue damage by 50%. Moe et al. [
8] found that, for a containership operating in the Pacific Ocean, the contribution of wave induced vibration caused about 50% of the total fatigue damage. Wang et al. [
9] investigated, also both numerically and experimentally, the wave-induced vibration of a 156,800 m
3 LNG ship; it was found that, for the full loading condition, the numerical results obtained by the hydroelasticity method for the fatigue damage was about 1.4 times larger than those by treating the hull as a rigid body. Li et al. [
10] carried out an investigation of an ultra-large ore carrier. The adopted three-dimensional hydroelasticity method also showed good agreement with the experimental results. It has also been found that fatigue damage while considering springing responses is 1.36 times larger than that caused only by wave frequency load. It suggests that springing may have a significant influence on the fatigue damage of the large ship hull structure.
Slocum and Troesch [
11] investigated, experimentally and numerically, the linear and nonlinear springing response, and analyzed the influence of a variety of parameters on the excitation force and the response, including the ship speed, wave length, and encounter frequency. According to the comparison between the theoretical results and the full-scale measurements, Storhaug et al. [
6] found, generally, the springing predictions are lower than seen in the measurements for the omission of non-linear terms. Based on a second order strip theory formulation and comparison with measured data, Vidic-Perunovic and Jensen [
12] found that the second-order terms could improve the numerical accuracy of springing calculation. However, there are also many cases where the adopted nonlinear method performs worse than the linear one in terms of accuracy. Shao and Faltinsen [
13] analyzed the contribution of the second-order velocity potential and the quadratic velocity terms in the Bernoulli’s equation on the spring effect. It was found that the second-order springing wave excitation is higher for the blunt ship than the slender one.
There exists an amount of studies on nonlinear springing and some advances have been made. However, the comparison with experiment shows that the existing nonlinear springing methods do not guarantee better accuracy [
12].
Although many researches have been carried out in the past decades, most of them dealt with ships of certain sizes, and some meaningful findings were presented, the size effect on the hydroelasticity and fatigue damage due to linear springing has not been systematically investigated and remains unclear. Aiming at a systematic parameter study, the responses of four ultra large ore carries of different size are investigated. In addition to the natural frequency and structural damping, a variety of loading conditions, speeds, and sea states are discussed in order to reveal the size effect on the fatigue damage of the structure.
2. Method of Solution for Hydroelasticity
The seminal and the most representative hydroelasticity method to numerically simulate springing was done by Bishop and Price [
14]. The hull model is represented by a Timoshenko beam and the hydrodynamics by the strip theory [
15]. In order to investigate the behavior of non-beam like structures, three-dimensional (3-D) hydroelasticity methods were devised by Wu [
16] and also by Price and Wu [
17]. In the early 1990s, hydroelasticity theoO MV Derbyshire [
18,
19]. The 3-D hydroelasticity method has now been widely applied in the analyry was applied to investigate structural failures of bulk carriers, such as the Onomichi Maru and OBsis of wave load and structural response of large ships and floating structures [
20]. Adenya et al. [
21] studied the motion and the load response of a 550,000 DWT ore carrier, both numerically and experimentally. The hydroelasticity theory and experiment were generally in good agreement, except that the rigid-body theory overestimates responses in the low frequency range. The hydroelasticity theory is more preferable when investigating the elastic effect on the wave load of large ships.
By taking a similar approach [
22], a linear 3-D hydroelasticity method has been implemented in this study and applied to analyze the contribution of springing to fatigue damage to structures [
10,
21]. The free-surface Green’s function is adopted to satisfy the linear free-surface condition, and the speed effect is accounted for by means of encounter frequency. In order to calculate the motion and wave load response of the flexible structures, the diffraction potential
and the radiation potential
are solved with the generalized fluid-structure coupling boundary conditions [
23]:
where
is the incident wave potential,
the wave natural frequency,
the vector normal to the surface,
the central point displacement on the surface under the
rth mode, and
the constant velocity around the hull
where
is the unit vector in the direction of the
axis and
is the navigation speed.
The equation of motion for a flexible body travelling with forward speed
in regular deep-water waves can be written as
where
is the encountered frequency;
,
, and
, of dimensions
, are the generalized masses, structural damping, and stiffness matrices, respectively;
,
, and
, of dimensions
, are the generalized added inertia, hydrodynamic damping, and fluid restoring matrices, respectively;
, of dimension
, is the generalized wave excitation vector, consisted of the contributions of both the incident wave and the diffraction;
, of dimension
, is the principal coordinate vector.
,
, and
can be calculated by
and
by
where
is the generalized radiation force and it takes the form:
is the fluid density, the material density of structure, the average wetted body surface, the volume of ship, the vertical deflection of mesh central point on the body surface under the kth mode, the gravitational acceleration vector, the angle deformation of hull under the kth mode, and the incident wave amplitude.
The elements of and can be obtained while using the orthogonality of the system mass matrix and the stiffness matrix of the modal function using the transfer matrix method. The structure damping matrix can be determined experimentally or empirically.
Once the principal coordinates in regular waves are obtained, the distortions and section loads, such as bending moments, shear forces can be calculated by modal superposition. Then, the displacement including both the rigid body motion and the elastic deformation is
the vertical bending moment at a cross-section of the hull is
and the vertical shear force at a cross-section of the hull is
In these expressions, , , and denote, respectively, the modal vertical displacement, vertical bending moment and shear forces for the rth mode shape with the corresponding principal coordinate evaluated in regular waves from Equation (3).
4. Numerical Configurations and Parameters
To systematically investigate the influence of the calculation parameters on fatigue damage, four realistic ultra large ore carriers (OC) are used, namely, 250,000 DWT OC, 300,000 DWT OC, 388,000 DWT OC, and 400,000 DWT OC, and their principle particulars are given in
Table 1. The mass distributions and the structural properties are shown in
Figure 3.
In the present study, the sea states of the North Atlantic are adopted to calculate the fatigue damage to the ultra-large ore carriers. According to the results obtained for the vertical bending moment, the range of frequency 0.157 rad/s to 3.14 rad/s, which encompasses the two-node wet-mode natural frequency, is adopted for calculating the response of wave induced vibration. The wave induced vibration originates from the coupling effects of wave encounter frequency and the first order natural frequency of the ship hull. When the encounter frequency coincides with the natural frequencies of the hull, linear springing will occur. Since the encounter frequency is dependent of the ship speed, so are the numerical results. The actual ship speed is, in general, reduced in high seas. The speeds in relation with the significant wave heights explored in the present study are shown in
Table 2, where
is the service speed.
In the calculation of the fatigue damage, inclusion of more lower-order elastic modes will improve the numerical accuracy while decrease the efficiency, the same thing goes with the mesh refinement for the hull. In order to improve the computational efficiency at the cost of negligible loss of accuracy, a convergence study has been conducted for both, and the results are presented in the
Appendix A, see
Figure A1,
Figure A2,
Figure A3,
Figure A4 and
Figure A5.