# Influence of Linear Springing on the Fatigue Damage of Ultra Large Ore Carriers
## 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-20000 TEU 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, [PERSON] et al. [5] showed that the wave-induced vibrations caused 44% of the fatigue damage. The hydroelasticity model test that was conducted by [PERSON] et al. [6] on a large ore carrier also showed that 56% of the damage was caused by springing and whipping. [PERSON] 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 showedthat 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%. [PERSON] 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. [PERSON] 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. [PERSON] 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. [PERSON] and [PERSON] [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, [PERSON] 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, [PERSON] and [PERSON] [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. [PERSON] and [PERSON] [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.