## 1 Introduction  
Shaft alignment is influenced by hull deformation, engine output, propeller force and propeller eccentric thrust. These factors lead to an unevenly distributed load on the bearings, and subsequently, they increase the risk of damage [1]. It is necessary to ensure the distances between the bearings and to set the specific height for each bearing with respect to the fair curve alignment theory [2; 3; 4]. Therefore, the load can be shared evenly for every bearing [5]. Even so, the after-stern tube bearing damage can still occur under sudden changes in the stern water flow field, such as rapid ship turning [6; 7; 8; 9; 10; 11; 12]. It is known that the major cause of rapid ship turning is the unstable behavior of the shaft due to the change in eccentric thrust of the propeller. This is a result of the change in the countercurrent distribution at the rear of the stern, which temporarily applies excessive load to the stern tube bearing. In the cases of newly built ships, it has become more difficult to ensure the stability of the shafting system [13; 14; 15] due to the current green and eco-friendly trend in marine transportation. Accordingly, the low-speed, ultra-long stroke engine has been widely applied to enhance ship operation efficiency [16; 17]. To maintain the ship's speed at a lower engine speed, the propeller diameter is slightly increased compared to the previous design. This design change not only increases the propeller mass but also the load and pressure rise, which are applied to the after-stern tube bearing in terms of shaft alignment, making it easier to achieve the safety tolerance recognized by the manufacturer. In this context, this study concerns the bending moment behavior of the propeller shaft in a medium-range tanker of 50,000 deadweight tonnage (DWT). Due to the friction of the ship's hull moving in water while sailing, a boundary layer of water is created around the ship and moves at the same speed as the hull. The thickness of this layer increases from the fore-to-aft of the ship. The displacement of water by the ship also induces wake waves. Under rapid turning conditions, the quick change in direction of the ship's heading causes a sudden change in the wake field [18]. This was determined by the non-uniform water flow in the propeller working area behind the ship's hull. Many studies have been conducted in the wake field. The study by [PERSON] et al. [19] used computational fluid dynamics (CFD) analysis to investigate the nominal wake fields of a container ship in regular waves with wavelength equal to the ship length in five different headings. The result reveals significant fluctuation in the nominal wake fraction in wave conditions. [PERSON] et al. [20] created a framework for more than 1300 different model tests for three ship hulls of different geometry and hydrodynamic characteristics that focuses on the added resistance and drift forces, propulsion and rudder force tests in waves and the assessment of maneuverability of ships in waves. A series of measurements carried out by [PERSON] and [PERSON] [21] showed that all ship motion components have a clear effect on the additional wave resistance. [PERSON] et al. [22] used the wake field data of multiple two-dimensional planes obtained by scanning particle image velocimetry (PIV) measurements to reconstruct the three-dimensional, three-component (3D-3C) wake field. [PERSON] et al. [23] conducted computational studies using the Reynolds Averaged Navier-Stokes (RANS) method on viscous wake fields and the propulsion performance of a four-screw ship. Their results provided the behavior of vortices and turbulent structures in the hull. The wake field caused fluctuation amplitudes of the loads on the propeller. Another study was conducted by [PERSON] and his team [24]. They performed experiments and computations for a containership in calm water and regular waves. They found that that a significant fluctuation in the wake field can affect propeller/engine performance.