In Section 2, the motion responses of the vessels were analyzed under wave conditions without considering mooring systems. However, during actual port operations, vessels are securely moored prior to conducting cargo-handling or bunkering tasks. This section investigates vessel responses with mooring systems in place. To accurately compute the loads acting on the mooring lines, it is essential to account for the effects of nonlinear forces. Therefore, a time-domain analysis was performed to capture both the motion responses and the associated nonlinear load effects.
3.1 Time-domain Equation of Motion
For the time-domain analysis, Cummins’ equation based on the time-memory function was adopted (
Cummins, 1962). The equation is typically expressed as:
Here, mjl is the mass, ajl is the added mass at infinite frequency, Xj is the motion repsonse, Ljl is the retardation function (time-memory function), cjl is the hydrostatic restoring coefficient, Fwave is the wave excitation force, Fdrift is the drift force, and Fmooring is the mooring restoration force. The vessel motion responses are calculated via time integration based on a quasi-static analysis of the mass–damping–stiffness system described on the left-hand side of the equation.
In port operations, the mooring system generally consists of two main components: fenders, which prevent direct collision between the ship and quay wall, and mooring lines, which suppress dynamic motions and prevent separation from the quay wall. For bunkering operations, an additional mooring system is installed between the bunkering ship and the receiving ship. As with quay mooring, this system also includes fenders and mooring lines to ensure stability between the two vessels.
3.2 Fender System
To prevent collisions between the ship and the quay during berthing, fenders are installed between the hull and the quay wall. Fenders absorb the kinetic energy of the vessel through elastic deformation. The fender system is designed such that the berthing kinetic energy is less than the energy absorption capacity of the fender (
PIANC, 2002). According to PIANC (The World Association for Waterborne Transport Infrastructure), the berthing energy (
Ef) of a vessel is defined as:
Here, M is the vessel mass, V is the berthing velocity, Ce is the eccentricity coefficient, Cm is the added mass coefficient, Cs is the softness coefficient (default 1.0), and Cc is the contact point coefficient (default 1.0). Recommended berthing velocities vary by ship type and displacement. For this study, a recommended berthing velocity of 0.15 −0.2 m/s was applied for vessels with a displacement below 10,000 t. Using PIANC-specified coefficients and the principal dimensions of each ship, the eccentricity and added mass coefficients were calculated as follows: For the receiving ship: Ce = 0.876, and Cm = 1.61. For the bunkering ship: Ce = 0.776 and Cm = 1.55.
This study assumed that two pneumatic fenders of sufficient size were positioned within the inter-vessel and quay gaps. When vessel motion reduces the gap distance, the fender deforms accordingly, generating a reactive force (
Fig. 8). The reactive force is calculated as:
Here, ky is the fender stiffness, ly0 is the original fender diameter, and ly is the deformed fender diameter.
The performance of a fender is commonly indicated by its Guaranteed Energy Absorption (GEA), which is defined as the energy absorbed at 60% deformation. This GEA value also forms the basis for calculating the reactive force of the fender. In this study, the stiffness values of the fenders were determined using the GEA force and the corresponding deformation rate. The selected fender model featured a diameter of 1 m, which is appropriate for maintaining a gap within 2 m. According to typical manufacturer data, pneumatic fenders generate reactive forces in the range of approximately 180–300 kN at GEA deformation, depending on the fender length. Accordingly, the stiffness was set to 500 kN/m between the receiving ship and the quay wall, and to 350 kN/m between the receiving ship and the bunkering ship.
3.3 Wave Drift Force and Motion Response with Fender System
To analyze the impact of waves on the mooring system, the motion-induced responses of the fenders under regular wave conditions were examined. First, wave drift forces were calculated to determine the steady forces acting on the ships. Using the pressure integration method, the wave drift force can be expressed as:
Here, ζ is the wave elevation, δ is the displacement vector, nH is the horizontal normal vector, F(1) is the linear wave load, and ξR is the rotational motion. * indicates a complex conjugate, Re{} is a real part, and Wp is the waterplane area.
Using
Eq. (5), the frequency-dependent wave drift forces acting on each vessel under beam wave conditions are presented in
Fig. 9. The left figure shows the drift force on the receiving ship, which peaks at a wave period of approximately 6 s. In contrast, the bunkering ship experiences a maximum negative drift force at 6 s, which then transitions to a positive peak around 4.8 s. A negative drift force indicates that the force acts in the opposite direction of wave propagation, meaning the bunkering ship is pushed away from the direction of the waves, and thus opposite to the receiving ship.
This study assumed that two pneumatic fenders were installed in parallel between the quay wall and the receiving ship, and an additional two were installed between the receiving and bunkering ships (
Fig. 10). Three wave periods under beam wave conditions were simulated using regular waves, and vessel motion responses along with fender reaction forces were examined through time-domain analysis.
Figs. 11–
13 illustrate the relative motions between the ships and the corresponding fender reaction forces for each wave period.
Fig. 11 represents long waves with a period of 10.5 s. In this case, both ships experience minimal drift forces. As a result, the relative motion does not reduce the gap sufficiently to trigger contact, and no fender reaction force is generated–even in the absence of mooring lines.
Fig. 12 shows the results for a wave period of 6.28 s. At this frequency, the receiving ship experiences a maximum positive drift force, while the bunkering ship experiences a maximum negative drift force. The bunkering ship is thus pushed in the opposite direction to the receiving ship, preventing the gap between them from narrowing enough to activate the fenders. However, the receiving ship is pushed toward the quay wall with considerable force. Once this force reduces the separation beyond a certain threshold (as indicated by the green box), fender reaction force is generated between the ship and the quay.
Fig. 13 presents the case of a shorter wave period of 4.83 s, during which both vessels experience positive drift forces. As the ships move in the same direction, slight pitching motions also occur, resulting in non-horizontal contact angles with the fender. This leads to more complex motion responses and more frequent fender reactions. Despite the relatively modest magnitude of drift forces at higher frequencies, their combined effect with vessel motions can result in frequent activation of the fenders.
3.3 Mooring Arrangement and Working Safety
Ships berthed at a quay wall maintain their position by connecting mooring lines to bollards. Mooring lines are generally classified into four types based on their function: head lines, breast lines, spring lines, and stern lines (
Fig. 14). According to the guidelines of the Oil Companies International Marine Forum (OCIMF), it is recommended that all mooring lines be constructed from the same material. In practical applications, however, achieving complete uniformity across all lines is difficult; therefore, it is advised that mooring lines serving the same function employ the same material and possess consistent mechanical properties. In this study, all mooring lines are assumed to have identical stiffness. Because the vessels considered in this analysis are relatively short, breast lines–which typically run perpendicular to the hull–were omitted.
The material properties of mooring lines vary depending on the size of the vessel. Larger vessels require mooring lines with higher stiffness, whereas smaller ships may use lines with lower stiffness. This variation arises from the size-dependent nature of vessel motions and the corresponding loads imposed on the mooring lines. In this study, because both vessels are categorized as small to medium in size, mooring lines composed of synthetic fiber materials with relatively low stiffness were employed.
The mooring system between the two ships was simplified by placing mooring lines only at the bow and stern of the bunkering ship, which is sufficiently small. These inter-ship mooring lines were modeled with lower stiffness than those used for the receiving ship (
Fig. 15).
Each mooring line develops proportional tension according to its elongation from the initial length. Pretension was applied by adjusting the initial line length to ensure a baseline tension under static conditions (
Fig. 16). This can be mathematically described as follows:
Here, (x0, y0, z0) and (x, y, z) represent the pre- and post-deformation directional displacements. The length of each mooring line was calculated as
l0=x02+y02+z02 and
l=x2+y2+z2, respectively.
One important consideration in the mooring arrangement is the minimization of the vertical angle of the lines between the ship and the quay (
OCIMF, 2018). If there is a significant elevation difference between the ends of a mooring line, its effectiveness decreases due to reduced force transmission efficiency.
Fig. 17 illustrates mooring configurations under high and low tide conditions, as well as various bollard locations on the quay. During high tide, the freeboard of the ship increases, resulting in steeper vertical angles and diminished mooring effectiveness.
Fig. 18 presents the effect of varying freeboard height differences–from 2.4 m to 4.0 m–between the quay and the receiving ship, while maintaining the same pretension. As the vertical angle increases from 5.6° to 14.7°, the motion responses of the ship also increase, indicating a decline in mooring performance. In
Fig. 19, the bollard location is shifted laterally inward while maintaining the same freeboard height, thereby reducing the vertical angle to 12.5° and improving mooring effectiveness. The OCIMF guidelines recommend that the vertical angle of mooring lines should not exceed 25°.
Many existing studies on moored ship motions are based on frequency-domain or static analysis (e.g.,
Lee et al., 2003;
Kwak and Moon, 2014;
Kwon et al., 2021). Additionally, research on operational safety under port conditions typically incorporates the combined effects of waves, currents, and wind loads (e.g.,
Cho, 2017;
Kim et al., 2016;
Kim et al., 2017). This study, however, focuses exclusively on a time-domain analysis of wave-induced motions as an initial investigation into the mooring arrangement.
A numerical free-decay test was conducted to identify the natural period of the coupled sway mode between the two vessels, determined to be approximately 23 s. This test involved displacing the bunkering vessel (located on the seaward side) by 1.0 m and simulating its free response in the absence of external forces. The initial pretension in the mooring lines was set to be less than 200 kN between the quay and the ship and less than 100 kN between the two vessels. Because synthetic fiber lines are used, these values are well below the maximum breaking load (MBL), and the tension variations caused by vessel motions remain small. For the environmental conditions, irregular beam waves were generated using a peak period of 4 s and significant wave heights (Hs) of 0.5 m, 0.75 m, and 1.0 m, representative of the wave climate in Busan Port.
To evaluate the results, we referenced existing motion criteria. The
Ministry of Oceans and Fisheries (2024) provides port design guidelines that include two standards for working vessels.
Table 2 presents the limiting significant wave heights for cargo handling operations, categorized by vessel size: small ships (<500 GT), ultra-large ships (>50,000 GT), and medium-to-large ships (everything in between).
Table 3 outlines the allowable motion response criteria based on ship type and cargo equipment. Most values are based on peak-to-peak motion, except for sway, which uses 0-to-peak. Gas carriers, in particular, are assessed based on the motion tolerance of the loading arm. However, bunkering vessels typically use flexible hoses instead of loading arms (
Fig. 20). While gas carrier standards were used as a reference, the motion tolerance for bunkering vessels is expected to be more lenient.
Figs. 21 and
22 plot the sway motion responses of the receiving ship and the bunkering ship, respectively, under varying wave heights. At H
s = 0.5 m and 0.75 m, both vessels exhibit sway motions of less than 0.2 m. At
Hs = 1.0 m, the responses increase sharply—up to 0.5 m for the receiving ship and 0.7 m for the bunkering vessel. Consequently, the relative sway motion between the ships increases significantly at
Hs = 1.0 m (
Fig. 23). As sway motion is highly influenced by drift forces, the response tends to rise sharply beyond a certain wave height. Nevertheless, both vessels satisfy the sway motion criteria for gas carriers listed in
Table 3.
Figs. 24 and
25 illustrate the heave motion responses. Although gas carriers have no explicit criteria for heave, the receiving ship’s motion remains within 0.2 m (peak-to-peak). On the other hand, the bunkering ship exhibits heave motions reaching 1.0 m, which may be considered large when compared to general cargo vessel standards. However, as LNG bunkering ships use flexible hoses rather than loading arms in STS operations, a 1.0-m heave motion is unlikely to pose significant operational risk.
Figs. 26 and
27 present the roll motion responses. For the receiving ship, roll amplitudes range from 1.3° to 1.9° (peak-to-peak) across wave heights, approaching but not exceeding the 2.0° limit for gas carriers. In contrast, the bunkering ship exceeds this limit at
Hs =0.75 m and above, with responses of 1.9°, 3.5°, and 7.5°, respectively. According to
Table 2, the bunkering ship qualifies as a medium-to-large vessel, with a handling wave height limit of 0.5 m, under which both vessels remain within acceptable motion criteria. It should be noted that the roll motion in this study was analyzed under beam wave conditions, with an assumed viscous damping term. To further validate these findings, future studies should include independent verification of roll motion for standalone vessels.
Other motion responses–surge, pitch, and yaw–were significantly smaller and are therefore not presented. Overall, motion amplitudes tended to be larger for the smaller bunkering ship, despite both vessels being subjected to the same environmental conditions. As previously noted, LNG bunkering ships do not utilize rigid loading arms, and roll motion alone is unlikely to impose direct constraints on STS bunkering operations. Currently, port approval standards for moored vessel operations are typically based on single-ship criteria. In contrast, LNG bunkering involves STS configurations where both ships are berthed and operating together. Furthermore, because the bunkering ship is considerably smaller, its motion responses under wave conditions are more pronounced. Therefore, approval standards for STS LNG bunkering operations should be adjusted and relaxed relative to those of conventional gas carriers, and further research is necessary to establish such criteria.