Based on the loading characteristics of the FMC structure of the lock, the upper structure of the mooring column is assumed to be a simply supported overhanging beam, as presented in Figure 2.

For any loading point D on the FMC of the lock, the corresponding total strain includes the axial tensile strain and the bending strain. Based on strain superposition, the strain at point D on the surface of the FMC of the lock can be stated by: ε = ε 1 + ε 2 = F z E A + F x y L 1 − h L 2 / L 1 E I d + F z 2 R / π E I d (1) where ε ₁ represents the tensile strain, ε ₂ denotes the bending strain, A is the cross-sectional area of the FMC, E is the elastic modulus of the FMC, L ₁ is the length of the simply supported section of the FMC, L ₂ denotes the length of the outstretched arm of the FMC, h is the distance between the strain measurement position and the fulcrum point of the FMC, R represents the circle radius of the cross-section of the FMC, I is the moment of inertia of the annular section of the FMC, and d denotes the vertical distance between the strain measurement position and the neutral plane.

Within the range of the actual change of the mooring angles, the force analysis of the actual structure of the floating mooring column is carried out, and it is determined that the surface of the hollow cylinder near the vertical plane of the inner side of the lock wall line is always the tension zone, that is, the strain values are positive, and the bending strain generated by the lateral component force and the strain generated by the vertical component force can be avoided by choosing this area as the strain measurement location. Three strain measurement positions (i.e., B, C, and D) are selected on the surface of the axial cross-section of the dock FMC, and the factor h is set to 600 mm. Then, welded strain gauges are used to conduct measurements at the positions presented in Figure 3, which the standard range of welded strain gauges is ±3000με, the resistance deviation is less than or equal to ±0.1%, the sensitivity coefficient dispersion is less than or equal to ±2%, and the creep is less than or equal to 2 μm/m. The L ₁ section containing these measurement positions represents the non-mooring work area of the mooring column to prevent the strain gauge from being damaged by cables during operation. The angle between measurement positions B and C and the circle centre is denoted by μ, which is set to 10°. Furthermore, the angle between measurement positions C and D and the circle centre is represented by φ, which is set to 10°. The angle between the gate wall line and the line connecting measurement position B and the circle centre is represented by θ, which is set to 80°.

Based on the characteristics of the mooring load, the horizontal angle between the cable and the gate wall line is represented by α, and the vertical angle between the cable and the horizontal plane is represented by β. Then, F x y = F f ⋅ cos ⁡ β (2) F z = F f ⋅ sin ⁡ β (3)

As presented in Figure 3, the strains of monitoring points B, C, and D are ε _B , ε _C , and ε _D, respectively. Using Eqs 1–3, the total strains at these points are evaluated as: ε B = F y ⁡ sin ⁡ β E A + F y ⁡ cos ⁡ β L 1 − h L 2 / L 1 E I d B + F z ⁡ sin 2 R / π E I d B (4) ε C = F y ⁡ sin ⁡ β E A + F y ⁡ cos ⁡ β L 1 − h L 2 / L 1 E I d C + F z ⁡ sin 2 R / π E I d C (5) ε D = F y ⁡ sin ⁡ β E A + F y ⁡ cos ⁡ β L 1 − h L 2 / L 1 E I d D + F z ⁡ sin 2 R / π E I d D (6) where d B = R ⁡ sin α + θ − 90 ∘ , d C = R ⁡ sin α + θ + μ − 90 ∘ and d D = R ⁡ sin α + θ + μ + φ − 90 ∘ .

On the basis of Eqs 2–6, the following parameters can be readily evaluated: the horizontal angle between the mooring cable and the gate wall line (α), the vertical angle between the mooring cable and the horizontal plane (β), and the applied mooring load F _f at the specified height.

The rationality of the mooring load calculation method proposed in this paper is verified by a generalized physical model test of the load response. According to the true size and material of the upper structure of the FMC, a generalized physical model is established, and strain gauges are welded at the specified strain measurement positions. The mooring load of the FMC and the exerted loads with various mooring cable angles are simulated. The test process is depicted in detail in Figure 4.

Under a variety of loading conditions, the strain results are analysed by employing Eqs 4–6, and then the mooring load and mooring cable angle are calculated. The calculated data are compared with the standard values for the mooring load. The considered loads are presented in Table 1.

Figure 5 illustrates that the errors of the calculation results are all less than 10% compared to the standard values of the physical model test. The results include the mooring load at the specified height of the dock FMC, the horizontal angle between the cable and the lock wall line, and the vertical angle between the cable and the horizontal plane. This comparison procedure therefore verifies the proposed calculation method of the mooring loads of the FMC.

To further check the accuracy of the mooring load calculation model, a finite element numerical simulation is conducted. The main aim is to compare the calculation results of Eqs 4–6 with those of the numerical simulation. For this purpose, a three-dimensional (3D) numerical simulation model of the upper structure of the FMC is established by employing the ANSYS Workbench and the true size of each component of the FMC, as illustrated in Figure 6.

Each component of the upper structure of the FMC comprises three materials: cast steel (ZG310-570) in the hollow cylinder column, stainless steel (1CR18Ni9Ti) in the three plates of the fixing hollow cylinder, and Q235 steel in the other components. The material properties are presented in Table 2.

In these numerical simulations, the permanent load acting on the upper structure of the dock FMC is considered as follows.

(1) The structure weight is specified as G = 8.1554 kN, with a vertically downward action direction.

The numerical simulation conditions of the dock FMC are given in Table 3.

The three-dimensional numerical simulation model of floating mooring column of lock was meshed by hexahedron meshing method provided by ANSYS Workbench, the single mesh size is a parallelepiped with sides of 10 mm, and the total number of meshed cells was 69,796. According to the previous research results of our team, the research works related to grid-independence tests were carried out, and compared the calculation results of the simplified calculation model of floating mooring column with the numerical model with the same mesh structure and mesh size in this paper. The relative error of the two results is only 3.6% (Liu M W et al., 2020), which is sufficient to show that the mesh structure and mesh size selected by the model in our study can provide sufficient calculation accuracy.

The stress distribution diagram of the upper structure of the FMC is evaluated on the basis of the numerical simulation, as shown in Figure 7.

Figure 7 indicates that the strain within the hollow cylinder of the FMC is greater than that of other positions. Particularly at the welding position of the three stainless steel plates near the hollow cylinder, the stress distribution is more noticeable. Such a distribution prevents the strain gauge from being damaged by the cable in actual operation.

The strain at each measurement position of the FMC is obtained by the numerical simulation, as presented in Figure 8A.

To validate the accuracy of the simulation model under the action of load, strain simulations are carried out using Eqs 4–6 to evaluate the ship mooring load, the horizontal angle, and the vertical angle. The obtained results are then compared with the pre-set standard values of the numerical simulation test, as presented in Figure 8B.

According to the proposed mooring load calculation method, the mooring force is calculated by measuring the tension and strain in the simulated mooring environment and combining with the actual ship operation conditions. It is found that the change of mooring force calculated by the simulated mooring environment under common mooring conditions is basically consistent with the experience of shiplock operation. The following is the simulation curve of mooring load (from the ship entering the lock to the opening of the opposite lock after water release) on different mooring positions (after spring line, head line, stem line) of a shiplock, as illustrated in Figure 9.

The performed analyses reveal that the initial peak of the mooring load is mainly produced during the berthing braking of the ship. The force peak can be improved through better system management. The irregular fluctuations with small amplitudes in the water delivery process are produced by the flow force and the operation barrier of the FMC. The magnitude of these fluctuations is commonly trivial in normal operating conditions. The final force peak occurs when the gate is opened after water delivery, causing the mooring column force to magnify and exceed the upper limit of the design. This issue guides us to the conclusion that the load setting must be properly optimized.

Based on the pertinent design specifications (JTJ 306-2001, 2001), the maximum load of the FMC largely relies on the longitudinal horizontal component of the mooring load, including the gradient force and the wave force acting on the ship. Essentially, the rate of the applied force exhibits no influence and is related to inertial superhigh linearity. The corresponding theoretical calculations are given as: G r a d i e n t   f o r c e : P i = φ ρ g x h 4 − 4 h q 1 2 g 4 h − h 4 − 4 h q 2 2 g 4 h (7) W a v e   f o r c e : P b = Δ Q Δ t ⋅ W a ω n − x + 2 Q 2 W 1 − a g 1 C ω n − x B n (8) I n e r t i a l   s u p e r h i g h   l i n e a r i t y : d = μ 2 ω L n p C (9)

The parameters and coefficients given in Eqs 7–9 have previously been introduced and defined in Ref. (JTJ 306-2001, 2001).

The exerted loads on the FMC are complex in real operating conditions. In addition, the shiplock operation-monitoring program opens the gate before the massive inertia reaches its maximum value, as shown in Figure 10. This decision is made to lessen the damage caused by the massive inertia on the gate hoist and the motor. The error of the theoretical calculation under static loading is large. Hence, the theoretical calculation is not suitable for determining the operational state.

Therefore, the horizontal and longitudinal forces of the FMC can be rationally chosen as an indicator of the safety state. The longitudinal horizontal component of the mooring load is denoted as P _L, while the upper design limit of P _L is assumed to be 15 t. The protection system is denoted as k. The upper design limit of the massive inertia (d) is assumed to be 0.25 m. The judgement condition of the safety state of the FMC at the end of the water delivery is given as: G r a d i e n t   f o r c e : P L = k ⋅ F x y ⋅ cos ⁡ α = k ⋅ F f ⋅ cos ⁡ α ⋅ cos ⁡ β ≤ 15 t (10)

An optimization criterion is commonly used to maintain F _y and d values below the safe mooring thresholds and to lessen the water delivery time as much as possible. Thus, a distinct progressive prototype debugging approach can be implemented as explained in the following.

During the water filling process of the large dock, including the moving water shut-off valve mode, the parameters to be optimized are: the residual water head (△H), the opening degree of the moving water shut-off valve (N), and the waiting time of the flush (△t). The following initial values are taken into account for the operating parameters: △H = 6.5 m and N = 0.70. The first debugging value of △H is set as △H ₁ = △H ₀ + 0.5. The first debugging value of N is also set equal to N ₁ = N ₀. In addition, the debugged values are set as follows: △H=(△H ₀, △H ₁, △H ₂, … , △H _m), △t = (△t ₀, △t ₁, △t ₂, … , △t _m), and N = (N ₀, N ₁, N ₂, … , N _a).

Additional factors are set up as follows. The valve opening from fully closed to fully opened is assumed to range from 0 to 1. The parameter values of x and y are assumed to be integers between 1 and 5, and the factor d ₀ is set equal to 0.25 m.

The debugging method was itemized per the following descriptions.

I. Open the water-filling valve to opening 1. In the case of △H = △H _m, start to close the water shut-off valve and stop closing the valve as soon as N = N _a. Continue to fill the water until the water level of the gate chamber is flush with the upstream water level. Evaluate the massive inertia, namely, d, and the waiting flush time, i.e., △t.

To keep the valve closed or open, with the value of N ₁ remaining unchanged. The debugging processes are repeated, and △t and d are simultaneously computed.

V. In the optimized plan, the following conditions are imposed:

To keep the residual water head △H ₂ of the moving water shut-off valve unchanged and repeat the debugging process.

VI. In the optimized plan, when △t = 0 and d = d ₀, debugging is completed. Subsequently, △H and N represent the final operating parameters.

A double-sided water-filling operation is chosen for validation testing. The moving water shut-off valve process is adopted. The test ship is the largest 6,000 t class standard single ship (with a large aspect ratio) on the Yangtze River. The initial parameters are set as △H = 6.5 m and N = 0.70. The numerical simulations are conducted according to the above debugging method to preliminarily monitor multiple groups of shut-off valve process parameters, as presented in Table 4.

The problem is appropriately calculated and analysed for multiple operating conditions. The operation process is selected as △H = 7.5 m and N = 0.45 based on a comprehensive consideration including the massive inertia and water delivery time. The massive inertia is measured to be d = 0.1 m ≤ 0.25 m.

The monitoring data of the mooring load (P _L) are illustrated in Figure 11.

Figure 11 reveal that the dock with double-side water filling operates in a normal manner. For the single ship of 6,000 t class in the initial processes of water filling and levelling, the maximum mooring load P _L is 300 kN (30.6 t) (see Figure 11A), noticeably exceeds the upper design limit of the load, which is 15 t for the FMC. After process optimization, the maximum mooring load decreases to 136.5 kN (13.9 t) (see Figure 11B), lower than the upper design limit with a noticeable reduction in the maximum mooring load; furthermore, the corresponding time of water delivery slightly increases. As a result, the process optimization method proposed in the current exploration is capable of effectively enhancing the safety of the ship mooring and suitably achieving a favourable balance with respect to effective water delivery.