1. Introduction
The construction of water conservancy project is used to solve the problems of water resources, while it also produces the pollution of river, disaster of flood and the damage of natural environment and groundwater [
1,
2,
3,
4]. It is necessary to establish a water conservancy project that has little impact on the environment. Floating structures are widely used in new types of sluices in flood control, tidal power stations, pumping stations, and oil production platforms, and they have begun to play an important role as environmentally friendly facilities [
5,
6,
7]. The stability of the structure is an issue of concern because such stability affects both construction and operation.
Research in recent years has focused on the stability of floating structures and the interaction between water and floating structures in infinite water, still water, and water under the action of wave motions. Roy and Ghosh [
8] proposed a simple model to estimate the horizontal force and moment of a floating plate against waves at different depths and found that the force and moment converged with increasing wave period and depth. Hayatdavoodi and Seiffert [
9,
10,
11,
12] conducted laboratory experiments and numerical simulations of conidial waves propagating over a submerged floating plate. The vertical force acting on the submerged floating plate appeared to scale linearly with the wave height, while the horizontal force varied nonlinearly with wavelength. They noted that the water depth and the wave submergence depth had an effect on the safety and that the interaction between a wave and a floating bridge contributed significantly to the force magnitude. Lee and Hong [
13] adopted experimental tests and the Marker and Cell numerical method to verify the nonlinear wave effect on a large floating structure. The test configuration reduced the wave force and improved the stability with the installation of baffles. Belibassakis and Athanassoulis [
14] analyzed the hydrodynamic characteristics of the floating structure by hybrid technology in any water depths. The seabed model with different slopes and curvatures was shown to have an influence on the hydrodynamic characteristics and the response of floating structures under the action of waves. Liu et al. [
15] predicted the wave force and the momentum of the structure for infinite water and finite water depths using the three-dimensional time-domain Green’s function method. In addition, they investigated the motion response under the action of waves by means of numerical simulations [
16,
17], concluding that the water depth had an effect on the overturning phenomenon.
The studies noted above considered the response of floating structures in still water or under the action of waves in finite water, in which the momentum and its influencing factors were analyzed. Such structures have similar characteristics to ships, offshore platforms and sluices. However, the hydrodynamic conditions are different from that of ships in infinite waters and offshore platforms [
18,
19]. The force leading to the overturning of structures in currents consists of an inertia force, hydrodynamic pressure and frictional resistance. Lu [
20] analyzed the stability of a floating bulkhead and its influencing factors in still and flowing water. He also compared the different characteristics between hydrostatic stability and dynamic stability and discussed improvement measures of the stability of a floating bulkhead. Based on the principle of force balancing, Fu et al. examined the actors affecting the overturning of floating bulkheads and developed a stability formula describing these structures [
21]. They found that the inertia force and the hydrodynamic pressure in currents had an effect on the overturning phenomenon and proposed a method to improve bulkhead safety. Huang et al. [
22] introduced a numerical simulation to study the relationships among water depths, flow velocities, closing angles and the floating bulkhead. Based on a neural network, Fan and Suo [
23,
24] established a database for floating transportation structure technology, which was used to calculate the stability of tidal power plates in the process of floating and sinking. This method overcomes the limitations of stability formulas and is easy to use. Johnson et al. [
25] compared three models to calculate the dynamic responses of floating breakwaters under the action of waves and currents, wave heights, wave lengths and currents. Rey and Touboul [
26] devised experiments of a submerged floating plate related to the currents acting on it. The current had a strong effect on the reflection coefficient and horizontal force. Karmakar and Guedes Soares [
27] analyzed the bending moment and shear force on a moored finite floating elastic plate under gravity waves in the case of finite water depth, concluding that the effect of lateral pressure on the plate should not be neglected. Cui [
28,
29] adopted numerical simulation and experimental tests to study the force acting on the floating structures, the sensitivity of influencing factors were analyzed and the force formula was given. No uniformity of force acting on structures led to the overturning; however, the variation tendency of overturning and its influencing factors were not studied.
Previous studies have analyzed the influencing factors of the overturning of floating structures in currents and addressed the relationships between the factors and different positions. The influence of hydrodynamic pressure on the overturning of a floating structure is the motivation to develop a method to accurately calculate the overturning moment via hydrodynamic pressure. Based on a large number of hydraulic model tests, the theory of hydrodynamics and equilibrium analysis are employed to analyze the hydrodynamic factors acting on the capsized features of a floating structure at different positions in currents. The micro-integral method is used to calculate the structure’s overturning moment. The hydraulic parameters, i.e., the floating ratio of the structure, water level, velocity and vertical position, are investigated to evaluate the stability through a theoretical analysis; also, the relationship between overturning and parameters is also described. In light of the experimental results, a formula of overturning moment in terms of the hydrodynamic pressure in currents is given; the results from the formula are in good agreement with the experimental tests. The findings provide a useful reference for the stability determination of floating structures.
3. Calculation of the Overturning Moment
The change of the hydrodynamic pressure and the distance between adjacent measuring points is relatively small; therefore, in order to get the exact value, the overturning is calculated by measuring the surface pressure using the micro-integral method. The dynamic pressure on the structure was collected, and time-averaged values were used to calculate the overturning moment by means of the micro-integral method.
Figure 4 shows the calculation schematic of the overturning moment and the pressure profile of the measurements.
The upstream face of the floating structure is selected and depicted in
Figure 4. The profile is parallel to the flow direction;
C is the center of the floating structure.
Li1 and
Li2 are the horizontal distances of measuring points
A and
M from the measuring point to
C, respectively;
l is the horizontal distance from measuring points
A to
M,
hi1 and
hi2 are the water depths of the measuring points, respectively, and
ei is the pressure center of the computational domain.
Although the flow velocity near structures is relatively large, the hydrodynamic pressure acting on the surface of the structures changes smoothly along the flow direction. The distance between adjacent measuring points is small, so it is reasonable to assume that the pressure distribution is linear between them. The pressure is calculated as follows:
where
g is the acceleration of gravity, 9.81 m/s
2;
ρ is the water density, 1000 kg/m
3;
dl is the differential length of
M, m;
h(
l) is the submerged depth of
M, m, and
Pi is the pressure between two measuring points, N.
The distance between
ei and
C is
Lei:
The overturning moment in terms of the time-average pressure distribution is defined as
MPi:
Therefore, the total overturning moment is calculated as follows:
The direction of MPi is negative in the clockwise direction and positive in the counter-clockwise direction. The value of MP is related to the unit width dynamic pressure (B = 1 m).
The overturning moment of the floating structure varies according to the floating ratio of the structure, water level, velocity and vertical position based on experimental observation and analysis. Dimensional analysis is used for analyzing the relationships between influencing factors and overturning, which makes the obtained results harmonious in dimension and universal in meaning. It found that
MP [ML
2/T
2] and the other relevant physical quantities, namely,
L [L],
v [L/T],
g [L/T
2],
a [L],
ΔH [L],
e [L],
H [L],
ρ [M/L
3] and dynamic viscosity
μ [MT/L
2],, are related as follows:
The effect of
μ can be discussed in terms of the sheer force. It is observed that the effect of viscosity on the floating structure can be neglected compared with the hydrodynamic pressure [
29]. According to the
π theorem,
H,
g and
ρ are selected as the basic parameters. Two dimensionless Pi-groups are given:
π2 = L/H and
π4 = a/H. The shape of floating structures is a factor that affects the overturning moment. The dimensionless parameters were rearranged logically to yield, the shape ratio to obtain:
π2/π4 = (
L/
H)/(
a/
H)
= L/a. The equation is derived as follows:
where
is the shape ratio. Given the definition of the overturning moment of floating structures’ and
MKP, we obtain
5. Verification and Error Analysis
To evaluate the accuracy of the estimate Equation (13), the adjusted multiple correlation coefficients (AMCC) and the standard error of estimation (SEE) were used. AMCC is the modified version of the multiple correlation coefficients; it gives the percentage of variation explained by only those significant variables that in reality affect the predicted value. The value of AMCC indicates the goodness of the fit of the
MKP values, it was used to verify the prediction is right and the regression model is satisfactory. The value of SEE evaluates the reliability of the data, smaller values are better, which indicates that the observations are closer to the fitted line. The quantities of AMCC and SEE are defined as follows:
where
R2 is the coefficient of determination,
K is the size of the data set,
J is the number of dimensionless independent variables, and
yk is the measured value of
MKP. Here,
is the estimated value of
MKP of Equation (13), and
is the mean value of
yk.
It is determined that AMCC and SEE for Equation (13) are 0.90 and 0.02, respectively, thus proving that the significance is strong.
The model can be further tested to judge its applicability of the regression equation and the fitting effect. A 95% confidence interval was selected; the corresponding value of F is 395.10. Autocorrelation affects the ability to conduct valid statistical tests. Autocorrelation affects the ability to conduct valid statistical tests. The autocorrelation of the model was evaluated by using Dubin-Watson (DW) test which is commonly used. The significance test of variables is meaning and the prediction of the model is valid if the value of DW is smaller than 5. The value of DW was 1.84, that is, no autocorrelation exists in the model. The correlation coefficient of the formula between the variables is low; the independent variables are not correlated. Multicollinearity produces large standard errors in the related independent variables and it causes imprecise estimates of coefficient values, it also results in the imprecise and instability the model predictions. To quickly eliminate the instability of the model, the variance inflation factor (VIF) method was used to perform multicollinearity diagnosis. The maximum value of VIF was 5.60; thus, there is no multicollinearity problem between the variables and
MKP. The F-test, DW test and VIF test formulas are given as follows:
where
is the mean value of
,
et is the error term at time
t, and
Ri2 is the multiple coefficient of determination of the independent variable.
The standard residuals need to satisfy the requirements of randomness and normality to prove the correctness of the obtained formula. The following results were obtained by analyzing the standard residuals.
Figure 9 illustrates the corresponding statistical analysis of the residuals of Equation (13) and the histogram of the residuals with normal probability curve; the residuals present a normal distribution. The 95% distribution of standardized residual is between −2 and +2, which suggests that the model assumptions are reasonable.
Figure 10 shows the scatter plot of the standardized residual.
As for
MKP, the comparison between the measured and calculated results is shown in
Figure 11, suggesting a good agreement. Thus, the calculation formula of overturning moment is obtained.
6. Conclusions
The overturning moment of the floating structure influenced by currents was assessed by means of theoretical analysis and physical model tests. A calculation formula was proposed by using dimensional analyses between the overturning moments and the contributing parameters: structural shape ratios, structural vertical positions, flow velocities and water levels.
The structure shape ratio influences the stability of the structure. The stability deteriorates as the ratio increases; the range of overturning moment shows a decreasing trend with the increase of structural vertical positions. The stability of structures is influenced by Fr2. If the value of Fr2 is small (≤1), then the overturning moment decreases significantly with increasing Fr2. If the Fr2 value is greater than 1, then the range of overturning moment converges to a straight line, and the effect of Fr2 on overturning moment is not easily detectable. A lower water level leads to smaller overturning moment and improves the stability of structures.
Nonlinear regression and multi-linear regression were used to obtain the formula for calculation of the overturning moment. Statistical indices were used to quantitatively investigate the accuracy of the formula; the values of AMCC and SEE were 0.90 and 0.02, respectively, and the residuals of the formula presented normal distribution. The calculation formula is in agreement with the experimental data, thus supporting the validity of the stability evaluation. Compared with the action of hydrodynamic pressure, the role of shear force is small. This study did not consider overturning generated by shear stress; In order to study the systematical and accurate value of overturning moment in prototype engineering, the moment and the proportion generated by shear force can be studied in the future. Thus, the influence of shear stress on the overturning phenomenon and its proportion are recommended for evaluation in the future.