Offshore oil platforms are important facilities for offshore oil exploration. Underwater pile grippers play an important role in the construction and installation of platforms. The gripper body is evenly distributed with several hydraulic cylinders in the circumferential direction, and an external oil source provides the hydraulic oil for the hydraulic cylinders. Under the action of hydraulic pressure, the pressure block connected with the plunger holds the pipe pile to provide a horizontal clamping force for the pipe pile, so as to meet the underwater working requirements. When the pile body is supporting the hydraulic cylinder in the working process, the pile body should not only meet the strength requirements, but also ensure that the body is deformed within a certain range [1]. Therefore, the optimum design of the body structure is of great value to ensure that the pile gripper works safely and reliably. In recent years, scholars have conducted finite element analysis on the gripper from the structure of the body, the sealing performance of the hydraulic cylinder, to the gripping manner of the clamping jaws. Sun and Meng [2-3] used ANSYS software to optimize the design of the pinch teeth of the pile gripper, where the tooth profile of the pressure block was optimized, and the maximum stress was reduced by 22.98%. Liang [4] used the ANSYS software to design the gripper. The number of hydraulic cylinders suitable for the structure of the gripper was obtained, and the results of the satisfactory structure of the body were obtained. Jiang, Fang, and Wang [5-7] studied the gripping performance of the underwater gripper, and the relationship between the friction coefficient and gripper angle for the gripper and the pipe pile was obtained. The size of the press block, the optimized structure characteristics of the press block, and the stressed state were improved. Jiang [5,8] analyzed the load on the offshore platform, the working principle of the hydraulic system of the gripper, and designed a hydraulic circuit that met the needs of holding. Finally, a control system that satisfies the requirements using a PID control algorithm was obtained. Sun [8-9] used the finite element software to design and optimize the hydraulic cylinder snap ring. The contact finite element design method was used to simulate the force condition of the snap ring and optimize the design of the structure of snap ring. The snap ring equivalent stress is reduced by 7MPa. The design and optimization of the gripper are mainly realized by using finite element software. The existing research results involve optimization design of the body of the pile gripper, and the main analysis focuses only on the diameter of the piles. In this paper, ANSYS software was used for structural design and size optimization of the body of the gripper. Under the condition of satisfying the structural rigidity requirements, the requirements for minimum equivalent stress and minimum volume of the pile body were achieved.

Figure 1 shows the installation of the pile gripper. Figure 2 shows the structure of the pile gripper.

As shown in Figure 1, the pile gripper is sleeved in the pipe pile and the upper end of it is welded to the lower part of the skirt pile sleeve. During the installation of the jacket platform, the leveling machine is used to level the pipe pile, and then the pile gripper starts to work and clamps the pipe pile to provide guarantee for the smooth operation of the subsequent grouting operation. As shown in Figure 2, the underwater skirt pile gripper consists of a body, some hydraulic cylinders, a lifting lug, and a control panel of the Remote Operated Vehicle (simplified to ROV). The body is circumferentially opened in several holes to install and fix a plurality of hydraulic cylinders. In order to facilitate the installation of the pile gripper, the body is uniformly welded with four lugs in the circumferential direction, which facilitates the lifting of the pile gripper when the project is applied. The ROV control panel is welded to the specified location on the body.

The material of the body is ZG06Cr16Ni5Mc, the elastic modulus E is 200GPa, the Poisson’s ratio is 0.3, the yield limit is 600MPa,the safety factor is 1.5, and the allowable stress is 400MPa. In the working process, the hydraulic cylinders are uniformly distributed on the body clamping the pipe pile. The reaction force of the pipe pile on the hydraulic cylinder acts on the step surface of the body through the flange surface of the hydraulic cylinder. A force diagram of the body of a pile gripper is shown in Figure 3. A diagram of the trapezoidal hole of the body is shown in Figure 4.

The underwater pile gripper provides 1500 tons of clamping force for the pipe pile in the working process. Here, n hydraulic cylinders are evenly distributed around the body. The working pressure of each hydraulic cylinder is P. The press block of pile gripper is connected to the front end of the hydraulic cylinder plunger. When the hydraulic cylinder is working, the press block is pushed out to clamp the pipe pile. According to the working principle of the hydraulic system, the relationship between the number of hydraulic cylinders, working pressure, and clamping force that pile gripper is provided:

WhereF is the clamping force and equals 1500t, and S is the working area of a hydraulic cylinder.

Substituting the data into Equation (1), the relationship between the number of cylinders and the working pressure of the hydraulic cylinder can be obtained with $P=\frac{15}{Sn}$(MPa).

According to Newton’s third law, Equation (2) can be obtained.

Where ${{P}_{N}}$ is the force of the trapezoidal hole of body and ${S}'$ is the area of the trapezoidal hole.

In order to meet the need of grouting, the size of the trapezoidal hole can be preliminary designed: ${{D}_{1}}=\phi 305\text{mm}$ and ${{D}_{2}}=\phi 350\text{mm}\text{.}$ The depth h of the two small holes is the same, which is half the thickness of the body. According to the above parameters, it can calculated by ${S}'=0.023{{\text{m}}^{\text{2}}}$. It can be obtained by

In order to ensure that the forces around the pipe pile are uniform, n should be an even number. The cylinder number nwas chosen as 6, 8, 10, 12, and 14. The working pressure corresponding to different cylinder numbers is shown in Table 1.

The pipe diameter of the 84-inch pipe pile is $\phi 2134\text{mm,}$ and the body of the pile gripper cannot contact the pipe pile. According to the grouting requirements, the dimensions of the main body of the pile gripper were preliminarily designed as ${{D}_{3}}=\phi 2190\text{mm,}$${{D}_{4}}=\phi 2330\text{mm,}$ and H=700mm. The finite element model of the body is shown in Figure 5.

According to the force diagram shown in Figure 3 and the load ${{P}_{N}}$ shown in Table 1, the structure of the body with different number of open holes was analyzed. The corresponding stress cloud diagrams and strain cloud diagram of different numbers of holes in the body are shown in Figure 6.

When there are few openings in the pile gripper, the trapezoidal surface of the hole of the body is subject to greater stress. With an increasing number of opening holes, the forceis reduced. It can be seen from Figure 6 that when the number of cylinders n increases from 6 and so on, the stress and strain gradually decrease. When n=12, the effective stress of the body is 286.65MPa, and the equivalent strain is 0.68mm. When the numbers of holes continue to increase, the external force of the trapezoidal surface is reduced, but the distance between each hole decreases and the structural strength decreases. It can be seen from Figure 6 that when n continues to increase, the equivalent stress and strain increase again. Therefore, when n=12, the force of the body structure is reasonable.

Even though the development of the pile gripper has been made, the requirements for specific engineering applications are often not fully satisfied. This requires structural optimization after the preliminary design of the pile gripper. The body is evenly distributed with 12 trapezoidal holes in the circumferential direction. A schematic diagram of a trapezoidal hole is shown in Figure 4. The trapezoidal hole of the body is subjected to the external force and is prone to stress concentration. In theory, the greater the difference between ${{D}_{1}}$ and ${{D}_{2}}$, the larger the annular area of the hole of the body and the smaller the ${{P}_{N}}$, which is beneficial to the force of the trapezoidal hole. However, the increase in ${{D}_{2}}$ will reduce the structural strength of the body. Therefore, it is necessary to optimize the size of the trapezoidal holes. For the body, the increase in the inner diameter ${{D}_{3}}$, outer diameter ${{D}_{4}}$, and height H is advantageous for structural strength, but the increase in the size of the structure will reduce the rigidity of the structure.It is not conducive to achieving the goal of a lightweight structure. In order to obtain a more reasonable structure of the pile body, an optimized design is required for the inner diameter, outer diameter, and height of the body [4,10].

The circumferential deformation of the body is f₁. In order to ensure the safety of the hydraulic cylinder, the equivalent strain of the body needs to meet the stiffness requirement, i.e., ${{f}_{1}}\le 0.01{{D}_{3}}$. In the previous design, the outer diameter of the hydraulic cylinder is $\phi 260\text{mm}$ and the outer diameter of the hydraulic cylinder flange is $\phi 320\text{mm}$. To ensure that the hydraulic cylinder can be installed in the small hole of the body, the conditions $260\text{mm}<{{D}_{1}}<{{D}_{2}}$, ${{D}_{2}}>320\text{mm}$ must be satisfied. The size of the diameter of the 84-inch pipe pile is $\phi 2134\text{mm,}$ and then ${{D}_{3}}$ satisfies the condition ${{D}_{3}}>2134\text{mm}\text{.}$ The minimum value of the maximum equivalent stress ${{\sigma }_{ea}}$and the minimum value of the bulk volume Vwere taken as the objective function. The diameter of the trapezoidal hole ${{D}_{1}}$, major diameter ${{D}_{2}}$, inner diameter of the body ${{D}_{3}}$, outer diameter ${{D}_{4}}$, and height Hare the optimal design variables.

The mathematical optimization model of the body of the pile gripper is:

Where ${{\sigma }_{ea}}$and V are target variables,X is the design variable, and ${{f}_{1}}$ is the state variable.

The sensitivity of the optimization model refers to the degree of influence of the design variables on the output of the model [11]. In this paper, before the optimization calculation, it is necessary to verify that the design variables involved in the optimization model have a significant impact on the ontology equivalent stress and the volume of the ontology, which was used to ensure the selected design variables are reasonable.

The relationship among the target variable V and each design variable is

From Equation (5), it can be seen that the volume of the body is related to the design variables mentioned in Equation (4). The inner diameter, outer diameter, and height of the body are larger, and the influence on the body volume is larger. The size of trapezoidal holes is small and has a little effect on the volume of the body. The above five design variables are valid for the target variable V.

For the target variable ${{\sigma }_{ea}}$, the influence of each parameter quantity is not a simple linear relationship, and it is difficult to obtain a specific functional equation. The optimal gradient method in ANSYS software is the proper method to obtain the maximum equivalent stress sensitivity of the design variables to the body. The sensitivity curves of the parameters to the maximum equivalent stress of the body structure are shown in Figures 7,8,9,10,and 11.

From Figures 7 to 11, it can be seen that as the diameter of the small diameter ${{D}_{1}}$ increases, the effective stress of the body increases. Because the increase in ${{D}_{1}}$ will reduce the area of the annular surface of the hole, this makes the annular surface stress of the hole ${{P}_{N}}$ increase. When the diameter ${{D}_{2}}$ increases, the area of the annular surface of the small hole increases, the force on the annular surface decreases, and the maximum effective stress of the body decreases.

However, with an increase in the diameter ${{D}_{2}}$ of the hole continuously, the strength of the body structure will decrease. The maximum equivalent stress starts to increase, which is consistent with the previous force analysis results. The change in the small hole size has a significant effect on the maximum equivalent stress of the body. Therefore, ${{D}_{1}}$ and ${{D}_{2}}$ could be selected as the design variables for reducing the maximum equivalent stress.

It can be seen from the above figures that the increase of the inner diameter ${{D}_{3}}$, outer diameter ${{D}_{4}}$ and height H of the body causes the maximum equivalent stress of the structure to decrease. However, the influence of these three variables on the maximum equivalent stress is relatively small. Since the dimensions of the inner diameter ${{D}_{3}}$, outer diameter ${{D}_{4}}$, and height H of the body are large, the influence on the volume of the body is large. Therefore, the inner diameter ${{D}_{3}}$, outer diameter ${{D}_{4}}$, and height H of the body can be selected as the design variables for the lightening of the structure.

According to the optimization model of Equation (4), the body structure of the pile gripper is optimized. After several iterations, six groups of non-inferior solutions are obtained. The data before and after the optimization are shown in Table 2.

Since the influence of each parameter on the objective function is inconsistent, it is necessary to adopt an appropriate method to select the best design solution. This paper uses the fuzzy matter-element method [12] to select the best solution to obtain the equivalent stress and minimum volume that meet the requirements. In order to satisfy the calculation, the parameters were defined. ${{x}_{1}}$,${{x}_{2}}$,${{x}_{3}}$, and ${{x}_{4}}$ represent the small diameter of the hole ${{D}_{1}}$,large diameter of the hole ${{D}_{2}}$,internal diameter of the body ${{D}_{3}}$, and outer diameter of the body ${{D}_{4}}$ respectively, and ${{x}_{5}}$ represents the height of body H. The concrete steps of the fuzzy matter element method are as follows.

(1) According to the known dezsign results, the judgment matrix R of this method was first obtained.

Where ${{M}_{i}}$(i =0, 1, ⋯, 6) represents the evaluation index and ${{A}_{j}}$(jz=0, 1, ⋯, 6) represents the program.

(2) After the matrix R was obtained, the parameters were converted to dimensionless blur values. In this paper, range conversion is the most appropriate method.

The benefit indicatoris

Where ${{x}_{ij}}$ is the corresponding value of ${{M}_{i}}$ in scheme ${{A}_{j}}.$

The cost index is

For the optimized parameters, a smaller maximum equivalent stress and volume of the body are better, and they were selected as the cost index; in other parameters, a small diameter ${{D}_{1}}$ and large diameter ${{D}_{2}}$ mainly affect the equivalent stress, and the impact on the volume of the body is small. An increase in ${{D}_{1}}$ will increase the maximum equivalent stress and therefore should be as small as possible, and it should be selected as the cost index. When ${{D}_{2}}$ increases, the maximum equivalent stress decreases first and then increases. As can be seen in Figure 8, when the inner diameter of the body ${{D}_{2}}$ satisfies the relationship ${{D}_{2}}\le 380\text{mm}$, the maximum equivalent stress decreases with an increase in ${{D}_{2}}$. When the relationship ${{D}_{2}}>380\text{mm}$ is satisfied, the maximum equivalent stress increases with an increase in ${{D}_{2}}$. In the design, the size of ${{D}_{2}}$ satisfies the relationship ${{D}_{2}}\le 380\text{mm,}$ and the maximum equivalent stress decreases with an increase in ${{D}_{2}}$, whichis selected as the benefit index.

The internal diameter ${{D}_{3}}$, outer diameter ${{D}_{4}}$, and height H of the body have little influence on the equivalent stress, while they have a greater influence on the volume of the body. It can be seen from Equation (5) that the body volume increasesas ${{D}_{3}}$ and H increase. Therefore, the smaller the two parameters, the better. The two parameters were selected as the cost index. When ${{D}_{4}}$ increased, the volume of the body was reduced. Therefore, the bigger the parameter, the better the result, and it was selected as the efficiency index. The corresponding membership matrix ${R}'$ is as follows.

(3) The membership matrix was obtained, and then the entropy weight method was used to obtain the weight of each design scheme, which was denoted as a vector ${{W}_{j}}$.

The entropy weight method was used to obtain the weight of each parameter. The weight represents the importance of each parameter in the decision. The greater the weight, the more important the representative parameter. Using this method, the pros and cons of each solution can be analyzed more objectively [13-14]. According to the matrix R and ${R}'$ obtained, the entropy value is calculated according to the definition. If the evaluation object n=7 is known, the entropy of the i^th evaluation index is as follows.

Where

${{p}_{ij}}=\frac{1+{{r}_{ij}}}{\sum\limits_{j=1}^{n}{(1+{{r}_{ij}})}}$

In the equation, ${{r}_{ij}}$ is the value in matrix${R}'.$

According to the definition, the available matrixcan be calculated:

${{g}_{i}}=[\begin{matrix} 0.8450 & 0.8324 & 0.8452 & 0.8451 \\ \end{matrix}$$\begin{matrix} 0.8451 & 0.8451 & 0.8449{{]}^{T}} \\ \end{matrix}$

The entropy weight can be obtained:

The number of indicators m=7, from which the vector shown below can be obtained:

${{W}_{j}}=[\begin{matrix} 0.1413 & 0.1526 & 0.1411 & 0.1412 \\ \end{matrix}$$\begin{matrix} 0.1412 & 0.1412 & 0.1414] \\ \end{matrix}$

(4) Calculate the performance vector of each solution:

$P={{W}_{j}}{R}'=[\begin{matrix} 0.126 & 0.132 & 0.135 & 0.132 \\ \end{matrix}$$\begin{matrix} 0.139 & 0.129 & 0.146] \\ \end{matrix}$

Based on the above results, it can be known that the sequence of optimization of the pile gripper body optimization programis ${{A}_{6}}$, ${{A}_{4}}$, ${{A}_{2}}$, ${{A}_{1}}$, ${{A}_{3}}$, ${{A}_{5}}$, ${{A}_{0}}$. Compared with the original structure scheme, these six parameters have been optimized, which shows that the optimization parameter selection is reasonable and the optimization method is feasible. Scheme 6 is the optimal scheme of the optimal design. After the optimization parameters were rounded, the optimization parameter values as shown in Table 3 were obtained.

According to the optimal parameter value, the body of the pile gripper was modelled. Staticanalysis was completed, and the equivalent stress diagram and equivalent strain diagram of the body optimizedare shown in Figure 12 and Figure 13.

From Figure 12 and Figure 13, it is known that the maximum equivalent stress of the body is 244.37MPa, which satisfies the strength requirement, and the stress decreases significantly. The maximum equal effect f₁ is 0.52mm, which meets the stiffness requirement ${{f}_{1}}\le 0.001{{D}_{3}}=2.21\text{mm,}$ and the maximum equivalent strain hasbeen significantly reduced before optimization.

The ANSYS software was selected to analyze and optimize the body of the pile gripper. Through the structural optimization design, it was determined that the 12-holes sleeve body of the 84-inch pile gripper has the best form of stress. Sensitivity analysis was used to determine the choice of smalldiameter of hole ${{D}_{1}}$, large diameter of hole ${{D}_{2}}$, internal diameter of body ${{D}_{3}}$, outer diameter of body ${{D}_{4}}$, and height of body H as the optimal variables. Through optimization analysis, six groups of solutions were obtained, and each parameter was normalized to a dimensionless value using the fuzzy matter-element method. Finally, the optimal design program was completed. After the optimization design of the body, the maximum equivalent stress of the structure was 244.37MPa, which is 14.75% smaller, and the volume of the body was 0.317m³, which is 4.8% smaller. The maximum equivalent stress of the body was reduced by the optimization design in this paper, achieving a lighter and a more reliable guarantee for the safe operation of the pile gripper. The optimization method in this paper provides reference for the design and optimization of the follow-up pile gripper structure.

This research is supported by the Central University Fund (HEUCFP201727), Industrialization Application of the Gripper of Jacket Skirt Pile, and Harbin Engineering University Qingdao Ship Science and Technology Co., Ltd.