Weight Reduction of a Ship Crane Truss Structure Made of Composites

Abstract

Weight reduction remains a relevant topic for various ship structures, as it can improve their seagoing capacity, especially for high-speed and light-craft vessels, and make deck equipment more efficient. The introduction of new lightweight composite materials and the development of new processing technologies and computer modeling tools open up new opportunities in equipment design, but also require new design approaches, including those based on optimization techniques. This article presents an engineering technique for optimizing the design of a ship’s crane manipulator boom based on modern computer simulation tools and numerical optimization methods. The loads on the crane arising due to the ship’s motion were obtained and specified, and the requirements for the finite element model for strength and stability analysis were derived. Constructive, technological, and resource constraints were then derived to reduce the number of independent optimization parameters. The mass reduction problem was set and solved by combining the screening method and the Jaya algorithm. Applying this developed technique for optimizing the crane boom structure using 465 parameters, an 18% reduction in mass was achieved.

1. Introduction

One of the features of modern structural design is the use of advanced manufacturing technologies and new types of structural materials. The use of composite materials makes it possible to achieve required mechanical and operational properties at the design stage [1]. The introduction of new technologies in design leads to the complex problems of structural mechanics.

The general goal in designing ship cranes is to develop a structure that performs all specified operations and uses a minimum amount of material. The use of fiber-reinforced plastic (FRP) as a structural material [2,3], as well as modern technological processes for their production [4,5], removes a number of substantial restrictions, increasing the number of design options and making the development process more complicated. That makes the use of modern computer simulation and optimization tools crucial for designing hoisting mechanisms.

There are also certain difficulties associated with the use of FRP. In particular, it is well known that in case of FRP, the material and structure are formed simultaneously in the same technological cycle. Therefore, the mechanical properties of the finished structure are inextricably linked with the process technology, which requires experimental confirmation of the declared characteristics as they may strongly depend on the process conditions and on the equipment used.

However, despite the difficulties of designing structures made of FRP in comparison with those made of metals, the use of FRP can be justified by decisive advantages in operation. One example of such an application is a ship’s crane manipulator (CM). The advantages of FRP that may strongly increase the performance of certain types of cranes include low weight, resistance to extremely low temperatures, resistance to aggressive climatic and chemical influences, and low sensitivity to physical fields (low magnetism).

The use of FRP in marine engineering should be regulated by appropriate documents of classification societies, for example, Det Norske Veritas (DNV) [6]. These regulations cover hull structures and some ship equipment from FRP, but do not cover ship cranes. Despite that, the design of cranes could benefit from the replacement of some solid metallic parts with truss structures made of FRP. In such a design, cross sheets (frames) can be also made of FRP [7].

Several books and articles are devoted to optimization problems for truss structures.

Paper [8] presents algorithms suitable for optimization of the truss structure with discrete admissible diameters of the rods. The authors emphasize the convergence problems of new methods in relation to specific engineering problems. Papers [9,10] present efficient integer programming models for truss topology optimization problems. However, a crane boom has a standardized topology structure. Paper [11] describes a new machine-learning-combined method for constructing a composite framework. Despite the efficiency of the method proposed by the authors, its parameters adjustment and implementation in engineering practice is quite tricky. Articles [12,13] present methaheuristic algorithms for solving optimization problems of truss structures. However, there is no algorithm that could be used for efficient optimization of the entire crane boom structure. The use of the considered algorithms and methods to solve the optimization problem for such structures is difficult due to the large number of variables, unknown parameter correlation functions, several calculation cases, and the large number of additional constraints and dependencies imposed by the production technology.

So, a universal methodology for engineering use is needed. In this paper, we present an engineering technique for optimizing a design of the CM boom made of FRP.

In the first part, the description of the crane design and a comparison of the boom made of metal and of FRP are presented. In the second part of the work, the constraints for the optimization problem that allow for the reduction of the number of parameters and speeding up of the solution of the problem are obtained. In the third part, a technique for optimizing the truss structure of the FRP boom, which combines the use of screening and Jaya algorithm, is presented.

2. Description of the Design and Technology

2.1. Design

Conventionally, CM is a welded metal structure consisting of a boom, secondary boom, column, and crane pad. In this article, we consider the design of the telescopic boom, which is the most challenging part of the CM when it comes to its design. We have developed an FRP version of the boom, shown in Figure 1. Two types of elements are used in the novel FRP design: sheet elements and rod elements. The sheet elements are manufactured by vacuum infusion or contact molding technology, which are traditional for composite materials.

The CM boom consists of three sections. The first section consists of 11 frame spacings, the second of 15 frame spacings, and the third of 17 frame spacings. The developed FE model of the CM is shown in Figure 2.

2.2. Technology

For the rod elements we consider the robotic winding technology developed by BaltiCo GmbH for producing structural core elements of highly loaded composite products. The key element of this technology is the process of laying pre-penetrated fibre roving strands on previously created support structures. The strands are then hardened, which allows for the production of complex 2D or 3D structures made of composite rods. The laying process is realized by a robotic system specially developed for that purpose.

There are certain benefits and special features associated with this technology and the structures produced by it. For instance, the diameter of the rods can be optimized for the given loads, and the truss structure can be designed in a way that the main load acts along the fibre direction. That enables an optimum utilization of the carbon fibres and allows for the reduction of material consumption. Another benefit of the technology is that it uses a low-refined material like fibre roving instead of a prepreg material, which eventually allows for a certain cost reduction.

2.3. Properties of Material

To calculate the strength, the following safety factors were taken: the strength safety factor equal to 3, and the stability safety factor equal to 3. Properties of the materials are shown in

Table 1. Properties of materials.

Material	Carbon-Fiber Composite	Glass-Fiber Composite
Part	Rovings	Frames
E	112 GPa	110 GPa
$\nu$	0.3	0.3
$\rho$	1330 kg/m${}^3$	1540 kg/m${}^3$
${\sigma }_u$	447 MPa	603 MPa

${\sigma }_u$ is the ultimate compressive strength in 0${}^{\circ }$ direction.

.

3. Structural Analysis

Given the number of possible geometric variables and the complex loading state of carbon fiber structures for offshore applications, it is convenient to use the finite element method (FEM) to calculate strength and buckling instead of analytical methods. FEM makes it possible to describe deformed state of the boom with required accuracy taking into account complex nonlinear dynamics, material properties, etc. In addition, the use of FEM in the design process increases its speed due to the natural integration with CAD models.

Due to the general specifics of truss structures, only tension or compression occurs along the length of the truss element. Therefore, in this case of the gloabal analysis, we can ignore the anisotropic properties. We use a single-axis rod type of a finite element without bending moments for the truss element. Thus, in the local coordinate system of the fiber, we use only one axis of physical properties.

3.1. FE Model

To perform structural analysis, we built a linear FE model in accordance to requirements of DNVGL-CG-0127. The FE mesh was built using Beam 188 and Shell 181 linear elements with an average size of 100 mm. The mesh size and its refinement were chosen using both the stress convergence criterion and the energy error criterion. The mesh was built using Ansys meshing tools where scpecified mesh quality criteria were implemented. Multi-layer orthotropic structure of frames was modeled using SECDATA APDL commands. The rovings were modeled with isotropic material. To connect elements, we used a conformal mesh. The sections of the boom were connected using Fixed Joints MPC 184.

We also followed the requirements of DNVGL-CG-0127 for finite element analysis.

3.2. Load Cases

During lifting operations in the port and the sea, the CM is subject to the influence of loads from the weight of the cargo, inertia due to the motion of the vessel, wind loads, etc. The design and strength calculation of offshore cranes must be carried out in accordance with the rules of the maritime classification societies, for example, DNVGL-ST-0377 or EN 13852-1:2004(E). The rules include several basic design cases that take into account the above loads in the form of dynamic factors, heel angles, acceleration values that depend on the ship’s basic measurements and wave loads, and wind pressure. In general, more than ten combinations of loads, accelerations and positions of the CM are obtained.

Dynamic loads were converted to static loads with corresponding dynamic coefficients. An equivalent static load set is defined as a static load set which generates the same displacement field as that from a dynamic load at a certain time for different crane positions.

DNV rules consider 11 load combinations, which are divided into three groups: regular loads caused by the lifting and movement of cargo on the high seas and in the port, irregular loads caused by wind, temperature and environmental influences, and special loads caused by emergency situations. Ultimate loads are considered in the third group. From 11 load combinations we chose 6 the most dangerous.

To evaluate strength and stability of the structure, three characteristic positions of the CM (Figure 3) were considered in accordance with regulatory documents [6]. The load combinations include forces $S_k$ and P (Figure 4) representing the combined action of gravity, cargo weight, wind pressure, inertia forces caused by the ship’s motions, loads from the roll and trim, and dynamic loads caused by the operation of drives and mechanism, calculated for a given roll angle $\vartheta$. The calculation scheme is shown in Figure 4.

4. Constraints and Relationships of the Optimization Problem Parameters

Direct parametrization of the presented structure requires an extremely large number of parameters and makes using stochastic algorithms inefficient due to convergence issues [14,15,16]. To reduce the number of optimization parameters, it is necessary to introduce design, technological and resource constraints. Design constraints reflect the operating conditions and design principles of lifting mechanisms. Technological constraints reflect the specifics of production and material characteristics. Resource constraints reflect the results of preliminary design calculations, which allow for a significant reduction in calculation time.

As a result of analyzing the structure, including performing preliminary calculations as well as specifics of the considered manufacturing technology, we obtained recommendations for the choice of constraints that are presented in

Table 2. Design constraint.

	Imposed Constraint	Description
1	Implementation of the “fishing rod” principle. Starting from the root zone of the section, each successive diameter of the roving is less than or equal to the previous one	Considered loads allow identifying two characteristic types of deformation—longitudinal deformation and transverse bending of a cantilever beam. That leads to the conclusion that an optimal design should satisfy the “fishing rod” principle, according to which parameters of the cross-section decrease along the section’s length
2	Accounting for the symmetry, cross-section diameters of the rods are identical within a single frame spacing in the following groups: Upper flange of longitudinal rovings (3 rovings); Lower flange of longitudinal rovings (3 rovings); Longitudinal flange of rovings in the area of the neutral axis (2 rovings); Diagonal flange of rovings above the neutral axis (4 rovings).	Due to given the probabilistic nature of the loads, the boom structure may bend in different planes during operation, we neglect the insignificant difference in distance from the neutral axis
3	In the area of overlapping of the sections of the boom, constraint No. 1 is overridden. Diameter values in this area are not required to satisfy the “fishing rod” principle. However, within each section in this area, the diameters are considered to be the same.	Preliminary strength calculations have shown the localization of stresses and their excess of the limit values in the area of sections overlapping. In this regard, it is necessary to reinforce the structure locally in this area

,

Table 3. Technological constraint.

	Imposed Constraint	Description
1	Within each frame spacing, the ratio of the diameters of the longitudinal and adjacent diagonal rovings is 2:1 (the longitudinal rovings diameters are twice the diameters of the diagonal ones)	This constraint is due to the specifics of the robotic winding technological process. In addition, this relationship significantly reduces the number of parameters
2	The roving diameter increment step is 1 mm	The increment is chosen based on the diameter of one winding strand of the selected material. The multiplicity is determined by the diameter of the cured bundle with a fixed binder content value

and

Table 4. Resource constraint.

	Imposed Constraint	Description
1	The range of admissible values for the roving diameters is withing one half of the roving diameter from the initial approximation	Initial approximations are proposed based on the results of preliminary design calculations. Limiting the range of admissible values is required to use computational resources rationally

.

5. Optimization Technique

There are many books and articles devoted to problems of discrete parametric optimization and mathematical programming. In the mathematical formulation, it is a problem of hill climbing in the finite-dimensional space vector, bounded by a set of inequalities. Heuristic stochastic search algorithms, such as the genetic algorithm, are popular for solving such problems. These algorithms have proven to be efficient in engineering and scientific research [17,18]. However, with a large number of variables, the efficiency of these methods is significantly reduced due to the large number of particles (individuals) in the solution space. Large admissible regions of parameters and large number of local minima also complicate finding the solution. The proposed structure optimization algorithm represent an attempt to solve these difficulties by combining random search methods and heuristic algorithms.

The idea behind optimizing the structure of the CM boom is to reduce the weight of the boom without losing strength and stability while providing the necessary stiffness. Mathematically, the problem can be expressed as (1). It is important to emphasize that the restrictions in (1) must be satisfied for all considered load combinations.

Only maximum stresses from all design cases are taken into account. It also helps to simplify the objective function and the formulation of the optimization problem.

$$\begin{array}{cc}\hfill & MinimizeM=\sum _{i}m{r}_i+\sum _{j}m{f}_j+m_0,\hfill \\ \hfill & m{r}_i={\rho }_{roving}\frac{\pi d_i^2}{4}{l}_j,\hfill \\ \hfill & m{f}_j={\rho }_{frame}{S}_j{t}_j,\hfill \\ \hfill & {\sigma }_{ref1}⩽{\sigma }_1^i,{\sigma }_3^j,⩽{\sigma }_{ref2},\hfill \\ \hfill & {\sigma }_{ref3}⩽{\sigma }_1^i,{\sigma }_3^j,⩽{\sigma }_{ref4},\hfill \\ \hfill & k_{ref}⩽k\hfill \end{array}$$

(1)

where $m{r}_i$ is the mass of $i_{th}$ circular roving with diameter $d_i$, $m{f}_j$ is the mass of $j_{th}$ frame with thickness $t_j$, $m_0$ is the mass of other parts of the boom, ${\sigma }_1^i$, ${\sigma }_3^i$, ${\sigma }_1^j$ and ${\sigma }_3^j$ are maximum and minimum principal stresses in the $i_{th}$ roving and $j_{th}$ frame, respectively, k is the buckling load factor, $k_{ref}$ is the minimum buckling load factor, ${\sigma }_{ref1}$ and ${\sigma }_{ref3}$ are permissible minimum principal stresses for rovings and frames, ${\sigma }_{ref2}$ and ${\sigma }_{ref4}$ are permissible maximum principal stresses for rovings and frames, and $S_j$ is the cross-sectional area of the roving.

5.1. The Proposed Technique

The robotic winding technology used makes it possible to provide 465 different diameters of rovings using various paths and directions of continuous winding. The purpose of the optimal design procedure is the calculated selection of diameters that ensures the minimum use of material while ensuring the strength and reliability of the CM. It must also be carried out in accordance with the current regulatory rules for the design of ship cranes.

After calculating the constraints, the optimization of the design was carried out in three stages. During the study, various optimization algorithms were tested: screening, the multi-objective genetic algorithm (MOGA), and the adaptive multi-objective algorithm [19,20,21].

At step 1, each section was divided into three zones along its length: the root zone, the middle zone, and the overlapping zone. The length zones of the second section are shown in Figure 5. Within each section, the constraints presented in Table 1 were implemented. For top diagonal and low diagonal rovings within each of the zones, a separate parameter was used to describe diameters. Lateral longitudinal rovings were supposed to have a diameter per length zone. Top longitudinal and low longitudinal rovings were characterized by a parameter per frame spacing within the middle zone, and a parameter per length zone within the root zone and overlapping zone. Thus, the number of parameters was reduced by a factor of 7 from 465 to 67. Figure 5 shows the second section of the boom with the description of the rovings.

Based on the preliminary research involving various algorithms, the screening method (quasi Monte Carlo) was chosen for further solution. This method, compared to the others, showed consistently shorter times to find the best value. The method is effective at determinint the most probable area for finding extremal functions. The Hammersley distribution was chosen as a quasi-random distribution [22]. The Hamersley distribution is two dimensional. In the first dimension, it represents Vander Corput’s distribution of several equal intervals. Consistent use of this method allowed the determination of ranges of admissible parameter values.

At step 2, the number of parameters was first increased up to 168 to refine the solution. Each section was divided into the same three zones as used in the first step, but, contrary to the first step, top diagonal and low diagonal rovings of the middle zone were supposed to possess various diameters within each frame spacing. The same was assumed for longitudinal rovings in the middle zone.

However, to improve convergence and reduce calculation time, the following parameters of diagonal and longitudinal rovings were linked:

• Low diagonal to low longitudinal.
• Top diagonal to top longitudinal.
• Low diagonal to low longitudinal.

Diameters of diagonal rovings within each spacing were assumed to be equal to half the diameter of the longitudinal rovings in the same frame spacing. Using these links, the number of parameters was reduced to 89.

The imposition of links between diagonal and longitudinal rovings led to the need of increasing lower and upper bound values for the parameters to ensure strength and stability by at least 25%. The search method used was screening.

Step 2 of the optimization procedure results in the weight of the CM boom being 1154 kg, which is a weight reduction of 16% with respect to the initial structure.

At step 3, the Jaya algorithm [23] was applied to refine the parameters values, taking the results of step 2 as the initial configuration. The Jaya algorithm is an optimization method that has been rapidly developing in recent years. It combines the advantages of both evolutionary algorithms and methods based on swarm intelligence. It is based on screening out the worst solutions and replacing them with the best ones. The method is iterative and shows high efficiency in solving optimization problems.

An ANSYSWorkbench project was used in a form of a batch file using a Python library that allows working with project settings, i.e., the diameter values. Using the results of previous steps as initial approximations improved convergence of the algorithm of the problem in step 3.

Steps 1 and 2 above are preparatory, and are required to obtain a rational solution to the crane design problem in a reasonable time during the final step.

The main goal of the proposed technique is to simplify the design procedure without losing the accuracy of the result. It is necessary to note that the proposed method of the calculation of design parameters is strongly related to this type of structure (truss structure) and the load cases of the CM.

A schematic of the proposed algorithm is shown in Figure 6.

5.2. Restrictions

The proposed technique is quite efficient for engineers at early design stages. However, the structural analysis of theCMincludes only global analysis and does not account for local stress concentrations. Therefore, to finalize the design, an additional detailed structural analysis is required. Such an analysis should be based on a more detailed FE model of the CM, or on detailed models of its fragments and sub-modeling techniques.

For optimization, the calculation of the initial approximations of the parameters according to the strength conditions was carried out in accordance with the DNV rules. We recommend to avoid random initial approximations.

The considered models of materials for FE analysis in the study were linear. However, composite materials can behave non-linearly. Taking into account non-linear behavior of the structure can significantly increase the calculation time.

5.3. Results

Application of the proposed technique reduced the mass of the CM boom by 18%. At the same time, the number of calculation iterations using the screening and Jaya algorithm together was 250. The proposed optimization technique was compared with multi-objective genetic algorithm (MOGA), adaptive multi-objective algorithm and Jaya algorithm. Comparison of the results of applying the algorithms was based on the reduction in the mass of the arrow and the value of the objective function. The optimization calculation for each of the algorithms was limited to 24 h. Proposed technique showed a more efficient solution by 7–10% in terms of weight reduction. The total calculation time was about 16 h on a standard desktop with an Intel Core i7 10700K processor, which is quite acceptable for engineering use at the design stage. The weight reduction per iteration of the algorithm is shown in Figure 7.

The comparison of the original and optimized design is shown in Figure 8.

The direct stresses of rovings for one of the design cases are shown in Figure 9.

Numerical results of applying the proposed optimization technique at each step are shown in

Table 5. Results.

	Original Design	Step 1	Step 2	Step 3
Mass, kg	1370	1262	1154	1125
Weight reduction	-	8%	16%	18%
Level of permissible stress	85%	87%	90%	95%

.

Considering rather large number of parameters, finding the absolute minimum of the objective function may be extremely time-consuming and is not usually set as a practical goal. In engineering practice, a weight reduction of 18% is usually considered as a satisfactory one. Further weight reduction in the studied case leads to a dramatic increase in the required number of iterations, and is not thought reasonable.

6. Discussions

In general, the main difficulty of designing optimized structures made of modern materials by means of novel technological processes is the formulation of the optimization problem and its constraints taking into account all relevant factors (such as design, process, and resource factors). In practical calculations, a designer cannot achieve their goals using universal optimization algorithms. Additionally, using complex mathematical approaches may be too time-consuming for industrial engineering processes. Because of those reasons, optimization procedures are often abandoned, and geometric parameters of a structure under design are defined by a simple choice on the basis of strength analysis results. That results in designs that do not fully use the advantages of modern materials and the possibilities of technological design. In this study, we propose an engineering technique which allows for a reasonable quasi-optimal design for the considered type of structures to be obtained.

The proposed technique can be expanded by taking into account other design factors. One of the ways to improve its efficiency is including additional design parameters, such as the thicknesses of FRP plates. It is also possible to account for the manufacturing cost, introducing it as an optimization parameter. Other ways for further improvement of the optimization approach include sensitivity analyzing and taking into account optimal paths of the robot during the process of manufacturing.

7. Conclusions

An engineering technique for optimizing the CM truss structure was developed and implemented using ANSYS Workbench and Python software (version number 3.8). As part of the study, design, technological and resource constraints that allow reducing the number of independent parameters were formulated and used. The developed technique was tested and allowed to reduce mass of the CM boom from 1370 kg to 1125 kg, corresponding to a weight reduction of 18% with respect to the initial design developed during preliminary design calculations. The total required number of iterations was 250, which makes the technique quite efficient for engineering implementation.

8. Future Works

In this study, the proposed optimization technique was implemented using 3D finite-element computations with commercial Ansys software (version number 2020R2). However, there are many other approaches to solving problems of this type, such as reduced-order models, including those that allow accounting for nonlinear behavior of materials. Such approaches may be a subject for future research in this area.

The next step is to check the connection nodes with additional local strength analysis. In this article, the main emphasis was placed on the initial design stage, which includes the choice of the geometric dimensions of the main connections, taking into account the wide possibilities of this technology. The next step requires local analysis of nodes and connections with a final verification calculation. With regard to fatigue strength, it is usually taken into account in the first stage of design in safety factors (DNV rules). It can be noted that the technology described in the article allows introducing local amplifications if necessary when analyzing nodes.