Mathematical Problems in Engineering

Volume 2015, Article ID 828930, 12 pages

http://dx.doi.org/10.1155/2015/828930

## Reliability-Based Design Optimization for Crane Metallic Structure Using ACO and AFOSM Based on China Standards

Mechanical Engineering College, Taiyuan University of Science and Technology, Taiyuan, Shanxi 030024, China

Received 9 July 2014; Revised 4 January 2015; Accepted 19 January 2015

Academic Editor: Paolo Lonetti

Copyright © 2015 Xiaoning Fan and Xiaoheng Bi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The design optimization of crane metallic structures is of great significance in reducing their weight and cost. Although it is known that uncertainties in the loads, geometry, dimensions, and materials of crane metallic structures are inherent and inevitable and that deterministic structural optimization can lead to an unreliable structure in practical applications, little amount of research on these factors has been reported. This paper considers a sensitivity analysis of uncertain variables and constructs a reliability-based design optimization model of an overhead traveling crane metallic structure. An advanced first-order second-moment method is used to calculate the reliability indices of probabilistic constraints at each design point. An effective ant colony optimization with a mutation local search is developed to achieve the global optimal solution. By applying our reliability-based design optimization to a realistic crane structure, we demonstrate that, compared with the practical design and the deterministic design optimization, the proposed method could find the lighter structure weight while satisfying the deterministic and probabilistic stress, deflection, and stiffness constraints and is therefore both feasible and effective.

#### 1. Introduction

As tools for moving and transporting goods, cranes are used in various settings to aid the development of economy and undertake the heavy logistical handling tasks in factories, railways, ports, and so on. Metallic structure is mainly made out of rolled merchant steel and plate steel by welding method according to certain structural organization rules. Crane metallic structures (CMSs), also called crane bridge, bear and transfer the burden of crane and their own weights. CMSs are mechanical skeleton and form the main components of cranes. Their design qualities have direct impact on the technical and economic benefit, as well as the safety, of the whole crane.

Generally, a CMS requires a large quantity of steel and consequently weighs a considerable amount. Its cost accounts for over a third of the total cost of the crane. Thus, under the condition of satisfying the relevant design codes, improving the performances of the CMS, saving material, and reducing weight have important significance in terms of cost-savings. As a consequence, various scholars have researched on this problem, and current optimization methods mainly include finite element method [1], neural networks [2], and Lagrange multipliers [3], amongst others [4–6]. However, these methods are all based on deterministic optimum designs and do not consider the effect of randomness in the structure and/or load. Studies have shown that the loads effected on the CMS, and the materials and geometric dimensions of the structure itself are uncertain. Deterministic optimum designs are pushed to the limits of their constraints boundaries, leaving no room for uncertainty. Optimum designs obtained without consideration of such uncertainties are therefore unreliable [7, 8]. Recently there have been a few reports about reliability-based design of crane structure [9, 10]. Literature [9] researched on the reliability-based design of structure of tower crane based on the finite element analysis (FEA) and response surface method (RSM). Meng et al. [10] analyzed the reliability and sensitivity of crane metal structure by means of BP neural network based on finite element and first-order second-moment (FOSM) method. Nevertheless, design only considering reliability could not guarantee the best work performance and the optimal design parameters. The aim of a design is to achieve adequate safety at minimum cost under the condition of meeting specified performance requirements. Hence, the optimization based on reliability concepts appears to be a more rational design philosophy, which is why reliability-based design optimization (RBDO) has been developed. RBDO incorporates the optimization of design parameters and reliability calculations for specified limit states. At present, it is attracting increased attention, both in theoretical research and practical applications [11–13]. Despite advances in this area, few RBDO approaches specific to CMS have appeared in the technical literature. Therefore, this paper develops an RBDO methodology for optimizing CMS that both minimizes the weight and guarantees structural reliability. The main structural behaviors are modeled by the crane design code (China Standard) [14] based on material mechanics, structural mechanics, and elasticity theory.

CMSs are engaged in busy and heavy work. It must have sufficient strength, stiffness, and stability under complex operating conditions. Their design calculations involve the hyperstatic problem of space structures. Therefore, both the calculating model and design calculation are very complicated. Furthermore, in practical production, structural dimensions are usually taken for integral multiples of millimeters and the specified thickness of the steel plates [15]. Due to these manufacturing limitations the design variables cannot be considered as continuous but should be treated as discrete in a large number of practical design situations, which means that the CMS design optimization is a constrained nonlinear optimization problem with discrete variables, known as NP-complete combinatorial optimization. To solve such problems, recent studies have focused on the development of heuristic optimization techniques, such as genetic algorithms (GAs) [16, 17], particle swarm optimization (PSO) [18–20], ant colony optimization (ACO) [21, 22], big bang-big crunch (BB-BC) [23, 24], imperialist competitive algorithm (ICA) [25], and charged system search (CSS) [26]. These algorithms can overcome most of the limitations found in traditional methods, such as becoming trapped in local minima and impractical computational complexity [27, 28]. In view of the simple operation, easy implementation, and the suitability of ACO for computational problems involving discrete variables and combinatorial optimization, the optimization process of BRDO described in this paper is performed using ACO with a mutation local search (ACOM) [29].

Structural reliability can be analyzed using analytical methods, such as first- and second-order second moments (FOSM and SOSM) [30] and advanced first-order second moment (AFOSM) [31], or with simulation methods such as Monte Carlo sampling (MCS) method. The FOSM method is very simple and requires minimal computation effort but sacrifices accuracy for nonlinear limit state functions. The accuracy of the SOSM method is improved compared with that of the FOSM, but its computation effort is greatly increased and this makes it not frequently used in practices. MCS method is accurate; however, it is computationally intensive as it needs a large number of samples to evaluate small failure probabilities. The AFOSM method, a more accurate analytical approach than the FOSM method, is able to efficiently handle low-dimensional uncertainties and nonlinear limit state functions [32] and is applied in most practical cases. It is used to calculate the reliability indices of RBDO in this paper.

The paper is organized as follows. Section 2 outlines the general formulation of discrete RBDO and then Section 3 constructs the RBDO model of an overhead traveling CMS (OTCMS). Section 4 develops the ACOM algorithm used for the optimization process of the RBDO and Section 5 describes the AFOSM method applied for reliability analysis. The RBDO procedure is illustrated in Section 6. Some applied examples that demonstrate the potential of the proposed approach for solving realistic problem are presented in Section 7, followed by concluding remarks in Section 8.

#### 2. Formulation of Discrete RBDO

In contrast to deterministic design optimization (DDO), RBDO assumes that quantities related to size, materials, and applied loads of a structure have a random nature to conform to the actual one. The parameters characterizing these quantities are called random variables, and these need to be taken into account in reliability analysis. These random variables may be either random design variables or random parameter variables. In optimization process, the mean values of the random design variables are treated as optimization variables. The formulation of discrete RBDO problem is generally written as follows: where and are the mean value vectors of the random design vector and random parameter vector , respectively; is the objective function (i.e., the structure weight or volume); and are the deterministic and probabilistic constraints; denotes the probability of satisfying the th performance function and this probability should be no less than the desired design reliability are the number of deterministic and probabilistic constraints, respectively. can only take values from a given discrete set , where and are the number of random design and parameter vectors, respectively.

#### 3. RBDO Modeling of an OTCMS

Cranes are mechanically applied to moving loads without interfering in activities on the ground. As overhead traveling cranes are the most widely used, a typical OTCMS is selected as the study object. As shown in Figure 1, it is composed of two parallel main girders that span the width of the bay between the runway girders. The OTCMS moves longitudinally, and the two end carriages located on either sides of the span house the wheel blocks. Because the main girders are the principal horizontal beams that support the trolley and are supported by the end carriages, they are the primary load-carrying components and account for more than 80% of the total weight of the OTCMS. Therefore, the RBDO of OTCMS mainly focuses on the design of these main girders. The crane’s solid-web girder is usually a box section fabricated from steel plate, for the main and vice webs, top, and bottom flanges, as shown in Figure 2. Therefore, given a span, its RBDO is to obtain the minimum dead weight, that is, the minimum main girder cross-sectional area, which simultaneously satisfies the required deterministic and probabilistic constraints associated with strength, stiffness, stability, manufacturing process, and dimensional limits [14]. In this paper, a practical bias-rail box main girder is considered. The mathematical model of its RBDO is given in Table 1.