Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1964165, 10 pages

https://doi.org/10.1155/2017/1964165

## Reliability Evaluation of Bridges Based on Nonprobabilistic Response Surface Limit Method

^{1}School of Resource and Civil Engineering, Wuhan Institute of Technology, Wuhan 430073, China^{2}School of Civil Engineering & Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China

Correspondence should be addressed to Xiaoya Bian

Received 14 October 2017; Accepted 10 December 2017; Published 28 December 2017

Academic Editor: Xiangyu Meng

Copyright © 2017 Xuyong Chen et al. 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

Due to many uncertainties in nonprobabilistic reliability assessment of bridges, the limit state function is generally unknown. The traditional nonprobabilistic response surface method is a lengthy and oscillating iteration process and leads to difficultly solving the nonprobabilistic reliability index. This article proposes a nonprobabilistic response surface limit method based on the interval model. The intention of this method is to solve the upper and lower limits of the nonprobabilistic reliability index and to narrow the range of the nonprobabilistic reliability index. If the range of the reliability index reduces to an acceptable accuracy, the solution will be considered convergent, and the nonprobabilistic reliability index will be obtained. The case study indicates that using the proposed method can avoid oscillating iteration process, make iteration process stable and convergent, reduce iteration steps significantly, and improve computational efficiency and precision significantly compared with the traditional nonprobabilistic response surface method. Finally, the nonprobabilistic reliability evaluation process of bridge will be built through evaluating the reliability of one PC continuous rigid frame bridge with three spans using the proposed method, which appears to be more simple and reliable when lack of samples and parameters in the bridge nonprobabilistic reliability evaluation is present.

#### 1. Introduction

There are many unavoidable uncertainties in practical structure engineering. Traditionally, the probability model is utilized in the structural reliability analysis [1]. Probabilistic reliability analysis strongly depends on the probability distribution function, which relies on a large number of statistical data [2]. However, for some important and complicated structures, many uncertain parameters have little or no statistical data, which causes difficulties in accurate description of parameter distribution. In addition, probabilistic reliability is very sensitive to variations of model parameters. Small errors in statistical data can lead to considerable errors in the structure [2, 3].

Because of inadequate data, the probabilistic reliability is not useful for solving these practical problems. However, nonprobabilistic reliability can effectively deal with reliability problems when only few statistical data can be obtained. A nonprobabilistic convex model was first proposed in the 1990s by Ben-Haim [4, 5]. Ben-Haim and Elishakoff [6] proposed a nonprobabilistic safety factor to measure the nonprobabilistic reliability index using an interval theory. The nonprobabilistic reliability theories presented by Ben-Haim [4, 5] and Ben-Haim and Elishakoff [6] were not involved in probability at all and could overcome the inextricable difficulties faced by the traditional probability model. Therefore, nonprobabilistic reliability is an appropriate method when the available data of uncertainties are limited or absent. This was illustrated by Guo et al. [7, 8], who contrasted the probabilistic and nonprobabilistic reliability methods through modeling concepts, model construction, and formulations for computation. Nonprobabilistic reliability of structures has become a new, exciting research direction, and the corresponding research approach has also aroused wide attention from theory and engineering circles.

In summary, nonprobabilistic reliability analysis was generally based on the interval model [9, 10] or the convex model [11–13]. The nonprobabilistic reliability index based on the interval model is actually the minimum norm of the coordinate vector in the standardized space, and solving the reliability index is an optimum problem with the equality constraint. For the linear performance functions, the analytical expression of the nonprobabilistic reliability index can be easily obtained. However, the performance functions are generally nonlinear in practical engineering.

For simple nonlinear performance functions, Guo et al. [7, 8] suggested definition approach, transfer approach, and optimization approach to solve the nonprobabilistic reliability index. For complexly and strongly nonlinear performance functions, researchers frequently used an optimized iterative algorithm. For the hyperellipsoidal model, the nonprobabilistic reliability index is in accordance with the probabilistic reliability index in definition, so the design point method, successfully applied to the probabilistic reliability analysis, can be used for the nonprobabilistic reliability analysis [14]. For the reliability index defined by the Euclidean norm, the Most Probable failure Point (MPP) can be obtained along the normal direction of the limit state surface. For the nonprobabilistic reliability index based on the interval model defined by the infinite norm, the MPP might not be along the normal direction of the limit state surface.

In order to simplify the search process, other researchers suggested the one-dimensional optimization method [15, 16] and the space search algorithm [17]. These two methods are correct only for the linear performance function, because only part of the probable failure points is searched for the nonlinear performance function. If the performance functions are the normalized quadratic expression, the Sequence Quadratic Programming (SQP) can be used to solve the nonprobabilistic reliability index based on the interval model [18]. Recently, the Gradient Projection Method (GPM) was proposed to solve the nonprobabilistic reliability index [19]. GPM is the general method that is most suitable for the nonprobabilistic convex model, although the convergence process during the iteration needs special treatment. The interval model was used in the structural reliability optimization design [20, 21], although this issue is not discussed in this article in detail.

When the limit state function is not easily obtained, the above methods are not applicable. Jiang et al. [22] presented the nonprobabilistic response surface method based on the interval model, which contributed to solving the nonprobabilistic reliability index for the implicit performance function. Chen et al. [23] used the response surface method to build the explicit performance function, accepted the interval model and the ellipsoidal model to compute the nonprobabilistic reliability index, and compared the results from these two methods.

Until present, nonprobabilistic reliability analysis based on the convex model and the interval model has achieved some positive results and been applied in engineering practice. A great obstacle is solving the nonprobabilistic reliability index. The traditional nonprobabilistic response surface method is prone to lengthy and oscillating iteration processes, which leads to difficultly solving the nonprobabilistic reliability index. For these problems, this article will propose the nonprobabilistic response surface limit method based on the interval model and build the nonprobabilistic reliability evaluation process for bridges through evaluation of the reliability of one PC continuous rigid frame bridge with three spans.

#### 2. Nonprobabilistic Reliability Index Based on the Interval Model

Suppose is the set of the basic interval variables for the structures, where (). The performance function is given as

When is the continuous function of (), also is the interval variable. From the perspective of the nonprobabilistic reliability theory, the hypersurface is the failure surface of the structures. When , the structure is in the failure state; when the structure is safe.

If and represent the mean and dispersion of , respectively, the nonprobabilistic reliability index is estimated as

Based on (2), when , for any , which means the structure is in the safe state; when , for any , which means the structure is failed; when , and maybe occurs for any , which means the structure may be safe or at risk. The greater* η*, the safer the structure.

Standardize the interval variable through the following transformation formula:where and , which represent the mean and dispersion of , and and are the upper and lower values of , respectively. Therefore, the interval variable sets can be transformed into the standardized interval variable sets . The region of variation of the standardized interval variable is , and the extended range of is [15]. Substitute into the failure surface ; then the continuous performance function can be transformed into the standardized performance function:

In practice engineering, the performance function is generally expressed aswhere is the resistance and is the load.

*(1**) For the Linear Performance Function*

Standardize and : , , and then the linear performance function will be transformed intoand the nonprobabilistic reliability index is estimated as [24]which can be also expressed using the geometric illustration in the two-dimensional surface (Figure 1).