Reliability Evaluation of Bridges Based on Nonprobabilistic Response Surface Limit Method

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.


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][12][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.
Standardize the interval variable   through the following transformation formula: where    = (   +    )/2 and    = (   −    )/2, which represent the mean and dispersion of   , and    and    are the upper and lower values of   , respectively.Therefore, the interval variable sets x = { 1 ,  2 , . . .,   } can be transformed into the standardized interval variable sets  = { 1 ,  2 , . . .,   }.The region of variation of the standardized interval variable   is [−1, 1], and the extended range of   is [−∞, +∞] [15].Substitute  = { 1 ,  2 , . . .,   } into the failure surface (x) = 0; then the continuous performance function can be transformed into the standardized performance function: In practice engineering, the performance function is generally expressed as where  is the resistance and  is the load.(1) For the Linear Performance Function Standardize  and :  =   +   ,  =   +   , and then the linear performance function will be transformed into and the nonprobabilistic reliability index is estimated as [24] which can be also expressed using the geometric illustration in the two-dimensional surface (Figure 1).
(2) For the Nonlinear Performance Function.The nonprobabilistic reliability index based on the interval model is defined as [16,24] Here the interval variables u satisfy ( 1 ,  2 , . . .,   ) = 0, where ‖u‖ ∞ is the infinite norm of the vector u; namely, In practice, the nonprobabilistic reliability index is the minimum distance from the origin of coordinates to the failure surface measured by the infinite norm in the regularized standard space.The above nonprobabilistic reliability index for the nonlinear performance function can be expressed using the geometric illustration in the two-dimensional surface (Figure 2).
Figure 2: Reliability index of nonlinear function.

Reliable region
Failure region

Nonprobabilistic Response Surface Limit Method
3.1.Limit Method Basic Principle.Fan and Chen [25] pointed out that all MPPs only exist among the extreme points and the root points of the limit state function.Obviously, the tangent line must exist at the extreme points of the failure surface.
Taking the two-dimensional case as an example, the square is used to approximate the performance function curve when the length of side of the square, with the origin of coordinates as its center, is increasing from 0. With increased length of the side of the square, one side of the square will be tangent with the performance function curve, or one vertex of the square will intersect with the performance function curve.The maximum of the absolute values of coordinates for the earliest intersection is the desired nonprobabilistic reliability index.
For two-dimensional case, the MPPs may be in the region above the 45 ∘ line, in the region below the 45 ∘ line, or on the 45 ∘ line (the boundary of these two regions).The limit method will also be discussed for these three cases, respectively (Figures 3, 4, and 5).[26] and was mainly used to fit the limit state surface when the performance function was not known in practical engineering.Myers and Montgomery [27] perfected the proposed basic principle of the response surface method, which was the theoretical basis for the development of the response surface method.

Response Surface Method. The response surface method was initially presented by Box and Wilson
For linear function, the first-order response surface is generally selected to approximate the real limit state function.The established response surface function is as follows: For high-order nonlinear function, the quadratic response surface is selected to approximate the real limit state function.The established response surface function is as follows: In ( 11) and ( 12), ,   ,   , and   are the undetermined coefficients.

Nonprobabilistic Response Surface Limit Method.
The task of the nonprobabilistic response surface limit method is to solve the upper and lower values of the nonprobabilistic reliability index.Compared with the traditional nonprobabilistic response surface method, not only will the oscillating and nonconvergent phenomenon not appear in the iteration process of the nonprobabilistic response surface limit method, but also this iteration process is simpler and more rapid.The nonprobabilistic reliability index is finally obtained by iterating step by step.The basic steps are follows.
(3) Solve all the nonprobabilistic reliability indices  1 1 ,  1 2 , . . .,  1   using the Matlab software.Then obtain the maximum  max 1 which is not larger than the initial value of the parameter  (if  1  1 ,  1 2 , . . .,  1  are all larger than the initial value of the parameter , the selected initial  value is small and should increase the  value).This process is also called the first iteration using the nonprobabilistic response surface limit method, as shown in Figure 6.
(4) Determine the new parameter points using the obtained  max 1 and then compute the nonprobabilistic reliability index until reaching the maximum  max 2 , which is not larger than  max 1 .This process is called the second iteration.(5) Perform this iteration until the equation has no solution or the solution of the equation is not smaller than  max −1 .Then the nonprobabilistic reliability index  min (namely,  max  ) will be obtained.Actually here  min is equal to  max −1 .This process is called the th iteration.
This final iteration process is also called the ( + 1)th iteration using the nonprobabilistic response surface limit method, as shown in Figure 7. Note.Relative error is obtained by comparing the nonprobabilistic reliability index calculated using other methods with that calculated using the analytical method.

Failure domain
Reliable domain
Standardize the limit state function as shown by (14), so the performance function can be expressed as the following form again: First, the global optimal solution method is used, and the nonprobabilistic reliability index can be computed; that is,  = 1.7191.Next, the nonprobabilistic reliability index will be computed using the nonprobabilistic response surface limit method again.Letting  = 2, after selecting the initial point, the fitted performance function can be given as follows: After the regularization of the above fitted performance function, the nonprobabilistic reliability indices are solved:  = (−9.6727,−4.7119, 2.2084, 20.6377, 1.7191, 26.512).The maximum of the nonprobabilistic reliability indices, which are smaller than  = 2, is 1.7191; that is,  = 1.7191.Then letting  = 1.7191, after the iterative computations,  = 1.7191 can be computed again.This result satisfies the condition of convergence, so the nonprobabilistic reliability index is finally obtained:  = 1.7191.
The nonprobabilistic reliability index computed using the traditional nonprobabilistic response surface method and that computed using the presented nonprobabilistic response surface limit method are summarized in Table 1.
Standardize the limit state function as shown by (17), so the performance function can be expressed as the following form again: First, the Gradient Projection Method [19] is used, and the nonprobabilistic reliability index can be computed; that is,  = 2.0397.Next, the nonprobabilistic reliability index will be computed using the presented nonprobabilistic response surface limit method again.After selecting the initial point, the computation process indicates that there is no solution with  = 2. Then let  = 3 again, and after four iterations the fitted performance function can be given as follows: With the above function, the results  min = 1.962 and  max = 2.178 can be obtained.Meanwhile, the nonprobabilistic reliability indices can be also solved:  = (−2.5383,−2.7875, −3.339, −2.0397).Due to the condition of  min <  <  max , the nonprobabilistic reliability index is  = 2.0397.Then letting  = 2.0397, after the iterative computations,  = 2.0397 can be computed again.This result satisfies the condition of convergence, so the nonprobabilistic reliability index is finally determined:  = 2.0397.
The nonprobabilistic reliability index computed using the traditional nonprobabilistic response surface method and that computed using the nonprobabilistic response surface limit method are summarized in Table 2.  Table 2 indicates that, in solving the high nonlinear problems, the usage of the proposed nonprobabilistic response surface limit method will greatly reduce the iterative steps compared to the traditional nonprobabilistic response surface method.The computational efficiency is significantly improved when there is no difference in the solving accuracy.

Example 3.
As shown in Figure 8, the concentrated loads  1 and  2 are applied on the cantilever beam, and the distances of these two concentrated loads from the fixed end are  1 and  2 , respectively.Suppose the cantilever beam is failed when | max | ≥  cr , where  cr is the critical ultimate bending moment and  max is the maximum of the bending moment of beam.The basic interval variables are 5, 5.5] m, and  cr ∈ [32, 40] kN⋅m.These parameters will be used in the estimation of the nonprobabilistic reliability for the structure.
Let  = 2; the performance function is established using the primary response surface: The nonprobabilistic reliability index computed using the one-dimensional optimization algorithm [15,16] is  = 1.7446.Finally, the nonprobabilistic reliability indices, computed using the one-dimensional optimization algorithm, the quadratic form response surface method, and the primary response surface method, are summarized in Table 3.
Table 3 indicates that the forms of the response surface functions have great influence on the final nonprobabilistic reliability index.The response surface function built greatly approximates the real limit state function, and the nonprobabilistic reliability index obtained greatly approximates the real value.

Example 4.
A PC (prestressed reinforced concrete) continuous rigid frame bridge with three spans is investigated.Three spans are 75 m + 130 m + 75 m, respectively, and the substructure is twin thin-wall pier.

Nonprobabilistic Reliability Evaluation
Process.The proposed nonprobabilistic response surface limit method will be used to evaluate the bridge reliability.The measured data is obtained for building the Finite Element Model (FEM).The nonprobabilistic reliability index is solved based on Note.Relative error is obtained by comparing the nonprobabilistic reliability index calculated using other methods with that calculated using the analytical method.the processed data using the Matlab software.The specific evaluation process is shown in Figure 9.The sensitivity analysis indicates that concrete compressive strength and area of the single prestressed tendon have great influence on the bending moment.Therefore, the reliability analysis will be conducted by selecting concrete compressive strength  and area of the single prestressed tendon  as the variables.The Finite Element Model is established using the Midas software, as shown in Figure 10.The maximum resistance and effect values of the control section will be also computed using the Midas software.The nonprobabilistic reliability analysis of the structure will be performed by selecting the midspan section as the control section.
4.4.3.Reliability Evaluation before Reinforcement.Let  = 2; the initial parameters of the model, the resistance (), and the effect () computed using the Midas software are summarized in Table 4. Now the proposed nonprobabilistic response surface limit method is utilized to solve the nonprobabilistic reliability index based on the parameters in Table 4.The fitted performance function can be given as follows: After the regularization of the performance function, the nonprobabilistic reliability indices are solved:  = (4.218,132.733, 15.425, 0.686).The maximum of the nonprobabilistic reliability indices, which are smaller than  = 2, is 0.686; that is,  = 0.686.Then letting  = 0.686, after the first iterative computations, the iterative results are summarized in Table 5.
Next the proposed nonprobabilistic response surface limit method is again utilized to solve the nonprobabilistic reliability index based on the parameters in Table 5.The fitted performance function can be given as follows: ( After the regularization of the performance function as shown by (23), the nonprobabilistic reliability indices are obtained:  = (12.36,131.24, 46.64, 0.628).The maximum of the nonprobabilistic reliability indices, which are smaller than  = 0.686, is 0.628; that is,  = 0.628.Then letting  = 0.628, after the second iterative computations, the iterative results are summarized in Table 6.
Next the presented nonprobabilistic response surface limit method is again utilized to solve the nonprobabilistic reliability index based on the parameters in Table 6.The fitted performance function can be given as follows: After the regularization of the performance function as shown by (24), the nonprobabilistic reliability indices are computed:  = (19.768,−82.202, 58.167, 0.619).The maximum of the nonprobabilistic reliability indices, which are smaller than  = 0.628, is 0.619; that is,  = 0.619.Then let  = 0.619, after the third iterative computations, the iterative results are summarized in Table 7.
Next the proposed nonprobabilistic response surface limit method is again utilized to solve the nonprobabilistic reliability index based on the parameters in Table 7.The fitted performance function can be given as follows: After the regularization of the performance function as shown by (25), the nonprobabilistic reliability indices are computed:  = (21.680,−133.413,67.108, 0.618).The maximum of the nonprobabilistic reliability indices, which are smaller than  = 0.619, is 0.618; that is,  = 0.618.Here (0.619 − 0.618)/0.618= 0.0016 < 0.01, which satisfies the condition of convergence.In summary, the nonprobabilistic reliability index is finally determined, which is  = 0.618.

Reliability Evaluation after Reinforcement.
According to  = 0.618 < 1, the bridge is unreliable.Then 16 bundles of the external prestressed tendon are adopted to reinforce the unreliable bridge, and the nonprobabilistic reliability index for the reinforced bridge is solved again.
Let  = 2; the initial parameters of the model, the resistance (), and the effect () computed using the Midas software are summarized in Table 8.
Next the proposed nonprobabilistic response surface limit method is again utilized to solve the nonprobabilistic reliability index based on the parameters in Table 8.The fitted performance function can be given as follows: After the regularization of the performance function as shown by (26), the nonprobabilistic reliability indices are computed:  = (4.19, 131.191, 14.579, 1.791).The maximum of the nonprobabilistic reliability indices, which are smaller than  = 2, is 1.791; that is,  = 1.791.Then letting  = 1.791, after the first iterative computations, the iterative results are summarized in Table 9.

Conclusions
This article is the first to successfully propose a nonprobabilistic response surface limit method to perform nonprobabilistic reliability analysis for the structures based on the interval model.Using this method, nonprobabilistic reliability analysis can be performed with the conditions of unknown performance function and data shortage.Compared with the traditional nonprobabilistic response surface method, there are two other main advantages of the proposed method: (1) it effectively avoids the tedious and oscillating iteration process; (2) it greatly enhances computational efficiency and precision and obtains the nonprobabilistic reliability index rapidly.The complete nonprobabilistic reliability evaluation process for the bridge was illustrated and built through evaluating the reliability of one PC continuous rigid frame bridge with three spans.The results indicated that the reliability level of the selected PC continuous rigid frame bridge with three spans was very low, so that reinforcement measures were needed to improve the performance of this bridge.The further reliability estimation for the reinforced bridge demonstrated that the reliability level of the reinforced bridge was greatly improved, and the reinforcements were effective.In summary, this study illustrated the effectiveness and feasibility of the proposed nonprobabilistic response surface limit method in the bridge nonprobabilistic reliability estimation through theoretical analysis and example verification.

Figure 1 :
Figure 1: Reliability index of linear function.

( 6 )
Establish the new limit state function based on [ min ,  max −2 ].If and only if there exists  +1 or the condition of convergence can be satisfied,  +1 is the final value of the nonprobabilistic reliability index.The condition of convergence is as follows:      +1 −  max      <       max      ( = 0.01) .

Table 1 :
Comparison of calculation results.

Table 2 :
Comparison of calculation results.Note.Relative error is obtained by comparing the nonprobabilistic reliability index calculated using other methods with that calculated using the analytical method.

Table 3 :
Comparison of calculation results.