A hybrid approach combining uniform design and support vector machine to probabilistic tunnel stability assessment

Structural Safety 61 (2016) 22–42

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A hybrid approach combining uniform design and support vector machine to probabilistic tunnel stability assessment Xiang Li a,⇑, Xibing Li a, Yonghua Su b a b

School of Resources and Safety Engineering, Central South University, Changsha, Hunan 410083, China College of Civil Engineering, Hunan University, Changsha, Hunan 410082, China

a r t i c l e

i n f o

Article history: Received 18 December 2013 Received in revised form 4 March 2016 Accepted 6 March 2016

Keywords: Tunnel stability Probabilistic assessment Limit state function (LSF) Response surface method (RSM) Support vector machine (SVM) Uniform design (UD)

a b s t r a c t Applications of the reliability-based method to stability evaluation of tunnel structures have become an ever-increasing concern over recent years. One critical challenge in conducting such a task is the implicit nature of the limit state function (LSF). To address this issue, the focus of this study is on, among others, the use of response surface method (RSM) by considering both the selection of the sampling method and the choice of the response surface form (as two major factors affecting the RSM’s performance). In this context, the current paper develops for tunnel-reliability analysis a hybrid approach combing an experimental design called uniform design (UD) and a regression device known as support vector machine (SVM). For the proposed hybrid approach, the UD is used to generate sampling points and then the SVM is employed to construct the response surface approximating the original inexplicit LSF. Such an approach integrates the merits of both UD and SVM used for complex nonlinear modelling. Three carefully selected tunnel examples are illustrated: one for a typical tunnel under relatively simplified tunnelling conditions and the other two for real-life tunnels. Comparisons are made to validate the computational accuracy and efficiency of the present approach. In particular, for the tunnel example where the LSF is known only implicitly through the numerical analyses (which is the scenario of many real-world applications in tunnel community), the obtained results further demonstrate the efficiency of this approach: it can be much more economical to achieve reasonable accuracy than the conventional RSMs when a small number of sampling data is used. Such comparisons made in this work verify the application potential of the developed hybrid approach for probabilistic tunnel stability assessment involving the implicit LSF. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction In geotechnical engineering analysis and design, the existence of inherent uncertainties and their importance for the problems of decision making, risk evaluation and management have widely been recognized since the pioneering work reported in the 1960’s [1,2]. The conventional deterministic approach to assessing the performance of geotechnical structures involves calculation of a factor of safety. Since such a method cannot explicitly and sufficiently characterize uncertainties and may sometimes be inclined to yield misleading results, it is desirable to employ a more logical and realistic treatment within a probabilistic framework (also called the reliability-based method) to cope with uncertainties in geotechnical structures [3]. The assessment of tunnel stability (or

⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (X. Li), [email protected] (X. Li), [email protected], [email protected] (Y. Su). http://dx.doi.org/10.1016/j.strusafe.2016.03.001 0167-4730/Ó 2016 Elsevier Ltd. All rights reserved.

safety including strength and serviceability) is one important geotechnical subject dominated heavily by uncertainties. One of the earliest suggestions to utilize the probabilistic principles in tunnels was made in 1983 [4]. Since the beginning of the 1990’s, a number of probabilistic studies of tunnels and other structures related to underground excavations have been published in the literature and the readers may refer to [5–14], for example. These valuable contributions set the foundation for illustrating the implementation and benefits of reliability-based method in this field. For the stability evaluation in tunnel community, owing to the innate complexity (e.g., the ground-support interaction) and the multiple effects of diverse kinds of factors (e.g., rock mass conditions, excavation techniques and support structures, see some classical monographs [e.g., 15–18]), the mechanical model (corresponding to deterministic modelling) exhibits to a large extent a highly nonlinear behavior in many situations. As such, it may hence be not possible in the subsequent reliability analysis

X. Li et al. / Structural Safety 61 (2016) 22–42

to obtain an explicit, closed-form limit state function (LSF); instead, the resulting LSF can only be expressed implicitly. In this environment, estimating the probability of failure by the direct computation of a multifold integral could often become intractable, since the computational challenge in determining the integral lies in the multiple evaluations of LSF [19,20]. Among other various approximation methods, the so-called ‘‘fast probability integration” method [21] (i.e., the first- and second-order reliability methods) may be viewed as one of the most extensively used tools. When employing such a method for tunnel structures, calculation of the LSF’s derivatives (which is essential to the reliability scheme) can be readily performed for the simple and explicit LSF [e.g., 22], whereas it could be hindered by the complicated and implicit LSF. To circumvent this problem, one possible way is to resort to the Monte-Carlo technique [e.g., 11], whose results obtained are known to be quite accurate except for the tremendous computational cost. With some improvements (like importance sampling [23], directional simulation [24], antithetic variates [25] and conditional simulation [26]) and further rampant growth in technology and availability of computational resources in the near future, wide application of this technique may be no longer infeasible for many practical problems. Yet the calculation at present is still quite timeconsuming [20,27]. Several other methods could also be pursued for the purpose. They can mainly be divided into three categories: (a) method where the computation of the derivatives is avoided, including the point element method [28,29] and the spreadsheet algorithm [13,14]; (b) method where estimation of the derivatives is conducted via simple approximations, involving the rational polynomial approximation [30] and the difference approximation (proposed by the author and co-workers [31]); and (c) method where the original complex and implicit LSF is replaced by the simple and explicit function (called the response surface method (RSM)), typically the polynomial-based RSM [e.g., 7,32–34], the artificial neural network (ANN)-based RSM [e.g., 9,35,36] and more recently, the Hermite polynomial-based RSM [e.g., 37]. For the aforecited RSM, it has become one well-established class of methods to solve probabilistic tunnel stability problems with the implicit LSF. Generally, the RSM’s performance is largely dependent on two factors: one is relevant to the sampling method for the location and number of data points selected to identify the response function and the other is pertinent to the response function shape adopted for fitting. Considering in tunnel-reliability analysis with complicated and inexplicit LSF, the polynomialand ANN-based RSMs are currently the most two representatives, we take such two types of RSM as an example to elucidate, respectively, the two factors as follows: First, for the former factor (i.e., the sampling method): in the context of polynomial-based RSM, the conventional factorial designs (e.g., the central composite design) may lead to the unacceptably high computational efforts with increasing the number of random variables for complex systems and even become more time consuming than direct Monte-Carlo method [38,39]. On the other hand the widely used interpolation scheme in sampling for the complicated LSF is hindered by the choice of an arbitrary parameter, whose extremely low variations may, however, trigger wild and unexpected fluctuations of the calculation results [40]. It is worth noting that the analyses of tunnel-reliability in [32] indicated another difficulty when using this interpolation scheme with the symmetrical pattern, and thus developed a modified sampling strategy in a nonsymmetrical manner. In the context of ANN-based RSM, one disadvantage of the frequently used random method in sampling ([9,41,42]) is that the randomly selected sampling points without uniformly covering the design space may lead to erroneous results, particularly when the number of variables is large and the number of sampling points is relatively small [27,43]. Additionally, since there is no-well defined criterion to determine

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the optimal network architecture, it is relatively difficult to identify the number of sampling points for building a good ANN model [44]. At this point, a case study for tunnel excavations in [45] proved the major effect of the number of sampling points used for the ANN model on how to reliably and completely predict the unknown relationship of weak geological zones. Secondly, for the latter factor (i.e., the response surface): in the polynomial-based RSM, the main limitations reside in like the existence of false design points [46], the difficulty in handling a large number of random variables, especially for mixed or statistically dependent ones [9,41], the biased approximation of results for cases without conforming to the true LSF’s nonlinearities [47] and the severe oscillations with increasing the polynomial order [20]. Particularly, the probabilistic stability analysis for tunnels in [32] showed that using the polynomial model fails to converge to the design point, and thus had to suggest a sophisticated function by modifying the polynomial model. In the ANN-based RSM, some issues are needed to be further examined like the difficulty in designing the network architecture, under- or over-fitting, local minimum and less generalization ability [48]. Several of them were documented in [49] when the ANN model had application in tunnelling for the ground movement prediction. Considering the potential inadequacies of the current RSMs in some applications and meanwhile the intrinsic complexity of tunnel stability issue, examination of a suitable approach that provides results with reasonable accuracy and also alleviates the computational cost as effectively as possible could always be desirable for the RSM’s application in tunnel structures. Motivated by this, we propose in the RSM framework a hybrid approach combining uniform design (UD; an experimental design corresponding to the sampling method) and support vector machine (SVM; a regression device corresponding to the response surface). The reason for developing such a hybrid approach is attributed to the appealing properties of UD and SVM both in modelling the complex nonlinear relationship. By integrating the merits of both UD and SVM, we here make an attempt to suggest an RSM to handle the implicit LSF in tunnel-reliability analysis. Although for either UD or SVM, so far each has been found for a range of engineering applications, the hybridization of UD and SVM in the RSM context for probabilistic tunnel stability assessment has yet to be investigated [50]. The remainder of this paper is framed as follows. In Section 2, some basic concepts of UD and SVM are, respectively, summarized, and this summary is followed by the concise presentation of the analysis procedure for the proposed hybrid approach. In the sequel, our emphasis is placed on its detailed applications in the context of tunnel structures in Section 3, where three selected tunnel examples are illustrated to demonstrate the accuracy and efficiency of our approach. Then Section 4 reviews the results obtained from the three examples, respectively. The conclusions are finally given in Section 5.

2. Proposed hybrid approach combining UD and SVM 2.1. Allocation of sampling points by UD To find an approximate model for the implicit LSF in tunnelreliability analysis, some merits of the UD provide us the most important motivation to use such a sampling strategy. They are concisely listed as follows (see [51,52] for detailed information): (a) help users in modelling with a small number of experiments; (b) accommodate the largest possible number of levels (i.e., representatives values taken) for each factor among many experiment designs; and (c) impose no strong assumption on the underlying model (i.e., the UD’s performance is robustness against changes of the underlying model).

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X. Li et al. / Structural Safety 61 (2016) 22–42

Generally, when modelling the nonlinear relationship between output (response) and factors, it would be desirable to accommodate multiple levels for each factor in order to obtain as good results as possible. Expanding on this notion in [53,54] manifested that a design allowing multiple levels for each factor is considered to be of importance to complex nonlinear modelling, since the actual situations for the underlying model may be bypassed when the levels are not taken adequately. On the other hand, to make the model-building not computational-intensive, use should be made of a design at the cost of as a small number of experiments as possible even in the circumstance of a large number of levels required for each factor. The UD is precisely such an experiment scheme. The fundamental idea behind the UD stems from the numbertheoretic (NT) method. Such a method is also regarded as the quasi-Monte Carlo method [51,55]. The reason is that the NT method provides a more efficient generation of a small number of sampling points in a deterministic manner than the generation based on the random sampling as in the Monte-Carlo method. Those sampling points produced via the NT method are called the NT-net, which are uniformly scattered with high representativeness on the investigated domain. Note that uniformity can be viewed as an important property for experimental designs [56]. In this context, as emphasized in [51,52], the above basic idea of NT method is introduced to the experimental design and then the UD is developed with the aid of NT-nets. The application of UD can be efficiently performed with tabular procedures. A series of UD tables have offered great convenience to users. The form of UD tables is represented as Um(qs), where U signifies the terminology of ‘‘uniform design”; m, q and s are the number of experiments, the number of levels for each factor and the maximum factors to be considered, respectively. It should be pointed out here that we consider a widely used type of UD tables with m ¼ q (i.e., the number of experiments equals the number of levels for each factor). For instance, Table A1 (see Appendix) denotes an U29(298) UD table, from which we can consider up to 8 factors (see ‘‘No. of columns” from Column 1 to Column 8). Each factor has 29 levels (the entries from 1 to 29 in each column emerge once equally), yet only 29 experiments (corresponding to the entries in the column under ‘‘No. of experiments”) are required. To employ Table A1, a recommendation regarding the choice of different columns can be made by using an accessory table. Table A2 in Appendix is precisely the accessory table of Table A1. From Table A2, one chooses the columns 1 and 6 for two factors to be considered (s = 2); one chooses the columns 1, 3 and 6 for three factors to be considered (s = 3), and so forth. It can be further observed from Table A2 that the notation ‘‘D” stands for ‘‘Discrepancy” in the last column with different values. For example, if s = 4, then the columns 1, 2, 5 and 7should be chosen and the resulting D equals 0.1596. This manifests that when considering 4 factors, the choice of columns 1, 2, 5 and 7 can guarantee the uniformity of the sampling points scattered on the experimental domain (i.e., a set within which the factors could possibly change) and the resulting discrepancy takes the minimum value of 0.1596. In the UD analysis, the discrepancy is an important index, which can be used to measure the degree of uniformity of the sampling points [51,52]. Usually, the smaller the value of discrepancy the more uniform those sampling points scattered over the domain will be. For more detailed exposition of UD tables, please refer to [57] or visit the UD-website at www.math.hkbu.edu.hk/UniformDesign. 2.2. Approximation of LSF by SVM Knowing those sampling points by the UD, the response surface model can be constructed within the experimental domains. This corresponds to the task of function approximation. We here adopt the SVM, whose main merits are as follows (see [58,59] for more

details): (a) highlight the small samples property especially suitable to most real problems with sampling limitations; (b) bypass the curse of dimensionality and learns the nonlinear relationship effectively; and (c) overcome the over-fitting problem and exhibit excellent generalization performance. We now summarize the underlying principles of SVM for function approximation (a thorough treatment can be found in [60,61]). For the linear case with the input vector x ¼ ðx1 ; x2 ; . . . ; xi ; . . . ; xn Þ, ði ¼ 1; 2; . . . ; nÞ, the representation can be written as f ðxÞ ¼ ðx  xÞ þ b, x; xi 2 Rd , b 2 R, where (xx) indicates the dot product of x and x; xi is the ith component (defined as a d-dimensional vector) of x; Rd and R are the d-dimensional and one-dimensional vector spaces, respectively; b and x are a scalar bias and an adjustable weight vector, respectively. The aim of function approximation is to seek an optimal function of f (x) that gives a deviation parameter e from the actual output and meanwhile is as flat as possible. The SVM uses a penalty parameter C to determine the trade-off between the flatness and the amount up to which the deviations larger than e are tolerated. Notice that the flatness for f ðxÞ ¼ ðx  xÞ þ b implies that one should obtain a small x. This can be fulfilled by minimizing the Euclidean norm ||x||2. Such an issue is equivalent to a convex quadratic optimization problem, which is solved by Lagrangian multipliers ai and ai⁄. For the nonlinear function approximation, the key idea is how to transform the nonlinear case into a linear one. The SVM uses the concept of kernel substitution. In brief, a kernel function K (xix) is employed to map the input data into a high-dimensional feature space and then a linear approximation is performed in such a mapped space. Hence the nonlinear case is expressed as follows (we here call it the SVM model):

f ðxÞ ¼

n X ðai  ai ÞK ðxi  xÞ þ b

ð1Þ

i¼1

To apply Eq. (1), besides the determination of the bias value b, deviation parameter e and penalty parameter C mentioned above, a kernel function K (xix) has also to be identified. Empirically, the lower complexity of kernel functions may be selected for the SVM [62]: for instance, two kernel functions commonly considered are the polynomial kernel: K ðxi  xÞ ¼ ðcxi x þ 1Þd , ðc 2 R; d 2 NÞ   and the radial basis kernel: K ðxi  xÞ ¼ exp ckx  xi k2 , ðc 2 RÞ, where both d and c are called the kernel parameters. Note also that in Eq. (1), some of Lagrangian multipliers ai and ai⁄ may vanish. This suggests one does not need the total sampling points for function approximation. Geometrically, the sampling points with nonzero Lagrange multipliers (i.e., ðai  ai Þ – 0) are referred to as the ‘‘support vectors”, which can be viewed as the critical elements of the total sampling points. The remaining points corresponding to ðai  ai Þ = 0 are regarded irrelevant to the final solutions (i.e., sparseness property). At this rate, the complexity of Eq. (1) is independent of the dimensionality of the input space, but pertinent mainly to the number of support vectors. For the above SVM analysis, a number of computer codes are readily available online. Among them, we choose the commonly used LIBSVM tool [63], which can be found at the website: www.csie.ntu.edu.tw/~cjlin/ libsvm. Now, we adopt one example from [9] to succinctly demonstrate the capability of the SVM model employed to map a complicated nonlinear function. Such a function has the form y ¼ x1 x2 þ ðx3 Þ2  ðx4 Þ1=4 , where x1, x2, x3 and x4 are four input variables and y is a single output. As shown in [9], the input samples randomly generated consist of 20 training samples and 10 testing samples, respectively (see Table 1). A comparison of the error estimations in [9] indicates the advantage of the ANN model in constructing the complicated nonlinear function,

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X. Li et al. / Structural Safety 61 (2016) 22–42 Table 1 Comparisons of results estimated by different modelling methods. x1

x2

x3

x4

Actual y

Computed y SVM model in this study

ANN model in [9]

Polynomial model in [9]

Training dataset 2.5 4.6 2.4 3.5 2.9 2.0 3.3 2.4 2.2 5.0 4.0 4.1 1.1 2.0 1.8 1.8 4.0 4.7 4.1 4.6 4.1 4.8 1.6 2.1 2.2 5.0 2.4 4.5 4.2 3.8 4.8 2.2 4.3 1.1 1.2 3.0 1.6 4.7 4.3 2.7 Mean square error (MSE)

3.7 2.1 1.5 4.2 2.2 1.8 1.2 4.9 2.1 3.1 2.6 4.1 3.1 3.9 3.7 1.7 2.8 4.5 4.2 3.2

1.3 3.0 1.0 3.0 3.7 1.2 4.3 1.9 4.6 3.3 2.4 3.6 1.4 4.1 3.5 4.2 2.4 2.7 1.5 1.8

24.12 11.49 7.05 24.24 14.45 18.59 2.20 26.08 21.75 27.12 25.20 18.79 19.52 24.59 28.28 12.02 11.33 22.57 24.05 20.69

24.1538 11.4621 7.0190 24.2756 14.4844 18.6090 2.1687 26.0448 21.7142 27.1537 25.1641 18.7610 19.5538 24.5555 28.2507 12.0496 11.3564 22.5998 24.0220 21.1858 0.0131

16.07 11.28 13.74 21.99 12.91 29.32 24.17 10.57 13.76 20.70 24.89 19.26 20.25 25.62 28.48 10.94 12.83 23.01 23.80 21.75 0.77

24.00 12.33 6.89 25.08 16.01 18.17 0.99 25.33 20.60 25.57 23.35 16.82 19.60 25.59 27.99 14.48 11.96 22.98 25.86 20.54 1.41

Testing dataset 3.3 1.5 2.9 2.4 2.5 3.0 5.0 2.5 2.8 1.2 4.3 4.4 5.0 2.4 4.2 2.2 3.4 4.0 2.5 1.5 Mean square error (MSE)

3.5 2.5 2.6 3.3 3.4 3.7 3.7 1.8 1.7 4.3

1.5 3.8 3.7 1.6 2.5 2.1 4.6 1.6 3.5 3.2

16.09 11.81 12.87 22.27 13.66 31.41 24.23 11.36 15.12 20.90

16.9099 12.0402 13.2253 22.3122 14.0904 30.8391 23.4675 11.4236 15.0861 20.7176 0.1964

16.07 11.28 13.74 21.99 12.91 29.32 24.17 10.57 13.76 20.70 0.86

14.57 11.39 12.63 22.44 11.29 29.89 25.07 12.63 16.55 19.77 1.62

because in Table 1 the mean-squared error (MSE) (as a common criterion for performance evaluation) of the polynomial model (i.e., 1.41 and 1.62 corresponding, respectively, to the training and testing samples) is about two times that of the ANN model (i.e., 0.77 and 0.86 corresponding, respectively, to the training and testing samples). Based on the same training and testing samples, we give in Table 1 the computed y values of the SVM model (here the radial basis kernel is taken as an example). It is mentioned that the values of squared correlation coefficient (R2) (as another metric assessing the model’s performance) obtained in the training and testing phases are, respectively, 0.9997 and 0.9973. This manifests the sufficiently high degree to which the computed y values from the SVM model match the actual y values of the original function. In fact, from Table 1 the MSE of the SVM model (i.e., 0.0131 and 0.1964 corresponding, respectively, to the training and testing samples) is alarmingly lower than that of either the ANN model or the polynomial model. Hence in such an investigated example, the SVM model outperforms both the polynomial and ANN models in [9]. 2.3. Reliability analysis procedure for proposed hybrid approach In this work, the procedure for tunnel-reliability analysis involving the implicit LSF can be outlined as follows: Step (1): identify the random variables for the geomechanical and construction parameters of tunnel structures, and mean-

while provide their associated statistical information (like the mean values and standard deviation). Step (2): employ the UD to generate the sampling points: (a) define the factors (i.e., input random variables obtained in Step (1)), the experimental domain and the number of levels for each factor; (b) choose an appropriate UD table and its accessory table (see Appendix; more tables can be found in [57] or at the website: www.math.hkbu.edu.hk/UniformDesign); and (c) record the outputs of experiments (‘‘experiment” corresponds to the evaluation of function values for the original limit state under tunnelling conditions). Step (3): use the sampling points produced in Step (2) to extract the SVM model of Eq. (1) (i.e., the response surface function) within the experimental domain (as mentioned in Section 2.2, this process can be implemented using the LIBSVM [63], whose computer codes are available online at www.csie.ntu.edu.tw/ ~cjlin/libsvm). Step (4): adopt the standard reliability method to compute the reliability index (or the probability of failure) through the SVM model obtained as the approximated explicit LSF. Here we consider the widely-used first-order method for this purpose, in which the distribution types and correlated properties of random variables are both considered. For non-normal distributions, this involves a transformation to the equivalent normal distributions; For correlated properties, this involves a transformation to uncorrelated variables by obtaining the eigen-values/ vectors. Correspondingly, Appendix lists the essential computer program coded with MATLAB by referring to [64].

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X. Li et al. / Structural Safety 61 (2016) 22–42

Before following the above procedure, it is essential to realize that the purpose of RSM is to approximate as sufficiently well as possible the boundary between failure and safe domains which are divided by the LSF; in particular in a very narrow region around the design point that really contributes most to the probability of failure [20] (also called the ‘‘region of most interest” [65]). This reflects the ‘‘local approximation” feature of RSM [66,67]. Such a feature, also as one key issue of our approach in the RSM framework, will be clarified in the next section. 3. Illustrative examples In this section, we will illustrate the applications of our approach to probabilistic tunnel stability assessment. Owing to the restricted space, three examples selected carefully are intended for illustration purposes. The first example is a representative one for the analysis of tunnel behavior, where the LSF is indeed simple and presented in an explicit manner. Using our approach for such an example offers no particular advantage. However, it should be stressed that the aim of this example is to demonstrate clearly all the steps of the calculation process (especially to graphically explain the local approximation feature in the RSM context) involved in the proposed hybrid approach, and then to discuss the usage of this approach in a form which can readily be understood. Of course, to further assess the effectiveness of our approach when the LSF is not explicitly known (since this is the main scope we employ the RSM), the complexity of LSF relevant to tunnel behavior increases gradually in the subsequent analyses. That is, implicit LSFs are included in both the second and third examples of real-life case study applications. 3.1. Example 1 – Consider hypothetical tunnel analyzed in [8] The first example pertains to tunnel stability calculations conducted by Hoek in the influential work [8]. A typical mechanical model was used to explain the basic concepts of rock support interaction. As illustrated in Fig. 1, such a model involves a circular tunnel (of radius r0) excavated in an elastic-perfectly plastic rock mass subjected to hydrostatic in situ stresses p0 and a uniform support pressure ps. When failure of the rock mass surrounding the tunnel occurs, there is a plastic zone of radius rp around the tunnel. On this basis, one LSF was defined as follows when the plastic zone extent (as quantified by the ðr p =r 0 Þ ratio) reaches the limiting (or permissible threshold) ratio L [13]: 1  k1 rp 2ðp0 ðk  1Þ þ rcm Þ gðxÞ ¼ L  ¼ L  ¼0 ðk þ 1Þððk  1Þps þ rcm Þ r0

ð2Þ

since k ¼ ð1 þ sin uÞ=ð1  sin uÞ and rcm ¼ ½cðk  1Þ= tan u, where g(x) is the performance function (or state function) and x is a

p0

rp

p0

p0 r0

ps

plastic elastic

p0

Fig. 1. Configuration of typical model for tunnel behavior (Example 1).

Table 2 Statistical information obtained from [8] for Example 1. Random variable

Mean value

Standard deviation

Angle of friction u (°) Cohesion c (MPa)

lu = 22.85 lc = 0.23

ru = 1.31 rc = 0.068

random vector; rcm, u and c are the uniaxial compression pressure, the angle of friction and the cohesion of the rock mass, respectively. Note that the performance function g(x) considered here is akin to the serviceability limit state, albeit it can also be means to control ultimate limit state [33]. Additionally, although the LSF of Eq. (2) is simple and expressed explicitly, such a closed-form solution can provide useful insights on designing realistic tunnels (e.g., how support operates) [8,13,68,69]. For the selection of the limiting ratio L, we give a brief explanation as follows. It has been pointed out in [8] that in a 5 m diameter tunnel (i.e., r0 = 2.5 m), a plastic zone diameter of 15 m (i.e., rp = 7.5 m) would certainly cause visible signs of distress (corresponding to ðr p =r 0 Þ = 3.0 considered as a maximum permissible value in this case) and the installation of a relatively simple support system is very effective in controlling the size of the plastic zone. To be consistent with the maximum of ðr p =r0 Þ ratio mentioned above, the limiting ratio L in Eq. (2) was assigned a value of 3.0 in [13] to illustrate the reliability analysis for this tunnel. 3.1.1. Analyses and results To check the feasibility and validity of our approach and meanwhile conveniently make comparisons between the results in this study and [13], we also use the LSF of Eq. (2) with L = 3.0 to perform the analysis of tunnel-reliability. The data in the calculations can all be traced to [8,13]: (a) three parameters r0, p0 and ps were assumed to be deterministic inputs, where r0 = 2.5 m, p0 = 2.5 MPa and ps was provided with different values from the construction perspective; (b) other two parameters u and c were regarded as random variables, whose statistical values (mean value lu and standard deviation ru for u; mean value lc and standard deviation rc for c) obtained via the Monte Carlo analyses (please refer to [68] for more details) are listed in Table 2; and (c) for different cases (which will be discussed below), u and c were assumed to be normally/non-normally distributed and negatively correlated with varying quc (where quc is the correlation coefficient between u and c). 3.1.1.1. Demonstration of calculation steps. Now, we take this typical tunnel with vanishing support pressure (ps = 0.0 MPa) and normal distributions for u and c with quc = 0.5 as an example to demonstrate our approach. In conformity with Section 2.3, the calculation steps can be explained below. Step (1): the angle of friction u and the cohesion c are two random variables, whose statistical information is shown in Table 2. Step (2): The UD is used to produce the sampling points, which involves: (a) Two random variables u and c are designated as two factors and the ðr p =r 0 Þ ratio in Eq. (2) is viewed as the output. For the experimental domains within which the factors u and c could possibly vary, the lower and upper values of realizations for each factor can, according to the calculated distributions for u and c in [8,68], be selected symmetrically around the mean and a multiple of 3 times the standard deviation. That is, the experimental domain for u is selected to fall in the range of [lu  3ru, lu + 3ru] = [18.92, 26.78] (°) and that for c in the range of [lc  3rc, lc + 3rc] = [0.0260, 0.4340] (MPa). As described in Section 2.1, the UD is suitable

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X. Li et al. / Structural Safety 61 (2016) 22–42 Table 3 Arrangement and output in uniform design (UD) analysis. No. of experiments

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Table 5 Values of Lagrangian multipliers ai, ai⁄ and corresponding difference (ai  ai⁄).

Factors (corresponding to two random variables)

Output

u (rad)

c (MPa)

rp/r0

0.3302 0.3351 0.3400 0.3449 0.3498 0.3547 0.3596 0.3645 0.3694 0.3743 0.3792 0.3841 0.3890 0.3939 0.3988 0.4037 0.4086 0.4135 0.4184 0.4233 0.4282 0.4331 0.4380 0.4429 0.4478 0.4527 0.4576 0.4625 0.4674

0.2883 0.1280 0.4049 0.2446 0.0843 0.3611 0.2009 0.0406 0.3174 0.1571 0.4340 0.2737 0.1134 0.3903 0.2300 0.0697 0.3466 0.1863 0.0260 0.3029 0.1426 0.4194 0.2591 0.0989 0.3757 0.2154 0.0551 0.3320 0.1717

2.7982 5.4151 2.1212 3.0194 7.1418 2.2105 3.3199 12.0557 2.3189 3.7571 1.8611 2.4544 4.4674 1.9220 2.6299 5.8846 1.9944 2.8691 10.8023 2.0823 3.2208 1.7137 2.1922 3.8065 1.7640 2.3350 5.0602 1.8236 2.5312

i

ai

ai⁄

(ai  ai⁄)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

22.7632 276.9006 0.0000 0.0000 0.0000 0.0000 0.0000 352.2816 0.0000 512.0000 0.0000 176.0892 0.0000 232.0015 0.0000 0.0000 0.0000 0.0000 191.8417 0.0000 0.0000 0.0000 311.8731 494.3282 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 24.2290 0.0000 512.0000 4.1407 434.3641 0.0000 0.0000 0.0000 49.6682 0.0000 0.0000 0.0000 0.0000 512.0000 314.3934 206.1078 0.0000 0.0000 0.0000 5.2940 0.0000 0.0000 0.0000 251.5305 256.3513 0.0000 0.0000

22.7632 276.9006 24.2290 0.0000 512.0000 4.1407 434.3641 352.2816 0.0000 512.0000 49.6682 176.0892 0.0000 232.0015 0.0000 512.0000 314.3934 206.1078 191.8417 0.0000 0.0000 5.2940 311.8731 494.3282 0.0000 251.5305 256.3513 0.0000 0.0000

Table 6 Reliability results obtained during iteration. Table 4 Results obtained in solving support vector machine (SVM) model. Related parameters

e

C

c

b

0.0313

512

32

8.0182

Squared correlation coefficient, R2

Number of support vectors

0.9966

20

for using as many levels as possible to deal with complex nonlinear modelling, yet at the cost of a small number of experiments. With this in mind, we here attempt to take 29 levels for each factor during test calculations, since the corresponding 29 ‘‘experiments” (the number of experiments equals the number of levels, see Section 2.1) conducted via Eq. (2) (i.e., determination of the output ðrp =r0 Þ values) are not too many. (b) Given 29 levels for both u and c, the U29(296) table and its accessory table (see Tables A3 and A4 in Appendix, respectively) can then be used to produce the sampling points (whose effectiveness will be verified by subsequent reliability calculations). (c) Owing to two factors (u and c) involved, the columns 1 and 4 with s = 2 and D = 0.0520 should be chosen (see Table A4). On this basis, the ranges for u and c can be both split equally into 29 levels which are coded from 1 to 29. Then the obtained levels can, respectively, be located properly by following the entries in Column 1 (corresponding to u) and Column 4 (corresponding to c) in U29(296) (Table A3). The resulting arrangement and output are shown in Table 3 (the unit of u becomes radian by unit conversion (1 rad  57.3248°)). Step (3): After obtaining the 29 sampling points, the SVM model in Eq. (1) is used as the form of the response function to map the actual LSF of Eq. (2). This process can be implemented through

No. of iteration

Iterative design point, (u (rad), c (MPa))

Value of performance function, g(x)

Reliability index, b

1 2 3 4

(0.40284, (0.40242, (0.40245, (0.40246,

4.82600  102 3.36164  104 8.66400  108 2.11286  1010

0.7758 0.6944 0.6938 0.6938

0.17952) 0.18481) 0.18481) 0.18481)

the LIBSVM tool mentioned in Section 2.2, where a kernel function   K ðxi  xÞ ¼ exp ckx  xi k2 (i.e., the radial basis kernel) is adopted (the kernel function’s selection will be further addressed in the following subsection). Table 4 offers the results of the optimal combination for those parameters involved in the SVM model (e, C, c and b). In this table, the squared correlation coefficient (R2), as one metric used to assess the performance of the obtained SVM model, takes the value of 0.9966. This suggests the sufficiently high degree to which the ðrp =r0 Þ values obtained through the SVM model within the experimental domains match those acquired via the real LSF of Eq. (2). In addition, Table 5 lists the values of the Lagrangian multipliers ai and ai⁄, where ðai  ai Þ = 0 when i = 4, 9, 13, 15, 20, 21, 25, 28 and 29. This indicates that such 9 sampling points corresponding to vanishing ðai  ai Þ are considered irrelevant to the final solutions in the SVM analysis, and only the remaining 20 sampling points with no vanishing ðai  ai Þ (called ‘‘support vectors”) really contribute to the SVM model-building in Eq. (1). That is why the last column in Table 4 shows 20 under ‘‘Number of support vectors” (which embodies the sparseness property in the SVM analysis, as observed in Section 2.2). Step (4): Knowing the values of b, c and ðai  ai Þ in Step (3), the SVM model in Eq. (1) can be formulated. Then the standard reliability method (say, the widely used first-order reliability method) is adopted to carry out the calculations. Appendix provides the computer program coded for the analysis of tunnel-reliability. Notice that since u and c are negatively correlated with a

28

X. Li et al. / Structural Safety 61 (2016) 22–42

Table 7 Comparisons of reliability results for the case of ps = 0.0 MPa and normal distributions for u and c with quc = 0.5. Method

Reliability results

Proposed hybrid approach Li and Low method [13] Monte-Carlo simulations [13]

Reliability index, b

Probability of failure, Pf (102)

Design point, (u (°), c (MPa))

0.6938 0.6933 –

24.39 (29) 24.41 24.57 (1  104)

(23.05902, 0.18481) (23.01845, 0.18551) –

Note: the final acceptable design-point value of u taking on 0.40246 with the unit of ‘‘radian” (see Table 6) has been re-converted into 23.05902 with that of ‘‘degree”. The figure in the parentheses of the column under ‘‘Probability of failure, Pf (102)” refers to the number of sampling points required in the computations, which is equivalent to the total number of g(x)-calls of the actual limit state in Eq. (2).

correlation coefficient quc = 0.5, the transformation of correlated variables into uncorrelated ones should be considered (see [70] for more details). The reliability index and the corresponding design point obtained during the iteration are recorded in Table 6. Up to now, we provide a complete elucidation of the calculation steps when applying our approach to this tunnel example. Table 7 compares the results obtained from the proposed hybrid approach and the other two methods in [13] (the Li and Low method and Monte-Carlo simulations). From Table 7, for the computational accuracy, the agreement between the results (whether the reliability index, probability of failure or design point) calculated in this study and those obtained in [13] is excellent. For the com-

putational efficiency, as mentioned in [13], the number of trials (i.e., the number of sampling points generated randomly by a random generator) needed by the direct Monte-Carlo simulations is 1  104 (see the figure recorded in the parentheses of the column under ‘‘Probability of failure, Pf (102)” in Table 7). This means that the total number of g(x)-calls of the actual limit state (or the computing times for g(x)) in Eq. (2) is up to 1  104. For the proposed hybrid approach, it achieves similar accuracy with 29 sampling points, indicating that the total number of g(x)-calls of the actual limit state in Eq. (2) is only 29 (also see the figure in the parentheses of the column under ‘‘Probability of failure, Pf (102)” in Table 7). Evidently, the proposed hybrid approach

0.8 real limit state function (LSF) approximated limit state function (LSF) sampling points design point

X2: Cohesion (MPa)

0.7 0.6 0.5

(0.3302, 0.4340)

(0.4674, 0.4340)

0.4 0.3 experimental domains

0.2 0.1 0.0 0.0

(0.3302, 0.0260)

0.1

0.2

0.3

(0.4674, 0.0260)

0.4

0.5

0.6

0.7

0.8

X1: Angle of Friction (rad) (a) Approximated limit state function (LSF) using proposed hybrid approach within experimental domains. 0.45 0.40

X2: Cohesion (MPa)

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.32

0.36

0.40

0.44

0.48

X1: Angle of Friction (rad) (b) Enlarged figure from Fig. 2(a) within mentioned experimental domains. Fig. 2. Comparison between real limit state function (LSF) and its approximation.

29

X. Li et al. / Structural Safety 61 (2016) 22–42

in the RSM context is at the expense of dramatically lower computational cost compared with Monte-Carlo simulations. 3.1.1.2. Examination of local approximation. Next, let us examine the ‘‘local approximation” feature in the RSM context, as stressed in Section 2.3. To do so, we proceed to use the above example to graphically demonstrate the sampling points produced by UD and the LSF approximated by SVM. Fig. 2(a) plots the real LSF of Eq. (2), the design point, the total of 29 sampling points generated within the experimental domains by UD and the resulting LSF approximated by SVM. In this figure, the angle of friction u is designated as the horizontal axis X1 and the cohesion c as the vertical axis X2; the experimental domains for u: [lu  3ru, lu + 3ru] = [0.3302, 0.4674] and for c: [lc  3rc, lc + 3rc] = [0.0260, 0.4340] are located inside of the closed rectangular border, where the coordinates of four right angles are read clockwise (from foot left): (0.3302, 0.0260), (0.3302, 0.4340), (0.4674, 0.4340) and (0.4674, 0.0260), respectively. It can be seen from Fig. 2(a) that the 29 sampling points produced by UD are concentrated around the design point, which characterizes the region that contributes most to the probability of failure. Hence the approximated LSF agrees well with the real LSF in this important region. To better understand this issue, let us consider Fig. 2(b). It is a figure enlarged from Fig. 2(a) within the mentioned experimental domains. Clearly, the real LSF and the approximate LSF in Fig. 2(b) are in very close agreement in the neighborhood of the design point, thus verifying the accuracy of the response surface around the design point (i.e., the ‘‘region of most interest” in forming the response surface). This illustrates the ‘‘local approximation” feature in the RSM framework for the proposed hybrid approach and also explains why such an approach yields reasonably good results in Table 7. Interestedly, in Fig. 2(b) the 29 sampling points produced by UD are distributed uniformly over the given experimental domains. Let us recall the description in Section 2.1. UD emphasizes the uniformity of sampling points chosen in the investigated domains, which makes it generate sampling points with high representativeness. At this rate, there is a relatively high chance to obtain satisfactory results in developing the SVM model. As mentioned in Table 4, the squared correlation coefficient R2 used to evaluate the performance of the obtained SVM model is larger than 0.99 (R2 = 0.9966). 3.1.1.3. Reliability calculations for different cases. To further study the applicability of our approach for different cases in tunnelreliability analysis, four typical cases in this example are taken as an illustration to perform the following analyses: (I) The case with no support pressure (ps = 0.0 MPa), where u and c are normally distributed with varying quc. Table 8 lists the results when quc is varied from 0.9 to 0.0.

Table 8 Comparisons of reliability results considering various correlation coefficients. Correlation coefficient between u and c,

Proposed hybrid approach

Li and Low method [13]

Reliability index, b

Probability of failure, Pf (102)

Reliability index, b

Probability of failure, Pf (102)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.5883 0.6061 0.6253 0.6462 0.6690 0.6938 0.7208 0.7502 0.7822 0.8168

27.8169 27.2239 26.5882 25.9061 25.1744 24.3899 23.5508 22.6561 21.7062 20.7015

0.5848 0.6030 0.6242 0.6455 0.6727 0.6933 0.7171 0.7543 0.7943 0.8429

27.93 27.32 26.62 25.93 25.06 24.41 23.66 22.53 21.35 19.97

quc

(II) The case with different support pressures ps, where u and c are normally distributed with quc = 0.5. Here, it has to be stressed that due to the variations in ps, the resulting ðrp =r0 Þ values in Eq. (2) will change. Thus when using the U29(296) UD table (Table A3), the experimental results (i.e., the output ðr p =r 0 Þ values) shown in the last column of Table 3 will change in response to the varying ps. This indicates that the corresponding 29 sampling points produced in this way are surely different from those generated in the above case with no support pressure. Accordingly, the SVM model should be re-built, which in the long run leads to variations of the reliability results. Table 9 exhibits the results when ps = 0.1, 0.2, 0.3, 0.4 and 0.5 MPa, respectively. (III) The case with different support pressures ps, where u and c are non-normally distributed (lognormal distribution) with quc = 0.5. Here the reliability analysis involves a transformation from the lognormal distribution to the equivalent normal distribution (see [70] for detailed information). Also, the SVM model should be re-built with varying ps, as explained above. Table 10 displays the results when ps = 0.0, 0.1 and 0.2 MPa are, respectively, taken for illustration. (IV) The case with a multivariable reliability problem, where besides u and c, two more parameters, i.e., the in situ stresses p0 and support pressure ps are also considered to be random variables. Note that when using the U29(296) UD table (Table A3), the choice of columns 1, 3, 4 and 5 with s = 4

Table 9 Comparisons of reliability results considering different support pressures. Support pressure, ps

Proposed hybrid approach

Li and Low method [13]

Reliability index, b

Probability of failure, Pf

Reliability index, b

Probability of failure, Pf

0.1 0.2 0.3 0.4 0.5

1.5699 2.3722 3.2064 4.1177 5.2853

5.8220  102 8.8407  103 6.7211  104 1.9136  105 6.2733  108

1.5294 2.3720 3.2192 4.1126 5.2859

6.31  102 8.84  103 6.43  104 1.96  105 6.26  108

Table 10 Comparisons of reliability results involving non-normal distribution at different support pressures. Support pressure, ps

Proposed hybrid approach

Li and Low method [13]

Reliability index, b

Probability of failure, Pf

Reliability index, b

Probability of failure, Pf

0.0 0.1 0.2

0.6172 1.8730 3.2698

2.6856  10–1 3.0532  10–2 5.3806  10–4

0.6109 1.8139 3.2851

2.71  10–1 3.48  10–2 5.10  10–4

Table 11 Comparisons of calculation results of multivariable problems for tunnel-reliability. Description

Consider four random variables u, c, p0 and ps; u and c are normally distributed with quc = 0.5

Case

Support pressure ps with mean value of 0.3 MPa and standard deviation of 0.05 MPa Support pressure ps with mean value of 0.8 MPa and standard deviation of 0.133 MPa

Reliability index, b Proposed hybrid approach

Li and Low method [13]

2.8913

2.85

4.8417

4.85

X. Li et al. / Structural Safety 61 (2016) 22–42

It is timely to examine the reliability results recorded in Tables 8–11 as in Table 7. From these tables, whether for the correlated/ uncorrelated properties (Table 8 with varying quc), the changes of the SVM model (as the form of response surface) due to variations of the input parameter in modelling (Tables 9 and 10 with varying ps), the normal/non-normal probability distributions (Tables 8 and 9 for normal distributed random variables; Table 10 for lognormal distributed ones), or for the bivariate/multivariable reliability problems (Tables 8–10 with the 2-random-variable case (u and c); Table 11 with the 4-random-variable case (u, c, p0 and ps)), a close look at all these tables sheds light on the fact that the proposed hybrid approach does provide reasonably good results consistently. Moreover, in this example, since the same U29(296) table used to generate 29 sampling points in the UD analysis is applicable to all the above conditions, an implication may possibly be made that the UD is, from the viewpoint of robustness, insensitive to the assumptions imposed on the underlying model. That is, the UD is robust against changes (i.e., various conditions in the above reliability analyses) of relationships between the factors (u, c, p0 and ps) and the output (the ðr p =r 0 Þ ratio). At this point, it is precisely such an UD’s salient feature outlined in Section 2.1. 3.1.2. Discussion of proposed hybrid approach In accordance with Section 2.3, the emphasis for our approach to tunnel-reliability analysis is mainly placed upon two issues: one aims to adopt the UD to prepare the sampling points, and the other intends to employ the SVM model as the response surface to approximate the original LSF within the experimental domains. In this context, it is therefore necessary to further discuss the influence that the usage of UD and SVM has, respectively, on the final results of this typical example. Such a discussion could then be taken for reference to extend our approach to real-life case study applications in the following two tunnel examples. 3.1.2.1. Basic usage of SVM. For the SVM’s usage, particular attention here is paid to the selection of kernel functions involved in the SVM model of Eq. (1). As indicated in Section 2.2, two kernel functions frequently employed are the polynomial kernel and the radial basis kernel. With such two types of kernel, we use the mentioned case with vanishing support pressure (ps = 0.0 MPa) (where u and c are normally distributed with quc = –0.5 and the U29(296) table (Table A3) is used in the UD analysis) to examine their influences on the SVM model’s performance, and then on the final results of tunnel-reliability. Table 12 summarizes the error estimations of MSE and R2 for the obtained SVM model using the polyno-

Table 12 Comparisons of performance of support vector machine (SVM) model with different kernel functions. Polynomial kernel

Radial basis kernel

Polynomial degree

Mean squared error (MSE)

Squared correlation coefficient, R2

Mean squared error (MSE)

Squared correlation coefficient, R2

d=2 d=3 d=4 d=5

1.2583 0.4729 0.2086 4.6447

0.8657 0.9477 0.9685 0.5948

0.0234

0.9966

13 12

Computed rp/rc by Eq. (1)

should be made in its accessory table (Table A4) owing to 4 factors (u, c, p0 and ps) involved. Table 11 presents the results when the mean value and standard deviation for p0 are designated as 2.5 MPa and 0.25 MPa, respectively, as well as the mean values and standard deviations for ps are assumed to be 0.3 MPa, 0.05 MPa and 0.8 MPa, 0.133 MPa, respectively.

use of polynomial kernel with d = 4

11

2

R = 0.9685

10 9 8 7 6 5 4 3 2 1 1

2

3

4

5

6

7

8

9

10

11

12

13

Actual rp/rc via Eq. (2) (a) Performance of SVM model by use of polynomial kernel. 13 12

Computed rp/rc by Eq. (1)

30

use of radial basis kernel 2 R = 0.9966

11 10 9 8 7 6 5 4 3 2 1 1

2

3

4

5

6

7

8

9

10

11

12

13

Actual rp/rc via Eq. (2) (b) Performance of SVM model by use of radial basis kernel Fig. 3. Comparison of Performance of support vector machine (SVM) model using different kernel functions.

mial and radial basis kernels, respectively. As seen from this table, for the polynomial kernel itself, the polynomial degree d has a significant effect on the performance of the SVM model. Among the values of d varying from 2 to 5, taking d = 4 for the polynomial kernel has the best estimations of MSE (the least value of 0.2086) and R2 (the largest value of 0.9685). In Table 12, for the radial basis kernel, there are a lower MSE (0.0234 < 0.2086) and a larger R2 (0.9966 > 0.9685, see the scatter plot for the sampling points in Fig. 3a and b) compared with such error estimations for the polynomial kernel with d = 4. This indicates that the radial basis kernel presents somewhat better performance of the SVM model over the polynomial kernel (d = 4). To further address this issue, Table 13 compares the estimations of MSE and R2 for this tunnel example with other support pressure ps = 0.1, 0.2, 0.3, 0.4 and 0.5 MPa, respectively. Clearly, in terms of the values of MSE (which are relatively small) and R2 (most of which are larger than 0.99) in this table, both the polynomial kernel (d = 4) and the radial basis kernel make the SVM model in each case show good performance (meaning that such a developed model matches the actual LSF of Eq. (2) reasonably well within the sampling regions). Correspondingly, it could be anticipated that reasonably good results in the subsequent reliability analyses will be achieved based on such two types of kernel function. This can be verified by comparisons of the design points tabulated in Table 13 and the reliability indexes

31

X. Li et al. / Structural Safety 61 (2016) 22–42 Table 13 Comparisons of calculation results for support vector machine (SVM) model with different kernel functions when varying support pressure. Support pressure, ps

0.0 0.1 0.2 0.3 0.4 0.5

Proposed hybrid approach (SVM model with polynomial kernel (d = 4))

Proposed hybrid approach (SVM model with radial basis kernel)

Li and Low method [13]

Mean squared error (MSE)

Squared correlation coefficient, R2

Design point, (u (°), c (MPa))

Mean squared error (MSE)

Squared correlation coefficient, R2

Design point, (u (°), c (MPa))

Design point, (u (°), c (MPa))

0.2086 0.0066 0.0020 0.0014 0.0402 0.0038

0.9685 0.9959 0.9969 0.9956 0.9815 0.9953

(23.0882, (23.1911, (23.2019, (23.2480, (23.0147, (22.5992,

0.0234 0.0012 7.6758  10–5 1.5205  10–5 0.0259 0.0026

0.9966 0.9993 0.9998 0.9999 0.9901 0.9971

(23.0590, (23.1678, (23.2240, (23.1694, (23.1415, (22.6059,

(23.0185, (23.1695, (23.2576, (23.2774, (23.1225, (22.7161,

0.1822) 0.1319) 0.0849) 0.0315) 0.0064) 0.0013)

5.5 5.0

Reliability index ß

4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

Proposed hybrid appraoch with radial basis kernel Proposed hybrid appraoch with polynomial kernel d=4 Li and Low method [13]

0.5 0.0 0.0

0.1

0.2

0.3

0.4

0.5

Support pressure ps Fig. 4. Comparisons of reliability index via proposed hybrid approach when using different kernel functions for support vector machine (SVM) model.

shown in Fig. 4, where there are no major differences between both in this study. It is worthwhile pointing out that there are no precise guidelines or theory at present for the selection of kernel functions in the SVM analysis. The work on limiting kernels using prior knowledge can be found in some monographs [e.g., 62]. From the above discussion, the performance of the SVM model is indeed sensitive to the degree in the context of polynomial kernel. However, once a suitable degree for the polynomial kernel is chosen through test calculations, it seems that there is not pronounced difference in the SVM model’s performance between the radial basis kernel and the polynomial kernel, although the former kernel has in a strict sense slightly better performance than the latter one. These findings would be consistent with those outlined in some empirical analyses [e.g., 60,62] and engineering applications [e.g., 71,72]. The reason may probably be attributed to the fact that the nature of the optimization problem in the SVM analysis makes the problem kernel-insensitive, as emphasized in [59].

0.1848) 0.1304) 0.0816) 0.0334) 0.0076) 0.0014)

0.1855) 0.1326) 0.0807) 0.0301) 0.0071) 0.0018)

3.1.2.2. Basic usage of UD. For the UD’s usage in this example, special effort will be made towards the following two aspects provided the number of sampling points has been determined (in the next section particular attention will be paid to the issue on how to reasonably identify the number of sampling points): (a) the discrepancy used to measure the uniformity of the sampling points, and (b) the experimental domains chosen for the factors. First, to investigate the effect of the discrepancy on the final reliability results, the above case with ps = 0.0 MPa (where u and c are normally distributed with quc = 0.5) is used again. For this case, we have intended to use the U29(296) UD table (Table A3) and its accessory table (Table A4) to prepare the sampling points (the columns 1 and 4 should be chosen due to s = 2). We now use another type of the UD table: U29(298) (Table A1) and its accessory table (Table A2). It is worth noting that the number of levels for each factor (or the number of experiments) in the U29(298) table is the same as that in the U29(296) table (both equal 29), yet the choice among the columns and the corresponding discrepancies in the U29(298) table is completely different from that in the U29(296) table. Specifically, when using the U29(298) table and its accessory table, the columns 1 and 6 have to be selected and the discrepancy D takes the value of 0.0663 (see Table A2), albeit the number of factors (u and c) equals 2 all the same (i.e., s = 2). Table 14 shows the results in the analysis of tunnel-reliability via the U29(296) and U29(298) tables, respectively. From this table, the reliability index b and the corresponding probability of failure Pf obtained through the U29(296) table with D = 0.0520 are very close and almost identical with those obtained in [13], whereas those acquired via the U29(298) table with D = 0.0663 yield comparatively large errors. This reveals that the obtained results via the UD table with a lower discrepancy seem to have a higher accuracy in the examined case. To clarify this issue, it is useful to recall the ‘‘Discrepancy” (see Section 2.1): the smaller the value of discrepancy, the more uniform a set of sampling points scattered on the domain will be. This contributes actually to better quality of the selected sampling points, thus leading to a higher degree to which the fitted SVM model in Eq. (1) (as the form of response surface) within the experimental domain matches the real LSF of Eq. (2). At this point, the comparisons of MSE and R2 in the column under ‘‘Performance of SVM model” of Table 14 demonstrate that using the UD table with a smaller value of discrepancy

Table 14 Effect of discrepancy pertaining to uniform design (UD) table on reliability results. Proposed hybrid approach

Li and Low method [13]

UD table (accessory table for use)

Discrepancy, D

Performance of SVM model (assessed by mean squared error (MSE) and squared correlation coefficient, R2)

U29(296) (choose columns 1 and 4) U29(298) (choose columns 1 and 6)

0.0520 0.0663

MSE = 0.0234 MSE = 0.0516

R2 = 0.9966 R2 = 0.9885

Reliability index, b

Probability of failure, Pf (102)

Reliability index, b

Probability of failure, Pf (102)

0.6938 0.8438

24.39 19.94

0.6933

24.41

32

X. Li et al. / Structural Safety 61 (2016) 22–42

Table 15 Reliability results for variations in experimental domains when support pressure ps = 0.0 MPa. Proposed hybrid approach

Li and Low method [13]

h

Experimental domain of u, [lu  hru, lu + hru] (degree)

Experimental domain of c, [lc  hrc, lc + hrc] (MPa)

Design point (u, c)

Reliability index, b

Design point (u, c)

Reliability index, b

3 2 1

[18.92, 26.78] [20.23, 25.47] [21.54, 24.16]

[0.0260, 0.4340] [0.0940, 0.3660] [0.1620, 0.2980]

(23.06, 0.1848) (23.02, 0.1850) (23.01, 0.1861)

0.6938 0.7020 0.6856

(23.02, 0.1855)

0.6933

Table 16 Reliability results for variations in experimental domains when support pressure ps = 0.3 MPa. Proposed hybrid approach

Li and Low method [13]

h

Experimental domain of u, [lu  hru, lu + hru] (degree)

Experimental domain of c, [lc  hrc, lc + hrc] (MPa)

Design point (u, c)

Reliability index, b

Design point (u, c)

Reliability index, b

3 2 1

[18.92, 26.78] [20.23, 25.47] [21.54, 24.16]

[0.0260, 0.4340] [0.0940, 0.3660] [0.1620, 0.2980]

(23.17, 0.0334) (22.92, 0.0236) (23.57, 0.0100)

3.2064 3.4770 4.2680

(23.27, 0.0301)

3.2192

(0.0520 < 0.0663) makes the resulting SVM model have a lower MSE value (0.0234 < 0.0516) and a larger R2 value (0.9966 > 0.9885). Thus the subsequent reliability calculations based on the SVM model with better performance could obtain more precise results. From the above analysis, in order to obtain the results as accurately as possible, an UD table with a relatively small discrepancy seems to be generally desirable among the possible design tables for a given number of sampling points. Secondly, to study the effect of the experimental domains chosen for the factors on the final reliability results, we take two cases with ps = 0.0 MPa and 0.3 MPa analyzed above as an example (where u and c are normally distributed with quc = 0.5 and the U29(296) UD table (Table A3) is used). Notice that the experimental domains selected for u and c in the above two cases fall within the ranges of ±3 times the standard deviation away from mean (i.e., u: [lu  3ru, lu + 3ru] = [0.3302 rad, 0.4674 rad] and c: [lc  3rc, lc + 3rc] = [0.0260 MPa, 0.4340 MPa]). They are on the whole consistent with the limits of physical admissibility for u and c reported in [8,68]. Thus the selected ranges could become the maximum experimental domains for u and c in this study, respectively. To reflect the variations within such domains, several sub-domains can then be subdivided (by convention in a symmetrical manner). For instance, for the factor u, one obtain [lu  2ru, lu + 2ru] and [lu  ru, lu + ru]; for the factor c, one obtain [lc  2rc, lc + 2rc] and [lc  rc, lc + rc]. For convenience, let u belong to [lu  hru, lu + hru] and c belong to [lc  hrc, lc + hrc] where the coefficient h is assumed to be varied as 1, 2 and 3, respectively. Table 15 (ps = 0.0 MPa) and Table 16 (ps = 0.3 MPa) display, respectively, the results for tunnel-reliability when accounting for the changes in the experimental domains. From Table 15, for each case of the chosen domains of u and c (h = 1, 2 and 3), the design point calculated in this study (its values of u and c both lie well within such domains) agrees remarkably well with the ‘‘exact” (or actual) design point recorded in [13]. This indicates that the selected domain for each case can sufficiently well cover the important region contributing most to the probability of failure, within which the fitted SVM model (as the response surface) provides reasonable approximations to the original LSF in the vicinity of the design point (similar to the situation graphically demonstrated in Fig. 2). That could be why the resulting reliability index computed in this study for each case in Table 15 is very close with that documented in [13]. Next, let us proceed to consider Table 16. Also, for the chosen domains of u and c when h = 3, the design point ((u, c) = (23.17°, 0.0334 MPa)) and the reliability index (b = 3.2064) computed by our approach coincide satisfactorily with the ‘‘exact”

design point ((u, c) = (23.27°, 0.0301 MPa)) and the reliability index (b = 3.2192) in [13]. The reason for this could be attributed to the same explanation (i.e., the chosen domain in this case can sufficiently well cover the region around the design point which contributes most to the probability of failure) as in Table 15 just mentioned. Further, compared with the ‘‘exact” design point falling well within the chosen domains in Table 16 when h = 3, its value of c (=0.0301) is conspicuously not inside of the chosen domain for both [0.0940, 0.3660] when h = 2 and [0.1620, 0.2980] when h = 1. This means that such selected domains do not cover the most likely failure region around the ‘‘exact” design point. Hence, the obtained points (u, c) = (22.92°, 0.0236 MPa) when h = 2 and (u, c) = (23.57°, 0.0100 MPa) when h = 1 may be actually viewed as the ‘‘false” design points. In this wise, the resulting reliability indexes b = 3.4770 (when h = 2) and b = 4.2680 (when h = 1) are then considered distorted and unreliable. From the above discussion, a decision must be made with some precaution for the determination of the experimental domain when using UD to generate sampling points. Because improper choice of the experimental domain may make it difficult to obtain the ‘‘exact” design point, it is necessary during test calculations to roughly estimate the design point (regarded as a candidate or tentative point) so as to consider whether the experimental domain should be adjusted. As experienced in Table 16, a possible strategy can be implemented in the following line. To begin with, we may choose a relatively

Fig. 5. Entrance of Liziping tunnel (Example 2).

33

X. Li et al. / Structural Safety 61 (2016) 22–42

narrow experimental domain. Once the candidate point (if obtained) fails to lie well within such a chosen domain, it can be inferred that the obtained point is regarded as the ‘‘false” design point and the ‘‘exact” one to be obtained does not fall into the pre-chosen domain. In this case, it may then be advisable to progressively increase the width of the experimental domain (i.e., a gradually increment of h is recommended to consider in subsequent sampling, as in Table 16). By doing so, the experimental domain chosen beforehand may hence be possible to include the ‘‘exact” design point. On this basis, the chosen domain incorporating most part of the likely failure region around the design point (i.e., the ‘‘region of most interest” contributing most to the failure of probability, as stressed in Section 2.3) makes the candidate point have a possibility to be accepted as the desirable result. 3.2. Example 2 – Consider implicit LSF for real-life tunnel analyzed in [31] By means of the above typical example, we have offered a detailed elucidation of our approach in the RSM context to assist in an overall understanding for its preliminary application. Now, let us extend this approach to tunnel-reliability problems with the inexplicit LSF. To do so, the second example is pertinent to the probabilistic stability evaluation of the Liziping tunnel (see Fig. 5) in China. The associated reliability analyses for this tunnel have been conducted by the author and co-workers, who used the difference approximation to deal with the implicit LSF [31]. The limit state relevant to the collapse mode of failure of the primary support represents the ultimate limit state. Such a state was assumed to emerge when the supporting resistance provided by the sum of support pressures Pi,shot (corresponding to shotcrete lining) and Pi,bol (corresponding to rockbolts) reaches the rock pressure Pi,rock (i.e., the load bearing capacity of the rock mass around the perimeter of the tunnel). Since all formulas derivations for Pi, shot, Pi,bol and Pi,rock have been specifically described in [31], we now give their expressions directly such that the emphasis will be placed on the examination of our approach to handling the inexplicit LSF. As analyzed in [31], the LSF gðxÞ ¼ 0 can be defined as

gðxÞ ¼ Pi;shot þ Pi;bol  Pi;rock ¼ 0 since

P i;rock ¼ cr 0

(

ð3Þ

) 1sin u ð1  sin uÞðc cot u þ P 0 Þ 2 sin u 1 ð1 þ sin uÞðc cot u þ Pi;rock Þ ð3aÞ

Pi;shot ¼ K s ður0  u0 Þ=r 0 h Pi;bol ¼

ð3bÞ

i   ur0  ua0  urc  ua0 rr0c Eb Ab

ð3cÞ

ðrc  r 0 ÞSc Sl

   K s ¼ Es d= r 0 1  m2s ur ¼

ð3dÞ

 1sin u r20 ð1 þ mÞðP0 sin u þ c cos uÞ ð1  sin uÞðc cot u þ P0 Þ sin u  Er c cot u þ Pi;rock ð3eÞ

where g(x) is the performance function and x is a random vector involving the mechanical and geometrical parameters; c, u, c, E and v are, respectively, the cohesion, the angle of internal friction, the unit weight, the Young’s modulus and the Poisson’s ratio of the rock mass; P0 is the in-situ stress; Ks is the support stiffness modulus; d, Es and ms are, respectively, the layer thickness, the Young’s modulus and the Poisson’s ratio of the shotcrete; Ab, Eb, Sc and Sl are, respectively, the cross-sectional area, the Young’s modulus, the circumferential spacing and the longitudinal spacing of the rockbolts; r0 is the radius of tunnel excavated; rc is the radius denoting a distance from the tunnel center to the inner face of the anchoring zone (r c ¼ r0 þ L, L is the length of the rockbolts); ur0 and urc are, respectively, the radial displacements ur at radius r0 and rc (calculated by Eq. (3e) when the radial distance r ¼ r0 and r ¼ rc ); u0 is the original radial closure before installation of the shotcrete lining; ua0 is the radial displacement of the tunnel wall when the support installed by the rockbolts is soon completed. Note that the cohesion c for the rock mass involved in Eq. (3c) is modified as c0 ¼ c þ sb Ab =ðSc Sl Þ due to the phenomenon of interventions in the bolting zone, where sb is the shear strength of the rockbolts. From Eqs. (3b), (3c) and (3e), Pi,shot and Pi,bol are both in connection with Pi,rock. Meanwhile, Pi,rock is not explicitly known, as shown in Eq. (3a). Thus the LSF of Eq. (3) is rendered implicitly in a complex way. Following [31], six parameters c, u, P0, E, u0 and d were considered random variables, whose statistical information is listed in Table 17. These random variables were assumed to be independent (uncorrelated) with normal distributions. The values of those rest parameters were known deterministically as: r0 = 5.9 m, ua0 = 1.92 cm, c = 26.5 kN/m3, m = 0.5, Es = 28 GPa, ms = 0.167, Eb = 210 GPa, L = 3.0 m, Ab = 380.13 mm2, sb = 312 MPa and Sc = Sl = 1.0 m. In the course of reliability analyses, it is useful to refer to the calculation steps (see Section 3.1.1) and the associated findings

Table 17 Statistical information obtained from [31] for Example 2. Random variable

c (MPa)

u (degree)

P0 (MPa)

E (GPa)

u0 (cm)

d (cm)

Mean value Standard deviation

0.507 0.0675

28.700 2.3000

9.975 0.7111

4.370 0.5244

3.200 0.4000

20.300 2.0000

Table 18 Comparisons of reliability results for different methods. Method

Probability of failure, Pf (102)

Reliability index, b

Proposed hybrid approach Su et al. method [31] Monte-Carlo simulations [31]

0.4059 (28) 0.4000 0.4023 (1  106)

2.6471 2.6521 –

Design point (c (MPa), u (degree), P0 (MPa), E (GPa), u0 (cm), d (cm)) (0.5487, 32.0091, 9.3114, 5.0478, 3.7661, 20.0172) (0.5489, 32.0165, 9.3205, 5.0539, 3.7700, 20.0183) –

Note: as in Table 7, the figure in the parentheses of the column under ‘‘Probability of failure, Pf (102)” refers to the number of sampling points required in the computations. This is equivalent to the total number of g(x)-calls of the actual limit state in Eq. (3).

34

X. Li et al. / Structural Safety 61 (2016) 22–42 Table 20 Variation of Discrepancy in uniform design (UD) analysis at different number of sampling points.

5.0 4.5

Number of sampling points, M

Reliability index ß

4.0

UD table for use

3.5 3.0

U12(1210) U16(1612) U22(2211) U26(2611) U28(288) U30(3013) U37(3712)

12 16 22 26 28 30 37

2.5 2.0 1.5

Discrepancy, D (involved in each accessory table of UD table) Number of factors (s = 6)

Number of factors (s = 7)

0.2670 0.2518 0.1930 0.1828 0.1578 0.1621 0.1929

0.2768 0.2769 0.2195 0.1967 0.1550 0.1924 0.2245

Proposed hybrid approach Su et al. method [31]

1.0 0.5 5

10

15

20

16

25

calculated value by proposed hybrid approach "Exact" value from Monte-Carlo simulations

Fig. 6. Variation of reliability index with different coefficient of variation (COV) using proposed hybrid approach.

(see Section 3.1.2) in Example 1. For the usage of SVM, we again adopt the radial basis kernel for the SVM model in Eq. (1). For the usage of UD, six factors (the mentioned random variables c, u, P0, E, u0 and d) are considered and the experimental domain (i.e., the lower and upper values of realizations) for each factor is also chosen to fall within the range of multiple times the standard deviation around the mean. Then an attempt is made to choose the U28(288) UD table (see Table A5 in Appendix) and its accessory table (Table A6, choose the columns 1, 2, 3, 5, 6 and 7 with s = 6) to produce 28 sampling points (whose feasibility will be further addressed in this section). Now, let us examine the reliability results obtained by means of the proposed hybrid approach. As shown in Table 18, for the computational accuracy, the reliability index and the corresponding design point show good agreement between calculations based on our approach and Su et al. method [31]. Also, the probability of failure inferred from our approach with 28 sampling points is in close agreement with that obtained by 1  106 direct Monte-Carlo sampling, which demonstrates again the computational efficiency of the proposed hybrid approach. Next, to investigate the sensitivity of the proposed hybrid approach to the uncertainties in the random variables, the influence of the coefficient of variation (COV) on the calculated results is analyzed. Fig. 6 depicts the typical trend that the calculated reliability index decreases with increasing COV when the COV of all the random variables is assumed to be varied as 5%, 10%, 15%, 20% and 25%, respectively. Moreover, it is clear from this figure that the reliability index computed by our approach corresponding to each value of the COV coincides well with that obtained by Su et al. method [31]. Correspondingly, as tabulated in Table 19, there is good agreement between the calculations of the design points from such two methods for different values of the COV. Here it should be emphasized that we perform the above analyses of tunnel-reliability provided the number of sampling points has been pre-specified (i.e., there are 28 sampling points used to

-2 Probability of failure Pf 10

Coefficient of variation (COV)(%) 14

12

10

8

6

4 12

16

20

24

28

32

36

Number of sampling points, M Fig. 7. Variation of probability of failure with different number of sampling points using proposed hybrid approach.

solve the SVM model). However, since such a number is actually not known in advance (which is the situation for most real-world applications), one critical question is how to provide a possible guideline in the UD analysis for suitably choosing the number of sampling points. In this sense, it is of interest to further explore the influence that the consideration of different number of sampling points has on the final results of tunnel-reliability. To clarify this issue in an appropriate manner, we now take the mentioned case with COV = 15% as an illustration to carry out the following analysis. To begin with, the ‘‘exact” value of the probability of failure in this case is obtained as 5.4553  102 using 106 direct Monte-Carlo simulations. Subsequently, various numbers of sampling points M (=12, 16, 22, 26, 28, 30 and 37, i.e., increasing M from the initial 12 sampling points) produced by the corresponding UD tables are listed in Table 20 (owing to space limitations of this paper, except for the U28(288) table given in Table A5 for the above reliability analyses, detailed information for other UD tables can be found in [57] or at the UD-website: www.math.hkbu.edu. hk/UniformDesign). Notice further that in Table 20, the third

Table 19 Comparisons of design points with varying coefficient of variation (COV). Coefficient of variation (COV) (%)

Proposed hybrid approach Design point: (c (MPa), u (degree), P0 (MPa), E (GPa), u0 (cm), d (cm))

Su et al. method [31] Design point: (c (MPa), u (degree), P0 (MPa), E (GPa), u0 (cm), d (cm))

5 10 15 20 25

(0.5253, (0.5269, (0.5282, (0.5258, (0.5266,

(0.5261, (0.5262, (0.5285, (0.5270, (0.5258,

32.5663, 32.8270, 33.0197, 32.6315, 32.7655,

8.6209, 8.7778, 8.9538, 8.7091, 8.8251,

4.7659, 4.7980, 4.8230. 4.7747, 4.7900,

3.5049, 3.5345, 3.5602, 3.5147, 3.5282,

20.0153) 20.0181) 20.0171) 20.0180) 20.0134)

32.7014, 32.7075, 33.0431, 32.8368, 32.6390,

8.7327, 8.7248, 8.9901, 8.8208, 8.7110,

4.7817, 4.7829, 4.8274, 4.7989, 4.7736,

3.5194, 3.5206, 3.5649, 3.5355, 3.5131,

20.0146) 20.0161) 20.0156) 20.0152) 20.0155)

X. Li et al. / Structural Safety 61 (2016) 22–42

column under ‘‘Number of factors (s = 6)” (due to 6 random variables involved in this example) corresponds to ‘‘Discrepancy, D”. As observed in Section 2.1, the smaller the value of discrepancy, the more uniform those sampling points will be. Accordingly, constructing the SVM model in virtue of the generated sampling points with better quality in the UD context can yields more accurate results in the subsequent reliability analysis. With this in mind, Fig. 7 plots the relation between the number of sampling points M generated by UD to establish the SVM model and the probability of failure Pf estimated by the proposed hybrid approach. For the sake of cross-reference, let us consider Table 20 and Fig. 7 together. First, as seen from Fig. 7, there is a relatively large error in Pf between the ‘‘exact” value and the calculated one when M = 12, suggesting that using such 12 sampling points is too small and thus not enough to build an accurate SVM model. This is because when M = 12, the corresponding D = 0.2670. This is the largest magnitude among the values of the discrepancy D in Table 20. As M increases gradually (from 16 to 26), the error in Pf exhibits a decreasing trend, meaning that the accuracy of the reliability results is improving. The reason is that D has a tendency to diminish with increasing M, as shown in Table 20. When M increases to 28, the calculated Pf (=5.3841  102) is very close with that ‘‘exact” one (=5.4553  102) (the relative error is only ca. 1.31%). The good agreement could be attributed to the fact that D = 0.1578 (corresponding to M = 28) becomes the smallest value of discrepancy (see Table 20). With a further increase of M (from 30 to 37), the error in Pf behaves, however, an increasing trend. This may be a result of an increasing magnitude of the discrepancy D in Table 20, implying that the chosen sampling points scattered on the experimental domain tend to become less uniform. Such a fact explains why less accurate reliability results may be obtained even if M is increasing. From the above analyses, the discrepancy involved in UD plays a crucial role in identifying the number of sampling points. Because a smaller number of sampling points corresponds to a larger discrepancy, using the increasing number of sampling points with a diminishing discrepancy indeed improves the accuracy of reliability results. Nevertheless, when the number is relatively large, a substantial computational effort could be demanded. More importantly, a further increase of the number of sampling points may make the discrepancy have a tendency to increase, thereby leading to the deterioration in accuracy of the reliability results. This reveals that the generation of sampling points should carefully be planned to obtain a proper balance between the number of sampling points and the corresponding discrepancy. During test calculations, it seems that a proper strategy is to initiate the sampling with a relatively small number of sampling points and then gradually increase the number of sam-

Fig. 8. Entrance of Jingzhushan tunnel (Example 3).

35

Fig. 9. Schematic of cross-section of Jingzhushan tunnel.

pling points. When the trial number increases to a certain extent, it is desirable to have as small value of discrepancy as possible in choosing the UD table. 3.3. Example 3 – Consider implicit LSF for real-life tunnel via numerical method The intention of the third example is also to illustrate the application of our approach to the tunnel-reliability problem with the implicit LSF. In this example, the considered LSF is only implicitly known through the numerical method (which is the scenario of many real-world applications in tunnel community). Such an example involves the evaluation of the reliability of the serviceability performance of a road tunnel, called the jingzhushan tunnel (see Fig. 8) recently built in China. It is a tunnel with a span of 12.04 m and a height of 9.402 m (see the cross-section in Fig. 9), which was excavated and constructed at a maximal depth of ca. 200 m. A major proportion of the rock mass through which it is mined is silty slate. During the preliminary design stage, concrete lining combined with rockbolts were planned to be installed as support structures based on the geotechnical investigations. The reliability level of this tunnel design will be assessed with respect to the serviceability limit state of excessive downward vertical displacement. The LSF gðxÞ ¼ 0 can be given by

gðxÞ ¼ U max  UðxÞ ¼ 0

ð4Þ

meaning that the positive value reflects safe situation of the tunnel with relation to the corresponding limit state, while the negative value corresponds to conditions of limit state violation. In Eq. (4), g(x) is the performance function and x is a random vector composed of the mechanical and geometrical parameters; Umax denotes the prescribed maximum allowable displacement, which generally occurs at the crown of the tunnel; U(x) denotes the displacement as a function of those stochastic parameters. For such an example, Umax is set to be 80 mm in light of the tunnel’s behavior based on in situ observations and the associated design guidelines (say Code for design of road tunnel used in China [73]). Here a closed-form expression for the displacement function U(x) is not available and thus it has to be calculated via the numerical method. Common numerical methods can be classified as continuum methods (e.g., the finite element method and the finite difference method) and discontinuum methods (e.g., the distinct element method and the discontinuous deformation analysis). Usually, the selection of a continuum or discontinuum approach rests on the size or scale of the discontinuities in regard to the size or scale of the problem to be solved [35]. On the other hand there

36

X. Li et al. / Structural Safety 61 (2016) 22–42

are no universal quantitative guidelines to determine when one method should be used instead of the other [74]. In this study, the finite difference code FLAC [75] commonly considered in tunnel (or underground) structures is used for numerical analyses to determine the magnitude of the displacement function U(x) in Eq. (4). To simplify the computations for illustrative purpose, a typical numerical model can be built up based on plain-strain conditions with the size of 100 m in both horizontal and vertical directions. Fig. 10 shows the mesh illustrated diagrammatically for this example. For the displacement boundary conditions, the bottom boundary is assumed to be fixed and the vertical boundaries are constrained in the normal direction. For the material models adopted in this example, the rock mass is represented by a conventional elastic-perfectly-plastic model based on the Mohr–Coulomb failure criterion. The concrete lining and rockbolts are, respectively, simulated by the shell and links of linear elastic behavior. Note that the excavation of the tunnel and the installation of the support structures are assumed to be applied synchronously, which results in a significant simplification regarding the simulation of tunnelling construction. During the computations, a stress control method is used to account for the excavation steps and also avoid the erroneous results that may be caused by a displacement control method [32]. It is timely to identify now the random variables for the jingzhushan tunnel. For the Young’s modulus E, angle of friction u,

Fig. 10. Plot of mesh for numerical model of Jingzhushan tunnel.

cohesion c and density q of the rock mass, together with the horizontal stress Ph and vertical stress Pv of the in-situ stress field, these parameters are generally varied within a relatively wide range in realistic geological environments, as discussed in [8]. For thickness t of the concrete lining, it is also changed markedly since the in situ measurements may have a relatively high error margin and the over-excavation phenomenon that is difficult to avoid occurs frequently during construction. Hence such 7 parameters can be considered random variables, whose statistical information is listed in Table 21 (where E, u and c are considered to be lognormally distributed, and Ph, Pv, q and t normally distributed; u and c are assumed to be negatively correlated with a correlation coefficient 0.5). Variability of the rest magnitudes including the Poisson’s ratio m0 of the rock mass, the uniaxial compressive strength rc, elastic modulus E1 and Poisson’s ratio m1 of the concrete lining, along with the length L, diameter D, circumferential spacing Sc, longitudinal spacing Sl, elastic modulus E2 and Poisson’s ratio m2 of the rockbolts are relatively small compared with those random variables considered. Thus the influence of their variations may be negligible for simplifying computations. For this reason, their values can be fixed in a deterministic manner, which are presented by m0 = 0.3, rc = 25 MPa, E1 = 23 GPa, m1 = 0.17, L = 3.0 m, D = 25 mm, Sc = 1.25 m, Sl = 1.75 m, E2 = 210 GPa and m2 = 0.15. For reliability calculations in this example, the failure domain of Eq. (4) can be found through repeated point-by-point numerical analyses in FLAC and then the probability of failure is found to be 10.8685  102 using 1  104 direct Monte-Carlo simulations. When applying our approach combining UD and SVM, the values of displacement function U(x) in Eq. (4) are computed by FLAC at sampling points generated by UD, and then a closed-form LSF for Eq. (4) can be formulated by SVM. For the usage of SVM, the radial basis kernel is still chosen for the SVM model in Eq. (1). For the usage of UD, 7 random variables E, u, c, Ph, Pv, q and t are viewed as 7 factors, U(x) is designated as the output, and the experimental domains for the factors are obtained based on the ranges of values in Table 21. Next, to determine the appropriate number of sampling points when choosing the UD table, let us again consider Table 20. The last column displays different values of the discrepancy D corresponding to ‘‘Number of factors (s = 7)”, where D takes a relatively small magnitude (=0.1550) when the number of sampling points reaches M = 28. According to the explanation given in Example 2, it may hence be possible to choose the U28(288) UD table (see Table A5 in Appendix) and its accessory table (see Table A6; select the columns 1, 2, 3, 5, 6, 7 and 8 corresponding to s = 7). The corresponding reliability results are shown in Table 22. Again, the present approach can offer the result which is found to be in satisfactory agreement with that inferred form

Table 21 Statistical information obtained for Example 3. Random variable

E (GPa)

u (degree)

c (MPa)

Ph (MPa)

Pv (MPa)

q (kg/m3)

t (cm)

Mean value Standard deviation Range of values Distribution type

3.5 0.3812 [2.37, 4.59] Lognormal

30 2.7240 [27.20, 38.16] Lognormal

0.6 0.0892 [0.34, 0.67] Lognormal

10.4 0.8320 [8.15, 11.97] Normal

5.8 0.58 [4.52, 6.65] Normal

2216 229.1344 [2398, 2682] Normal

22 2.7104 [15.6, 26.8] Normal

Table 22 Comparisons of reliability results for different methods with given number of sampling points. Method

Number of sampling points

Reliability index, b

Probability of failure, Pf (102)

Relative error of Pf (%)

Polynomial-based response surface method (RSM) Artificial neural network (ANN)-based response surface method (RSM) Proposed hybrid approach Monte-Carlo simulations

175 70 28 28 1  104

1.3421 1.2607 1.3268 1.2463 –

8.9782 10.3708 9.2287 10.6327 10.8685

17.39 4.58 15.09 2.17 –

X. Li et al. / Structural Safety 61 (2016) 22–42

Monte-Carlo simulations. This implies that the SVM model in Eq. (1) as the response surface form within the experimental domain in the UD analysis could fit the displacement function U(x) in Eq. (4) reasonably well. In theory, this can be verified by bringing the obtained response surface form into comparison with the real LSF in the region around the design point, which contributes most to the probability of failure, as illustrated previously in Fig. 2. Obviously, for the 7-dimensional space (corresponding to 7 random variables) in this example, it is impossible to visualize such a space, but certain features could still be identified. Probably, one practical and transparent strategy can be developed in the following line: (I) Note that the obtained response surface form (i.e., the SVM model in Eq. (1) used to approximate the displacement function U(x) in Eq. (4)) involves 7 random variables: E, c, u, Ph, Pv, q and t. It could be used as a function of the two random variables which are varied while the remaining five random variables are set at the design point. Here, an attempt may be made to choose E and c as the two parameters to be varied, due to their significant effects on the displacement [e.g., 8,32,68] and relatively high coefficients of variation (about 11% and 15% for E and c, respectively, according to Table 21). As a result, the other five random variables: u, Ph, Pv, q and t are kept at their design point values. (II) Let us consider a set of trial values within the experimental domain for E, which is assumed to be varied as 2.8, 3.0, 3.2, 3.4 and 3.6 GPa, respectively. It may hence be possible to, through the obtained response surface form, back-calculate a series of values for c that would give a displacement of 80 mm (owing to Umax = 80 mm, as in Eq. (4)). In this way, the corresponding values for c can be estimated as 0.5467, 0.5329, 0.5124, 0.4936 and 0.4673 MPa, respectively. These obtained values of E and c, along with those of u, Ph, Pv, q and t fixed at the design point can further be used to compute the magnitude of the displacement via numerical analyses in FLAC. Interestedly, if there are no approximation errors, i.e., a sufficiently high degree to which the fitted response surface form matches the displacement function U(x) in the neighborhood of the design point, then the FLAC computed displacement would be 80 mm, which is equal to Umax in Eq. (4). By doing so, the resulting values of the displacement are 80.0214, 80.0183, 80.0107, 80.0269 and 80.0335 mm, respectively. As anticipated, these values calculated for the displacement using FLAC are very close with the target displacement of 80 mm. Therefore, the response surface form obtained by the current approach could be a good approximation within the sampling region to the actual displacement function calculated with FLAC. Correspondingly, one can consider the proposed approach as sufficiently accurate for the computation of reliability results. These findings obtained in this study match with those analyzed in [e.g., 9,32,38]. To further demonstrate the accuracy and efficiency of our approach when the LSF is rendered implicitly via numerical method, the polynomial- and ANN-based RSMs (as two main approaches to handling the implicit LSF) are also used to perform the reliability analyses for this tunnel example, respectively. In the polynomial-based RSM, the most widely used second-order polynomial with interpolation sampling strategy (stemming from [38,39]) is adopted in this work. In the ANN-based RSM, the well-known back-propagation algorithm in conjunction with the frequently employed random method in sampling (see applications to structural reliability [27,43] and geotechnical reliability [9,41,42]) is presented to build the ANN model. There are seven input neurons denoting seven random variables E, u, c, Ph, Pv, q

37

and t, and one output neuron representing U(x). Additionally, the hyperbolic tangent (Tanh) function (producing slightly better results than the sigmoid function for this example) is used to transfer the values of the input layer nodes to the hidden layer nodes, and then the linear transfer function is employed to transfer the values from the hidden layer to the output layer. Since there is no general rule to determine the number of neurons in a hidden layer [44], we determine by convention the optimal number (corresponding to 12) of hidden neurons using a trial-and-error process. Here, it is noteworthy that to fairly compare the efficiency of our approach in the RSM context with that of the polynomialand ANN-based RSMs, the metric for comparisons should be carefully identified. Notice that for reliability analysis of this real-life tunnel, the main purpose we employ the RSM to replace the implicit LSF of Eq. (4) is to reduce the computational cost in conducting repeated point-by-point numerical analyses (to predict the downward vertical displacement occurred at the crown of this tunnel). Since the number of runs of the numerical method normally equals the number of sampling points used to perform multiple limit state analyses via Eq. (4), one possible metric for measuring the efficiency for this example could be the number of sampling points needed for establishing the SVM model. At this point, such a definition is consistent with that given in the same way for large and complicated structures as in [e.g., 27,43]. Table 22 lists the reliability results with different numbers of sampling points obtained from the proposed hybrid approach, the polynomial-based RSM and the ANN-based RSM, along with the direct Monte-Carlo simulations. The comparisons can be carried out by integrating the ‘‘exact” probability of failure (=10.8685  102) obtained by 1  104 direct Monte-Carlo simulations. Typical features observed in this table are described as follows: (I) Among three methods in the framework of RSM, the largest value of the relative error (=17.39%) of Pf is achieved when using the second-order polynomial-based RSM. From the viewpoint of computational accuracy, such an RSM gives the biased approximation of the probability of failure for cases where the quadratic response surface is applied to complicated nonlinear problems [47]. In fact, expanding on the analyses also for tunnel-reliability in [32] found difficulty in employing a quadratic approximation for the implicit LSF, and then had to recommend one refined form by adjusting the second-order polynomial. For the computational efficiency of the polynomial-based RSM, the final quadratic response surface adopted for this example can be obtained after 11 cycles of iteration. In each cycle, (2n + 1) = 15 sampling points (n = 7 corresponding to 7 random variables) are required to update the polynomial results. Hence the total number of sampling points shown in Table 22 touches 11  (15 + 1)  1 = 175, which is greater than those used by both our approach and the ANN-based RSM. Actually, the efficiency of the polynomial-based RSM diminishes greatly with increasing the number of random variables, which is the main limitation emphasized in [27]. To improve the accuracy and efficiency of the polynomialbased RSM, we may refer to some recent studies like [47,76] whose papers concentrate entirely on them. (II) Using our approach with 28 sampling points, the relative error of Pf between the derived solution here (=10.6327  102) and the ‘‘exact” one (=10.8685  102) with 1  104 direct Monte-Carlo sampling is found to be 2.17%. Clearly, our approach provides a satisfactory result at considerably smaller computational cost than Monte-Carlo simulations. This is similar to the situation in Table 7 for Example 1 and that in Table 18 for Example 2. On the other hand, it is interesting to learn of the fact that the ANN-based

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X. Li et al. / Structural Safety 61 (2016) 22–42

RSM under the condition of the same 28 sampling points generated as in our approach yields a probability of failure Pf = 9.2287  102. This corresponds to, however, a relatively large error (=15.09%) in Pf, implying that for the same number of patterns, the SVM model involved in our approach shows better performance than the ANN model. Such a situation is also consistent with the results (see Table 1) from the example used to demonstrate the capability of the SVM model in Section 2.2. This seems to due to the fact that such 28 sampling points may be not enough to construct an accurate ANN model, and thus more number of sampling points is required for the ANN-based RSM in this tunnel example. Also, this would coincide with the sentiment outlined by [45] for the prediction problem encountered during tunnel excavation (where it is important to have adequate data points for building an ANN model with acceptable precision). In this wise, the proposed hybrid approach could be more accurate than the ANN-based RSM when the sample size is relatively small. When the number of sampling points prepared for the ANN-based RSM is increased to 70 (generated by the widely used random method, as mentioned earlier), the relative error of Pf between the approximated one (=10.3708  102) and the ‘‘exact” one (=10.8685  102) is obtained as 4.58%. This suggests that the performance of the ANN-based RSM is improved with the increasing number of sampling points and then the comparable accuracy is achieved as in the proposed hybrid approach. From the above analysis, although there is no major difference in the accuracy between our approach (Pf = 10.6327  102 using 28 sampling points) and the ANN-based RSM (Pf = 10.3708  102 using 70 sampling points), the ANN-based RSM requires more distinct sampling points than the developed approach in this work. This implies that our approach could be particularly advantageous in the case where the number of sampling points is relatively small. Here, it is useful to recall that on the one hand, the small samples property of the SVM has been mentioned in Section 2.2. At this point, as elucidated in [60], the SVM technique carries out the model estimation (i.e., function approximation in this study) with the aid of the structural risk minimization principle, which focuses on dealing with the small samples problem. Note that the abovementioned polynomial or ANN techniques perform the model estimation relying mainly on the empirical risk minimization principle, which is actually suitable for the large samples scenario (or in theory when the number of samples tends to infinity). From this perspective, the SVM may be a promising tool for model-building with a small set of samples, especially for those real-world applications that have sampling limitations. On the other hand, for the UD, it intends to assist users in modelling with a small number of sampling points (see Section 2.1). As such, the proposed hybrid approach combining UD and SVM provides a balance between theoretical background and practical application, thus showing its potential for tunnel-reliability problems posed by the implicit LSF. 4. Comments on results It has been shown that in the examined range of tunnel examples, the current approach combining UD and SVM can be used successfully for their reliability analyses involving the implicit LSF. To better understand the preceding work done for demonstrating the feasibility and validity of such an approach, it is of interest in this section to succinctly review the results obtained from Examples 1, 2, and 3, respectively. First, with regard to Example 1, it should be emphasized that we intended to select a very typical example for the analysis of tunnel behavior, albeit the corresponding LSF is indeed simple

and explicit. Using our approach for such an example offers no particular advantage. However, it has been well discussed via other reliability methods in previous literature [e.g., 8,13,68] and thus become rather familiar to the readers. In this sense, Example 1 could be considered as a reliable metric when illustrating our approach. By dint of this example, it would be convenient to: (a) demonstrate clearly all the steps of the calculation process involved in the current approach, in particular to graphically explain and validate the local approximation feature in the RSM context; (b) assist users with an overall understanding for the applicability of the present approach to various aspects of tunnel-reliability analyses, including the correlated/uncorrelated properties, the changes of underlying model used as the response surface form, the normal/non-normal distributions and the bivariate/multivariable reliability problems. Notice that for all the mentioned aspects, the developed hybrid approach does provide reasonably good results consistently, thus verifying its robustness; and (c) discuss the basic usage of both SVM and UD in a form which can be understood readily. For the SVM involving the selection of kernel functions, there is not pronounced difference in the model’s performance between the radial basis kernel and the polynomial kernel with a suitable degree. For the UD in the case of a given number of sampling points, an UD table with a relatively small discrepancy seems to be generally desirable in order to obtain the reliability results as accurately as possible; on the other hand, experience has shown that initiating the UD procedure with a relatively narrow experimental domain and gradually increasing its width in subsequent sampling could be a proper strategy to guarantee satisfaction to the reliability results (e.g., a rough estimation of the design point). Next, we applied the proposed hybrid approach, based on the results obtained from Example 1, to the other two examples, i.e., Examples 2 and 3 for real-life tunnels. In the two examples, both LSFs in probabilistic analyses are rendered implicitly in a complex fashion of the random variables: (I) In Example 2, the sensitivity analysis of our approach to the uncertainties in the stochastic parameters was first carried out. This verifies the typical trend that the obtained reliability index gradually decreases as the coefficient of variation increases. It should be pointed out that a key issue addressed in this example is on how to suitably choose the number of sampling points in the UD context, since such a number is actually not known in advance for most practical problems. It has been shown that a possible guideline is to start the sampling with a relatively small number of sampling points and then progressively increase the number of these points during test calculations, When the trial number increases to a certain extent, it is desirable to have as small value of discrepancy as possible in choosing the UD table. That is, the generation of sampling points in the UD analysis should carefully be planned to obtain a proper balance between the number of sampling points and the corresponding discrepancy. By doing so, the effectiveness of the UD technique for suitably selecting sampling points has been clarified. (II) Our approach was further extended to Example 3, where the LSF involved in the tunnel example is only known implicitly through the numerical analyses (corresponding to the scenario of many real-life applications in tunnel community). As indicated in this example, one can consider our approach as sufficiently accurate for the computation of reliability results. Particular attention for such an examined example was paid to the comparisons made with the conventional RSMs (we consider the most two representatives, i.e., the polynomial-based RSM and the ANN-based RSM) to demonstrate the advantage of the proposed hybrid approach. The

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X. Li et al. / Structural Safety 61 (2016) 22–42

results manifest that the current approach yielding reasonably good results can be more efficient than the polynomial-based RSM and the ANN-based RSM in the case where a relatively small number of sampling points are used. The reason for this may probably be attributed to the fact: the UD intends to help users in modelling with a small number of sampling points and the SVM emphasizes the small samples property for model-building. Through Example 3, the application potential of the developed hybrid approach is verified for tunnel-reliability problems involving the implicit LSF. In passing, it is mentioned that for Example 3, the failure of probability computed is around 10%. This indicates a relatively high probability that the limiting tunnel displacement would be exceeded when the support structures are provided only by aforementioned concrete lining combining with rockbolts. Therefore substantial support is required to prevent possible violation of tunnel safety situation. A technique that is frequently attempted in such a case is to install a passive system of steel sets, cables or lattice girders (or some combination of these systems). Nevertheless, such a technique with the installation of heavy passive support may, as emphasized in [68], introduce some practical problems. Hence the practical solution adopted in the actual case is to use like sliding joint top hat section sets, or a more elaborate system relying on grouted fiberglass dowels and grouted forepoles with a protective umbrella. Details of the support requirements will not be further extended in this paper, please refer to [68] for more comprehensive analyses.

various [response surface] approaches as presented depends on the specific problem under consideration. Therefore it appears to be difficult to provide a general recommendation on methods to be used in applications” (pp. 161). At the conclusion of this paper, future work pertaining to UD and SVM are also needed to be concisely mentioned. In the UD context, it may be difficult to use the analysis of variance (ANOVA) for data analysis because it is impossible to evaluate all main effects of the factors for a relatively small number of experiments [51]. For the SVM, some limitations reside in like a heuristic process entailed to determine the model parameters [72] and the interpretable results with confidence measures required owing to limited information of the relationship between the variables [77]. Acknowledgements Grateful acknowledgements are expressed to the Editor and two anonymous reviewers for their constructive criticism on the earlier versions of this paper and offering fruitful and thoughtful suggestions that contributed to its substantial improvement. This research is a part of the work carried out by General Financial Grant from the China Postdoctoral Science Foundation (No: 2014M552159) and the National Natural Science Foundation of China (No. 11472311). The support provided by the National Natural Science Foundation of China (No. 51478479) is also appreciated. Finally, special thanks to the support of the specialized research fund from Central South University for enabling the first author to work as a postdoctoral research fellow. Appendix

5. Conclusions The reliability analysis of tunnel structures in many situations is hindered by the implicit nature of the LSF. To handle this issue, a hybrid approach in this paper combining UD and SVM has been developed in the framework of RSM. The proposed hybrid approach involves the generation of the sampling points by UD and the subsequent approximation of the original implicit LSF by SVM. Such an approach integrates the merits of both UD and SVM used for complex nonlinear modelling. In particular, the UD can help the users in modelling with a small number of sampling points, and the SVM also emphasizes the small samples property that underlies the model-building. This reveals that the proposed hybrid approach combining UD and SVM provides a balance between theoretical background and practical application. Three carefully selected examples were analyzed: one for a typical tunnel under relatively simplified tunnelling conditions and the other two for real-life tunnels, to illustrate the application and effectiveness of the proposed hybrid approach. Comparisons were further made to assess its computational accuracy and efficiency. In the investigated examples, such an approach yields results which were found to be in satisfactory agreement with the ‘‘exact” results and expends dramatically less computational effort than Monte-Carlo simulations. Furthermore, through the tunnel example involving a numerical method-based reliability analysis, it has been shown that compared with the conventional RSMs (say, the polynomial-based RSM and the ANN-based RSM), the proposed hybrid approach is much more economical to achieve reasonable accuracy when a relatively small number of sampling points are employed. It should be stressed that the developed hybrid approach is not intended to replace any of the existing RSMs. Rather such an approach could be viewed as an alternative or complementary strategy for tunnel-reliability problems with the implicit LSF. In this sense, as concluded in [20], ‘‘the relative accuracy of the

MATLAB codes for reliability calculations As indicated in Section 2.3, the following essential computer program is coded in the context of the first-order reliability

Table A1 Uniform design (UD) table for U29(298). No. of experiments

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

No. of columns 1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

7 14 21 28 6 13 20 27 5 12 19 26 4 11 18 25 3 10 17 24 2 9 16 23 1 8 15 22 29

9 18 27 7 16 25 5 14 23 3 12 21 1 10 19 28 8 17 26 6 15 24 4 13 22 2 11 20 29

16 3 19 6 22 9 25 12 28 15 2 18 5 21 8 24 11 27 14 1 17 4 20 7 23 10 26 13 29

20 11 2 22 13 4 24 15 6 26 17 8 28 19 10 1 21 12 3 23 14 5 25 16 7 27 18 9 29

23 17 11 5 28 22 16 10 4 27 21 15 9 3 26 20 14 8 2 25 19 13 7 1 24 18 12 6 29

24 19 14 9 4 28 23 18 13 8 3 27 22 17 12 7 2 26 21 16 11 6 1 25 20 15 10 5 29

25 21 17 13 9 5 1 26 22 18 14 10 6 2 27 23 19 15 11 7 3 28 24 20 16 12 8 4 29

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Table A2 Accessory table of U29(298). Number of factors, s

Recommendation of column no. chosen

Discrepancy, D

2 3 4 5 6 7

1 1 1 1 1 1

0.0663 0.1128 0.1596 0.1987 0.2384 0.2760

6 3 2 2 2 2

6 5 4 4 4

7 78 678 5678

Table A3 Uniform design (UD) table for U29(296). No. of experiments

No. of columns

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

13 26 9 22 5 18 1 14 27 10 23 6 19 2 15 28 11 24 7 20 3 16 29 12 25 8 21 4 17

17 4 21 8 25 12 29 16 3 20 7 24 11 28 15 2 19 6 23 10 27 14 1 18 5 22 9 26 13

19 8 27 16 5 24 13 2 21 10 29 18 7 26 15 4 23 12 1 20 9 28 17 6 25 14 3 22 11

23 16 9 2 25 18 11 4 27 20 13 6 29 22 15 8 1 24 17 10 3 26 19 12 5 28 21 14 7

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Table A4 Accessory table of U29(296). Number of factors, s

Recommendation of column no. chosen

Discrepancy, D

2 3 4 5

1 1 1 2

0.0520 0.0914 0.1050 0.1730

4 34 345 3456

method to perform reliability calculations for this study after the SVM model (i.e., the response surface function viewed as the approximated explicit LSF) is obtained. For the purpose of illustration, the tunnel example corresponding to ‘‘Determination of calculation steps” in Section 3.1.1 of Example 1 is used to undertake this task.

Calculation results: the above computer program will result in the design point (u (°), c (MPa)) = (23.05902, 0.18481), the reliability index b = 0.6938 and the probability of failure Pf = 0.2439, as tabulated in Table 7.

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X. Li et al. / Structural Safety 61 (2016) 22–42 Table A5 Uniform design (UD) table for U28(288). No. of experiments

No. of columns

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

7 14 21 28 6 13 20 27 5 12 19 26 4 11 18 25 3 10 17 24 2 9 16 23 1 8 15 22

6 3 19 6 22 9 25 12 28 15 2 18 5 21 8 24 11 27 14 1 17 4 20 7 23 10 26 13

18 7 25 14 3 21 10 28 17 6 24 13 2 20 9 27 16 5 23 12 1 19 8 26 15 4 22 11

20 11 2 22 13 4 24 15 6 26 17 8 28 19 10 1 21 12 3 23 14 5 25 16 7 27 18 9

23 17 11 5 28 22 16 10 4 27 21 15 9 3 26 20 14 8 2 25 19 13 7 1 24 18 12 6

24 19 14 9 4 28 23 18 13 8 3 27 22 17 12 7 2 26 21 16 11 6 1 25 20 15 10 5

25 21 17 13 9 5 1 26 22 18 14 10 6 2 27 23 19 15 11 7 3 28 24 20 16 12 8 4

Table A6 Accessory table of U28(288). Number of factors, s

Recommendation of column no. chosen

Discrepancy, D

2 3 4 5 6 7

1 1 1 1 1 1

0.0545 0.0935 0.1074 0.1381 0.1578 0.1550

4 2 2 2 2 2

5 5 3 3 3

7 78 567 5678

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