Probabilistic-based assessment of existing steel-concrete composite bridges – Application to Sousa River

9 This paper presents a framework to assess the safety of existing structures, combining deterministic 10 model identification and reliability assessment techniques, considering both load-test and complementary 11 laboratory test results. Firstly, the proposed framework, as well as the most significant uncertainty sources 12 are presented. Then, the developed model identification procedure is described. Reliability methods are 13 then used to compute structural safety, considering the updated model from model identification. Data 14 acquisition, such as that collected by monitoring, non-destructive or material characterization tests, is a 15 standard procedure during safety assessment analysis. Hence, Bayesian inference is introduced into the 16 developed framework, in order to update and reduce the statistical uncertainty. Lastly, the application of 17 this framework to a case study is presented. The example analyzed is a steel and concrete composite bridge. 18 The load test, the developed numerical model and the obtained results are discussed in detail. The use of 19 model identification allows the development of more reliable structural models, while Bayesian updating 20 leads to a significant reduction in uncertainty. The combination of both methods allows for a more accurate 21 assessment of structural safety.


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Structural assessment comprises all activities required to evaluate the condition of structures for future use, 28 namely their safety. Several authors have used probabilistic-based procedures to assess the safety of existing 29 structures, having shown that conclusions can be dramatically different from those obtained by using existing 30 codes [1][2][3][4][5][6]. When assessing an existing structure, the available information regarding materials and geometry is 31 usually limited. In order to overcome this drawback, model identification techniques may be used to estimate 32 structural parameters based on measured performance, such as deflections. More recently, Bayesian inference was 33 introduced to improve the quality of the probabilistic models for both resistance and effect of loadings, by using 34 data collected from the structure under analysis [7,8].

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In this work, a probabilistic-based structural assessment framework, combining deterministic model 36 identification and reliability assessment, is presented and tested on a composite steel-concrete bridge subjected to 37 a performance load test. In the first step, a sensitivity analysis is used to identify the most influential parameters 38 on the overall structural response at both service and ultimate loading conditions. Then, these parameters are found 39 considering a model identification algorithm, which consists in an optimization procedure minimizing the 40 difference between observed performance (e.g. vertical displacements collected during the load tests) and 41 performance predicted using a non-linear numerical model. A convergence criterion which addresses the expected 42 accuracy of experimental and numerical results, is considered [9]. This procedure yields a set of near optimal 43 solutions, from which the best model is selected considering the probability of each solution occurring based on 44 previous knowledge, followed by an engineering judgment procedure. A reliability assessment algorithm is then geometrical properties of the structure, which is fundamental to define the probabilistic distributions of structural 56 parameters that will be used in the reliability assessment of the structure.

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The model identification procedure can be computationally expensive due to the need to evaluate a large 58 number of NL-FEM models. To minimize the impact of this, a sensitivity analysis is used to identify the critical 59 parameters, minimizing the complexity of the model identification procedure [10,11]. This analysis consists of 60 evaluating the fitness function variation with each input parameter [12]. An importance measure, bk, is obtained 61 for each parameter as, where bk is the importance measure of parameter k, ∆yi,k is the variation in structural response parameter, ∆xi,k is 63 the variation of input parameter around its average value xm,k, ym,k is the average response, n is the number of data  75 If more than one measurement is made, independently of being the same type or not, then the standardized values 76 should be added and divided by the total number of measurements, in order to obtain a final standardized value.

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As such, by normalizing the value of each parameter, it is possible to use different transducers, measuring different 78 parameters in any section of the structure and load case (LC).

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In order to limit the probability of overfitting, optimization is conducted, not to find the best solution, but 4 a group of solutions associated with a fitness under a given threshold. It is assumed that when computing the 81 difference between numerical and experimental data, results associated with a fitness below the expected 82 amplitude of errors are considered as optimal. The threshold value, ε, is calculated using the law of propagation of 83 where u(xi) is the error associated with each source of uncertainty and ∂f/∂x is the partial derivative of the fitness 94 function in order to each component x.

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In safety assessment, a comparison between resistance, R, and loading, S, distributions is performed [17].

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Accordingly, the failure probability, pf, corresponds to the case in which the structural resistance is lower than the 127 applied load. In this situation, the limit function may be defined by Z(R,S) = R -S. The correspondent reliability 128 index, β, is given by β = -Φ -1 (pf), being Φ -1 the inverse cumulative distribution function for a standard Normal 129 distribution.
6 structural safety assessment [14,18] consists of computing the obtained reliability index and comparing it to a 135 target value, βtarget, proposed by existing standards [12].

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The abutments are independent from the rest of the structure, as expansion joints allowing longitudinal movement 156 due to temperature and other environmental effects, were introduced in the bridge ends ( Figure 1b) Figure 3a shows the location of all sensors. A frequency of 10 Hz was designed for registering 177 the vertical displacement data. Two displacement transducers were installed in span A1 -C1, designated by VD1 178 and VD2, and other two at span C1 -C2, denoted as VD3 and VD4 (Figure 3a). The load was applied using four 179 identical vehicles (with four axles), each vehicle loaded with sand in order to obtain a total weight close to 32 tons.

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Three different load cases (LC) are considered, as represented in Figure 3b: (1)

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The precast slab presents a non-uniform geometry ( Figure 1c). This non-uniformity is considered in the 201 numerical model by introducing several concrete layers. As the number of layers increase, the model's geometry 202 becomes more accurate, but also more complex. For the purpose of this paper, two rectangular layers are used.

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The reinforcing steel is considered to be embedded in precast concrete slab [12].

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There are two distributions of headed studs along the bridge: low (groups of 6 studs) and high (groups of 210 10 studs). The space between each group of studs, 0.50 m, is the same for both low and high stud densities

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The evaluated parameters and corresponding CVs and standard deviations (σ), used to compute the 319 importance measures are given in Table 3.   Table 4.

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The fitness value criterion establishes that its improvement (Δf) should be less than or equal to a threshold 359 value (ε), which can be understood as the model identification procedure precision [12,37]. By applying the law The model identification procedure is executed five times, considering different randomly generated 362 starting points, as to limit the probability of underperforming results. Each analysis provides a final population of 363 10 models, resulting in a total of 50 models. Based on the principle that the most suitable model is that with smaller 364 deviation from the initial mean values (see Table 5), unless some accidental situation is detected, the best model 365 is that which presents the highest likelihood value [12,37]. Figure 9 presents the obtained normalized values for 366 the maximum likelihood for all the selected models. In this situation, model 20, from the second analysis, presents 367 the highest value, being thus the selected model. 368 Table 5 indicates initial and identified values for the critical parameters considered in the model 369 identification procedure, showing that the used concrete presents a higher quality than initially expected. With 370 respect to horizontal spring stiffness results indicate that the identified k1 and k2 values are respectively lower and 371 higher, respectively, than the initial prediction. The slab height (hslab) is slightly higher, around 3%, than the design 372 value. The concrete self-weight (conc) is practically unchanged. However, the obtained pavement load (ppav) is 373 15% higher than the design value. This might be due to the irregularity in bituminous thickness.

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The comparison between the fitness function, considering initial and identified critical parameters value 375 shows a reduction in error from 67.33% to 53.74%, being an improvement of more than 20%.

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The results obtained are shown in Table 7. Regarding concrete, obtained results indicate that the quality is 394 slightly superior than the predicted, confirming the model identification results. With respect to reinforcing steel, 395 results confirm the steel quality considered in the design phase. The steel profile material quality is slightly superior 396 to that considered in design.  Table 3. The updated model is respectively 414 based on the initial one, but considering the mean values (μ) obtained from model identification (Table 5). In this 415 case, the CVs are obtained in a similar way by using the CVs from Table 3.

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Bayesian inference procedure was developed by considering an informative and a non-informative (Jeffrey's) prior, being the adopted posterior PDF that with the lowest standard deviation [12,37]. Table 8 gives   418 the probabilistic models for the critical parameters resulting from each of the analysis performed.

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The analysis of these results confirms the complementary tests, once the obtained mean is higher than the 420 initial prediction. The uncertainty is lower than the initial one, once the CV has been reduced.  randomly generated models. Then, a curve fitting procedure is developed to compute the resistance PDF 436 parameters. Obtained mean (μ) and standard deviation (σ) values are given in Table 9, being then computed the 437 failure probability, pf, and the corresponding reliability index, β, as a comparison between the resistance and the 438 effect of loads curves.

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An overall analysis of those results allows to conclude that, for the considered LCs, and for the developed 440 numerical model, the overall bridge resistance is substantially higher than the applied load model, by comparing 441 the resistance PDF mean of each analysis, from Table 9, with the loading PDF mean (4939.40 kN). By comparing 442 the obtained resistance PDF for the four probabilistic models, it is possible to conclude that model identification 443 practically did not change the obtained results. This is due to the fact that the majority of assessed parameters in 444 model identification, in service phase, do not influence the bridge behavior up to failure. The application of a Bayesian inference procedure leads to an increase in the failure load, confirming an additional structural resistance 446 capacity which was not initially identified. When evaluating the CV, it is possible to conclude that both initial and 447 model identification models provide similar results. A slight decrease on this value is verified with the Bayesian 448 inference procedure. This is due to a decrease on the standard deviation value of some of the updated parameters 449 (Table 8).

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The values obtained with this safety assessment procedure, respectively, the probability of failure, pf, and 451 the reliability index, β, are indicated at Table 9

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According to fib Task Group 5. 1 [46], and considering that an overall analysis of the structure is developed, 456 it is possible to conclude that the assessed bridge is in very good situation (8 < β ≤ 9). This is in agreement with 457 Tabsh and Nowak [47] guidelines, which indicate that a β-value higher than 5-6 corresponds to a structure with a 458 very good performance.