Probabilistic-based assessment of a masonry arch bridge considering inferential procedures

Engineering Structures 134 (2017) 61–73

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Probabilistic-based assessment of a masonry arch bridge considering inferential procedures Vicente N. Moreira ⇑, José C. Matos 1, Daniel V. Oliveira 1 ISISE, Department of Civil Engineering, University of Minho, Guimarães, Portugal

a r t i c l e

i n f o

Article history: Received 18 January 2016 Revised 3 October 2016 Accepted 29 November 2016

Keywords: Safety assessment Bayesian inference Masonry arch bridges Limit analysis Uncertainty sources Non-destructive tests Railway bridges

a b s t r a c t Considering the safety assessment requirements of masonry arch bridges, different levels of reliability, based on uncertainty, may be distinguished, whose core objectives are to accurately analyse the ultimate load-carrying capacity and the serviceability structural response. Within this framework, a simplified full-probabilistic methodology for the safety assessment of existing masonry arch bridges is proposed, which combines both structural analysis and Bayesian inference procedures. The proposed framework aims to determine the ultimate load-carrying capacity (Ultimate Limit State) of masonry arch bridges, by using probabilistic procedures and Limit States principles. Geometric, material and load characterization, as well as inherent uncertainties will be also considered. In order to determine the ultimate loadcarrying capacity, a limit analysis approach, based on the mechanism method, will be employed. Due to the high computational costs required by a probabilistic safety assessment framework, a sensitivity analysis will then be introduced. The incorporation of new information from monitoring and/or testing will be performed by the application of Bayesian inference methodologies. Based on the information collected, two reliability indexes will be computed and compared, one with data collected from design documentation and literature and the other with data collected from testing, emphasizing the importance of testing and the advantages of Bayesian inference procedures. The probabilistic framework developed is tested and validated in a Portuguese railway masonry arch bridge from the 19th century. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The highest expansion period of railway transport in Europe occurred in the early twentieth century [1]. Many of these bridges are over 100 years old, being the largest stock composed of masonry arch bridges (MAB) [2]. In Portugal, it was verified, between 1951 and 1973, the development of oriented programs for infrastructure maintenance [3]. Accordingly, MAB were renewed and/or strengthened for the new demands required by society’s needs [4]. Nowadays, many in-service bridges are submitted to much higher loads compared to those used in their design project. In addition, maintenance investments in this field are scarce, and, therefore, an advanced deterioration process is commonly identified as a result of several years in use without intervention. Thus, it is of utmost importance

⇑ Corresponding author at: ISISE, Dep. of Civil Engineering, University of Minho, Campus de Azurém, 4800-058 Guimarães, Portugal. E-mail addresses: [email protected] (V.N. Moreira), jmatos@civil. uminho.pt (J.C. Matos), [email protected] (D.V. Oliveira). 1 Address: ISISE, Dep. of Civil Engineering, University of Minho, Campus de Azurém, 4800-058 Guimarães, Portugal. http://dx.doi.org/10.1016/j.engstruct.2016.11.067 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.

to assess the performance of existing bridges, in order to ensure its safety [5]. Most procedures for safety assessment of existing structures are based on the partial safety factor method. The major disadvantage of this methodology is the non-explicit consideration of uncertainties, resulting in a safety assessment procedure that does not reproduce the assessed structure behaviour with liability [5,6]. Therefore, the development of safety assessment methods, which are easy to apply and give accurate results, is of extreme importance. Accordingly, some methodologies that explicitly consider the existing uncertainty when computing the reliability index were recently proposed [7,8]. In this work, a probabilistic-based assessment framework combined with Bayesian inference procedures is presented. As a first step, the bridge’s geometry and material characterization are performed according to literature and design documentation, allowing to define the deterministic numerical model. Since the number of structural variables involved in safety assessment is typically high and, consequently requiring costly computational and time resources, a sensitivity analysis is introduced, being obtained the critical structural parameters. Thus, probability density functions (PDF) will be assigned only to these parameters. In order to

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compute the failure probability, simulation methods are used. Latin hypercube sampling (LHS) is chosen, due to its reduced required resources [9,10]. Some structural parameters may also present some correlation between each other. This correlation is considered by proper coefficients, according to literature. Thus, the developed framework combines a LHS sampling procedure with a built-in Iman and Conover algorithm [9,11], which allows the sampling of correlated random variables. After generating all numerical models, the results are then statistically analyzed. The structural analysis is performed through the Upper Bound (Kinematic) Theorem of Plasticity. The bridge’s safety is evaluated by comparison of its resistance with the effect of loadings, being obtained the failure probability and corresponding reliability index. Once data acquisition is common in safety assessment evaluations, Bayesian Inference procedures are introduced. Bayesian Inference procedures consist in updating and reducing the statistical uncertainty, through incorporation of gathered external data into the data analysis process. In the present paper, external data regarding geometry is collected by photogrammetric methods, being complemented by conventional measures. Additionally, material characterization tests are performed in order to gather masonry’s physical and mechanical properties. Lastly, the framework proposed in chapter 2 is applied and tested with an in-service Portuguese stone masonry arch railway bridge. The comparison of two reliability indexes, obtained before and after Bayesian updating, shows the importance of data acquisition when assessing MAB and that Bayesian inference is a key procedure to incorporate gathered data into the numerical analysis, providing a more reliable safety assessment. 2. Probabilistic-based structural safety assessment of masonry arch bridges The probabilistic-based structural assessment framework proposed here departs from a deterministic numerical model. Then, the influence of each structural parameter is evaluated by a sensitivity analysis. Afterwards, the MAB structural performance is assessed according to a reliability assessment procedure, by incorporating randomness into structural parameters. Later, Bayesian Inference techniques are employed to update the initial PDFs, by considering the collected information from visual inspection, characterization tests and/or monitoring systems. After the updating process, a set of numerical models are respectively computed, being obtained the failure load factors. This way, it would be possible to compute the resistance PDF before and after the inferential procedure. 2.1. Masonry arch bridge structural behaviour MAB construction is not currently practiced, but many of these structures are still in service and playing an important role in the railway network. These type of structures were conceived as gravity structures for which mass and geometry were the design criteria [12,13]. Fig. 1 presents the typical elements and typology of masonry arch railway bridges [14]. The arch is the structural element responsible for transposing the span and transferring the loads from the fill material to abutments and piers, while the fill material disperses the live loads, confers additional compression to the arch and provides passive pressure when the arch tends to move against it, enhancing the ultimate load-carrying capacity. In the case of multi-span MAB, piers’ geometry is conditioning since they may be involved in the collapse mechanism due to their slenderness, resulting local or global collapse mechanisms [13,15]. Thus, MAB structural behaviour is highly dependent on the interaction between the fill material, arches and piers [16], which is

presented in detail in [17–21]. A detailed review and description of MAB failure modes, load effects and geometrical and material issues is presented elsewhere [17,22–24]. 2.2. Data acquisition Data acquisition is very important due to the fact that safety assessment of MAB depends on the liability of the input parameters. Hence, if possible, it is important to perform geometric and material characterizations to obtain geometry details, current physical and mechanical parameters, thus improving the predicted safety level assessment [12,13]. 2.2.1. Visual inspection The first inspection method to gather data for condition assessment should be visual inspection. The external visual inspection includes the identification of structural changes (e.g. settlements, deformations), missing geometric characterizations (e.g. piers thickness and height or arch width and thickness), defects (e.g. cracks), deterioration (e.g. masonry or ballast deterioration, mortar loss) and damaged structural elements due to accident situations (e.g. impact loads). Visual inspection may be complemented by non-destructive tests (NDT), being possible the detection of micro cracks, damages sources and progression [25]. A complete description of these methods is presented in [25]. 2.2.2. Geometry The first step in order to create a numerical model is the definition of the MAB geometry. The project documentation is usually lost or, in case it exists, rarely present any drawings of construction details. There are several techniques to perform geometry surveys of MAB. The most used methods are: (i) conventional methods, such as tape measure, level or laser meter; (ii) terrestrial laser methods, such as 3D laser scans; and (iii) photogrammetric methods, such as close range photogrammetry (under 100 m from the structure) and specific software. These methods may complement each other. When performing MAB geometry characterization, it is important to give special attention to the following elements, since they are the most relevant geometrical parameters in the overall structural behaviour [15,17,22]: (i) arch thickness and width; (ii) fill depth at arch crown; (iii) rise at mid-span; and (iv) span-length. These parameters are typically described by a Normal PDF [12,17,22]. According to measurements performed by [17,22], a coefficient of variation (CoV) of 10% and 5% may be assigned to describe the variability of the thickness and width of the arch, respectively. Regarding the piers thickness and height, since literature is very scarce and this geometric detail is vital for defining the collapse mode of multiple-span MAB (local or global) [12,13], it is assigned a CoV of 10% as in the case of arch thickness [12,13]. All these parameters are here assumed to be described by a Normal PDF [12,13,17,22]. 2.2.3. Materials Besides the geometric configuration, which plays a very important role on stability, the structural behaviour of MAB is highly dependent of the mechanical properties of the materials, namely masonry and fill material [12,15,17,18]. Moreover, the type and quality of materials used for the arch, piers and fill material may be different even in the same MAB. The main characteristics of masonry are its heterogeneity, anisotropy, moderate compressive strength and reduced tensile strength [26], while the homogeneity of the fill material depends on the materials used [23]. NDT are the most used ones to characterize structurally MAB components. When selecting the most suitable material characterization tests, priority should be given to those that provide

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Fig. 1. Masonry arch railway bridges terminology and elements (adapted from [14]).

information about the following mechanical and physical properties, once they are the most relevant ones in the overall structural response [15,17,22]: (i) density, compressive and tensile strengths and endurance limit to fatigue of masonry; (ii) density, compressive strength and elasticity modulus of fill material; and (iii) masonry shear strength. The material uncertainty is very dependent on each MAB, once they were typically built with locally available materials. Therefore, the statistical information about these properties is scarce and very particular for each MAB, for which in situ characterization tests are recommended. 2.3. Structural analysis 2.3.1. Limit analysis Collapse of masonry arches is attained when a sufficient number of hinges are formed, turning the arch into a mechanism. Recognizing the nature of collapse mechanism of masonry arches, Pippard [27] and Heyman [28] formulated the following hypothesis to single unreinforced masonry arches: (i) arch barrel idealized as an assemblage of rigid blocks (negligible changes in geometry); (ii) arch masonry has infinite compressive strength; (iii) no tensile strength at masonry arch joints (even if there is any); and (iv) no sliding between masonry blocks (sufficient inter-surface friction to prevent sliding). Gilbert [29] extended the basis of basic rigidblock method described previously, allowing the inclusion the following hypothesis in the analysis: (i) sliding between adjacent blocks; (ii) multiple spans and multiple arch rings; and (iii) finite masonry strength. Limit State’s software RING [30] is based on the Mechanism method and it incorporates Gilbert’s hypothesis. This software will be adopted in this research work due to its numerical robustness reduced computational costs. Within this software, the MAB is idealized as series of rigid blocks, which may collapse due to loss of equilibrium, formation of a sufficient number of internal releases, such as plastic hinges or sliding planes between units, or a mix of both. Material crushing is another

possible collapse mode, but it is not common, due to the low stresses on the arch blocks. Fill material between the arch and the ballast, if present, although not modelled explicitly, is considered to disperse live loads and to provide passive pressures, which is computed according to the classical lateral earth pressure theory [30]. Spandrel walls are not modelled as their contribution to the ultimate load capacity is typically of low importance as they suffer damage much before the arch does. Furthermore, it is common that spandrel walls are detached (common defect) from the arch [17,25,30], being their contribution to the increase in the ultimate load-carrying capacity very small. Masonry blocks, at failure, may rotate and slide at all block joints. In this way, plastic hinges may develop in any block joint, including at the piers base. The live load dispersion is also considered in the transversal direction, based on the concept of the effective bridge width, i.e., the transversal width of the masonry arch that resists to the applied loads. In order to compute the failure load factor, i.e., the critical position of each plastic hinge, RING software uses a linear optimization programming technique. Once the failure load factor is found, the collapse mechanism and load position is obtained. A full description of RING software is provided in [30]. 2.3.2. Sensitivity analysis The number of variables involved in a safety assessment procedure is high, resulting in a huge computational cost. In order to reduce this number of variables, a sensitivity analysis can then be carried out [7,13,31,32]. During this analysis, the parameters with most influence on the structural response, i.e., the critical parameters, are identified. The importance measure, obtained from Eq. (1), quantifies the contribution of a given parameter, i.e., the influence that each parameter has in the structural response [7,12,13,31–33],

bk ¼

n X ðDyi;k =ym;k Þ=ðDxi;k =xm;k Þ  CoV i¼1

ð1Þ

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being bk the importance measure of parameter k, Dyi,k the variation in structural response, Dxi,k the variation of input parameter mean value xm,k, ym,k the average response, n the number of generated parameters and CoV the coefficient of variation of the assessed parameter. Each model parameter importance measure is evaluated by adding or subtracting a standard deviation value to the mean parameter value, keeping the remaining parameters values with their mean values. Afterwards, each set of assessed parameter values is analyzed through structural analysis software, being then applied Eq. (1) to obtain the parameter importance measure. This procedure is applied to all model parameters. Then, obtained importance measure values are normalized with respect to the highest one and represented in a bar plot. In order to select the critical parameters, a limit importance measure is defined, blim. Accordingly, only the parameters with importance measures higher than the defined limit should be considered in the safety assessment. Additionally, characterization tests should be performed to evaluate the critical parameters rigorously, predicting the overall structural response more accurately. 2.4. Performance-based safety assessment 2.4.1. Randomness The probabilistic analysis is performed by introducing randomness in selected model parameters. Accordingly, after gathering information to characterize the uncertainty of structural variables obtained from sensitivity analysis, probabilistic distribution laws are assigned to each critical parameter. Some parameters may present some dependence among them, which can be expressed by adequate correlation coefficients. In the particular case of MAB, the statistical information regarding geometry and material parameters is rather scarce, or even null. Therefore, characterizations tests are recommended. 2.4.2. Bayesian inference Bayesian inference techniques allow the introduction of information gathered during characterization process into the structural model [5,7,8,13,33]. This procedure starts with an initial distribution, the prior distribution, whose parameters may be assigned based on literature, experience or from previous safety assessment procedures. The prior distribution reflects the uncertainty about a specific variable. Hence, based on Bayes Theorem, which is the basis of the updating process, the prior information and the new data are weighted, passing from the prior distribution to the posterior distribution. Regarding the modelling process, the choice of a prior distribution is a crucial step, once it has an impact on the posterior distribution, being always necessary to check its stability. A noninformative prior distribution may be convenient, although it is always required to verify if the computed posterior distribution is proper [34]. A common non-informative prior is the Jeffrey’s prior distribution. If any information, regarding the interest parameter is available, an informative prior distribution may be chosen instead. If the prior and the posterior distribution belongs to the same parametric form, it is verified the conjugacy property. Conjugacy is mathematically advantageous once the parametric form of the posterior distribution is known, being the obtained results easy to interpret and may be presented in its analytical form. The consideration of a Normal likelihood has the advantage of either conjugate or non-informative priors resulting in proper posteriors [34]. 2.4.3. Safety assessment Reliability analysis is performed based on the Limit State (LS) principle. In this case, the LS establishes the safe and the failure

regions of structural performance. LS is mathematically represented by a function g(X), as follows expression (2),

gðXÞ ¼ ZðR; SÞ ¼ R  S

ð2Þ

where X is the vector of random variables, Z is the safety margin variable, R is the resistance and S the effect of loads. The failure probability, pf, reads expression (3),

pf ¼ pðgðR; SÞ < 0Þ

ð3Þ

being the reliability index, b, expressed by Eq. (4),

b ¼ /1 ðpf Þ

ð4Þ

where /1() is the inverse function of the Normal standard distribution function [35]. The structural failure probability, pf, and corresponding reliability index, b, are obtained by a sampling procedure. As a last step, safety evaluation of the assessed structure is performed by comparison of the computed reliability index against a target one, which is presented in literature [36–40]. It is worth to note that MAB typically reach a Ultimate Limit State (ULS) not by structural capacity (ULS:STR), but by the formation of sufficient internal releases, such as plastic hinges or sliding planes, leading to structural collapse (ULS:EQU) [12,13,17,22]. Thus, the LS function formulation depends of the failure mode. 3. Durrães viaduct The Durrães viaduct, also locally known as the Dry Bridge (Fig. 2a), was built in 1876 and is located at the Minho railway line (41° 630 5,9100 N; 8° 660 9,3400 W), connecting Durrães and Barcelos. It consists of sixteen semi-circular arches of 0.70 m thick and 9.00 m span and fifteen piers with 11.50 m high, making a total height of 18.00 m and a total extension of 178.00 m. Two of the piers have approximately the double thickness of the remaining piers, located at 1/3 and 2/3 of extension length. The building material is granitic masonry, with rough and dry joints. The deck has a longitudinal slope of 1.45% with a width of 5.25 m [13], housing a single railway line composed by UIC 60 bi-block sleepers, see Fig. 2b. The depth of fill material at arch crown is 0.60 m and the thickness of the ballast layer is 1.00 m. At the piers and abutments there is backing, which height reaches 3.00 m. The thickness of the spandrel walls is 0.30 m, while inner spandrel walls do not exist. The backing occupies the full bridge width and reaches a height of 3.00 m. 3.1. Data acquisition 3.1.1. Visual inspection A visual inspection of Durrães viaduct was performed with the purpose of obtaining its main characteristics, including constituent materials and detection of any existing damage (cracking, settlements, open joints, etc.). Regarding material characterization, it was possible to ascertain that the viaduct was built with granite with rough and dry joints. With respect to its condition, it was identified vegetation between joints of blocks, infiltrations on piers and arches, open joints on piers, and cracks on some blocks on the arches, which are presented in Fig. 3. A few cracks formed due to the detachment of spandrel walls from the arch barrels were also detected. Also, the cracks and opening joints width varied between negligible (easily treatable: <0.1 mm) to severe (expensive repair: 15–25 mm) [41]. 3.1.2. Photogrammetric survey The geometric characterization of Durrães viaduct was accomplished by project documentation and inspection reports, kindly provided by the Portuguese railway authority REFER. Inner

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Fig. 6 presents the photogrammetric geometric model and geometric details (longitudinal direction) of Durrães viaduct. The Durrães viaduct was also studied by Arêde et al. [42]. These authors performed GPR profile reads in order to obtain the geometry of inner structural elements, such as inner thickness of the arch, piers components and the ballast layer thickness, fill material homogeneity and possible structural damages. This geophysical method consists in the emission of electromagnetic waves into the structure, in the frequency band of 10 MHz–2.5 GHz, see [25] for details. Arêde et al. [42] performed a total of 25 profile reads, determining that the arch inner thickness is 0.70 m, being the arch thickness constant along the bridge width. Also, from GPR measurements it was determined that the ballast layer is 0.50 m thick. By comparing these measures with those presented in design project documentation, it is verified that the real thickness of the ballast layer is half of that predicted by design project. This fact highlights the importance of in situ surveys when assessing MAB.

Fig. 2. Durrães viaduct: (a) perspective; and (b) railway track line [13].

structural elements, such as arch thickness and piers backing height, were possible to characterize through the design project documentation, see Fig. 4. Fig. 5 presents the longitudinal view of Durrães viaduct. This geometric information, obtained from previous documents, may not reproduce, in a reliable way, the Durrães viaduct geometry. Hence, close range photogrammetric analysis and conventional measurements were performed. A photogrammetric model was built based on photos taken during the visual inspection, in a total of 120 photos, defining a minimum of ten common points between each photo. The geometric model obtained from the photogrammetric measurements was then complemented and validated by conventional measuring methods, such as tape and laser meter.

3.1.3. Material survey In the context of Durrães viaduct material characterization four monotonic uniaxial compression tests were carried out to determine the stone masonry compressive strength and elasticity modulus. Also, tested blocks were previously measured and weighted in order to obtain its density. For the evaluation of masonry mechanical properties, stones were carefully sawn in the form of blocks with dimension of 150  150  100 mm3 so that their faces keep without any apparent irregularities, corresponding to a total height of 300 mm (prism with three stacked blocks) [13]. Obtained values for elasticity modulus, E, density, cm, uniaxial compressive strength, rpeak, and corresponding strain, epeak, for each prism are given in Table 1. The strikethrough values were obtained from a prism with a natural defect. In this case, only the density value was considered in the analysis, since this property was not affected by stone defects. Arêde et al. [42] also performed characterization tests for the Durrães viaduct characterization. These authors have performed tests to masonry and ballast materials. With respect to masonry, a mean value of 25.01 kN/m3 with a CoV of 6.7% for the density masonry, cm, were obtained, confirming the results here obtained. Regarding the ballast material, its density, cb, was evaluated based on extracted cores, being obtained a mean value of 22.62 kN/m3 with a CoV of 4.2%.

Fig. 3. Visual inspection: vegetation (green), cracks (red), efflorescence (dark blue) and open joints (cyan): (a) south elevation view; (b) plan view; and (c) north elevation view. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 4. Historical drawings of the Durrães viaduct: details of piers, arches and abutments.

3.2. Load model 3.2.1. Static effects The railway load model LM71, from UIC [43] and adopted by the standard EN 1991-2 [44], has been used in this work. The LM71 load model is characterized by four single loads with 250 kN of magnitude, Qvk, spaced of 1.60 m, and two uniform distributed loads, qvk, of 80 kN/m, with an undefined length. Moreover, the uniform distributed load can be only applied in some spans, being some areas load-free. The loading scheme and characteristic values are shown in Fig. 7. An important issue regarding this model is the inclusion (or not) of these uniform distributed loads in the analysis as they are optional and may, in some circumstances, generate favourable

Fig. 5. Durrães viaduct: elevation and plan views.

effects. Santis [45,46] evaluated the inclusion of such loads in the analysis of multi-span MAB, concluding that the presence of uniform distributed loads have favourable effects, resulting in a higher

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Fig. 6. Geometry of the Durrães viaduct: (a) rendered photogrammetric model; and (b) geometric details in longitudinal direction obtained from photogrammetry, GPR and conventional measurements [13].

Table 1 Physical and mechanical properties of stone masonry [13]. Prism P1 P2 P3 P4 Mean (l) CoV (%)

cm

[kN/m3]

rpeak

epeak

[MPa]

[‰]

E [GPa]

25.55 25.60 24.67 25.29 25.28 1.69

41.64 47.99

6.10 5.69

10.45 13.07

43.82 44.48 7.25

5.68 5.83 4.12

10.92 11.48 12.17

Fig. 7. Load Model LM71 [44].

failure load factor. This is due to an increase in the compression provided by the uniform distributed load when arches tend to move upward, preventing the formation of the collapse mechanism. Thus, only the four single loads should be considered in the multi-span MAB analysis.

3.2.2. Dynamic effects The moving railway loads induce vibrations that enhance their static effects, and if not suitably considered may lead to the weakening of the arch due to the formation of cracks, loss of mortar or opening of joints, culminating in the diminishing of MAB ultimate load-carrying capacity or even in its collapse. Thus, railway loads are dynamic in nature, causing the amplification of the structure response, comparatively to the static one. Such amplification is primarily due to [44]: (i) the mobile nature of loads; (ii) the successive crossing of carriages with regularly speed axes; (iii) a frequency that may approach the natural frequencies of the bridges, leading to the resonance phenomena; (iv) damping; and (v) the irregularities of railway track, rail and wheel defects. EN 1991-2 [44] proposes to quantify the static response amplification due to dynamic effects for carefully maintained railway tracks, through the dynamic factor, /3, as follows, expression (5),

1:44 /3 ¼ pffiffiffiffiffi þ 0:82 L/  0:2

ð5Þ

being L/ the determinant length. L/ is the span length for simply supported bridges or an equivalent span length for other support conditions and bridge types, which are provided in [44]. According to EN 1991-2 [44], the determinant length, L/, for MAB is twice the clear opening. Also, in the case of MAB, a reduction factor, given by Eq. (6), may be applied in order to reduce the dynamic effects [44],

red/3 ¼ /3 

h1 10

ð6Þ

being red/3 greater or equal to 1,0 and h is the height of the cover (in meters) including the ballast from the top of the deck to the top of the sleeper or, in case of arch bridges, from the crown of the extrados. The Durrães viaduct span is of 9.00 m, representing a determinant length of 18 m [44]. Thus, a dynamic amplification factor of 1.18 is obtained according to Eq. (5). Then, a reduced dynamic factor of 1.16 is given by Eq. (6), which will multiply the static load values, in order to consider the dynamic effects in safety assessment.

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3.3. Deterministic analysis. Failure modes In order to obtain the ultimate load-carrying capacity of Durrães viaduct, a numerical model was developed in RING software [30]. In this software, the MAB is idealized as an association of rigid-plastic blocks, being computed a failure load factor and the related collapse mechanism. The consideration of transversal effects may reduce the applied stress on the arch, allowing higher ultimate load-carrying capacity. Hence, the masonry arch width in the transversal direction that resists the applied loading, i.e., maximum effective bridge width, was considered to be equal to the width of the bridge (5.25 m) as no longitudinal cracks were found on the arch, with a transverse angle of distribution through ballast of 15° and a transverse angle of distribution through fill of 30° [25,30]. In the longitudinal direction, the live load dispersion is performed according to the Boussinesq theory, with a longitudinal angle of distribution of 15° and 30°, through ballast and fill material respectively. In order to compute the failure load factor, the four concentrated mobile live loads are considered for each 200 mm along the bridge length. For each position a collapse load factor is obtained, being the failure load factor the lowest value of all obtained collapse load factors. Any possible settlements of abutments and piers are not considered within this framework, due to an unavailable geotechnical information regarding foundations and soil properties. Fig. 8 illustrates the failure mechanism, obtained for a failure load factor of 2.78. A seven plastic-hinge mechanism has formed, mobilizing the 14th and 15th spans and the 14th pier. It is noted that numerical model was built with information gathered from literature [13,17,22,39] and design project, being considered for each model parameter its mean value, without considering any parameter uncertainty, see Table 2. 4. Reliability analysis 4.1. Loading curve Regarding the loading curve, the load model LM71, presented by EN 1991-2 [44], was adopted. This model may be employed for the design of new bridges and assessment of existing bridges, for local and global verifications. The characteristic values of load magnitude are defined for the 98th percentile of a Normal PDF, for a return period of 50 years [31]. Thus, an average value of 207.40 kN was obtained. A CoV of 10% is also recommended for the load model [13,47,48], resulting in the following load curve parameter, see Eq. (7):

LM71  Nð207:40; 20:742 Þ

ð7Þ

Regarding the load curve, the average value of load model LM71 [44] (207.40 kN) is normalized in order to correspond to a unitary failure load factor. This value is then affected by the dynamic amplification coefficient. Thus, by applying the dynamic factor red/3, proposed by the European standard [44], the load curve used in safety assessment is represented by a Normal PDF with the following statistical parameters, see Eq. (8),

S  Nð1:16; 0:122 Þ

ð8Þ

4.2. Resistance curve 4.2.1. Sensitivity analysis In order to reduce the computational time of analysis, a sensitivity analysis is performed before the sampling procedure in order to reduce the number of involved random variables. Accordingly, the model parameters listed in Table 2 were analyzed, considering a mean value, l, obtained from the design project documentation

and an appropriate standard deviation, r, given in literature [13,17,22,39,49]. However, for some parameters (e.g. fill depth at crown, hf, and piers height, hp) it was not possible to find such information, being an expert judgement employed. In this analysis, critical parameters are those with an importance measure equal or higher than 35%. The sensitivity analysis, depicted in Fig. 9, pointed out as critical parameters the density of masonry, cm, and fill material, cf, the fill depth at crown, hf, the ballast height, hb, the track load, Tl, the backing height, hh, the arch thickness, ta, and the piers thickness, tp, being the latter the most influent parameter. The obtained results were expected, since MAB are gravity structures, being necessary for its stabilization the mass provided by masonry, fill material and track load. Regarding the structural resistance, the arch thickness is very important, since it sustains the material’s self-weight and live loads in longitudinal direction. The fill depth at crown and the ballast height are very important, once they disperse the live loads, diminishing stresses on the arch and, therefore, increasing the ultimate load-carrying capacity. Lastly, the pier thickness is the most influent parameter since it is the ‘key’ element for the formation of global or local failure mechanisms. Thus, from the initially considered 14 structural parameters, only 8 of them are identified as critical, reducing the required resources. Accordingly, a PDF will be only assigned to these parameters, and considered into the reliability analysis. 4.2.2. Sampling procedure Variance reduction techniques increase the efficiency and accuracy of structural reliability assessment evaluations, by using relatively few simulations, when compared to pure simulation methods, such as Monte Carlo method [7,31,35]. Thus, in order to perform the safety assessment, the LHS technique was chosen [9,10]. In this case study, 100 simulations are generated. Another relevant fact is that some structural parameters have some dependence among them, which should be taken into consideration when assessing a structure. Iman and Conover [9,11] developed an algorithm that incorporates the dependence among random variables involved in safety assessment, when generating random values. This algorithm preserves the input parameters PDF and can be used with any sampling type. Correlation values among model parameters are based on literature [12,13]. In this case, it is only considered the correlation between the arch thickness and the piers thickness, with a correlation coefficient equal to 0.61 [12,13]. 4.2.3. Data analysis. Reliability index based on literature and design project After generating the random values for each critical parameter based on information from literature [13,17,22,39] and design project, see Table 2, the RING software [30] computes the failure load factors for each set of values. Once again, seven plastic-hinge mechanism is always observed. A fitting curve procedure to the obtained histogram is then performed. A Normal PDF is the one that better fits the computed histogram according to the obtained Chi-square Goodness-of-the-fit (GOF) test value, v2GOF, see Fig. 10. The outliers were computed based on robust methods, namely the median absolute deviation (MAD) and the Grubbs test [50,51]. The resistance curve is, thus, described through a Normal PDF with the following statistical parameters, see Eq. (9),

R  Nð2:72; 0:222 Þ

ð9Þ

Thus, once the resistance and load curves are obtained, it is possible to compute the failure probability pf, and, therefore, the corresponding reliability index, b. Also, according to LS function employed in the present analysis and defined by Eq. (2), the corre-

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Fig. 8. Collapse mechanism of Durrães viaduct (partial elevation view).

Table 2 Model parameters considered in sensitivity analysis.

Material

Masonry [12,13,15,17,22,39]

Fill material [12,13,15,17,22,39]

Ballast [12,13] Track [12,13,17,49] Geometric

Backing [12,13] Piers [12,13] Arch [12,13,17,22]

bk

hf

γf

2.50 5.00 0.12 2.00 6.00 25.00 0.06 1.77 0.10 0.14

Height, hh (m) Thickness, tp (m) Height, hp (m) Thickness, ta (m)

Normal Normal Normal Normal

3.00 1.40 11.50 0.60

0.30 0.14 0.12 0.06

lation between the structure resistance, R, and load, S, curves was considered as null. Thus, a failure probability, pf, of 1.78  1010 with a corresponding reliability index, b, of 6.27 are obtained.

γm

hb

(61%)

4.3. Bayesian updating

(61%)

(50%)

(46%)

hh

γb

40,0%

(39%)

(33%)

fc

(8%) 0,0%

r

25.00 25.00 0.58 20.00 30.00 0.00 0.60 17.66 1.00 1.42

(100%) (100%)

80,0%

20,0%

l

Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal

ta tp

100,0%

60,0%

PDF Density, cm (kN/m3) Compressive strength, fc (MPa) Friction coefficient, l (–) Density, cf (kN/m3) Friction angle, / (°) Cohesion, c (kPa) Depth at crown, hf (m) Density, cb (kN/m3) Height, hb (m) Track load per unit area, Tl (kN/m2)

μ

(0%)

φ

(11%)

hp

(33%)

blim (35,0%)

Tl

c

(4%)

(0%)

Parameters Fig. 9. Standardized importance measures.

Fig. 10. Obtained values and Normal fitted curve.

Bayesian inference is applied to update each model parameter PDF, reducing, consequently, the related statistical uncertainty, i.e., the uncertainty in statistical parameters. The Bayes theorem, which is the basic tool for the updating process, is employed to obtain the posterior distribution based on the weighting of the prior distribution and the collected data (likelihood), for each model parameter. Regarding prior distributions, both Jeffrey’s and conjugate priors are considered in this analysis. In this case, the inference procedure is applied to masonry and ballast densities, arch and piers thicknesses. The ballast layer thickness will not be updated since no discrete data was available, being its mean value changed to 0.50 m according to recent GPR measurements, developed by Arêde et al. [42], and its CoV was kept the same as above (10%). The prior distribution, for these parameters, is a Normal PDF, described by a mean, l, and a standard deviation, r, statistical parameters. For the likelihood function, a Normal PDF is considered, which allows the use of conjugate initial distributions, achieving very good results. Accordingly, the posterior distribution, due to the conjugacy property, is also a Normal distribution. In this situation, both materials and geometric parameters are updated with data collected from characterization tests and photogrammetric and GPR methods, respectively. Used Bayesian inference methodology is exemplified here with an application to density of masonry, cm, for the case when both mean and variance are unknown. In the situation of a Jeffrey’s improper prior, the following joint prior distribution is employed, expression (10),

pðl; r2 Þ /

1

r2

; 1 < l < 1;

r2 < 0

ð10Þ

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V.N. Moreira et al. / Engineering Structures 134 (2017) 61–73

According to Bayes Theorem, the posterior distribution given all observations xi, is computed by expression (11),

pðl; r

2



1

"

1=2

1 jxÞ /  exp   2 r2   1 S  exp   2 2  r2



2 #  ðn1Þþ1 2 1  2

l  x pffiffiffi r= n

r

ð11Þ

being S = R(xi  x)2 and n the number of samples (in this case, it is considered twice the number of characterization tests in order to increase the likelihood data weight in the posterior distribution, resulting in n = 2  4 = 8). Thus, the posterior distribution of l conditional on r2 is given by a Normal PDF with mean x (in this case equal to 25.28, see Table 1) and variance r2/n, Eq. (12),



r ljr2 ; x ! N x;

2



n

  r2 ) ljr2 ; x ! N 25:28; 8

ð12Þ

with the marginal posterior distribution of 1/r2, an inverse v2 distribution, expression (13),

ðn  1Þ  s2

r2

! v2n1 )

1:29

r2

! v2n1

ð13Þ

being s2 = R(xi  x)2/(n  1) the sample variance (in this case, equals to 0,43). As r2 appears in conditional distribution l|r2, it is possible to conclude that l and r2 are dependent. Therefore, the parameter distribution values may be computed by simulation, through the application of expressions (12) and (13), or by analytical solutions [34]. In the case of the natural conjugate prior, the following joint prior distribution is employed, Eq. (14),

pðl; r2 Þ /

   v 0 =2þ1 n0 1  exp   ðl  l0 Þ2  2 r r2 2  r0   S0  exp  2  r2  n 1=2 0 2



r2



n0

  r2 ) ljr2 ! N 25:00; 4

ð15Þ

being the prior distribution of 1/r2 a Gamma PDF with parameters m0/2 and S0/2, Eq. (16),

1

r2

  v 0 S0 1 ) 2 ! Gammað2; 9:38Þ ! Gamma ; 2 2 r

ð16Þ

The posterior distribution of l conditional on r2 is a Normal PDF with mean l1 and variance r2/n1 (r(l1)2), with n1 = n0 + n the total samples size, expression (17),



ljr2 ! N l1 ;

r2 n1



  r2 ) ljr2 ! N 25:19; 12

ð17Þ

Being the marginal posterior distribution of 1/r2 a Gamma PDF, Eq. (18),

  v 1 S1 1 ) 2 ; x ! Gammað5:5; 10:12Þ jx ! Gamma ; r2 2 2 r 1

n0  n ðx  l0 Þ2 ) S1 ¼ 20:24 n0 þ n

ð19Þ

Therefore, the posterior sum of squares, S1, combines the prior, S0, and the sample sums, s2, with the additional uncertainty given by the difference between the sample and the prior mean. This means that l and r2 are dependent once r2 appears in conditional distribution l|r2. Thus, the parameter distribution values may be computed by simulation, through expressions (17) and (18), or by analytical solutions [34]. Table 3 and Fig. 9, respectively, present the posterior distributions parameter values and plot, obtained from the Bayesian inference analysis. The obtained results for the posterior population mean are approximately the same. However, concerning the posterior population standard deviation, rpop, once the initial information presented in the prior distribution is considerably different than the one contained in the likelihood, the conjugate prior leads to a posterior distribution with higher uncertainty, i.e., a higher posterior population standard deviation, rpop, when compared to the noninformative prior (Jeffrey’s). As such, the Jeffrey’s prior will be that employed in reliability analysis, see Fig. 11. Table 4 presents the posterior parameters of all updated structural parameters, considering a Normal PDF for the likelihood data. With respect to material parameters, the posterior mean values are close to the prior ones and corresponding standard deviation values were considerably reduced. Regarding geometric parameters, the posterior mean values have increased and the standard deviation values were significantly diminished. In a global way, uncertainty related to all model parameters have been reduced and their mean value increased. Thus, it is expected that the updated resistance curve has a higher mean value and a reduced standard deviation as a result of Bayesian inference, thus providing a higher reliability index.

ð14Þ

being n0 the initial sample size (in this case, n0 = 4), S0 = R(xi  x)2 the prior value for S, obtained from the prior standard deviation r0 (equals to 2.50, see Table 2), and l0 the prior mean value (equals to 25.00, see Table 2). Thus, the prior is a Normal-Gamma PDF, i.e., the product of an inverted Gamma PDF, with argument r2, and m0 (m0 = n0  1) degrees of freedom, by a Normal PDF with argument l, being the variance proportional to r2. Therefore, the prior distribution of l conditional on r2 is a Normal PDF with prior mean l0 and variance r2/n0 (r(l0)2), expression (15),

ljr2 ! N l0 ;

S1 ¼ S0 þ ðn  1Þ  s2 þ

ð18Þ

with m1 = m0 + n the posterior degrees of freedom. The posterior value for S is thus obtained from expression (19),

4.3.1. Reliability index based on Bayesian inference results Since some PDF structural parameters were updated, new values are generated by LHS, in order to obtain the updated resistance curve. Accordingly, a new set of failure load factors is obtained for these generated values, where failure was always attained with a seven plastic-hinge mechanism. Then, a Normal PDF is fitted to the computed histogram according to Chi-square GOF test value, v2GOF, see Fig. 12. Again, the same outliers methods described in Section 4.2.3 were applied [50,51]. The updated resistance curve is described by the following statistical parameters, Eq. (20),

R  Nð2:82; 0:192 Þ

ð20Þ

As expected, the updated resistance curve provides a higher mean and diminished the standard deviation value. Fig. 13 illustrates both resistance curves, before (design project and literature data) and after Bayesian inference (experimental data from characterization tests), showing the impact of experimental data in reliability analysis. By comparison of the new resistance curve with the load one, a failure probability, pf, of 1,46  1014 and a corresponding reliability index, b, of 7.60 are obtained. 4.4. Comparison of reliability indexes and safety assessment According to structural reliability theory [35], safety is verified if the following Eq. (21) is observed,

b P bT

ð21Þ

where b is the reliability index and bT is the target value for the reliability index. The target values, recommended by current standards, are given in Table 5. Table 5 also presents the computed reliability indexes, before and after Bayesian inference, b1 and b2 respectively.

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V.N. Moreira et al. / Engineering Structures 134 (2017) 61–73 Table 3 Posterior distributions, considering different prior distributions. Parameter

Jeffrey’s

Conjugate

l0 r0 l1 r (l1) r1 r (r1) lpop rpop

– – 25.28 0.19 0.51 0.17 25.28 0.57

25.00 2.50 25.19 0.43 1.46 0.35 25.19 1.56

Fig. 13. Resistance curves: before and after Bayesian inference.

Table 5 Target values for the reliability index for large failure consequences and high safety measures costs [13].

Fig. 11. Bayesian inference results.

Table 4 Posterior parameter values. Parameter

cm cf ta tP hb a a

PDF

[kN/m3] [kN/m3] [m] [m] [m]

Normal Normal Normal Normal Normal

Posterior values

a

l

r

25.28 22.62 0.70 1.46 0.50

0.57 1.14 0.015 0.032 0.05

Only mean value changed (no Bayesian inference performed).

Fig. 12. Obtained values and Normal fitted curve.

b

Standard

bT

b1 = 2.86

b2 = 6.86

EN 1990 [40]a ISO 2394 [37]b ISO 13822 [38]b PMC [39]a

4.30 3.10 4.30 2.60

✗ ✗ ✗ ✗

U U U U

fib [36]a

Unacceptable b < 4.60 Satisfactory 4.60 6 b < 6.00 Good 6.00 6 b < 8.00 Very good 8.00 6 b < 9.00

✗

U

✗

U

✗

U

✗

✗

Reference period of 50 years. Whole life reference period (remaining/residual life time).

The ISO 2394 [37], the EN 1990 [40], the Probabilistic Model Code (PMC) [39] and fib [36] have proposed values for the design of new structures, while the ISO 13822 [38] is dedicated to the safety assessment of existing structures. Aside from fib [36], whose target reliability indexes are oriented for a global safety assessment (i.e., considering the structural system) and the nonlinearity of the structural behaviour, the remaining target values are oriented to safety assessment, regardless the structural system (i.e., safety assessment at local level without considering the nonlinearity on the structural behaviour). According to Table 5, considering the literature and design project information, the Durrães viaduct does not check the safety requirements for any standards. By consideration of experimental data into the reliability analysis through the Bayesian inference procedure, the reliability index has substantially increased when compared to the initial one, from 2.86 to 6.86. This shows the importance of experimental data when assessing MAB, since the data provided in literature covers a wide range of possibilities, not necessarily reflecting the in situ data. The difference between the computed reliability indexes was expected to be higher. However, the mean value of ballast is reduced in half after Bayesian updating, leading to significantly reduction of live load dispersion. Once it is an important parameter in structural response (61% influence), the ultimate load-carrying capacity is, therefore, significantly reduced. As such, even with the diminishment of the

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uncertainty, since one of the most influent parameters mean is reduced in half, the ultimate load-carrying capacity does not increase substantially, and, therefore, the reliability index. Thus, when applying Bayesian inference techniques, uncertainty was reduced with the amount and quality of available information, leading to more realistic and assertive results and in considerably additional reserve strength, which otherwise it was not possible to consider. Also, this result pointed out that MAB have a high ultimate load-carrying capacity when assessed with proper procedures, as confirmed by previous studies [17,22].

Acknowledgments This work was partly financed by FEDER funds through the Competitivity and Internationalization Operational Programme COMPETE and by national funds through FCT – Foundation for Science and Technology within the scope of the project POCI-010145-FEDER-007633. Also, the collaboration and information provided by the Portuguese railway network (REFER) is gratefully acknowledged. Finally, the information made available by Dr. Cristina Costa is kindly acknowledged. References

5. Conclusions This paper presents a methodology for the probabilistic structural assessment of railway masonry arch bridges (MAB). The structural analysis of MAB is attained by the Limit Analysis Theory, through RING software. Since this framework requires high computational resources, a sensitivity analysis is introduced in order to overcome this. This framework also incorporates external information collected from characterization tests or monitoring systems, allowing the updating of parameters PDF through Bayesian inference methodologies. This framework was applied to the case study of the Durrães viaduct. In order to obtain a reliable safety evaluation, material and geometric surveys are proposed. In respect to geometric characterization, photogrammetry proved to be a very valuable method to acquire the MAB external geometry. Recently, GPR and physical and mechanical characterization results, obtained from literature, were also used to characterize the MAB materials, masonry and fill materials [25,42]. Obtained relevant information on the geometry and mechanical and physical properties of masonry and fill materials were taken into consideration through Bayesian inference techniques. By applying Bayesian techniques, it was found that the uncertainty associated with each parameter was significantly reduced and its mean value increased, being possible to asses an additional strength capacity. Additionally, different characterization techniques may be employed to collect data regarding structural parameters, proving the flexibility of Bayesian techniques. Additionally to this work, the consideration of the MAB internal geometry may also be of importance in terms of Bayesian inference, deserving future attention. Two reliability indexes were obtained, before and after the application of Bayesian inference. The first reliability index only considers information collected from design project documentation and literature. The second one also includes the experimental data, obtained from characterization tests, representing thus an updated performance indicator which reflects the real condition of the assessed MAB. By comparison of both indexes, a considerably additional reserve strength was obtained. Indeed, the adopted distribution has impact on obtained results. However, at this point, no other information leads us to conclude that another distribution is the most suitable. On the contrary, considering the existing bibliography [12,13,17,22] and the performed statistical test Chi-Square for collected data (likelihood), the Normal distributions is the most adequate. Thus, due to MAB structural response and intrinsic sources of uncertainty, it is complex and hard to predict its safety condition. Hence, by applying the present framework, which incorporates gathered data from the assessed MAB, it is possible to predict its ultimate load-carrying capacity in a more accurate way. Therefore, the obtained reliability index from the developed framework provides a more robust basis for decisions on future structural repairs of existing MAB.

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