Calibration of the numerical model of a stone masonry railway bridge based on experimentally identified modal parameters

Engineering Structures 123 (2016) 354–371

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Calibration of the numerical model of a stone masonry railway bridge based on experimentally identified modal parameters C. Costa a,⇑, D. Ribeiro b, P. Jorge c, R. Silva c, A. Arêde c, R. Calçada c a

CONSTRUCT-LESE, Polytechnic Institute of Tomar (IPT), Tomar, Portugal CONSTRUCT-LESE, School of Engineering, Polytechnic of Porto, Porto, Portugal c CONSTRUCT-LESE, Faculty of Engineering (FEUP), University of Porto, Porto, Portugal b

a r t i c l e

i n f o

Article history: Received 16 November 2015 Revised 22 May 2016 Accepted 25 May 2016

Keywords: Stone masonry Railway bridges Numerical modelling Vibration testing Operational modal analysis Genetic algorithm

a b s t r a c t This paper focuses on the calibration of a numerical model of a stone masonry arch railway bridge using dynamic modal parameters estimated from an ambient vibration test. The developed 3D numerical model is based on the finite element method, featuring a realistic representation of the bridge structural components and materials. The calibration methodology relied on a genetic algorithm strategy which allowed estimating and updating numerical model parameters, particularly the elastic properties of materials. The validation of the updated bridge material properties’ parameters was based on the results of material testing, on existing bridge design data and on observations resulting from in situ visual inspections. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The need to identify exploitation limits and conditioning effects of regular traffic operation on stone masonry arch railway bridges led to the development of the StonArcRail research project. This study comprised experimental and numerical research activities on the effects of rail traffic in the structural behaviour of bridges. It includes identifying the vibration effects caused by traffic action and the influence of its parameters (speed, type of train and track irregularities) on the dynamic response of the bridge, track and train subsystems, aiming at assessing the structure and the track safety as well as the passenger comfort. The large number of existing and in operation masonry arch bridges on the rail network across Europe justifies the need to study this type of bridges. According to data reported in UIC [1], which several European have contributed to, about 60% of railway bridges are arched ones or culverts. In Portugal there are about 11 746 such cases which amount to 90% of the total existing railway bridges. The report also concludes that 80% of these Portuguese bridges have spans lower than 5 m and 70% are aged between 100 and 150 years. The European project ‘‘Sustainable ⇑ Corresponding author. E-mail addresses: [email protected] (C. Costa), [email protected] (D. Ribeiro), [email protected] (P. Jorge), [email protected] (R. Silva), [email protected] (A. Arêde), [email protected] (R. Calçada). http://dx.doi.org/10.1016/j.engstruct.2016.05.044 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.

Bridges” [2] which involved about 220 000 bridges in Europe, also concluded that about 41% are arched bridges, of which 35% are over 100 years old and 62% have small span. Due to their age, the characterization of these bridges’ conservation status is essential to allow assessing their structural behaviour and identifying the exploitation limits for future rail networks. For this purpose, experimental campaigns on these bridges should be made, mainly involving non-destructive testing, as presented by McCann and Forde [3] in a review of such type of experimental methods applicable to and used on concrete and masonry structures. Orbán and Gutermann [4], refer the UIC project results relative to methods of inspection and testing of railway bridges in arched masonry. Also, Olofsson et al. [2], present the results of the European project ‘‘Sustainable Bridges” describing methods to upgrade existing railway bridges in the European network. Roberts and Hughes [5] introduce a monitoring task through the use of accelerometers in the upper part of the superstructure. Brencich and Sabia [6,7] investigated masonry bridges both in service conditions and in various demolition stages, having developed an experimental program involving lab tests on material samples, as well as in situ activities, namely flat jack and sonic tests to characterize material properties and dynamic tests for the identification of modal shapes and frequencies. Arêde et al. [8] presented the results of a project which involved the implementation of an instrumentation system on a stone arch bridge during the construction (2004–2005) and its structural response monitoring

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during a load test. This project also included ambient vibration [9] and lab testing of the materials used in the bridge construction, namely classic tests for material strength characterization in stone specimens and mortar, triaxial tests on filling material samples and shear-compression test on samples representative of the joints [10]. Srinivas et al. [11] present experimental methodologies using flat jacks to assess the longitudinal internal forces on the bridge due to the increased weight of freight wagons. The evaluation of structural operation and load capacity of masonry arch bridges always lacked appropriate numerical tools for their analyses. The first scientific approaches were based on the concepts of static and failure mechanisms experimentally observed, such as the works of Philippe de la Hire [12], Pippard [13] and Heyman [14]. Later on, thanks to the rapid evolution of computational technologies, new methodologies based on the finite element method (FEM) and the discrete elements methods (DEM) are developed and increasingly used, some of which are addressed in the following paragraphs. Fanning et al. [15] used a 3D FEM model in which masonry, filling and pavement were discretized with continuous solid elements using commercial software ANSYS [16]. Frunzio et al. [17] studied a Roman arch bridge adopting a 3D FEM model developed also in ANSYS, wherein the nonlinear material behaviour was considered using the Drucker-Prager criterion for all materials. Cavicchi and Gambarotta [18] used a 2D FEM where the arch and backfill interaction was considered, the arches and piers were modelled with 1D elements with perfect elasto-plastic material behaviour, ductile in compression and with no tensile strength. The backfill was simulated by 2D triangular finite elements interconnected by interface elements ruled by a modified Mohr-Coulomb criterion with tension ‘‘cut-off”. Arani and Zandi [19] also studied the conditions of a three span railway bridge, using a FEM based model with SOLID-2D plain strain elements. Detailed FEM modelling strategies were adopted by Costa et al. [20] for simulating the structural response of stone arch bridges under road traffic loading, resorting to the computer code CAST3M [21]. In the adopted models the bridges’ masonry components are represented by FEM micro modelling strategies using solid elements, to define the individualized blocks, and zero thickness joint elements at their interfaces. The backfill is also modelled with solid elements connected to zero thickness joint elements in the interfaces between the infill and blocks of the masonry structure. Nonlinear constitutive models supported by experimental calibration were considered assuming a nonlinear Mohr-Coulomb friction model without dilatancy for the joint elements and using the Drucker-Prager model for the infill material. In the sequence of this work, DEM models were also used for comparative purposes to evaluate the load-carrying capacity of one of the case studies [22]. Anderson [23] developed a comprehensive 2D model of a railway concrete arch bridge and a 3D model of the three spans on the north side of the bridge, using the SOLVIA03 program [24]. For the structural modelling, the concrete was considered as a continuous nonlinear distributed cracking material and the backfill behaving the Drucker-Prager criterion. The 3D model involved the separation between the various constituent elements, namely arches, spandrel walls, backfill, piers and foundations. Domede et al. [25] studied an arched masonry railway bridge using a 3D FEM based damage model developed in CAST3M [21]. The masonry was simulated as a nonlinear homogenized solid material by means of a damage model in which the masonry behaviour in compression and tension is handled separately. The backfill was modelled with a nonlinear homogeneous material following the Drucker-Prager criterion. Comparing the numerical results with the experimental ones, it is normal to have differences that should be minimized by optimizing the bridge numerical models [26]. Bayraktar et al. [27,28] present the numerical modelling and calibration of roadway stone

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masonry arch bridges for which people walking was adopted as the dynamic excitation during vibration testing; in this case the numerical model calibration only involved changes of the bridge boundary conditions. In line with previous studies, this paper presents the outcome of a very recent work involving 3D FEM modelling of a stone masonry arch railway bridge over 100 years old. The model was generated in ANSYS [16], with refined individualization of the various bridge components in order to allow assigning different material parameters to each of them. These parameters were considered on the basis of an extensive experimental campaign involving in situ and laboratory tests. Aiming at assessing the bridge response, ambient vibration tests were performed from which natural frequencies, vibration modes and damping coefficients were obtained. A good correlation was obtained between the numerical and experimental results, for which an optimization procedure was adopted to improve the numerical model based on data obtained from the ambient vibration tests. The calibrated model was developed to allow performing analyses involving the bridge-train interaction, from which the response of both can be obtained. 2. Durrães railway bridge 2.1. Description The Durrães bridge (Fig. 1) dates back to late 19th century and is located at km + 64.344 of the Minho line which constitutes the rail link between the Porto and Valença cities. The bridge presents a structural system made of granite masonry arches and is part of a single-track section in Iberian gauge currently allowing the circulation of freight and passenger trains with maximum speeds of 100 km/h and 120 km/h, respectively. The bridge length is about 178 m, having a longitudinal rectilinear profile deck with 1.45% slope and 5.3 m width. It consists of 16 arches with approximately 9 m span, supported by 15 piers and two abutments. The spandrel walls, vertically supported on the outer faces of arches and piers, are formed by horizontal rows of carved stone. The maximum gap between the bridge ground level and the railroad is approximately 22 m. Fig. 2 shows the west view of Durrães Bridge obtained from a topographical survey made within the StonArcRail project activities. The arches have 0.7 m uniform thickness and the piers’ height, measured between the top face of the foundations’ blocks and the arches’ bases, range from 11 m to 12 m. The two piers located at about 1/3 and 2/3 of the total length of the bridge (between arches A5 and A6 and arches A11 and A12) have about twice the cross section area of the other piers. The shallow foundations are based on a rocky base at a variable depth between 5 m and 10 m. The railway line consists of bi-block type sleepers and UIC60 rails, laid on a ballast layer approximately 0.50 m thick. The side guards are made of granite stone blocks. 2.2. Preliminary experimental campaign An experimental campaign was carried out aiming at studying and evaluating the physical and mechanical parameters of the structural components of two stone masonry bridges, including the herein addressed Durrães bridge, described at length (including presentation of results) in another paper prepared by some of the present paper authors [29]. The characterization of the Durrães bridge materials consisted of in situ tests, namely, flat jack (FJ) and Ménard Pressuremeter (PMT) tests, and laboratory tests on samples of stone blocks and joints extracted from the bridge. Tests with Ground Penetrating Radar (GPR) were also carried out in order to study the geometric

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Fig. 1. Durrães bridge.

Fig. 2. Durrães bridge – longitudinal West view.

constitution of the sections of the structural elements and the terrain’s foundation profile. For the latter purpose, Dynamic Probing Super Heavy tests (DPSH) were also made. In total two FJ tests were performed in some piers’ facing stones, as well as four PMT tests (in two piers and two spandrel walls) and four DPSH tests on four ground locations between piers. Fig. 3 shows the marked locations of these tests. GPR tests were performed to obtain GPR profiles from outer facings of the bridge located in abutments, piers and spandrel walls and along the terrain profile between P1 and P15 piers. 2.2.1. GPR (Ground Penetrating Radar) and DPSH tests Radargrams’ analyses obtained with the GPR tests allowed identifying the thickness of facing stones and the position of foundation soil layers. GPR tests were performed using a Malå GeoScience AB equipment which included a control and registration unit, a power source and two antennas with 250 and 500 MHz of frequency. To perform the radargrams’ analyses the following computational programs have been used: GroundVision (MALÅ GeoScience AB) and Reflex2DQuick (Sandmeier geophysical software) to read and treat the radargrams obtained in situ and GPRSIM and GPRSLICE (Geophysical Archaeometry Laboratory) to simulate geometric

DPSH1 to DPSH4 tests

models. Using GPRSIM simulation software, synthetic radargrams have been generated based on the geometry and dielectric constants of the materials involved which revealed to be consistent with the radargrams obtained in situ. Fig. 4a shows a measuring phase of a pier profile and Fig. 4b includes the respective radargram. Good agreement was obtained between the results estimated based on the interpretation and processing of the measured radargrams and other geometric data available, namely, from the design drawings, DPSH tests and observation of layers of masonry and infill materials extracted from drilled cores in the bridge [29]. The determination of the firm depth was verified with the results of the DPSH tests. In Fig. 5a it can be seen an overview of the used equipment, while Fig. 5b shows the in depth plot evolution of the number of blows (N20) and dynamic resistance of the tip (qd) obtained from the DPSH ground tests [29]. By observing the evolution of N20 in the various DPSH tests, it was found the presence of a firm layer at about 10 m depth in most of the bridge length, where the piers’ foundations are sit on. The DPSH4 test results show the presence of a rocky outcrop near the South abutment, which raises the possibility of lower depth foundations at that level.

PMT1 to PMT4 Pressurmeter tests

FJ1 to FJ2 Flat-jack tests

Fig. 3. Locations of the in situ experimental tests in the Durrães bridge.

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Fig. 4. GPR test: (a) measurements with 500 MHz antenna and (b) radargram.

Fig. 5. DPSH test: (a) equipment used for DPSH tests and (b) in depth evolution of the number of blows (N20).

Fig. 6. In situ FJ2 deformability test: (a) test setup and (b) experimental curves.

2.2.2. In situ flat-jack testing (FJ) In situ stress state and deformability properties of masonry were estimated by single and double flat-jacks tests, respectively.

These tests were performed in masonry facing stones, in accordance with the test procedure described in [30] and the applicable standards [31,32]. Fig. 6a shows one of the double tests performed

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and the layout of displacement measuring devices (LVDTs), four vertical and one horizontal. Fig. 6b shows a graph with the loading/unloading cycles obtained from a double test [29], from which it is possible to estimate elastic modulus of that masonry zone (consisting of stone blocks and joints) by calculating the average slope of the stress-strain plot in each vertical alignment. For the installed in situ stress, a value of 1.5 MPa was estimated, resulting from the analysis of the results of single FJ tests, while the masonry deformability modulus was found in the range 7–23 GPa which results from double FJ tests. A large variability in results was found and the upper limit value 23 GPa is clearly overestimated, for which a very plausible reason is the type of masonry found in the Durrães bridge piers with very large regular stones (0.5 m high and more than 1 m long), which does not fit in the applicable standards for this type of tests. 2.2.3. Pressuremeter tests (PMT) Ménard pressuremeter tests (Fig. 7a) allowed estimating the deformability characteristics of the backfill. The tests were performed according to the French standard NF-P94-110 [33]. Based on pressuremeter curves obtained from testing it was possible to calculate the deformation parameters designated by pressuremeter deformation modulus (GPMT) and pressuremeter modulus (EPMT). Fig. 7b presents a pressuremeter curve regarding a test on a pier [29]. For each test it was determined the range of variation between the minimum and maximum value of the pressuremeter modulus (EPMT). For the tests PMT1 and PMT2, in pier P11, the range of values was between 450 and 680 MPa, for PMT3 test, on the spandrel wall, the range was 280 the 400 MPa and for the PMT4 test, in the abutment, the interval was between 190 and 480 MPa. The elastic modulus (E) of the backfill material can be estimated by dividing the pressuremeter modulus (EPMT) by an appropriate coefficient as described in Briaud [34], for which it was adopted the 0.5 value, often adopted for unweathered rock material in the foundation design framework. 2.2.4. Laboratory tests The mechanical characterization of the stone extracted from the bridge was based on laboratory tests for determining the compressive and tensile strength as well as the elastic modulus in compression. Table 1 presents the results of these tests. The parameters were evaluated according to the rules applicable to each type of test with the reference standard included in Table 1. Laboratory shear and compression tests were also made in samples of

Table 1 Physical and mechanical parameters of granite stone blocks and masonry joints. Material

Parameter 3

Granite stone Unit weight (kN/m ) blocks Compressive strength

Value

Standard

25.9–26.5 34.8–59.4

– NP EN 12504-1 NP EN 12390-3 NP EN 12390-6

Tensile strength by diametric 3.7–5.4 compression (MPa) Elastic modulus (GPa) 20.0–23.5 Joints

Normal stiffness (MPa/mm) Shear stiffness (MPa/mm)

1.652 0.728

NP EN 14580 – –

masonry joints taken from the bridge and the obtained parameters, in terms of shear and normal stiffness, are also included in Table 1. 2.2.5. Visual inspection A visual inspection to Durrães bridge was carried out aiming at identifying damage and defects. According to the observations made, two types of defects were differentiated: defects related to environmental, physical and chemical actions and defects related to mechanical actions. The first feature in a wider form in the structure and, in general, are independent of the structural behaviour involving problems associated with the presence of water, biological pollution and erosion as shown in Fig. 8. It was observed that in general the bridge facing stones have water flowing, efflorescence, some stalactites, black films, vegetation, moss and lichens. In areas not affected by the presence of vegetation it is common to see block erosion. The lack of mortar in the joints and the existence of open joints is more visible in the lateral west surface of the bridge and at higher levels. In blocks of the piers’ faces closer to the ground, land deposits, moss and humidity are observed. The defects related to mechanical actions are associated with the presence of cracks and deformations resulting from the structural behaviour due to static or dynamic loading. There are problems such as: (i) longitudinal crack in the intrados of the arches alignments underneath the spandrel walls, (ii) longitudinal cracking on the arches intrados along the central axis of the bridge, (iii) block fracture (iv) cracking at the joints on the interface between the mortar and blocks or within the mortar and (v) joint opening. In Fig. 9 some examples of such anomalies are shown. In arches A3, A5, A10, A12, A13, A15 and A16 it can be observed longitudinal cracking on the arch intrados near the spandrel walls (east and/or west) and in some cases also in the axis of the bridge. Occasionally cracking is observed in the mortar and/or in the

Fig. 7. PMT test: (a) Ménard pressuremeter and (b) experimental pressiometric curve.

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(a) Water flowing, efflorescence, stalactites and vegetation, on the arch intrados.

(b) Vegetation and black films on the west face on the connections between the top cornice and the spandrel wall.

(d) Block erosion on the connections between the top cornice and the spandrel wall.

(e) Rising damp; dirt deposits and moss in piers’ facing blocks close to the ground.

(c) Moss and lichens on a pier.

(f) Lack of mortar in pier joints.

Fig. 8. Defects associated with environmental, physical and chemical actions.

(a) Longitudinal cracking in the intrados of the arch.

(b) Block fracture.

(c) Cracking at the joints.

(d) Joint opening.

Fig. 9. Defects associated with mechanical actions.

connection between the mortar and the blocks, noticing that the joints most affected by this anomaly are located generally at higher levels along the arch crown. Fig. 10 shows the location of some of the observed defects, in both east and west elevations.

2.3. Numerical modelling The numerical modelling of Durrães bridge was carried out using a 3D FEM model developed in the ANSYS program [16].

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Fig. 10. Anomalies on the bridge: (a) west side; (b) east side.

Fig. 11 shows the numerical model of Durrães bridge with the identification of the various structural components. This model was based on a simplified initial model previously presented in another publication [35]. In the final model the geometry of the spandrel wall and the depth of the foundations were updated from the initial model. The arches, piers, spandrel walls, backfill, abutments, foundations, embankment, ballast and sleepers, were modelled using volumetric finite elements of triangular prismatic and parallelepiped shape with 6 and 8 nodes, respectively. The rails were modelled using beam finite elements. The side guards were modelled

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Abutments Foundations Base of the piers Piers Arches Spandrel walls Lower backfill Upper backfill Embankment Ballast Sleepers Rail

through mass elements distributed over both spandrel walls. The model’s geometry was based on a topographic survey carried out on the bridge, as well as data from the experimental campaign, including the tests with GPR and DPSH, and details available in design drawings. The boundary conditions were established using rigid supports to block all displacements of the nodes of the finite element mesh located at the base of foundations and abutments. Table 2 outlines the geometric and mechanical parameters of the numerical model of the bridge, including the designation, the adopted value in the initial model and the respective unit. In addi-

A16 A15 A14 A13

9

A12 A11 A10 A9

12

A8

11

A7 A6

5

10 8

A5 A4

7

A3 A2

6

A1

1

Fig. 11. 3D numerical model of Durrães bridge.

4

3

2

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C. Costa et al. / Engineering Structures 123 (2016) 354–371 Table 2 Characterization of the parameters of the numerical model of the Durrães bridge. Parameter

Designation

Initial value

Low.

Up.

Efund

Elastic modulus of masonry foundations Volumetric mass of masonry foundations Elastic modulus of the masonry of arch, spandrel walls, piers and abutments Volumetric mass of the masonry of arch, spandrel walls, piers and abutments Elastic modulus of the backfill Lower zone Upper zone Volumetric mass of the backfill Lower zone Upper zone Elastic modulus of the embankment Mass density of the embankment Elastic modulus of ballast Volumetric mass of ballast Elastic modulus of concrete (sleepers) Volumetric mass of concrete (sleepers) Elastic modulus of steel (rail) Volumetric mass of steel (rail)

15.0 2497.5 10.0

– – 7

– – 23

GPa kg/m3 GPa

2497.5

2396.3

2525.5

kg/m3

1.00 0.60 2303.8 2201.8 0.75 2000 0.145 2000 36.0 2890 210.0 7850

0.980 0.416 2302.7 2199.2 – – – – – – – –

1.470 1.026 2438.5 2378.2 – – – – – – – –

GPa GPa kg/m3 kg/m3 GPa kg/m3 GPa kg/m3 GPa kg/m3 GPa kg/m3

qfund Eg

qg Ee,inf Ee,sup

qe,inf qe,sup Eat

qat Eb

qb Et

qt Ea

qa

tion there are also indicated the lower and upper limits of the identified parameters in the experimental campaign presented in Section 2.2. It should be noted that there was a slight delay between the definition of the initial model and the processing of data from the tests, which is why the parameter values of the materials adopted in the initial model do not all strictly coincide with the average values obtained in the tests. 2.4. Modal parameters Fig. 12 presents the frequency values and the corresponding modal shapes of some of the bridge vibration modes obtained from the numerical model developed based on the adopted values of the parameters listed in Table 2. Modes 1–3, 5–7 and 9 involve transverse bending of the piers which, by structural compatibility, cause transverse movements of the deck. Modes 4 and 8 involve the longitudinal bending of the piers. The vibration modes shown are those that will be used later in the calibration phase of the numerical model of the bridge. 3. Ambient vibration test The ambient vibration test of Durrães bridge aimed at the experimental identification of its modal parameters, namely, the natural vibration frequencies, mode shapes and damping coefficients. 3.1. Measurement setup This test was performed based on a technique that considers fixed reference points and mobile measurement points, where there were used 20 PCB piezoelectric accelerometers, model 393B12, with a sensitivity equal to 10 V/g, a range of measurement of ±0.5 g, a broadband resolution equal to 6.0 lg rms and a frequency range between 0.15 Hz and 1000 Hz. The ambient vibration response was evaluated in terms of the accelerations in the longitudinal (x) and transversal (y) directions, in a total of 32 measurement points, organized in two measurements setups. Each setup involves the use of 4 reference sensors, placed on the deck (positions 4, 8 and 14) and on pier P7 (position 24), and 14 mobile sensors, that changed their position from one setup to the other. In the first measurement setup the mobile accelerometers were positioned on the deck, on one of the side guards, in transversal direction. Otherwise, in the second setup, the accelerometers were

Exp. campaign values

Units

installed at half-height of the piers and positioned in longitudinal direction. Fig. 13 shows the location of the reference accelerometers and the position of the mobile accelerometers for measurement setups 1 and 2. Data acquisition was performed using a cDAQ-9172 system from NI (National Instruments) equipped with 24-bit resolution IEPE type analogue input modules (NI 9234). The time series have been acquired during periods of 10 min, with a sampling frequency of 2048 Hz, and subsequently decimated to 256 Hz frequency. The sampling frequency equal to 2048 Hz corresponds to the lowest sampling rate allowed by NI 9234 module, which, posteriorly, enabled a proper decimation of the signal. The connection of accelerometers to the side guards and piers was carried out by means of steel plates or metallic angles glued or bolted to the stone surface. Fig. 14a and b shows the details of accelerometers located on the side guard of the deck and on the front of one of the bridge’s piers, respectively. The accelerometers placed on the piers were installed with the help of a platform lift with extendable arm. The typical extreme values of the deck’s accelerations, in transversal direction, are approximately equal to ±0.25 mg, and at the piers, in longitudinal direction, are approximately equal to ±0.12 mg. 3.2. Modal parameter identification The identification of modal parameters of the bridge was performed by the application of the Enhanced Frequency Domain Decomposition method (EFDD) available at ARTeMIS software [36]. The EFDD method allows identifying the mode shapes by locating the peaks of the singular values decomposition plots calculated from the spectral density functions of the measured responses [37]. In EFDD method, the Single Degree-of-Freedom (SDOF) Power Spectral Density function, identified around a peak of resonance, is taken back to the time domain using the inverse discrete Fourier transform. The natural frequency is obtained by determining the number of zero-crossing as a function of time, and the damping by the logarithmic decrement of the corresponding SDOF normalized auto correlation function [37]. Each SDOF function is estimated using the shape determined by peak picking and a reference singular vector is set for a further correlation analysis based on the Modal Assurance Criterion (MAC). Therefore, the MAC value between the reference singular vector and a singular vector for each particular neighbourhood frequency

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Mode 1 (f1T = 1.94 Hz)

Mode 2 (f2T = 2.25 Hz)

Mode 3 (f3T = 2.58 Hz)

Mode 4 (f1L = 2.89 Hz)

Mode 5 (f4T = 3.14 Hz)

Mode 6 (f5T = 3.81 Hz)

Mode 7 (f6T = 4.57 Hz)

Mode 8 (f2L = 5.12 Hz)

Mode 9 (f7T = 5.33 Hz) Fig. 12. Bridge frequencies and modal shapes of numerically obtained vibration modes.

line, is computed. If the MAC value of this vector is above a userspecified MAC rejection level, the corresponding singular value is included in the description of the SDOF function. The lower this MAC rejection level is, the larger the number of singular values included in the identification of the SDOF function will be. A more detailed explanation of EFDD method can be found in reference [37]. Fig. 15 presents the curves of the average normalized singular values of the spectral density matrices of two experimental configurations of the ambient vibration test that include the accelerations measured in transverse (Fig. 15a) and longitudinal

(Fig. 15b) directions, and where the peaks corresponding to 9 of the global vibration modes of the bridge can be identified. Fig. 16, illustrates, in perspective, the identified modal configurations with an indication of the average values of frequencies (f) and damping coefficients (n). The analysis of mode shapes allows identifying joint bending movements of the deck and piers in transverse (1T to 7T) and longitudinal (1L and 2L) directions with very good definition. Mode shapes 1L and 2L were defined at deck level using an approximate linear extrapolation from the modal coordinates obtained at half-height of the piers. The average values of the damping coefficients are between 1.34% and 3.61%.

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Fig. 13. Measurement setups 1 and 2: location of the measurement points on the deck and piers.

Fig. 14. Details of the installation of accelerometers: (a) on the side guard of the deck, (b) on the front of the pier.

4. Calibration of the numerical model To minimize the differences between numerical and experimental modal responses, a calibration process was performed. This process involved two stages: first, a sensitivity analysis, in order to identify the numerical parameters that more influence the modal responses, and second, an optimization, which involved the application of an iterative method based on genetic algorithms, originally developed by Ribeiro et al. [38] and adapted to the case of Durrães bridge.

4.1. Methodology The iterative calibration method based on a genetic algorithm involves the use of three software packages: ANSYS [16], Matlab [39] and Optislang [40]. Fig. 17 shows the flowchart illustrating the computational implementation of the method with an indication of the operations performed by each software package. In the ANSYS environment the FEM numerical model is defined based on a set of initial parameter values h1, h2, . . ., hk, where k is the number of individuals in each generation. The selection of calibration parameters is based on the results of a previous sensitivity analysis. The sets of parameter values of generation 1 are randomly generated by applying the Latin Hypercube Sampling method. After the generation of each model and based on their numerical parameters a modal analysis is performed, from which the fre-

quencies of vibration and the corresponding mode shapes are obtained. The realization of this last step in ANSYS environment constitutes an upgrade in relation to the original methodology proposed by Ribeiro et al. [38] and allowed a significant enhancement on its computational efficiency. In the Matlab software, based on the experimental and numerical modal information, the mode pairing between numerical and experimental modes is performed using MAC parameter [41], and the values of the natural frequencies and the corresponding MAC values are exported in text format. Finally, the OptiSlang software, based on an objective function and on the application of an optimization technique supported by a genetic algorithm, estimates a new set of parameters focused on the minimization of the objective function residuals. The generation of these new individuals is based on crossover, mutation and replacement operations. This procedure is repeated iteratively until the maximum number of generations is reached. The objective function f comprises two terms, the first related to the natural frequencies and the second related to the MAC values:

 n modes X f exp i

f ¼a

i¼1 exp

num 

 fi

exp fi num

þb

nX modes

   MAC /exp ; /num  1 i

i

ð1Þ

i¼1

where f i and f i are the experimental and numerical frequencies for mode i, /exp and /num are the vectors containing the experimeni i tal and numerical modal information regarding mode i, and a and b are weighing factors of the terms of the objective function.

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Fig. 15. Average normalized singular values of the spectral density matrices obtained from the experimental setup with mobile sensors located in the (a) transverse and (b) longitudinal directions.

4.2. Sensitivity analysis The sensitivity analysis intent was the selection of parameters that most influence the values of frequencies and MAC, and therefore should be considered in the subsequent phase of optimization. In this study a global sensitivity analysis was used, where all parameters vary simultaneously, allowing to obtain, in a single attempt, the sensitivities between the parameters and responses from a set of samples generated by Latin Hypercube Sampling method [42]. In Latin Hypercube Sampling method the domain associated to each parameter is divided into intervals with equal probability of occurrence. Each interval is defined by the value of the parameter corresponding to its centre of gravity and in accordance with the correspondent probability density function, which, in this case, were all assumed as uniform. The choice of each of the intervals is performed randomly, so each interval is counted only once for sampling generation [43]. The sensitivity analysis results are based on Spearman correlation coefficients (rSxy) that express the correlation between two vectors of samples, x and y, based on their rank order vectors, R(xi) and R(yi), by the expression [40]:

r Sxy

  Pn  i¼1 Rðxi Þ  RðxÞ Rðyi Þ  RðyÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pn  2ffi Pn  i¼1 Rðxi Þ  RðxÞ i¼1 Rðyi Þ  RðyÞ

ð2Þ

  where RðxÞ and RðyÞ are the average values of the rank order vectors R(xi) and R(yi), respectively, with i = 1, 2, . . ., n, where n is the number of samples of the variables. The sensitivity study involved three tests, performed sequentially, each one associated with different approaches regarding the number and types of numeric parameters considered. All analyses were performed using a stochastic sampling technique based on 1000 samples generated by the Latin Hypercube Sampling method and samples with MAC values below 0.50 were skipped.

4.2.1. Approach 1 The first calibration approach involved considering eight of the numerical model parameters, namely: the elastic modulus of the masonry superstructure elements (Em), the corresponding lower and upper bounds for the backfill (Ebl and Ebu) and the elastic modulus of the embankment (Ee); the densities of the masonry (dm), the corresponding lower and upper for backfill (dbl and dbu) and that of the embankment (de). The ranges for each parameter were considered according to the lower and upper limits given in Table 2. Fig. 18 shows the results of the Approach 1 sensitivity analysis through a matrix of linear Spearman correlation coefficients. The correlation coefficients situated in the range [0.30; 0.30] were excluded from graphical representation.

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Mode 1T f = 1.85 Hz | ξ = 2.83 %

Mode 2T f = 2.08 Hz | ξ = 2.43 %

Mode 3T f = 2.41 Hz | ξ = 2.40 %

Mode 1L f = 2.50 Hz | ξ = 3.61 %

Mode 5T f = 2.79 Hz | ξ = 3.37 %

Mode 6T f = 3.31 Hz | ξ = 2.51 %

Mode 7T f = 3.83 Hz | ξ = 1.81 %

365

Mode 2L f = 4.11 Hz | ξ = 2.36 %

Mode 8T f = 4.33 Hz | ξ = 1.34 % Fig. 16. Experimentally measured modal parameters and vibration mode shapes.

The correlation matrix shows that the masonry elastic modulus of superstructure elements, including arches, spandrel walls, piers and abutments is the most sensitive parameter to responses, especially to vibration frequencies, with most values of the correlation coefficients near 1.0. On its turn the embankment elastic modulus also has a significant influence on most of the MAC parameter values. The high values of correlation coefficients associated with these parameters, especially the masonry elastic modulus, foresee the possibility of a stable estimated value at a later optimization stage, but it can render difficult estimating the values of the less sensitive elastic parameters. The use of a single elastic parameter for all masonry elements is also clearly insufficient to adequately

reproduce the different anomalies observed in visual inspection. The parameters associated with the density of the bridge constituent materials have a reduced sensitivity to all responses and therefore may be excluded from the optimization phase. 4.2.2. Approach 2 The second approach was performed by replacing the elastic modulus of masonry by specific elasticity parameters of each structural element of the bridge. Thus four new numerical parameters were created representing the elastic modulus of the abutments (Eam), the piers (Ep), the spandrel walls (Es) and the arches (Ea). The ranges of these new parameters are in correspondence

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Fig. 17. Flowchart of the iterative calibration methodology based on genetic algorithm.

Fig. 18. Matrix of linear Spearman correlation coefficients obtained from Approach 1.

Fig. 19. Matrix of linear Spearman correlation coefficients obtained from Approach 2.

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with the lower and upper limits of the masonry elastic modulus shown in Table 2. With this approach it is intended to take advantage of the high sensitivity of the masonry elastic modulus, so that consistent estimates are achieved for that parameter values for the various bridge structural elements. Fig. 19 shows the results of the Approach 2 sensitivity analysis through a matrix of linear Spearman correlation coefficients. From this analysis, the parameters associated with the density of the constituent materials of the bridge have been excluded. The analysis of the results allows the following conclusions: – The elastic parameters of the piers and spandrel walls similarly contribute to the modal responses, in particular for the majority of the transverse frequencies of the bridge, and therefore in the optimization phase there can be different combinations of these parameters that lead to the same solution in terms of optimization of the problem. – The elastic parameters of the arches and abutments contribute less significantly to the modal responses of the bridge in the transverse direction. Regarding arches, given their location and layout, the influence of their elastic parameters should be more relevant to the modal parameters in the vertical direction. In relation to the abutments, and given their location at the ends of the bridge, the influence of their elastic parameters is restricted mainly to the mode shapes with higher amplitudes in these areas. – The elastic parameters of both backfills present low values of the correlation coefficients with the responses. For the lower backfill its inability to influence the modal parameters of the bridge in the transverse direction is due: (i) to its location inside the pier section which is unfavourable to control the transverse stiffness of the masonry-filled composite section, and (ii) to their elastic properties being of a much lower order of magnitude to those of the masonry.

Fig. 20. Matrix of linear Spearman correlation coefficients obtained from Approach 3.

367

– The embankment elastic parameter had a significant reduction of the values of the correlation coefficients with the responses in terms of modal parameters. This reduction is related to the fact that some of the new masonry parameters are now able to locally change the shape of some vibration modes contrary to what happened in Approach 1. 4.2.3. Approach 3 In the third approach new numerical parameters were defined which allowed improving some of the limitations identified in Approach 2, namely: – The predictable inability to estimate the elastic parameters associated with each structural element of masonry and backfills, resulting in the need to aggregate back its elastic properties. – Incorporate the variability of the materials’ parameters over the bridge development, reflecting the anomalies observed in the visual inspection described in Section 2.2. Thus, eight new parameters have been defined (K1 to K8) representing the correction factors of the elastic modulus of the elements in masonry, including the arches, spandrel walls, piers and abutments, located in different areas of the bridge. For this purpose, eight zones were defined corresponding to the grouping of

Fig. 21. Values of the numerical parameters for the optimization runs GA1 to GA4.

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masonry elements belonging to each two consecutive spans. It was also defined a new parameter (Kf) representing the correction factor of the elastic modulus of the upper and lower backfill throughout the length of the bridge. The K1 to K8 and Kf parameters have a variation range between 0.40 and 1.40, and the correction factors with values equal to 1.00 correspond to the situation wherein the elastic parameters assume initial values listed in Table 2. Fig. 20 shows the results of the Approach 3 sensitivity analysis through the corresponding matrix of linear Spearman correlation coefficients. From the analysis of the correlation matrix it can be concluded that, except for the correction factor of the backfill elastic modulus, all other parameters show to be sensitive to the responses with similar values of correlation coefficients between them, situated in the interval 0.40–0.80, allowing, therefore, to predict a stable optimization process. 4.3. Optimization The optimization phase allowed obtaining the parameters values that minimize the differences between the numerical and experimental modal responses, and involved the application of an optimization technique based on a genetic algorithm as described in [38]. The optimization of the bridge model involved the use of 8 design variables and 18 modal responses, 9 frequencies and 9 MAC parameters. The genetic algorithm was based on an initial

population consisting in 30 individuals and 150 generations, for a total of 4500 individuals. The initial population was randomly generated by Latin Hypercube Sampling method. In this algorithm the number of elites was equal to 1 and the number of substitute individuals was also defined equal to 1. The crossing rate was considered equal to 50% and the mutation rate was set equal to 15% with a standard deviation, variable along the optimization, between 0.10 and 0.01. The objective function weights a and b were considered equal to 1.10 and 0.90, respectively. The optimal values of the parameters were obtained based on the results of four independent optimization runs (GA1 to GA4) with different initial populations. Fig. 21a and b represents plots of ratios between values of numerical parameters K1 to K8 and the corresponding limits (indicated in Section 4.2.3) for the optimization runs GA1 to GA4. A ratio of 0% means that the parameter coincides with the lower limit. A ratio of 100% means that it coincides with the upper limit. The obtained values for the numerical parameters are also indicated in brackets next to the corresponding plots. Optimization results show that, in most areas, there is a trend of reducing the structural elements stiffness. This finding is in line with the results of visual inspection in which several anomalies have been identified throughout the bridge development, as shown in Fig. 10. The bridge areas where there is a greater tendency of stiffness degradation are associated with the parameters K3, K5 and K8, which correspond to the spans 5–6, 9–10 and 15–16, respectively. Some of these areas include piers and arches where more serious anomalies were identified (see Section 2.2) from

Fig. 22. Errors between the numerical frequencies, before and after updating, in relation to the average value of the experimental frequencies.

Fig. 23. Comparison of the MAC values, before and after the updating of the numerical model.

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the visual inspection, fact that contributes for consistency of the herein obtained results. The grouping of structural elements with anomalies together with other elements where no significant anomalies were identified complicates the interpretation of results, especially the correlation between the optimal parameters’ values and the visual inspection results.

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In general, the results from the different optimization cases are very close together, with variations lower than 16%, demonstrating the stability and robustness of the genetic algorithm. Fig. 22 shows the error values between numerical and experimental frequencies, before and after calibration, taking as reference the values of the experimental frequencies. The results after calibration are referred to the optimization run GA3, which is asso-

Mode 1 (1.85 Hz | 1.73 Hz)

Mode 2 (2.08 Hz | 1.98 Hz)

Mode 3 (2.41 Hz | 2.35 Hz)

Mode 4 (2.50 Hz | 2.54 Hz)

Mode 5 (2.79 Hz | 2.79 Hz)

Mode 6 (3.31 Hz | 3.38 Hz)

Mode 7 (3.83 Hz | 3.97 Hz)

Mode 8 (4.11 Hz | 4.43 Hz)

Mode 9 (4.33 Hz | 4.63 Hz) Fig. 24. Comparison between the experimental and numerical mode shapes, after updating and considering optimization run GA3.

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ciated to the lowest residual of the objective function. The average error of the frequencies of global modes decreased from 14.5%, before calibration, to 4.0% after calibration. Fig. 23 shows a comparison of the MAC values before and after calibration of the numerical model. The average MAC value passed from 0.901, before calibration, to 0.926 after calibration, and all MAC values are higher than 0.890 after calibration. Finally, Fig. 24 shows the comparison of experimental and numerical mode shapes after calibration. In the figure there are also indicated, in brackets, the values of the experimental and numerical natural frequencies, respectively, for optimization case GA3.

zation of Structural Behaviour of Stone Masonry Arch Bridges under Railway Traffic Loading - Application to Existing Portuguese Bridges (StonArcRail)”. The authors which to thank engineers Ana Isabel Silva, Hugo Patrício and Nuno Lopes from REFER for their collaboration and for the precious help on providing information about the bridges. The authors’ gratitude is also expressed to Dr. Nuno Pinto and Mr. Valdemar Luís, both technicians of the LESE laboratory, for their indispensable assistance during the preparation and execution of experimental tests.

5. Conclusions

[1] Orbán Z. UIC project on assessment, inspection and maintenance of masonry arch railway bridges. In: Proceedings of the 5th international conference on arch bridges (ARCH 07), Madeira, Portugal, 2007. [2] Olofsson I, Elfgren L, Bell B, Paulsson B, Niederleithinger E, Jensen J, Feltrin G, Täljsten B, Cremona C, Kiviluoma R, Bien J. Assessment of European railway bridges for future traffic demands and longer lives – EC project ‘‘Sustainable Bridges”. Struct Infrastruct Eng 2007;1:93–100. [3] McCann D, Forde M. Review of NDT methods in the assessment of concrete and masonry structures. NDT and E Int 2001;34(2):71–84. [4] Orbán Z, Gutermann M. Assessment of masonry arch railway bridges using non-destructive in situ testing methods. Eng Struct 2009;31(10):2287–98. [5] Roberts T, Hughes T. Serviceability of masonry arch railway bridges. In: Proceedings of the 5th international conference on arch bridges (ARCH 07), Madeira, Portugal, 2007. [6] Brencich A, Sabia D. Experimental identification of a multi-span masonry bridge: the Tanaro Bridge. Constr Build Mater 2008;22(10):2087–99. [7] Brencich A, Sabia D. Structural identification of masonry arch bridges. In: Proc. int. operational modal analysis conference 2009 (IOMAC 2009). [8] Arêde A, Costa P, Costa A, Costa C, Noites L. Monitoring and testing of a new stone masonry arch bridge in Vila Fria, Portugal. In: Proceedings of the 5th international conference on arch bridges (ARCH 07), Madeira, Portugal, 2007. [9] Costa C, Arêde A, Costa A, Caetano E, Cunha A, Magalhães F. Updating numerical models of masonry arch bridges by operational modal analysis. Int J Archit Heritage 2015. http://dx.doi.org/10.1080/15583058.2013.850557. [10] Costa C, Arêde A, Costa A. Mechanical characterization of the constituent materials of stone arch bridges. In: Proceedings of the 9th international masonry conference (9IMC), Guimarães, Portugal, 2014. [11] Srinivas V, Sasmal S, Ramanjaneyulu K, Ravisankar K. Performance evaluation of a stone masonry-arch railway bridge under increased axle loads. J Perform Constr Facil 2014;28:363–75. [12] De La Hire P. Sur la construction des voûtes dans les édifices. Mémoires de mathématique et de physique de l’Académie royale des sciences; 1712. [13] Pippard AJS. The approximate estimation of safe loads on masonry bridges. Civ Eng War 1948;1:365–72. [14] Heyman J. The safety of masonry arches. Int J Mech Sci 1969;11(4):363–85. [15] Fanning PJ, Boothby TE, Roberts BJ. Longitudinal and transverse effects in masonry arch assessment. Constr Build Mater 2000;15(1):51–60. [16] ANSYSÒ Academic Research, Release 15.0, Help System, ANSYS Inc; 2014. [17] Frunzio G, Monaco M, Gesualdo A. 3D FEM analysis of a roman arch bridge. Hist Constr 2001:591–8. [18] Cavicchi A, Gambarotta L. Collapse analysis of masonry bridges taking into account arch-fill interaction. Eng Struct 2005;27(4):605–15. [19] Arani K, Zandi A. Protection of an old stone masonry arch bridge against railway impact. In: Protection of historical buildings (PROHITECH 09), Rome, Italy. [20] Costa C, Arêde A, Costa A. Detailed FEM modelling of stone masonry arch bridges under road traffic moving loads. In: Proceedings of the 3rd ECCOMAS thematic conference on computational methods in structural dynamics and earthquake engineering (COMPDYN), Corfu, Greece. [21] CEA, Presentation et utilization de Cast3M by E. Le Fichoux, Commissariat à l’Énergie Atomique; 2011. Available at: . [22] Costa C, Arêde A, Morais M, Costa A. FEM and DEM detailed modelling of stone masonry arch bridges for the assessment of load-carrying capacity. In: Proceedings of the 1st international conference on structural integrity (ICSI1). Funchal: ESIS; 2015. Procedia Eng 2015;114:854–861. [23] Andersson A. Load capacity assessment and strengthening of a railway arch bridge with backfill. In: Proceedings of the IABSE Spring Conference; 2013. [24] SOLVIA Finite Element System. User manual. Vasteras, Sweden: SOLVIA; 2003. [25] Domede N, Sellier A, Stablon T. Structural analysis of a multi-span railway masonry bridge combining in situ observations, laboratory tests and damage modelling. Eng Struct 2013;56:837–49. [26] Friswell M, Mottershead J. Finite element model updating in structural dynamics, vol. 38. Springer Science & Business Media; 1995. [27] Bayraktar A, Altunisßik A, Birinci F, Sevim B, Türker T. Finite-element analysis and vibration testing of a two-span masonry arch bridge. J Perform Constr Facil 2009;24(1):46–55. [28] Bayraktar A, Birinci F, Altunısßık A, Türker T, Sevim B. Finite element model updating of Senyuva historical arch bridge using ambient vibration tests. Baltic J Road Bridge Eng 2009;4(4):177–85. [29] Arêde A, Costa C, Morais M, Topa Gomes A, Silva R. Avaliação experimental do comportamento dos componentes e dos materiais de uma ponte ferroviária

This paper addressed several aspects about the development and experimental calibration of a finite element method (FEM) based numerical model of a stone masonry arch bridge existing in normal service conditions in the Minho railway line connecting Porto to the up-northwest of Portugal. The experimental identification of modal parameters was based on an ambient vibration test, which allowed estimating natural frequencies and mode shapes of several vibration modes in transverse and longitudinal directions resorting to a frequency domain identification technique. The experimental results obtained from material mechanical characterization tests made in situ and in laboratory was a quite important and useful task to ensure a better estimate of the initial values of the material mechanical parameters adopted in the FEM model. The sensitivity analysis revealed the importance of an adequate strategy for defining the number and types of calibration parameters especially in situations with limited experimental information and where several parameters similarly contribute to the modal responses, therefore rendering more difficult the estimate of stable optimal solutions. The model calibration results evidenced quite good agreement between numerical response and experimental measurements. In fact, comparing with the results of a numerical model defined prior to the calibration, the calibrated model has shown significant improvements on simulating modal parameters, as proven by smaller deviations of numerically obtained frequencies relative to experimental ones and by increased MAC coefficients of numerical and experimental modal configurations. The optimization results show a tendency of reducing the stiffness of the structural elements of the bridge which is in accordance with the results of visual inspection in which several anomalies have been identified throughout the bridge development. In future studies, the calibrated numerical model of the bridge will be used to assess the dynamic behaviour of the train-track coupled system, particularly for freight trains, in order to draw conclusions regarding the performance of the bridge in terms of structural safety and track safety, concerning the track and wheel-track contact stability. These dynamic studies are based on advanced train-track-structure interaction models that also include a calibrated model of the freight train and track irregularities measured by a track inspection vehicle. Additionally it is planned to perform a dynamic test under traffic actions in order to validate the dynamic responses of the train-bridge system. Acknowledgements This work includes research carried out with the financial support of the Portuguese Science Foundation (FCT – Fundação para a Ciência e Tecnologia) through the project PTDC/ECMEST_1691/2012 entitled ‘‘Experimental and Numerical Characteri-

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