EXPERIMENTAL ANALYSIS AND MODEL UPDATING OF A BOWSTRING ARCH RAILWAY BRIDGE

Transcription

1 EXPERIMENTAL ANALYSIS AND MODEL UPDATING OF A BOWSTRING ARCH RAILWAY BRIDGE Diogo Ribeiro 1*, Pedro Almeida 1, Rui Calçada 1 and Raimundo Delgado 1 1 Faculty of Engineering University of Porto R. Dr. Roberto Frias s/n, Porto, Portugal dribeiro@fe.up.pt Keywords: Railway bridge, bowstring arch, experimental analysis, model updating. ABSTRACT This paper focuses on the development and experimental updating of a numerical model for the dynamic analysis of São Lourenço railway bridge under traffic loads. The preliminary experimental campaign consisted of an ambient vibration test and a dynamic test under traffic loads. The comparison between numerical and experimental natural frequencies and mode shapes revealed important limitations of the initially developed numerical model. The study of the influence of several parameters in the dynamic properties of the structure enabled to identify the mass of the deck, modulus of elasticity of the concrete and the vertical stiffness of the supports as key parameters for the numerical model updating. The updated numerical model was used for the simulation of the dynamic response of the bridge under traffic loads and a good agreement between numerical and experimental results was obtained. 1. INTRODUCTION Railway bridges are structures subjected to high intensity moving loads, where the dynamic effects can reach significant values. At present, these effects are being given greater importance due to the increase of the circulation speed, not only in conventional lines but also in new lines, such as the high speed railway lines. In high speed railway lines the dynamic effects tend to increase considerably as a result of the resonance phenomena that occur due to the passage of trains composed by regularly spaced groups of axles. The procedures required for the assessment of these effects were included in the latest version of EN [1], which reflects the most recent research undertaken by the experts of the D214 [2] commission from ERRI regarding the effects of high circulation speed on railway bridges. In structures with complex behaviour, such as bowstring arch bridges, the evaluation of these effects is necessarily performed by means of dynamic analyses.

2 In terms of its structural scheme, a bowstring arch bridge consists of a deck slab suspended by lateral arches. These arches use the deck as a tie in order to avoid horizontal forces from being transmitted to the supports and consequently to the piers and foundations. This structural arrangement has a high planar stiffness which is of great importance towards facing the demanding deformation criteria imposed on high speed railway lines. Furthermore, since the horizontal supports are not necessary in terms of the structural stability, the construction stage is facilitated and can be performed by launching, assembly with crane or rotation. In this paper, the development of a dynamic behaviour numerical model is presented for the case of a bridge of this structural typology for the passage of railway traffic, the São Lourenço bridge, which is located at the Northern line of the Portuguese railways. The results of a preliminary experimental campaign are also presented, which involved an ambient vibration test and a dynamic test performed for the passage of railway traffic. The results of the experimental work were used for the updating and validation of the numerical model of the bridge. 2. SÃO LOURENÇO RAILWAY BRIDGE São Lourenço railway bridge is located at km of the Northern line of the Portuguese railways, in a recently upgraded section for the passage of the Alfa Pendular tilting train which can travel at a speed of 220 km/h. The bridge is a bowstring arch consisting of two half-decks with 42 m span, each one carrying a single track [3]. Each deck consists of a 40 cm thick prestressed concrete slab suspended by two lateral arches. The suspension is performed by means of metallic hangers and diagonals in the proximity of the support blocks of the arches. The arches are linked in the upper part by transversal girders with rectangular hollow section and diagonals in double angles that assure the bracing of the arches. The deck seats in the abutments by means of pot bearing supports. The distance between the supports is 38.4 m, and the extremities of the deck slab work as cantilevers with 1.8 m span. In Figure 1 are presented two general views of São Lourenço bridge. Figure 1. General views of São Lourenço bridge 3. NUMERICAL MODELLING The dynamic analysis of the São Lourenço bridge was performed using a hybrid threedimensional model including the track, as shown in Figure 2. The deck slab was modelled with solid finite elements. The arches, hangers, diagonals and bracings were modelled with beam elements. The track was modelled in an extension

3 corresponding to the bridge length and in a distance of about 10 m from each abutment, in order to simulate the support of the track on the adjacent embankments. The rails were modelled by means of beam elements levelled with the centre of gravity axis, and the sleepers and the ballast layer were modelled using solid finite elements. The influence of the decktrack composite effect in the dynamic properties of railway bridges, due to the transmission of shear stresses between the deck and the rails at the ballasted layer, has been previously demonstrated by the authors [6] and by other researchers [7, 8, 9]. For modelling the structure, particular attention was given to the connection between the ends of the arches and the support blocks in order to guarantee a monolithic linkage. As the solid finite elements of the support blocks do not have rotational degrees of freedom, the beam elements of the arch were extended inside the solid elements in order to guarantee the continuity of the rotations in these connections. Identical procedure was adopted for the connections between the hangers and diagonals with the deck slab. In order to correctly reproduce the deformability length of the hangers and diagonals, rigid elements were introduced in the extremities of the beam elements, between the insertion points of these elements with the arches and the axis of the beam elements of the arch. The supports were initially considered rigid in the vertical direction. Figure 2. Hybrid three-dimensional model including the track The densities of the concrete, steel and ballast were considered equal to 25, 77, and 18 kn/m 3, respectively. The modulus of elasticity of the steel was taken as 210 GPa. The modulus of elasticity of the concrete was estimated on the basis of the value of the modulus of elasticity of a concrete of C35/45 strength class at the age of 28 days (34 GPa), corrected in order to consider the age of the concrete at the time of the experimental campaign ( 500 days), which resulted in 35.6 GPa. The modulus of elasticity of the ballast was considered to be 70 MPa. The masses of non-structural elements such as coatings, lateral parapets, footway slabs, etc. were calculated and added on the nodes of the finite element mesh in correspondence with the locations of each of those elements. In Table 1 are presented the values of the natural frequencies in correspondence with the first three vertical bending modes (1V, 2V e 3V) and the first torsion mode (1T) of the deck. In Figure 3 are illustrated the mode shapes in correspondence with the identified frequencies.

4 Mode 1V V T V 8.94 Table 1: Natural frequencies of vibration based on the three-dimensional numerical model. 1 st Bending mode (1V) f = 3.99 Hz 2 nd Bending mode (2V) f = 5.99 Hz 1 st Torsion mode (1T) f = 6.80 Hz 3 rd Bending mode (3V) f = 8.94 Hz Figure 3. Mode shapes of the first three vertical bending modes and the first torsion mode of the bridge 4. PRELIMINARY EXPERIMENTAL TESTS In order to update the numerical model of the bridge, a preliminary experimental campaign was performed which involved an ambient vibration test and a dynamic test for the passage of railway traffic. 4.1 Ambient vibration test The ambient vibration test allows for the identification of the dynamic properties of the structure, namely its natural frequencies and mode shapes. In this test, two seismographers (models GSR-18 from Geosig) were used, equipped with triaxial force-balanced accelerometers, controlled and synchronized by a laptop computer. The ambient response was measured in successive setups, considering a fixed reference point (REF) and 21 mobile measurement points (Figure 4a) located in the axis of the main girders of the deck slab. In Figure 4b) it is possible to visualize one of the seismographers placed at the fixed reference point.

5 In each setup were acquired acceleration-time series with 8 minutes of duration at a sampling frequency of 100 Hz. 1/3 span 1/4 span REF Fixed reference point Mobile measurement point Figure 4. Ambient vibration test: a) measurement points; b) seismographer placed on the reference point These acceleration series were analysed and post-processed in order to obtain estimates of the average normalised power spectrum density functions (ANPSD) in the measurement points, and transfer functions, which relate the response in each mobile measurement point and the response in the fixed reference point. The estimates of the spectra were calculated by the application of the Welch s method at each individual record, considering an overlapping of 50% and the application of a Hanning time window [4]. Figure 5 shows two of these spectrums obtained for points located at 1/2 span and 1/3 span of the deck. The peaks corresponding to the frequencies of 4.49 Hz, 6.05 Hz, 9.96 Hz and Hz are in correspondence with the first four vertical modes of vibration. The peak at the frequency of 7.13 Hz is associated to the first torsion mode. Looking at the curve of the power spectrum estimate at 1/2 span, the amplitude of the vibration mode 1V (anti-symmetric) is null, as expected. 4.E E /2 span 1/3 span Amplitude 2.E-06 1.E E Figure 5. Average and normalised estimates of the accelerations measured at 1/2 span and 1/3 span of the bridge deck Table 2 shows a comparison between the values of the natural frequencies obtained from the three-dimensional numerical model and from the experimental results. The observation of the table allows for the identification of the existence of important differences between numerical and experimental natural frequencies, except for the frequency of the 2 nd vertical bending mode. The relative errors are indicated in parenthesis, taking the experimental values of the frequencies as reference.

6 Mode Numerical Experimental 1V 3.99 (-11%) V 5.99 (-1%) T 6.80 (-5%) V 8.94 (-10%) 9.96 Table 2: Numerical and experimental natural frequencies. In Figure 6 are presented the numerical and experimental mode shapes for vibration modes 1V, 2V and 1T. The figure evidences a good agreement between the numerical and experimental configurations for all the analysed vibration modes. 1 st Bending mode (1V) 2 nd Bending mode (2V) Numerical Experimental Numerical Experimental 1 st Torsional mode (1T) Numerical Experimental Figure 6. Numerical and experimental mode shapes associated with vibration modes 1V, 2V and 1T 4.2 Dynamic test for the passage of railway traffic The dynamic test for the passage of railway traffic enables to obtain several acceleration records in different locations of the bridge deck. Figure 7a) shows, as an example, filtered and unfiltered acceleration records at the reference point for the passage of the Alfa Pendular train at a speed of 155 km/h. The digital filter applied is a 20 th order low-pass Chebyshev (type II) filter with a cut-off frequency in accordance with the recommendations of EN1990-A2 [5], that is, 30 Hz. The figure enables to verify that the vertical peak acceleration is g, for the unfiltered record, and g, for the filtered record. The application of a filter allows a reduction of the peak value acceleration by 7%. Figure 7b) presents a power spectrum density estimate of the acceleration. The figure enables to verify that the acceleration is clearly dominated by a frequency in correspondence with the frequency of passage of regularly spaced groups of axles with 25.9 m spacing (f = v/d = 155/3.6/25.9 = 1.66 Hz) and with the frequency of the first vertical bending mode. It is also possible to observe peaks at frequencies higher than 25 Hz, possibly associated with

7 contributions from the track irregularities and flat wheels, which cause excitation of the axles or bogies of the vehicles Non Filtered Filtered 30Hz 3.0E E E+04 Acceleration (g) Amplitude 1.5E E Train passage Free vibration 5.0E Time (s) 0.0E a) b) Figure 7. Dynamic test for the passage of railway traffic: a) filtered and unfiltered acceleration series; b) power spectrum density estimate of the acceleration, corresponding to the passage of the Alfa Pendular train at 155 km/h The modal damping coefficients were determined through the logarithmic decrement method, using part of the acceleration records corresponding to the free vibration response [6]. The values of the modal damping coefficients obtained for the modes 1V, 2V and 3V are indicated on Table 3. An inspection of the table enables to observe that the values of the damping coefficients calculated considering the initial zone of the free vibration response are higher than those calculated considering an intermediate zone. This result corroborates the trend of growth of the damping with the increase of the level of vibration. It can also be pointed out that the values of the modal damping coefficients are higher than the value specified in EN for bridges with composite steel-concrete deck, which is 0.5%. Mode Damping coefficient ξ (%) Initial zone Intermediate zone 1V V V 2.40 (*) 1.79 (*) (*) Approximate value attending the low participation of the 3 rd mode in the response. Table 3: Damping coefficients as a function of the level of vibration. 5. NUMERICAL MODEL UPDATING The updating of the numerical model developed for São Lourenço bridge involved a sensitivity analysis and manual adjustment of the model. Following these procedures, the updated numerical model was validated by the comparison of the numerical and experimental dynamic responses under traffic loads. 5.1 Sensitivity analysis The main objective of the sensitivity analysis was to identify parameters that justify the differences between the numerical and the experimental natural frequencies.

8 In this analysis, the influence of the mass of the deck, the modulus of elasticity of the concrete, the modulus of elasticity of the ballast and the vertical stiffness of the supports in the dynamic properties of the structure were taken into account. Figure 8 shows the sensitivity of the first three vertical modes and the first torsion mode of the deck in relation to the variation of the mass of the deck (M), the modulus of elasticity of the concrete (E c ), the modulus of elasticity of the ballast (E b ) and the vertical stiffness of the supports (K s ). The figure also shows the values of the experimental frequencies as a reference to the other values. The interval of variation of the mass of the deck is due to the variability of the densities of concrete and ballast and to the inclusion of geometric tolerances related with the cross-section of the deck. The vertical stiffness of the supports depends essentially on the integrity of the rubber and the confinement of the rubber in the steel pot of the bearing. Mode 1V K s (kn/m) x 10 3 Mode 2V K s (kn/m) x f 1V,exp = 4.49 Hz Ec 3.74 M Eb 3.73 Ks % -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 3.99 Parameter variation (%) f 2V,exp = 6.05 Hz Ec M 5.30 Eb Ks % -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% Parameter variation (%) 6.08 Mode 1T K s (kn/m) x f 1T,exp = 7.13 Hz % -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 6.80 Parameter variation (%) Ec M Eb Ks Mode 3V K s (kn/m) x % -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 8.94 Parameter variation (%) f 3V,exp = 9.96 Hz Ec M Eb Ks Figure 8. Sensitivity of the frequencies of the mode shapes 1V, 2V, 1T e 3V, to the variation of the mass of the deck (M), the modulus of elasticity of the concrete (E c ), the modulus of elasticity of the ballast (E b ) and the vertical stiffness of the supports (K s ). The figure shows that the frequencies of the mode shapes are strongly influenced by the mass of the deck and the modulus of elasticity of the concrete but not by modulus of elasticity of the ballast. As for the vertical stiffness of the supports, when it decreases so do the frequencies, especially in the symmetric vertical vibration modes (2V and 3V). 5.2 Manual adjustment The manual adjustment of the numerical model was made through a search of values of the mass of the deck, the modulus of elasticity of the concrete, the modulus of elasticity of the ballast and the vertical stiffness of the supports that minimize the objective function (ε), which is defined as a sum of the absolute values of the relative differences between the

9 experimental (f i exp ) and numerical (f i num ) frequencies, taking as reference the experimental frequencies of the four modes described above (1V, 2V, 1T and 3V). The objective function is given by the following expression: exp num 4 fi f i ε = (1) exp f i= 1 i The minimum value of the objective function was obtained for a vertical stiffness of the supports of kn/m, a decrease of the mass of the deck of 16% and an increase of 20% of the modulus of elasticity of the concrete, as shown in Figure f Figure 9. Evolution of the objective function as a function of the mass of the deck mass and the modulus of elasticity of the concrete, for K s = kn/m Table 4 shows a comparison between the numerical and the experimental frequencies after the manual adjustment. The results show that an excellent agreement of results, except for the case of the torsion vibration mode, with a 4% deviation. This table also shows the numerical frequencies with rigid supports (K s = ), to prove the importance of considering the deformability of the supports in the updating of the numerical model. Mode Numerical Numerical K s = K s = kn/m Experimental 1V 4.56 (+2%) 4.48 (-0.2%) V 6.60 (+9%) 6.04 (-0.2%) T 7.49 (+5%) 7.45 (+4.3%) V (+3%) 9.94 (-0.2%) 9.96 Table 4: Comparison of the natural vibration frequencies obtained by numerical (after the adjustment) and experimental methods 5.3 Validation M (%) E C (%) After the updating, the numerical model was used for the prediction of the dynamic

10 response of the bridge under traffic loads. These results were compared with those obtained in the experimental campaign. The method used for the dynamic analysis was the modal superposition method, in which the experimentally measured damping coefficients were adopted. According to 4.2, the estimation of the power spectrum density of the experimentally measured acceleration (peaks with frequencies higher than 25Hz) should be associated with the contribution of the wheel flats or track irregularities that generate excitation of the axles or the bogies of the vehicles. The numerical model developed at this stage did not consider the bridge-train dynamic interaction. For that reason, the dynamic analyses were limited to the contributions of the vibration modes with frequencies up to 20 Hz. Figure 10 shows a comparison between the numerical and the experimental results in terms of vertical acceleration of the deck obtained for the passage of the Alfa Pendular train at 155 km/h. The experimental record is filtered using a low-pass digital filter with a cut-off frequency of 20 Hz. The figure shows that, despite the gap at the end, there is a good match between the experimental and numerical results Experimental_Filtered_20 Hz Numerical_Filtered 20Hz Acceleration (g) Time (s) Figure 10. Comparison between the numerical and experimental results in terms of vertical acceleration of the deck obtained for the passage of the Alfa Pendular train at 155 km/h 6. CONCLUSIONS In this paper, the development and the experimental updating of a numerical model for the dynamic analysis of São Lourenço bridge under railway traffic was presented. The preliminary experimental campaign consisted of an ambient vibration test and a dynamic test for the passage of railway traffic. The comparison of the numerical and experimental natural frequencies revealed limitations in the initially developed numerical model. The sensitivity analysis identified the mass of the deck, the modulus of elasticity of the concrete and the vertical stiffness of the supports as important parameters for updating the numerical model. The updated numerical model was used to predict the dynamic response of the bridge under traffic loads. The results obtained with the updated model provided a good match with the results obtained in the preliminary experimental campaign. This research work is part of a broader project including a PhD thesis by the first author. As sequence to this, further research will be carried out with the objective of developing models to take into account the bridge-train dynamic interaction. The model updating will

11 require experimental campaigns in the bridge, in the train, and also measurements of track irregularities. The application of finite element model updating techniques will allow for an automated adjustment of the parameters which can influence the dynamic response of the bridge-train system, and this is also an objective for the research project. The updated model will be used in the numerical simulation of the dynamic response for high-speed railway traffic in order to draw conclusions regarding the performance of the bridge in relation to structural safety (dynamic amplification and fatigue), track safety (track and wheel-track contact stability) and passengers comfort. ACKNOWLEDGMENTS The present work has been funded by the Portuguese Foundation for Science and Technology (FCT), in the context of the Research Project with reference PTDC/ECM/69697/2006. The first author, Ph.D. student, acknowledges the support provided by the European Social Fund, Programa Operacional da Ciência e Inovação The authors also would like to thank all the collaboration and information provided by Eng.º Tiago Abecasis, designer of this bridge, and Eng.ª Ana Isabel Silva from REFER. REFERENCES [1] EN1991-2, Actions on Structures Part 2: General Actions Traffic loads on bridges. European Committee for Standardization (CEN), Brussels, [2] ERRI D214/RP9, Railway bridges for speeds >200 km/h. European Rail Research Institute (ERRI), Final Report, Utrecht, [3] REFER, E.P Rede Ferroviária Nacional, Substituição da ponte de S.Lourenço ao km 158,662. Projecto de execução Memória descritiva e justificativa, Lisboa, Janeiro de [4] F. Magalhães, Identificação modal estocástica para validação experimental de modelos numéricos. Dissertação apresentada à Faculdade de Engenharia da Universidade do Porto para obtenção do grau de Mestre em Estruturas de Engenharia Civil, FEUP, Porto, [5] EN1990-A2, Annex A2: Application for bridges. European Committee for Standardization (CEN), Brussels, [6] D. Ribeiro, R. Calçada, R. Delgado, Calibração experimental de um modelo numérico da ponte ferroviária de São Lourenço. 6º Congresso de Construção Metálica e Mista, Porto, [7] C. Rigueiro, C. Rebelo, L. Simões da Silva, Numerical assessment of the vibrations in railway viaducts for real traffic. Computational Methods in Structural Dynamic and Earthquake Engineering (COMPDYN), Crete, [8] V. Zabel, M. Brehm, C. Bucher, Seasonal changes of the dynamics of railway bridges with steel girders embedded in concrete. Experimental vibration analysis for civil engineering structures (EVACES), Porto, 2007.

12 [9] G. Chellini, W. Salvatore, Updated models for steel-concrete composite HS railway bridges. Experimental vibration analysis for civil engineering structures (EVACES), Porto, 2007.