Dynamic analysis of a cable-stayed deck steel arch bridge

Dynamic Analysis of a Cable-Stayed Deck Steel Arch Bridge P. Galv´ın, J. Dom´ınguez ∗ Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain Abstract A theoretical and experimental research work in relation to Barqueta cable-stayed bridge is described in this paper. Barqueta Bridge, across Guadalquivir river, links the city of Seville with the Scientific Park Cartuja 93. At jam hours cars may cover one half of the bridge lanes for more than one hour. Full-scale tests were carried out to measure the bridge dynamic response. The experimental program included the dynamic study for two different live load conditions: the bridge with one half of it lanes full of cars, and the bridge empty of cars. Modal parameters estimations were made based on the acquired data. Ten vibration modes were identified in the frequency range of 0-6 Hz by different techniques, being two of these modes very close to each other. The traffic-structure interaction is also studied. Experimental results were compared with those obtained from a three-dimensional finite element model developed in this work. Both sets of results show very good agreement. Finally, a damage identification technique has been applied to determine the integrity of the structure. Results obtained from a test developed in July 2005 have been correlated to experimental results obtained in October 2006 using the damage index method. Key words: arch bridge, damage detection, operational modal analysis, vehicle-structure interaction 1 Introduction Experimental tests constitute the most reliable method to obtain the dynamic properties (natural frequencies, mode shapes and damping ratios) of actual structures and to validate, from these results, numerical models used for their ∗Corresponding author. Tel.: +34 954487293; fax: +34 954487295. Email address: [email protected] (J. Dom´ınguez). Preprint submitted to Elsevier Preprint 5 November 2006 analysis. They permit also to asses the state of damage of a structure by comparison with the dynamic properties obtained in previous analyses. Dynamic identification methods from ambient vibration have been used in complicated structures like dams [1], offshore platforms [2], sports stadia [3], bridges [4], etcetera. A theoretical and experimental research work in relation to Barqueta bridge is described in this work. It is a steel arch bridge with cable-stayed deck that links the old town of Seville with the Scientific Park Cartuja 93. At certain hours, a traffic jam occurs on the bridge and cars cover one half of the bridge lanes (Fig. 1) for more than one hour. The experimental program developed in this work includes the dynamic characterization of the structure for two situations under normal conditions and when it is covered by traffic. The modal properties of the bridge are identified from ambient vibration and the results obtained for both situations of the bridge are compared. By this analysis, the effect that vehicles have on the dynamic behaviour of the structure is studied. The idea presented in [3], where changes of the modal properties of a sport stadium were determined at times when different activities took place at the stadium, is explored in this paper for the bridge case. The authors of that paper concluded that the dynamic properties of the structure depend to some extent on the type of the activity celebrated in the stadium. The possibility of these changes taking place in other types of structures is studied in this work. The obtained experimental results are compared with those from a threedimensional finite element analysis. Finally, a damage identification methodology is applied to verify the structural integrity of the bridge. Experimental results obtained with a time difference of one year and a temperature difference of 10oC are analyzed. 2 Description of the structure The aesthetic functions and symbolic values of the structures are more and more important every day, in particular when these structures are built in urban zones [5,6]. Barqueta bridge was built in Seville for the 1992 International Exhibition [7,8]. It is an innovative structure, designed by JJ. Arenas and M. Pantale´on, with a flying central arch rising from the vertex of two lateral triangular frames. Fifteen years after its construction, Barqueta is an indispensable piece of the urban landscape of Seville. Barqueta is a bowstring steel bridge. The 168 m span structure rests on two sets of two vertical supports spaced 30 m in the transverse direction located at the banks of the Guadalquivir river (Figs. 1-3). The cross-sections of the arch and inclined legs include deep grooves that produce enough local inertia to avoid the need of any internal 2 longitudinal stiffener. The deck cross-section is shown in Fig. 2. It is 16 m wide and 2.4 m deep. The total wide of the bridge is 21 m with two cantilever pedestrian decks. The deck cross-section includes two vertical webs separated by a distance of 1 m (see Fig. 2). The hangers are anchored between then. The hangers have variable inclination. The deck is supported on the extremes by transversal beams, with variable depth, which rest on the vertical supports. 3 Finite Element Model A three dimensional finite element model (FEM) has been developed for the numerical analysis of the structure using as-built drawings of the bridge and some double-check in-situ measurements. Modal analysis was carried out using ANSYS [9]. The arch, supports, and the internal stiffener were represented as two-node beam elements (BEAM44) with 6 degrees of freedom per node. This element permits the end nodes to be offset from the centroidal axis of the beam. The hangers were modeled as truss elements (LINK10) with 3 degrees of freedom per node. The deck slab was modeled using eight-node shell elements with 6 degrees of freedom per node (SHELL93). The two extreme beams and the vertical supports were connected by spring elements (COMBINE14). A detailed model of all the bridge elements was intended. As a consequence, the number of degrees of freedom is high. The full model consists of 10328 beam elements, 17 truss elements, 15672 shell elements and 8 spring elements, resulting into 26025 elements and 47024 nodes. Fig. 3 shows a full 3-D view of the finite element model of the bridge and details of the deck cross-section. 4 Full-scale testing Dynamic properties can be obtained by measurement of vibrations produced by ambient loads. This technique is simpler for civil engineering structures than classical modal analysis because it is not necessary to excite the structures by shakers. In addition, the structure can be used during the testing process. The experimental program, carried out during July 15 2005 and October 11 2006, includes dynamic characterization of the structure in normal conditions and when a half of the bridge is covered by traffic. The response of the structure was measured at 16 selected points (Fig. 4) using Endevco (Model 86 and 3 Model 4370) accelerometers. Preliminary results obtained from a FE dynamic analysis were used to determine the optimum location of the sensors. Since nine Model 86 accelerometers were available for the testing and two of these sensors (at locations 1 and 2) were held stationary for reference during the test carried out in 2005 and one of them (at location 2) during the test developed in 2006, two set-ups were required to cover the 16 measurement points. It is worth to mention that in output-only modal analysis, where the input force remains unknown and may vary between the set-ups, the different measurements setups can only be linked if there are some sensors in common. The reference accelerometers were chosen in order to be able very carefully to measure all global modes of the bridge. The hangers were not instrumented in the first test because, according to preliminary numerical analysis and preceding experimental studies [10,11], significant interaction between the stay-cables and the rest of the structure was not expected. This interaction is significant when the lowest natural frequencies of the structure and the natural frequencies of the cables are close. In the present case, cables are short, estimating their first natural frequencies around 6 Hz. There are at least, 10 global modes of the structure below this values. Therefore it is not expected that cables have an important participation in the global modes of the bridge. This assumption was validated by the second test, where the cables were instrumented (Figure 5a.). The power spectral density of the cable’s response is shown in Figure 6 where it can be observed that the first bending mode of the cable is around 6 Hz. During the first test, data of the response of the structure were acquired when the structure was under fluid traffic conditions, in locations 1 to 9, and when cars cover one half of the lanes of the bridge, in points 1,2 and 10 to 16. In order to obtain the mode shapes of the structure, the response at all points have been used. Traffic-structure interaction is not expected to cause changes in structural mode shapes. Natural frequencies and damping ratios, were determined using each one of the set-ups independently. In the second test, the response of the structure was acquire in all locations for both situations: when the structure is under fluid traffic conditions and when cars cover one half of the lanes of the bridge. Ambient vibration response was acquired during 1000 seconds per channel and per set-up. The data were sampled to 64 Hz. Data were decimated (order 3) to carry out data analysis in the frequency range of interest (0 to 10 Hz). Data records were Hannning-windowed with 66.67% overlapping for spectral averaging. Acquired data are available for interested reader by sending an e-mail message to the authors. 4 5 Data Analysis Different procedures to obtain modal parameters from ambient vibration data have been used in this work. In output-only modal analysis, also called operational modal analysis, the applied forces are unknown and, therefore, neither the frequency response function nor the impulse response function can be obtained to determine modal parameters as in classical modal analysis [12]. The signal at one of the fixed transducers is used as a reference to determine the frequency response function and the impulse response function. Four complementary identification methods have been considered in the present work: three of them based on frequency domain analysis and one on time domain analysis. The study has been developed using MATLAB [13] and ARTEMIS [14] software. The first identification method employ is Peak-Picking (PP), which has been used with success in many other applications [5,11,15,16]. This method is based on the fact that when the frequency response function reaches a peak at a certain frequency, it can be associated to the force or to a resonance frequency of the structure [17]. Natural frequencies are identified from peaks of spectral densities function: ωdi =q1−ξ2 i·ωni (1) This procedure produce a good estimations of natural frequencies for weakly damped structures. To distinguish between peaks associated to the excitation and those associated to resonance frequencies of the structure, mode shapes can be used [17]. The response values at all points of a weakly damped structure for one of its resonance frequencies, are in phase or out of phase by 180o, depending on the mode shape. Peaks of spectra density function associated to the excitation normally present a phase difference of the cross-spectra function between two measurements points different from 0oor 180o. In addition, the coherence function between two signals has a value close to one for the resonance frequencies of the structure, due to the high relation response-noise at those frequencies. This fact helps to decide which of the frequencies really are natural frequencies of the structure. The PP method is based on the assumption that the dynamic response of the structure at resonance peaks is determined for each mode. This is valid when modes are well separated. Therefore, it is difficult to identify modes very close to each other using this method. The auto-spectra, cross-spectra and coherence functions obtained from the first test, are shown in Fig. 7. The natural frequencies were identified from resonance peaks in auto-spectra and cross-spectra functions. Coherence function peaks are the same that the peaks of the previous functions, and the phase value of the cross-spectra function for those peaks are 0oor 180o, providing 5 additional evidence that these peaks correspond to natural frequencies of the structure. The second identification technique used in the present study is the so called Averaged Normalized Power Spectral Densities (ANPSDs) which is a practical implementation, developed by Felber [18], of the PP method. In this case, autospectra functions are normalized and averaged to obtain an average spectra density function that, normally, shows all resonance frequencies of the system. ANPSD(ω) = 1 l l X i=1 PSD(ω) Pn j=1 PSD(ωj)(2) where lis the number of measurement locations. The natural frequencies of the structure are obtained from the simple observation of the peak in the ANPSDs diagram. This method was used successfully in the dynamic characterization of an arch bridge in reference [19]. The ANPSDs diagrams for both situations of the bridge obtained from the first test are shown in Fig. 8. The third identification technique used, also in the frequency domain, is called Enhanced Frequency Domain Decomposition (EFDD) [20] which, from a simple form, introduces significant improvements to Peak Picking Technique. This method is based on a modal decomposition realization of the spectral density matrix, being one of the advantages of the method the possibility to identify very close modes. The Power Spectral Density matrix (PSD) of the mmeasured responses can be expressed, for a lightly damped structure, as: Gyy(jω) = X k²Sub(ω) dkφkφT k jω −λk +dkφkφT k jω −λk (3) where dkis a scalar constant, φkis the mode shape vector, λkis the pole of the output PSD, and Sub(ω) is the limited number of modes that will contribute to the response at frequency ω. The first step of the EFDD is to estimate the output PSD matrix at discrete frequencies, and to carry out the Singular Value Decomposition (SVD) of the matrix ˆ Gyy(jωi) = UiSiUH i(4) where the matrix Uicontains the singular vectors uij and Siis a diagonal matrix with the scalar singular values sij. Close to a peak, where the kmode is dominant, there will be only one mode in Sub(ω) and, therefore, the first singular vector uk1is an estimate of the mode shape; i.e., ˆ φk=uk1, and the corresponding singular value is the auto power spectral density function of the singular degree of freedom system. This power spectral density function is identified around the peak comparing the estimated mode shape ˆ φkwith the 6 singular vectors from the frequency lines around the peak. If the MAC value obtained from the singular vector and ˆ φkis higher than a reference value close to one, the singular value belongs to the auto power spectral density function. Once the auto power spectral density function has been obtained around the peak, the natural frequency and the damping ratio are estimated by Inverse Fast Fourier Transform. The singular values decomposition of the spectral density matrix is shown in Fig. 9. Peaks representing vibration modes have been selected. It can be observed that two modes very close to 2 Hz exist for the bridge under study, one of them not been determined by the previous methods. This type of modes can be easily identified with EFDD, by observation not only of the highest singular value but also of the next one. The last technique is a more elaborated mathematical procedure that works directly with time domain acquired data. It is called Stochastic Subspace Identification (SSI). The interested reader can find details of the mathematical approach in references [21,22]. SSI is a powerful tool (perhaps the most advanced identification method that exists up to day) that has been used for dynamic characterization of many structures [21,23,24]. A brief review of the main ideas of this method is presented in the following. The dynamic behaviour of a structure is described by: M¨ U(t) + C˙ U(t) + KU(t) = F(t) (5) where M,C, and Kare the mass, damping, and stiffness matrices of the structure, respectively. U(t) and F(t) are the displacement and input force vectors respectively. This equation can be written as a state space equation: ˙x(t) = Acx(t) + Bcu(t) (6) where the state vector x(t)=[U(t),˙ U(t)]Tand the state matrix Ac, and the system control influence coefficient matrix Bc, are defined by Ac=   0I −M−1K−M−1C   Bc=   0 −M−1   where F(t) = B2u(t) (7) The output vector y(t) can be expressed as y(t) = Ccx(t) + Dcu(t) (8) where Ccis the output influence coefficient matrix and Dcis the output control influence coefficient matrix. These equations constitute a continuous-time state space model of a dynamic system. 7 Since experimental data are discrete, a discrete-time state space model can be obtained by sampling the continuous-time state space model: xk+1 =Axk+Buk(9) yk=Cxk+Duk(10) where xk=x(k∆t) is the discrete-time state vector containing the sampled displacements and velocities; uk,ykare the sampled input and output, respectively; A= exp(Ac∆t) is the discrete state matrix;and B= [A−I]A−1 cBc is the discrete input matrix. Including the stochastic components; i.e., the noise due to disturbances and modeling inaccuracies (wk) and the noise due to sensor inaccuracy (vk), the discrete state space model can be written as: xk+1 =Axk+Buk+wk(11) yk=Cxk+Duk+vk(12) The process noise wkand measurement noise vkare assumed to be zero-mean, white noise, statistically independent, and with covariance matrices E    wp vp   µwT pvT p¶ =   Q S STR   δpq (13) where Eis the expected value operator and δpq is the Kronecker delta. Q,R and Sare process and measurement noise covariance matrices. In ambient vibration testing, only the responses of the structure are measured, obtaining: xk+1 =Axk+wk(14) yk=Cxk+vk(15) where the input ukis modeled by the noise terms. This equation constitutes the basis for time domain modal identification from ambient vibration testing. There are several techniques to carry out modal identification based on this equation. Fig. 10 shows a stabilization diagram obtained by applying SSI. System order and stable poles can be found in these diagrams which provide modes of the structure. 8 6 Dynamic behaviour of the bridge The dynamic behaviour of Barqueta Bridge is governed by vertical bending and torsional modes, in the frequency range of 0−6 Hz. Ten modes have been identified in this frequency range. Table 1 shows the obtained natural frequencies for the bridge under fluid traffic conditions whereas Table 2 corresponds to the situation when the bridge is jammed. Damping ratios obtained for both bridge conditions are shown in Tables 3 and 4. It is observed in both tests that the natural frequencies of the structure change very little due to the traffic conditions on the bridge. The first test was carried out on July 2005, while that the second one took place on October 2006, with an ambient temperature difference of 10oC. Very little changes appear in the natural frequencies obtained from the two experimental tests as can be seen from the values shown in Tables 1 and 2. Damping ratios may increase up to 200 per cent when the bridge is jammed with vehicles, as compared to the empty situation. Increments in damping ratios of the same order were observed in both series of tests. A very similar conclusion was reached in [3], where it was concluded that the damping ratios in a sport stadium increases to a significant extent when it is crowded. Mode shapes of the structure identified from experimental modal analysis and from numerical analysis are shown in Figs. 11 and 12.A complete agreement between both sets of mode shapes can be observed. In order to quantify this agreement, experimental results have been compared with the finite element results by using the Modal Assurance Criterion (MAC) [25]. MAC values vary from 0 to 1; a value of one implies perfect correlation of the two modes vectors (one vector is proportional to the other), while a value close to zero indicates no correlated modes (orthogonal modes). The MAC value is defined as: MAC(φA,k, φB,j) = (φT A,kφB,j)2 (φT A,kφA,k)(φT B,jφB,j)(16) where φC,k is the k-mode of data set C, and Tmeans transpose matrix. The obtained MAC matrix between the results obtained from the test carried out in 2005 and those obtained from the test carried out in 2006, are shown in Fig. 13. The mode vector correlation using MAC seems quite good, finding the highest difference in the fifth mode shape. The MAC matrix between the numerical and experimental results have also obtained. Fig. 14 shows that there is a quite good agreement between experimental and numerical results. Therefore, the numerical model can be used to represent the dynamic behaviour of the bridge. 9 (a) (b) Fig. 3. The 3-D FE model of Barqueta Bridge (a) and detail of the deck cross-section model (b). 16 Fig. 4. Measurement locations. 17 (a) (b) Fig. 5. Accelerometers at cables and arch member 18 0 5 10 15 20 0 1 2 3 4 5 6 7x 10-3 Frequency [Hz] PSD [m/s2/Hz] Accelerometer at cable Fig. 6. Power spectral density of the cable’s response 19 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 x 10-5 Frequency [Hz] Auto-spectra magnitude [m/s 2] Transducer 1 Transducer 2 Transducer 5 Transducer 8 Transducer 9 (a) Auto-spectra function 0 1 2 3 4 5 6 0 0.5 1 1.5 x 10-5 Frequency [Hz] Cross-spectra Magnitude [m/s 2] Reference Transducer 2 Transducer 4 Transducer 7 Transducer 12 Transducer 14 Transducer 16 (b) Cross-spectra function (magnitude) 0 1 2 3 4 5 -150 -100 -50 0 50 100 150 Frequency [Hz] Cross-spectra Phase [º] Reference Transducer 2 Transducer 4 Transducer 7 Transducer 12 Transducer 14 Transducer 16 (c) Cross-spectra function (phase) 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency [Hz] Coherence Reference Transducer 2 Transducer 4 Transducer 7 Transducer 12 Transducer 14 Transducer 16 (d) Coherence function Fig. 7. Auto-spectra function, cross-spectra function (magnitude and phase) and coherence function. 20 0 1 2 3 4 5 6 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 Frequency [Hz] ANPSDs Normal Traffic jam Fig. 8. Average Normalized Power Spectral Densities. 21 d B | ( 1 . 0 s ¯ ² ) ² / H z F r e q u e n c y [ H z ] 0 2 4 6 - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 F r e q u e n c y D o m a i n D e c o m p o s i t i o n - P e a k P i c k i n g A v e r a g e o f t h e N o r m a l i z e d S i n g u l a r V a l u e s o f S p e c t r a l D e n s i t y M a t r i c e s o f a l l D a t a S e t s . Fig. 9. Singular Value Decomposition of the Spectral Densities Matrices. 22 State Space Dimension Frequency [Hz] 0 2 4 6 Stabilization Diagram Data Set: Measurement 2 CVA [Data Driven] 10 20 30 40 50 60 70 80 Markers Stable Modes Unstable Modes Noise Modes Fig. 10. Stabilization Diagram. 23 (a) Mode 1 (b) Mode 2 (c) Mode 3 (d) Mode 4 (e) Mode 5 (f) Mode 6 (g) Mode 7 (h) Mode 8 (i) Mode 9 (j) Mode 10 Fig. 11. Mode shapes from experimental modal analysis. 24 (a) Mode 1 (b) Mode 2 (c) Mode 3 (d) Mode 4 (e) Mode 5 (f) Mode 6 (g) Mode 7 (h) Mode 8 (i) Mode 9 (j) Mode 10 Fig. 12. Mode shapes from numerical modal analysis. 25 Table 4: Damping ratios. Traffic jam on the bridge. ξEF DD[%] 13 ξEF DD[%] 14 ξSSI [%] 15 ξSSI [%] 16 1.178 2.196 2.26 3.527 1.748 2.266 4.119 2.054 1.663 2.629 1.665 2.601 2.674 2.843 2.053 3.132 0.809 1.434 1.296 1.117 1.161 1.183 1.703 2.667 1.272 2.8 1.48 2.58 1.208 1.583 1.76 2.459 1.186 1.515 1.185 1.503 1.462 1.302 2.048 1.498 13 Test carried out in 2005 14 Test carried out in 2006 15 Test carried out in 2005 16 Test carried out in 2006 32 View publication statsView publication stats