540 T.H.T. Chan et al. density of 7335 kg / m 3 and a flexural stiffness EI = 29.97kN/m 2 . The first three theoretical natural frequencies of the main beam bridge was calculated as f~ = 4.5 Hz, f2 = 18.6 Hz, and f3 = 40.5 Hz. Figure 2. Experimental setup for moving force identification The U-shape aluminum track was glued to the upper surface of the main beam as a guide for the car. The model car was pulled along the guide by a string wound around the drive wheel of an electric motor. The rotational speed of motor could be adjusted. Seven photoelectric sensors were mounted on the beams to measure and check the uniformity of moving speed of the car. Seven equally spaced strain gauges and three equally spaced accelerometers were mounted at the lower surface of the main beam to measure the response. Bending moment calibration was carried out before actual testing program by adding masses at the middle of the main beam. In addition, a 14-channel type recorder was employed to record the response signals. Where Channels 1 to 7 were for logging the bending moment response signals from the strain gauges. Channels 8 to 10 were for the accelerations from the accelerometers. The channel 11 was connected to the entry trigger. In the meantime, the response signals from Channels 1 to 7 and Channel 11 were also recorded in the hard disk of personal computer for easy analysis. The software Global Lab from the Data Translation was used for data acquisition and analysis in the laboratory test. Before exporting the measured data in ASCII format for identification calculation, the Bessel IIR digital filters with lowpass characteristics was implemented as cascaded second order systems. The Nyquist fraction value was chosen to be 0.05. PARAMETER STUDIES For practical reason, one parameter was studied at a time. The examination procedure was to examine one parameter in each case to isolate the case with the highest accuracy for the corresponding parameter and then another parameter was examined. The parameters, such as the mode number, the sampling frequency, the speed of vehicle, the computational time, the sensor and sensor locations were considered as variables to examine their effect on the accuracy of force identification. There are two ways to check this kind of effects. One is that the identified results are checked directly by comparing the identified forces with the true forces. However, because the true forces are unknown, it is difficult to proceed. The other way is that the identified results are checked indirectly by comparing the measured responses (bending moments, displacements or accelerations) with the rebuilt responses calculated from the identified forces. The accuracy is quantitatively defined as Equation (9), called a Relative Percentage Error (RPE). RPE = EJftn, e - ~'dentl x 100% (9) El/.el Equation (9) is also used to calculate the relative percentage errors between the measured and rebuilt responses from identified forces instead of comparing the identified forces with the true forces directly. The measured response (R d) and rebuilt response (Rreb,,izt)are herein substituted for the true force (ftr,,e)and identified force (fide,,)in equation (9) respectively. In the present parameter studies, most of results were from the comparison of the relative percentage errors between the measured and rebuilt response only for the bending moment response. Regarding the results associated with the accelerations, they will be reported separately. Parameter Studies of Moving Force Identification in Laboratory Effects of Mode Number 541 For comparing the effects of different Mode Number (MN) on identified results in the TDM and FTDM, it was assumed that the sampling frequency (f~) and vehicle speed (c) were constant, and the case of fs = 250Hz, c = 15 Units (1.52322 m/s) was chosen. The data at all the seven measurement stations for bending moments were employed to identify the moving forces. The mode number was varied from MN=3 to MN=10. The identified forces were calculated first, and then the rebuilt responses from the identified forces were then computed accordingly. The Relative Percentage Error (RPE) for both the TDM and FTDM are shown in Figure 3. Figure 3. Effects of mode number For the TDM, the RPE values at the middle measurement stations are always less than the ones at the two end measurement stations. This is associated with the signal noise ratio of various measurement stations because there are bigger responses at the middle stations than those at the two end stations. It is found that when the mode number is equal to or bigger than four, the relative percentage errors are reduced dramatically. This means the TDM is effective if the required mode number is achieved or exceeded, otherwise, the TDM will be failed. The minimum RPE value case is of MN=5, the maximum RPE value case is of the biggest mode number involved (MN=I 0). This also shows that the case MN=5 is the most optimal case in this kind of comparisons. Similar conclusions are drawn for the FTDM. However, the biggest difference from the TDM is that the RPE is independent of the increment MN after MN=5. By comparing the identification accuracy by the TDM and FTDM in Figure 3, it can be seen that the results are very close to each other when the MN is equal to 5 and 6 respectively, especially at the middle measurement stations. However, it can be seen from the identified forces in Figure 4 that the FTDM is worse than the TDM because it has components with higher frequency noise. Figure 4. Identified forces (MN=5, fs = 250Hz, c = 15 Units ) Effects of Sampling Frequency The sampling frequency fs should be high enough so that there is sufficient accuracy in the discrete integration in equation (4) and (5) [Law et al 1997]. In the present study, the data was acquired at the sampling frequency 1000 Hz per channel for all the cases. This sampling frequency was higher than the practical demand because only a few of lower frequency modes were usually used in the moving force identification. Therefore, the sequential data acquired at 1000 Hz was sampled again in a few intervals in order to obtain a new sequential data at a lower sampling frequency. Here, a new 542 T.H.T. Chan et al. sequential data at the sampling frequency of 333,250, and 200 Hz would be obtained by sampling the data again every third, fourth and fifth point respectively. For the case of the vehicle running at 15 Units, the bending moment data was acquired at the different frequencies of 200, 250 and 333 Hz respectively. The RPE results between the rebuilt and measured bending moment responses are calculated and listed in Table 1 for both the TDM and FTDM. TABLE 1 EFFECTS OF SAMPLING FREQUENCY (c = 15 Units) Sta. TOM FTDM No. MN=3 MN=4 MN=5 MN=3 MN=4 MN=5 I II III I II III I II III I II III I II III I II III 1 13.8 783. 412. 6.44 6.05 6.11 8.86 5.32 3.75 188. 1541 1724 6.35 16.1 445. 3.38 4.39 2005 2 7.15 609. 244. 2.81 2.74 2.69 3.34 2.61 2.40 167. 1618 1636 2.68 10.3 419. 2.51 2.16 1370 3 6.50 358. 185. 2.74 2.10 1.95 2.87 2.10 1.94 163. 1679 1647 2.09 8.50 427. 2.01 2.14 1230 4 3.61 216. 216. 3.15 2.96 2.80 3.72 2.71 2.12 165. 1686 1678 2.69 8.05 433. 2.09 2.22 1321 5 6.27 359. 189. 3.16 2.74 2.53 3.58 2.68 2.44 164. 1676 1657 2.61 8.62 430. 2.42 2.13 1238 6 7.41 614. 245. 4.74 4.58 4.32 5.42 4.31 3.45 169. 1615 1654 4.22 10.0 424. 2.89 2.70 1387 7 17.3 780. 420. 6.61 5.84 5.94 9.36 5.19 4.05 187. 1514 1717[ 5.89 16.0 446. 3.92 4.30 2095 Case I, II, and III is for 200,250 and 333 Hz respectively. For completely comparing the effects of different sampling frequency, the effect of mode number on identification accuracy is also incorporated in the study. It is found that the higher the sampling frequency is, the lower the RPE values are for all the measurement stations in the TDM. This shows that the higher sampling frequency is better than the lower sampling frequency, and the TDM method has higher identification accuracy if the response is acquired at a higher sampling frequency. In Table 1, it is shown that the FTDM method is failed when the sampling frequency fs = 333 Hz and mode number MN=3 because the RPE values are too big to accept for all the measurement stations. However, The FTDM method is still effective for the case in which the mode number is bigger than 3, f~ = 200 Hzandf, = 250 Hz respectively. By comparing the RPE values at a lower sampling frequency f~ = 200 Hz with that at f, = 250 Hz, it is found that the identification accuracy at fs = 200 Hz are higher than one at f, = 250 Hz. It shows that the identified results are acceptable and useful if more mode number and suitable sampling frequency is determined in FTDM method. Effects of Various Vehicle Speeds In this section, some limitations on identified methods TDM and FTDM should be considered firstly. In particular, necessary RAM memory and CPU speed of personal computer are required for both the TDM and FTDM. Otherwise, they will take very long execution time due to the bigger system coefficient matrix B in equation (7), or they cannot execute at all due to inefficient memory. As the mode number, the sampling frequency and bridge span length had not been changed for this case, a change of the vehicle speed would mean a change of the sampling point number, namely change of dimensions of matrix B in equation (7). Therefore, in order to make TDM and FTDM effective and to analyze the effects of various vehicle speeds on the identified results, the case of MN=4 and fs = 200 Hz was selected. When the test was carried out, the three vehicle speeds were set manually to 5 Units (0.71224 m/s), 10 Units (1.08686 m/s), and 15 Units (1.52322 m/s) respectively. After acquiring the data, the speed of vehicle was calculated and the uniformity of speed was checked. If the speed was stable, the experiment was repeated five times for each speed case to check whether or not the properties of the structure and the measurement system had changed. If no significant change was found, the recorded data was accepted. The RPE values between the rebuilt and measured bending moment responses are calculated and listed in Table 2. It shows that the TDM is effective for all the three various vehicle speeds. The RPE values tend to reduce for each measurement station as the vehicle speed increases. But, the RPE values are close to each other in the case of 10 Units and 15 Parameter Studies of Moving Force Identification in Laboratory 543 Units. It shows that the identification accuracy for the faster vehicle speed is higher than that at slower vehicle speed. However, the FTDM is not effective in the case of lower vehicle speed 5 Units, but the identified results are getting better and better as the vehicle speed increases. Fortunately, the identified result is acceptable in the case of 15 Units in the FTDM. TABLE 2 EFFECTS OF VEHCILE SPEEDS (MN=4, fs = 200Hz ) Station TDM FDTM No. 5 Units 10 Units 15 Units 5 Units 10 Units 15 Units 1 6.81 5.40 6.45 1045.69 101.53 6.35 2 5.54 2.49 2.81 708.18 46.57 2.68 3 5.88 3.03 2.74 621.36 24.83 2.09 4 8.67 3.01 3.15 562.90 45.11 2.69 5 4.50 2.76 3.16 586.98 23.98 2.61 6 4.66 3.93 4.74 647.38 44.70 4.22 7 6.56 7.94 6.61 965.75 94.26 5.89 Effects of Various Measured Station Number This section estimates the effects of measurement station number ( N t ) on the identified accuracy. The N t was set to 2, 3, 4, 5 respectively while the other parameters MN=5, f, = 250 Hz, c = 15 Units were not changed. The RPE values between the rebuilt and measured responses are given in Table 3. The results in Table 3 show that the TDM is required to have at least three measurement stations to get the two correct moving forces for the front and rear wheel axles respectively. But the FTDM should have at least one more measurement station, i.e. 4, to get the same moving forces. However, the errors are increased obviously when the measurement station number is equal to 5 for the FTDM. TABLE 3 EFFECTS OF MEASUREMENT STATIONS TDM Station No. 2 3 4 1 (L/a) * * * 2 (2L/8) * * 1.50 3 (3L/8) 2003.03 2.15 2.08 4 (4L/8) * 2.27 * 5 (5L/8) 2029.36 2.48 2.04 6 (6L/8) * * 2.38 7 (7L/8) * * * Asterisk * indicates the station is not chose. 5 1.91 2.21 2.62 2.39 2.82 FTDM 2 3 4 5 * * 2.67 27.06 1192.66 86.92 2.35 14.82 * 115.42 * 27.95 1198.49 87.52 2.49 14.73 * * 2.93 26.62 Comparison of computational time The computational time consists of three periods, i.e., i) forming the system coefficient matrix B in equation (7), ii) identifying forces by solving the equation and iii) reproducing the responses. The above parts are same for the TDM and FTDM. The case described here is of MN=5, f, = 250 Hz, c = 15 Units, N t = 7 by using a Pentium II 266 MHz CPU, 64M RAM computer. The total sampling points are 700 for bending moment response at each measurement station and the total sampling points are 604 for each wheel axle force in the time domain. Therefore, the dimensions of matrix B are (7 x 700, 2 • 604). The execution time recorded is listed in Table 4 for the comparison on each period of the TDM and FTDM in details. It shows that the FTDM takes much longer than the TDM method in forming the coefficient matrix B. The execution time in other two parts is almost the same for the two methods. The TDM takes shorter time than the FTDM from the point of view of the total execution time. 544 T.H.T. Chan et al. TABLE 4 COMPARISON OF COMPUTATION TIME (in Second) PERIOD TDM Forming coefficient matrix B Identifying forces Rebuilding responses Total FTDM 332.69 1059.57 1837.97 1834.07 55.04 53.99 2225.7 2947.63 CONCLUSIONS Parameter studies on moving force identification in laboratory test have been carried out in this paper. These parameters include the mode numbers, the sampling frequencies, the vehicle speeds, the computational time, the sensor numbers and locations. The study suggests the following conclusions: (1) A minimal necessary mode number is required for both the TDM and FTDM. It should be equal to or bigger than 4. If first five modes are determined to identify the moving forces, the identification accuracy is the highest in the cases studied. (2) The TDM has higher identification accuracy when the higher sampling frequency is employed. However the FTDM is failed if adopting the higher sampling frequency and the lower mode number. (3) The faster car speed is of benefit to both the TDM and FTDM, but FTDM method is not suitable for the slower car speed case. (4) At least three and four measurement stations are required to identify the two wheel axle forces for the TDM and FTDM respectively. (5) The TDM takes shorter time than the FTDM. (6) Both the TDM and FTDM can effectively identify moving forces in time domain and frequency domain respectively, and can be accepted as a practical application method with higher identification accuracy. (7) From the point of view of all the parameter effects on the identification accuracy, the TDM is the best identification method. It should be firstly recommended as a practical method to be incorporated in the future developed Moving Force Identification System (MFIS). ACKNOWLEDGMENT The present project is supported by the Hong Kong Research Grants Council. REFERENCES 1. Briggs J.C. and Tse M.K. (1992). Impact force Identification using Extracted Modal Parameters and Pattern Matching. Int. J. Impact Engineering 12:3, 361-372. 2. Chan T.H.T. and O'Connor C. (1990). Wheel Loads from Highway Bridge Strains: Field Studies. Journal of Structural Engineering 116:7, 1751-1771. 3. Chan T.H.T. Law S.S. Yung T.H. and Yuan X.R. (1999). An Interpretive Method for Moving Force Identification. Journal of Sound and Vibration 219:3, 503-524. 4. Fryba L. (1972). Vibration of Solids and Structure under Moving Loads, Noordhoff International Publishing, Prague. 5. Hoshiya M. and Maruyama O. (1987), Identification of Running Load and beam system. Journal of Engineering Mechanics ASCE, 113, 813-824. 6. Law S.S. Chan T.H.T. and Zeng Q.H. (1997). Moving Force Identification: A Time Domain Method, Journal of Sound and Vibration, 201:1, 1-22. 7. Law S.S. Chan T.H.T. and Zeng Q.H. Moving Force Identification-Frequency and Time Domain Analysis, Journal of Dynamic System, Measurement and Control (accepted for publication) 8. Moses F. (1984). Weigh-In-Motion System using Instrumented Bridge, Journal of Transportation Engineering ASCE, 105(TE3), 233-249. 9. O'Connor C. and Chan T.H.T. (1988). Dynamic Wheel Loads from Bridge Strains, Journal of Structural Engineering, 114:8, 1703-1723. 10.Stevens K. K. (1987). Force Identification Problems-An Overview, Proceeding of SEM Spring Conference on Experimental Mechanics, 838-844. SEISMIC ANALYSIS OF ISOLATED STEEL HIGHWAY BRIDGE Xiao-Song LI 1 and Yoshiaki GOTO 2 1 Research Associate, 2 Professor Dept. of Civil Engineering, Nagoya Institute of Technology Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan ABSTRACT Seismic isolators with dissipation devices have been widely used for highway bridges in Japan, because they may effectively absorb energy and reduce inertia force induced by earthquake. The main factors that influence the response of the isolated bridge are initial stiffness and yield force of the isolator. These quantities should be appropriately designed. Besides, introduction of the isolators leads to an interaction between the bridge pier and the isolator and increases the computational difficulty due to the nonlinearity that occurs in both the pier and the isolator. The purpose of this paper is to investigate the seismic response of the isolated bridges subjected to ground motions, where we examine how the behaviors of the bridge pier are influenced by the initial stiffness and yield force of the isolator and the elongation of natural period of the bridge. Then, the applicability of the 'Displacement Conservation Principle' for predicting the maximum responses of the piers of the isolated steel bridges is numerically examined. The numerical results show that the application of the 'Displacement Conservation Principle' may be reasonably safe and accurate for the practical design of isolated steel piers. KEYWORDS seismic isolation design, nonlinear dynamic analysis, steel highway bridge 545 546 INTRODUCTION X S. Li and Y. Goto After the great earthquake happened in Kobe, in 1995, seismic isolators with dissipation devices have been widely used for highway bridges in Japan. Due to the significant increase of the natural period, the isolators may effectively absorb energy and reduce inertia force induced by earthquake. The main factors that influence the response of the isolated bridge are initial stiffness and yield force of the isolator. These quantities should be appropriately designed. Besides, introduction of the isolators leads to an interaction between the bridge pier and the isolator and increases the computational difficulty due to the nonlinearity that occurs in both the pier and the isolator. The purpose of this paper is to investigate the seismic response of the isolated bridges subjected to ground motions, where we examine how the behaviors of the bridge pier are influenced by the initial stiffness and yield force of the isolator and the elongation of natural period of the bridge. Then, the applicability of the 'Displacement Conservation Principle' for predicting the maximum responses of the piers of the isolated steel bridges is numerically examined. The numerical results show that the application of the 'Displacement Conservation Principle' may be reasonably safe and accurate for the practical design of isolated steel piers. ANALYTICAL MODEL A typical isolated bridge is used as an analytical model, as shown in Fig.1. The total weight of the bridge M=1067ton consists of the weight of the deck Mb=0.95M and the weight of the pier Mp=0.05M. For the height of piers, two values are considered, that is, H=13m for Model-l, and H=11m for Model-2. The corresponding fundamental natural periods for the two piers are Tl=0.705s and T2=0.549s. Fig.l: Analytical Model A lead-rubber bearing (LRB) is assumed as an isolator with dissipation device that has bilinear yield stiffness as shown in Fig.2. In Fig.2, Qy and Uby are the yield force and yield displacement, Seismic Analysis of Isolated Steel Highway Bridge 547 respectively. K~,~ and Kb2 are the elastic stiffness and post-yield stiffness with a relation of Kbz=KbJ6.5; Kr~ is the equivalent stiffness and UBe=0.7Ub which are suggested by the 'Manual of Menshin (isolation and dissipation) Design of Highway Bridges' (1992). Fig.2: Hysteresis Behavior of Isolator ANALYTICAL METHOD A numerical method that considers both geometrical and material nonlinearity is used to carry out the dynamic analysis (Li and Goto, 1998). The post-yield modulus of material is assumed to be Ep=E/100. The effect of damping is considered by a mass-proportional damping matrix. The damping coefficient is set to h=0.01 for elasto-plastic analysis and h 0.05 for elastic analysis. Two standard ground accelerations suggested by Japan Road Association are used for the analysis. One is Type 2 at Soil Group II (hard soil site), the other is Type 2 at Soil Group Ill (soft soil site). Both acceleration waves are illustrated in Fig.3. The time interval adopted in the numerical integration is 0.01s. Fig.3: Ground Accelerations In order to investigate the effect of the initial stiffness and yield force of the isolator, the calculation is carried out by changing Qy/Py and I~I/K p from 0.2 to 0.9 that is the possible range in practical design, where Py=2( cr y-Mg/A)/(HB) and Kp=3EI/H 3 are the yield force and elastic stiffness of the pier. 548 NUMERICAL RESULTS X S. Li and Y. Goto Typical responses of an isolated bridge are shown in Fig.4. It can be seen from Fig.4 (a) that the displacement of the pier is much smaller than that of the deck due to the isolator. Furthermore, the pier is damaged little and the energy induced by seismic wave is almost dissipated in the isolator, as illustrated in Fig.4 (b). In the following, the effects of the initial stiffness Kb~ and yield force Qy of the isolator and the elongation of natural period of the bridge are investigated. Then, the applicability of the 'Displacement Conservation Principle' for predicting the maximum responses of the piers of the isolated steel bridges is numerically examined. Fig.4: Responses of Model-1 with Qy/Py=0.6 and Kbl/I~ =0.5 Subjected to Wave Type 2-111 Effect of K~l and Qy on Pier and Isolator With the designated yield force ratios Qy/Py=0.3, 0.5 and 0.7, the maximum response displacements of the pier and the isolator are obtained by changing the initial stiffness ratio Kbl/Kp from 0.2 to 0.9. The relations of ductility factors//p and/1 b of the pier and the isolator vs. the initial stiffness ratio Kb~/K p of the isolator are shown in Fig.5 for Model-1 and Model-2 subjected to waves Type 2-II and Type 2-lit. In this figure, the ductility factors for piers and isolators are defined as /1 p=Upmax/Upy (with solid line) and /1 b=Ubmax/Uby (with dotted line), where Upmax=maximum displacement of pier, Upy=yield displacement of pier, and Ubmax=maximum displacement of isolator and Uby=yield displacement of isolator. It should be noted that the/1 p may be considered as the maximum response or the ductility factor of the pier, while/.t b denotes only the ductility factor of the isolator because the Uby varies with Qy/Py or Kb~/K p. Similarly, the relations that vary with the yield force ratio Qy/Py are shown in Fig.6 for the designated initial stiffness ratios Kbl/Kp=0.3, 0.5 and 0.7. From Fig.5, it can be seen that the both ductility factors ~t p and /~ b of the pier and the isolator increase with the increase of the initial stiffness ratio Kbl/K. p of the isolator for all cases. However, Seismic Analysis of Isolated Steel Highway Bridge 549 some influences caused by the wave types can be found as follows. The maximum responses of the pier subjected to wave Type 2-11" almost linearly increase as Kbl/Kp increases from 0.2 to 0.9 as shown in Fig.5(a), while those subjected to wave Type 2-m" increase a little shapely when Kbl/I~ >0.6. Furthermore, the values of It p for Model-2 with a smaller ratio of Qy/Py=0.3 become greater than those with Qy/Py=0.5 and 0.7 when Kbl/Kp~0.6, as shown in Fig.5(c). The relations between the ductility factor It b of the isolator and Kbl/Kp exhibit a different tendency depending on the value of Qy/Py. That is, It b with Qy/Py=0.3 exhibits a large increase, while that with Qy/Py=0.7 shows just a small increase. Fig.5: Effect of Initial Stiffness Ratio Kbl/Kp of Isolator on Ductility Factors It p and It b Fig.6: Effect of Yield Force Ratio Qy/Py of Isolator on Ductility Factors It p and It b . Moving Force Identification-Frequency and Time Domain Analysis, Journal of Dynamic System, Measurement and Control (accepted for publication) 8. Moses F. (1984). Weigh -In- Motion System using Instrumented. shown in Fig.5 for Model-1 and Model-2 subjected to waves Type 2-II and Type 2-lit. In this figure, the ductility factors for piers and isolators are defined as /1 p=Upmax/Upy (with solid line). was sampled again in a few intervals in order to obtain a new sequential data at a lower sampling frequency. Here, a new 542 T.H.T. Chan et al. sequential data at the sampling frequency of