3英文

Evaluation of dynamic loads on a skew box girdercontinuous bridgePart II: Parametric study and dynamic load factor AbstractStudies on dynamic loads are important for bridge engineering as well as pavement design.A large number of research studies have indicated that bridge dynamic loads increase road surface damage by a factor of 2–4. Although the field test is the best available approach to understanding actual vehicle-induced dynamic loads on bridges, according to pervious studies there is only a limited amount of field data available on skew box girder continuous bridges. This paper presents an evaluation of vehicle-induced dynamic loads, based on a field test that was carried out on a skew box girder continuous bridge as reported in a companion paper (Part I). The effects of different parameters such as the weight, speed, type, number of axles and position of vehicles on dynamic loads are investigated. Based on the statistical analysis, the use of the dynamic load factor (DLF) is proposed. The dynamic load factor obtained in this study is less than the values provided by most bridge design codes.Keywords: Skew bridge, Dynamic load factor, Dynamic load allowance, Normal traffic condition1. IntroductionV ehicles that are expected to cross a bridge are accounted for in the design or evaluation of the bridge through a static design loading and a certain prescribed fraction of it that is referred to as an impact factor, a dynamic load allowance, or a dynamic increment [1].As opposed to static design loading which is a tangible entity that can be formulated from the static weights of actual or foreseen vehicles, determining the dynamic load is not a straightforward procedure because of the complex nature of the interaction between the bridge and moving vehicles.Numerous studies on bridge dynamics have been carried out. It has been discovered that the magnitude of the dynamic load depends on several factors, including bridge dynamic behaviors,road roughness, vehicle dynamic characteristics, vehicle speed, type, weight, number of axles, axle spacing, the position of vehicles on the bridge, and so forth. Although the exact magnitude of this dynamic load can be reasonably estimated based on the static load, it has never been ascertained as it varies from case to case.Most studies have used the analytical approach to study the bridge–vehicle interaction problem and estimated the dynamic load factor (DLF). Even though the field test is the best available approach to understanding the actual bridge–vehicle interaction to estimate the DLF, the amount of data available from the field on dynamic loads is limited [2]. Billings [3] carried out field measurements of dynamic loads on 27 bridges of various kinds. He found that the dynamic load allowance ranges from 0.05 to 0.10 and from 0.08 to 0.20 for prestressed concrete and steel bridges, respectively. Tests carried out by Cantieni [4] on 226 bridges in which most of the bridges were loaded with the same vehicle, under the same load, and with the same tire pressure showed that a dynamic load allowance as high as 0.7 could be obtained from the field. Chan and O’Connor [5,6] have found maximum value of 1.25 for a dynamic load allowance. Recently, on the basis of the analytical simulations and field tests, Nowak et al. [7] pointed out that the dynamic load allowance is considerably lower than code-specified values. According to their findings, the maximum simulated and measured dynamic load allowances do not exceed 0.17 for a single heavy truck, and 0.10 for two trucks moving side-by-side. It can be understood that, given the limited number of field tests, most of the studies on vehicleinduced dynamic loads on bridges have ended up with different outcomes. This shows the need to conduct more field tests on various bridges using unbiased random samples of vehicles.In this study, a field test was conducted on a skew box girder continuous bridge to collect vehicle-induced dynamic response data on the bridge and information from various kinds of vehicles under normal traffic conditions. The details about the tested bridge such as the cross section, location of sensors, roadway width are presented in a companion paper [15]. A parametric study was carried out to determine the factors that affect the DLF. To determine the appropriate DLF value for design purposes, a statistical analysis was conducted.2. Dynamic load factor2.1. Definition of the dynamic load factorV arious studies have been carried out to determine and evaluate the magnitude of thedynamic load factor. Past studies have used several definitions to represent this factor. Bakht and Pinjarkar [8], in their state-of-the-art review of bridge dynamic tests, have summarized eight definitions of dynamic load factor. After comparing and analyzing the eight definitions, they suggested the following equation for computing the dynamic load factor:DLF = 1 + DLA (1) where DLF is the dynamic load factor, and DLA is the dynamic load allowance given bystat statdyn R RR DLA -=(2) where, Rdyn is the maximum dynamic response and Rstat is the maximum static response.V arious researchers (such as Cantieni [9], Chan and O’Conner [5,6], Kim and Nowak [10], Laman et al. [11]) have used these equations to compute the dynamic load factor, and the equation has also be used for this purpose in design codes (Ministry of Transportation of Ontario [12], AASHTO [13]).2.2. Dynamic load factor from the responses of a bridgeMost of the previous studies on the dynamic load factor used the measured responses of a bridge as their input. The dynamic responses of a bridge are mainly measured in terms of the bending moment or the deflection at selected locations of the bridge structure. Then, the dynamic load factor can be computed by dividing the measured maximum dynamic response to the known static response by using Eqs. (1) and (2). According to Eq. (2), Rstat, the static response can be acquired from a theoretical analysis, by measuring or filtering out the dynamic component of the measured dynamic responses, whereas Rdyn can be directly obtained from the peak measured dynamic response. The next section will present the method of estimating the static response Rstat from the measured dynamic responses.2.3. Design of the filterAs for the analysis of a large amount of field data on the dynamic responses of the bridge owing to normal traffic, the fastest and the most convenient method to estimate the static responses of a bridge due to a moving vehicle is to use an appropriate filter. However, no universally accepted method yet exists for designing a digital filter specifically for the study of bridge dynamics. The method currently used, to choose the best filter, is based on atrial-and-error approach.In selecting the low pass cutoff frequency to filter out the dynamic portion of the responses and estimate the static response of the bridge due to moving vehicles, careful attention should be taken to ensure that nearly all static frequency responses must occur below the cutoff frequency and nearly all dynamic frequency responses should be filtered out above this cutoff frequency[14].A low frequency “body bounce” may influence the calculated dynamic load factor values if this dynamic mode is below the low pass threshold and remains among the presumed extracted responses. According to Billings [3] and Chan and O’Conner [6], a typical vehicle “body bounce” occurs in the range between 2 and 5 Hz frequencies. A vehicle body bounce, if present, will result in higher calculated static responses, which in turn will cause the calculated dynamic load factor to decrease.The typical frequency spectrum graphs from both acceleration and strain responses of the bridge under moving vehicles are shown in Fig. 10 of the companion paper [15]. During the selection of appropriate filter parameters, the cut off frequencies greater than the frequency range that affects the static strainand less than the first natural frequency of the bridge were considered. In addition, in order to avoid the body bounce effect of a vehicle, cutoff frequencies in between 2 and 5 Hz were avoided. A low pass Butterworth digital filter was used in this study. This type of filter ideally retains all frequency responses below the cutoff frequency and omits all frequency responses above the cutoff frequency [14]. The other important parameter which needs consideration in filter design is the order of the filter. As indicated by Johnson et al. [16], by keeping the low pass cutoff frequency constant, the measured response improves as the filter order increases.The dynamic responses of the bridge induced by the control vehicle (calibration truck) as reported in [15] were filtered by applying different low pass cutoff frequencies and filter orders. The results were compared with the known measured static strain responses of the bridge caused by the calibration truck as well as with theoretically computed bending moment responses.The comparison of the measured dynamic and the corresponding filtered responses with the measured and the theoretically computed responses was carried out on the selected strain gauge location of the bridge. The selected strain gauges were strain gauges 7, 8, 9, and 22 (refer to Figure 2 in the companion paper [15] for the locations of the strain gauges). The first three strain gauges were selected from the middle of each web of the first span so that the maximum responsecould be obtained from those locations. As to negative bending moment responses, strain gauge 22 was selected to study the responses due to negative bending moment responses at the bridge deck near the support. In addition to the criteria presented in the above paragraphs, the selection of the filter parameters was based on the good agreement between the filtered responses and the measured responses from the known loads. It was found that a Butterworth filter with a low pass cutoff frequency of 1.4 Hz and filter order 10 was fit for the given criteria. Some of the results of the filtered responses with measured and theoretical values are shown in Fig. 1. From the four figures in Fig. 1, it can be noticed that, in general, the filtered responses agree well with the measured and theoretical responses.(a) Maximum measured static, and dynamic and filtered (b) Maximum the oretically obtained static, and dynamic andstrain and responses: fast lane. filtered measured bending moment responses: fast lane.(c) Maximum measured static, and dynamic and filtered (d) Maximum the oretically obtained static, and dynamic andstrain responses: slow lane. filtered measured bending moment responses: slow lane.Fig. 1. Comparison of responses before and after filtering.Typical dynamic and static responses for computing the dynamic load factor obtained from strain gauge 9 (refer to Figure 2 in [15] for the location of the sensor) for five axle trucks moving at highway speed are given in Fig. 2. After estimating the static responses (in this case the static strain responses) by filtering, it is a straightforward procedure to compute the dynamic load factors using Eqs. (1) and (2).(a) Dynamic response. (b) Static response.Fig. 2. Typical responses for computing dynamic load factor.3. Factors affecting the dynamic load factorAs mentioned in the preceding sections, many factors affect the magnitude of the dynamic load factor. When carrying out dynamic measurements in the field, in addition to strain and acceleration responses, information on the speed, weight, number of axles, axle spacing, and position of the vehicles on the bridge was acquired. The influence of these parameters to the dynamic load factor was studied and is presented in the following sections:3.1. W eight of the vehicleIt was indicated in previous studies (Chan and O’Conner [5], Nassif and Nowak [17], Laman et al. [11], Nowak et al. [7]) that the dynamic load factor is dependent on the weight of the vehicles. In this study, the strain responses of the bridge were recorded for each case having a single vehicle traversing on the bridge. According to the recorded data, in 5 days, 309 good cases (a single vehicle at a time) were selected. From the collected data, it was noticed that most of the single vehicles (more than 95%) during the test choose Lane 2 (the slow lane) to cross the bridge. As for the selection of the locations of the strain response measurements, those locations with the expected maximum response measurement points were included in the analysis. Therefore, the strain gauge sensors along the lines m3 and d2 as shown in [15] (strain gauges 7–9, 18 and 19, refer to Figure 2 in [15] for the locations of the strain gauges) were expected to have their maximum positive bending moment in the first span, and strain gauge 22 was expected to have its maximum negative moment at the support which is near the first pier.(a) At strain gauge 9. (b) At strain gauge 19.(c) At strain gauge 22.Fig. 3. DLA versus static live load strain.The computed DLA versus the corresponding static live load strain at strain gauge locations of 9, 19, and 22 are given in Fig. 3. It is clear from the figure that the dynamic load allowance decreases as the static load increases in all strain gauge locations. The maximum static strains are found to be recorded in the strain gauges 9 and 19, which are located on the first web of the bridge deck under the slow lane [15]. The mean calculated DLA for each strain gauge location of 7–9, 18, 19, and 22 are 0.233, 0.208, 0.193, 0.182, 0.165, and 0.149, respectively. Although the strain recorded in the strain gauge at location 22 is found to be the smallest, the mean computed DLA is not the largest of all. On the contrary, the mean DLA computed from this location is the smallest of all. In contrast to the other strain gauge locations mentioned here, strain gauge 22 recorded negative bending moments near the support.3.2. Number of axlesSome bridge design codes like OHBDC [12] and CHBDC [18] relate the dynamic load factor with the number of axles of the design vehicle. In this study, an investigation has been carried out to determine if there is a correlation between the dynamic load factor and the number of axles of the vehicles. From the data samples that were collected of single vehicles, it was found that the number of axles of the vehicles ranged from two to six. Vehicles with five axles make up the largest proportion in the collected database followed by vehicles with three, four, two and six axles.The DLA versus the number of axles for some of the strain gauge locations are given in Fig.4. The computed means of the DLAs are summarized and given in Fig.5. From Fig. 5, it is noticed that the maximum mean DLA obtained from strain gauges 7–9, 18 and 19 are all from vehicles with two axles, whereas the minimum DLA for the same strain gauge locations are from six-axle vehicles. The minimum mean DLA for strain gauge 22 was obtained from a three-axle vehicle. In general, the mean DLAs obtained from strain gauge locations 7–9, 18 and 19, for the three-axle vehicles, are lower than that of other vehicles except for the six-axle vehicles. The reason for this, according to the video data collected from the field, is that most of the single three-axle vehicles recorded in the threeaxle vehicle category were double-decker buses (which are very common in Hong Kong), carrying a more or less similar load on them. In other words, the samples of the three-axle vehicles collected are more uniform in type than the others.(a) At strain gauge 9. (b) At strain gauge 19.(c) At strain gauge 22.Fig. 4. DLA versus number of axles.Fig. 5. Number of axles versus mean DLA.3.3. Speed of vehiclesThe effect of the speed of vehicles on the dynamic load factor has been studied both analytically and experimentally. The study carried out by Hwang and Nowak [19] reported that the lightest vehicles induced an increase in DLA with an increase of speed, however, the heavier vehicles induced a smaller DLA value as speed increased. Whereas, based on data obtained from field measurements, Laman et al. [11] concluded that there is no correlation between vehicle speed and DLA values, while Schwarz and Laman [20], in their investigation involving a limited number of samples collected from the field, indicated that DLA increases with speed. Chan et al.[21] concluded that, for the same axle spacing parameter (ASP = 1.0), the dynamic load allowance increases with vehicle speed.4. Statistical analysis of the dynamic load factorA large scatter of the values of dynamic load factors can be obtained from field measurements, even when the bridge and the vehicle are the same. Bakht et al. [1] concluded that the dynamic load factor is not a deterministic quantity. To obtain a single value of this factor for design purposes it is necessary to know the statistical properties of the scatter data. Nassif and Nowak [17], and Kim and Nowak [10] have used statistical approaches to derive the cumulative distribution function and other statistical parameters of the values of dynamic load factors collected in the field.In this study, a large amount of data on bridge responses and vehicle information was acquired through a field test. Therefore, it is important to introduce a statistical analysis to obtain the appropriate value of DLA. In the previous sections it was found that DLA is dependent on the static weight of a vehicle. It was also found that the speed of vehicles can affect the DLA.From the identified parameters, it was determined that DLA is strongly affected by the static weight of the vehicles. It was also shown that the DLA from a single vehicle at a time is larger than that of vehicles moving side by side on the bridge. Therefore, it was decided that the DLAs obtained from a single vehicle moving on the bridge would be the governing DLA for the statistical analysis. However, more than 95% of the single vehicles crossed the bridge using Lane 2 (slow lane, refer to [15]). The remaining 5%, of which most were light vehicles, crossed the bridge using Lane 1 (fast lane, refer to [15]). Thus, the locations of maximum response were found near strain gauges 9 and 19. For these reasons, the DLA values obtained from these twostrain gauge locations will be included in the statistical analysis for maximum responses in the middle of the first span, and the DLAs from strain gauge 22 will be used in the statistical analysis to determine the appropriate DLA for negative bending moment responses near the support.5. Comparing different bridge design codes5.1. American associations of states highway and transport officialsIn the standard AASHTO code [22], to allow for dynamic, vibratory and impact effects, the live load stress in the superstructure of the bridge due to the H and HS loadings should be multiplied by a DLA as defined by ,1.384.15+=L DLA (3)where L is the loaded length in meters, and the DLA does not exceed 0.3.5.2. Canadian highway bridge design codesIn the newly introduced Canadian Highway Bridge Design Code [18], the DLA is practically equal to 0.25 for all heavy trucks. Larger DLAs of 0.4 and 0.3 are applied to single and dual axle vehicles, respectively.5.3. EurocodeIn the Eurocode [23], the dynamic amplification established for a medium pavement quality and pneumatic vehicle suspension has been included for traffic load models, which depends on various parameters such as road roughness and location of the load under consideration.6. ConclusionsIn this study an evaluation of vehicle-induced dynamic load that was carried out on a skew continuous bridge is presented. The bridge response data acquired in the field test were mainly measured strains from strain gauge sensors. As the strains are proportional to the bending moment of the bridge, therefore, the computed DLFs values obtained in this study are based on the bending moment of the bridge.On the basis of the results obtained from this particular field study the following conclusions, which may be applicable for similar kinds of bridges, are drawn:(1) The dynamic load factor was found to depend on the weight of the vehicles, as theweight of the vehicles increases the dynamic load factors (DLFs) decrease.(2) Weak correlations were found between the speed of the vehicles and the DLF.(3) No relationship can be found between the number of axles and the DLF.(4) The mean DLF obtained from vehicles moving side by side in a given sensor location was found to be less than the mean DLF obtained from a single vehicle.(5) The DLF obtained based on the statistical analysis in the study was found to be less than the DLF values provided by most bridge design codes.20。