10th International Conference on Short and Medium Span Bridges Quebec City, Quebec, Canada, July 31 – August 3, 2018 EXTREME LIVE LOAD EFFECTS ON SHORT AND MEDIUM SPAN CONTINUOUS BRIDGES FROM MEXICAN WIM DATA García-Soto, A David1,2, Hernández-Martínez, Alejandro1, Valdés-Vázquez, J Gerardo1, and Vizguerra-Alvarez, R Alejandra1, University of Guanajuato, Mexico adgarcia@ugto.mx Abstract: Extreme value theory is frequently used to estimate the live load effects on bridges associated to certain return periods This information can be used for design, reliability and code calibration purposes In recent years the use of weigh-in-motion (WIM) data has become very common to estimate live load effects, especially for simple spans or limited cases of double spans or multiple span continuous bridges An extensive WIM database is used to compute the live load effects for many bridge configurations with between two and five continuous spans Since commercial software is not efficient when a large amount of bridges and/or heavy trucks are involved (as in the present study), a program was expressively codified for this study The computed load effects are used to carry out an extreme value analysis and to fit the results to a probability distribution function (PDF), which could be used to obtain the loads effects associated to certain return periods for short and medium span continuous bridges A discussion on the fitting procedure is outlined, including the implications in extrapolating the loads effects for several continuous bridges Comparisons with loads effects for simple span bridges, some trends found when varying the configurations and number of spans for continuous bridges, as well as possible implications in design and reliability are reported, and some conclusions are drawn The results and conclusions could give some guidance to practitioners, code developers and those interested in deriving live load models for continuous bridges INTRODUCTION The study of live load effects of bridges subjected to vehicular loads is a long-standing research topic Many of the studies in the last decade deal with simple span bridges (e.g., Liu et al., 2017) and/or few cases of continuous bridges; this is sometimes justified by arguing that most bridges correspond to simple spans in some countries (e.g., Tapia-Hernández et al., 2017) While many recent works focused on live load effects generated by vehicles from WIM data, including multiple presence statistics (e.g., Gil and Kang, 2015) or multiple girders for simple span bridges (e.g., Anitori, et al 2017;), other studies deal with particular bridges (e.g truss bridges) subjected to live loads from code models or testing trucks (e.g., Laurendeau et al., 2015).Recent studies dealing with continuous bridges have found that live load models developed by using simple spans could be adequate to reproduce the loads effects for continuous bridges under the same traffic conditions, however some deviations are found and several cases of continuous bridges should be studied to assess possible differences (García-Soto et al., 2015) Other recent studies include simple-span and two-span continuous bridges for carrying out calibration of live load factors for Michigan Highway bridges (e.g Eamon et al., 2016) Eamon et al (2016) extrapolated the live load 29-1 effects for a return period of interest using several techniques, but it was concluded that the empirical data followed a cumulative Normal distribution and the extreme behavior could be represented for the Gumbel distribution without significant differences if using other distributions Studies have been also conducted for live load factors by considering continuous bridges with more than two spans, although for only a couple of three-span configurations, subjected to the special load case of extremely heavy mining trucks, which loads effects are probabilistically modeled as Normal (Oudah et al., 2017) Soriano et al (2017) proposed a simplified probabilistic model by fitting the tail end of WIM data results with an equivalent normal distribution, considering only a single span bridge and moments at midspan Most of the recent studies focus on live load calibration, reliability analysis, multiple presence and other issues; however, not much details about the extreme value analysis are discussed except in some works (e.g., Soriano et al., 2017), and certainly not that much is said about the implications of different thresholds for fitting the upper tail of the extreme value distribution, especially when extensive number of continuous bridges of more than two spans are considered Therefore, the main objective of this study is to carry out an extreme value analysis of load effects for continuous multiple span bridges with two to five spans, discussing the implications of selecting different thresholds in the upper tail of the distribution for the fitting, depending on the bridge configuration and considered load effect, and mentioning possible implications in reliability, code calibration and codified design 2.1 STRUCTURAL ANALYSIS OF LIVE LOADS ON CONTINOUOS BRIDGES WIM data in Mexico and selected bridges As a previous step to carry out the structural analysis to obtain load effects on bridges, the input data must be selected, namely the loads and bridge configurations Regarding the loads, a WIM database is used to extract sequences of loads and spacings (i.e., a mathematical abstraction of a recorded vehicle) and run them over the selected continuous bridge configurations and obtain the maximum live load effects for single vehicle passage, to be used in the analysis of extremes in the next part of the study The WIM data was recorded on 2009 and has been described, employed and cited somewhere else (e.g., García Soto et al., 2015) New WIM data was recorded during 2017 as part of a research project financially supported by the Mexican National Council of Science and Technology (CONACYT, by its acronym in Spanish; Project “Problemas Nacionales 2014” No 248162) However, this information is presently being analyzed and no load effects from such databases are included here Nevertheless, it can be stated that similar information to that of 2009 has been found, and that very heavy trucks with gross vehicular weights (GVW) over 1300 kN has also been found for the same highway; in fact, the existence of even larger GVWs is being discussed, to disregard it as recording errors or to validate such possibility A preliminary set of histograms from part of the information for the three studied highways in 2017 is shown in Figure 1; only part of the histograms is shown, for GVW over arbitrarily selected values, so that only information regarding heavy trucks is depicted On top of Figure a histogram for GVW and only one lane (left lane, for one direction only), for the same 4-lane highway (Irapuato – La Piedad Highway, Class A4) where the WIM data was collected in 2009, is shown for GVW > 343 kN It can be observed in Figure that trucks with GVW over 1000 kN were recorded in the Irapuato – La Piedad highway (in Guanajuato State, Central Mexico), and even over 1300 kN for one truck; this is consistent with other WIM databases (García-Soto et al., 2015) and with static surveys carried out during the 90s (Rascon, 1999); even higher GVWs for other lanes where found, but as stated previously, undergoing research will try to answer whether these recordings are feasible and reliable For the selected lane in the Irapuato – La Piedad Highway a total of 57,678 vehicles were recorded in one week In the middle of Figure a histogram for GVW from another Class A4 highway (Queretaro – Irapuato Highway), again for only the left lane in one direction only, exhibits a similar behavior and comparable results as those for the Irapuato – La Piedad Highway, and although there are some trucks with GVWs over 1000 kN, the extreme values are not as large as those shown on top of Figure 1; for this case a total of 27,053 vehicles were recorded in one week for the selected lane, and only trucks with GVW > 149 kN were used for the figure Finally, at the lower part of Figure a histogram for the GVW > 49 kN of the two-lane Guanajuato – Los Infantes Highway (both ways, Class C highway) is shown; this highway does not allow as heavy traffic as the A4 29-2 highways (heavy 9-axles trucks are not permitted), which explains the much lower GVWs for the total of 127,218 vehicles recorded during one week for the two lanes Figure 1: Histograms of WIM data recorded during 2017 at several highways in Guanajuato 29-3 Figure implies that traffic can be site specific and this will have an impact in developing live load models for bridge design, in carrying out reliability studies and in performing code calibration tasks These issues will be reported in future studies For the present study information only from 2009 was used (García-Soto et a., 2015) for continuous short and medium span bridges with several lengths Schematic configurations of the selected bridges are shown in Figure Although an extensive number of cases was analyzed (others are currently being calculated), we selected only spans with lengths L= 10m and L=40 m for the discussion and scope of the present study, plus one more example for a four-span continuous bridge which will be described below Although not shown in Figure 2, simple spans are also used for comparison Figure 2: Bridge configurations used Variables a, b, c and d denote possible values of the span length L 2.2 Software formulation The structural analysis to compute the load effects for the selected bridges implies a large amount of calculations, since most of the vehicles in the extensive database (171 days with daily traffic from around 8,000 to around 30000 vehicles) is used for the analyses; moreover, each vehicle is run over the whole length of the bridges with multiple spans at small increments (tens of centimeters) in both directions, to capture the maximum flexure moments and shear forces Therefore, to deal with this issue, it was decided to modify and use an existent software named AMER 1.0 ©, which was developed as a complementary aid to the notes for the course Structural Analysis II for the Civil Engineering program at the Department of Civil Engineering of the University of Guanajuato, in Mexico Consequently, the source code of the originally academic software was modified and extended, to cope with the extensive number of required analyses within a reasonable time The software is based on the direct stiffness method, which requires the establishment of relations between forces and displacements given by the Eq below, where [K] represents the stiffness matrix, {D} the unknown displacements vector at the structures nodes, and {F} the forces applied to the nodes [1] [ K ] { D} = { F } Aimed at speeding up the computing, the equations systems are solved by means of the LDL T decomposition for the stiffness matrix [K], and stored in a semi bandwidth to optimize the required memory, and consequently reduce the time to solve the system given by Eq The modified and extended software will be freely available in the website www.di.ugto.mx/GEMEC/ We hope that the next version of the software AMER 2.0 © will be available to download in the website in 2018; the new version will include several storing characteristics, methods for solving the equations systems, among other 29-4 features Further details of the computing and envelopes of maximum bending moments and shear forces are not given for brevity; instead, some features of these load effects are discussed from a probabilistic standpoint in the following section 29-5 EXTREME VALUE ANALYSIS The results by using the software described in the previous section are employed to inspect the empirical cumulative distribution function (CDF) of the live load effects on bridges The positive and negative flexure moments for a set of two to five continuous multiple span bridges, with equally spaced spans of 40 m, are arbitrarily selected to show the empirical CDF (absolute values) in different probability papers and are depicted in Figure (Normal, Lognormal, Gumbel and Weibull); the legends represent the number of continuous spans and its length in meters (i.e., 40-40 stands for a double span with L = 40 m, 40-40-40 for a triple span with that same span length, 40-40-40-40 for a four-span bridge, and so on) In Figure results for a simple span with the same length (legend 40, for L = 40 m) are also shown To obtain the CDFs in Figure 3, samples of the maximum loads effects computed from each of the 171 recorded days for four-lane traffic are used, as has been carried out before for simple and double spans (García-Soto et al., 2010; García-Soto et al., 2015) The empirical probability distribution will be transformed into empirical probability distribution of annual extreme truck load effects, as will be explained later For the assessment of the empirical probability distribution of daily extreme truck load effects, the pairs ( xi, i/(n+1)) are used, where n (equal to 171) is the number of samples, and xi denotes the ordered data from the smallest to the largest values Figure 3: Empirical CDFs in different probability papers for daily flexure moments due to truck loads 29-6 In Figure the first legend (simple span with L=40 m) corresponds to the values for maximum positive flexure moment in the far-right side of the figure for each probability paper; as expected, they are the largest From right to left, the following four sets of values (and the following four legends from the top) correspond to the maximum positive moment for double, triple, quadruple and fivefold continuous span bridges, respectively; in fact, only the moments for double spans are clearly appreciated and are significantly smaller than those of simple spans and slightly larger than those for continuous bridges with equally spaced three, four or five spans, which differences are almost negligible and undistinguishable in Figure This feature is possibly linked to the fact that for spans longer than long heavy trucks, the maximum flexure moments are caused by the same trucks for single vehicle passage, which can be advantageous for developing live load models, since the model based on only simple, double and triple spans will be enough for capturing the behavior for bridges with more than three spans; this may not be necessarily the case for short spans, other load effects (e.g., shear) and non-equally spaced spans (García-Soto et al., 2018) In fact, the general trend of the empirical CDFs is preserved, just with different magnitude, as can also be observed in Figure 3, implying that the analysis of extremes employed for simple and double spans (García-Soto et al., 2010; García-Soto et al., 2015) is expected to lead to analogous results for bridges with a larger number of spans The previous discussion also applies for negative flexure moments, which are also shown in Figure in absolute values (obviously excluding simple spans) being the set to the far left the corresponding to maximum for double spans (40-40 legend in the 6th position from top going down, negative moments for three spans, 40-40-40, in the th, etc.), followed to the right by the set of negative moment values for triple equally spanned bridges, which are larger than those for double spans, and this trend (which is reversed as compared to positive moments) leads to slightly larger negative moments for increasing number of spans (with differences also marginal and unnoticeable in Figure for four and five spans) It is noted that arrows and text are added on Figure and its legends (and in the next figure when it applies) to distinguish between positive and negative flexure moments In the following, the decision on the fitting to extreme values procedure is described; it is essentially based on previous studies (García-Soto et al., 2010; García-Soto et al., 2015), since as it was indicated before, the trends for triple, quadruple and fivefold span bridges are analogous to those of simple and double span bridges; nevertheless, a brief discussion of some further aspects of the fitting are also included From Figure it can be observed that the normal and lognormal PDFs best follow the trend of the empirical data, this and other considerations (see details in García-Soto et al., 2010; García-Soto et al., 2015) leads to adopt the Gumbel distribution for fitting the information to the annual maximum flexure moments and shears Once the Gumbel distribution is selected for the extreme value analysis, a direct probability distribution fit to the empirical distribution of annual extreme given by ((xi, (i/(n+1)) 376.25/4) is performed The annual maximum of the truck load effect is modeled as a Gumbel variate given by, ( FLE (x) = exp − exp ( −α ( x − u ) ) ) where the parameters α and u are determined using the least square method for the distribution fitting For the same set of continuous span bridges shown in Figure 3, the empirical probability distributions for annual extreme are shown in Figure 4a In the fitting the largest 20 points of the annual empirical distribution were considered, and the results are shown by the fitted lines in Figure 4b (steeper slopes); the effect of a change in this consideration is discussed shortly after The number of adequate points for the extreme value analysis (Castillo, 2005; Stuart, 2001) is basically intended to capture the extreme behavior of the upper tail of the probability distribution, because this is the relevant region for engineering purposes; to answer which threshold (i.e., the selected number the largest values in the upper tail) is more adequate, is a compromise between capturing the most possible information without loosing focus on the extreme values, as indicated by an empirical approach described in the literature (Castillo, 2005) and other study (García-Soto et al., 2018) Such a threshold does have an impact in the extrapolation of load effects to selected return periods (and therefore in any associated reliability analysis and live load 29-7 model development) as can be observed in Figure 4b where the fitting by using all the data points is included (lines with softer slopes) Note that the fittings shown in Figure 4b for double spans, are also very adequate to fit the maximum bending for triple, quadruple and fivefold continuous bridges; this feature could be important for simplifying the developments of live load models and reliability studies, since the checking of the double span case will cover those of more spans just as well; nevertheless, this is the case for equally spaced spans of 40 m, and may not necessarily be so for other cases and/or load effects (García-Soto et al., 2018) 29-8 Figure 4: a) Empirical distribution of annual extreme; b) Fitting to annual extreme for double span (L=40 m) and positive and negative flexure moments; c) Shear force case for L=40 m; d) Flexure moments case for L=10 m; e) Fitting for double span bending, L=10 m; f) Shear force case for L=10 m Continue… …Figure (continuation): g) Asymmetric bridge (bending); h) Asymmetric bridge (shear) Figure 4c shows the results for the same continuous bridges with L=40 m spans, but for other live load effect, the maximum annual shear under the assumption of a Gumbel empirical CDF; it is observed that unlike the case of bending, the shear is always smaller for simple spans, and that neither significant differences are observed between the maximum shear for double spans in relation to triple span continuous bridges, nor a clear trend in very small but systematic differences for increasing number of spans; there seems to be a general trend of decreasing shear for increasing number of spans, but this is not always the case, as shown by the lack of uniformity in the shape of the CDFs compared to the bending case, which may be due to different trucks or subconfigurations leading to the maximum shear, unlike the flexure moments, which seem to be often generated by the same truck for L=40 m (span long enough to accommodate the longest Mexican trucks) Analogous to Figure 4b, in Figure 4c a couple of fittings for double spans are also shown; it can be seen that the fitting for shear is not as good as for bending, for the considered bridges To inspect the probabilistic behavior for shorter spans, the same procedure is followed to obtain the Gumbel probability papers for the annual maximum flexure moment and shear but considering continuous 29-9 bridges with L=10 m spans The results are shown in Figures 4d and 4e for flexure moment, and in Figure 4f for shear In Figure 4d it is shown that, in absolute terms, the flexure moment is still the largest for the simple span bridge, however, for continuous bridges the differences between double and triple spans (or mores spans) are smaller, in fact, the negative flexure moments for continuous multiple span bridges can be of the order of the positive flexure moments (unlike the case of L=40 m) and they can even be larger in the lower tail of the distribution; this can be more clearly observed in Figure 4e, where the results for only the double span bridge are extracted Also in Figure 4e, the fitting for the upper tail of the distribution and for the whole set of data are shown with lines; in this case the fitting is alike, as can be inferred from the slop of the fitted curves which are similar with the ordinate to the origin slightly shifted This means that the adequate selection of the threshold for the fitting of the upper tail (the relevant for extremes) depends on the span length, and also on the considered load effect, as shown in Figure 4f, which includes the empirical annual Gumbel CDFs for shear and where the fitting is quite different depending on the considered threshold (the last points selected for the fitting) Figure 4f shows that the CDF is practically the same for continuous bridges with any number of equally spaced spans; this means that unlike the L=40 m case (where differences were more accentuated in shear), for these short spans is not the shear the one exhibiting dissimilarities in the empirical probability distributions, but the flexure moment Note also that in contrast to the L=40 m case, when L=10 m the Gumbel distribution fits better to the shear force than to the flexure moment The previous discussion implies that when carrying out an analysis of extremes for continuous bridges, attention should be paid to the differences due to span length and the considered load effect; it could be expected that this can also have an impact if reliability analyses and code calibration tasks are of interest and further studies are recommended to inspect this A couple more of results are included in Figure 4g and Figure 4h for bending and shear, respectively; these are aimed at inspecting possible differences of continuous bridges with equally spaced spans in relation to continuous bridges with unsymmetrical configurations (i.e., non-equally spaced spans) For this purpose, a four-span continuous bridge with spans equal to 10, 30, 40 and 20 m (in this order, and denoted as 10-30-40-20) is arbitrarily selected to show the CDFs for load effects From Figures 4g and 4h it can be observed that this unsymmetrically spaced continuous bridge leads to loads effects values (in statistical terms) located between those of equally spaced span bridges with L=10 and L=40 m, and the general behavior of the curves and the fitting is closer to the symmetrical 40 m case, possibly because long spans dominate the configuration of this particular selected bridge (i.e., 10-30-40-20) Nevertheless, these differences should be inspected for more cases, to assess the adequacy of live load models developed for only simple spans and/or continuous bridges with equally spaced spans Further studies are also recommended to assess other important issues (e.g., multiple presence) It is noted that the selection of 10 m and 40 m for the above discussion is somehow arbitrary and limited As mentioned before more research is required; in fact, a much more extensive and comprehensive study was recently submitted for possible publication (García-Soto et al., 2018) and, if it is hopefully published, more interesting and far-reaching significant conclusions will be available CONCLUSIONS A discussion on the analysis of extremes of load effects generated on continuous multiple span bridges (two to five spans) by heavy trucks from a WIM database is given Even though new WIM data from 2017 is described for a brief comparison, the load effects are computed using information from previous recordings; nevertheless, such information had never been used before for load effects on continuous bridges with more than three spans, as it is the case in the present study The new WIM data from 2017 shows that traffic can be site specific, which will impact the development of future live load models for bridge design, as well as reliability studies and code calibration tasks, but it is currently being investigated and will be reported in future works For the studied cases, it is found that the number of selected points for the extreme value analysis is very likely to have an impact in the extrapolation of load effects, and in turn in any reliability analysis and live load model development, which could be conducted for continuous bridges It was also found that the fitted values using the Gumbel distribution for double spans, are also very adequate to fit the maximum 29-10 bending moments for triple, quadruple and fivefold continuous bridges, which could be advantageous for simplifying the development of live load models and reliability studies, since the checking of the double span case, will simultaneously cover those of more spans just as well However, while this is the case for equally spaced spans of 40 m, it may not be so for other cases or other load effects Therefore, i t is concluded that when carrying out an extreme analysis for short and medium span continuous bridges, attention should be paid to the differences due to span length and the considered load effect Further studies are recommended to inspect this it was observed that when an unsymmetrically spaced continuous bridge is considered, loads effects values exhibit a behavior in between those of equally spaced span bridges, with lengths corresponding to the minimum and the maximum spans of the unsymmetrical bridge It is concluded that further research is desirable for more cases of non-equally spanned continuous bridges Acknowledgements The financial support by the National Council of Science and Technology of Mexico, CONACYT (Project “Problemas Nacionales 2014” No 248162), and by Universidad de Guanajuato is gratefully acknowledged We are thankful to the Technical Committee and to two anonymous reviewers for their comments and suggestions which helped to improve this study References Anitori, G., Casas J.R., and Ghosn M 2017 WIM-based live-load model for advanced analysis of simply supported short- and medium-span highway bridges Journal of Bridge Engineering, ASCE, 22(10): 111 Castillo, E., Hadi, A S., Balakrishnan, N., and Sarabia, J M 2005 Extreme value and related models in engineering and science applications John Wiley & Sons, New York, NY, USA Eamon, C.D., Kamjoo V., and Shinki K 2016 Design live load factor calibration for Michigan highway bridges Journal of Bridge Engineering, ASCE, 21(6): 1-14 García-Soto A.D., Gómez R and Hong H.P 2010 Basis for Truck Load Model for Bridge Design in Mexico 8th International Conference on Short & Medium Span Bridges 2010, Canadian Society for Civil Engineering, Niagara Falls, Ontario, Canada 10 p paper in CD proceedings García-Soto, A.D., Hernández-Martínez A., and Valdés-Vázquez J.G 2015 Probabilistic assessment of a design truck model and live load factor from weigh-in-motion data for Mexican Highway bridge design Canadian Journal of Civil Engineering, 42: 970-974 García-Soto, A.D., Hernández-Martínez A., and Valdés-Vázquez J.G 2018 Probabilistic assessment of live load effects on continuous span bridges with regular and irregular configurations and its design Implications Canadian Journal of Civil Engineering, (submitted) Laurendeau, M., Barr P J., Higgs A., Halling M.W., Maguire M., and Fausett R W 2015 Live load response of a 65-year-old Pratt truss bridge Journal of Performance of Constructed Facilities, ASCE, 29(6): 1-9 Liu, Y., Zhang H., Deng Y., and Jiang N 2017 Effect of live load on simply supported bridges under a random traffic flow based on weigh-in-motion data Advances in Structural Engineering, 20(5): 1-15 Oudah, F., Norlander G., and El-Hacha F 2017 Live load factor and load combination for bridge systems conveying extremely heavy mining trucks Journal of Bridge Engineering, ASCE, 22(4): 1-13 Rascón, O A (1999) Modelo de cargas vivas vehiculares para diseño estructural de puentes en México Instituto Mexicano del Transporte y Secretaria de Comunicaciones y Transportes, publicación técnica 29-11 No 118, Sanfandila, Qro (in Spanish) Soriano, F., Casas J.R., and Ghosn M 2017 Simplified probabilistic model for maximum traffic load from weigh-in-motion data Structure and Infrastructure Engineering, Maintenance, Management, LifeCycle Design and Performance, 13(4): 454 - 467 Stuart, C 2001 An introduction to Statistical Modeling of Extreme Values Springer-Verlag, London, Berlin, Heildelberg Tapia-Hernández, E., Perea T., and Islas-Mendoza M.A 2017 Design assessment of short-span steel bridges Int J Civ Eng, 15(2): 319-332 29-12 ... differences of continuous bridges with equally spaced spans in relation to continuous bridges with unsymmetrical configurations (i.e., non-equally spaced spans) For this purpose, a four -span continuous. .. ANALYSIS OF LIVE LOADS ON CONTINOUOS BRIDGES WIM data in Mexico and selected bridges As a previous step to carry out the structural analysis to obtain load effects on bridges, the input data must... analysis of extremes of load effects generated on continuous multiple span bridges (two to five spans) by heavy trucks from a WIM database is given Even though new WIM data from 2017 is described