Current fatigue load models (FLMs) for road bridges are typically calibrated based on steel and reinforced-concrete bridges. These models are not likely applicable to fiber-polymer composite bridges, primarily because the damage rate—defined as the slope of the fatigue load versus life curve of the composites—is significantly lower for composites than for steel. To address this limitation, a methodology is proposed to derive FLMs for composite road-bridge superstructures, and its application was demonstrated via a case study of a composite bridge deck. The adhesive deck joint was identified as the fatigue-critical location. Given the typically short spans of current composite bridges, a double-axle FLM was adopted, and the axle loads were calibrated using a composite road bridge that has been in service in Switzerland since 2012. Three different slopes of the F-N curve were assumed, all representative of composites. Calibration was based on a series of real traffic loads from the three countries, obtained from weigh-in-motion (WIM) measurements. The FLM axle loads were calibrated at specified numbers of cycles such that the damage generated by the FLM was equal to that in real traffic. Large axle loads were the predominant contributors to damage even at low cycle counts. The relationship between the axle loads and the number of cycles was established as a reference for testing across the three slopes. Characteristic axle loads increased with slope and decreased with cycle number. The results of this study represent an initial step toward developing an FLM for composite road bridge superstructures. Further calibration of bridges with varied geometries and composite materials is required to derive a generally applicable, parametrized model.
Fatigue failure typically initiates structural discontinuities and stress concentrations, such as the bonding interface between reinforcement bars and concrete in reinforced concrete bridges, or at bolted and welded connections in steel bridges. The fatigue assessment of reinforced concrete and steel bridges under traffic loads is based on fatigue load models (FLMs), as specified in EN 1991-2 (Eurocode 1) [1], American Association of State Highway and Transportation Officials (AASHTO) Load and Resistance Factor Design (LRFD) Bridge Design Specifications [2], and Swiss Society of Engineers and Architects (SIA) 261 [3]. These FLMs were calibrated based on the principle of equivalent damage, wherein the damage induced by the FLMs equals to that caused by real traffic. The cumulative damage was calculated based on the hypothesis of linear damage accumulation proposed by Palmgren and Miner—commonly referred to as the Palmgren-Miner rule—in conjunction with the stress-fatigue life (S-N) curve. The classical power-law relationship expressed in Eq. (1) is typically used to characterize S-N curves of structural steel components (within the range of 104 to 5 × 106 cycles):
where N denotes the number of cycles to failure, C is a constant representing the influence of the structural detail, Δσ is the constant amplitude stress range, and m is the slope coefficient of the S-N curve in a double-logarithmic scale.
In the calibration of FLMs presented in EN 1991-2 [1], the relevant S-N curve with a slope coefficient m = 3 or 5 [4] was selected for steel bridges. Traffic load measurements of 1986 on the Auxerre motorway in France—among the heaviest trafficked roads in Europe at the time—were used. Five different FLMs (FLM1-FLM5) were defined in Ref. [1] for various applications. Among these, FLM3 is widely employed to assess fatigue in most European countries. It comprises a predefined set of axles with a scaling factor, the so-called damage equivalent factor, to consider bridge- and traffic-specific effects, including the critical influence length [5]. Furthermore, the axle loads were multiplied by a dynamic amplification factor of 1.2. The model was calibrated by equating the damage produced by FLM3 during a specified number of cycles (two million) to the cumulative damage generated by real traffic over the design service life (100 years). Current investigations on FLMs focus primarily on modifications of the damage equivalent factor [6], [7], [8], [9] for the application of FLM3 and development of a new FLM [10], [11], [12], [13] to improve the accuracy of the load model.
Fiber-polymer composite bridge decks have attracted significant attention owing to their high strength-to-weight ratio, excellent fatigue and corrosion resistance, and low-cost transportation and construction [14,15]. These composite decks are designed to provide both transverse load-carrying capacity and longitudinal top-chord function for the girders, with composite deck-to-girder connections ensuring full composite action and demonstrating excellent fatigue resistance. Fatigue damage typically initiates within composite decks, particularly at adhesive bond lines in deck connections or web flange junctions [14,16]. However, current FLMs cannot be applied for fatigue assessment of composite bridges because the S-N curve of composites is significantly flatter than that of steel, with the slope coefficient m varying from 10 to 20. The S-N curve slope of the composites depends on several factors, including material composition—such as fibers, polymers, and core materials in sandwich panels; loading conditions, such as the stress ratio and loading history; and environmental effects, such as temperature and moisture [17], [18], [19], [20], [21], [22]. For instance, carbon composites exhibit a flatter S-N curve than glass composites [20], the fatigue life of flexible adhesives decreases more rapidly than that of stiff adhesives [19], and the slope coefficient increases as the stress ratio increases [21].
Currently, no FLM has been developed for the fatigue assessment of composite road bridges that calculates the load action effects on bridge members or joints as defined in Ref. [23]. Therefore, the present study first aims to establish a general methodology for deriving an FLM for composite bridges, incorporating the specificities of composites, such as variable material compositions and damage rates. Second, the methodology is applied in a case study to develop an FLM for a specific composite bridge. The paper is organized as follows: Section 2 presents the proposed methodology for calibrating an FLM. Section 3 introduces the weight-in-motion (WIM) database used for traffic input. Section 4 applies the methodology to calibrate the FLM through a detailed case study of a composite bridge. Section 5 discusses the findings and their limitations, followed by conclusions in Section 6.
2. Methodology of calibration of FLM
A methodology for the calibration of an FLM for composite road bridges was developed, consistent with the calibration of FLM3 [24,25], as shown in the flowchart of Fig. 1. The process begins with the selection of a composite bridge and the identification of the critical location where damage or failure can potentially initiate. This identification was performed at the member or joint level according to Ref. [23]. The influence surface of the critical internal forces at this location was derived (e.g., bending moment). The flowchart is divided into two branches. On the left side, the equivalent internal force range (ΔFeq,WIM) generated by real traffic measurements is determined, whereas the internal force range (ΔFFLM) attributed to the proposed FLM with axle loads (Q) is obtained on the right side. Continuing on the left side, a series of real traffic loads is run across the influence surfaces to obtain the internal force (F) history, and the corresponding spectrum is derived by performing a cycle count employing the rainflow method [26]. For fatigue resistance, the slope of the F-N curve is either determined via fatigue testing at the structural member or joint level for composite bridges or estimated based on prior data and engineering judgment if experimental testing is not performed. The equivalent internal force range for a specified number of cycles (e.g., two million cycles or the total number of axles during the design service life) is computed based on the principle of equivalent damage, employing the linear damage accumulation hypothesis and the F-N curve. The internal force range generated by the FLM is obtained on the right side. Finally, the axle load of the FLM is calibrated by equating the internal force ranges resulting from the FLM with the equivalent force ranges generated by real traffic loads.
3. WIM database
3.1. Data selection
The traffic records used to calibrate the load model were obtained from WIM measurements of gross vehicle weights (GVWs), axle weights, number of axles, and axle spacing [27]. The gross vehicle and axle weights were calibrated to static weights in the WIM database. Representative WIM databases from motorways in three European countries were selected to calibrate the FLM, which varied in terms of recorded years, observation periods within each year, and overall traffic volume. An overview of the WIM databases is presented in Table 1 [5,6,9,10].
The traffic records from motorway A6 in Auxerre, France, in 1986 were selected because they were previously used to calibrate the Eurocode FLMs, and they contained heavy European continental traffic with the highest number of large axle weights recorded at that time. The motorway featured three lanes for each traffic direction, and the traffic in the slow and first fast lanes in one direction was recorded; the records for two days were considered representative [4]. Records from motorway A16 near Moerdijk in the Netherlands, near the Port of Rotterdam—one of Europe’s largest ports—were selected owing to the high prevalence of heavy loads. Similarly, the traffic of the slow and first fast lanes was recorded among three lanes in one traffic direction, with records spanning multiple periods, namely April 2008, April 2018, and January to October 2015. The accuracy of the WIM database was validated using strain gauge measurements of a bridge structure, as reported in Ref. [10]. The final set of traffic records was obtained from the two most heavily used motorways in Switzerland, and the measurements were continuously recorded for varying periods of 16-22 years. Records were collected from a section of the A1 motorway near Zurich, including both slow and fast lanes, in one direction from Oberbüren to Zurich. Additional data were selected from two sections of the A2 motorway: one near Monte Ceneri, where both slow and fast lane data were recorded from Monte Ceneri to Bellinzona, and the other near Gotthard, where data from only one lane was recorded from Gotthard to Wassen.
3.2. Filtering criteria
Traffic databases obtained from WIM measurements must be examined and filtered prior to their use in calibration because they can contain incorrect or inaccurate data [28,29]. Although the GVW or axle weight is limited to 500 and 12 kN, respectively, in most European countries [29], these values are frequently exceeded by permanent or special permissions and illegal heavy vehicles. Furthermore, the minimum axle spacing cannot be below 0.6 m based on the vehicle geometry, as described in Ref. [30]. Light vehicles with a total GVW of less than 35 kN or axle weight of less than 8.8 kN can also be excluded from the calibration because they do not significantly contribute to the cumulative damage [9,24]. Consequently, the raw traffic data measurements were filtered based on the following three criteria:
(1) 35 kN < GVW < 1000 kN;
(2) 8.8 kN < axle weight < 250 kN, axle weight < 150 kN, and axle weight < 200 kN for two- and three-axle vehicles;
(3) Axle spacing > 0.6 m.
The average daily traffic after filtering was calculated for all motorways as listed in Table 1. By comparing the resulting database with the databases in Refs. [9,10], the selection of the filtering criteria was validated by good agreement.
Traffic categories can be defined based on the number of heavy vehicles, Nobs (GVW > 100 kN) per year and per slow lane, as specified in Table 4.5 in Ref. [1]. Therefore, the traffic categories in Auxerre (France), Moerdijk (the Netherlands), and Switzerland were determined to be of ① Category 2 (Nobs > 0.5 × 106, roads and motorways with medium flow rates of lorries), ② Category 1 (Nobs > 2.0 × 106, roads and motorways with two or more lanes per direction with high flow rates of lorries), and ③ Category 2, respectively (Table 1).
3.3. Traffic composition
To characterize traffic composition, vehicles can be categorized according to the number of axles. Accordingly, the percentages of axles in the slow and fast lanes for the different motorways are shown in Fig. 2(a). Vehicles with up to eight axles were found, of which two-, four-, and five-axle vehicles were the three most common. Five-axle vehicles dominated all motorways except for the fast lanes in Switzerland, where two-axle vehicles were dominant. The proportion of two-axle vehicles was significantly higher in the fast lanes than in the slow lanes in all databases.
The cumulative distribution function (CDF) of GVW for all records is shown in Fig. 2(b) as a semi-logarithmic plot. The GVW of vehicles in slow lanes was significantly larger than that in fast lanes, particularly on motorways in Switzerland. The records from the Auxerre motorway contained the heaviest vehicles within the weight limit of 500 kN, whereas the Moerdijk motorway A16 in the Netherlands had heavier vehicles than the A1 and A2 motorways in Switzerland. In the slow lanes of the Auxerre motorway, a significant proportion (10.0%) of the recorded GVW exceeded the weight limit, whereas smaller fractions (1.0% and 0.1%) on the Moerdijk motorway and motorways in Switzerland exceeded this limit. However, the maximum GVW of vehicles on the Auxerre motorway was smaller than that on the Moerdijk motorway and motorways in Switzerland because of the increase in extremely heavy vehicles since the 1980s and also because of the short measurement period of Auxerre traffic.
A CDF plot of the axle weight is shown in Fig. 2(c). Similarly, the axle weight in the slow lanes was larger than in the fast lanes on all motorways, particularly on the motorways in Switzerland. The Auxerre motorway records had the largest axle weights within the weight limit of 12 t, whereas the proportion of axle weights in the slow lanes of the Moerdijk motorway was similar to that in Switzerland. A significant proportion (18.2%) of the axle weights on the Auxerre motorway exceeded the weight limit, whereas a smaller proportion (0.7%) on the Moerdijk motorway and motorways in Switzerland exceeded this limit. The maximum axle weight on the Auxerre motorway was lower than that in the latter case for the same reasons mentioned above.
A CDF plot of the axle spacing is shown in Fig. 2(d). No significant differences were observed between the slow and fast lanes. The majority (approximately 90%) of the axle spacings were less than 6 m, and the most frequent axle spacing was approximately 1.25 m, which corresponded to the axle spacing of the tandem- or tridem-axle group of vehicles. The maximum axle spacing on the Moerdijk motorway and motorways in Switzerland was significantly higher than that on the Auxerre motorway, reaching 16 m, for the same reasons.
Notably, only traffic measurements in the slow lane of the Auxerre motorway were used for the calibration of the FLMs in EN 1991-2 [1] owing to the high frequency of heavy traffic compared with that in the fast lane (Fig. 2(b)). Accordingly, the traffic records in slow lanes from the three countries were used in the following calibration.
4. Calibration of FLM
4.1. Proposed FLM
Because composite bridges currently have relatively small spans, a simple double-axle system was selected to represent the FLM with an axle load of Q and an axle spacing of 1.2 m, as shown in Fig. 3. The assumed geometry was similar to that specified for FLM1 or one of the two tandems of FLM3 in EN 1991-2. The selected axle spacing was close to the most frequently occurring axle spacing (1.2 m) observed in traffic records (Fig. 2(d)). Tire contact surfaces are not yet specified for composite bridge decks; they depend on the type of composite bridge deck (web-core or homogeneous-core decks) and surfacing. Guidelines for determining contact surfaces in specific cases are provided in Ref. [23]. In the following numerical analysis, point loads were assumed.
4.2. Composite bridge
4.2.1. Identification of critical locations
The two-lane Avançon Bridge, built in Switzerland in 2012, was selected for the FLM calibration. The 11.45 m span bridge is composed of a composite sandwich deck slab adhesively bonded onto two steel main girders [14]; the design service life is 100 years. The plan view of the bridge was simplified, that is, skew angles of 65° at the supports were neglected, as shown in Fig. 4. This modification did not significantly affect the internal forces at the critical point (see below). The composite deck, which spans over 7.50 m in the transverse direction, comprises three similarly sized sandwich panels (Panels 1-3) connected using adhesively bonded Z-shaped joints, that is butt joints in the core and scarf joints in the face sheets (Fig. S1(a) in Appendix A). Each sandwich panel had a thickness of 285 mm and was composed of two 22 mm thick glass composite face sheets and a 241 mm thick balsa wood core that provided shear resistance to the panel. The two main girders were fixed at both ends to two concrete cross beams, which were placed onto the stone abutment walls, and the transverse deck edges were supported by these cross beams [14]. Further information on the fiber architecture and mechanical properties of the composite components can be found in Ref. [14].
To validate the design of the transverse panel-to-panel adhesive joints, four-point bending experiments were performed on full-scale beams using the general setup shown in Fig. 4. The beam dimensions were 4.85 m in length, 0.6 m in width, and had the same depth as the bridge deck. The adhesive joint was arranged orthogonally to the beam axis at mid-span. The load versus deflection response at mid-span exhibited an almost linear behavior, with the ultimate failure load reaching 190 kN per loading point, at which fiber tear failure occurred in the face sheet laminates of the joint [14] (Fig. S1(b) in Appendix A). The static bending moment resistance of the joint (460 kN·m·m−1) was determined to be the critical sectional resistance rather than the shear resistance because the shear force reached only 54% of the shear resistance of the balsa core at beam failure [14]. Therefore, the FLM was calibrated for damage resulting from the bending moment across the joint. The mid-span location (point “A”) was selected where the bending moment reached its maximum value. However, any other location on the bridge deck was selected for calibration. The characteristic value (5% fractile) of the bending moment resistance (320 kN·m·m−1) was derived based on EN 1990 [33], assuming a coefficient of variation of 15% and considering that only one test was conducted for this configuration [14].
4.2.2. Moment vs number of cycles curve
To calculate the cumulative damage in the following analysis, the critical moment vs number of cycles to failure curves (M-N), according to Eq. (1), must be determined. Parameter C can be approximated using the static resistance [34,35], representing a “fatigue” experiment that sustained a half cycle with a moment amplitude corresponding to the static resistance, since the load ratio R in the lightweight composite bridge deck is close to zero. The design value for the fatigue resistance was derived to be 160 kN·m·m−1 by dividing the characteristic value by the partial factor of γMf = 2.0, as specified in Ref. [23]. Three representative values for the slope coefficient were further assumed, namely, m = 10, 15, and 20, representing the range observed in the experiments [17], [18], [19]. Therefore, the moment versus number of cycles to failure curves can be determined using three different slopes, as shown in Fig. 5.
4.2.3. Influence surface
As derived above, the transverse adhesive deck joint was identified as the critical location for the fatigue resistance, and the maximum response was identified to occur in the middle of the transverse joint area (i.e., point “A”) (Fig. 4). To obtain the influence surface of the bending moment at point “A” in the span direction, a bridge model was developed using SAP2000 software (Computers & Structures, Inc., USA), taking the orthotropy of the composite face sheets into account. Frame elements were used to model the steel girders, and shell elements were used to represent the sandwich panels, with the composite face sheets and balsa wood core represented by the layer definition in the shell elements. The material properties of each component were specified according to Ref. [14]. Joint restraints were defined to simulate the boundary conditions, as described in Section 4.2.1, that is restraints were applied to restrict the vertical translation of the transverse deck edges, and rigid constraints were applied to both ends of the two main girders. A moving load, with a unit value of “1,” was defined in the load case to obtain internal force responses. The resulting moment influence surface of point “A” was extracted after analysis in SAP2000 and is shown in Fig. 6(a).
4.3. Damage accumulation in bridge joint
The traffic was assumed to run in one lane of the bridge with its center on the lane axis (i.e., the steel girder axis), referring to the first loading pattern for the calibration of the traffic load model on bridges [36], as shown in Fig. 4. The variable transverse positions of the vehicle were not considered and the loads were assumed to be point loads (see above) to simplify the calibration. For the convenience of the following calculation of bending moment of point “A” based on the axle loads recorded by WIM, the moment influence line of point “A” is shown in Fig. 6(b) along the designated load path (on the center of one lane of the bridge), which was derived from the influence area by applying a set of unit axle loads. Therefore, the moment responses at point “A” were obtained by running the WIM-recorded axle loads with corresponding axle spacings for each vehicle across the influence line at the center of one lane, resulting in a series of variable moment histories. A MATLAB program (MathWorks, USA), was developed to perform these calculations by loading a series of traffic records containing axle load and spacing data. A representative moment response of point “A” generated by the traffic records on the Auxerre motorway is presented (Fig. S2 in Appendix A).
4.3.1. Cycle count
The rainflow method was used for cycle count to generate the moment range spectrum, which converted the variable moment responses into i equivalent sets of constant amplitude moments, ΔMi, with their respective number of cycles, ni. This process can be directly implemented in MATLAB using a signal-processing toolbox. The moment spectrums, that is the number of cycles (ni) versus the moment range (ΔMi), are shown in Fig. 7, which were obtained from the selected traffic records in three countries. Notably, the cycle count results were established for one year by replicating the daily and monthly results accordingly for the Auxerre and Moerdijk motorways. The moment range varied from 0.3 to 11 kN·m·m−1, and the spectrum generally reflected that of the axle loads, as high axle loads resulted in high moment responses owing to the short influence length. The distributions exhibited a significant decreasing trend with an increasing moment range because large axle loads were less frequent (Fig. 2(c)).
4.3.2. Damage distribution
The damage occurring in the panel-to-panel adhesive joints, Di, caused by the corresponding moment range per year, was obtained based on the relationship Di = ni/Ni, where Ni is the number of cycles at failure of this moment range, which was obtained from the M-N curve (Fig. 5). Notably, creep was not considered in damage accumulation since creep deformations, owing to traffic loads in composite bridges, normally recover during unloaded periods [23]. Therefore, the damage distributions generated by each cycle count of the moment range derived from the selected traffic records in the three countries (Fig. 7) are shown in Fig. 8, in which the results obtained from slope coefficients m = 10 and 20 are compared. The damage exhibited a contrary distribution compared with the moment spectra (Fig. 7), which demonstrates that larger axle loads contributed more to the damage despite a lower cycle count. This was more significant when the slope coefficient m was changed from 10 to 20. A steeper slope (i.e., m = 10) resulted in a substantially higher damage level for a similar moment range.
4.3.3. Cumulative damage
The cumulative damage Dtot was calculated based on the Palmgren-Miner rule, as shown in Eq. (2).
The cumulative damage for the three slopes of the M-N curves generated by traffic records over one year from different motorways is shown in Fig. 9. These values correspond to the summation of the bar heights shown in Fig. 8. They were used to quantify the damage level relative to the ultimate damage threshold of “1.” As indicated in Section 4.3.2, damage can initiate from either a small number of high-axle loads or from a larger number of light-axle loads (Fig. 8), and then propagate owing to subsequent traffic flows in real scenarios. The damage accumulated in adhesive joints followed three stages: an initial rapid increase, a steady increase in the second stage, and a final rapid increase until failure [34]. The largest annual damage was obtained from the measurements of the Auxerre motorway owing to the high number of large axle loads (Fig. 2(c)), even though the number of heavy vehicles (Nobs) (Table 1) was lower than those of the Moerdijk and Switzerland motorways. The damage resulting from the traffic records of the Moerdijk motorway was slightly higher on average than that of motorways in Switzerland because of the higher number of heavy vehicles (Fig. 2(b)) and higher frequency of high-axle loads (Fig. 2(c)). A flatter slope resulted in less cumulative damage in all cases. However, the cumulative damage level exhibited a larger variation at the flattest slope because the damage was determined more by large axle loads (Fig. 8(b)), which showed more differences among the countries.
Assuming a design service life of 100 years for the composite bridge, the mean value of total cumulative damage over this period generated by traffic records from the three countries for the three slopes was determined, as listed in Table 2. Among the motorways, the highest cumulative damage in the bridge joint was observed at the steepest slope for the Auxerre data because of the large number of large axle loads; however, the damage was still far below the threshold value of 1.0. The lowest cumulative damage was obtained at the flattest slope for the Swiss data because the total number of vehicles was lower than that of the Moerdijk motorway (Table 1).
4.4. Equivalent moment ranges from WIM databases
The equivalent moment range, ΔMeq,WIM, at a specified number of cycles, nspecified, obtained from real traffic records over the design service life of 100 years, was calculated using Eq. (3) based on the rule of equivalent damage [6].
Two million cycles were specified for FLM3 in Ref. [1] over the design service life (usually 100 years) and the FLM in Swiss code SIA 261 [3], that is, nspecified = 2 × 106, whereas the total number of vehicles, extrapolated from the average daily traffic number, over the design service life (75 years) was used for nspecified in AASHTO LRFD [2]. For comparison, two million cycles were used for calibration. However, the cycles that accounted for the total number of axles over the design service life were used because damage to composite bridge decks is caused locally by single-wheel loads in most cases (and the number of single-wheel loads is identical to the number of axles). The designed service life was 100 years for all three cases.
4.5. Moment range from FLM
The moment response of the FLM was obtained by running it across the influence surface (Fig. S3 in Appendix A). The moment range, ΔMFLM, was obtained from the different values between the maximum and minimum moment response (Fig. S3), and expressed as a function of the axle load (Q).
4.6. Determination of axle load of FLM
The axle load (Q) of the FLM for a specified number of cycles was determined by equating ΔMeq,WIM = ΔMFLM. The resulting axle loads of the FLM, derived from the traffic records of the different motorways, are shown in Fig. 10 for the three slopes of the M-N curves and the selected specified number of cycles, that is nspecified = 2 × 106 and nspecified = total number of axles over the design service life. Notably, the multiple data points shown in Fig. 10 represent the results of extrapolating one year of traffic measurements to a design service life of 100 years, with the daily and monthly results replicated accordingly for the Auxerre and Moerdijk motorways, based on data availability, as indicated in Section 4.3.1. This approach allowed for subsequent statistical analyses to determine the characteristic values of the axle load of the FLM.
The calibrated axle loads were almost constant for nspecified = 2 × 106 (Fig. 10(a)), whereas they increased with the slope coefficient when nspecified was considered as the total number of axles over the design service life (Fig. 10(b)). Substantially higher axle loads were obtained from the traffic records of the Auxerre motorway compared with the other databases, which was attributed to the high number of large axle loads in the former (Fig. 2). To obtain the characteristic values (i.e., 95% fractile) of the axle loads (Qk) [37], the probability density distributions of the axle loads Q of the FLM for the three slopes and two values for nspecified are derived in Fig. 11. The characteristic values of the axle loads for the FLM were determined from the fitted normal distributions (Fig. S4 in Appendix A). The relationships between the characteristic axle loads and the slope were linear for both cases. The FLMs determined for different slopes in both cases are shown in Fig. 12. The obtained characteristic axle loads were smaller than the axle loads from the recorded Auxerre data (Fig. 10) because of the high number of large axle loads in the latter case.
5. Discussion
5.1. Fatigue assessment
The fatigue design assessment of composite bridge structures can be performed at the structural member or joint level based on Eq. (4) [23].
where Ed is the design value of the moment in the joint caused by the FLM, γFf is the partial factor for the FLM, Qfat is the characteristic FLM axle load, Rf,k is the characteristic value of the fatigue resistance of the joint obtained from joint testing at a constant amplitude fatigue test load, and ƞc is a conversion factor to consider the effect of temperature and moisture on the material property [23].
The difficulty in this assessment is that, according to the derived fatigue load model, Qfat and Rf,k depends on the number of cycles that must be selected. This selection is primarily based on the number of cycles to be applied in testing (to obtain Rf,k), Qfat can then be determined accordingly. Owing to the detrimental fatigue-creep interaction effects existing in composites, particularly the dominance of creep under high-cycle fatigue [34,38] and the absence of a fatigue limit [39,40], the selected number of cycles for fatigue assessment must not be substantially small to avoid overestimating the fatigue resistance. To establish the relationship between the axle loads and the number of cycles, as a basis for testing, 5, 10, and 50 million cycles were selected to obtain the corresponding characteristic axle loads of the FLM. Therefore, the axle loads versus the selected number of cycles for the three slopes on a double-logarithm scale are shown in Fig. 13(a) and can be fitted by a classic power-law function. Furthermore, the axle loads versus the three slopes for different numbers of cycles are shown in Fig. 13(b); they exhibit an increase with an increase in the slope coefficient.
Therefore, fatigue assessment according to Eq. (4) can be obtained as follows:
(1) The slope of the F-N curve of composite bridges was first either experimentally determined or estimated based on prior knowledge.
(2) The number of testing cycles was determined based on the tradeoff between the allowable testing time and accuracy requirements.
(3) The axle loads of the FLM (Qfat) were determined based on the previously determined slope and selected number of testing cycles (Fig. 13(a)).
(4) The action effects—that is, internal forces—induced by the determined axle loads of the FLM were calculated at the critical fatigue bridge locations.
(5) The fatigue test setup and the fatigue test load were defined to reproduce the same action effects in the test specimen as those in the critical bridge location. The specifications for fatigue testing given in Ref. [23] were followed.
(6) The characteristic value of the fatigue resistance (Rf,k) was determined from the fatigue test results, as specified in Ref. [23], and Eq. (4) was validated.
For example, for a slope coefficient of m = 15 and selected five million cycles, the corresponding characteristic axle load of the FLM applied to the bridge is 115 kN (Fig. 13). According to the specifications in Ref. [23], three fatigue tests are required. Assuming a loading frequency of 2 Hz, the total duration required to complete the fatigue testing is approximately 90 d. The fatigue test load depended on the experimental setup, as described previously.
5.2. Representative data for calibration of the FLM
For economic reasons, the WIM databases used for calibrating FLMs [1], [2], [3] are typically extrapolated from short-period traffic measurements, which are considered representative of the design service life. One-year measurements in Ref. [2] and one-month measurements in Ref. [10] were considered representative of the FLM calibration for steel bridges; however, representative periods for composite bridges have not yet been evaluated. Therefore, the resulting axle loads using one-year databases were first evaluated and subsequently those for one-month databases. The axle loads obtained at two million cycles were selected for this evaluation because the largest variation was observed in this case (Fig. 10). The slope coefficient m was varied from 5 to 20 for comprehensive comparisons.
The axle loads obtained from the records by replicating one-year traffic measurements of the Ceneri (Monte Ceneri-Bellinzona) and Gotthard (Gotthard-Wassen) motorways in Switzerland were selected because of their long-period measurements (Table 1). The ratio of the resulting axle load to the average value obtained from the entire duration versus different slope coefficients is shown in Fig. 14(a). This difference increased with the slope coefficient; however, the maximum variation remained within 20% of the average value when the slope coefficient increased to 20.
The axle loads obtained from the records by replicating one-month traffic measurements were selected from the Moerdijk (2015, the Netherlands) and Gotthard (1999, Switzerland) motorways because significant differences were observed in these two years compared with others on the same motorways (Fig. 10); the corresponding comparison is shown in Fig. 14(b). Similarly, the difference increased with an increase in the slope coefficient. This variation remains within 20% when m = 5. However, it exhibited a larger scatter, approximately within 30%, when m varied from 10 to 20, which was greater than that obtained from the one-year measurements.
Considering a variation of 20% as a limit, a WIM database size of one month is not sufficient to represent long-term traffic to calibrate the FLM for composite bridges. Therefore, extending the database size to one year is recommended.
5.3. Limits of the calibrated FLM
The Palmgren-Miner linear damage accumulation rule can significantly over- or underestimate the fatigue life of composites [41], [42], [43]; consequently, it is not considered applicable for the composites in Ref. [23] because it does not consider the loading sequence effects in the loading spectrum. The effects of the loading sequence on the fatigue life are attributed to the varying residual strength levels at different load levels and the sensitivity of the composites to the loading history owing to their viscoelastic properties [44]. Investigations showed that a “high-low” sequence in a two-block loading is less damaging than a “low-high” sequence [41,42]. In addition, frequent transitions in the loading sequence exhibit significant damaging effects [45]. Consequently, nonlinear Miner damage accumulation models [42], residual strength models [41], and stiffness degradation models [46] have been developed to better consider the effect of the loading sequence on the fatigue life. However, these models were mostly validated and applicable to two-stage loading conditions, and were fitted to specific cases. For variable-amplitude fatigue loading, such as in the case examined in this study, involving strongly variable moment histories, these models are not applicable. Because no alternative exists for the calculation of cumulative damage, the Palmgren-Miner rule was applied in this study.
The FLM calibration was performed for one specific bridge with one specific bridge deck type; therefore, it required validation for other dimensions and deck types. In further steps, dynamic amplification factors and damage equivalent factors considering the effects of the span length, traffic category, and design service life must be derived, as they are available for concrete and steel bridges. The effects of the distributions of the transverse axle positions and tire patch shape also need to be addressed.
6. Conclusions
A methodology for deriving an FLM for composite bridges was proposed, and this methodology was applied to a specific bridge to demonstrate the process. A double-axle fatigue model was developed and calibrated using existing composite road bridges and full-scale experimental data. The transverse adhesive deck joint was identified as the critical location for fatigue resistance in the composite bridge. Three different slope coefficients m = 10, 15, and 20 of the M-N curves were assumed, and a series of real traffic loads from France, the Netherlands, and Switzerland were used to calibrate the model. This calibration was performed by defining the cumulative damage produced by the FLM at a specified number of cycles as equal to that generated by real traffic over the design service life. The conclusions of this study are summarized as follows.
(1) Traffic records from the Auxerre motorway in France contained a high number of large axle loads compared with motorways in the Netherlands and Switzerland, whereas traffic measurements on motorways in the Netherlands contained large numbers of heavy axles and heavy vehicles.
(2) The resulting damage was primarily caused by large axle loads, even with small cycle counts, and was relatively more significant with an increase in the slope coefficient. The maximum cumulative damage over the design service life was observed for the traffic records on the Auxerre motorway because of the high number of large axle loads, followed by the damage results from the Moerdijk motorway.
(3) The characteristic axle loads (i.e., 95% fractile) of the FLM for the different slope coefficients and number of cycles were determined from the probability density distributions of the axle loads from the three countries. These loads showed an increase with the slope coefficient of the M-N resistance curves and a decrease with the increasing number of cycles. The relationship between the axle loads and the number of cycles was established for the three slopes as a basis for testing.
(4) A WIM database size of one month is not sufficient to obtain a representative traffic spectrum for calibrating an FLM for composite bridges; therefore, extending the database size to one year is recommended.
The derived FLM is valid for this specific calibrated bridge. To further develop the FLM towards a generally valid parameterized model for composite bridges, other composite bridges with different geometries and deck types must be analyzed accordingly.
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