Bridge rating modification to incorporate fatigue damage from truck overloads

Abstract

As the number of requests for permits to use bridges by heavy trucks increases, there is a concern over the potential for rapid fatigue damage to the affected bridges. The damage may especially become critical for bridges that have been designed for lower truck loads than those currently used in practice. To limit damage, and to indirectly impose a cap on the number of overload permits, a modification factor in the bridge rating equation to incorporate the fatigue damage potentials from overloads is introduced. This factor is derived based on the amount of damage that a specific overload may cause and is found to be mainly affected by the current bridge age and the percentage of overloads in the entire truck load population. For single and continuous span steel girder bridges with welded cover plates, simple forms of this modified equation have been proposed and introduced in the paper.

Keywords

Load rating fatigue damage truck overloads special permits steel bridge

1 Introduction

1.1 Truck load population and overloads

For a given class of trucks, the intensity and frequency of load occurrences in the population vary to a great extent. Load populations mostly exhibit an inconsistent pattern - often with two or more distinct peaks. The reason being attributed to the variety of loads in the population; and in most cases, the appearance of two peaks may be because of a combination of loaded and empty trucks in the mix. Furthermore, load data also contain overload frequencies that may be significant in some cases resulting in multiple peaks in the population. It is noted that the term overload, as used here, refers to truck weights in excess of the 356-kN (80-kip) limit. Studies on the effect of truck load on highway systems have shown that truckload spectra often contain occurrences of overloads that in certain cases may actually be rather frequent [1 –3]. Specific to 5-axle trucks, truck load data gathered in several stations in Illinois, using the weigh-in-motion (WIM) system, indicates that the percentage of overloads can be as high as 10% of the total population. Figure 1 shows the distribution of truck load data, designated as the gross vehicle weight (GVW), for three stations in Illinois. As seen in these data sets, the distributions appear with two peaks with some loads in the population in excess of the 356-kN (80-kip) limit.

Fig.1

Distribution of gross vehicle weight (GVW) for three stations in Illinois.

For a given bridge, a few passages of overloads are not expected to contribute significantly to fatigue damage. However with frequent applications of overloads on a given bridge, one may want to have a closer look at the overloads and their damage potentials and perhaps impose a limit on the number of overload trucks that use the bridge. Since overload trucks use a given bridge through permits and a detailed bridge rating analysis, a method to limit fatigue damage from overloads would be to impose a limit on the number of permits issued for overload trucks. In this paper, a modification in the bridge rating equation is proposed in an effort to account for the fatigue damage effects from overloads and as a means of limiting the number of overload permits. A major step in estimating the fatigue damage potentials of overloads is to develop models that can be used to mathematically describe the load distribution on a bridge including the effects of overloads. Using data from several WIM stations, mixed-type probability distribution models were obtained and used in the investigation of the fatigue damage potentials of overloads. However, this paper only focuses on describing the underlying concept and the process used in developing the proposed method for bridge rating modification.

1.2 Potential for fatigue damage

Fatigue damage in steel bridges appears as cracks in structural components and grows with repeated load applications. In load populations where overloads make up a sizeable proportion, fatigue cracks grow more rapidly with an increase in the intensity and the frequency of overload applications. As the number of requests for permits to use bridges by heavy trucks increases, there is a concern over the potential for rapid damage to bridges. This is especially true if the number of such permits is issued frequently. The fatigue damage may especially become critical for bridges that have been designed for lower truck loads than those currently used in practice. Examples of these include older bridges on secondary roads or feeder ramps, which may be along the road networks utilized by overload trucks. These bridges may have a shorter life than expected and thus become subject to failure, if the loading spectra contain many occurrences of overloads. To limit fatigue damage, and to indirectly impose a cap on the number of overload trucks, a modification factor in the bridge rating equation to incorporate the fatigue damage potentials from overloads is introduced. This factor is derived based on the amount of damage that a specific overload may cause and is found to be mainly affected by the current bridge age and the percentage of overloads in the entire load population on a given bridge. A steel girder bridge with welded cover plates is selected from the fatigue detail category in AASHTO specification which is category E since the detail is the most critical among the category and was broadly used for the existing steel girder bridges. Therefore, for single and continuous span steel girder bridges with welded cover plates, simple forms of this modification equation have been proposed and introduced in the paper. Illustrative examples are presented demonstrating how the modified bridge rating can be effective in modifying the bridge rating values depending on the current bridge age and expected percentage of overloads in the truck load population.

2 Modification of bridge rating – model description

2.1 An overview of bridge rating equation

The bridge load rating is used to determine the capability of a component or connection in carrying the truck load. The current load rating equation using the Load and Resistance Factor Rating (LRFR) method as given in the AASHTO’s Manual for Bridge Evaluation [4] is: $RF = \frac{φ R_{n} - γ_{DC} DC - γ_{DW} DW}{γ_{L} LL (1 + IM)}$ (1) in which φ= capacity reduction factor; R_n = nominal capacity corresponding to the type of load for which the rating is being conducted; γ _DC = dead load factor for structural components and attachments; γ _DW = dead load factor for wearing surfaces and utilities; γ _L = live load factor for the rating vehicle; DC = dead load effect due to structural components and attachments; D W = dead load effect due to wearing surfaces and utilities; L L = live load effect due to the rating vehicle; and I M = impact factor (dynamic load allowance). For the purpose of the permit-load rating, the rating factor represents the safety and serviceability of the bridge by evaluating the effect of passage of the overload truck. Generally, unless otherwise specified, a rating of greater than 1 is considered satisfactory.

2.2 An overview of fatigue damage potentials of overloads in steel girder bridges

As it is well known, fatigue damage in steel structural members of a bridge is caused by the cyclic stress ranges developed as a result of periodic truck load applications on the bridge. As expected, the damage is affected by (1) the intensity; and (2) the frequency of stress ranges. Steel girder splices and sections containing cover plates are especially vulnerable to fatigue damage. In a given bridge, as explained earlier, the load population is a mix of various load intensities; and often the population contains overloads. In cases where the overloads happen infrequently, the accumulated damage is not expected to be significant. However, as overload frequencies increase, and especially at higher loads, the damage potentials of overloads may become significant. In this paper, we are especially focusing on the damage that can occur to bridges in cases where overloads become a frequent occurrence.

To compute the fatigue damage from overloads, the Miner rule is used, which defines damage as: $D = \sum_{i = 1}^{k} \frac{n_{i}}{N_{i}}$ (2) in which D = total damage accumulated; k = the number of stress ranges developed from the load population on a given bridge; n_i = the number of applied cycles for i^th stress range; and N_i = the total number of cycles to failure of the i^th stress range. The value of N_i for the i^th stress range is defined by the S-N relation (which is component-dependent): $N = \frac{C}{S^{m}}$ (3) in which N = the number of cycles to failure corresponding to stress range S; and C and m = constants specific to the structural detail (see for example, Ang & Munse [5]). The computation of the stress range, at a fatigue-critical location, requires a structural analysis of the bridge when it is subjected to the truck axle loads. It is noted that the Miner damage assumes a constant load amplitude and a linear formulation for the computation of fatigue damage. These are idealizations that are not consistent with the actual situations. Our formulation uses the Miner damage rule in a probabilistic formulation in which the variations in the stress range, fatigue damage and S-N relationship are considered. Furthermore, with the probabilistic approach, a distribution model is considered which allows the randomness in the number of stress range cycles, thereby, treating each stress range with a different number of corresponding cycles.

Among various kinds of fatigue-critical components, our study is focusing on welded cover plates. When fatigue damage because of overloads becomes significant, the service life of the affected bridge will shorten. This in turn may require costly expenses associated with premature aging of the bridge. As a preventive measure, and to limit fatigue damage from overloads, one needs to consider imposing a limit on the number of overload permits for the bridge. Alternatively, a modification in the rating equation may be considered to account for the damaging effect of overloads. This modification is expected to result in a lower rating value than otherwise is obtained when no modification is used. And as such, the lower ratings will result in certain number of overload permits becoming disqualified, which in turn will provide for an indirect limit on the number of overload permits. The proposed modification rating is explained in the next section.

2.3 Modified bridge rating equation

The modification of the bridge rating equation in proposed in the following equation using the Load and Resistance Factors Rating (LRFR) philosophy, $RF = \frac{φ R_{n} - γ_{DC} DC - γ_{DW} DW}{γ_{L} LL (1 + IM) (1 + α_{FLE})}$ (4) in which α _FLE = a modification factor to account for the significance of the fatigue damage potentials of the overloads. In essence, this modification factor is related to fatigue life expended (FLE), which can be estimated by using Equations 2 and 3. And as such, the factor is mainly affected by (1) the type of fatigue-critical component; (2) stresses ranges from overloads and number of stress cycles; and (3) the damage that has already accumulated in the component. The latter is directly related to the current age of the bridge through a series of factors which describe the volume of truck traffic, traffic growth rate, etc. It is emphasized that Equation 4 is the form suggested for rating following the LRFR philosophy. However, in finding the estimate for the modification factor, we use the rating for the live load without the load factors as explained later.

The Manual for Bridge Evaluation [4, 6] provides an equation that can be used for estimating the remaining life of a bridge in terms of these parameters. In essence this equation uses the fatigue damage as a means of estimating the remaining life. The estimated remaining fatigue life can then be used as a convenient method for estimating α _FLE . The AASHTO equation for the estimation of the finite fatigue life of bridge is given below:

$\begin{matrix} Y = \\ \frac{log [\frac{R_{R} A}{365 n [(ADTT)_{SL}]_{PRESENT} [(Δ f)_{eff}]^{3}} g (1 + g)^{a - 1} + 1]}{log (1 + g)} \end{matrix}$ (5)

in which Y = finite fatigue life of bridge, a = current age of bridge; g = annual traffic volume growth rate; [(ADTT) _SL ] _PRESENT = present average number of trucks per day in single lane; R_R = resistance factor; A = fatigue strength coefficient; n = number of stress cycles per truck crossing; and (Δf) _eff = effective stress range at the fatigue-critical component. The remaining fatigue life of bridge can be determined by subtracting the current age from the finite fatigue life (i.e., Y-a). The following paragraph summarizes the procedure in estimating α _FLE . It is emphasized that the procedure we have used in estimating α _FLE is bridge-specific – although it may also be used for a group of similar bridges. Our effort in developing a procedure that can be used more universally for a larger group of bridges continues.

The underlying concept for estimating the α _FLE is that, conceivably, this modification factor should result in a smaller rating factor, for an overload such as P, than otherwise computed without the modification factor. However, in computing the rating factor for the purpose of quantifying the modification factor, we need to use the loads without load factors. This is because fatigue damage is obtained for a fatigue limit state, which corresponds to a service live load (i.e., the live load as is). If the rating without the modification factor is RF, the one with modification will be RF’, where RF’ < RF. This inequality suggests that if modification factor is not used, then RF’ would be corresponding to a larger load (say P’), which, if applied, produces a larger stress range in the fatigue-critical component. To obtain this larger stress range, we will need the amount of damage that the overload P is causing. This can be decided based on the amount of fatigue damage that we would like to allow for the affected component. Since fatigue damage is related to the remaining bridge life, instead of selecting a limit for fatigue damage, we can select how much we would like to allow the remaining life to be shortened. For example, we decide that we would like to limit the total reduction in fatigue life to no more than 10% of the total life. This reduction will result in a modified value for Y. Then using this modified value in Equation 5, and knowing the characteristics of the critical component, truck traffic on the bridge, traffic growth, etc., the value of the stress range (Δf)_eff that is corresponding to the modified Y is computed. This stress range corresponds to the load P’. Now if we consider the RF and RF’ are equal, the value of α_FLE can be computed from the relationship between {RF and {RF’ in terms of the live load effect. Figure 2 shows the procedure for finding the one of alpha_FLE as a flowchart. This process is then continued for a host of overload values and different levels of fatigue damage allowed to establish a larger data base, and a more simplified equation, for determining α_FLE for practical use.∥The selection of the limit for shortened remaining life, which can be allowed for a bridge, can be made based on how much fatigue damage overloads induce in a given bridge or in a group of similar bridges. Historical data and truck load populations, such those compiled for Illinois WIM data reported in this paper, can be used for this purpose.

Fig.2

Procedure for finding the one of α_FLE.

2.4 Selection of parameters affecting α_FLE

The data base generated for the modification factor can be used in identifying key parameters that would affect it most. In generating the data base, our study looked into significance of such parameters as the type of structural details, truck traffic, the distribution of traffic to roadway lanes, traffic growth, span length, the current age of bridge, and the type of bridge (redundant or non-redundant system). Among these, and specifically for single-span bridges with welded cover plates in their steel girders, it was found that the current age of bridge, the gross vehicle weight (GVW), and the percentage of overloads in the truck load population are affecting α _FLE significantly and indeed can be used as those in a proposed equation for the modification factor. Specific to the data compiled for truck weights as reported in this study and as an example, Figure 3 is developed, which shows the variation of α _FLE in terms of GVW and bridge current age for three WIM data sets in Illinois. Further investigations into the role of the three parameters indicated that statistical equations for α _FLE can be developed using GVW and the percentage of the overloads in the truck load population as independent variables. However, different equations need to be developed for different possible values of the bridge current age to make the equation applicable for a wider variety of bridge types.∥In a generic form, the equation for α _FLE can be written as: $α_{FLE} = f (a, OL)$ (6) in which a = current age of bridge and OL = percentage of overloads in the truck load population. The value of current age of bridge in generating the modification factor is used as a percentage, for example, 0.01 and 1.0 are representing the 1 year and 100 years of fatigue life of bridge, respectively. Based on the values generated for α _FLE, a bilinear model was found as a suitable function to represent α _FLE. Figure 4 shows the bilinear model for several OL values. The location of the common point for each case is selected based on the data fit and trend in the values of α _FLE. As an example, for OL = 0.10, Equation 6 will be split into two as summarized below with the common point at a = 0.63.

Fig.3

Variation of modification factor, α _FLE, with the current age of bridge and the gross vehicle weight.

Fig.4

The bilinear model for α _FLE for several OL values.

$α_{FLE} = 0.3183 \times a - 0.0022 for a \leq 0.63$ (7a) $α_{FLE} = 1.8526 \times a - 0.9651 for a > 0.63$ (7b)

3 Limitations of model

3.1 Limits on parameters of rating equation

The equation proposed for α _FLE necessitates imposing certain limitations on the two independent variables. These limits are needed to avoid obtaining unreasonable results and are explained below.

The current age of the bridge is 1 year as a minimum and 100 years as maximum (which is considered an accepted fatigue life for steel girder bridges). The 100-year upper limit is adopted from AASHTO [4] and Bowman et al. [6], where the maximum fatigue life of a bridge is stipulated as 100 years as the “Fatigue Serviceability Index” is introduced in the Manual for Bridge Evaluation.

For the percentage of overloads in the truck load population, a lower bound equal to one percent; and an upper bound equal to 30% are used. In analyzing truck load population from several sets of WIM data in Illinois and Michigan, we found the maximum percentage of overloads in the truck load population was less than 11%. We believe 30% presents a reasonable upper bound value for this variable. Higher percentages of overloads are unlikely since they severely damage bridges.

When values of α _FLE are generated by simulation, depending on the bridge age and overloads applied, mathematically negative values for the modification factor may be obtained. Thus a lower bound for α _FLE is selected at zero.

3.2 Limitations in the α_FLE equations

The model equations proposed for α _FLE are based on typical overload populations observed in WIM data analyzed in this study. Furthermore, the trends in α _FLE were investigated for a limited class of bridges, i.e., those with single spans and welded cover plates. Thus the proposed equations (Equations 6 and 7) are limited to bridges with these conditions.

In our model, when computing the damage already accumulated in a component, the load history is assumed to be the same as the current one. This is an approximation since the past load history and truck traffic growth rate in previous years are different from those in the current data. Using an accurate estimate of the past history of truck traffic on a given bridge is indeed challenging in the sense that the exact nature of the load history may not accurately be known. Approximate methods have been considered by the authors by trying to adjust the past truck load history based on the design load and the known values for traffic growth in the past. This method may offer a more reasonable estimate of the past load history yet still has approximations. However, we emphasize that the objective of using the modified rating value is to have a comparative measure to assess the condition of one bridge versus the other. And as such, we believe in this sense using some reasonable method to incorporate the past history of truckload in the formulation should be acceptable as long as the method is consistently used across all cases. Nevertheless, it is imperative to research for a more accurate method of using the past load history in the formulation.

4 Illustrative examples and discussions

4.1 Examples

For illustrative purposes, a simply supported steel girder bridge with a welded cover plate (Category E in fatigue details) is considered. The span length of the bridge is 21.34 m (70 ft). Two different ADTT cases are considered in this example. The other parameters used in this example are: a = 0.65 (65 years), ADTT1 = 2623 trucks/day, OL1 = 0.105 (10.5 %), ADTT2 = 3182 trucks/day, OL2 = 0.034 (3.4 %), φ= 0.95, γ _DC = 1.25, γ _DW = 1.5, γ _L1 = 1.441 (for ADTT1), γ _L2 = 1.455 (for ADTT2), and I M = 1.2 (for minor surface deviations). In this study, these parameters were used for rating with and without the modification factor. The example bridge was analyzed using the 5-axle truck configurations for illustrative purposes only. Structural analyses were conducted to obtain the bending moment and the corresponding stress range at the critical component in the steel girders. The modification factor, α _FLE , was determined from the proposed function and then applied to the modified rating equation. Figure 5 presents the results for the two cases of ADTT and OL values.

Fig.5

Values of rating factor for the example bridge.

4.2 Discussion

The values of the modification factor, α _FLE , were determined to be 0.2492 for case 1 and 0.0758 for case 2. In this example bridge, the rating values are satisfactory without the modification factor if the gross vehicle weight is less than 645-kN (145-kip) and 534-kN (120-kip) for case 1 and 2, respectively. However, the rating values with considerations for the fatigue damage result in less than 1 for overloads in at 556-kN (125-kip) and 512-kN (115-kip), which represent case 1 and 2, respectively. As is evident from Figure 5, using the modification factor has resulted in certain number of overloads be disallowed. This in turn indirectly imposes a limit on the number of permits.

4.3 Conclusions

The following presents the main conclusions from this study.

When the entire population of truck load in a given WIM station is analyzed, the distribution often appears with two distinct peaks indicating that a mixed distribution function is perhaps needed to represent the data.

Fatigue damage due to overloads may become significant, if no limits on overload permits are imposed.

A modification factor can be used in the rating equation as a means to control the number of permits for overloads.

For a class of bridges made up of single spans and steel girders with cover plates, the modification factor is mainly a function of the current age of bridge and the percentage of overloads in the entire truck load population.

Footnotes

Acknowledgments

The assistance provided by David C. Thompson of the Illinois and Thomas Foltz of the Michigan Department of Transportation, respectively, in making the WIM data available to the authors is appreciated.

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