Abstract
Based on realistic modeling of 44 bridges in national bridge inventory using its actual layout and dimensions, a theoretical framework is defined to calculate the tolerable differential settlement limits based on the over-bending moment or over-shear force generated by settlements and the moment or shear capacity of the bridge. First, the derived over-bending moment or over-shear force is validated through literature results. Second, the differential settlement limits of bridges are derived from the bridge strength capacities. Finally, based on the established differential settlement limits, the relationship of bridge differential settlement limits with its span length, number of spans, bridge skew, and the bridge type is discussed and compared with American Association of State Highway and Transportation Officials specification (2010). A statistics-based differential settlement limit criterion is also suggested for steel and pre-stressed concrete bridges and compared with the field survey results. From the research outcomes, it is concluded that extra caution is needed when implementing the 0.4% differential settlement limits of bridge foundations in American Association of State Highway and Transportation Officials.
Keywords
Introduction
Excessive differential settlement of bridge foundations causes undesirable damage to the superstructure of continuous span bridges. Both strength and serviceability concerns can arise from differential settlements, depending on the bridge actual dimensions, which ranges from ride quality to structural distress and to the functionality and safety of the bridge. Types of damage caused by differential movement include the following: damage to abutments; damage to piers; distress in superstructure such as development of cracks; damage to railings, curbs, and sidewalks; damage to bearings, sheared anchor bolts, and cracking of concrete at the bearing; and poor riding quality (Moulton et al., 1985).
Foundation movement limits were first introduced to regulate the movement of building foundations. Skempton and MacDonald (1956) recorded the first settlement criteria for buildings from a survey of buildings with experienced settlements. Later, researchers noted that the established criteria for buildings were not applicable to bridges since the stiffness of buildings and bridges is different (Grover, 1978). Separate movement criteria would be needed for bridges. The early research mainly consisted of surveying damage associated with bridges that had experienced movements (Bozozuk, 1978; Walkinshaw, 1978). One of the most influential projects on movement criteria for highway bridges was completed for the Federal Highway Administration in 1985 (Moulton et al., 1985). An extensive field survey of 204 bridges that experienced movements provided the researchers with a database of movements and associated damages from the movements. The data were categorized by type of damage, type of movement, type of bridge, magnitude of movement, and whether the damage was considered to be tolerable or intolerable. The decision on whether the movement was considered to be tolerable or not was dependent on the personal judgment of the inspector. This method of determining tolerance to movement is subjective in nature and uniform movement criteria cannot be realized. However, the surveyed data provided the first-hand settlement effect and promoted the 0.4% foundation settlement limit in the specification.
Various types of analysis were also performed on the data collected in the project (Moulton et al., 1985). The influence of substructure variables on abutment and pier movements was checked, including soil condition, type of abutment, type of foundation, and height of approach embankment. The effect of settlement on bridge superstructures was also summarized with respect to different types of span, bridge construction material, abutment type, and different number of spans. And finally the tolerance of the bridge to the experienced movements was summarized with respect to each variable.
Wang et al. (2011) conducted a similar research project. In the project, a statistical analysis on the effect of foundation settlements on bridges was performed. A pool of data for 21 bridges was used to perform the statistical analysis. A reliability index was defined and calculated to study the effect of different factors on the reliability index. Deterministic methods were also adopted to study the settlement effect on the change in positive and negative moments due to settlement at different supports. For the statistical analysis, the following random variables were considered: settlement, dead load, live load, and material resistance in terms of shear and moment capacity. The limiting settlement to reach a reliability index of 3.5 was finally determined for different design scenarios.
A continuation of Moulton et al.’s work was conducted recently (Schopen, 2010). Moulton et al.’s work was reproduced and further expanded to establish settlement criteria at different conditions. However, the research considered only vertical differential settlements. The study was based on numerical modeling using a finite element package STAAD.PRO without verification. Girder types of continuous steel plate, pre-stressed I-girder, and rolled steel girders types were used. The settlement limits were computed by comparing the increase in bending moment due to differential settlement with the reserved capacity for 24 different bridges. It was found that the ratio between the section moment of inertia and span length used by Moulton et al. (1985) to establish a relationship between settlements induced stress and span stiffness was not as accurate as using the ratio of the section moment of inertia and the span length squared. Also the percentage of increase in positive bending moment at center support due to center support settlement is generally larger than that at the mid-span. Higher grade steel results in less increase in the moment since smaller sections are needed and smaller bending stiffness resulted. Settlement of exterior support was found to be more critical than that of the interior one. Therefore, negative bending moment determines the tolerable differential settlements. A methodology to design bridges taking into account the differential settlement-induced stresses was also proposed.
In this article, a realistic bridge model is adopted based on actual bridge layout and dimension. A logical procedure to find the differential settlement limits for each bridge is presented. The created bridge model is first validated through hand calculations and literature results, and then used to capture the differential settlement limits of 44 bridges through comparing its load demand over its capacity. The derived differential settlement limit database is tested to follow a log-normal distribution. Finally, statistics-based settlement criteria are suggested for different types of bridges, and the relationship of differential settlement limits with different bridge parameters is summarized and compared with American Association of State Highway and Transportation Officials (AASHTO) specifications (2010), which will provide an accurate differential settlement limit for bridges and a rational and scientific basis for these settlement limit articles in the AASHTO specifications (2010). It is worth to mention that typically settlement happens gradually and through creep of soil and superstructures. Creep of soil or gradually happened settlement does not change the mechanical responses or over-bending moment if a linear model is adopted. However, if the creep is induced in the superstructure, creep will reduce the stiffness of these superstructural elements, sometimes their capacities as well, which will lead to reduced over-bending moment and reduced moment capacity. It will increase the tolerable differential settlement limit only if the stiffness effect is considered; otherwise, it could increase or decrease the tolerable differential settlement limits depending on which degradation is more severe comparing the stiffness degradation and the capacity degradation. The same logic could be applied to slab cracking, but it needs to be mentioned that the effect of creep and slab cracking was not simulated in this article.
Methodology in calculating tolerable differential settlement limits of bridges
Steel and pre-stressed concrete bridges are modeled in the CSIBridge program based on their actual highway bridge plans, and the deterministic differential settlement limits are investigated for settlement of a support. The support associated with the shortest span of the bridge is chosen to settle since the shortest span will generate the most severe over-bending moment, based on the over-bending moment and I/L2 relationship. A subsequent analysis is then preformed to determine whether the girders have exceeded the allowable differential settlement limits for flexure and shear at the strength 1 limit states. It needs to point out that the span given here is the longitudinal span. No lateral span has been considered.
The CSiBridge Modeling Program (2012) is a parametric modeler where a layout line is defined along the length of the bridge, and actual bridge sections and properties are defined along this layout line. Support structures, such as piers or bents, are then added in along the layout line at the appropriate locations. Each span could be set to include any sections with parametric variations that follow linear, quadratic, or exponential variation paths. The concrete deck is modeled compositely with the girders by default, while staged construction can be used to simulate non-composite action. Steel reinforcing and pre-stressing strands are then specified after the bridge sections have been defined along the layout line. The steel tendon can be modeled inside or outside of the bridge sections. The self-weight of the components is calculated by the program using its specific weight, and the vehicle loading is applied in accordance with the AASHTO LRFD Bridge Design Specifications (2010).
The precast girders are typically made of high-strength concrete and are pre-tensioned with high-strength steel strands. The strands are modeled as element objects with pre-stressed loads calculated as forces acting along the tendon profile. The tendons are discretized into smaller segments with discretization points located at either end of the discretized segment. When the discretization point lies within a structural element, an interpolation constraint will connect that point to all joints in the element. The pre-stressing load is then converted into an equivalent strain and the strain load is transferred to the structure.
The pre-stressed concrete bridge superstructure is considered to consist of simple spans for dead load and continuous spans for live load. Staged construction load cases were used to model different span types and the loads applied to those spans. For the simply supported structure, the dead load effect of the girders and deck was computed using the stiffness of the non-composite pre-stressed girders. The concrete deck was modeled as having zero stiffness and its full self-weight. For the live load and the weight of the wearing surface, the composite stiffness of the girders and the deck was used to calculate its load effect. The main load cases considered in the modeling can be seen in Table 1.
Primary load cases considered in the modeling.
FWS: friction wearing surface.
The stiffness and loading scenario for each load case is modeled using staged construction analysis. Structural objects that contribute to the stiffness and loading calculations can be arranged to be inserted into the model as desired. For determining the stiffness of the composite section, the following process was followed. The substructure and bearing assignments are added into the model space and the girders are joined with their self-weight ignored first. The tendon elements are then superimposed and pre-stressing force is applied in separate stages to ensure that the pre-stress is applied to each girder separately. The self-weight of the girders is then added to the model. The concrete deck is overlaid with zero stiffness and its self-weight is added to simulate the effect of wet concrete. The stiffness of the concrete deck is then gradually increased to simulate the concrete hardening. The final process is to add continuity links at the joints to link the girders and deck together making a continuous structure. The loading cases that act on the composite section can now be added to the end of this sequence of construction events.
The steel superstructure is considered to have continuous non-composite spans for the self-weight of the girders and concrete deck. For the live load, barrier and future wearing surface load, the superstructure is continuous with a composite concrete deck.
The bearing condition at one of the abutments is considered to be unrestrained for rotations and restrained for linear displacements. The other abutment and intermediate piers are considered to be restrained in the transverse and vertical direction and unrestrained for rotations. Each girder has a set of bearing locations, resulting in double bearing locations at each pier.
The load combinations in the strength 1 limit states according to AASHTO LRFD Bridge Design Specifications (2010) are used to determine the demand loads on the bridge. The program creates an envelope of these load combinations to determine the maximum demands on the bridge. A representation of one concrete bridge from the finite element model is shown in Figure 1. The demand for an interior girder of one of the bridges modeled is shown in Figure 2.
FE model of example bridge 1 (1 in = 25.4 mm, 1 ft = 0.3048 m).
Moment envelope of demands with zero settlement (diamond stands for positive bending moment envelop and square stands for negative bending moment envelop, 1 in = 25.4 mm, 1 kip-in = 113.0 N-m).
Figure 2 shows the envelopes of the maximum and minimum moments from all the load cases. At any section, the controlling load combination is represented in the envelopes as shown in Figure 2. Please note that the abrupt change in moment at pier locations is due to the hyperstatic action of pre-stressing, since the tendon force will change its direction at each pier location and cause opposite additional rotation along the left and right spans (Aalami, 1998).
The maximum moment due to differential settlement always acts at a pier location. The moment from the settlement of a pier is linearly related to the amount of settlement for a particular bridge since a linear analysis is adopted in this study. This means only two settlement analysis cases are needed to find the maximum allowable settlement. The moment due to settlement is shown in Figure 3 for three different settlement values. From the maximum and minimum values on the graph, the linear relation of over-bending moment with settlement is clearly seen. Due to the continuous bridge layout, the maximum over-bending moment will be generated at the settlement location.
Over-moment due to settlement calculated by CSiBridge (1 in = 25.4 mm, 1 kip-in = 113.0 N-m).
The total factored moment envelope for the example bridge is discussed next to find the maximum differential settlement limits (Figure 4). It is found that the maximum positive bending moment is exceeded at the middle of span 1 for 1 in of settlement. A simple interpolation procedure is used to find the differential settlement limit.
Demand and resistance of an interior girder (1 in = 25.4 mm, 1 kip-in = 113.0 N-m).
The bending moment at the mid-span for 1 in of settlement is 38577.957 kip-in and 34418.98 kip-in for a half-inch settlement. The moment resistance at that location is 35404.8 kip-in. From a simple interpolation, the differential settlement limit for this example bridge could be calculated as
If the analysis is performed again with 0.62 in of settlement, the maximum moment at that location is 35,375 kip-in, closely reaching to the resistance of 35404.8 kip-in as expected. Please note that only the flexural moment capacity is checked in this article; however, this could expand to include the shear capacity and their interaction capacity checking.
Validation of the numerical bridge models
To validate the numerical bridge modeling, the results obtained from the program are compared to hand calculations and the results found in the literature. One of the bridges modeled in the NUTC-R237 project (MoDOT A3101) was adopted in this study, which is a two-span continuous bridge with each span of 120 ft. For each interior steel girder, the moment of inertia is 68,532 in4 from 0 to 82 ft and from 158 to 240 ft, and 116,536 in4 from 82 to 158 ft. Their modulus of elasticity is E = 29,000 ksi. The calculated settlement-induced moment from the NUTC-R237 report, and the simulated results from the CSiBridge software are compared with each other in Figure 5.
Validation of settlement-induced forces (1 in = 25.4 mm, 1 kip-in = 113.0 N-m, 1 kip = 4448 N).
Figure 5 shows good agreement of the forces in the girders from the settlement. Validation is also conducted for a bridge with three spans (Figure 6). The bridge is included in the Federal Highway Administration (FHWA) LRFD design specification as an example with all the parameters listed (Grubb and Schmidt, 2012). A comparison of the dead load effects and live load effects in a three-span bridge is plotted in Figures 7 and 8 respectively. Please be aware that CSiBridge gives less moment at mid-spans and piers (about 10%) because the optional live load of the second truck was not considered in CSiBridge.
FE model of example bridge 3(1 in = 25.4 mm, 1 ft = 0.3048 m).
Validation of the bending moment generated by the dead load only (1 ft = 0.3048 m, 1 kip-ft = 1356.0 N-m, 1 kip = 4448 N).
Validation of the bending moment generated by the live load only (1 ft = 0.3048 m, 1 kip-ft = 1356.0 N-m, 1 kip = 4448 N).
Validation is also conducted on the capacity calculation in the CSiBridge program with the manual calculation. The reinforcement information of the bridge deck could be found through Grubb and Schmidt (2012) and its section properties are calculated and compared with the properties predicted by the CSIBridge program. Moment capacities are also calculated following AASHTO LRFD Bridge Design Specifications (2010) and compared with the same program. Tables 2 and 3 show the comparisons between the manual calculations and the CSiBridge calculations of the section properties and moment capacities, respectively.
Section properties (1 in. = 25.4 mm).
SncTop is the elastic section modulus of the bridge portion above the neutral axis, while SncBot is the elastic section modulus of the bridge portion below the neutral axis.
Moment capacities (1 kip-ft = 1356.0 N-m).
The differences in the section property calculations are always negligible, and the difference in bending moment capacity ranges from 0% to 10%. With the agreeable results found in the validation study, the confidence in the bridge modeling and loading process used in CSiBridge is warranted.
Analysis of bridge settlement data
Based on the validated bridge modeling tool, the data collected from the numerical modeling of bridges in this study will be used to study the effect of bridge parameters on its differential settlement limit. The parameters reported to have influence on the tolerable settlement limit include the following: span length, magnitude of skew (defined as the angle between the girder line and the abutment line), number of spans, girder type, and the material type the girder is made of. A total of 44 bridges from the national bridge inventory database (FHWA, 2016) are simulated, and their layout dimensions are shown in Table 4.
Simulated bridge cases selected from national bridge inventory (1 ft = 0.3048 m, 1 in = 25.4 mm).
The current AASHTO LRFD Bridge Design Specifications (2010) recommend limiting the amount of differential settlement of a continuous span bridge to an angular distortion value of δ/L (in/in) = 0.004, where δ is the amount of differential settlement between two adjacent supports, and L is the distance between the supports. Figure 10 shows the data of all bridges from the numerical modeling with the AASHTO (2010) recommendation criterion to compare the general agreement in the trend of tolerable settlement.
Figure 9 demonstrates the distribution of tolerable differential settlement versus span length of the bridge. Also, it could be noted that the angular distortion limit shown in the figure represents an upper bound for the tolerable settlements. Further analysis of the data by bridge type, steel and pre-stressed concrete, shows the difference in tolerable differential settlement amounts in terms of span length between the two.
Comparison of differential settlement limits of this study with those of AASHTO recommendation (2010) (1 in = 25.4 mm, 1 ft = 0.3048 m).
The comparison of the steel and pre-stressed concrete bridge data suggests that pre-stressed concrete bridges have a smaller tolerance to differential settlements, which is consistent with AASHTO (2010) and literature results (Figure 10). However, it indicates that AASHTO specification (2010) allows larger settlements which may not be conservative. Please note the settlement limits for the two bridges of 140 and 150 ft spans are much less compared to other cases, which may due to their special dimension or configuration.
Differential settlement limit of steel (left) and pre-stressed concrete (right) bridges versus span length (1 in = 25.4 mm, 1 ft = 0.3048 m).
The number of spans in a bridge also affects the amount of settlement that the bridge can tolerate. The data from the numerical analysis show a general decrease in tolerable settlements with an increase in the number of spans. In this study, bridges with two, three, and four spans are studied to show this general trend.
From Figure 11, we could observe that the range of tolerable settlements for each number of spans is wide (due to different bridge types and span lengths), but the general trend can be seen to decrease with the addition of more spans in the bridge.
Effect of number of spans on differential settlement limits of bridges (1 in = 25.4 mm).
The degree of skew is also a parameter that can affect the amount of settlement a bridge can tolerate. It is found that skew changes the response of the bridge, and smaller values of tolerable settlements are reached with an increase in the degree of skew in general, since skew of bridge changes its dead and live load moment and shear distribution to different girders. Figure 12 shows the data for differential settlement limits in terms of skew from the numerical modeling results.
Effect of degree of skew on the differential settlement limits of bridges (1 in = 25.4 mm).
Statistics-based bridge tolerable settlement limits
Since the tolerable differential settlement limits are distributed widely over the variable domain, a statistics-based method is suggested to process the data. The statistical software Minitab is used to perform the distribution hypothesis analysis. The software performs a goodness-of-fit test using the following hypothesis test. The null hypothesis is Ho: the model adequately describes the data, and the alternative hypothesis is H1: the model does not adequately describe the data. The data are compared to different statistical distributions to find the best fit. At the α value of 0.05, the null hypothesis for the normal and exponential distribution is rejected. The log-normal distribution results in the acceptance of the null hypothesis because of the significant p-value. The probability plot of the differential settlement data following the log-normal distribution is shown in Figure 13.
Probability plot of numerical modeling data (1 in = 25.4 mm).
Further analysis is conducted to determine whether the data from the bridges by material type also follow the same distribution. The goodness-of-fit test for the steel and pre-stressed concrete bridges also concludes that both portions of the data independently follow the log-normal distribution. It is clearly indicated that the settlement limit for steel bridges is larger than that of pre-stressed concrete bridge (Figure 14).
Probability plot for steel and pre-stressed concrete (1 in = 25.4 mm).
With the appropriate statistical distribution identified, statistical analysis can be performed to find tolerable differential settlement limits for both steel and pre-stressed concrete bridges. Statistical analysis of the organized data results in the differential settlement limits for steel and pre-stressed concrete bridges at different reliability levels. Table 5 shows the differential settlement limits of bridges at different reliability levels, where β is the reliability level.
Statistical parameters and differential settlement limits (1 in = 25.4 mm).
Table 5 shows that the differential settlement limits for pre-stressed concrete bridges are lower than the differential settlement limits for steel bridges, which is consistent with the surveyed results in Moulton et al.’s (1985) work.
Comparison with literature results
A comparison is also made with the survey results (Moulton et al., 1985) and Zhang and Ng’s (2007) conclusion. A comparison of probability distribution of the current results and Moulton’s survey results is shown in Figure 15.
Comparison of the probability distribution of the present results and the survey result conducted by Moulton et al. (1985) (1 in = 25.4 mm).
From Figure 15, we can see that both the results follow the log-normal distribution very well, with the survey data giving larger settlement limits and broader band.
A fragility curve is also generated for the current steel and pre-stressed concrete and compared with the survey results on all bridges (Zhang and Ng, 2007). Figure 16 shows less severe/failure percentage for the survey results, which is reasonable due to two reasons. The first reason is small cracking/damage is allowed in practice, which is taken as tolerable in the survey; and the second reason is the large intervals used in the surveyed data reduce its accuracy.
Comparison of the fragility curve of the present steel and pre-stressed concrete bridge results and the survey result conducted by Moulton et al. (1985) (1 in = 25.4 mm).
Summary and conclusion
The process of modeling the effect of differential settlements on steel and pre-stressed concrete bridges in the CSiBridge program is described. A linear relationship exists between the over-bending moment in a bridge girder and the increase in settlement amount. Tolerable differential settlement limits of a critical support are found for steel and pre-stressed concrete bridges. The data collected from the numerical modeling are organized by bridge type, span length, number of spans, and skew that are identified to have an influence on the amount of differential settlement a bridge can tolerate. The data are also studied to determine the general trends of settlement limits related to these parameters. The tolerable differential settlements are found to follow a log-normal distribution for bridges, and a statistical analysis is suggested to find settlement limits at different reliability levels. When compared with the field survey results, the present tolerable differential settlement limits follow the same trend and distribution type, but with a less settlement limit magnitude. It is found that AASHTO specification (2010) on tolerable settlement limits derived from the field survey results is not conservative in some cases, especially for concrete bridges.
Footnotes
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.