1. Introduction
A decisive factor when selecting a bridge type is the required span distance. Reinforced concrete slab bridges are chosen for short spans because avoiding girders can reduce labor and formwork costs [
1]. Despite their limitation in span length, slab bridges are widely used. For example, in the U.S. 2020 National Bridge Inventory, nearly 10.5% of all highway bridges are classified as concrete slab bridges [
2].
Slab bridges can be straight or skewed. In straight slab bridges, the main longitudinal direction of the bridge is perpendicular to the support line. In skewed slab bridges, there is a deviation of the main longitudinal direction away from the vertical axis (see
Figure 1). In skewed slab bridges, the force flow is significantly more complex than in straight slab bridges [
3]. However, skewed slab bridges are common when urban or geographical constraints prevent the design of straight geometries. Actually, the number of skewed bridges is growing in developing and urban cities, and the number of cases of slab bridges with a skew angle of more than 45 degrees is increasing as well [
4].
Numerous design codes cover the design of slab bridges [
5,
6,
7,
8]. In this work, the focus lies on the AASHTO LRFD Bridge Design Specifications (AASHTO LRFD). AASHTO LRFD is employed for design, evaluation, and rehabilitation of highway bridges [
5]. The safety philosophy of this American standard is Load and Resistance Factor Design (LRFD), which is a reliability-based methodology that uses statistics to determine the appropriate safety factors for loads and resistance of components [
5]. AASHTO LRFD allows the design of simply supported solid slab bridges with main longitudinal reinforcement parallel to the direction of the traffic using simplified procedures [
5]. AASHTO LRFD does not prescribe the maximum skew angle for which the simplified design procedures can be applied [
5].
Limited attention has been paid to skewed solid slab bridges. They were simply treated as one-way slabs where the main longitudinal moments are carried by the longitudinal reinforcement and the transverse moments are handled with empirical expressions [
9]. In 2006, the collapse of the Concorde Overpass [
10,
11] resulted in concerns with regard to the capacity of existing reinforced concrete slab bridges. The event caused five fatal casualties and the injury of six people. This failure drew attention to the shear strength of skewed solid concrete slab bridges [
12]. In the same decade, the shear capacity of existing reinforced concrete slab bridges was questioned in the Netherlands [
13]. Adopting the Eurocodes [
14,
15,
16] resulted in higher sectional shear forces and lower shear capacities than those used in the Dutch national codes [
17,
18], so that assessment of these bridges became a priority.
In recent years, efforts have been geared towards determining the effects of skew on slab bridges. This paper provides practical and relevant insights on the effects of skewness to the designer using AASHTO LRFD. More specifically, the aim is to answer the research question: How does skew influence the amount of reinforcement and its layout in reinforced concrete skewed slab bridges? To do so, the selected method is a parametric study, where AASHTO LRFD simplified procedures are compared to more refined linear finite element analyses (LFEA). This parametric study results in the main longitudinal and transverse bending moments, as well as shear forces at the obtuse corner using both approaches. These design magnitudes are then translated into reinforcement layouts. Comparing the reinforcement layouts from both methods, we can identify when larger, equal, or smaller amounts of reinforcement are found using the AASHTO LRFD hand calculations as compared to LFEA. In addition, the reinforcement layout is further translated into total steel weight to evaluate the cost. In parallel, the moment distribution capacity of the slab bridges is assessed. This is performed to determine when the same spacings or bar diameters for the main longitudinal reinforcement can be provided over the entire width of the bridge, which enhances ease of construction. As such, this work produces a tangible response on how skew influences the design of reinforced concrete skewed slab bridges, and goes a step further than research from the literature, which focused on the design moments and shear forces.
This paper is divided in two main sections following the literature review. First, the case study bridges are described as well as the two methodologies used for analysis (AASHTO LRFD simplified procedures and LFEA) and their application towards design. Second, the results of the parametric study are presented. These are condensed within six subsections: main longitudinal bending moment, distribution width, secondary transverse bending moment, shear, influence of materials, and reinforcement comparisons based on weight of steel.
2. Literature Review
Several methods for understanding the effect of skew are reported in the literature. One of these is through load testing of existing bridges. Davids load tested 14 bridges and compared the rating factors obtained with AASHTO LRFD and FEA. The study showed that the rating factors increased up to 37.6% for bridges with skew angles between 15° and 20° when using FEA [
19]. Other load tests of slightly skewed slab bridges have shown that the procedures for rating existing reinforced concrete slab bridges using the European codes are conservative for both shear and bending moment [
20,
21]. Load testing of highly skewed concrete bridges is rare. However, Bagheri developed an artificial intelligence model that can predict nondimensional frequency parameters related to the vibration modes of a slab bridge. It operates in the ranges of 0° to 60° skew angles. The input parameters are span length, deck width, deck thickness, and skew angle. With the nondimensional frequency parameters, one can calculate the flexural rigidity. This magnitude is used in the load rating and nondestructive evaluation of existing bridges; thus, the neural net is useful where structural information is incomplete [
22].
Skewed slab bridges can also be studied through computational models. For example, nonlinear analysis has been used in the past [
23] and in recent years [
24] to study the behavior of skewed slab bridges at the ultimate limit state. Cope [
23] determined that the first load that generates cracking drives the response of the slab, so that nonlinear analysis can only yield approximations of the slab’s actual behavior. Hassan [
24] also studied cracking load and observed that for skew angles up to 30°, the cracking load remained the same as for straight bridges, but there was a decrease when the skew reached 45°. Additionally, computational approaches have also been combined with probabilistic approaches to determine the seismic fragility of various types of skewed bridges [
25].
Experimental work on skewed slab bridges is limited. Laboratory testing dating back to the eighties focused on the effect of shear in reinforced concrete slabs. One of these studies was conducted at the University of Liverpool and considered specimens with skew angles ranging from 30° to 60°. One of the main objectives of the study was to determine how to predict shear forces and evaluate shear capacity of skewed slabs. The study showed that Mindlin plate theory, with appropriate mesh refinements, can predict skewed slab behavior to a certain extent. The experiments showed that the failure mode changes from flexure to shear, and then to punching as the skew angle increases [
26]. Another study, with a much more limited scope, tested two 50° skew angle scaled bridge models. The failure mode for the first specimen was flexure, and that of the second specimen, which had increased flexural reinforcement, was punching shear. Additionally, this study determined that thick plate theory could predict the initial distribution of shear stress at the obtuse corner [
27]. More recently, Sharma developed a theoretical formulation to predict the ultimate flexural strength of skewed slab bridges. The outcome of the formulation was compared to results obtained from scaled test specimens with skew angles from 15° to 60°, and yielded accurate results [
28].
Parametric studies provide an additional way of comprehending the response of skewed slab bridges. Some parametric studies have focused on the development of skew factors. For example, Théoret conducted a parametric numerical study on 390 simply supported slabs. This study resulted in a series of expressions for moment reduction factors and shear magnification factors as a function of the skew angle. These factors compensate for skewness when using the simplified analysis procedures from AASHTO LRFD [
9]. Similarly, skew factors that increment load effects were developed in the Netherlands for bridge assessment [
29].
Other parametric numerical studies have focused on force distribution and concentration in skewed slab bridges. Menassa analyzed 96 case study bridges using the AASHTO Standard Specifications, AASHTO LRFD, and LFEA. The research, which focused on bending moments, confirmed that skewed slab bridges can be designed as straight for skew angles smaller than 20° [
30]. Likewise, Hulsebosch developed a parametric study with a focus on the influence of skew towards the magnitudes of bending moments and shear forces. He determined that the addition of ATS (additional triangular segments) adjacent to the free edges of the slab bridge reduces the governing shear forces at the obtuse corners [
12], and recommended this practice for the design of new skewed slab bridges. Additionally, Fawaz analyzed 96 case study bridges with a special attention on the influence of railings on bending moments. The parametric study showed that the presence of railings, on top of the skew angle, can further reduce the main longitudinal bending moments obtained with AASHTO LRFD in skewed slab bridges [
31].
From the literature review, we identified the research gap as a parametric study on the resulting reinforcement layout in reinforced concrete skewed slab bridges. By focusing on the resulting reinforcement, this paper provides the designer with practical insights. Additionally, since there is no clear consensus on how to evaluate the shear capacity of slab bridges [
14], the application of a new approach is presented herein. The selected procedure comes from Lipari, who proposed variations to extend shear design code provisions for straight geometries to skewed geometries [
32]. These procedures will be elaborated in
Section 3.2.2.
5. Discussion
Slab bridges are widely used and form an important element in our infrastructure. Even though straight geometries are preferable, urban or geographical constraints may make the selection of skewed geometries necessary. This paper presents the results of a parametric study to determine the applicability of AASHTO LRFD for simply supported reinforced concrete skewed slab bridges. Ninety case study bridges with different span lengths, skew angles, number of lanes, and material properties were used in this study. The bridges were analyzed with the AASHTO LRFD simplified procedures using hand calculations, and with LFEA using SCIA Engineer 20 [
38]. In a final step, the required longitudinal, transverse, and shear reinforcement were designed according to the AASHTO LRFD for both analysis methods.
Our observed reductions in maximum longitudinal bending moments for skewed geometries are aligned with the parametric study conducted by Menassa. Menassa found a 50% bending moment reduction for the 50° case study bridges [
30]. For the bridges analyzed in our parameter study, a 45% and 65% bending moment reduction was obtained for the 45° and 60° cases, respectively. A main difference between our study and the work by Menassa lies in the magnitude of the bending moments for the exterior strips calculated with AASHTO LRFD. In Menassa’s work, these moments were smaller than those of the interior strips [
30]. Contrarily, in our work, all exterior strip bending moments were higher than those in the interior strip. This difference is explained by the fact that Menassa did not consider concrete barrier loading. We considered concrete barrier loading and assumed that it acted solely on the exterior strips when using AASHTO LRFD, as recommended by Rodríguez [
36].
The moment reduction coefficients developed by Théoret are not directly comparable to the moment reductions observed in this study because Théoret’s values were computed for the relation of width over length [
9]. We can observe, however, that both Théoret’s work and ours indicate a significant reduction of the longitudinal bending moment reduction as the skew angle increases.
Based on the width factor determined with LFEA, we can conclude that the assumption of the barrier acting solely on the exterior strips can be overly conservative for exterior strip design when the simplified procedures are used. The results from
Section 4.2 show the large distribution capacity for bending moments of concrete slab bridges. The reader should note here that this distribution capacity is based on linear elastic calculations. In reality, after cracking, the distribution capacity of these bridges is even larger. In consequence, we recommend distributing the weight of the barrier over the entire slab width when using simplified code procedures.
A common trend is observed for the transverse bending moments in the parametric studies from Théoret and Menassa, which used more refined analysis than AASHTO LRFD. While these moments are practically negligible for straight geometries, they increase significantly as the skew increases [
9,
30]. In our study, we also found that the transverse bending moments are negligible for straight geometries. The magnitude of these moments increased from 0° up to 30° or 45°, depending on the width and span length. We observed a significant decrease at 60°. This difference in trend can be explained by how the width of the bridge is accounted for in the studies. Both Théoret and Menassa keep the overall width constant as the skew increases. We considered the width to be dependent on the skew as it is related to the actual lane layout and driving direction. In consequence, when the skew increased, the trajectories of the principal stresses shifted towards the secondary direction. However, they did not concentrate as much because the larger width allows for more distribution.
For shear design, there is no clear agreement [
32]. This study followed the design provisions from AASHTO LRFD based on the Modified Compression Field Theory [
37], and applied the recommendations from Lipari [
32] and Lantsoght [
42]. We extended the proposals from Lipari [
32] to quantify the influence of his recommendations on the shear reinforcement in simply supported reinforced concrete slabs. None of the 80 case study bridges analyzed require shear reinforcement. The LFEA procedure can capture the stress concentrations close to the critical shear section at the obtuse corner in skewed bridges. The design procedure from Lipari was also validated [
32] for all case study bridges. Our approach can be used in practice for the design of simply supported reinforced concrete skewed slab bridges.
Our parametric study aimed at comparing AASHTO LRFD simplified procedures and LFEA in a practical way. To quantify the differences between both approaches, we analyzed the weight of steel resulting from each design. Such a practical comparison is not provided in the parameter studies from the literature. In general, using LFEA allows us to reduce the amount of longitudinal steel while fulfilling the transverse bending moment demand in skewed slab bridges.
We identify a few topics of future research. The first topic to explore further is the influence of the width. We used a practical lane layout to determine the width and defined the width in terms of driving direction. Our parameter studies can be extended by looking at geometries in which the width is defined parallel to the support line so that the support line width remains constant. The results from this study suggest that transverse moments as well as shear forces would further increase with this modification. A second topic for future research is the critical position of the truck for shear. We used a position at 600 mm from the barrier, as suggested in AASHTO LRFD. However, concentrated loads closer to the edge result in larger stress concentrations in the obtuse corner. Therefore, it would be interesting to verify the shear demand with a more critical truck positioning. Third and most important, experimental research on skewed slab bridges is very limited. Therefore, we recommend conducting experiments on skewed slabs to better understand the behavior at the ultimate limit state. Such studies can validate if LFEA is adequate for skewed slab bridge analysis or if a more refined analysis would be justified. The experimental results can be used to verify the assumption of the shear distribution width of 4 d, which was derived for straight slabs.