Modification of Railway Soil-Steel Bridge's Minimum Depth of Cover Using Three Dimensional Finite Element Analysis

1. Soil-steel structures recently have been used extensively in different countries as a highway or railroad bridge. Due to the licity and ease of construction method, these structures were successful variants in the road-railway unleveled crosses’ projects. Common standard formulas for minimum depth of cover are based on highway loading pattern with significance of preventing soil tensile and wedge sliding failures. Modification of minimum depth of cover for railway bridges is needed not only in respect to different modes of soil failure but also for non-uniform soil settlement which endangers the safety of railway track. Therefore, regarding permissible track settlement and wall buckling control, new pattern of minimum depth of cover was developed for boxes and high-profile arches individually using 2D finite element analysis. However, three dimensional distribution of railroad could not be taken into account by 2D FE method. Therefore, modification of developed pattern with attitudes toward longitudinal distribution of railway load is the scope of this study. In this paper, regarding permissible track settlement, railway twist, longitudinal settlement of the structure and wall buckling control, the minimum depth of 32 structures with spans greater than 8 meters was determined by 3D finite element analysis. Then the resulted pattern was compared to those obtained from 2D analyses. Finally, the modification factors were calculated by least squares method and new formulas were established for boxes and high-profile arches individually. 2. The primary objective of the current study is to introduce a new set of minimum soil cover equations for long-span railway bridges based on the numerical interpolation of results of the 2D FE analyses. To determine the variation trend of the minimum depth of cover along with its governing parameters (geometry, length of span and the panel stiffness) through numerical analyses, the permissible settlement of the track, metal structure buckling and soil body failure criteria have been checked initially for each bridge structure for a 0.6 m depth of cover (the minimum limit of cover depth specified by CHBDC). When all of the defined criteria were not fulfilled simultaneously, the depth of soil cover above the crown was increased, and the analyses were then restarted for a new depth of cover. The minimum depth of cover in which all of the criteria were simultaneously fulfilled was chosen as the minimum depth of soil cover for a specific bridge structure. In this manner, the results of the 2D FE analyses present specific patterns for the calculation of the minimum depth of cover for box culverts and low-profile arches. In order to check the applicability of the proposed equations for minimum depth of cover in practical problems, a series of 3D finite element analyses with more realistic idealization of the railway superstructure components and the lateral slope of bridge embankment were carried out-of-plane the out of plan buckling in the steel plates. 2.1. FE modeling: The multi-purpose FEM-based software package, PLAXIS was used for the numerical modeling and analysis. Regarding to the special serviceability criteria for railway bridges, for computing the minimum depth of cover, the permissible settlement of the railway track and the buckling of the conduit walls in different sections along the longitudinal axis of the bridges in 3D analyses were controlled. For these cases, the spans from 8.07 m up to 13.46 m of box culverts and 14.13 m up to 23.40 m of low-profile arches using stiffened and non-stiffened deep corrugated panels have been considered. This study assumes that all the metal structures are buried in well-graded gravel (GW) as engineered backfill material [1]. The material nonlinearity of the soil and metal structure as well as the stage construction effects were accounted for in the numerical analyses, and the railway load model LM71 [2] was applied.3. Results and discussion3.1. The results of the 2D and 3D finite element analyses: As a result of various numerical analyses and satisfying the aforementioned criteria, the minimum depth of cover was evaluated against the span of railway box and low-profile arch bridges using various panel types (stiffness). This is due to the different structural geometry of the boxes and results in a different mechanism of behavior under the loading. In order to account for the moment of inertia, the speed and length of span which are the representative of panel stiffness, the effect of dynamic factor and geometry of metal structure, the basic form of the minimum depth of cover for the railway boxes and high-profile arches is introduced as:Where α, β, γ and µ are unknown constants. These constants were calculated separately for the boxes and low-profile arches by using the least-squares method (LSM) to determine the function of best fit.An approach similar to the 2D analysis was used for the results of the 3D FE method, which resulted in modifications to the previously developed equations. The major finding of the comparison between the results of the 2D and 3D FE analyses was that the resulting minimum depth of cover from the 3D FE analysis was smaller compared to the 2D FE results for the same conditions. This difference was due to the stiffening effects of the railway track due to the third dimension of the bridge that was considered for the analysis. Therefore, the difference between the 2D and 3D FE results can be accounted for by using the reduction factor in the (IVI/I) ratio. Consequently, Eq. (1) can be modified as:for boxes (2)for low-profile arches (3)where and were calculated by using the least-squares method for a train speed of 120 km/h. The R-squared values of Eqs. (2) and (3) are R2=0.84 and R2=0.97 for the box culverts and low-profile arches with panels (III) and (IV), respectively.3.2. Validity range of the derived equationsThe conformity of Equations (2) and (3) with 2D and 3D results was evaluated using an R-squared value that was calculated by using R2=1-SSE/SST, where SSE=  and SST=()-[(Yi)2]/n. For panels III and V, the R-squared values for Eqs. (2) and (3) were 0.67 and 0.88, respectively. Panels I and II were eliminated from the calculations because the 2D FE analysis demonstrated that these panels were not suitable for soil-steel railway bridges. The minimum depth of cover calculated from the 3D FE analysis of bridges with panel VI represented a depth of cover less than 0.6 m. A minimum of 0.6 m was maintained for railway track maintenance (complete removal of ballast layer in some operations). Therefore, the data for panel VI was not considered in the calculation of the R-squared values. Consequently, the limited amount of data for 3D FE analysis resulted in equations with less accuracy. The results of the 2D and 3D FE analysis were compared to the standard limits and are shown in Fig. 1. Regarding the maintenance operations, buckling criterion and AASHTO [3] and CHBDC [4] limits, a recommended minimum of 0.5 m and maximum of 1.5 m must be maintained for the depth of cover by using the following expressions: for boxes (120 km/h) (4) for low-profile arches (120 km/h) (5)4. The minimum depth of cover requirements given by different codes are typically based on vehicle loads, non-stiffened panels and only the geometrical shape of the metal structure to avoid the failure of soil cover above a soil-steel bridge. In this paper, the effects of spans larger than 8 meters (using stiffened panels under railway loads) are investigated using an FE analysis. For this study, 2D and 3D FE analyses of four low-profile arches and four box culverts with spans larger than 8 meters were performed to develop new patterns for the minimum depth of soil cover. Using the least-squares method to adopt the best-fit equation of the numerical data, two new sets of formulas were recommended. Based on the numerical results, the primary research findings are summarized as follows: 1) The minimum depth of cover increases exponentially along with an increase in the span of boxes and low-profile arches. 2) The efficiency of the stiffened panels in reduction of the required cover depth is more pronounced for large spans. 3) Different trends of the minimum depth of cover were determined for box bridges and low-profile arches. This difference is due to the various structural geometry of the boxes that resulted in a different mechanism of behavior under the loading. 4) The modified exponential forms of the minimum depth of cover for railway boxes and high-profile arches is applicable for a train speed of 120 km/h and exhibits relatively good conformity with 3D FE results (R2 > 0.6). 5) With respect to the permissible settlement criterion, panels with EI greater than 33062 (kN.m2/m) are found to be the only suitable panels that can be used for high speeds trains (greater than 160 km/h) for railway box and low-profile bridges.