Authors: Sitaram Vemuri, Srimaruthi Jonnalagadda
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In civil engineering practice, 1D modeling (also called line modeling) is often the most common method of modeling used for the analyses of structures. It is easy to create the model and saves time without loss of significant accuracy in the results. However, these line models are ideally not a true representation of the behavior of the structure as they are overly simplified versions. Though this simplification leads to quick modeling and reduces analyses-design cycle time, this over-simplification can lead to inaccuracies and over- design that can impact the economy of construction in repeat projects. Also, in the current era of precast modular construction, it would be prudent to analyze a structure as precisely as it can be so that optimal solution is achieved before designing as standard member that can be reused across an array of projects. In this study an effort is made to model a box bridge structure to analyze and compare the behavior of the bridge under railway vehicular train loading. A Road under Bridge (RUB) of 25m barrel length is considered and analysis is carried out by creating two models subjected to IRS vehicular rail axle loading using STAAD.PRO commercial software. A comparison of flexural forces in the culvert structure is made between results of 1D line model versus 3D box finite element model. It was found that flexure at the critical sections of the box based on line model analysis are much higher than those from 3D finite element analysis. In this particular example, a 14% excess of flexural reinforcement and a 40% excess shear reinforcement are required if we design the structure based on line model of analysis. Thus it is inferred that a design based on line model analysis can be overly conservative and uneconomical. It is suggested that detailed 3D finite element analysis be performed for the design of modular and reusable members especially for precast construction and engineering.
Modeling is the foremost step in the computerized structural analysis for any structure. Modeling refers to, generating the geometry of the structure in the analysis software and simulating similar loading environment and support conditions that the structure in real is expected to be subjected to. Proper modeling of the structure results in good analysis and yields accurate results. Poor modeling leads to erroneous results and may lead to unsafe design or, over conservative design which is not economical. Thus, modeling plays a key role in delivering cost optimized and safe designs especially in the field of civil engineering where the cost of mega projects like railway and road projects touch sky high.
Minor bridge is a bridge having a total length up to 60m . Box bridge comes under minor bridges category. Box bridge is an integral structure consisting of a top slab, bottom slab and side walls of definite thickness with a vent in it to allow passage of vehicles (Underpass) or water (Box culvert). The box is rested on the level ground. Road under bridge is an under pass where trains will be passing on the top slab of the box and road is laid beneath the railway track. IRC:5 – 2015 specifies that box culverts are minor bridges whose span, i.e., distance between outer faces of side walls, is less than 6m. Road under bridges are being constructed to bridge even up to 10m span due to their robustness and ease of erection . From mid 19th century, Structural analysis programs have been the most common tool for analyzing structures. Pertaining to box bridges, analysis can be performed in two ways. The first and the most commonly employed method is two dimensional (2-D) line modeling method. The second one is highly sophisticated method called finite element modeling method.
In 2-D line modeling method, the cross section of the box is modeled in a 2-D plane by joining the centre line of top slab, side walls and bottom slab. The width or the barrel length considered is 1m. Thus a 1m segment of barrel is considered for the analysis. Here, the 1m wide top slab, bottom slab and side walls are considered acting as beams. Loading is applied as uniformly distributed load (UDL) per unit width of the slab. Fig. 1 shows a typical line model and 1m width box segment from STAAD.PRO.
Invention of finite element analysis (FEA) dates back to early 1940’s. The invention of finite element analysis has ignited the fuel and lead to rapid development in the mechanical, civil and material science industry. Finite element analysis is mathematically intensive. The basic principle involved in finite element analysis is finding the solution of a differential equation. Every physical phenomenon in this earth at least, has a governing differential equation associated with it. The solution of the differential equation is also a function of some variable. So, in brief to say, in FEA, a solution function for the differential equation is assumed and that solution function is taken as sum of weighted functions at finite number of variable points which is nothing but called as interpolation function. This division at finite variable points is equivalent to division into finite elements. Followed by that, the assumed solution is substituted in the differential equation back to satisfy the equation. Since, the solution is assumed one, the equation leaves some residual. Then from method of weighted residuals, the weighted functions are solved and final solution is obtained. Thus FEM is highly useful in solving any differential equation, explicitly analyzing any physical phenomena. FEM is even used in simulating fluid flow which is used in wind analysis of tall buildings, popularly known as computational fluid dynamics [3, 4].
In finite element modeling, the entire box structure is modeled in 3-D space and reflecting the exact geometry of the structure. The top slab, bottom slab and side walls are considered as plate elements. The plates are again discretized into smaller plates to capture the analysis results accurately. The loads are applied in the form of uniformly distributed load per unit area. Fig. 2 shows the finite element model of the box structure. Anil K. Garg and Ali Abolmaali conducted a parametric study to develop design equations from a three dimensional verified finite-element model of culverts .
In the next section, details of a Road under Bridge (RUB) considered for structural analysis under railway loading are provided. Followed by that, structural analysis is carried out using 2-D line modeling method and then by FEM method. Later results and discussion follows and finally conclusions are presented.
II. DETAILS OF THE RUB CONSIDERED FOR ANALYSIS
An RUB is considered for analysis under railway loading. The clear vent height is 2.65m and width is 5.5m. The barrel length is 25m. The depth of top slab is taken as 500mm. Since, the vent clear height is less than the vent width, the depth of the side walls is limited to 350mm as that itself satisfies the ultimate limit state criteria of design. The cross section of the RUB is shown in Fig. 3.
The loads to be considered are Dead load that includes self weight of the structure, Superimposed dead load on top slab that includes weight of rails, ballast, sleepers, etc, super imposed dead load on bottom slab due to wearing coat, earth pressure on walls, dead load surcharge, live load surcharge and live load due to train. The earth pressure, dead load surcharge and live load surcharge are calculated using the formulations mentioned in IRS substructure code . The live load considered for the design is rail loading confirming to IRS Bridge rules . A 25T axle train is considered and its axle configuration is shown in Fig. 4. The coefficient of dynamic augment is a function of depth of cushion and is worked out from IRS bridge rules.
There is a maximum increase of around 88KNm moment for LMM over FEM at section 2-2. At the joint between the sections, 2-2 and 3-3, the moments obtained by FEM are not equal where as moments obtained from LMM are equal. It is due to the fact that in LMM, the model is considered two dimensional and the moment distribution along the other direction doesn’t take place. Where as in FEM moment distribution takes place in three directions as discussed previously. Obviously, corresponding area of tensile steel required for ultimate limit state of moment resistance per meter width of the slab is also higher for LMM over FEM. At section 5-5, the critical shear force from LMM exceeded the shear force from FEM by 70KN which is huge. Thus, shear reinforcement can also be reduced a lot by employing FEM. At section 3-3, there is difference of around 300mm2 of tensile steel area per meter width of the slab which clearly states that, the design using line modeling method is uneconomical and over conservative. Moreover, FEM also gives more realistic insight into the actual structural behavior of the box bridge.
In this paper, a Road under Bridge is considered and analyzed for 25T axle Indian Railway Specifications based rail loading in STAAD.PRO using 2-D line model method and finite element method. It is observed that, in this particular example, a 14% excess of flexural reinforcement and a 40% excess shear reinforcement are required if we simplify the structural analysis using 1D line model instead of developing a comprehensive 3D finite element model for the RUB culvert. The increasing cost of construction materials demands for economic structural designs [9,10]. Hence replacing finite element methods over conventional analysis methods can produce quite a considerable cost saving designs. For modular construction projects and in precast construction, it is imperative to achieve even slightest efficiency in form or design [11, 12] because these modules and designs are reused over multiple projects, so very small savings per member could still be large savings when implemented over multiple projects. A constant and continous improvement in efficiency of engineering and construction methods are of paramount significance in achieving goals of sustainable engineering and construction as recommended by Jonnalagadda et al  in his doctoral thesis. The models in this study did not include the haunch at the corners of these box culvert units. The haunch provides more rigidity at the wall-slab joint and hence more rotational restraint, so it would be interesting to study its effect on the design economy of these structures. This study can be extended further to include the effect of haunches on the behavior of minor box bridges and their design economy and efficiency.
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Copyright © 2023 Sitaram Vemuri, Srimaruthi Jonnalagadda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.