Transverse design of rc hollow box bridge girder Esmerald Filaj1, Enio Deneko2, Erdit Leka1, Arian Lako1 1 Polytechnic
University of Tirana, Faculty of Civil Engineering, Tirana, Albania
[email protected]
Abstract The design of a reinforced concrete hollow box bridge is usually done separately for the transverse and longitudinal directions. In this paper, only the transverse behavior will be discussed. At this aim, a simplified 2-D (two dimensional) plane frame model of unit length, for a considered bridge, is built and analyzed using SAP 2000 software, for two different cases: 1) using linear frame elements connected in rigid joints as specified by the software (truss model); 2) using shell elements and so considering quite ideal rigid joint regions. This model allows for load distribution to the webs and slab members, relative to their stiffness, but, the overall transverse behavior depends also on how the elements and joints are modelled. In order to study mainly the joint stiffness effects, especially the upper ones, further simplification regarding loading conditions are made, considering only the selfweight and the live load. The live load is applied in different positions, to account for the worst loading cases for the elements. Analysis results show that the joints stiffness influence on the transverse bridge behavior must be considered, for stresses and displacements evaluation also, aiming the rigid conditions that do represent better the state of the art and the literature recommendations. The model built with shell elements, gives more reliable results and is more appropriate to be used for this kind of analysis. Keywords:
rc bridge, hollow box girder, transverse behavior, shell element, 2-D frame model, 2-D shell model, SAP2000
1 Introduction A box girder bridge is a bridge in which the main girders, one or more, have a hollow box shape. These type of bridges are commonly used for highway flyovers and for modern elevated structures of light rail transport. On geometry basis, they can be classified as: monocellular, monocellular with ribs and struts, double-cell and composite multiple box girders. The box girders normally can either be prestressed reinforced concrete, structural steel, or a composite of steel and reinforced concrete, with a typical rectangular or trapezoidal cross section. Because of high torsional resistance, a box girder structure is particularly suited to bridges with significant curvature. This paper is focused on the transverse design of monocellular reinforced concrete hollow box bridges.
2 Design approach The design of a hollow box bridge, usually, is separately done for the transverse and longitudinal directions [1]. To correctly represent/ study the box girder (bridge), one would need to do a three dimensional analysis and incorporate all loads the box is subjected to, along with proper boundary conditions [2]. Due to the complexity of this type of analysis, in particular the application of prestressing, to the three dimensional systems, this is seldom done. In lieu of this complex analysis, it is common practice to model the box as a 2-D (two dimensional) plane frame of unit length, which allows for load distribution to the webs and slab members, relative to their stiffness. But, the overall transverse behavior depends also on how the elements and joints are modelled. Typical 2-D frame model is assumed to be supported at the lower end of the webs. A more accurate model is based on a partial 3D finite element model of the box girder. The term “partial” implies that the entire bridge superstructure need not be modeled; rather it should be interpreted as a partial length of the box that will be long enough to include three dimensional effects. From this model, influence lines can be generated at any section of interest. The first presented approach, can give good qualitative results regarding the joint stiffness influence on the bridge behavior, quite the main interest of this study, so it is the one chosen to for the analysis.
Figure 1. Hollow box bridges
Figure 2. 2-D frame model
3 Case study The monocellular reinforced concrete box girder bridge, concrete C35/45 and Figure 3. Partial 3-D model reinforcement S500s - Class B or C according to Eurocodes [3] requirements, with the typical trapezoidal cross section presented below, will be analyzed. The geometry is simplified not accounting for any variable increase of web or bottom slab thickness, neither a side longitudinal continuous vehicle barrier. Dimensions are in centimeters (SI Units).
Figure 4. RC box girder bridge geometry & loading conditions
4 Design load cases The design loads considered in transverse design include [4], but not limited to:
dead loads of (non) structural components, wearing surface and utilities live loads of vehicles and dynamic load allowance (DLA) primary prestressing forces; creep and shrinkage effect of concrete thermal gradient (inside-outside box girder temperature difference) accumulated locked-in force effects resulting from the construction
In order to study mainly the joint stiffness effects, some simplification regarding loading conditions can be made. So, only the self-weight of the structure is considered as a dead load. The live load, as a bi-component load block with load vectors in a 2m of distance (almost the standard transverse wheel distance of a design vehicle like a truck or a tandem [5], [6]), is applied in different positions, to account for the worst loading cases for the elements. Due to the fact that the side longitudinal continuous vehicle barrier is not considered as a part of the model (because of the barrier discontinuities and uncertain future quality, this edge stiffening effect is neglected & not recommended), the live load can be assumed to act even on the girder extremities, practically not possible (herein is assumed at least 50cm from the side). Figure 5. Live load configuration
The value of each load-vector is assumed P=50kN (a mean value between the maximum wheel load of a truck and a tandem according to EC2 and LRFD requirements [5], [6] - if load intensity is too small no significative differences can be noticed between considered analysis cases). However, it must be pointed that all the above simplifications, do not affect qualitatively the results of this study.
5 Analysis models Two different models are built using SAP 2000 software, to analyze the monocellular reinforced concrete box girder bridge: 2-D (two dimensional) plane frame of unit length, built using frame elements, with variable cross section when needed, connected in “rigid joints”. There are two alternatives to interpret the corner regions rigidity: by default software joint region parameters, which are function of the intersecting frame elements stiffness and geometry; for a better interpretation, the nodes in the joint region, created artificially, based on judgment and also geometry, can be “coupled”, constrained, so creating a rigid region - this alternative is closer to the real behavior of the bridge, but a good care should be made while creating the artificial “coupled” nodes, trying not to extend unnecessary the “coupling’” effect, meaning shortening the effective length of the intersecting frame elements.
Figure 6. 2-D plane frame analysis model with frame/shell elements Model behavior depends also on the frame elements geometry and intersection specifics. As stated above, some frame elements have been defined with variable cross section. The variation law is influenced by the length, meaning that to get closer to the real box girder geometry, some frame elements should be modeled with a number of shorter sub-elements - the model will be complicated. However, it can be noticed that variable girder elements have short length and the relative error in results derived using the simplified model (Fig. 6) can be neglected. Intersection specifics are related to the way/ manner how the frames intersect in the joint. This issue can be addressed to the use of
section cardinal points. In our case, it is assumed that the frames intersect by the longitudinal axis - cross section center of gravity. 2-D (two dimensional) plane frame of unit length, built using shell elements thus considering quite ideal rigid joint regions and no discontinuities, closer to the state of the art and the literature recommendations. For both of the models, the above presented loading conditions (unfactored) will be applied, and the most important sections will be discussed regarding normal stresses (bending moments) and vertical displacements. These sections, in theory, should correspond in both models, but in reality, due to the differences shown, this isn’t quite true. In the model built using frame elements, a studied point on the frame element longitudinal axis, represents also the section, meanwhile in the model built using shell elements a studied point, with the same abscissa, represents only a fiber (extreme one in our case) of the section. Although this physical differences, the inaccuracy level while comparing the results to fulfill the goal of this study, would be small.
LLC-1
LLC-2
LLC-3
LLC-4
6 Analysis results for frame elements model Analysis results for the 2-D (two dimensional) model, built using frame elements are presented below for each live load configuration (LLC).
Figure 8. Deck displacements (mm) and Normal Stress diagram daN/cm2
Figure 7. Bending diagram
moment
Figure 9. Deformed shape
7 Analysis results for shell elements model Analysis results for the 2-D (two dimensional) model, built using shell elements are presented below for each live load configuration (LLC). LLC-1
LLC-2
LLC-3
LLC-4
Figure 10. Normal stress S11 diagram (N/mm2)
Figure 11. Deformed shape (mm)
8 Comparison of analysis results As discussed, box girder corner “rigid” regions affect its overall transverse behavior. To evaluate somehow these effects and any possible qualitative or quantitative difference between the two models, the most influenced element of the section (the deck) is considered (because of its role and dimensions).For a more complete study, all the elements of the section should be analyzed.
Figure 12. Deck displacements (mm) and Normal Stress diagram daN/cm2
At this aim, a fine division of the deck is made at quite equal spaces - a total 33 points/ sections - including also the live load application points/ sections. Analysis results in terms of displacements and normal stress are shown. Bending moments of 2-D frame elements model are expressed as normal stress, basing of classical methods (Strength of Materials), and accounting for the section dimensions at each of the 33 stations. It can be noticed that deck vertical displacements follow almost the same variation law for both of the models. The shell elements model presents an increased stiffness compared to the model built with frame elements. Extreme numerical differences refer to the first and second LLC, respectively up to 55%/ 64% at midsection, and 72%/ 21% at sides. Deck normal stress variation is at a first look, different between the models. Stress variation of frame elements model follows the bending moment law which is composed by two parts cantilever and space - “joined” in the corner intersection region. So in theory normal stresses with two different values would be developed in the intersection corresponding section. Meanwhile, as expressed in (5), there isn’t a discrete intersection point/ section in the shell model, and this is why the stress variation is quite smooth along the deck. The influence of the corner region rigidity is more present for the space segment of the deck, Figure 13. Deck displacement comparison with no differences in the cantilevers, and this is quite logical based on the geometry conditions and also the fact that in the cantilever part the models are identical. So, the corner regions present an increased rotational stiffness, reducing space stress level.
Except for the first LLC, a good agreement can be seen in the normal stress distribution.
9 Conclusions Transverse design analysis of reinforced concrete hollow box bridges can be based on simplified 2-D models built with frame or shell elements. The overall transverse behavior depends on how the elements and rigid joints are modelled. The last alternative is more reliable while better including on the results the effects of rigid regions of the girder section. Thus it gives a more exact evaluation of normal stresses especially in rigid corner regions (in such a case cannot be exactly calculated using Bernoulli hypothesis), and displacements also. The model built with shell elements, is more appropriate to be used for this kind of design analysis.
Figure 14. Deck Normal Stress comparison
10 References [1] Rombach, G.A.: Finite element design of concrete structures, First edition, Thomas Telford Publishing, Cornwall, 2004. [2] Theryo, T. S.: Precast Balanced Cantilever Bridge Design Using AASHTO LRFD, Bridge Design Specifications, Major Bridge Service Center, U.S., 2005. [3] Eurocode 2, Design of concrete structures, Part 2: Concrete bridges - Design and detailing rules, European Standard, CEN, Brussels, 2004. [4] Eurocode 1, Actions on structures, Part 2: Traffic loads on bridges, European Standard, CEN, Brussels, 2004. [5] AASHTO LRFD Bridge, Design Specifications - Customary U.S. Units, American Association of State Highway and Transportation Officials, U.S., 2012. [6] Bridge Design to Eurocodes, Worked examples, European Commission Joint Research Centre, Luxembourg: Publications Office of the European Union, 2012 [7] Sanpaolesi L., Croce P.: Handbook 4, Design of Bridges, Leonardo da Vinci Pilot Project, “Development of skills facilitating implementation of Eurocodes”, Pisa, 2005 [8] Computer and Structures Inc.: CSI Analysis Reference Manual, For SAP2000, ETABS, SAFE and CSi Bridge, 2014
