Comparative Analysis of Post Tensioned T-Beam Bridge Deck by Rational Method and Finite Element Method

Transcription

1 Comparative Analysis of Post Tensioned T-Beam Bridge Deck by Rational Method and Finite Element Method Y. Yadu Priya 1 and T. Sujatha 2 (Post Graduate Student, V.R. Siddhartha Engineering College, Kanuru, Andhra Pradesh) (Assistant Professor, V.R. Siddhartha Engineering College, Kanuru, Andhra Pradesh) Abstract: In modern days in the field of bridge engineering, the enhancement of pre-stressed concrete bridge decks have been increased due to its better ability to carry live loads. The analysis is carried out using IRC codal provisions. T-beam bridge decks are one of the major types of cast in-situ concrete decks which consist of a concrete slab integral with girders. The problem in continuum mechanics is approximated by FEM (finite element method) in STAAD Pro, which is general method of structural analysis. In this study a single span two lane t-beam bridge is analyzed by varying the span of 25m, 30m, 35m, 40m where the width is kept constant. The bridge models are subjected to the IRC class AA and IRC 70R tracked loading system in order to obtain maximum bending moment and shear force. From the analysis it is observed that with the increase in the span, shear force and bending moment in the girder increases. It is also observed that the results of bending moments and shear forces obtained from both courbon s method and finite element method have no significant variation. Keywords: T-beam, I.R.C. Loadings, FEM, STAAD ProV8i I. INTRODUCTION Now day s conventional bridges are being replaced by pre stressed bridges because of their cost efficiency, better stability, serviceability. Pre stressed concrete bridges are of a technique which avoids concrete weakness in tension. These type of bridges can have the span range of 20-40m whereas the conventional bridge span is between 10-20m. The number of longitudinal girders depends on the width of the road. T-beam bridges are composed of deck slab 20 to 25cm thick and longitudinal girders spaced from 2 to 2.5m and cross beams are provided at 4 to 5m interval. Pre-stressed concrete is basically concrete in which internal stresses of a suitable magnitude and distribution are introduced so that the stresses resulting from external loads are counteracted to a desired degree. In reinforced concrete members, the pre-stress is commonly introduced by tensioning the steel reinforcement. In this study, for a post tensioned t-beam bridge deck analysis is done for four different spans 25m, 30m, 35m, 40m using rational method (Courbon s method) and finite element method(staad.prov8i). These four spans are analyzed for two different IRC loadings cases IRC Class AA tracked and IRC Class 70R tracked. Each span is provided with two lanes. Bending moments and shear forces for different spans are observed. Figure1: Typical section of a bridge showing various parts OBJECTIVES AND METHODOLOGY OF THE S TUDY Objectives In this project a comparative study of post tensioned T beam Bridge with different spans by rational method and finite element method is carried out under standard IRC loadings. The maximum bending moments and shear forces are calculated. Methodology Analysis of post tensioned T-BEAM Bridge is carried out by Rational method for different spans i.e is 25m, 30m, 35m,40m. Page 9

2 Analysis Of Rational method and FEM will be done by using IRC Codes. Analysis is done for IRC Class AA and 70R tracked vehicle loading. FEM Analysis of T-BEAM Bridge is carried out by using Staad Pro V8i Software for different spans. Comparison of rational method and FEM results from Staad Pro will be done. Loads acting on Bridge Dead Load Dead or permanent loading is the gravity loading due to the structure and other items permanently attached to it. It is simply calculated as the product of volume and material density. Live loads Live load means a load that moves along the length of the span. These loads are categorized based on their configuration and intensity. Classification of several loadings is: IRC class AA loading IRC class70r loading IRC class A loading IRC class B loading Loadings considered in this study are IRC class AA and class 70R tracked loadings which are almost similar IRC class AA loading Two different types of vehicles are specified under this category grouped as tracked and wheeled vehicles. The IRC Class AA tracked vehicle (simulating an army tank) of 700 kn and a wheeled vehicle (heavy duty army truck) of 400 kn. All the bridges located on National Highways and State Highways have to be designed for this heavy loading. These loadings are also adopted for bridges located within certain specified municipal localities and along specified highways. Alternatively, another type of loading designated as Class 70 R is specified instead of Class AA loading. IRC Class 70 R Loading IRC 70 R loading consists of the following three types of vehicles. a) Tracked vehicle of total load 700 kn with two tracks each weighing 350 kn. b) Wheeled vehicle comprising 4 wheels, each with a load of 100 kn totaling 400 kn. c) Wheeled vehicle with a train of vehicles on seven axles with a total load of 1000 kn. The tracked vehicle is somewhat similar to that of Class A A, except that the contact length of the track is 4.87 m, the nose to tail length of the vehicle is 7.92 m and the specified minimum spacing between successive vehicles is 30 m. The wheeled vehicle is m long and has seven axles with the loads totaling to 1000 kn. The bogie axle type loading with 4 wheels Figure2: IRC Loading Impact Load: For I.R.C. Class AA or 70R loading (i) For span less than 9 meters For tracked vehicle- 25% for a span up to 5m linearly reduced to a 10% for a span of 9m. For wheeled vehicles-25% Page 10

3 (ii) For span of 9 m or more For tracked vehicle- for R.C. bridges, 10% up to a span of 40m. For steel bridges, 10% for all spans. For wheeled vehicles- for R.C. bridges, 25% up to a span of 12m. For steel bridges, 25% for span up to 23 meters II. RATIONAL METHODS OF ANALYSIS OF BRIDGE It can be done in several procedures such as: 1. Courbon s Method 2. Guyyon Massonet Method 3. Hendry-Jaegar Method 4. This study has been done using the Courbon s Method Courbon s Method Among these methods, Courbon s method is the simplest and is applicable when the following conditions are satisfied: a) The ratio of span to width of deck is greater than 2 but less than 4. b) The longitudinal girders are interconnected by at least five symmetrically spaced cross girders. c) The cross girder extends to a depth of at least 0.75 times the depth of the longitudinal girders. Courbon s method is popular due to the simplicity of computations as detailed below: When the live loads are positioned nearer to the kerb the centre of gravity of live load acts eccentrically with the centre of gravity of the girder system. Due to this eccentricity, the loads shared by each girder is increased or decreased depending upon the posit ion of the girders. This is calculated by Courbon s theory by a react ion factor given by = distance of the girder under consideration from the central axis of the bridge W = Total concentrated live load N = number of longitudinal girders e = Eccentricity of live load with respect to the axis of the bridge. Analysis of t-beam Bridge by rational method Courbon s Method Analysis of PT t-beam bridge deck by IRC CLASS AA TRACKED LOADING for 30m Preliminary Details clear width of roadway=7.5m footpaths=1m wide thickness of wearing coat=100mm spacing of cross girders=5m c/c live load IRC class AA tracked vehicle materials: M-40 for deck slab M-50 for girders 7mm dia high strength strands with ultimate tensile strength at 1500MPa.Cable consists of 12 strands anchored at the end with a suitable dia meter anchor block Permissible stresses and design constants The permissible compressive stresses in the concrete at transfer and at working loads as recommended in IRC 18 are as follows: <0.5 =0.5 40=20 MPa Loss ratio=0.8 Permissible compressive stress in concrete under service loads( )=16.5 MPa Allowable tensile stress in concrete at initial transfer of prestress( )=0 Allowable tensile stress in concrete under service loads( )=0 Maximum Bending Moment due to Dead Load a) Weight of Deck Slab = 0.25 X = 6 KN b) Weight of Wearing Course = 0.1 X = 2.2KN c) Total Weight = 8.2KN Page 11

4 Longitudinal Girder and Cross Girder Design a) Reaction Factor Bending Moment in Longitudinal Girders by Courbons s Method for Class AA Tracked Vehicle Figure3: Showing eccentricity and clearance Minimum Clearance Distance: /2 = 1.625m e = 1.1, P =w/2 For outer girders =0.382W For inner girders =0.294W b) Dead load from slab for girder Dead load of deck Slab is calculated as follows Weight of 1. Parapet Railing KN/m 2. Footpath= (0.3 x1x24).7.2kn/m 3. Deck slab = (0.25x1.1x24)..6KN/m Total...=14KNm Total Dead load of Deck=(2 14)+( )=89.5KN It is assumed that dead load is shared equally by all girders Therefore, DL/girder=22.37KN Figure4: Influence Line for bending moment in Girder Reaction of W2 On Girder B = 63KN Reaction of W2 On Girder A = 287KN BM at center of girder=0.5( ) 700=4935KNm Impact factor (For class AA Loads) =10% Bending Moment including Impact and reaction factor for outergirder is=(4935x1.1x0.382) = KN Bending Moment including Impact and reaction factor for outergirder is=(4935x1.1x0.294)= kn Live Load Shear For estimating the maximum Live load shear in the girders, The IRC Class AA Load are placed Total load on Girder B =(350+63)=413 KN Maximum reaction in girder B = (413x28.2)/30=388KN Maximum reaction in girder A=(287x28..2)/30= 270KN Maximum live load shears with impact factor in Inner girder=(388x1.1)=427 KN Outer girder = (270x1.1)=297 KN Page 12

5 Figure5: Position of IRC CLASS AA TRACKED Load for Maximum Shear e) Dead load BM and SF in main girder. The depth of the girder is assumed as 1500mm Sectional properties of the girder: Top flange=1200mm 250mm Rib=800mm 200m Bottom flange=500mm 450mm Self weight per meter run of girder =10.2kNm Reaction of cross girder on Main girder=12kn Reaction from deck slab on each girder=22.37kn Total dead load/m on Girder=( )=32.74kN/m Mmax= kNm Dead load Shear at Support=520.5kN Figure6: Dead load on main girder f) Results of shear force and bending moment in KN and KN/m For the above results the pre stressing force calculated from the equation is 6870KN for classaa and 6816KN for class70r tracked loadings Finite element method of analysis The finite element method is a well-known tool for the solution of complicated structural engineering problems, as it is capable of accommodating many complexities in the solution. In this method, the actual continuum is replaced by an equivalent idealized structure composed of discrete elements, referred to as finite elements, connected together at a number of nodes. The finite element method involves subdividing the actual structure into a suitable number of sub-regions that are called finite elements. These elements can be in the form of line elements, two dimensional elements and three- dimensional elements to represent the structure. The intersection between the elements is called nodal points in one dimensional problem where in two and three-dimensional problems are called nodal lines and nodal planes respectively. At the nodes, degrees of freedom (which are usually in the form of the nodal displacement and or their derivatives, stresses, or combinations of these) are assigned. Models which use displacements are called displacement models and models based on stresses are called force or equilibrium models, while those based on combinations of both displacements and stresses are called mixed models or hybrid models. Displacements are the most commonly used nodal variables, with most general purpose programs limiting their Page 13

6 nodal degree of freedom to just displacements. A number of displacement functions such as polynomials and trigonometric series can be assumed, especially polynomials because of the eas e and simplification they provide in the finite element formulation. To develop the element matrix, it is much easier to apply a work or energy method. The principle of virtual work, the principle of minimum potential energy and castigliano's theorem are methods frequently used for the purpose of derivation of element equation. The finite element method has a number of advantages; they include the ability to: - Model irregularly shaped bodies and composed of several different materials. - Handle general load condition and unlimited numbers and kinds of boundary conditions. - Include dynamic effects. - Handle nonlinear behavior existing with large deformation and non- linear materials. STAAD Model of T-Beam Bridge For the modeling of the bridge structure STAAD PRO V8i is used. The bridge models are analyzed to conduct a comparative study of post-tensioned t-beam bridge deck with rational method and finite element method. The modeling involves the construction of t-beam bridge model with single span. The bridge models are simply supported at the two ends. Staad Pro model has been created and illustrated in the following diagram. Analysis of Staad Model for 30m is shown in as follows Figure7: 3-D model of bridge Figure8: Bending moment diagram due to prestress load Figure9: Shear force diagram for prestress load Page 14

7 Figure10: Bending moment diagram for vehicle, dead and prestress loads Figure11: Shear force diagram for vehicle, dead and prestress loads III. RESULTS The obtained values for different spans and different type of loading is shown in the following tables and chats Bending Moments (KN-m) For CLASS 70R tracked span DESIGNVALUES FEM VALUES 25 OG IG OG IG OG IG OG IG OG IG OG IG OG IG OG IG design values fem values Page 15

8 For Class AA Tracked span DESIGNVALUES FEM VALUES 25 OG IG OG IG OG IG OG IG Shear Forces (KN) span DESIGNVALUES FEM VALUES 25 OG IG OG IG OG IG OG IG IV. DISCUSSIONS Parametric study is carried out on two-lane bridge and Bending moment and Shear force values were obtained by two approximate methods i.e. Courboun s method for class AA Tracked vehicle and class 70R tracked. These values are also compared with STAAD-PRO results. The results obtained are presented in the form of tables and graphs. 1. For 35m span, the value of BM in rational method is 22.04% for OG for class AA higher than that of FEM analysis here we find the maximum variation Page 16

9 2. For 30m span, the value of BM in rational method is 9% for IG for class AA higher than that of FEM analysis here we find the minium variation 3. The value of shear force in rational method is less than that of FEM analysis as we considered load combinations in FEM analysis. 4. BM and SF values are validated by comparing STAAD-PRO results with the values obtained by approximate method for various spans of longitudinal girder and it is observed that there is significant difference between Courbon s method and finite element method. We get higher values in Courbon s method V. CONCLUSION The comparative study was conducted based on the analytical modeling of simply supported post tensioned T -beam bridge deck by rational method and Finite element method using Staad Pro. In this study Courbon s method and Staad Pro we analyzed the bridge deck by varying the span of the bridge deck, here I cons idered 25m, 30m, 35 and 40m spans, here I considered the class AA tracked load which is the worst case for any bridge and also 70R tracked load which gives the average result with respect to BM values in the longitudinal girder as compared to finite element method (STAAD.PRO) and there is significant difference for both methods. The design details can also be known clearly by finite element method of STAAD.PRO. Further we can check for different load combinations in STAAD.PRO. VI. REFERENCES [1] D. Johnson Victor, Essentials of Bridge Engineering, Oxford & IBH Publications, New Delhi [2] N. Krishna Raju, Design of Bridges, Oxford & IBH Publications, New Delhi, fourth edition (reprint)- 2010,pp-111 to 164. [3] T.R. Jagdish & M.A. Jairam, Design of bridge structures, Pentice hall of India private Limited,New Delhi [4] IRC: Design Criteria for Pre-stressed Concrete Road Bridges (Post Tensioned Concrete) (Third Revision). [5] IRC: Standard Specifications and Code of Practice for Road Bridges, Section-II Loads and Stresses. [6] IRC: Standard Specifications and Code of Practice for Road Bridges, Section-III Cement Concrete. Page 17