To Design a Code for the Building of Skew Bridges

Transcription

1 To Design a Code for the Building of Skew Bridges An interim project report submitted in partial fulfillment of the requirement for the degree of Masters of Technology (M.Tech) In Civil Engineering By A.K. Mahadevan (Roll No: 00 CE 3006) Under The Guidance Of Prof. J.N. Bandyopadhyay Department of Civil Engineering Indian Institute of Technology Kharagpur India December, 2005

2 CONTENTS 1. INTRODUCTION 1 2. REVIEW OF LITERATURE SIMPLIFIED METHODS RIGOROUS METHODS CRITICAL DISCUSSION 9 3. SCOPE OF THE INVESTIGATION THEORY ORTHOTROPIC BRIDGE ANALYSIS FLOWCHART FLOWCHART DETAILS THE PROBLEM INPUT FILE DETAILS RESULTS DISCUSSION PLAN FOR THE NEXT SEMESTER REFERENCES 25

3 1.0 INTRODUCTION The skew bridge is one whose longitudinal axis makes an angle less than or equal to 90 degrees. The skewness may be a way to avoid certain obstacles and thus create the most economically viable option. The topography of the area and the alignment of the roads are the deciding factor for the skewness of the bridge. There is very little literature available on skew bridges. Compared to its frequent installation, the IRC codes are surprisingly silent about the methods of design of skew bridges. It is said that if the skew angle is less than 20 then it can be designed as a right bridge. Nothing is said about bridges with a skew angle of more than 20 but the most common skew angle noted is 45 degrees. As a precursor to this a program to calculate the moments and the shear forces in a right bridge and to understand the working of a semincontinuum method is being undertaken at this stage of the work. 1

4 2.0 REVIEW OF LITERATURE General This chapter reviews the existing literature on the analysis of slab-on-girder type bridge decks briefly. These bridges are most commonly used when the span is less than 30 meters. The different methods of bridge analysis as well as comparative review have been dealt with in this section Existing Methods of Analysis The literature available on the analysis of bridge decks may be classified into two broad groups:- 1. Simplified Methods 2. Rigorous Methods Simplified Methods The simplified methods are derived and used for right bridges only. There is no such similar method for the design of skew bridge desks. However, according to the IRC, skew bridges with skew angle less than 20 degrees can be designed as a right bridge of the effective span. In fact, before the inception of micro computers, skew bridges were designed using the simplified methods only. Under the simplified methods, the following are of use 1. Leonhardt Andre Method 2. Courbon s Method 3. Hendry-Jaeger Method 4. Morice- Little Method 5. Cusen-Pama Method 6. AASHTO Method 7. Ontario Method Rigorous Methods The various types of rigorous methods are 1. Orthotropic Plate Method 2. Finite Element Method 3. Grillage Analogy Method 4. Finite Difference Method 5. Finite Strip Method 6. Semicontinuum Method 2

5 2.1 SIMPLIFIED METHODS Leonhardt Andre Method In earlier days, slab-on-girder bridges were idealized as an assembly of longitudinal beams and one tortionless transverse beam placed along the midspan of the bridge by Leonhardt and Andre. This worked well for timber bridges. This idealization neglected the tortional effects. Due to the fact that the shear modulus of timber is very small compared to the modulus of elasticity, the neglect of tortional rigidities of the members of the bridge affected the results only slightly. Concrete decks which are now used have substantial torsional rigidities and cannot be designed in this manner. Also the idealisation is very coarse to arrive at correct distribution of structural responses. But this method yielded moments within 7% of those obtained by rigorous methods for timber bridges. Courbon s Method The load distribution method developed by Courbon assumes a linear variation of deflection in the transverse direction. The deflection will be maximum on the external girder on the side of the eccentric load and minimum on the other girder. Courbon s method is only applicable when the following criteria are satisfied. 1. The ratio of span to width is between 2 and 4 2. The longitudinal girders are interconnected by at least five symmetrically spaced cross girders. 3. The cross girders extend to a depth of atlesat 75% of the depth of the longitudinal girders. Henry-Jaeger Method Henry and Jaeger estimated the load distribution between longitudinal girders assuming that the cross girders can be replaced can be replaced by a uniform continuous transverse medium of equivalent stiffness. Hence the distribution of loading in an interconnected bridge deck system depends on the 3 non-dimensional parameters. A 3 12 L neit = 4 π h EI 4 π h GJ F = 2n L EIT EI 1 C = EI 2 3

6 Where, L = Span of the bridge H = Spacing of longitudinal girders N = Number of cross girders EI, and GJ = flexural and torsional rigitdites of longitudinal pair. EI T = flexural rigidity of the cross girder. The subscript 1 and 2 denote the outer and inner girder respectively This is the distribution coefficient method. We will later see that this is based on an earlier version of semicontinuum method which is used in this project. That particular version neglected the torsional rigidity of the transverse system. Graphs giving the values of the distribution co-efficient for the different conditions of number of longitudinal (2-6) and two extreme values of F i.e. zero and infinity are available. Distribution coefficients for the intermediate values are obtained by interpolation. Morice-Little Method In this method orthotropic plate theory is applied to concrete bridge decks. Design curves and full set of graphs for distribution coefficients of this method are given by Rowe. The torsional parameter (α) and the flexural parameter (θ) as stated below forms the basis of these curves. α = θ = (D xy + D yx) 2 (DxyD yx) b Dx 4 L D y Where, b = half the width of the bridge L = length of the bridge. The distribution coefficient are given for 9 standard reference points and load postions across the bridge width, and plotted against values of theta. The graphs are given for α = 0.0 and α = 1.0. For the intermediate values the distribution coefficients are evaluated using interpolation function. 4

7 Cusen-Pama Method The distribution coefficient approach of Morrice-Little is improved by taking the 9 harmonics of the series. The revised torsional parameter is now given be α = (D xy + D yx + D 1 + D 2) 2 (DxyD yx) They also considered the case of torsionally stiff and flexurally soft bridge decks by extending the range of α to 2. AASHTO Method This method is extremely simplified and used for obtaining longitudinal bending moments and shears due to live loads. In this method, a longitudinal girder plus its associated portion of the deck slab, is isolated from the rest and treated as an I dimensional beam. This beam is subjected to loads comprising one line of wheels of the designed vehicle multiplied by a function S/D where D has a prescribed value, in units of length, depending on the type of the bridge. The blanket D values depend only on the bridge type. However, it is confirmed that the load distribution depends upon a number of factors like aspect ratio of the deck, flexural and torsional rigidities of deck in longitudinal and transverse direction, width of load w.r.t bridge width, vehicle edge distance etc. Ontario Method Ontario employed a D type parameter developed from orthotropic plate theory, for distribution of vehicle live loads between the longitudinal girders of the bridges. This method overcomes all the limitations of AASHTO method. 5

8 2.2 RIGOROUS METHODS Orthotropic Plate Method This method idealises the bridge deck as an orthotropic plate having constant thickness but different flexural and torsional properties in two mutually perpendicular directions. The deflections of an orthotropic plate are governed by a non-homogenous plate equation which is solved to obtain the solution in terms of deflection. The various plate responses are then calculated using suitable expressions. Finite Difference Method The finite difference method developed by Nielson and later by Westergaards was used by many in the analysis of bridge decks with complex shapes and complicated boundary conditions. Amongst them, Heins and Loney are notable. Cusen and Pama summarised the application of this technique to the solution of orthotropic plates. In this method, the deck is divided into grids of arbitrary mesh size and the deflection values at the grid points are treated as unknown quantities. The governing equation of the deck and the accompanying boundary conditions are expressed in terms of these unknowns. The resulting simultaneous equations are then solved for unknown deflections. Grillage Analogy Method The approximate representation of the bridge decks by a grillage of interconnected beams is a convenient way of determining the general behaviour of the bridge under loads. Henry, Jaegar and Lightfoot Sawko did pioneering work in this method. Hambly summarised the application of this technique to bridge deck analysis. Jaeger Bakht gave recommendations on this idealisation of bridge types as grillages. This method of analysis anvolves the idealisation of the given bridge deck as an assembly of one dimensional beams subjected to loads acting perpendicular to the plane of assembly. Both the flexural and torsional rigidities of beams are taken into account. Grillage beams in the longitudinal direction are made to coincide with the centre line. Finite Element Method The finite element method is the most powerful method of analysis arising from the direct stiffness method. Zienkiewicz, Desai Abel and Martin Carey did pioneering work in this field. The method can be briefly outlined into three basic categories: 1) Structural Idealization: - A mathematical model of the structure is formulated in which it is represented as an assembly of discrete elements. Bridge decks may be represented by one dimensional beam elements or continuum elements (plate, shell etc.) or a 6

9 combination of these elements. Each element has finite dimensions and properties to perform the analysis. 2) Evaluation of Element Properties:- The choice of a particular type of element depends upon a number of factors like Geometry of structure, position of loading vehicles, convergence properties of the Elements etc. Regions in which stress concentrations are expected, are covered with a Finer division of elements than other regions in the structure. Therefore correct division of the bridge superstructure into elements and evaluation of their properties should be carried out with utmost care. 3) Structural Analysis of the Element Assemblage.:- The following requirements are to be satisfied: i) Equilibrium of internally and externally applied at forces at each node of the Element. ii) Geometric fit or compatibility of the element deformations in such a manner. That they meet at the nodal points in the loaded configuration. iii) Internal force/ displacement relationships must be established with each. Element as dictated by the existing geometry and material properties. These requirements are satisfied by the usual matrix method of analysis, by the direct stiffness approach where an assumed displacement function is considered to represent the actual deformation of the region. Finite Strip Method The finite strip method is a hybrid procedure which combines some of the advantages of the series solution of the orthotropic plates with the finite element concept. In this method, bridge deck is represented by strip elements extending from support to support. This bridge superstructure may be idealized either as a three dimensional assembly of strips of equivalent orthotropic idealization of bridge structure. Simple displacement interpolation functions are then used to represent displacement fields within and between individual strips. The step involved in the finite strip is : a) To assume a displacement function b) To establish the continuity at the boundaries with adjacent strips for slopes and deflections to get the displacement functions constants. Here it is necessary to express the function for deflection in terms of displacement function and a sine term. Slopes and deflections in terms of the displacement function and a sine term. Slopes and deflections at each edge are the amplitudes of the sine function. c) The total energy of a strip, is given as a sum of internal strain energy caused due to stress resultants and the potential energy due to external loading on the strip d) The total energy of the plate is the sum of the energies of all the strips. e) From the principle of minimum potential energy, the total energy of the plate is differentiated and equated to zero to obtain a set of simultaneous equations which, when solved will give the solution in terms of displacement or force. 7

10 Semicontinuum Method This simplified yet accurate method can quite easily be modeled on a micro computer. This method gives accurate results, well comparable to other rigorous methods at a lesser solution time. Also, the idealization and interpretation are quite simple. In this method, the longitudinal bending and torsional stiffness of the bridge are concentrated in a number of one dimensional longitudinal beams, coinciding with the physical girder positions, while the transverse bending and torsional stiffness are uniformly spread along the length in the form an infinite number of transverse beams. 8

11 2.3 CRITICAL DISCUSSION The accuracy of any method of analysis for a particular structure is difficult to predict or even check. It depends on the stability of the idealized model to represent three very complex characters i.e. the behavior of the material, the geometry of the structure and actual loading. This aspect together with the method of analysis results in giving out the efficacy of solutions in each case. More recently with the advent of computers and development in numerical techniques, accuracy in solving the complex problems has been increased. The methods discussed above will be examined for the specific aim of solving concrete slabs on girder skew bridges with or without cross section girders for various kinds of loadings. Critical Discussion of Rigorous Methods The orthotropic idealization of the physical structure of the girder bridge deck (with or without cross girders) into an equivalent orthotropic plate having different flexural and torsional rigidities in two mutually perpendicular directions is not truly correct. This is because the longitudinal stiffness for flexure and torsion are uniformly distributed in orthotropic plates and represented as Dx and Dxy respectively. However in case of T girder bridge, longitudinal bending and torsional stiffness are concentrated at the actual position of longitudinal girders. So the computed bending moment and shear are subjected to significant error, especially when the bridge deck is wide and the load occupies only a portion of the width. The finite difference method which is employs a numerical technique to solve orthotropic plate equations has got the same limitations with respect to idealization of the physical structure. However this method has overcome the slow convergence problem of series solution of orthotropic plate equation and accurate solutions can be achieved in a reasonably short time. The stiffness methods described are powerful methods to analyze the bridge superstructure. These methods will be viewed with reference to idealization of structure, solution of matrices and time and cost required to do the analysis. These methods are critically analyzed here. The grillage analogy method idealized the structure by representing it as a plane grillage of discrete inter connected beams. But in transverse direction, when no cross girders are present, i.e., when traverse stiffnesses are more or less uniformly distributed, the results are not as accurate as they are when there is a significant number of cross girder. It is difficult to visualize for the designer what kind of loading will give correct results. The finite element method is the most versatile of all and for difficult decks it is sometimes the only valid form of analysis. The beauty of the method is that one can analyse any structure with it. Its only drawbacks are lengthy data preparation, careful selection of elements, expert idealization, and interpretation of results and requirements of considerable computer time for analysis. No layman can use this method. These factors lead 9

12 to high cost. Due to this problem, this method is not popular for designing low cost, small span bridges. The finite strip method, which is a special case of the finite element method combining the advantages of the series solution of orthotropic plate method reduces computational requirement considerably. The semicontinuum method, as against all the methods discussed above, is near perfect in the idealization of T girders bridges with or without cross girders. The considerations of harmonic greatly reduce the number of unknowns. Thus, this method can be successfully programmed in a computer and very little computer time is needed. Further, this method includes torsion in both the directions which are neglected in the past methods. 10

13 3.0 SCOPE OF THE INVESTIGATION The advantages of the investigation lie in the mere simplicity of the method. The simplicity of the method combined with the accuracy of the results mean that even a microcomputer can solve the system of equations that the problem leaves us with. The major advantages of the semicontnuum methods over the other methods are listed below :- Near perfect idealization of the structure Ease of formulation Simplicity of computation Lesser number of unknowns Very less processor power. 11

14 4.0 THEORY 4.1 ORTHOTROPIC BRIDGE ANALYSIS An orthotropic plate is defined as one which has different elastic properties in two different directions. In practice two forms or orthography may be identified Material Orthography Shape Orthography A common example of material Orthography is Ply Wood. Bridges however normally fall into the second category which is the shape Orthography. Governing Equations :- For the element of an orthotropic plate the governing equations are :- Ex σ x = [ ε x + ν yε y] (1 νν x y) Ey σ y = [ ε y + ν xε x] (1 νν x y) τ xy = G γ xy where σ x and σ y are the normal stresses in the x and y directions respectively and ε x and ε y are the normal strains in the x and y directions respectively; τ xy is the shear stress on the section perpendicular to the x direction and parallel to the y direction and γ xy is the corresponding shearing strain in the plane perpendicular to the z axis. The modulus of elasticity are E x and E y in the x and y directions respectively; and are the Poisson s ratios and G is the shear Modulus. It is assumed that E xν y = Ey ν x 12

15 Applying the strain displacement relationship, the stress- strain equation can be expressed in terms of the transverse deflection w in the form. σ x σ y 2 2 -Exz w w = + ν y 2 2 (1 νν x y) x y 2 2 -Eyz w w = + ν x 2 2 (1 νν y x) y x τ xy 2 w = -2Gz xy The moment resultants are obtained as follows w M = σ zdz = - D + D x 2 2 x x x 1 A 2 2 x w M = σ zdz = - D + D y 2 2 y y y 2 A 2 2 y M = - τ zdz = + D xy xy xy Ax M = τ zdz = D yx yx yx Ay 2 w xy 2 w xy w y w x The bending rigidities are defined by D x and D y, the coupling rigidities by D 1 and D 2, and the torsional rigidities by D xy and D yx. Owing to possible difference in shape of the section in the x and y directions, D 1 and D 2 are not necessarily equal but usually considered to be so. The bending moments per unit width in the x and y directions are M x and M y respectively and the twisting moments are denoted by M xy and M yx. 13

16 Consideration of the equilibrium of the moments and forces acting on the element leads to the following set of equations: - δ Mxy δ M y δx δy + Q y = 0 δmxy δmx + δy δx Q x = 0 δqx δqy + δx δy +p( x,y ) = 0 In which Q x and Q y are the shearing forces per unit width of the section. By eliminating these shearing forces, the equilibrium equation leads to Mx Mxy Myx My + + = p(x,y) 2 2 x x y x y y Substituting the values of the moment resultants, the governing equations of an orthotropic plate is obtained as Where w w w x x y y D x + 2H + D y = p(x,y) H = (D xy+ D yx + D 1 + D 2 ) The shearing forces may be expressed in terms of w as follows w w Q = D + (D + D ) x xy w w Q = D + (D + D ) y y x 3 3 x x yx x y xy

17 In plate problems it is frequently necessary to reduce the number of boundary conditions and this is done by representing the twisting moments as a vertical force system. The supplemented shearing forces which result from this representation are obtained as follows: V V y x 3 3 w w = D x + (D yx + D xy + D 1) 3 2 x xy 3 3 w w = D y + (D xy + D yx + D 2) 3 2 y yx 15

18 5.0 FLOWCHART The flowchart describes the analysis of a simply supported right bridge for lateral load distribution by the semi continuum method Start Read DATAFILE; Call Function MOM;(calc moment) Call Function RMATR;( R Matrix) Harmonic N = 15; Call CONST; calculates constants for N Call AMATR; calculates A matrix for N Call EQN; to solve for ρ vector For I = 1 to N Call CONST; Call AMATR; Call EQN; Call MSDIST; To calculate distributed Moments, Shears and deflections and to print the results Stop 16

19 5.1 FLOWCHART DETAILS The Program or SECAN11.m :- The program only performs the function of a controller. It takes the values from the datafile and then organizes the various functions in order. It prints the final result. THE Functions:- 1. AMATR.m This function calculates all the terms of the [A] matrix for each harmonic number according to a certain order. The size of the matrix is 2*NG * 2*NG where NG is the number of Girders. 2. CONST.m This function calculates the various constants namely k r, m r, c r, λ r, η r and μ r which are used in the construction of the [A] matrix. For each harmonic r varies from 1 to the number of girders for each of the above constraints. 3. DEFLEC.m This function calculates free deflections in a longitudinal beam due to applied point loads at specified reference locations. 4. EQN.m This function solves simultaneous equations :- [ A]{ ρ } = {R} 5. FINDEF.m This function calculates the deflections of the longitudinal beams after taking into account of the transverse distribution of loads. 6. MOM.m This subroutine calculates coefficient for free moment due to one longitudinal line of loads. The coefficients are stored in Array BM() separately for each harmonic number. Thus for a given line of loads the free bending moment ML is given by :- π x 2π x ML = BM(1)sin + BM(2)sin +... L L 17

20 7. MOMSER.m This subroutine calculates free moments and free shears due to one longitudinal line of loads in a beam at a specified distance from the left hand simple support. The moments and shears are stored in AMM and SHR respectively 8. MSDIST.m This subroutine calculates the transversely distributed moments and shears at specified reference points due to a stated number of harmonics according to provided equations. The function MOMSER.m is called from here. 9. RMATR.m This function calculates the 2NG terms of the {R} vector for all the lines of loads. 18

21 6.0 THE PROBLEM Span = 1016 : Girder Spacing = 99 : Slab Thickness = 7.5 : E = 3e6 : G = 1.5e6 I = 6.124e5 : J = 0.177e5 : Load details are given in the figure each each Details of a bridge and applied loading 19

22 6.1 INPUT FILE DETAILS Number of Harmonics N = 5; Number of Girders NG = 5; Total Bridge Span SPAN = 1016; Young s Modulus for the girders. E = 3e6; Shear Modulus for the girders G = 1.5e6; NGG = NG - 1; Spacing of the girders from the left extreme for I = 1:NG GS(I) = 99; End Moment of Inertia and Torsional Inertia for all girders starting from the left for I = 1:NG GMI(I) = 6.124e5; GTI(I) = 0.177e5; end Thickness of the slab T = 7.5; Young s Modulus for the slab EC = 3e6; Shear Modulus for the slab GC = 1.5e6; Number of Concentrated loads M = 3; Load Values of the Concentrated Loads W(1) = 16000; W(2) = 16000; W(3) = 4000; Spacing of the Loads DLS(1) = 380; DLS(2) = 508; DLS(3) = 635; Number of Load Lines NW = 2; Spacing of the loads line with respect to the left most girder DLG(1) = -20; DLG(2) = 52; Point of reference NREF = 2 XREF(1) = 380; XREF(2) = 508; 20

23 7.0 RESULTS Rigidities From Program From example in Jaegar & Bakht [1] DY e8 DYX e8 Bending Moments From Program From example in Jaegar & Bakht [1] K e6 7.1e6 K e e6 K e e6 K e e6 K e6-0.69e6 R Matrix From Program From example in Jaegar & Bakht [1] R(1) R(2) R(3) R(4) R(5) R(6) R(7) R(8) R(9) R(10) λ From Program From example in Jaegar & Bakht [1] λ λ λ λ μ From Program From example in Jaegar & Bakht [1] μ μ μ μ

24 α From Program From example in Jaegar & Bakht [1] α α α α A Matrix From the Program which is a perfect match (1 st Harmonic) ρ (For the 1 st Harmonic) From program From example in Jaegar & Bakht [1] ρ(1) ρ(2) ρ(3) ρ(4) ρ (for the 2 nd Harmonic) From program From example in Jaegar & Bakht [1] ρ(1) ρ(2) ρ(3) ρ(4) ρ (For the 3 st Harmonic) From program From example in Jaegar & Bakht [1] ρ(1) ρ(2) ρ(3) ρ(4) ρ(10)

25 ρ (For the 4 th Harmonic) From program From example in Jaegar & Bakht [1] ρ(1) ρ(2) ρ(3) ρ(4) ρ (For the 5 th Harmonic) From program From example in Jaegar & Bakht [1] ρ(1) ρ(2) ρ(3) ρ(4) Mid Span Moment From Program From example in Jaegar & Bakht [1] M e e6 M e e6 M e e6 M e e6 M e e6 Girder Shears From Program From example in Jaegar & Bakht [1] V V V V V

26 8.0 DISCUSSION 1. The code to do the analysis of a right bridge was written by me in Matlab successfully to solve the problem of an orthotropic bridge deck using a semicontinuum method. The algorithm for the code was from Jaeger and Bakht. 2. The results of the code were validated using an existing problem from the book Jaeger and Bakht and an agreeable match was obtained between the results arising from the code and that which was mentioned in the book. 3. A good understanding of the working of the semicontinuum method was obtained. 9.0 PLAN FOR THE NEXT SEMESTER 1. The first part of the work will involve converting the code written by me to accept an angle input and analyze the bridge with a skew angle. 2. I shall then get the design curves for various load conditions for a case of a skew bridge from the program. 3. I shall try to perform bridge deck analysis for various classes of loading based on the IRC code. 24

27 10.0 REFERENCES 1. Leslie G Jaeger and Baidar Bakht, Bridge Analysis by Microcomputers McGRAW- HILL, New York, A.R. Cusens and R.P. Pama, Bridge Deck Analysis, John Wiley & Sons, Baidar Bakht and Leslie G Jaeger, Bridge Analysis Simplified McGRAW- HILL, D. Johnson Victor, Essentials of Bridge Engineering Oxford Book House,