Numerical Analysis of Rail-Subgrade System

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1 Numerical Analysis of Rail-Subgrade System LUCACI Gheorghe Politehnica University of Timisoara, Civil Engineering Faculty, Ioan Curea 1, Timisoara, Romania, Abstract:- In the present work, a numerical method for deriving the internal forces along railway tracks is presented. The system rail-subgrade is simulated as a beam on elastic foundation. The formulation of the numerical procedure is based on application of Transfer Matrix method. Using this method the differential equation of the problem and the corresponding boundary values are incorporated into a linear algebraic system. A computer algorithm is developed and representative examples are included. Key-Words:- Railway track, Transfer matrices, RailUIC 60, Beam on elastic foundation. 1 Introduction During last decades, solutions of civil engineering problems are mostly based in computer algorithms due to their complexity. For this reason, the suitable for computers numerical methods have wide use today. The aim of the proposeork is the numerical modeling of the theory of the beam on the elastic foundation in order to perform analysis of the system railsubgrade. The rail is simulateith beam supported to elastic foundation [1,2] subjected to loads produced by the wheels of trains. The reaction of the foundation can be considered to be proportional with the beam deflection [2,3]. The elasticity constant is relateith the type of ground [1]. The data of the problem consist of the loads of the rail, the cross-section geometrical parameters, the modulus of elasticity and the elastic constants of the ground and sleepers. The requirement of the work is the numerical determination of bending moments, shear forces and deflection distribution along the rail. 2 Formulation of the method In order to develop the basic equation that describes the deflection w, bending moment M and shear force distribution Q along the rail, a beam on elastic foundation is considered (fig. 1). Using the equilibrium equations of an element of the beam, the following relations [] are obtained: dw = θ 2 EI = + M 2 3 EI = + Q 3 (1) (2) (3) EI = q + p * () where: w: deflection θ: slope q: distributed loading due to the foundation reaction p*: external distributed loading I: moment of inertia E: modulus of elasticity. ISSN: ISBN:

2 Due to the considered elastic behavior of the subgarde it can be written: q=kw (5) where k is the subgrade modulus. Then, the equation () takes the form: w(x i )=w i w (x i )=w i EIw (x i )=+M i (9) EIw (x i )=+Q i EI + kw = p * (6) Fig. 1: Beam on elastic foundation The solution [5] of latter differential equation is the following: and w = e e βχ ( A cos βx + A sin βx) + 1 p ( A3 cos βx + A sin βx) k βx * Where 2 (7) w(x j )=w j w (x j )=w j EIw (x j )=+M j (10) EIw (x j ) =+Q j κ β = (8) ΕΙ Substituting the following general boundary conditions into differential equation (6) and making some standard operations, the following system of algebraic equations is resulted: ISSN: ISBN:

3 [u i ] = {W ij } [u j ] (11) where Wi W ' i u i = M i (12) Qi 1 [ ] the state vector at x=x i Wj W ' j u j = M j (13) Q j 1 [ ] the state vector at x=x j Fig.2: Action of a concentrated load P and a concentrated bending moment M the equilibrium equations for this point lead to the transfer matrix {W c ij} which have the following form: = ij M (15) P { W c } {W ij }= R o R 1 R 2 /EI R 3 /EI -p*(1- R o )/k -β R 3 R o R 1 /EI R 2 /EI -p*β R 3/k -β EIR 2 -β EIR 3 R o R 1 -p*ei β R 2 /k -β EIR 1 -β EIR 2 -β R 3 R o -p*ei β R 1 /k (1) The matrix {W ij } is called local transfer matrix of the element i-j, because it relates the state vector of the node i with the corresponding state vector of the node j. The quantities R 0, R 1, R 2, R 3 are given in the appendix. When the nodes i, j are in both sides of a point in which a concentrated load P (or/and a concentrated bending moment M) is acting (fig. 2), The matrix {W c ij} is called local concentrated matrix. By application of the matrix equation (11) in all the nodes of a beam, the following matrix equation can be obtained: [u L ] = {Wg} [u F ] (16) where [u L ], [u F ] are the state vectors at the last and the first node respectively and {Wg} is the global transfer matrix method given by the equation: ISSN: ISBN:

4 n { Wg} = { k = 1 W k } (17) In equation (17) {W k } is the k-th local transfer matrix and n is the number of the local transfer matrices. In the x system of algebraic equations (16) the nodes L and F are considered as the reactions of the beam. Then, taking into account the boundary conditions, four additional equations can be used. Therefore, the state vector of the reactions can be determined by the following corresponding 8x8 linear system incorporating the x matrix equation (16) and the four boundary conditions. Following the calculation of the state vector of the reactions, the state vector of any point Y can be calculated by the matrix equation [u Y ] = {W YF } [u F ] (19) where [u Y ] is the required state vector of the arbitrary point Y, [u F ] is the state vector of the first reaction (which is calculated by the equation (18)) and {W YF } is the global transfer matrix between the nodes Y and F. W F W F M F Q F W L W L M L Q L α 11 α 12 α 13 α α 15 α 21 α 22 α 23 α α 25 α 31 α 32 α 33 α α 35 α 1 α 2 α 3 α α = (18) In matrix equation (18), the quantities α ij are the elements of the global transfer matrix {Wg}. In this equation, the 5 th and 6 th row represents the boundary conditions of the first reaction F, while the 7 th and 8 th one represents the corresponding boundary conditions of the last reaction L. 3 Results Above procedure is applied in a rail of type UIC60 [6-8] with Jx=3083x10 mm, Ε=2.1x10 5 N/mm 2, κ=50 N/mm, Ρ=120 ΚΝ. The distance between the pairs of the wheels is x 3 =0 m, while the distance between two neighboring wheels is x 23 =2.5 m. The resulted distributions of w(x), M(x), Q(x) are shown in figures 3(a)(b)(c). ISSN: ISBN:

5 ww (mm) x X (mm) (a) M HNmmL M (Nmm) x X (mm) (b) Q HNL Q (N) x X (mm) (c) Fig. 3: Distribution of (a) Deflection, (b) Bending moment, (c) Shear force in a UIC 60 rail ISSN: ISBN:

6 The maximum values of these distributions are summarized in the following table: Rail UIC 60 J x =3083x10 mm E=2.1x10 5 N/mm 2 k=50 N/mm P=120 KN Max{w} (mm) Max{M} (Nmm) Max{Q} (N) x In order to investigate the influence of the values J x, k, P on these distributions, the same problem is resolved by changing above parameters. Following are summarized the conclusions of this parametric study. Conclusions 1. Reduction by 31 % of the J x resulted 9.62 % increase in maximum deflection Max{w}. For the Max{M} and Max{Q} no valuable influence was shown. 2. Increase by 50 % of the k resulted (a) 0 % reduction of the Max{w} (b) 11.7 % reduction of the Max{M} (c) 2.19 % reduction of the Max{Q}. 3. Reduction by 50 % of the P resulted (a) 70.3 % reduction of the Max{w} (b) 56.3 % reduction of the Max{M} (c) 51.1 % reduction of the Max{Q}. References: [1] Esveld C., Modern Railway Track, second edition, MRT- Productions, [2] Boresi, A.P., Sidebottom, O.M., Advanced Mechanics of Materials, Wiley, [3] Gianakos K, Actions in railway tracks, Papazisis publs, Athens [] Vouthounis P., Engineering Mechanics Strength of materials, Athens [5] Oery H., Reimerdes H., Dieker S., Berechnung von interlaminaren spannungen in mehrschichtigen faserverbundwerkstoffen mit hilfe von uebertragungsmatrizen, Z. Flugwiss, Weltraumforsch 8 (198), Heft 6. [6] UIC CODE 86-5 O [7] PR-EN 13230: Railway applications/track-concrete sleepers and bearers [8] UIC CODE 860-O APPENDIX chcos =chβx cosβx chsin =chβx sinβx ISSN: ISBN:

7 R o =chcos R 1 =(chsin+shcos)/2β R 2 =shsin/2β 2 R 3 =(chsin-shcos)/β 3 shcos =shβx cosβx shsin =shβx sinsβx chβx=(e βx +e -βx )/2 shβx=(e βx -e -βx )/2 ISSN: ISBN:

CASE STUDIES IN RAILWAY CONSTRUCTION

MSC COURSE 2016/2017 AUTUMN SEMESTER CASE STUDIES IN RAILWAY CONSTRUCTION RAILWAY SUPERSTRUCTURE CALCULATION ZIMMERMANN-EISENMANN METHOD SZÉCHENYI ISTVÁN UNIVERSITY Zoltán MAJOR junior lecturer Conventional