Veveri 331/95, Brno, Czech Republic

Transcription

1 Advanced Materials Research Online: ISSN: , Vol. 969, pp doi: / Trans Tech Publications, Switzerland Probabilistic Simulation of the Mechanical Response of a Pre-Stressed Railway Sleeper: Initiation and Propagation of Cracks During Static Test Ladislav Řoutil 1,a, Václav Veselý 1,b and Zbyněk Keršner 1,c 1 Brno University of Technology, Faculty of Civil Engineering, Institute of Structural Mechanics, Veveri 331/95, Brno, Czech Republic a routil.l@fce.vutbr.cz (corresponding author), b vesely.v1@fce.vutbr.cz, c kersner.z@fce.vutbr.cz Keywords: pre-stressed railway sleepers, static test, FE modelling, crack width, statistical analysis, probability analysis Abstract. The paper deals with the mechanical response of a pre-stressed railway sleeper during a standardized static loading test - estimations of the variability of the sleeper s response and especially the probabilities of the occurrence of cracks of specific widths at specific load levels (as a consequence of random concrete parameters) based on the results of (3D) numerical models are presented. The paper follows on from and extends previous results obtained by the authors research team in cooperation with specialists from the sleeper manufacturing company ZPSV a. s. Introduction The presented paper is focused on the probabilistic simulation of a standardized static loading test [1] conducted on a pre-stressed monoblock railway sleeper of type B 91 S/1 (S/2) [2]. The test was performed on the mid-sleeper cross-section with particular focus on the prediction of the initiation, distribution and widths of cracks evolving at the sleeper s surface in dependence on the applied load during the entire loading process. The results of the statistical analysis and the probabilities of the maximum crack widths for selected loading levels (as a consequence of random concrete parameters) are presented. Note that the continual recording of these data during the whole duration of standardized static loading tests (load crack opening displacement diagram) involves certain difficulties and is not regularly carried out. Moreover, initiated cracks are not visible immediately even experienced technicians are not able to observe cracks thinner than 0.05 mm (with the naked eye, as is specified in [1]). As a consequence of these facts, a suitable numerical model can present a useful tool for the design and assessment of sleepers. Numerical Model of a Pre-Stressed Railway Sleeper Deterministic Model. The study of the standardized static loading test for the mid-sleeper crosssection (Fig. 1) was carried out in ATENA 3D software [3], enabling nonlinear FEM analysis with implemented principles from nonlinear fracture mechanics, plasticity and damage. More details about the geometry (Fig. 1), loading and boundary conditions, material parameters and nonlinear analysis (incl. FEM mesh details) are described in [4] and [5]. Note that the constitutive models used for the simulation were as follows: the 3D Nonlinear Cementitious 2 material model for concrete (a fracture plastic constitutive model based on the smeared crack approach; the values of the material parameters used in the study were set up for the concrete cubic strength f cu = 40 MPa according to CEB-FIB recommendations with respect to the specifications for the concrete used for the sleepers); the 3D Elastic Isotropic material model for the load-distributing pads; the Reinforcement material model for the pre-stressing reinforcement (with an experimentally obtained stress-strain diagram); and the Bond for Reinforcement material model, which describes the dependence between the bond stress and the bonding slip at the concrete reinforcement interface. The results of the simulation can be represented by a load deflection diagram (Fig. 2) and by a load crack opening diagram [4], [5]. Fig. 1 features graphical representations of crack propagation. Note that the displayed isoareas of the cracks widths are in better visual agreement All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (ID: , Pennsylvania State University, University Park, USA-11/05/16,21:01:39)

2 Advanced Materials Research Vol with the experimental observation than the crack patterns [4], [5]. As the widths of individual cracks resulting from the FEM analysis (based on the crack band model implementation) depend on the mesh density used, the sum of the crack widths is taken as a parameter for further analysis. Fig. 1 Photo of the standardized static loading test for the mid-sleeper cross-section (left, above); diagram of the geometry of the FEM model (bottom left, according to [4]); examples of the graphical output of the numerical model (according to [4]) crack patterns (right, above) and isoareas of the crack width (bottom right) Stochastic Model. The following dominant concrete parameters for the material model were considered random variables: Modulus of elasticity, compressive and tensile strength, and fracture energy. All of the input basic random variables involved, and the particular set of their statistical parameters (mean value, coefficient of variation (COV), and probability distribution function (PDF)) are summarized in Table 1. Statistical correlation among the input basic random variables was also considered (introduced using simulated annealing [6]); see Table 2. Correlation coefficients were set up based on previous experiments (e.g. [6], [7]) and compared to the literature [8]. The Latin Hypercube Sampling method [6], [9] was used for the generation of sets of input parameters, with 30 simulations of a railway sleeper s response being performed in total. The obtained results 30 load deflection diagrams are shown in Fig. 2. They (as well as load crack opening displacement diagrams) are used in the subsequent analysis. Table 1 Basic random variables of dominant material parameters Variable [Unit] Mean value COV PDF Modulus of elasticity E [GPa] Normal Compressive strength f c [MPa] Log-normal Tensile strength f t [MPa] Log-normal Fracture energy G f [J/m 2 ] Normal Table 2 Correlation coefficients between dominant material parameters Variable E f c f t G f E f c f t Sym G f 1

3 300 Structural and Physical Aspects of Civil Engineering Load [kn] Deflection [mm] Fig. 2 Simulated load deflection diagrams of standardized static loading tests conducted on the mid-sleeper cross-section Results of Probabilistic Analysis. A reliability analysis was performed based on the results of the statistical analysis the best fit PDFs of the total sum of crack widths at selected load levels. The action of the load was considered to be deterministic for the selected levels. The probability of the total sum of crack widths for the selected load levels was calculated using Cornell s PDF. The results of this reliability study are shown in Fig. 3. The graph can be interpreted as follows: from the intersections of the horizontal line, e.g. for the probability equal to 0.01, and from the probability graphs for the each load level, we can observe that with this probability the sum of the crack widths is equal to 0.1 (0.3, 0.75 and 2.35) mm for a load of 60 (70, 80, 90 and 100) kn. Note that a similar study can also be conducted for, e.g. the maximal crack width, but subjectivity is introduced to some extent in this way. Fig. 3 Graph of the probability of the total sum of crack widths for the selected load levels

4 Advanced Materials Research Vol Summary The paper deals with probabilistic modelling of a pre-stressed railway sleeper s response during a standardized static loading test for the mid-sleeper cross-section. Special attention is paid to the analysis of the initiation and propagation of the cracks. The results from both the statistical and the probabilistic analysis (taking into account the random nature of the material parameters of concrete) can be used for loading test management and evaluation, or for parametric studies focused on the optimal design of sleepers. Acknowledgement The paper was supported by the project CZ.1.07/2.3.00/ Support for the creation of excellent interdisciplinary research teams at Brno University of Technology. Research reported in this paper was also supported by Competence Centres program of Technology Agency of the Czech Republic (TACR), project Centre for Effective and Sustainable Transport Infrastructure (No. TE ). References [1] ČSN EN : Railway application Track Concrete sleepers and bearers Part 2: Prestressed monoblock sleepers. Czech standardization institute, 2004 (in Czech). [2] ŽPSV a.s., OHL Group Electronic catalogue of concrete products, [3] V. Červenka et al., ATENA Program Documentation, Theory and User manual. Cervenka Consulting, Prague, [4] J. Lahner, V. Veselý. Numerical study of failure of pre-stressed railway sleepers during evaluative bending test. In: Proceedings of Young Scientist 2009 (CD-ROM), Podbanske, Slovakia, April P. Platko, A. Eštoková, V. Kvočák (Editors), Technical University of Kosice, Faculty of Civil Engineering, 9 p., ISBN: [5] V. Veselý, Z. Keršner, L. Řoutil, Tool for estimation of crack distribution in face of prestressed sleeper s during check tests. Technical sheet and report, CIDEAS DVZ09_ ( Brno, [6] D. Novák, M. Vořechovský, D. Lehký, K. Bergmeister, R. Pukl, V. Červenka, Stochastic nonlinear analysis of concrete structures Part I: From simulation of experiment and parameters identification to reliability assessment. In 10th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP10, Tokyo, Japan, ISBN , [7] Z. Keršner, D. Novák, L. Řoutil, J. Podroužek, Stochastic Nonlinear Analysis of Concrete Structures Part II: Application to fiber-reinforced concrete facade panels. In 10th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP10, Tokyo, Japan, ISBN , [8] Probalistic model code. JCSS. < [9] D. Novák, M. Vořechovský, R. Rusina, FReET v. 1.5 program documentation, User s and Theory Guides, Brno,

5 Structural and Physical Aspects of Civil Engineering / Probabilistic Simulation of the Mechanical Response of a Pre-Stressed Railway Sleeper: Initiation and Propagation of Cracks during Static Test /