Identification of the degadation of ailway ballast unde a concete sleepe Qin Hu 1) and Heung Fai Lam ) 1), ) Depatment of Civil and Achitectual Engineeing, City Univesity of Hong Kong, Hong Kong SAR, China. 1) huqinsunny@gmail.com ABSTRACT The identification of ailway ballast degadation unde a concete sleepe is investigated following the Bayesian appoach. The ail-sleepe-ballast system is modeled as a Timoshenko beam with two added masses on an elastic foundation. In the model updating pocess, the stiffness of the ballast foundation is assumed to be continuous along the sleepe by using a polynomial. The accuacy of the identified ballast stiffness distibution stongly depends on the level of modeling eo and measuement noise. In the poposed method, Bayesian pobabilistic appoach is adopted to explicitly addess the uncetainties associated with measued modal paametes. A numeical case study is consideed in this pape to veify the poposed ballast damage detection method, and the analysis esults ae vey encouaging showing that it is possible to use the measued vibation data of the in-situ sleepe fo the pupose of ballast degadation identification. 1. INTRODUCTION Thee ae many methods developed fo monitoing the functional condition of the ails in the ailway tack systems. Howeve, the non-destuctive evaluation of ailway ballast unde the sleepes is still elying on visual inspection. It must be pointed out that visual inspection is only effective to obseve ballast damage on the suface, while ballast damage unde the sleepe cannot be obseved. The main function of the ballast is to suppot the sleepe and keep it in position. If the ballast is damaged, the stiffness that can be povided in suppoting the sleepe is educed. As a esult, the vibation chaacteistic of the ail-sleepe-ballast system is alteed. Theefoe, it is possible to detect the damage of ballast unde the sleepe based on the measued data of the in-situ sleepe. Timoshenko beam element was fist used to model the concete sleepe by Gassie (1995), a simple two-dimensional dynamic model was pesented to calculate 1) Gaduate Student ) Associate Pofesso 84
the dynamic chaacteistic paametes of the system. The poposed method was expeimentally veified using 1 types of sleepes. In 008, Dahlbeg used a vibating Timoshenko beam on an elastic foundation to analytically model the dynamic behavio of an in-situ concete sleepe. Dahlbeg obseved that the effect of ballast is significant fo lowe modes. This finding is consistent with the analysis esults in efeences (Lam 010, 01a). Lam (01b) conducted impact hamme tests on a segment of full-scale ballasted tack. It was concluded that the use of vibation data to detect damage of ailway ballast is feasible. It should be noted that these studies wee based on some discete models, which means the ballast foundation wee divided into seveal egions with equal size, the ballast stiffness in a egion is constant. Since the ballast stiffness vaiation is continuous along the sleepe, the discontinuous umps of ballast stiffness at the intefaces between two egions ae atificial. One of obectives of this pape is to extend the wok of efeences (Lam 01b; Hu 01) to ovecome this difficulty. In this pape, a class of models is developed by assuming that the distibution of ballast stiffness along the sleepe follows a polynomial. Uncetainty induced by the measuement noise and modeling eo is one of the most difficult poblems in vibation-based damage detection. Thus anothe new development in this pape is to employ the Bayesian model class selection (Beck 004) and the Bayesian model updating (Vanik 000) methods in ballast damage detection instead of following the deteministic appoach. The stuctue of this pape is as follows. In section, the poposed methodology is intoduced togethe with the elevant theoetical backgounds, such as the modeling of the ail-sleepe-ballast system, the Bayesian pobabilistic famewok and the Bayesian model class selection. In section, the numeical case study is pesented, which veifies and demonstates the poposed methodology. The conclusions and discussions ae given at the end of this pape.. PROPOSED METHODOLOGY AND BACKGROUND THEORIES.1 The ail-sleepe-ballast system model The main components of a ballasted tack ae ails, sleepes and ballast. In this pape, the ail-sleepe-ballast system is modeled as a Timoshenko beam with two additional masses on an elastic foundation with ballast stiffness k b as shown in Fig. 1. Accoding to efeences (Lam 01b; Hu 01), the two ails on the sleepe can be modeled as two diffeent additional masses which wee epesented by ml Lm and mr Rm, whee m is the nominal value of ail mass and L and R ae the two coesponding dimensionless scaling factos. Due to the aging poblem, the Young s modulus of the concete sleepe is not cetain, so anothe dimensionless scaling facto E is utilized to scale it in the model updating pocess. These thee dimensionless scaling factos, L, R and E, ae consideed as uncetain model paametes in model updating pocess. The efeence (Lam 01b) poposed to model the ballast with six egions of equal length, in each egion, the ballast stiffness is unifom, thus Lam employed six non-dimensional scaling factos to scale the coesponding stiffness value. This idea is simple but esulting in a elatively lage level of modeling eo, as the ballast stiffness along the sleepe should be continuous. This pape poposes to use polynomial 85
in appoximating the continuous vaiation of the ballast stiffness distibution. A -degee polynomial has moe uncetain model paametes (i.e., c 1 x + c x +c ), and an N-degee polynomial has N+1 moe model paametes. Theefoe, fo a N-degee polynomial model, a total of N+4 uncetain model paametes ae consideed in the model updating pocess, which ae gouped into a model uncetain paamete vecto = [ E, L, R, C1, CN+1 ]. Finite element method is employed in the analysis, and the sleepe is divided into 48 elements in this study. The classical beam theoies ae studied in the book (Kenk 001). The fomulations of Timoshenko beam on an elastic foundation is modified fom the Eule beam fomulations by consideing the shea effect, the element stiffness matix k is shown below: EI k (1 ) L 1 6 L 1 6 L 6 (4 ) 6 ( ) 1 6 L 1 6 L 6 ( ) 6 (4 ) 1 4 L 5L 4L 6L 4 1 4L 6L L 5L (1) whee E and I ae the elastic modulus and second moment of aea of the element, espectively. Effect of shea flexibility is descibed by the non-dimensional paamete, it is expessed as: 1EI GA L 0 () whee G is shea modulus, A 0 is the coesponding shea aea, which is smalle than the full coss-section aea. The coefficients i fo i = 1 to 6 can be expessed as: whee and 1 1 sinh cosh sin cos (a) 1 sin cosh sinh cos (b) 1 sinh sin (c) 4 L sin Lsinh L (d) 1 5 sinh cosh sin cos L L L L (e) 6 sin Lcosh L sinh Lcos L (f) sinh L L sin L (4) 86
k b 4 EI 1 4 (5) Consistent mass matix is employed in this study (Pzemieniecki 1985), the local element mass matix m is given by: 156 L 54 1L 4 1 m AL L L L L 40 54 1L 156 L 1L L L 4L (6) whee A, ρ ae coss-section aea and density of the element, espectively. System stiffness and mass matix ae assembled by the local element stiffness and mass matix. Modal paametes, such as natual fequencies and mode shapes, of the system can be calculated by solving the coesponding eigenvalue poblem. Fig. 1 Modeling of ail-sleepe-ballast system. Bayesian model updating and model class selection methods In most deteministic appoach, model updating poblem is teated as a minimization poblem to minimize the discepancy between the measued and calculated modal paametes (Lam 01b; Hu 01; Ruotolo 1997). The obective function in the minimization poblem usually consists of two pats, one is the diffeence in natual fequencies and the othe is the diffeence in mode shapes. Two diffeent weighting factos should be assigned to these two tems to show thei elative impotance in the model updating pocess. Nealy no commonly accepted method can be used to decide these two weighting factos unde the deteministic appoach. This pape poposes to follow the Bayesian pobabilistic famewok (Katafygiotis 000) to addess this poblem. This famewok tagets in calculating the posteio PDF of the model paametes ( in this pape) fo a given set of modal data and a given class of models. One can obtain the most pobable model by maximizing the posteio PDF which is equivalent to minimize the measue-of-fit function (Vanik 000): 87
n n n N θ φ Γ Ι ψ ψ Γφ θ m N T T T s J (7) 1 n1 Γφ whee N m is the numbe of inteest modes, N s is the numbe of data sets, is the standad deviation of the squaed cicula fequencies obtained fom the measued data, n is the -th mode measued natual fequency in the n-th set of data in ad/s, ( θ) is the -th mode calculated natual fequency fo a given θ, φ is the -th mode calculated mode shape. Note that Γφ and ψ ae nomalized to have unit nom. The selection matix Γ consists of only 1 and 0 that picks the obseved degees of feedom fom the model-pedicted mode shapes to match the measued ones, is the Euclidean nom. The uncetainty of the measued mode shapes is: 1 N s N s n1 ψ n n ψ ψ (8) whee ψ n is the -th mode shape of the n-th set of measuement and ψ is the aveage mode shape fo the -th mode fom the N s sets of data. The smallest value of J in Eq. (7) means the best fitting between the measued and calculated data, the coesponding θ is teated as the most plausible model. Fo a ail-sleepe-ballast system, many model classes can be consideed. Fo the pupose of damage detection, the identification of the best model class is essential. Fom the pobability theoy (Cox 1961), the pobability of a class of models conditional on the set of dynamic data is equied to detemine the plausible model class. Due to the limitation of space, only the most impotant fomula is pesented in this pape, which is the evidence of a model class conditional on a given set of data: N 1, θ θ θ p D M p D M p M fo = 1,, N M (9) whee θ epesents the most pobable model paametes in the model class M, and N is the numbe of uncetain paametes in θ. And θ is the Hessian matix of the function g(θ) (see Eq. (10)) evaluated at θ, it can be calculated by finite diffeence method at the most pobable model θ, the function g(θ) is given by:. NUMERICAL CASE STUDY θ ln θ θ, g p M p D M (10) A concete sleepe with length.4m and width 0.8m is employed to demonstate the poposed methodology, which is modeled as a Timoshenko beam with two masses on an elastic foundation. The nominal value of Young s modulus of the concete sleepe 88
takes the value of 8 10 9 N/m, the equivalent ballast stiffness and the additional mass ae 108 10 7 N/m and 4 kg, espectively. The density of concete sleepe is assumed to be 00 kg/m..1 Simulation of measued data The six equal ballast egions model (Lam 01b) is used to simulate the measued data. In ode to veify the poposed ballast damage detection method, damage is atificially intoduced to the system. The damage is simulated on the left hand-side of the sleepe and along / of the length of the sleepe, the stiffness eduction is 0%, thus the ideal ballast stiffness distibution is [0.7 0.7 0.7 0.7 1 1] multiplying the nominal value of ballast stiffness. To simulate the vaiation in ballast stiffness along the sleepe in eal situation, 1% andom fluctuation in the ballast stiffness is intoduced. The modal paametes can be easily calculated by solving the eigenvalue poblem though finite element method. The natual fequencies and mode shapes of the fist five modes wee consideed in the model updating pocess. To conside the effect of measuement noise, 1% RMS white noise is added to the calculated modal paametes to simulate the measued modal paametes. Ten sets of data ae consideed to fom multiple sets of measuements.. Bayesian model updating and model class selection Afte obtaining the measued data, Bayesian model updating can be used to find the most plausible model fo a given model class. Since thee is no idea about the degee of polynomial should be used, Bayesian model class selection method is used by calculating the evidences of a seies of model classes based on the given data. In this case study, 1 to 7-degee polynomial ae consideed, they fom seven model classes, which ae epesented by M, fo = 1,,7. The evidences of these seven model classes wee calculated and summaized in Table 1. Since the numeical values of the evidences ae vey lage, the logaithm is used to avoid numeical poblem. Table 1 Evidence of diffeent classes of models Class of models Logaithm of Evidence Logaithm of Likelihood facto Logaithm of Ockham M 1-494.5-445. -49. M 1917.9 1978.1-60. M 199.8 001.7-71.9 M 4 75.1 809.6-84.5 M 5 79.4 87.1-97.7 M 6 89.4 941.1-111.7 M 7 816.8 94.4-16.6 Fom Table 1, we can easily obseve that the 6-degee polynomial model with the lagest logaithm of evidence (89.4) is the most pobable model class. The logaithm 89
of evidence (79.4) of 5-degee polynomial model and logaithm of evidence (816.8) of 7-degee polynomial model ae both smalle than 89.4.. Ballast degadation detection Bayesian model updating was employed to obtain the set of most plausible uncetain paametes, which ae listed in Table. E is the scaling facto of the Young s modulus, L and R ae the scaling factos fo the left and ight additional masses, and C 1 to C 7 ae the coefficients of the 6-degee polynomial, which is used to appoximate the ballast stiffness distibution. E Table Updated model paametes of 6-degee polynomial model L R C 1 C C C 4 C 5 C 6 C 7 1.0001 1.000 0.9999 0.807 -.405 6.610-8.8515 5.96-1.719 0.764 Table states that the Young s modulus and two additional masses ae identified with high accuacy, the scaling factos ae close to unity. With the identified optimal model, the natual fequencies and mode shapes ae calculated and pesented in Fig. togethe with the measued modal paametes. It is clea that the matching between the measued and updated mode shapes is vey good. Fig. shows the identified ballast stiffness distibution togethe with the tue ballast stiffness distibution unde the sleepe. The ed solid line epesents the simulated ballast distibution, while the ed dashed line is the identified one. It tuly points out that the ballast along / length of the sleepe on left-hand side is damaged and the extent is about 0%. It can be concluded that the poposed method successfully identifies the ballast degadation in this case study. 4. CONCLUDING REMARKS This pape pesents the use of Bayesian model class selection and Bayesian model updating fo the detection of the ballast degadation utilizing measued vibation data of the in-situ sleepe. The numeical case study esults ae vey encouaging showing that the poposed methodology is feasible. This method can estimate not only the damage location but also the damage extent in the pesence of measuement noise. The esult also shows that polynomials can be used to epesent the continuous ballast stiffness distibution fo the pupose of ballast damage detection. Howeve, thee is a difficulty to be solved befoe this method can be put into eal application. It is not easy to tell whethe the sleepe o the ballast is damaged by consideing only the changes in modal paametes. In the poposed method, it is assumed that thee is no damage on the sleepe. Howeve, the concete sleepe may be damaged, because the damage on sleepe also can alte the modal paametes of the system (Lam 011), this may intoduce the complexity of ballast damage detection. Futhe eseach is equied in addessing this difficulty. 80
Fig. Matching between the measued and model-pedicted mode shapes Fig. Ballast degadation identification esults 81
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