**3. Results**

*Advances in Structural Health Monitoring*

**2.5 Numerical calculation method**

using the effective stiffness of the rails and pads.

In order to characterize the dynamics of concrete sleepers, many types of numerical models for concrete sleepers have been proposed. Grassie [5] proposed a simplified two-dimensional dynamic model in the free-free condition, with their analytical results having been verified by comparisons with 12 different types of sleepers. Dahlberg and Nielsen [27] developed a concrete sleeper model popularly denoted as the "Timoshenko Beam on an Elastic Foundation," for both free-free and in-situ conditions. Lam et al. [17, 18] also modeled the sleeper as a Timoshenko beam, the supporting ballast as discretized springs, and the rails as masses (with reference to previous studies [28, 29]). Furthermore, Kaewunruen and Remennikov [20] modeled the in-situ concrete sleeper as the sleeper, and the ballast and pads

In order to characterize affecting mechanisms of sleeper damage on modal characteristics, numerical calculations based on a finite-element model were performed in this study. **Figure 6** shows the subject numerical model for the concrete sleepers. Numerical analysis for the concrete sleepers was performed by LS-DYNA, version R8.0.0 [30]. A sleeper's concrete was modeled as hexahedral solid elements, and its steel wires and stirrups were modeled as beam elements. Supports and loading points for loading test analyses were modeled as rigid elements. A sleeper itself was modeled as a symmetrical model. Solid elements (concrete) and beam elements (steel wire and stirrups) share actual nodes to prevent slippage from occurring. **Table 4** shows the material properties of each element in the sleeper model. Young's modulus of concrete was set from the stress-strain curve. Uniaxial compression strength and uniaxial tensile strength of concrete was set from Young's modulus and Design Standards for Railway Structures and Commentary (Concrete Structures) [31]. For concrete, a material that can address cracking with tension softening and crushing was ultimately used [32]. Prestressing was reproduced by initial stress to the steel wires in the axial direction.

In order to reproduce the influence of bending damage upon modal characteristics, numerical simulations of loading and unloading under the same experimental bending conditions were first calculated, and then, the modal characteristics of damaged sleepers were investigated by eigenvalue analysis in each loading-unloading step.

*Numerical model of a concrete sleeper: (a) finite-element of concrete and (b) reinforced steel.*

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**Table 4.**

**Figure 6.**

*Element types and specifications of the numerical model.*
