2.2.2. Three-Point Bending Tests

According to EN 13230-2 [41], the standard requires positive and negative three-point bending tests for sleepers at the rail seat support. Only positive bending tests have been carried out due to the symmetrical shape of the samples. This means that the samples have the same positive and negative capacity. Also, the criterion requires articulated support and must be 100 mm wide, made of steel with Brinell: HBW > 240. A static load is applied at the mid-span to cause positive 3-point bending cracks and failure. Figure 4 shows the layout of the bending load process, also illustrates the excitation locations of the impact hammer, which have been strategically installed to perform two modal tests under different bending loads. Figure 5 demonstrates two pattern tests of the samples under different bending load conditions. The investigations are sufficiently performed in order to comply with BS EN 13230-1 standard [41].

**Figure 4.** *Cont*.

**Figure 4.** Testing arrangement of a full-scale FFU composite beam under bending loads.

**Figure 5.** Testing procedure of ultimate load test and repeated load test.

#### *2.3. Determination of Dynamic Elastic Modulus*

Dynamic elastic properties of a material can thus be calculated if the mass, geometry, and mechanical resonant frequencies of the test sample can be measured. This means that the dynamic Young's modulus can be identified utilizing the resonant frequency in either the bending or longitudinal mode of vibration, as given in Equation (1) [25,42], whilst the dynamic shear modulus, also known as modulus of rigidity, can be found by employing twisting resonant vibrations [23,43]. In this section, we only focus on the determination of dynamic elastic modulus in free-free boundary conditions (for future benchmarking purpose). This is because, based on the experimental results in the following part, it can be found that the first bending mode in a vertical plane obviously dominates the first resonant mode of vibration in free-free boundary conditions. By employing bending vibration modes, slender beams based on the Euler–Bernoulli theory of bending vibrations can be applicable to the test sample. The influences of rotational inertia and shear can be negligible generally. The equations derived on these assumptions are sufficient for relatively slender beams of lower modes. Nevertheless, this theory is likely to slightly overestimate the natural frequencies. According to Euler–Bernoulli's basic equations of flexure, the dynamic elastic modulus in bending of a beam can be assessed under forced free or bending free vibrations. The dynamic elastic modulus in bending of a beam can be expressed as Equation (1):

$$\left(\frac{E\_{dy}}{\rho}\right)\_n = \frac{\left(2\pi L F\_{f,n}\right)^2}{K\_n^4 \rho},\tag{1}$$

where *Edy* is dynamic elastic modulus (Pa), n is mode number, *L* is free length (m), ρ is stabilized density (kg·m−3), *Ff*,*<sup>n</sup>* is frequency of nth mode (Hz), and *Kn* is a coefficient related to the beam's support condition and mode number (e.g., *K*<sup>1</sup> is equal to 4.73 for a free-free end condition and 1.785 for a fixed-free condition [44], as given in Table 2). Finally, β is the square value of gyration radius divided by free length as provided in Equation (2):

$$\beta = \left(\frac{1}{L}\sqrt{\frac{I}{A}}\right)^2 = \frac{I}{L^2A}.\tag{2}$$

Herein, β denotes the square value of gyration radius divided by free length, *L*. *I* is the moment of inertia about the axis and *A* is the cross-section area. If no axis is specified, the centroidal axis is assumed.

**Table 2.** Dimensionless coefficients for computing the frequencies of a FFU composite beam in free-free conditions.


It is important to note that Equation (1) is a conceptual equation of vibration, which ignores the influence of rotational deformation and shear load in a simulation. Nevertheless, for an application of using this equation, it could be dominated by L/h ratios (i.e., more than 58 in a fixed-free end condition or more than 20 in a free-free end condition) [45]. In this paper, the modeling of FFU composite beam does not take into account shear deformation and rotational bending effects (as defined by the Timoshenko theory), due to the ratio of L/h ≥ 20 (thin beam). Additionally, both previous equations are limited to isotropic materials. It is noted that the FFU composite beam model was considered as an isotropic material. In fact, this material would be considered to be anisotropic, but we measure its dynamic responses only in the vertical direction. Thus, the material can be considered conceptually to be isotropic. The following section presents the numerical investigations of a FFU composite beam modeling using the dynamic parameters obtained from the experiments in order to determine the dynamic elastic modulus of the beam.
