Considering that the load transmission to a railroad track is achieved through the movement of a wheel over a long beam, one simple analysis approach involves the application of the Beam on Elastic Foundation theory. The BOEF theory can be applied to calculate rail and tie deflections, bending moments and shear forces in the rail, and rail seat force, ballast pressure and vertical stress between the tie and the
ballast. In the BOEF theory, the rail is simplified as an infinitely long Euler-Bernoulli beam supported by evenly distributed linear springs representing a continuous elastic foundation (the dense liquid or Winkler foundation).
Euler-Bernoulli Beam: Euler-Bernoulli beam theory, developed by Leonard Euler and Jacob Bernoulli in 1750s [25] is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. The beam under consideration extends from x = 0 to x = L, and has a flexural rigidityEI as shown in Figure 2.4.
Figure 2.4: Schematic Representation of an Euler-Bernoulli Beam Sub- jected to a Distributed Loading
Beam deflection (w) at a distance x as shown in Figure 2.4 is described by the differential equation in Equation 2.11.
EId
4w(x)
dx4 =−q(x) (2.11)
where q(x) is the distributed load on the beam.
Winkler Base Model: Conventional track systems consist of two parallel continuous beams (the rails), which are fixed at regular intervals on ties supported by the ballast bed from underneath and from the sides. The ballast bed rests on a subballast layer, which rests on the subgrade with the ultimate layer for the load transmission chain being a non-deformable formation. According to Winkler’s hypothesis (1867), at each
point of support, the compressive stress is proportional to the compression at that particular point [26],
Figure 2.5: Winkler Base Model
σ(x) = uw(x) (2.12)
where σ(x) is the compressive stress on the support, u is the foundation modulus, and w is the deflection of support, a shown in Figure 2.5.
For single point load P, the differential equation of the Beam on Elastic Founda- tion model is given by the following equation.
EId
4w(x)
dx4 +uw(x) = 0 (2.13)
whereEI is the flexural rigidity of the beam (rail or tie),w(x) is the vertical deflection at point x and u is the Winkler’s constant or track modulus.
The solution of the differential equation is given by Equation 2.14.
w(x) = P β 2ue
−βx
(cosβx+ sinβx) (2.14)
where β = (4EIu )1/4. The supporting line force along the beam is calculated as,
F(x) =−uw(x) (2.15)
The successive differentiation of the solution of the BOEF differential equation gives equations for slope (θ), bending moment (M), and shear force (V) at distance
x from the point load. θ(x) =−P β u e −βx (sinβx) (2.16) M(x) = P 4βe −βx (cosβx−sinβx) (2.17) V(x) = −P 2e −βx (cosβx) (2.18)
Multiple axles can be incorporated in this model using the principle of superposi- tion.
The rail seat load is calculated by the following equation.
Qn =FnS (2.19)
where Qn is the rail seat load for tie numbern, Fn is the supporting line force along the beam for tie number n, and S is the tie spacing.
Qn=FnS =uwnS (2.20)
where wn is the deflection of tie number n.
A.N. Talbot used the theory of BOEF for track design and analysis in the 1920s [9]. This is known as “Talbot’s Method”.
Limitations of the BOEF Approach
The BOEF theory is extensively used, and is one of the simplest and easiest ap- proaches for railroad track analysis under loadings. However, this analysis approach has the following limitations:
1. The track substructure is estimated in terms of one generic track modulus value in the BOEF formulation, but the substructure behavior is complex because there are multiple layers with varying properties. Hence, representation using
a single stiffness value eliminates the possibility for distinguishing between the responses of individual track substructure layers.
2. As per the BOEF theory, the rail is assumed to be supported continuously. However, in a conventional track, the rails are discretely supported by cross-ties.
3. According to the BOEF formulation, in the parts of the track that experience upward deflection, tensile forces will develop in the foundation. But in real- ity, the unbound nature of ballast layers means that no tensile forces can be mobilized to support any uplifting tendency of the superstructure.
4. The track response is assumed to be linear according to the Winkler formulation; however, geomaterials and most pads do not behave linearly [27].
5. The BOEF model does not take the weight of the rails and ties into account. This can, however, be justified somehow because the weight of rails and ties are negligible compared to the forces exerted by considerably large wheel loads.
6. Shear deformations in the rails are not included since the rail is modeled as Euler-Bernoulli beam.
7. In this model, the Winkler springs are independent of each other, and this model does not consider the interaction among the springs, which is unrealistic.
8. The deflection calculated from the BOEF formulation corresponds to static loading only; the dynamic loads are estimated by increasing the value of static load. Therefore, the effect of dynamic loading is only an approximation, and inertia effects are not taken into consideration during analysis.
Considering that the Beam on Elastic Foundation model does not take into ac- count the interaction among the Winkler springs, several models were developed by researchers in the past to overcome this limitation. Some of these models are discussed in the following sections.