CASE STUDIES IN RAILWAY

CONSTRUCTION

MSC COURSE

2016/2017 AUTUMN SEMESTER

RAILWAY SUPERSTRUCTURE CALCULATION

ZIMMERMANN-EISENMANN METHOD

SZÉCHENYI ISTVÁN UNIVERSITY Zoltán MAJOR junior lecturer

• Conventional track calculation is limited to quasi-static loading of the track structure, schematized as an elastically supported beam. To the static load is added a dynamic increment.

Winkler support model

• Conventional track consists basically of two parallel continuous beams, the rails, which are fixed at regular intervals onto sleepers supported from below and from the side by a medium which cannot be deformed, the ballast bed. In turn, the ballastbed rests on a formation which also cannot be deformed [295]. In elementary calculations it is usually presupposed that the Winkler hypothesis applies to track support; this hypothesis was formulated in 1867 and reads: at each point of support the compressive stress is proportional to the local compression. This relation can be written as:

σ=C*w • in which: • σ = local compressive stress on the support [N/m²]; • w = local subsidence of the support [m]; • C = foundation modulus [N/m³].

• This relation is illustrated in the figure:

Discrete rail support

• Let us consider the situation of a discretely supported rail. Between the vertical force F(xi) on a support number at x = xi with effective rail support area Ars and the deflection w(xi), the following relation exists according to Winkler:

• 𝐹(𝑥𝑖)=𝐶*𝐴𝑟𝑠*𝑤(𝑥𝑖)=𝑘𝑑*𝑤(𝑥𝑖)

• Hence the spring constant of the support is:

• 𝑘𝑑=𝐶*𝐴𝑟𝑠

• This relation is illustrated in the figure:

Continuous rail support

• In this case a distributed load p(x) between rail and the supporting structure will exist which is, according to Winkler, proportional to the deflection function w(x):

• 𝑝(𝑥)=𝑘*𝑤(𝑥)

• where:

• k = foundation coefficient [N/m/m]

• The foundation coefficient can be interpreted as a spring constant per unit length.

• In this case the contact pressure on the continuous rail support is:

• 𝜎𝑟𝑠(𝑥)=𝑝(𝑥)*𝑏𝑐

• in which bc is the width of the supporting strip under the rail.

Approximation of discrete rail support

• A discretely supported rail structure can be analyzed by numerical means. However, for a static assessment and simple configurations the continuous theory, may also be used as an approximation method for discretely supported rail structures.

• The equivalence with the actual discretely supported track then follows from:

• 𝑘≈𝑘𝑑*𝑎

• a = spacing between centers of discrete supports

• This relation is illustrated in the figure:

Beam on elastic foundation model

• Let us consider an infinitely long rail (CWR track) with bending stiffness EI which is continuously supported by an elastic foundation with foundation coefficient k and loaded a wheel load Q at x = 0. This beam calculation was first proposed by Zimmermann .

• This problem is illustrated in the figure:

where:

• Q = wheel load [N];

• EI = bending stiffness rail [Nm²];

• k = foundation coefficient [N/m/m];

• w(x) = rail deflection at x [m].

• To derive the formula for the deflection w(x) of the beam, we first write down the equilibrium conditions of a beam element. Since there is only one variable (x) here, differentials are indicated by 'd' (rather than '∂').

• Equilibrium conditions of a beam element :

Equilibrium requires:

• 𝑞*𝑑𝑥 + (𝑑𝐷/𝑑𝑥)*𝑑𝑥 = 𝑘*𝑤*𝑑𝑥

• 𝐷*𝑑𝑥 = (𝑑𝑀/𝑑𝑥)*𝑑𝑥

The constitutive equation is:

• 𝑀 = −𝐸𝐼*(𝑑2𝑤/𝑑𝑥2 )

• From these equations the differential equation of the problem can be derived:

• 𝐸𝐼*(𝑑4𝑤/𝑑𝑤4)+𝑘*𝑤=𝑞(𝑥)

• Since we deal only with point loads, the distributed load q(x) will not be considered here (q ≡ 0). The discrete wheel load Q will be introduced later as boundary condition.

After introduction of the short notation for spatial derivatives to x it follows:

• 𝐸I*𝑤𝐼𝑉+𝑘*𝑤=0

• The boundary conditions for the part x > 0 are:

• 𝑤(∞)=0;

• 𝑤′(0)=0;

• 𝑤"′(0)=𝑄/2𝐸I

After substitution of an exponential function for the deflection, the solution of the problem is:

• 𝑤(𝑥)=(𝑄*𝐿3)/(8*𝐸𝐼)*ƞ(𝑥)=𝑄/(2*𝑘*𝐿)* ƞ(𝑥)

The distributed reaction force of the foundation follows from equation:

• 𝑝(𝑥)= 𝑘*𝑤(𝑥)=𝑄/(2*𝐿 )*ƞ(𝑥)

The bending moment in the beam follows from equation:

• 𝑀(𝑥)=(𝑄*𝐿)/4 *μ(𝑥)

• The quantity L in these equations is the so-called characteristic length determined by:

• 𝐿= (4*𝐸𝐼/𝑘)0,25

• Furthermore, two shape functions are present:

• The functions η(x) and μ(x) determine the form of the elastic line and the moment distribution. The left portion (x < 0) of these lines results from symmetry considerations. These lines may also be used as influence lines for determining deflection and bending moment for x= 0 resulting from nearby wheel loads. The expressions represent heavily damped harmonic waves with wavelength 2πL. They are therefore also a good tools for the approximation of finite beams with a central wheel load, provided the length of the beam is greater than 2πL.

• The functions η(x) and μ(x):

• If there are several wheel loads, as in the case of bogies, then the resulting deflection and bending moment are found by means of superposition. At point x = 0 the following is found:

• In which ℓi is the distance of wheel load Qi to point x = 0.

Two-axle bogie - example

• Consider the situation where an infinitely long track is loaded by a two-axle with a half wheel-base ℓ = 1.25 m. The characteristic length L is fixed at 100 cm. UIC 54 rail with a bending stiffness of EI = 4.93 kNcm² is used. With the help of the ƞ-line and the μ-line, both deflection and bending moment for the rail are determined at points A and B.

• This problem is illustrated in the figure:

For a single wheel load Q = 125 kN the maximum deflection and the maximum bending moment are:

• 𝑤𝑚𝑎𝑥=𝑄*𝐿3/(8*𝐸𝐼)=0.32 𝑐𝑚

• 𝑀𝑚𝑎𝑥=𝑄*𝐿/4=3125 𝑘𝑁𝑐𝑚

For point A this reads: • 𝑤𝐴= (𝑄𝐿3/(8*𝐸𝐼))*(1+ƞ𝑐)=0.32 𝑐𝑚 • 𝑀𝐴= (𝑄𝐿/4)*(1+μ𝑐)=2766 𝑘𝑁𝑐𝑚

For point B the values become: • 𝑤𝐵=2*(𝑄𝐿3/(8*𝐸𝐼))*ƞ𝐴=0.23 𝑐𝑚 • 𝑀𝐵=2*(𝑄𝐿/4)* μ𝐴= −1125 𝑘𝑁𝑐𝑚 This calculation example demonstrates that negative bending moments of the order of 40% of the maximum positive bending moments may occur in the rail.

Alternative expressions for characteristic length L

• Based on equations, other useful expressions can be formulated for the characteristic length of the track:

in which: • EI = bending stiffness of rail [Nm²]; • k = foundation coefficient of continuous support [N/m/m]; • kd = spring constant of discrete support [N/m]; • a = spacing between centers of discrete supports [m]; • Crs = foundation modulus at contact area between rail and sleeper [N/m³]; • Cbs = foundation modulus at contact area between baseplate and sleeper

[N/m³]; • Csb = foundation modulus at contact area between sleeper and ballast bed

[N/m³]; • Ars = contact area between rail and sleeper for half sleeper [m²]; • Abs = contact area between baseplate and sleeper for half sleeper [m²]; • Asb = contact area between sleeper and ballast bed for half sleeper [m²].

Order of magnitude of elasticity constants

• The table lists the various quantities used to express the elasticity of the rail supporting structure together with their units. Moreover, global values are given which correspond to the qualifications 'poor' and 'good' to characterize the condition of the foundation.

Double beam model

• The Zimmermann model is sufficient to understand the general behavior of the track structure. In some cases a more accurate description is necessary, for instance when the individual behavior of rail pads is investigated or if the effect of a slab foundation under the track is to be assessed.

• A more realistic model is the double beam model. For discrete supports with relatively high spacing a two-layer model can be used based on FEM techniques. For continuously supported track a two-layer model is useful, which actually is a generalization of the standard Zimmermann model. Both models, loaded for instance by two equal vertical forces, are drawn in the next figure with the characteristic deflection lines.

• The upper beam in each model is the rail, whereas the other beam represents the continuity in the supporting structure which is mostly due to a slab track or other reinforcing structure.

• The model for the continuously supported rail is relatively simple and will be presented here. Without a distributed load on the rail the differential equations for this model are:

• 𝐸𝐼1*(𝑑4𝑤1/𝑑𝑥4)+𝑘1*(𝑤1−𝑤2)=0

• 𝐸𝐼2*(𝑑4𝑤2/𝑑𝑥4)+(𝑘1+𝑘2)*(𝑤2−𝑘1*𝑤1)=0

Dynamic amplification factor

• It is common practice to carry out a strength or fatigue calculation for a static load system, or often a single wheel load, using the longitudinal beam theory, according to which the dynamic effects are taken into account by a speed coefficient or dynamic amplification factor.

• The effect of running speed on load is in reality highly complex because of the dynamic interaction between vehicle and track. In view of the nature of the load, it is also more correct to carry out a fatigue calculation.

• Several simple formulas have been proposed which aim to assess the dynamic rail stress. These formulas are a rough approximation of reality because the geometric quality of the track and the mechanical characteristics of the track and the vehicle are not sufficiently taken into account. One such empirical calculation scheme, which has been accepted by European railway companies, was developed by Eisenmann and is based on the following observations and assumptions:

• Measurements on which this empirical method is based have shown that the stresses in the rail foot, from a statistical point of view, have a normal distribution as illustrated in the next figure;

• The mean value is independent of running speed V (studied up to 200 km/h) and can be determined with sufficient accuracy using Zimmermann’s longitudinal beam calculation;

• The standard deviation is dependent on the running speed V and the state of the track.

• The Eisenmann scheme to determine the DAF (Dynamic Amplification Factor) is dependent on the train speed, the track quality, and chosen factor t and reads as follows:

• DAF- formula:

• t = multiplication factor of standard deviation which depends on the confidence interval. Since the rail is so important for safety and reliability of rail traffic a value of 3 is recommended as the chance of exceeding the maximum calculated stresses is only 0.15%;

• ϕ = factor depending on track quality;

• V = train speed [km/h].

• Values of t and ϕ factors

Maximum bending stress in rail foot center

According to the Eisenmann method, the greatest expected dynamic bending tensile stress in the rail foot center can be determined from:

• σ𝑚𝑎𝑥=𝐷𝐴𝐹*σ𝑚𝑒𝑎𝑛

The mean value of the rail bending stress follows from :

• σmean=M/𝑊𝑦𝑓

• DAF = dynamic amplification factor

• M = bending moment [kNm];

• Wyf = section modulus, relative to the rail foot [m³]

Railway superstructure calculation

FEM method

• The classical concepts of railway track analysis, such Beam on Elastic Foundation (BOEF), Winkler’s theory or Zimmermann method are categorized as one-dimensional analysis of a railway structure and are simplification of a beam laid on a continuous support (soil’s subgrade or foundation). These methods are still very useful for analyzing a simple design and analysis of railway track systems. Unfortunately, for doing a complex analysis of a railway track, these methods have lack of capabilities, since they only take into account one dimensional system and neglect the actual discrete support provided by crossed sleeper, ballast, sub ballast mat and subgrade.

• Nowadays, the use of computer software for doing Finite Element Method (FEM) or Finite Element Analysis (FEA) of a structure is very common for engineers. FEA consists of a huge amount of complex calculations; therefore, a manual calculation by hand is almost impossible to be done. Hence, the use of computer software will be very useful in this manner. The applications of FEM using software also widen in the field of railway infrastructure design and analysis. There are many advantages of using FEM method using computer. However, related to its complexities, one should understands the concepts and “knows-how” to solve the problems, to idealize the structure into FEM model in computer, and to choose the suitable elements and its behaviours, and also the correct method.

The FEM model

E x I

Definition of the beam

Definition of the cross-section

Definition of the supports

Definition of the loads

Calculation

Results - deflection

Results - moments

Coenraad Esveld:

Modern Railway Track

Second Edition

MRT-Productions, 2001, Zaltbommel

REFERENCES