Computers & Srrucrurrs Vol 24. No 3. pp. 437441. lYX6 Pnnted I” Grca1 Brllaln

WS-7949:86 13.00 + 0.00 Pergnmon Journals Ltd

THE MECHANICAL BEHAVIOUR OF THE SLEEPER-BALLAST INTERFACE

V. PROFILLIDIS

Hellenic Railways Organization, Athens, Greece

P. PONIRIDIS

Royal lnstitute of Technology, Stockholm, Sweden

(Received 18 November 1985)

Abstract-V&o-elastic theories have been recently established for the study of the interface of various polymers. These theories are extended in the present paper; they are concerned with the study of the sleeper-ballast interface of the railway system. The thickness of this interface is calculated and the influence of the quality of the subgrade is analysed. This analysis offers the possibility to formulate an empirically well-known physical phenomenon, i.e. the existence of “dancing sleepers”.

1. INTRODUmlON

Mass transport has encouraged, during recent years, the construction or improvement of important rail- ways projects. The need to rationalize the design of such projects, which are expensive and difficult to modify when executed, has pushed forward the research on theoretical relations, based on the real mechanical behaviour of the various components of the system.

The problem of the study of track support struc- tures and the sub-grade (Fig. 1) has been treated by many authors and researchers in recent years [I, 2,4,6, 7,8, 1 I]. A delicate problem, however, is what happens at the interface between two successive layers. The most common method is to consider double nodes (Fig. 2), each one belonging to one layer; as in numerical models some components of the displacement must be considered continuous, this puts firm restrictions on the degrees of freedom of the system and reduces the possibility of having realistic values for the interfaces.

Strictly speaking, the reason for such consider- ations has been due, up to now, to the lack of a theoretical knowledge base concerning the behaviour of interfaces. Recent research, however, on reinforced polymers and composite materials [IO] can afford

today a sufficient background for the study of the various phenomena occurring at the interfaces, for which most of the studies make clear a visco-elastic constitutive law.

In the present paper it is shown how visco-elastic constitutive laws can be used for the study of the mechanical behaviour at the various interfaces of

track support structures, and particularly between sleeper and ballast.

2. VISCO-ELASTIC APPROACH OF THE MECHANlCAL BEHAVIOIJR OF INTERFACES

Most of the models used up to now to explain the mechanical behaviour of the system under study have a common characteristic, i.e. they consider the contact surfaces as perfect mathematical surfaces and also take into account continuity of displacements. In fact these surfaces are rough; from this roughness there results a stress concentration and an extended physical layer which will be designated as interphase. Mechanical stresses, high stress-gradients and stress singularities are developed there [9, lo].

The so-called interphase (or mesophase by some authors) is a layer consisting of an inhomogeneous material with progressively varying properties

track support structures 1

.-.-‘-.-““.-.-.-‘-.-.------/~

Fig. I. Track, track support structures and subgrade.

437

438 V. PKOFILLIDIS and P. PONIRIDIS

wheel load 7-A

K t sleeper. L L ballast

M c SlOeper: N6 ballast

P t sleeper, QC ballast

between the first and second layer. The interphase material consists mainly of the weakest layer, where most of the singularities occur.

There are two instances of interphase development: between sleeper-ballast and between the other couples of layers. In the first instance the interphase consists only of ballast material (Fig. 3) and in the second of both materials (Fig. 4).

If Ii&,(r) is the varying elasticity modulus of interphase, then the following relation is valid:

Fig. 2. The use of double nodes at the sleeper-ballast interface and the directions of continuity of displacements.

d-%,,(r) dr

= 0. r = r,n,

(El

El

E2 ---_--_--_

I. 1

0 ‘1 ‘int ‘2 (r)

Fig. 3. The sleeper-ballast interface. Ballast is a visco-elastic material, whereas sleeper has an elastic behaviour. The interphase consists only of ballast.

I I I I

.__~ _ .- - I theoretical

motcriol(0 (e.g. bollost) contact surface

r2 L--

! I _I 4-i k_ -----__-_-_____-~_~_~~ q__ __--_-------w--J

Ih

-11 interphase

motorial (2) (e.g. gravel ) 1 I I +-y-. I -- --------I I I

Fig. 4. The ballast-gravel interface. Both materials have a visco-elastic behaviour. The interface consists of both materials.

Moreover, hx the second case, we will have in addition to eqn (ljl the following condition

dE,,,fr) dr

= 0. I wrj

(3

In the present study only the first case is examined. According to Theocaris [lo], the variation in the elasticity modulus wilt be:

where Et and Et are the elasticity mod&i of sleeper and ballast and 2n is an adhesion coefficient between the two layers; n is always positive, tends to zero values when adhesian is bad and takes high values when adhesion is good. In eqn (3) the ~undary conditions are fulfified

Equation (I), using relation (3), yields:

The unknown parameters in the above equation are 2n and r,,,.

3. DETERMYNATION OF THE VALUES OF r, AND tn where

The adhesion between the surfaces is achieved by shear and normal stresses. Thus, the compliance J,, of the whole system can be derived from the compli- antes Ji, J2 and Ji,, [2]. In an electrical analogy scheme, the weighted moduli constitute capacitances connected in series.

The following relations are valid:

Ji,,,, JI$,, J; and J; being the storage and loss moduli of the whole system and ballast respectively [IO].

The sleeper is considered as purely etastic, Separat- ing, now, real from imaginary parts of eqn (1 l), we obtain

u I = ;, q,, 2x 5c.2” 12

u, _ r2 - f,,, (3

r2

where

Then

U, is the sleeper volume

Vie, is the interphase volume

r/, is the ballast volume.

where

For the interphase layer

from which

+(E,-Elf+-l-1)

and, after a Taylor series expansion,

where terms of higher power of B have been neglected. In fact, eqn (Sb) shows that values of g*” are very small and approximately equal to zero.

Then, by substituting eqn (IO) into eqn (6), we obtain

J:,, =: J, . U,

Jtb) = J;.&) - iJ;b,(w) (12af

J;(w) = J;(w) - iJ;‘(w), (12b)

J:0S=J,U,-2$(1 -#-B-‘)U;‘,, I

4(J;’ -J;‘)

U,(1 - 2n) & rr,,, f J; Y (13)

- U,(l -2n) E, Vt,, + J; U,, (14)

Since our studies cannot afford values for the loss moduli at present, we will not use eqn (14). From the results of Iaboratory experiments for v&co-elastic polymers, it can be deduced that

E”S 0.01 E’. (15)

440 V. PROFILLIDIS and P. PONIRIDIS

To the best of our knowledge, no publications are available today which confirm or deny the existence of solutions to the above systems.

4. APPLICATIONS AND RESULTS OF THE METHOD

The stress and strain fields at the various layers of the system presented in Fig. 1 have been studied by using a finite element approach [6, 71. It is common to consider various types of subgrade, depending on its quality:

subgrade of bad quality, named as QS, (E = 12.5 MPa)

subgrade of medium quality, named as QS, (E = 20-25 MPa)

subgrade of good quality, named as QS, (E = 30-35 MPa).

Although the above classification was established in French laboratories, it is actually adopted by the International Union of Railways and various International Associations.

The determination of the values of the elasticity moduli is explained in (61 and [7]. For the sleeper (consisting of reinforced concrete) it will be:

E rkpr = E; = 30. IO4 bars = 3.10 Nt/m2,

and for the ballast

E ba,,as, = E; = 1500 bars = 1.5. IO’ Nt/m*.

Using the values of the displacements at the sleeper

and subgrade level from [6] it is possible to evaluate the storage modulus of the sleeper-ballast system. We will examine the case of concrete sleepers (Fig. 1) with a height of 26cm.

In the case of QS, subgrade with a ballast thickness of 40 cm, it will be:

E;,,, = 2.5. IO8 Nt/m’;

in the case of QS, subgrade with a ballast thickness of 35 cm, it will be:

E;,, = 2.7. IO8 Nt/m2;

and in the case of QS, subgrade with a ballast thickness of 30cm, it will be:

El,, = 3. IO’ Nt/m2.

Subgrade

0”:: Qg,

Table 1.

rlnl fm) Lnph.u (mm) ~

0.265 5mm 377 0.271 11 mm 185 0.276 16mm 137

Fig. 5. Contact conditions at the sleeper-ballast interface.

For solution of the non-linear system consisting of eqns (4) and (13), the well known Newton-Raphson method was used.

The values deduced for the interphase and the adhesion coefficient are shown in Table 1.

5. CONCLUSIONS AND FURTHER RESEARCH

Many authors realized when studying the ballast behaviour that calculated values for the displacement at the sleeper level are systematically smaller when compared to the measured ones. The phenomenon is explained when considering the sleeper-ballast interface.

It is observed that under the influence of the railway loads the contact conditions soon become unilateral and a void of about l-10 mm for the first 1000 cycles and up to 10 mm for 1 ,OOO,OOO cycles is developed between sleeper and ballast. This phenom- enon is sometimes called “dance” since the sleeper is hanging from the rail and looks like “dancing”.

So, the interphase studied is this article has a physical significance, and the values deduced for h ,nlrwhav are close enough compared to measured values [7,8].

As observed, the present analysis shows that the quality of the subgrade has a significant influence on the thickness of the sleeper-ballast interface. This influence is partially due to the modification of the ballast thickness depending on the quality of the subgrade.

Furthermore, the present analysis can be used for the sleeper-gravel, gravel-sand and sand-soil inter- faces, where both materials have a visco-elastic behaviour.

REFERENCES

I. C. Chang, C. Adegoke and E. Sellig, Geotrack model for railroad track performance. J. G. E. Dir. Am. Sot. ciu. Engrs 106 (1980).

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3. J. D. Ferry, Visco-Elastic Properlies of Polymers. John Wiley, New York (1961).

4. A. Lopez Pita and C. Oteo Maze, Analysis de la deformabilidad vertical de una via ferrea mediante el metodo de elementos finitos. Rec. A.I.T. 15, (1977).

5. P. D. Panagiotopoulos, Convex analysis and unilateral static oroblems. fna. Arch. 45, 55-68 (1976).

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Mechanical behaviour of sleeper-ballast interface 4441

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9. R. Theocaris and P. Poniridis, Adhesion quality and extent of mesophase in particulates along the whole visco-elastic spectrum. J. appl. Polym. Sci. (1986).

10. P. Theocaris. The mesophase and its influence on the mechanical behaviour of composites. Ado. Polym. Sci. 66, 149-187 (1985).

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