dynamic models of a railroad track system

Dynamic models of a railroad track system

B. Prasad and V. K. Garg

Dynamic Research Division, Association o f American Railroads, 3140 South Federal Street, Chicago, lllhlois 60616, USA (Received 22 January 1979)

A continuum model with microstructure is presented for a railroad track system. The track is represented as a laminated structure with two layers continuously supported by springs and dampers. The differential equations advanced by Sun 24 for a composite system are used in the formulation. Solutions for deflections of the track system subjected to a moving wheel load are obtained in a close form using complex Fourier transforms for both damped and undamped foundations. Comparisons are made with the results obtained from quasi-static models of the track, which simulate the track as a single equivalent homogeneous beam on an elastic foundation with an effective modulus suggested by Voigt. 31 Results of the model are compared with experimental data in limiting cases. Critical velocities are calculated for the track system with conventional rails. The influence of foundation modulus on critical velocity is analysed in some detail. Critical velocities are also computed for an equivalent homogeneous track system using equations advanced by Kerr 1° for a beam. The results of this study agree quite well with those obtained using the continuum model with microstructure.

I n t r o d u c t i o n

The response of railroad tracks to moving loads has been investigated by various authors in the past. 4,16-17 Many models 21,3° have been proposed to predict the track response due to a moving (constant or fluctuating) load 16-17 An early investigation was performed by" Timoshenko 3° to calculate the dynamic stresses and deflections of a rail under the action of the moving wheel loads of a locomotive. In the analysis, the track was replaced by an elastic beam supported on a Winkler-type foundation.

Subsequently, many investigators carried out studies for the Euler-Bernoulli and Timoshenko beams placed on elastic foundations and subjected to moving loads. 1,3,9,22,23 In recent years, with the introduction of welded railroad track, considerable attention has been given to determining the effect of axial forces on critical velocitiesof the track, s,1° The response of an axially stressed infinite track without damping due to a moving load with constant velocity has been considered by Kerr. 1° Dokainish s has analysed a similar case by including the effect of damping.

Both the Euler-Bernoulli and Timoshenko beam models of a track structure do not account for load carrying

capacity and the inertia effect of ties and ballast which are present in the system. It has been noticed 11,14 that the portion of the crib-ballast underneath the rail section carries a significant amount of the vertical load. Ties also share the load partially tkrough bending and partially in direct compression.

The primary layer of ttie ballast is, however, non- homogeneous and granular in nature. Earlier investiga- torslO, 12,13,21 have neglected the effect of ties and ballast while considering a single running rail infinitely. It has been feltn, 14 that the layer consisting of ties and the alternating layers of crib-ballast fillers underneath the rail Cannot be neglected because they elastically support the toad and contribute to the inertia. Their inclusion in the analysis requires a significant detail in modelling. 26 A static model based on the finite element method has been developed recentlyfl 6 The model accounts for the ties and treats the primary ballast as the elastic materials underneath the rails. Stress-dependent material properties of the ballast, the sub-ballast and the subgrade obtained from triaxial tests are used in the model. A good agreement has been reported between the measured response at Section 9 of the Kansas Test Track 27 and that calculated using the finite element modelfl 6 No dynamic model

0307-904x/79/050359-08]$02.00 © 1979 IPC Business Press Ltd Appl. Math. Modelling, 1979, Vol 3, October 359

Dynamic models of a railroad track system: B. Prasad and V. K.

(either analytical or based on the finite element discretiza- tion) has been developed so far which includes the ties and ballast fillers and is subject to a constant moving wheel load. Previous attempts have been limited to an analyses of the rail as an infinite beam subjected to a moving or fluctuating load.9, m, 12, ~3

It is well known that homogeneity is not an absolute clmracteristic of matter, but is a term used relative to the scale of interest. When viewed on a sufficiently large scale, a heterogeneous medium can be considered as grossly homogeneous. 24 This has been the fundamental idea upon which classical continuum mechanics is based. The classical theory becomes inadequate when the characteristic length of deformation (wave length) is comparable to tile sizes of the inhomogeneities, e.g. the grains of a granular material or the fibres in a fibrous composite.

In recent years, following the work of Mindlin Is and Eringen 6 much effort has been made to develop a new microcontinuum mechanics, which takes into account the intrinsic motions of the microelements. In many heterogeneous materials, for example, the engineering com- posites, the material properties as well as the geometrical layout of the constituents, are known. It thus appears to be desirable if the gross material properties can be derived from those of the constituents. With the foregoing in mind, several continuum theories 2,24,2s have been proposed for a laminated medium. These theories 2,24,2s involve a smaller number of material constants and are simpler to use.

In 1961, Filippov 7 suggested a three-dimensional elastic half-space model for the track foundation in which the inertia effect of the ties was included but the bending rigidity was ignored. He solved the problem of an infinite beam resting on an elastic continuum subjected to a con- centrated load. Labra 12-° extended this work to study tile effect of an axial force, lie considered the track as an axially stressed elastic beam resting on an elastic half-space with inertia and subjected to a moving load of constant velocity.

The dynamic response of a composite or laminated structure subjected to a moving load has recently been analysed by Sve 26 using the continuum theory for a laminated medium 2s and by Prasad 28 using tile microstructure theory of a composite beam. 24 In Prasad, 18 dynamic response of a damped composite beam consisting of a large number of parallel alternating plane layers of two homogeneous, isotropic elastic materials has been considered. In this study the microstructure theory has been applied to study the response of a railroad'track system subject to a moving wheel load. The effect of tie spacing, foundation modulus, and damping on the critical velocity of a track system are analysed in some detail.

Descr ip t ion o f t rack m o d e l

The railroad track structure (Figure 1) usually consists of two parallel steel rails, wooden cross ties and several layers of coarse granular ballast materials all resting on the parent soil subgrade. Ballast is filled in and brought up flush with the rail bottom in between the tie spacingL Tile space occupied by crib-ballast (Figure la ) between any two adjacent ties varies from 12 to 14in (30.48 to 35.56 cm). A typical rail flange width measures about 3 to 4 in (7.62 to 10.16 cm) and the tie-depth usually varies from 7 to 8 in (17.78 to 20.32 cm). In service conditions, the space (3in x 12 in x 7 in) is fifily packed and compressed with the ballast materials. The media below the ties base acts like

Garg

a Tie spacing Ties Rail Crib-ballast

b . 4ft 8-1/2in(~435cm)-~ S°?

~7//Z,%,'g///~'x\v.///a',\v4///,,%,\'¢d///ZS u b g r a d e "///Ax\\'K///xq,\\"~///~\v,///,,%,\'g//2

Figure ! Typical sections of a conventional railway track. (a), longitudinal section; (b), transverse section

an elastic half-space and supports tile rails and the ties. The materials under the rails and in between tile ties deform as the rail deform due to moving wheel loads n,14 and as such they act together like a second infinite elastic layer.

In the present analysis, because of symmetry, only half of the track structure with a single rail is considered. The track system is assumed as a two-layered structure placed on an elastic foundation and subjected to a moving wheel load. The first layer is constituted by the original rail section. The second layer replaces tile ties and crib-ballast fillers (Figure la) , laid alternatively in the longitudinal direction of the track, by a layer of homogeneous material with equivalent moduli and mass density. The gross properties of tile equivalent homogeneous layer can be established in terms of its constituents. Two basic theories w,31 have been proposed. Tile first contribution was made by Voigt 31 who assumed that when a poly- crystalline specimen is subjected to a gross uniform strain, the crystals would be in the same state of applied uniform strain. Reuss 19 used a similar approach and assumed that all the crystals would be in the same state of uniform stress. For the track structure system considered here, the Voigt 31 model appears to be more appropriate and has been used in the present analysis.

For the composite medium with two materials namely tie (El , Gt, vt) and crib-ballast filler (E/-, Gf, v.t- ) distributed evenly along the longitudinal direction, the effective elastic and shear modulus of the equivalent micro elastic layer of material are written on the basis of Voigt 3~ as:

E 2 = UlEt +fb( l --/al)E.t- (1)

G 2 = IA1G t +fb(1 -- lal)Gf

where gl = twits; tw and t s denote the width and spacing of ties. fb is the fraction of ballasting (0 < f b < l ) ; fb = 1, corresponds to a case when the crib-ballast are fully packed in between the cross-ties and underneath the ra i l ; f b = 0 represents no ballasting.

The effective mass density of the equivalent layer is similarly established as:

P2 = P,/'ll +fb( 1 -- Id l )Pf (2)

where Pr and p/- are the mass densities of tic and crib- ballast, respectively. The equations similar to (1) and

360 Appl. Math. Modelling, 1979, Vol 3, October

Dynamic models of a railroad track system: B. Prasad and V. K. Gary

(2), have also been used by Sun 24 for comparing dis- persion curves for flexural waves of a laminated composite with microstructure and in Prasad 18 for comparing the dynamic response of a laminated beam due to a moving load.

The track model, therefore, consists of a layer of rail (El, GI, vl) and an equivalent layer underneath the rail (E2, G2, v2) replacing the ties and crib-ballast filler which are laid alternatively in the longitudinal direction (Figure Ia). The structural damping of the rail, the cushioning effect of ties and the airgaps between the ties, ballast sub-ballast and subgrade are simulated by the Winkler type foundation consisting of springs placed parallel to dashpots. The differential equations proposed by Sun 24 for a laminated system are modified for the track structure system characterized by equations (1) and (2) to analyse its dynamic response when subjected to a moving wheel load and placed on an elastic foundation with and without damping. The analytical formulation is presented in the following section.

Analy t ica l f o r m u l a t i o n

This section describes the analytical formulations for two models of the track support system. The first model is based on the microstructure theory and considers the track as a two-layered laminated system Whereas the other model is derived from an equivalent modulus theory.

Microstrttctttre theory

The microstructure theory for a laminated system proposed by Sun 24 yields a system of partial differential equations of motion in three dependent variables w, 4~ and $ as:

a2w O~p a¢ p 02w al - - - a 2 - - a 3 - - + - = a 4 - - ( 3 )

8 x 2 Ox OX ~ Ot 2

aw 02~ 02~ a2 ~xx + a5 - - - a 6 ~ t - a7 - - +a8~5 Ox 2 Ox 2

02~ 02~ = a 9 - - -- alO ~ (4) 0t 2 at 2

aw 02~ 02~ a3 Ox a7 Ox 2 + a s ~ + all OX 2 --al2~b

a2~ 0 2 ~ - - + a l 3 ' ( 5 )

= _ a m i)t 2 Ot 2

where w is the transverse displacement, ~ is the local rotation of the stiff layer and ff isZthe gross rotation; al, a2, . . . , a n are the constants which depend upon material and geometrical configurations of the composite beam and are defined in Sun. 24

If a load of constant magnitude F 0 moves with constant velocity v over the track, which is supported by springs placed parallel to dashpots, the quantity p in equation (3) becomes:

Ow = - - (6) p ( x , t ) F ° 6 ( x - v t ) - k w - ( 3 ° ~t

where k is the equivalent spring constant, fl0 is the damping coefficient.

Stipulating a steady-state solution with r = x - v t , equations (3 -5 ) can be rearranged as:

( a , - a 4 v2) d2w d~ d(ib 1 _ _ _ a 2 - - _ a 3 - - _ _ dr 2 dr dr

( dw '~_ F 0 5(r) x k w - f l o v d r / - - ' - ~

dw d2~ a 2 - - + (a 5 - a 9 v2) - - __ a6~

dr dr 2

dw a 3 - -

dr

(7)

(a 7 - -a lov 2) d25 - - - + a s ¢ = 0 ( 8 )

dr 2

(a7_a loV2) d2~ _ _ - - + a 8 l ] / dr 2

÷ (a, l -a13 v2) d2~b dr---- ~ - al2~b = 0 (9)

Tile above differential equations are in tile moving reference frame ( r ,y , z). r is the distance measured from the point of application of the moving load.

Introducing the non-dimensional variables:

w r F o II 7 = - - R = - V 0 = N/~2/P2 F* -

r l r I a l ~

kr~ floQ Vo Gl Pl e = /3- F = - - X = -

ai $ 2~a G2 P2

v d I h 0 = - - r / - - - } - - -

Vo dl +d2 dl +d 2 (lO)

and tile nondimensional constants:

al a 5 a7 all 2=a,r n3-ald (11)

a I a 5 a ? a l l

a, a4v2 a2 a4v~ % alev02 e 4 = a , 3 ~ (12)

simplification can be achieved. Tile solution may now be constructed with the use of Fourier transforms. The following pair is used:

co

= j ,(n) e - i ' n ~(s) I t g

oo

if u(R) = - - t~(s) e isR ds 2n

where s is a Fourier transform parameter.

(13)

1700 I " ~ j ~ ~ , ~ . ~ "" ~ .~16OO ~ zz~ 15OO

>- 14oo S 1 3 o o

~> 12oo ~11oo

' , , , ,

1OO OO 1OOO 15OO 2 0 0 0 2500 3 0 0 0 Foundation rnodutus, (Ib/in 2}

Figure 2 Effect of foundation modulus on critical velocity. 115 AREA rail section. ( ), laminated track model; (----), equivalent beam model based on Kerr's formulae

Appl. Math. Modelling, 1979, Vol 3, October 361

Dynamic models of a railroad track system: B. Prasad and V. K. Garg

Assuming that dependent variables and their derivatives approach zero at infinity, the equations ( 7 - 9 ) transform to:

D

[(a~ - a4v 2) s 2

isr 1 + kr~] -floV ~ ~ J

a2is

a3is

a2is a3is

- I n 6 + (as

-- a9V2) ~1~ ]

~8 + (aT

as + (aT

- -a loV2)r@]

-/a,2 +(ml - a,3v 2) ~]

,F

Fo[~

0 (14)

where W, ¢ and ff are the transformed variables. After simplification of (14) we obtain:

F*(sap4 + s2p2 + 1) • = O s )

A

F*is(s2ps + 1) = (16)

A

F*is(s2P6 + 1) - (17)

A

where:

and

F* = Fo[a l ~

A = (s2Pl -- 2flOis+a)(s4p4 +s2p2 or l) - - S2(s2p3 or 1) (18)

The coefficients p I, P2 . . . . . P6, in equations ( 1 5 - 1 8 ) depend upon the velocity parameter 0 and can be expressed in the foregoing nondimensional quantities (10 -12 ) . They are obtained as:

Pl = (l - 02/~1) (19)

174(I --02/0~4) -- 172(17"/'/1 -- 21"/--171 or 1) x (1 -- 0 % 2 ) -- 2r/173(1 -- 02/a3)

P2 = (20) 1 - - ( l - - 17)/171

p3 =

P4 =

174(1 --02/0t4) + 2/'/3(171 --1)(1 -- 0 % 3 ) or 172(171 - - 1)2( 1 - 02/a2)

(21) 17J(1 - ~) -- l

171 [,72174(1 - o % 2 ) ( I - 0 % , 0 - , 7 ] (1 - o ] / m ) 2]

17,/0 - 17) -- 1 (22)

(2.3) 1/1174( 1 -- 02[0:4) + 171173(171 -- 1)(1 -- 02/cx3)

p5 = m / ( 1 - - 17) - - 1

171172(171- 1)(1 --02[ot2) + 171173( 1 --02[°t3) P6 = (24)

m / ( l - o) - 1

The nondimensional constants 17i, ei, for i = 1, . . . , 4 defined in expressions ( 1 1 - 1 2 ) can also be expressed in terms of geometrical and material properties of tile laminated model of the railroad track. The independent parameters are %/3 and 0.

Tlie denominator A of equations (15--I 7) is a sixth- order polynomial in s with real and imaginary coefficients. Once the roots of the characteristic equation, A = 0, are deternfined, the transformed functions W--(s), fJ(s) and ¢(s) can be expanded into partial fractions; each term of which can be set in a general form of the type A[(s + a + ib) where A is a constant, - ( a + ib) is a root in which a and b can be positive, negative or even zero.

The inverse transform of W(s), ~(s) and qT(s) can be accomplished by recalling that:

I F -1 - - = - s g n ( b ) i ebRe -iaRH [-R sgn(b)]

s + a + i b (2Sa)

where F -z represents inverse Fourier transform, H[R] and sgn(b) are generalized functions defined by:

1 f o r b > 0 sgn(b) =

- 1 for b < 0 and: (25b)

1 f o r R > 0 H[R] =

0 f o r R < 0

As expected, the inverse transform of W, q~ and depends upon the characteristic roots of the equation (I 8) which in turn depends upon the specified values of the independent parameters a,/3 and 0. Tables 1 and 2 show the types of root which are encountered for the track model under various combinations of a, fl and 0 with and without damping, respectively. In a recent paper 18 the author has presented the solutions for dynamic deformatim~ of a composite beam due to a moving load under several possible cases, mentioned in Tables I and 2. These solutions were obtained for a composite beam having a large number o f parallel alternating plane layers of two homogeneous elastic materials. Since in the present track model we con-

Table 1 Types o f roots o f the character is t ic equa t i on (18) w i t h damp ing

T y p e General f o r m o f r oo t

Type I

Type II

Type III

Type IV

-+ b~ + ia~. +-b 2 -- ia2, + ia 3

-+ b~ + ia~, ± ia=, -- ia3, -- ia 4

-+ b , - - ia t , -+ in2, in3, ia 4

ia l, ia 2, ia s, - - i a 4 , - - i a s , - - i a 6

362 App l . Math. Mode l l ing , 1979, Vo l 3, Oc tober

Table 2 Types of roots of characteristic equation (18) without damping

General form of root Limiting forms

Typel -+ (al± ib~),±ib 2 a~=O

Typel l ± ib I,-~ib 2,-*ib~ b~=b2orb ~ = b 2 = b 3

Type III -+ a~,-+a2,-+ ib~ a~ =a 2

sider only two layers of materials, tile constants 77 and and other parameters in the solutions have been modified to comply with this requirement. The I section of the rail is replaced here by an equivalent rectangular section with depth d I and width b. Material properties expressed by equations ( I - 2 ) are later used to compute deformations and slopes at various points along the track. Numerical results are presented here in the plot forms for ease of comparison.

Determhtation o f critical velocities

It has been found that for some a with no damping (/3 = 0), the characteristics equation (18) may possess double roots: s = +a, -+a, and -+ib. This is a limiting form of case III mentioned in Table 2, which is in general of tile type: +a~, +a2, and -+ib3.

For this particular case the inversion integrals are of the form:

c o

fdx (x - a) 2 (x + a) 2 (26a) - - ¢ o I e ~

where x and a are real. It is well known that an integral of tile type in equation

(26a) does not exist even in the sense of a Cauchy prin- cipal value. It was discussed in Kenny 9 that such non- existence of a solution to a physical problem implies a resonance, in the sense that tile displacement becomes unbounded all along the beam. The value of the load velocity parameter 0, which yields an integral of the type in equation (26a) defines the critical load velocities.

Imposing the condition for double roots and eliminating b, one can find, using equation (18), a condition in terms of 0, which is:

33 '2 + 2e23' + e 3 = 0 (26b)

where % e2, e 3 and e 4 are defined as:

"7 = (e3e2 - 9ea)/(6e3 - 2e~) (26c)

e2 = (PIP2 + eP4 - P 3 ) / P l P 4 (26d)

e3 = (Pl + aP2 - 1)/PIP4 (26e)

e 4 = at/pip4 (26f)

Pl, P 2, - . . , P4 are functions of only the load velocity parameter 0. They are defined in equations (19-24) . Values of 0 for which equation (26b) is satisfied, are the resonant speeds. In general there can be several roots to equation (26b), but since only positive real roots are of physical significance, only those should be retained and the rest can be discarded.

Equivalent moduhts theory

In a gross sense, a multilayer elastic medium can also be considered as a homogeneous continuum without microstructure by calculating the effective moduli and material properties of an equivalent structural system. The dynamic

Dynamic models era railroad track system: B. Prasad and V. K. Gary

behaviour of such laminates can be obtained using tile classical solutions developed for a homogeneous beam placed on an elastic foundation and subjected to a constant moving wheel load. A widely used approach for evaluating the gross elastic properties of a laminated system is the law of mixtures, through which an effective modulus of the laminated system is expressed as the sum of the elastic moduli of the individual constituent materials weighted by their respective volume fractions.

For the track structure having two elastic layers, the effective Young's modulus and mass density are expressed as: 31

E=I~2EI +(1- - /12)E 2 (27)

p =/-12p I + ( 1 - - / 1 2 ) p 2

where la 2 =d l / (d l +d2); dl and d 2 a re the depth of rail and the equivalent tie sections, respectively.

The track structure is considered as a single honlo- geneous beam characterized by equation (27) and placed on an elastic foundation with or without damping.

The effective modulus is used in the approach outlined by Kerr ~° and in the classical theory of a beam on an elastic foundation to compute the approximate behaviour of the track system. In the absence of any axial load Kerr m suggested the expression for the critical velocity for a single beam with a constant moving load as:

Vcr = [4kEl] 114 L m - - T J (28)

where EI is the flexural rigidity of tile beam, k is the foundation modulus, and m the mass of the beam per unit length. The classical solution for a beam on an elastic foundation 2° provides an expression for the maxhnum deflection under a static load F as:

lgSa x F ( k ~,/4 = 2"k \4E-~] (29)

where EI and k have similar meanings. For comparing the results in the two cases, gross proper-

ties of the equivalent homogeneous track system as pro: posed in equation (27), are used in equations (28) and (29) to compute the effective values of the critical velocities and the maximum deflections under load, respectively. The equivalent moment of inertia l and mass density m of the track system required in the equations (28) and (29) are obtained as:

and:

- - + 1 2 I = I 1 + 1 2 +3 l d l (30)

m = A l p I +A2p 2

where the quantities with subscript I represent properties corresponding to the layer I and with subscript 2 represent properties corresponding to the layer 2.

Resul ts and discussions

In the previous section two models for a railroad track system are presented, one based on the microstructure theory for a laminated system, and the other based on the effective modulus theory. In this section, the results obtained from each model are discussed. Section properties of the rails used in the analysis are summarized in Table 3.

Appl. Math. Modell ing, 1979, Vol 3, October 363

Dynamic models o f a railroad track system: B. Prasad and V. K. Garg

Table 3 Section properties of.conventional rails. (1 in 2 = 6.45 cm2; I in 4 = 4 1 . 6 2 c m 4)

A r e n A Moment o f inertia Moment of inert ia (in 2) Izz* (in 4) lyy (in 4)

115 AREA 11.25"* 10.8 65.6** 1 1 9 C F & I 11.65"** 10.9 7 1 A * * * 132 AREA 12.95"* 14.6 88.2** 136 AREA 13;35"* 14.7 94.9"* 85 STD 8.34 623 29.4

100 ARA 9.84 9.6 48.9

* Measured graphical ly wi th an Amsler Integrator instrument. * * Amer ican Rai lway Engineering Associat ion (AREA) Manual o f Recommended Practices, pp. 4-1-3 to 4-1-5. * * * Co lorado Fuel and Iron Corp., Sect ion Engineering Dept.

Table 5 Comparison of static deformat ions fo r convent ional rail sections. (1 in = 2.54 cm; wheel weight = 25 000 Ib, Ib = 0 A 5 4 5 kg; t ie spacing = 22 in)

Max . static de fo rmat ion (in)

Effect ive Rail modulus Microst ructure Exper imenta l

section theory theory value 29

85 STD 0.222 0.259 0.305 100 ARA 0.204 0.;233 0.22 115 AREA 0.193 0.217 -- 119 CF& I 0.189 0 2 1 2 -- 132 AREA 0.182 0.202 -- 136 AREA 0.179 0.198 --

Table 4 Mater ia l properties of rail, t ie and ballast (1 psi = 6.9 kN/m2; 1 in = 0.0254 m)

Rail Tie Crib-ballast

Young's Modu- lus o f Elast ici ty 29 x 106 psi 1.5 x 106 psi 2.5 x 104 psi Sizes See Table 3 6 in deep 6 in deep

8 in wide Ful l ballasting 8 f t long

Effect ive densi ty 0.284 Ib/in 3 0.0218 Ib/in 3 0.07 Ib/in 3

Standard ties 6 in x 8 in x 8 ft at 22 in centre to centre spacing are used. Material properties of the rail, tie and crib-ballast used in the analysis are given in Table 4. Several values of the foundation modulus k measured experimentally before and after tamping, had earlier been reported by Talbot. 29 In order to show some comparisons between the analytical models outlined here and the existing experimental results obtained by Talbot, 29 a repre- sentative value of 920 lb/in 2 given for half-track structure and measured after tamping with 12 in of ballast layer under the tie level has been used for k. The same value o f k was previously used in So et aL 21 for comparison with experimental data.

A single moving wheel load of 25 000 Ib is considered in the analysis. The rail is assumed to be straight without any lateral or axial load.

Static deflections were found from the laminated track model using micro-structure theory without damping by setting the velocity parameter 0 = 0, in equations (3-5) . Static deflections were also determined from equation (29) for an equivalent track system using the values for the effective modulus and mass density as given by equation (27).

Table 5 compares the maximum static deflections of the track with different rail sizes under a single moving wheel load of 25 000 lb. It may be noted that the deflections predicted by the microstructure theory appear closer to the experimental values. A close agreement between the experiments and that predicted using the microstructure theory, may be attributed to the laminated nature of the proposed track model.

Based on the analyses of the Euler-Bernoulli and Timoshenko beams under moving loads, it has been shown shown],3,9,]0 that a critical speed of the load exists when the response of the beam Becomes unbounded. A similar behaviour has been observed in the case of a laminated model of the track. For an elastic beam, Kerr m has given an analytical expression for critical velocity of the beam subjected to a moving load. In order to compare the results,

Table 6 Comparison o f cr i t ical velocit ies fo r convent ional rail sections. (1 mi le/h = 1.61 km/h ;whee l weight = 2 5 0 0 0 Ib; 1 Ib = 0.4545 kg; t ie spacing = 22 in)

Cri t ical ve loc i ty (m/hr)

Kerr 's predic t ion based Laminated model Rail on effect ive modulus based on microstructure

section theory theory

85 STD 1243.4 1247.3 100 ARA 1265A 1284.2 116 A R EA 1254.7 1282.1 119 CF & I 1255.0 1284.8 132 AREA 1246.3 1282.2 136 AREA 1246.6 1284.7

0 2 7

0 2 6

0 2 5

~- 0-24

0 2 3 E~

u_ 022 E o

O21 a

02C

0-19

0 1 8 16

Figure 3 5 5 m p h

J J

. ~ . - - - - " ~ " ' ~ ST D rait

119 CFand I rai l

136 AREA rail

' 2' ' J6 18 2 0 2 24 Ti~' spocing, (in}

Effect of t ie spacing. Wheel wt = 25 000 Ib; speed =

the equivalent gross properties of the track structure have been used in the Kerr's expression (equation 28). Table 6 shows the critical velocities for various rail sections. In one case, critical velocities are obtained from the equivalent homogeneous beam model of the track based on the effective modulus theory, and in the other case, using the laminated model of the track based on microstructure theory. A good agreement can be observed.

Figure 2 shows the effect of foundation modulus on the critical velocity of the track. Two curves lmve been plotted, one is based on the laminated model of the track and the other obtained from tile KerFs expression for the beam when gross properties based on tile effective modulus theory are used. It may be noticed that for a track modulus of 1889 Ib/in 2 the two models predict the same critical

3 6 4 App l . Math . Mode l l i ng , 1979 , Vo l 3, O c t o b e r


- 0 2

b

"5 O1

0

13 = 0 0

o ,3 2'o 3'o 4'o s'o go ¢o 8'o 9'o 6o Sp~ed of moving whe~ti toad, (mph)

Figure 4 Effect of damping on track deflection (136 AREA rail section), fl is a fraction of critical damping

speed. For other values of the foundation moduhts, differ- ences are small.

Figure 3 shows the effect of the tie spacing. Three curves have been plotted corresponding to three standard rail sections. As expected, with the increase of the tie spacing, dynamic deflection under the load increases. For lighter rails, the effect is more pronounced than for the heavier rails.

Figure 4 shows the effect o f damping on dynamic response of a railroad track. The dynamic deflection of rail under a moving load is plotted against the speed of the load. It is observed that damping does not influence the magnitude o f critical speed;however, the magnitude of the deflection is reduced considerably.

Concluding remarks

An acceptable structural analysis of railway track support system (RTSS) cannot consider RTSS as composed only of rails supported on elastic foundation. A significant portion of the structural strength is derived from the tie, the crib-ballast and the ballast materials, i.e. the tie, the crib-ballast and the ballast also act as load carrying media and possess limiting (or allowable) response patterns.

The simple track model based on the effective modulus theory is intended to produce the gross effects of the track while the detailed model simulating the track as a laminated system is needed to determine the effects o f varying tie cross-sections, tie-spacings and partial ballasting. The results o f the laminated model compare well with the simple model and also with the existing experimental values.

Notation

al ,a2 , . . . ,a13

A AI ,A2

b d l , d 2

E

El , E2

Fo F* G

GI ,G2

Coefficients of composite beam equations (51 -63 ) defined in Sun 24 Total cross-sectional area Cross-sectional areas o f rail and equivalent tie, crib-ballast layer respectively Width of railroad track Thicknesses o f rail and equivalent tie, crib- ballast layer, respectively Effective Young's modulus Young's moduli of rail and equivalent tie crib-ballast layer, respectively Magnitude of ~:oncentrated force Nondimensional force parameter (F/at~) Effective shear modulus Shear moduli of rail and equivalent tie, crib-ballast layer, respectively

h I

k p(x, t) q, r

R rl, r2

s

ts tw t v w

X

~o

P K I , K 2

Pl, I)2

Pl,P2

Pt, P[

n,~,v,o~,3,0

a() H()

Total depth o f composite rail track Total moment of inertia o f the railroad track Spring constant (foundation moduhts) Distributed lateral load per unit span =is = x - v t Dimensionless length (r]rt) Radius of gyration o f rail and equivalent tie, crib-ballast layer, respectively Fourier transform parameter Tie spacing Tie-width Time Velocity o f moving load Transverse displacement of railroad track Dimensionless transverse displacement (w/rl) Exponential Fourier transform of IV Distance along railroad track Damping coefficient Local (micro) rotation of the rail layer Gross (macro) rotation Exponential Fourier transform of ~b Exponential Fourier transfornt of Effective mass density Shear correction coefficients for rail and equivalent tie, crib-ballast layer, respectively Poisson's ratios of rail and equivalent tie, crib-ballast layer, respectively Mass densities of rail and equivalent tie, crib-ballast layer, respectively Densities of the tie and crib-ballast filler layer, respectively Nondimensional parameters defined by equation (10) in text Dirac delta function Iteaviside unit functio~

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dynamic models of a railroad track system

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