ACTA MECHANICA

Acta Mechanica 30, 17--50 (1978)

@ by Springer-Verlag 1978

Analysis of Thermal Track Buckling in the Lateral Plane* By A. D. Kerr, P r i n c e t o n , New J e r s e y With 12 Figures

(Received May 27, 1976; revised July 23, 1976) S u m m a r y - Zusammcnfassung Analysis of Thermal Track Buckling in the Lateral Plane. The determination of the safe temperature increase in the rails of a continuously welded track, to prevent lateral buckling is presented. The criterion used is based on the post-buckling equilibrium branches of the track. The range of safe temperature increases is defined as T o < TL, where T L is the smallest value of the temperature rise at which a deformed state of equilibrium (thus, a buckled state) becomes possible. The post-buckling equilibrium states are determined analytically. To obtain a consistent formulation of the problem use is made of the principle of virtual displacements and the variational calculus for variable matching points. The obtained formulations arc nonlinear, but can be solved exactly. Solutions are presented for four buckled configurations. The results are presented graphically for a typical railroad track presently in use on main lines. The obtained results are compared with the corresponding results of other investigators. Analyse des thermischen Geleisebeulens in die Querebene. Der zul~ssige Temperatur. anstieg in den Schienen eines kontinuierlich geschweil~ten Gleises wird bestimmt, um seitliches Ausbeulen zu verhindern. Das verwendete tG'iterium basiert auf Nachbeulgleichgewichtskurven des Geleises. Der Bereich des zul/issigen Temperaturzuwachses wird definiert mit T o < TL, wobei TL der kleinste Weft des Temperaturanstieges ist, bei dem ein verformter Gleichgewichtszustand (ira ausgebeulten Zustand) mSgtich ist. Die Nachbeulgleichgewichtslagen werden analytisch bestimmt. Fiir eine klare Formulierung des Problems wird das Prinzip der virtuellen Verschiebungen und die Variationsrechnung fiir variable Endpunkte verwendet. Die erhaltenen Gleichungen sind nichtlinear, kSnnen aber exakt gelSst werden. L6sungen werden fiir vier Beulformen angegeben. Die Ergebnisse sind ffir eine typisehe Eisenbahnschiene, wie sic gegenw~irtig anf den I-Iauptlinien verwendet wird, graphisch dargestellt. Die erhaltenen Ergebnisse werden mit entsprechenden Ergebnissen anderer Untersuchungen verglichen.

I. Introduction Analyses for the in t e m p e r a t u r e , were These analyses m a y vertically a n d when

d e t e r m i n a t i o n of railroad t r a c k buckling, caused b y a rise conducted in the past several decades b y m a n y investigators. be grouped into two m a i n categories: when track buckles track buckles in the lateral plane. A l t h o u g h actual track

* Research sponsored by the Federal l~ailroad Administration, Office of R & D, with the Transportation Systems Center as Program 1Kanager, under Contract DOT-TSC-900 and by the National Science Foundation under Grant ENG 74-19030.

18

A.D. Kerr:

buckling m a y proceed in a more complicated manner, the choice of these two special modes of deformation was apparently made in order to simplify the resulting analyses. These specialized analyses were also suggested by observations made on buckled tracks. Namely, according to field and test observations, crosstie tracks when subjected to an excessive temperature increase, usually buckle in the lateral plane, as shown in Fig. 1. On the other hand when lateral motion is prevented, by an increased lateral rigidity of the rail-tie structure and/or an increased lateral resistance, the track will buckle out in the vertical plane.

Fig. t. Buckled tracks A critical survey of the analyses of thermal track buckling in the lateral plane, and a description and discussion of related test results, were recently presented by A. D. Kerr [1]. This survey revealed that the majority of the published results are not suitable for analyzing thermal track buckling problems, because they are based on formulations which do not describe correctly the physical phenomenon

Analysis of Thermal Track Buckling in the Lateral Plane

19

under consideration, Those few analyses which are conceptually on the right path, exhibit analytical shortcomings with an unknown effect on the final results. The purpose of the present paper is to present a mechanically reasonable and mathematically consistent analysis for the title problem. The obtained results are then compared with the relevant results published by other investigators.

II. The Thermal Track Buckling Phenomenon A uniform temperature increase, To, in a straight welded track induces in the two rails, due to constrained thermal expansions, an axial compression force (Fig. 2a).

Nt = EAc~To. In the above equation, which is valid when the rails respond elastically, E is u modulus, A is the cross-sectional area of the two rails, and ~ is the coefficient of linear thermal expansion. Thus, for a track with 115 lb/yard rails (E = 2.1 9 106 kg/cm e, A = 145 cm e, c~ = 1.05.10 .5 l/C~ a uniform temperature increase of 30 ~ (54 ~ induces in the rails an axial compression force of 96 tons (metric). For sufficiently large compression forces the tracks m a y buckle out. According to observations in the field and in track buckling tests [1], the continuously we/ded tracks presently in use buckle in the horizontal plane, as shown in Fig. 1. The observed buckling mode of a long straight track consists of a buckled region (of length 21 in Fig. 2) which exhibits large lateral deformations and the adjoining regions which appear to deform only axially. In the buckled region, a part of the constrained thermal expansions is released. This results in a reduction of the axial force to Nt, which in the literature is buckled state Top view

/

/ ,. . . . . . . . . . . . . . . . . . .

-----J--

llJ11111111771JlTll

undeformed state ~IIIII~IIIIIIIIIIIIIIIIIIIIIIIIIII111

JtFl lllt[ ,l F

(a} Axial compression force before buckling

~

I

I I I ~' I I I

I I

t I I

!

i

o

J

(b)Axi I compression force after buckling

Fig. 2. Distribution of axial compression forces before and after buckling. (.Note that in an actual track a is several times larger than l)

20

A.D. Kerr:

assumed to be constant. In the adjoining regions, because of ballast resistance to axial displacements of the tracks, the constrained thermal expansions vary; so does the axial force Nt <: N < Nt, as shown schematically in Fig. 2 (b). According to the above observations, thermal buckling of a long straight track appears to be a local phenomenon.

III. Analytical Preliminaries In the following analysis, the rail-tie structure is replaced by an equivalent beam of uniform cross-section, to be referred to as the track-beam, which is sym~ metrical with respect to the vertical x--z plane. The x-axis is placed through the centroid of the cross-section and is chosen as the reference axis. I t is assumed that the beam is subjected to a uniform temperature change1 T(x, y, z) = T0 = const.

(3.1)

and a uniformly distributed weight q per unit length of track axis. This weight consists of the unit weight of two rails and the averaged weight of the cross-ties and fasteners per unit length of track. To simplify the analysis, it is assumed that the vertical deflections of the rail-tie structure, prior and during buckling, are negligible. Thus, denoting b y w(x) the deflections in the vertical plane, it follows that

w(x) ~ O.

(3.2)

This assumption was made by all investigators of lateral track buckling reviewed in [l]. The lateral resistance exerted by the ballast on the rail-tie structure (due to lateral displacements) consists of the friction forces between the bottom surface and the two long sides of the ties and the ballast, as well as the pressure the ballast exerts against the front surface of the ties, as shown in Fig. 3 (a). For the following analysis it is assmned that the resulting lateral resistance is ~(x) (per unit length of track axis). This resistance acts at a distance el below the reference axis. However, because of the assumption that prior and during buckling the track deforms only in its plane, thus w(x) ~ O, the eccentricity el has no effect on the determined post-buckling response. Tests, in which track sections were displaced laterally, revealed that the corresponding resistance vs. displacement graph is non-linear [2], as indicated in Fig. 3 (a). However, as shown recently by A. D. Kerr [3], the simplifying assumption ~(x) = ~o0 = eonst. (3.3) m a y be sufficient for the determination of the saJe temperature increase. This finding will be utilized in the following analysis. The axial resistance exerted by the ballast on the rMl-tie structure (due to axial displacements) consists of the resistance between the ballast and the bottom surface of the ties and the pressure on the vertical tie surface exerted by the in a track, this change is measured from the installation temperature, (in theliterature often called the "neutral" temperatnre) at which the axial forces in the rails are zero.

Analysis of Thermal Track Buckling in the Lateral Plane

21

ballast in the cribs, as shown in Fig. 3 (b). In the following analysis it is assumed that the resulting axial resistance is r(x) (per unit length of track axis) and that it acts at a distance ez below the reference axis. Because of the assumption that. w(x) =- O, also for this case the e2-value has no effect on the determined postbuckling response. In this connection note the corresponding derivations presented in Ref. [4]. frail

+++ ;++++++ ;.~-

=~++~!,++

" ?++:++~!+

E

(a)

~, IOOC "~Q.

~lOOC - -

80C GOC

~ 40C

f

p:

F+

~ 40C

20C --

@..... . . . . . .

0I 0

L

-6 20C ,

, _ ,

,

,

,

2 4 6 8 I0 [2 ll4 16 Lateral displacement in mm

18

I

20

~

l Pi -

,

,

,

,

,

- 4 - 2 8 I 12 14 16 Axial displacement in mm

,

18 20

Fig. 3. Lateral and axial track resistances. (Test results according to [2]) Tests, in which track sections were displaced axially, revealed that the corresponding resistance vs. displacement graph is non-linear [2], as shown in Fig. 3 (b). Following the practice of a number of track buckling investigators (whose results will be compared in Section V with those obtained in the present paper), in the following analysis it is assumed that the resistance in the adjoining regions is r(x) = so ~- const.*

(3.4)

and that it is negligibly small in the buckled zone. This second assumption, which simplifies the solution, was made in all track buckling analyses reviewed in [1]. It may be partly justified by the observation made in the field that. lateral buckling is often initiated by a slight lift-off of the rail-tie structure (for example in front or rear of a wheel set) which eliminates the friction force between the ballast and the bottom surface of the ties. Furthermore, it is assumed that prior and during buckling the response of the rail-tie structure is elastic. The buckling analysis of a railroad track subjected to thermal compression forces consists of two parts: (1) the determination of all equilibrium states and (2) the inspection, which of the determined equilibrium states are stable and which are not. From the nature of the post-buckling equilibrium branches and their stability, established in l~efs. [5], it follows that the range of "safe" temperature increases to prevent track buckling may be determined solely from the post-buckling equilibrium branches. This concept is adopted in the following analysis. To insure a ]ormulation that is consistent, mechanically and mathematically, the equilibrimn equations for the track-beam are derived by utilizing the non* The justification o[ this assumption will be studied in a forthcoming paper.

22

A.D. Kerr:

linear t h e o r y of e l a s t i c i t y a n d t h e p r i n c i p l e of v i r t u a l displacements. To a v o i d t h e difficulties e n c o u n t e r e d b y other i n v e s t i g a t o r s (to be discussed in Section V) when m a t c h i n g t r a c k regions which are g o v e r n e d b y different differential equations a n d whose m a t c h i n g p o i n t s are n o t f i x e d a priori along t h e t r a c k axis, use is m a d e of v a r i a t i o n a l calculus for v a r i a b l e m a t c h i n g p o i n t s [6]. W i t h t h e n o t a t i o n of l~ef. [7] C h a p t e r I I I , t h e principle of v i r t u a l displacem e n t s m a y be s t a t e d as

- f f f F*.

dV - f f f *

V

aS = 0

(3.5)

S~

where l fF/"

U = 2

331

.

,

9

.

.

,

9

,.

(%.e.x ~ avvey v q- a~zez, q- a~ve. v T a~e~.~ T ov~ev.~) d g

(3.6)

V

is the elastic strain energy of t h e t r a c k - b e a m , V is its volmne in the u n d e f o r m e d state, g is t h e d i s p l a c e m e n t vector, a~ are generalized stresses, eij are L a g r a n g i a n strains, F * is t h e b o d y force of t h e t r a c k - b e a m , a n d / * is t h e t r a c t i o n force which acts on p a r t ~'~1 Of t h e t r a c k - b e a m surface S. The c o o r d i n a t e s y s t e m used a n d t h e p o s i t i o n of t h e t r a c k - b e a m , before a n d after d e f o r m a t i o n , are shown in Fig. 4. N o t e t h a t (x, y, z) are L a g r a n g e coordinates, (u, v, w) are t h e c o m p o n e n t s of t h e d i s p l a c e m e n t vector ~ of p o i n t (x, y, z), a n d ( ~ ) are variables which refer to t h e reference axis x. nx --4 • Wi

(

q

L *

)

x

Fig. 4. Notation and convention used in analysis W i t h t h e usual a s s u m p t i o n s of t h e bending theory of b e a m s (such as t h e p l a n e section h y p o t h e s i s ~, etc.) t h e expression for U reduces to X2

U - - f Tl [EA(~** -- .To) ~ + E I t : ''2] dx

(3.7)

x~

I t should be noted that the plane section hypothesi,s, although utilized by track investigators, is not satisfied for the lateral deformation of many tracks. However, because the cut-spike rail-tie fastener, currently used by U.S. railroads, exhibits only a very small rotational resistance, it appears justified to assume that for such tracks this hypothesis is valid for each rail and that the lateral bending rigidity of the track is the sum of the bending rigidities of the two rails with respect to their vertical axes; thus I = 2I~.. A thorough reexamination of the effect of the fastener rigidity on the lateral track response, will be contained in a forthcoming report.

Analysis of Thermal Track Buckling in the LaterM Plane

23

where A is the cross-sectionM area of the track-beam, I is its moment of inertia with respect to the vertical z-axis, ~T0 is the thermal strain, ~ , x = 4' + T 1 ~,2

(3.8)

and ( )' --d( )/dx. The derivation of the above relations (which also includes the order of magnitude estimates of the retained and neglected nonlinear terms) is identical to the one presented in t~ef. [4], except that w and v are interchanged.

IV. Analyses of Track Buckling Because of assumption (3.4), which states that the axial track resistance is constant, it follows that. in the adjoining track regions the axial track force varies linearly, as indicated b y the dashed line in Fig. 2(b). Therefore the length of these regions, which exhibit only axial displacements, is finite; namely a -: INt -- .~ti/ro. According to this scheme, beyond 2(1 q- a) the track does not deform due to buckling, which agrees with observations in the field. As shown in Fig. 1, the lateral buckling modes of a track are often symmetricM or antisymmetricM. Since the lateral resistance ~0 is always opposite to the direction of v(x), it follows that the simplest analysis for each of these modes is obtMned when the lateral displacements are assumed to be of form I or II, respectively, as shown in Fig. 5. In an actual track, the lateral displacements are not zero beyond 1 -- 1~. To study the effect this constraint has on the analytical results, it will be relaxed b y allowing the formation of additional half waves beyond l~ = l. This results in shapes I I I and IV shown in Fig. 5. The necessary analyses are more complicated, since more domMns governed by different differential equations have to be matched. I n the following, the post-buckling equilibrium states of the track--beam are determined for the deformation patterns I to IV, shown in Fig. 5. The obtained results are compared first with each other, in order to determine the effect of the constraints on the lateral displacements made in shapes I and II. They are then compared with the corresponding results of A. Martinet [8], K. N. Mishchenko [9], and M. N u m a t a [10].

IV.1 Analysis/or Symmetrical Dc/ormation Shape I Formulation of Problem For this analysis the track consists of five regions: the laterally buckJed zone of length 2l, two adjoining regions each of length a for which ~(x) ~ 0, and the two infinite regions in Ixl ~ l0 which are not affected by buckling, but which are subjected to the axial compression force N t = E A o c T o. Because of the assumed s y m m e t r y of the buckled zone, Eq. (3.5) m a y be written as follows:

jl 6

l0

co l0

}

= 0 F 1 dz § f Fa dx q- f ['~ dx § &G(I) § &%'(l) ~*

1

(4.1)

24

A. D, Kerr : Shape I -15

-I0

-5

'

"

I+

O

I0

~++~------+~+'--1--- --+,+'*~

I

\

I

I

IS

x in meters

L+.+/ . ~ }

r

a

5

I

t=l~

o

t=h

__

-~

Shape II

~ ,

k-2o ',

+

-15

i-I0 ]

-5

]

a

', ~ l u , ( x ) ~-20"*'~ 1"''

t =h

I 15 I x in meters

I ' __ _q~

i =h

a

x

[

--

[0

/

~-n

ShapeIII

I

1 I

a

/

x ]

uincml

=1!=

5

! !

l~

---i~

i.,

t

I

&

20F

h

1/

]~ in em _!_ h . . . . .

i

I j_

,.

IX in meters

.

t

t~

.

•

.

a__

Shape tV

-[,5

--

II[[

a

.2~

[ !

-r

-I

I2

-I

I I

5

I0

[I

_L

}u in em

J_

[1

- r

-

2+[-

J I

~1-

15

{

12

_I_

a

-'-

"I

Fig. 5. Lateral displacement sh+pes v(x)

where F 1 -~--

1

[EA(~I - - ~To) 2 § EI~;1 ''~] - - ~j(x - - l) ro~,1 4- oo~ 1

1

]

I (4.2V

| Fc~

1 --~ [EA(~+o - - ++To)2 4- EIv+o ,f2] - - ~)(x -- l) ro~ ~ 4- ,Oo~

]

J

s Since r 0 and ~oo are assumed not to be functions of the respective displacements, their directions h~ve to be prescribed ~s opposite to the corresponding anticipated displacements.

Analysis of Thermal Track Buckling in the Lateral Plane

25

and 1 n --- ?~n @

2

Vn ~2

n=

1, a, c~

s u b j e c t to t h e c o n s t r a i n t conditions in l _< x <-- l0 %(x) ~ 0 ~ %(x) = ~j(x) . . . . .

0 (4.3')

~(10) = 0

ec

and inloGx<

~(x) = 0 (4.3")

~(x)

(Note t h a t Nt = EAc~To ~= O) }

= 0

The s u b s c r i p t 1 refers ro t h e b u c k l e d region, t h e s u b s c r i p t a to t h e a d j o i n i n g regions, a n d t h e s u b s c r i p t oc to t h e x > lo region. ~](x -- l) is t h e unit step function a t t a c h e d to t h e a x i a l resistance r 0. N o t e t h a t r0 # 0 o n l y when u 4- 0 a n d t h a t ~o0 4= 0 o n l y when v ~= 0. The v a r i a t i o n a l Eq. (4.1) contains t h r e e L a g r a n g e multipliers. T h e y e n t e r because of t h e a s s u m e d c o n s t r a i n t va(x ) ~ 0 for l ~< x --< lo. The m u l t i p l i e r 21 is the l a t e r a l r e a c t i o n p r e s s u r e which m a y occur in I _< x ~< 10. The m u l t i p l i e r 22 is t h e a n t i c i p a t e d c o n c e n t r a t e d l a t e r a l r e a c t i o n force a t t h e p o i n t s x = l, which u s u a l l y occurs because of t h e use of t h e b e a m b e n d i n g t h e o r y a n d t h e s t i p u l a t i o n ~a(X) -- O.4 The m u l t i p l i e r ~ is a c o n c e n t r a t e d r e a c t i o n m o m e n t which m a y occur a t x = l. (If it does n o t occur, t h e analysis will yield )~ ---- 0.) Since x ~ I a n d x = l0 are variable m a t c h i n g points, Eq. (4.1) becomes, according to [6], l

l0

f

+ f Fo

0

I

co

+ f l0

+

[F l --

(4.4)

A~fla]l

+ [F~ - F ~ 0 ~lo + h [ ~ ( 1 ) + %'(1) ~l] + h [ ~ j ( 1 ) + %"(1) ~l] = o. P e r f o r m i n g t h e variations, t h e n i n t e g r a t i n g b y p a r t s , a n d g r o u p i n g t e r m s c o n t a i n i n g t h e s a m e v a r i a t i o n , Eq. (4.4) becomes 5 l

f [{(EIvl")" -- [EA(el -- o~To) Vx']' + ~o0}dvl 0

-- {lEA(el -- ~T0)]' + V(x - - l) r0} ~ul] dx lo

4- f [{(Elva")" -- [EA(% -- ~To) Va']'

- ~ OO - -

~1} dVa

l

- - {[EA(sa - - ~To)]' + ~(x - - l) r0} ~u~] dx 4 Note that the corresponding concentrated axial resistance was not included because the necessary "friction" coefficient is not known and also because its effect is not expected to be essential. In this connection note the related results for Shape III. 5 Since in (4.4) all variables refer to the reference axis, in the following the ( ~ ) symbol is dropped to simplify the presentation. Also note that because of (4.3), ~d(t) = 0 and ~d'(~) = 0.

26

A.D. Kerr: r

+ f [ { ( E I v ~ " ) " - - [ E A ( s ~ -- ~To) v~']' + eo} (~v~ Io -

I [ E A ( ~

--

~To)]' + ~(x -- l) ro} au~] dx

-~ { - - [ - - ( E I v l " ) ' + E A ( s l -- ~To) vl'] avl -- [ E I v / ' ] 5 v / -- [EA(el -- ~To)] (~ul}~:=o + { [ - - ( E l v l " ) ' ~- E A ( e l - - ~To) Vl'] 6vl -- [--(EIv/')'

- - ~ T o ) v~' - - 22] 6v~

-7 EA(e~

-r [ E I v / ' ] ~ v / -- [Elv~" -- 2~] 0%' J- [EA(el -- ~To)] 6ul - - [ W A ( s a - - o~To)] ~%a ~ - [ F 1 - - -Fa] ~ l } x : l

Jr {[--(EIv~")' + E A ( ~ -- ~To) v / ] ~va

+ [EIv~"] 6va' -- [EIv~] 6vs + [EA(s~ -- ~To)] 6u~

+ { [ - - ( E I v ~ " ) ' § E A ( e ~ -- aTo) v~'] ~v~

+ [EIv'g] avs + [ E A ( ~

- - ~T0)] ~ u ~ } ~ = ~ - : 0

(r

F r o m t h e a b o v e e q u a t i o n it follows t h a t t h e d i f f e r e n t i a l e q u a t i o n s for t h e t r a c k are : I n t h e buckled region, 0 ~ x <~ l, n o t i n g t h a t in this d o m a i n r/(x - - l) : 0, ( E [ v ( ' ) " -- [EA(el - - ~To)v~']' = --eo / [~A(~I

(4.6)

!

~"0)]' = 0

a n d in t h e adjoining region, 1 <_ x <~ lo, n o t i n g t h a t in this d o n m i n ~7(x - - l) ~ l, v~(x) ~ 0 a n d hence also ~o0 : 0,

;.I = 0

~

[EA(e~ -- ~T0)]' = --r0

!

(4.7)

where 8n ~

1

Un r ~ - ~

Vn '2

Yb ~

1~ a .

I n region l 0 ~ x ~< oo, as s t a t e d in (4.3"), u ~ ( x ) =- O, voo(x ) ~ O, a n d N t ~- E A s T o. F r o m Eq. (4.5) it also follows t h a t t h e b o u n d a r y c o n d i t i o n s a t x = 0 are ul(0) = 0 Vl'(O) = o

[--(EIv/')']o = 0

(4.8)

Analysis

of

ThermM Track Buckling

in

the Latera.1 Plane

27

Because of t h e c o n s t r a i n t conditions v~(x) =- 0 a n d u~(x) =- O, t h e b o u n d a r y t e r m a t x : ~ in Eq. (4.5) vanishes. The matching conditions at x = l a n d x = l0 are o b t a i n e d from t h e r e m a i n i n g t e r m s in (4.5) ; n a m e l y t h e b o u n d a r y b r a c k e t s a t x = I a n d x ~ l0. Because of t h e geometric c o n t i n u i t y conditions

v,(t) = %(1) Vl'(/) = Va'(/)

u~(l) = u S )

I

(4.9)

I

it follows t h a t a t x = l

~v~(l) = ~v~(/);

~Vl'(l )

=-

(~Va'(/);

(~Ul(/)- - ~ua(l).

Similarly, a t x = 10,

~v/(lo) = ~v~(lo),

OvAlo) = ~v~(/o);

~ua(lo)

~u~(lo).

Thus, t h e r e m a i n i n g t e r m s in (4.5) m a y be w r i t t e n as

{[--(EIVl")' 4- EA(sl - ~To) Vl' 4- (EIva")' -- EA(r 4-

[Elv,"

--

Elva"

4-

28] (~vl'

[EA(~,

4-

~To)

--

--

-- ~To) va' 4- 22] by1 EA(~

--

~To)] ~u~

4- [ E l - - A~a] (~l}x l

+ { [ - - ( E l v / ' ) ' + EA(e~ -- ~To) v+' 4- (EIvo~")' -- E A ( e ~ -- ~To) v•] ~v~ 4- [EIv~" -- E I v ~ " ] dVa" 4- [EA(e~ -- ~To) -- E A ( e ~ -- ~T0)] du~ 4- [ f ~ -

F ~ ] ~l~/~_t0 = 0.

(~.5')

H o w e v e r , in t h e a b o v e e q u a t i o n ~vl(l), 3vl'(l), etc. are n o t t h e v a r i a t i o n s of t h e v a r i a b l e end p o i n t l, as shown for v, in Fig. 6. According to Fig. 6, for t h e general case (needed because t h e c o n s t r a i n t conditions were t a k e n into c o n s i d e r a t i o n b y m e a n s of L a g r a n g e multipliers)

~v~(I)

--

~v,l

--

v/(l) ~l;

(~v~(lo)= (~V~o-- v/(lo) (~lo

3L

I

X i

1

+.

~vlll) : 8"vfl- ,,.+,[(u)3t

Fig. 6. 1%lations at the variable matching point x = 1

A. D. Kerr:

28 and similarly

dr1'(/) = dv'u - - vl"(1) dl; dul(t)=

dVa'(lo)

=

(~Vao - -

Va"(lO)

dl 0

d u u - - ul'(l) d/;

With these relationships, Eq. (4.5') becomes [--(EIv(')'+ -1- [ E l v l "

E A ( e l - - a T o ) vl' + ( E I v , " ) "

-- E A ( e ~ - - c~To) v~' + ;t2]l 6v,t

- - E I v ~ " -~- 2a]~ ~v'u + [ E A ( e l - - '~'~To) - - E A ( % - - ~T0)]i 6u~l

-- { [ - - ( E l v l " ) '

@ EA(sl

- - c~To) vl' 4- (ETv~")' - - E A ( e ~ - - c~To) v~' + Z~] vl'

+ [ E [ v 1 " - - E I v ~ " + )'a]

Vl H

-~- l E A ( e l - - ~To) - - E A ( s a - --

§ [--(EIv~")'

z

,~zz

+ E A ( e ~ -- a T o ) v~ § ( E I , J ~ )

t

-- EA(e~

.

-~ E A ( e . - - otto) v~' + ( E I v ~ ) '

[ E I v ~ " -- E I v ~ " ]

[l~ 1 --

-

(~l

,,

- - c~To) v~]zo dVao

-- EA(e~

va" -5 [ E A ( e a - - ~xTo) - - E A ( e ~

Fa]}I

- - c~To)]t~du~o

+ [ E I v ~ " - - E I v ~ " ] Z o dv~.o + [EA(e~ - - c~To) - - E A ( e ~ -- {[--(Elvo")'

aT0)] ul'

- - c~To) v s

Va'

- - c~To)] %'

[F~ - - f ~ l l ~ .

~zo = o .

(4.5")

In the above equation, all variations are independent. Thus, in addition to the geometrical matching conditions (4.9) subject to the constraint conditions in (4.3), which reduce to v,(1) = 0

|

(~.9')

vl'(1) = 0 Ul(1) = u.(l)

also the following conditions have to be satisfied at x = l u1'r

= uo'r

/

E I v 1 " ( I ) = --~a -- [(EIv,")']z

=

[

!

(4.1o)

- - 22 I

vl"(l) = 0

(transversality condition) J

and at x = lo, noting (4.3'), ,~o(lo) = 0 (4.11) %'(lo) = 0

(transversality condition) }

For a physical interpretation of the obtained boundary and matching conditions, it should be noted that the axial force, bending moment, and shearing

AnMysis of Thermal Track Buckling in the Lateral Plane

29

force in the lateral plane of the track-beam are expressed respectively as : Nn(x) = - - E A ( e n -- ~T0); M,(x) -

N > 0

compression

--EIv~" ;

n = 1, a

(4.12)

G ( z ) = - - ( E l v j ' ) " + E A ( ~ . -- c,~'o) v,' Because of the first transversMity condition v / ' ( l ) = 0, it follows from the second equation in (4.10) t h a t 2 a -- 0.

(4.13)

Thus, a concentrated reaction m o m e n t 23 does not exist at point x = 1. F r o m the third equation in (4.10) it m a y be concluded, noting t h a t vl'(1) = O, t h a t at x - l there acts a concentrated reaction force of magnitude

(4.14)

)~2 = [(E[vl")']l.

This reaction force occurs because of the use of the bending t h e o r y for the trackbeam and the constraint condition v~(x) --- 0. I t represents a concentration of the contact pressure in the close vicinity of x > I. F r o m the above derivations it follows t h a t the equilibrium formulation of the track in the lateral plane consists of the nonlinear differential equations in (4.6) and (4.7), the three b o u n d a r y conditions (4.8) at x = 0, the two conditions (4.11) at x = 4, and the five conditions at x = l, consisting of the three matching conditions in (4.9') and the two conditions from (4.10) /

u ( ( l ) = u~'(1) v('(1) : : 0

(transversality condition)

(4.10')

!

Thus, 10 conditions for the determination of the 8 integration constants and the 2 u n k n o w n lengths, I and l 0. Solution of Formulation for Shape I \

The differential equations in (4.6) and (4.7) are nonlinear. However, since the second equation in (4.6) when integrated yields E A ( e l - - C~To) -

const = --2~t

0 ~< x ~< 1

(4.15)

the first equation in (4.6) reduces, for E I = const., to E l v a ~v -~ N t v l " = --~oo

0 <~ x <_ l

(4.16)

a linear ordinary differential equation with constant coefficients. This analytical feature makes it possible to solve the derived nonlinear formulation for the lateral buckling of the railroad track exactly and in closed form. I t was utilized in [4] for the solution of the vertical track buckling problem. For the traclc beam, the coefficient E [ is constant, since in a railroad track the parameters of the rail-tie structure (such as the rail and tie characteristics, the gauge, the tie spacing, and the fastener type) usually do not vary.

30

A.D. Kerr:

According to (4.12), the left-hand side of Eq. (4.15) is the axial force in the buckled region. I t was denoted b y --N~. Thus, the axial compression force in the buckled region is (+-~t) and is constant. The general solution of Eq. (4.16) is vl(x) = A1 cos 2x + A2 sin ;~x + A a x 4- A a - - o* X2 2~2

(4.17)

where /

= [~-,

~o* -u-7 ~ o0

(4.17')

Since for the considered problem/Vt > 0, it follows that )0is a real number. F r o m the second and third conditions in (4.8) it follows that A 2 - A 3 : 0. The constants A I and A 4 are obtained using the first two conditions in (4.9').The resulting vl is

o'l{ [

v~tx)=~

x~

1

t~

2(c~ ix = i~ )g)]

z~sin~l

]"

(4.18)

The length l is as yet an u n k n o w n quantity. I t is obtained from the transversality condition in (4.10). Substituting (4.18) into this equation, it follows t h a t ib is satisfied when tg 21 :

)~l.

(4.19)

The roots of this equation are 2l ----0, 4.493 . . . . . . . .

(4.19')

The first root corresponds to the trivial ease. I t m a y be shown t h a t the second root corresponds to the symmetric deflection shape I, shown in Fig. 5% This root will be used in the following. I t should be noted t h a t the Vl(X) expression in (4.18), with 21 = 4.493, contains still one u n k n o w n ; n a m e l y the axial force Nt. For its determination we use the remaining equations of the above formulation (namely those in t e r m s of u). T h e y are: the second equation in (4.6) and (4.7), which are nonlinear, and the corresponding b o u n d a r y conditions in (4.8), (4.9'), (4.10) and (4.11). Since for the track-beam E A = const, and Va(X) =- O, the second equation in (4.7) reduces to the linear equation E A u a " = --re

I <~ x <_ lo.

(4.20)

I t s general solution is uAx) = -

re ) -~ X2 +

-~

B~x + G .

Using the first condition in (4.11), %(lo) : 0, we obtain B2

[2EA -- B~l~ "

6 Because of (4.IT), the expressiort ;~l = 4.493 may also be written as Nt = 20.19EI/l~. Note that in a number of references 1 = 1/2. Hence Nt = 80.7EI/l:.

Analysis of Thermal Track Buckling in the Lateral Plane

31

Thus Ua(X) = --2--E--A (x2 -- 102) @ Bl(X -- 10)"

(4.21)

Instead of the second equation in (4.6) we utilize its first integral given in (4.15); namely the nonlinear differential equation of the first order

(

EA

u-/ q- -~ vl '~ -- :~To

)

-= - - N t

0 <_ x <_ I.

(4.15)

Since, at this point of the analysis, v~(x) is a known function and is given in (4.18), the above equation reduces to a linear differential equation. Rewriting Eq. (4.15) as

u?(x) = (~,To

~

1 v,'~(x)

(4.22)

and integrating it from 0 to x, noting that according to (4.8) ul(O) = O, we obtain x

(4.28) 0

The three unknowns in (4.21) and (4.23) B~, 10, 2Vt are determined in the following from the remaining two matching conditions at / and the transversality condition at 10. Substituting expressions (4.21) and (4.22) into the matching condition u / ( l ) = %'(1) and noting that according to (4.9') vl'(1) -= 0, we obtain B I--

Nt/

r~

C~To-- E A ] + F~A"

(4.24)

Thus

u.(x)

::

[- ~

(z + a -

A/] r (~ T 0 - E
l) -

(4.25)

Substituting the u expressions from (4.23) and (4.25) into the matching condition ut(l) = Ua(1), we obtain l

(4.26) 0

Evaluation of the integral term, noting (4.18) and (4,19), yields l

f v,'2(x)dx =

51 {o~*l~ = 20.45 X 10-4~)'217. -C \ 7 1

(4.27)

0

With this expression Eq. (4.26) becomes

=X-X + 7~o ~& 1

re(1o -- 02 2EAl o

(4.28)

32

A.D. Kerr:

Substituting the Ua(X) expression given in (4.25) into the transversMity condition in (4.11), u~'(lo) = 0, we obtain (EA~xTo -- Nt) = ro(lo -- 1).

(4.29)

Since EAc~To ~ N t is the axial compression force in the undeformed track-beam, Eq. (4.29) may also be written as (4.29')

(Nt -- Nt) = ro(lo -- l).

Thus, the transversality condition at x = 10 yields the equilibrium equation oJ the ad]oininff reffion in the axial direction, as shown in Fig. 7.

L

r0 = CONS| a lo -i

Fig. 7. Mechanical interpretation of a derived condition

The exact solution is thus obtained. The displacements at equilibrium are given as follows: v~(x) by (4.18), ul(x) by (4.23), v,(x) =-- O, and u~(x) by (4.25). The relationship between To and 2Vt is given in (4.28), noting that according to (4.19') 2l = 4.193 for the symmetrieM mode [, that 10 is determined from (4.29), and that 2~ = N t / ( E I ) . To simplify the numerieM evaluation it should be noted that in Eq. (4.26) the unknown a = (lo -- 1) may be eliminated by utilizing Eq, (4.29). The resulting equation is l

(~Vt -- Nt) 2 @ 21re(Mr -- Nt) -- EAro f vl'~(x) dx = 0,

(4.30)

0

where Nt = E A a T o . Solving this quadratic equation we obtain

For a given track (thus, for known values of E, A, I, c~, re, ~o) the nmnericM evaluation of the obtained solution consists of the following steps: Choose a positive value for Nt and determine the corresponding value 2~ = ~ / ~ and / = 4.493/2. Next, obtain the corresponding To value from (4.31), noting (4.27). The corresponding displacements are then given by (4.18), (4.23) and (4.25). The numerical evaluation was performed for a track with 115 lb/yard rails on wooden cross-ties with cut-spike fasteners (o~ negligible rotational resistance).

Analysis of Thermal Track Buckling in the Lateral Plane

33

The following p a r a m e t e r s were used: A ~ 145 cm 2 (11.25 in s) I ---- 2/'~ = 899 cm 4 (2.1.6 in 4) E ~ 2.1 x 10 ~ k g / c m 2 (3'x 107 lb/in ~) cr -= t.05 x 10 -51/~ 20 = 600 k g / m (402 ]b/ft) re ---- 1000 k g / m (670.8 lb/ft) The corresponding graphs are shown i n F i g . 8 (as dashed lines), noting t h a t for shape I, v~nax ~ v(0). The obtained graphs are of the same t y p e as the ones obtained for the m u c h simpler t r a c k model analyzed in Ref. [5] (Fig. 11). According to Fig. 8, for the used track p a r a m e t e r s the safe temperature increase is 5~L = a,3.7~ Note t h a t for a n y uniform t e m p e r a t u r e increase T o > T~ there correspond three states of equilibrium : The (stable) 'straight state, the (unstable) equilibrimn state on branch A L and the (stable) equilibrium state on branch LB. Thus, when the track buckles at a t e m p e r a t u r e increase T o > T~ it will go over to the corresponding laterally deformed equilibrium configuration on branch LB.

.S

I

20

. . . . .

.......

Shopel " II " III " IV

E

r

V m f l x in c m I

20

I I

8'o

40;

I ,oo

i

i 200

i i

]60

' \ I

\ I|5 Ib/yard

track

po=6OO kg/m' to= IO00kglm

,.20 t20

80

4O ~IllQX I n CF/t

~

I

I

go

I

J

,oo

Fig. 8. Comparison of post-buckling.equilibrium branches and the corresponding axial forces Nt, for shapes I to IV 3

Acta MedL 30/1--2

34

A.D. Kerr:

I n this connection note the large drop of the axial force due to buckling and the corresponding values for a a n d l: Temperature increase T Oin ~

Axial force in straight state N t -~- E A s T o in tons

Axial force in buckled region 2~t in tons

a -- N~ -- Nt r0 in meters

l in meters

T L ----43.7 T ~- 50.0

139.7 160.0

90.0 62.8

49.7 97.2

6.5 7.8

I V.2 A n a l y s i s / o r A n t i s y m m e t r i c a l Deformation S h a p e I I

F o r m u l a t i o n of Problem The general form of the deformation shape I I is shown in Fig. 5. Also for its . analysis the track consists of five regions, as for shape I. Because of the antis y m m e t r y of shape I I , it is sufficient to consider only the p a r t for x ~ 0. The formulation is therefore identical to the one for shape I, except for the b o u n d a r y conditions at x -~ 0. Thus, the differential equations for the track are given, as before, by (4.6) and (4.7). ~ r o m Eq. (4.5) it follows t h a t the b o u n d a r y conditions at x = 0 are: v~(0) = 0;

v ( ' ( 0 ) = 0;

u d 0 ) = 0.

(4.32)

The conditions at x = l and x = 10 are the same as before. N a m e l y

vs(1)

=

o

Ul(I ) = Ua(1) Vl"(l) -----0

(4.9')

v,'(l) = o

and

U((1) = Ua'(1)

(transversality condition)

(4.10')

and

Ua(10) = 0 Ua'(lo) -~ 0

/ (tranversality condition)

!

(4.11)

Solution of F o r m u l a t i o n for Shape I I Because the differential equations are the same as for shape I, it follows t h a t the general solutions are the same. N a m e l y : vs(x) = A1 cos )~x ~- A~ sin ),x + Aax + A4 - - - - x 2 2~2

(4.17)

x

us(x) = (sTo -

x --

f

(4.23)

vl'(x)

0

va(x) = 0

u~(x) = - -

ro (x 2 2EA

lo~) + B l ( x -

lo).

(4.21)

The integration constants As to A4 are determined from the first two conditions in (4.32) and in (4.9').

AnMysis of Thermal Track Buckling in ghe Lateral Plane

35

T h e y are : 0*.

A

~* (1 -- cos 21) ~- (2l)2/2 -- 21 sin 21 ]

cos ~tl) -~ (;tl/2 cos ,tl -- sin 2l) 21 A

Q*

/

(4.33)

Substituting the obtained v~(x) into the transversality condition, v~"(l) ~ O, it follows t h a t it is satisfied when

(4.34)

2(1 -- cos 2/) = 21 sin ,~l provided tg 21 4= ,~l. Thus, the roots of this equation are ,~l ~ 2x, 8.987 . . . .

(4.34')

I t m a y be shown t h a t the first root corresponds t o . t h e a n t i s y m m e t r i c a l deform a t i o n shape I I , shown in Fig. 5. W i t h ;,1 = 2z, the expressions for the integration constants simplify to A1 =

p./4 . 16•4, e*!s 8~ 2;

Aa=

A2 =

~,l 4 (4.33')

= ~*l~ A~ 16~i

and v a(x) becomes

T h e a b o v e expression Vl(X) contains still one u n k n o w n ; n a m e l y the axial force 2~t. F o r its determination w e utilize the equations for u(x) of the above formulation. Since t h e y are the same as those used in the analysis of shape I, the steps are also the same. Therefore, the additional equation needed for the determination of 2Vt is

EA~To--Nt~-Iro

--1 +

~-12ro?

~ dx

(4.31)

where vl(x) is given b y (4.35) and the corresponding integral is l

f vl '~ dx == 17.430 • 10-5~'217.

(4.36)

o

Also for shape I I , the length of the adjoining region is

a = (EA~To -- Nt)/ro.

(4.29)

The exact solution for the deformation shape I I is thus obtained. I t was numerically evaluated for the same t r a c k p a r a m e t e r s as used for shape I. The results are shown in Fig. 8, as solid lines. Vm~x was calculated b y first forming 3*

36

A.D. Kerr:

dVl/(~,)J

=

0 which yielded x / l = 0.3464 and then b y substituting this value into

v~(x). The obtained graphs are very similar to the corresponding graphs of shape I. The drop of the axial force in the buckled region and the corresponding a and l values, at To ~ T~, are given in the following table: Temperature increase T Oin ~

Axial force in straight state ~Vt = E A a T o in tons

Axial force in buckled region Nt in tons

a = 2Vt - - N t

l

re in meters

in meters

T L =: 42.0

134.3 160.0

89.0 58.0

45.3 102.0

9.2 11.3

T = 50.0

IV.3

Analysis/or

Symmetrical

Deformation

Shape III

F o r m u l a t i o n of P r o b l e m The general form of shape I I I is shown in Fig. 5. For the following analysis the t r a c k consists of seven regions, since the buckled zone now contains three regions (instead of one, as for shape I)

Because of the assumed symmetry of the buckled track, Eq. (4.1) becomes

F1 dx + f F~dx + j"F~d.,§ f F~ ex + Z,~o(0 § X~vo'(0 = 0, Ii

t

(~.a7)

lo

where F1, Fa, Foe are given in (4.2), 1 [ E A ( s 2 - - otTo) ~ § E I v 2 ''~] - - y ( x - - I) rou ~ - - Qov2

(4.2')

subjec~ to the constraint conditions (4.3') and (4.3"). Proceeding as for deformation shape I, noting t h a t 11, 1 and 10 are variable end points, the following formulation results: The nonlinear differential equations: (EIv/')"

-- [EA(q

: (EIv(')"

- - O~To) v ( ] ' = - - ~ o ]

[EA(q --

lEA(e2

--

0 _< x _< 11

(4.38)

l~ ~ x ~ 1

(4.39)

l ~ x ~ lo

(4.r

- - ~To)]' = 0

~To) v ( ] ' = +~Oo /

[ E A ( e 2 - - ~To)]' = 0

!

~1 = 0

|

[ E A ( e a - - ~To)]' = - - r e

!

where en=un

,

§

1

,2. ,

n=l,

2, a

the constraint equations u~(x) -= 0;

v~(x)

=- 0

(4.r

Analysis of Thermal Track Buckling in ~he Lateral Plane

37

and the boundary and matching conditions

v((0) = 0 ;

v("(0) = 0 ;

u~(0) = 0

(4.42)

Vl(/I) --= v2(ll)

Vl"(ll)

= V2H(ll)

I

v~'(ll) = v~'(l~)

v/"(l~) = v~'"(lA

/

I

(4.43)

I

Vl(ll) = 0

[thus, also v2(11) --- O]

v2(l) = o

v~'(1) = o

u~(1) = uo(1)

u~'(l)

|

uj(t)

v~"(l) = 0

and u~(10) = o;

(4.44)

J

%'(I0) = o.

(4.45)

Thus, 17 conditions for the determination of the 14 integration constants and the 3 unknown lengths l~, I and l 0. Solution of Formulation for Shape I I I Integrating once the second equation in (4.38) and in (4.39), we m a y write EA(s~ -- ~To) = const. = - - ~ t l

0 ~ x ~ l~ l

EA(e2 -- ~T0) = const. = --Nt2

11 ~ x ~ 1

(4.46)

J

Noting the matching conditions for u' and v' in (4.43) it follows t h a t 2~tl = 2~t2 = -~t.

(4.46')

Thus, for E I =- const., the first equation in (4.38) and in (4.39) reduce to the linear differential equations with constant coefficients EIVl iv @ l~tvl" = --Oo

O~x~--ll}

EIv2 iv @ Ntv2 '' ~- @~o

ll ~ x ~ l

(4.47)

Their general solutions are vl(x) = A1 cos ,ix -~ ,A2 sin ,ix ~- A~x ~ A~ -- ~

x2

(4.4s) v2(x) = A~ cos ,ix ~- A6 sin ,ix + A~x ~- As ~- ~9* x 2

38

A.D. Kerr:

Using the 8 b o u n d a r y and matching conditions for v in (4.42), (4.43) a n d (4.44), the constants At to As are obtained as: 2~o*l9 At=~[~--cos2/~]; A4

=

A2=As=0

.0./4

2(2/) ~ 1(2l~)2 _ 4[~b -- cos 2l~] co s 2l~}

(4.49) A5 =

2~q * l ~

~b; A6 =

2q*14 (Z/)'

2q*l~

As = ( - ~ [--~b cos 21

-

-

sin Z/t;

A7

--

2q*l~ (Z/)a/ )Llt

sin 2l sin 21~ + 2lt21

-

-

(2/)2/4]

where ~b

-=

1 (cos 21 sin 21x -- 2lt -[- 2l/2). sin Zl - -

For the determination of 1t and 1 we utilize the two transversality conditions

v2(ll) • 0 and v2"(1) = 0, and obtain A~ cos 2I1 -t- A~ sin 2ll § AT11 -~ As ~- o~*14(Zlt)~/[2(2I)4] = 0 /

(4.50)

!

A5 cos 21 + A6 sin 2l -- o*l~/(21) 4 = 0

where Aa to As are given above. The lowest roots 211 and 21 of the two simultaneous algebraic equations in (4.50), which correspond to the shape I I I shown in Fig. 5, were found using a numerical trial and error approach. T h e y are 21t = 2.918;

)J = 7.551.

(4.50')

Thus, for shape I I I , l/ll = 2.59 For these roots the integration constants At to As become At = 1.20 • 10 s0*/a;

A2 = Aa = 0

}

A4 = 2.48 • 10-aQ*/4;

A5 = 0.60 • 10-S~o*/a

I

A 0 = 0,14 • 10-So*/4;

A7 = --1.36 • 10-2~*/3

/

As = 4.48 • 10-a~o*l~

/

(4.493

The expressions for vt(x) and v2(x) contain still one u n k n o w n ; n a m e l y the axial force 2~t. This u n k n o w n is determined in the following from the equations for u(x) of the above formulation, in a similar m a n n e r as done for shape I. F r o m the equations in (4.46) it follows, noting (4.46'), t h a t

ul'(x) = @To -- ~A ) -- -~ vt'2(x)

O~x~lt

u2'(x) = (~To -- -~A) -- + v2'2(x)

l~ ~ x ~ l

(4.51)

Analysis of Thermal Track Buckling in the L~teral Plane

39

Integrating the first equation from 0 to x we obtain, noting that ul(0) ~-- 0

Ul(X)

~-

~xTo -- ~

x-

-~

vl

(4.52)

dx.

0

Integrating the second equation in (4.51) from l, to x we obtain

(

~t) (x-t~)- -~~ /

u~(x) = u~(ll) + ~To -- - ~

(4.53)

v~'~dx.

l~.

The differential equation for Ua in (4.40), is the same as the one in (4.7) for shape I. Also the corresponding boundary and matching conditions are the same, as for shape I, except that ui(x) is replaced by the adjoining u2(x). Thus, the solution of the second equation in (4.40) subjected to the conditions ua(lo) -- 0 and u2'(l) ~ u~'(l) is, as before,

[

ua(x) -~ o'er~ (x ~- a - - l) - -

From the matching condition ul(ll)

~

(~xTo -

-x ) "

EA]J

(4.25)

U2(11) we obtain Ii

i u~(ll) = (~'To -- ~ i ) ll -- -~ f v(~dx

(4.54)

0

and from the matching condition u2(l) - u~(1)

~T~) -- -EA

-~

[;

vl '2 dx, -~

0

/ ] v2 '~ dx

-- 2EA "

(4.55)

l,

Eq. (4.55) is the additional equation needed for the determination of Nt. Substituting the u~(x) expression given above, into the condition ua'(1) -~ 0 we obtain, as before, (4.29')

(Nt - - Nt) : roa

where a = (10 -- l). Next, we eliminate the variable a from Eq. (4.55) by using (4.29'). The resulting equation may be written as

(N t-

Nt) ~ ~- 2 1 r o ( N t - N t ) -

EAro

f vl '~ dx ~0

v2 '~ dx I1

=0

(4.56)

40

A.D. Kerr:

or, in solved torm, E A ~ T o : Nt 45 lro -- 1 @

@ 12r-~

(

Vl '2 dx @ 0~

v2'2 (ix

.

(4.57)

l~

The integral expressions which appear in (4.57) were evaluated, noting Eqs. (4.48) and (4.49') and t h a t according to (4.50'), I/ll = 2.59 T h e y are 21

f vl '2 dx - : 3.09 • 10-%*~ll 7 1 o

(4.5s)

f v2 '2 dx -~ 6.25 • 10-6~'2/7 ]

The solution for shape I I I is thus obtained. I t was numerically evaluated for the t r a c k p a r a m e t e r s used before. The results are shown in Fig. 5 as dash-dot-dash lines, noting t h a t for the present case Vma~ ~ v~(0). The drop of the axial force in the buckled region and the corresponding values of a, l 1, and l, at To ~ TL, are given in the following table: Temperature increase T Oin ~

Axial force straight state N t = EAo~To tons

Axial force buckled region Nt tons

a -- -~rt -- Nt r0 meters

[1 meters

1 meters

T L = 4:1.2 T = 50.0

131.8 160.0

88.8 56.5

43.0 103.5

4.3 5.3

11.0 13.8

I V . 4 Analysis ]or Antisymmetrical De/ormation Shape I V F o r m u l a t i o n of Problem

The general form of shape I V is shown in Fig. 5. For its analysis the track consists of seven regions, as for shape I I I . Because of the a n t i s y m m e t r y of shape IV, it is sufficient to consider only the p a r t for x ~ 0. Therefore, the formulation for shape I V is identical to the one derived for shape I I I ; except for the b o u n d a r y conditions at x = 0. These three b o u n d a r y conditions are those of shape I I and are stated in Eq. (4.32). Solution of F o r m u l a t i o n for Shape I V Because the differential equations are the same as for shape I I I , it follows t h a t the general solutions are also the same. N a m e l y v1(x ) = A 1 cos ~x 45 A2 sin 2x 45 A3x -~ A4 -- Q'x2/(22 ~) ) v2(x) ~ A~ cos ).x 45 A 6 sin 2x 45 ATx -k As 45 e'x2/(222) Va(X ) ~ O;

Yea(X) ~ 0

J

(4.59)

Analysis of Therma.l Track Buckling in the L~teral Pla,ne

41

The integration constants A~ to A8 for shape IV are: 1 7 o*l*; A1 = -- (;~l)-1

As=~

1

1

[q~ - - 2 sin )J1] ~0.~4 1

[O sin itl + 2211-- 21--~bcos2l]~o*la;

A ~ = ~ e *O/ ~ ; A7 = ~

A2 =

A4=~o*l

4

(4.60)

A 6 = ~ . r o,la

[O sin ,tl -- 21 -- r cos 21] ~*l a

1E

A s = --(2/)--~ O (cos 2l + 21 sin 2l) + r

21 -- 21 cos )J) +

~o*l4

where 0 = 2 cos 211 - - 1 ('~ll sin'~l + c~ 2/1) O + ;tll ( 3 2 / 1 - 2/ r =

~l~) (4.60)

(2/1 cos 2/-- sin 2/1)

The two conditions for the determination of Ii and I are the same as those for shape I I I and are given in (4.50). For the An constants given above, the lowest roots of these two equations, which correspond to shape I V were found to be 211 = 5.31;

21 = 8.54.

(4.61)

Thus for shape IV, Ill1 = 1.61. F o r the above roots the constants A1 to As become AI -

--0.19 • 10-~*/i;

A4 -= 0.19 • 10"-8~o*/4;

A~ = , 0 . 5 7 • 10-S~o*/~;

As -- 4.89 • 10-a~o*/3

A5 = 0.02 X 10-~o*/~;

As = 0.26 • 10-~

(4.603

A7 -= - - 12.16 • 10-s~*/3; As = 5.12 x 10-3~o*l 4. The remaining derivations are the same as for shape I I I . The results are:

ul(x) =

~T~ -- E A / x -- -~

v, '2 dx 0

us(x)-=

aTo

--

EA]

x --

2

d~

v( 2 dz

(N, - 27~) = r0a

+

l,

v( s

dx

(4.62)

42

A.D. Kerr:

and the equation for the determination of Nt

[

E A c d ' o = N t + lro - - 1 4-

1 4- l,r---~

Vl'2 dx +

v2 '~ dx

;

Ii .

(4.6a)

The integral expressions which appear in the above equations were evaluated noting (4.59) and (4.60'). T h e y are, noting t h a t 1Ill = 1.61, It

l

f vl '2 dx = 5.49 • 10-4~'2l, 7 ; 0

f v l ' ~ dx = 1.91 • 10-7~'2l 7

(4.64)

I1

The solution for shape I V is thus obtained. The results of the numerical evaluation are shown in Fig. 5 as dash-dot-dot-dash lines, vm~X was calculated b y first forming dvl/dx = 0 which yielded x/l, -~ 0.40 and t h e n b y substituting this value into vl(x). The drop of the axial force in the buckled region and the corresponding values for a, l, and I, at T O ~ T~, are given in the following table: Temperature increase T Oin ~

Axial force straight state Nt = EAaTo tons

Axial force buckled region 2Vt tons

a -- Nt -- 2~t r0 meters

41.0 T = 50.0

131.2 160.0

88.0 56.3

43.2 103.7

T L ~

Iz meters

1 meters

7.8

12.5

9.7

15.6

I V . 5 R e m a r k s on Obtained Results A comparison of the post-buckling equilibrimn branches, and the corresponding axial forces, shown in Fig. 8, reveals t h a t the results for shape I to I V are very close, especially with regard to the T~-value. A graphical comparison of the stable lateral displacements v for the 115 lb/yard track at To = 50~ is shown in Fig. 5. I n each of the four graphs the vertical scales and the horizontal scales are the same. The shown wave lengths and amplitudes are of the order observed in tests and discussed in [1]. Note, t h a t according to the above tables the a-values are several times larger t h a n the corresponding/-values. F r o m Fig. 5 it follows t h a t as more lateral waves are included in the analysis (shapes I I I and IV), the length of track affected b y buckling ~ 210 = 2(l-~ a), increases noticeably. This is not the ease, however, in the actual problem. The increase of 2/o in Fig. 5 is caused partly b y neglecting the axial resistance r0 in the laterally buckled region, thus also in region 11 --< x _< l; a practice a d o p t e d b y all investigators who determined postbuckling equilibrium branches for shapes I I I and/or IV. Note that x is a Lag-range variable and that v(x) is only the lateral component of the displacement vector. Also the length of the buckled region is not 210,but 2[/o @ u(lo)]. However, for the problems under consideration u(lo) ~ lo .

Analysis of Thermal Track Buckling in the Lateral Plane

43

The above finding suggests t h a t for shapes I I I , I V or higher, the resistance r should be included, at least for x > 11. The resulting formulation for shape I I I or I V remains the same as derived above, except for the second differential equation in (4.39) which becomes

lEA(e2 -- ~T0)]' = --r0;

ll ~< x --< l.

(4.39')

I t is anticipated t h a t the effect of this correction on TL will be relatively small. For practical purposes the use of the results based on shapes I I and I I I appears to be sufficient.

V. Comparison With Results of Other Investigators A review of the analyses of thermal track buckling in the lateral plane was recently presented by K e r r [1]. One finding of this survey was t h a t only a small number of the published analyses, namely those which took into account the drop of the axial force 37t in the buckled region, are conceptually correct. However, these few analyses, although based on the s a ~ e fundamental assumptions (such as the replacement of the track by an elastic beam in bending, and the assumption that the axial and lateral resistances are constants) utilized different methods of solution with an unknown effect on the accuracy of the final results. In this section the well known analyses of Martinet [8], Mishchenko [9], and N u m a t a [10], which represent the different methods of solution, are briefly discussed and the obtained results are compared with those derived in the present paper. Martinet [8] described the lateral response of the buckled track region b y the linear differential equation of classical beam theory

E I d4v d2v dx----4+ ~ t dx~--= ~:~0

(5.1)

and used, at x = 11 and l, the same boundary conditions as derived in the present paper. Since the linear differential equation for v, for example (4.16), which was obtained from the nonlinear equations in (4.6), is identical with Eq. (5.1), the resulting solutions for the expressions v are also the same. For the determination of the unknown axial force -~t, Martinet set up a separate compatibility equation for axial displacements at the juncture of the buckled and adjoining regions. For shapes I and I I he obtained an expression which is identical with Eq. (4.31). For shape I I I his compatibility equation is identical with Eq. (4.57). I t is indeed noteworthy that, although Martinet used, a priori, a linear differential equation for v and derived the compatibility equation, heuristically, not making a distinction between Euler and Lagrange coordinates etc., he obtained results for shapes I, I f and I I I which are the same as the corresponding results derived in the present paper. (In this connection refer to the corresponding questions raised b y Kerr [11].) Since in the present paper the formulation for the entire track was obtained in a unified and consistent manner and the obtained solutions of the resulting nonlinear formulations are exact, it m a y be concluded t h a t t h e objections raised b y Mishchenko ([9] pp. 64--65) regarding the accuracy of Martinet's solution are not valid.

4:4

A.D. Kerr:

Mishchenko [9] and N u m a t a [10] used different variants of the energy approach. The effect of the drop of Nt to 5~t (for which Martinet used a displacement compatibility equation) Mishehenko took into consideration in the expression of the total potential energy

//=

H(t, l),

(5.2)

where / is an amplitude of lateral deflections and 21 is the length of the buckled region, and by a displacement compatibility equation. The equilibrium relations Mishchenko obtained from the conditions ~H

OH

--=0;

--=o.

(5.3)

Approximating shape I by the expression s vl(x) = 0.294 • f •

(

cos -4.492x - 7 - -- 2.190

q- 2.407

)

(5.4)

where ] -- vl(0), the equilibrimn equations obtained b y Mishchenko are 2.77

- - 33.56 l--~ - - 0.8589o/= 0

(5.5) (Nt2EA--~t)2

0.692 =~/2__+ 25.17 EI[2l ~ ~ 0.429]@0 ~- 0

and the derived compatibility relation m a y be written as (N t -- Nt) ~ @ 2lro(N t -- , ~ ) -- 1.,384 EAr~ l

• O.

(5.6)

Approximating shape I I by two expressions, each valid in a different region, Mishchenko obtained the equilibrium equations 11.06 -1-~t] -- 254.4 EI/ - 7 -- 1"0129~ = 0 (N , -

~ t) ~

2EA

(5.7) 2,764 `~t/212 q- 190.8 EIPli -- 0.506~ooJ = 0

and the corresponding compatibility equation (37t -- Nt) ~ + 2lro(Nt -- Nt) -- 5.528 EAr~ 1

-- 0

(5.8)

where ] = (Vl)max. The above equations were evaluated numerically and the results are shown in Fig. 9 and Fig. 10. s To simplify the reading and enhance comparisons, the following equations utilize the notations of the present paper.

Analysis of Thermal Track Buckling in the Lateral Plane

45

IOC

8C

u

6c

~_o 40

B

Presen! paper, Martinet [8]

g 20

Numata [103 Mishchenko[gJ

r

0

0

I

I

I

I

20

40

60

80

..... I

I00

Vmax in r

:Fig. 9. Comparison of post-buckling equilibrium branches for shape I by Martinet, Mishchenko, Numata, and Kerr. (The corresponding Nt-curves are, within the accuracy shown in :Fig. 8, essentially identical)

lO0

8(

~ 4e "-

fPresent paper, "l.Martinet [ 8 ]

"5 ~' r 2C E #,

Numata [10]

Mishchenko [91

o

I

2o

go Umaxin

.

i

-I ,oo

cm

Fig. 10. Comparison of post-buckling equilibrium branches for shape I I by Martinet, Mishchenko, Numata, and Kerr. (The corresponding Nt-curves are, within the accuracy shown in Fig. 8, essentially identical)

46

A.D. Kerr:

N u m a t a [10], using a different variant of the energy method and using different approximating functions for v, obtained for shapes I to IV the following equations

I =

fi(zt), eo

}

7c2{~t (n-t-_l)'7~2EI[ I (5.9)

2~, = 2t

~-i

J

and

Nt -- ~.t :- rol - - 1 +

(5.10)

V i l l - SOo,#>((En,lq. rol~i(Nt)Ti2 )

80,

/

/

////~ ~9

//

~_ 4G 5Presentpaper LMartinet [8] E

~. 2c

~

Numata[lO]

' 920

' 40' 60 .Vmax in cm

8'o

1 100

160 - ' t ~

120 t~

8c c o

I

0

20

I

I

60 Urnaxin cm

40

Y 80

[

[00

Fig. 11. Comparison ol post-buckling equilibrium branches and .~t-curves for shape I I I by Martinet, Numuta, ~nd Kerr

Analysis of Thermal Track Buckling in the Lateral Plane

47

where the coefficients n, fl, and # are given in the following table: Buckled shape

n

fi

#

I II III IV

1 2 3 4

1.000 0.2425 0.1685 0.097 7

8.8857 7.9367 11.7867 16.3004

The above equations were evaluated numerically and the results are shown in Fig. 9 to Fig. 12. F o r shapes I and I I the agreement between the shown graphs 8O

/

.-= 60

/

~

//

/~

"- .4G P

Present paper

E I~ 2C

Numota[10]

I

0

20

I

40

t

60

I

80

J

I00

ureax in cm

160 ./I

g

/

i.

%

"-%..

0

20

40

60

80

I00

ureax in cm

Fig. 12. Comparison of post-buckling equilibrium branches and 2~t-curves for shape by Numata and Kerr

IV

48

A.D. Kerr:

is very close. However, for shapes I I I and IV the N m n a t a results deviate noticeably from the exact ones and yield TL values which are about 10% higher. This deviation appears to be caused by the a priori assumption of the values for (Vl)m~/(V2)ma,~ and the position of (V2)ma~, which constitutes an additional analytical constraint on the solution. [For example, whereas N u m a t a stipulated for shape I I I (vl)m~ = 1.7 and

/--

-- 2

the corresponding values which result from the exact solution, derived in the present paper, are (Vl)m~ __ 4.8

and

t

-- 1.8.

Thus, according to the exact solution, the lateral displacements " d a m p out" more rapidly than assumed by Numata. This feature is even more pronounced for shape IV, as shown in Fig. 5. l~egarding the suitability of analytical methods for solving the thermal track buckling problem, it should be noted t h a t the claim made by Mishchenko ([9] p. 63) and S. P. Pershin ([12] p. 42), that the differential equation approach is not suitable for a complete analysis 9 of the thermal track buckling problem, is not justified. As shown in the present paper, the differential equation approach is capable of a complete determination of the post-buckling displacements and forces of the railroad track and thus for the determination of the range of safe temperature increases T o < T~. VI. Conclusions and Recommendations A study for the determination of the safe temperature increase in the rails of a straight track, to prevent lateral track buckling, was presented. The criterion used is based on the post-buckling equilibrium branches of the track. The range of the safe temperature increases was defined as

T o < T~ where T~ is the smallest value of the temperature rise at which a deformed state of equilibrium (thus, a buckled state) becomes possible. I t is shown that, contrary to the claims made in the literature, this problem can be formulated completely in terms of differential equations and the corresponding matching and boundary conditions. I t is also shown that the obtained non-linear formulation, because of a special 9 A track buckling analysis is referred to in the literature as complete when the drop of N t to ~t due to buckling is taken into consideration. As pointed out in [1], those numerous analyses, which do not take into consideration this drop should not be considered as analyses of track buckling caused by constrained thermal expansions.

Analysis of Thermal Track Buckling in the Lateral Plane

~t9

analytical feature of the obtained differential equations, can be solved exactly, in closed form. The obtained solutions for deformation shapes I to IV reveal that, for the track parameters used, the determined TL values are very close to each other. From the presented comparison of results for shapes I to IV and the following discussidn it may be concluded that for the determination of T~, an analysis based on deformation shape II may be sufficient for engineering purposes. According to Fig. 8, the obtained range of safe temperature increases for a track consisting of 115 lb/yard rails which is attached to wooden ties by means of cut-spikes is To < TL ~- 42"C. Note, however, that the above Tz-value is based on the used parameters and the made assmnptions. So, for example, a lower lateral resistance o0 caused by track maintenance activities will lower the Tz-value, whereas the effect of rail-tie fasteners which exhibit a torsional resistance will increase the range of safe temperature increases. Note also that the assumption that the lateral resistance ~o ~- ~o = const, was made in order to simplify the analyses. In actuality ~o is of the shape shown in Fig. 3a. Thus, although the post-buckling equilibrium graphs shown in Fig, 8 indicate that the straight state is always stable (with decreasing stability for increasing To), for an actual track there always exists a temperature increase Ter beyond which even the perfectly straight track may buckle ([3] Fig. 5). Also note that the assumptions ~o(x) = ~o0 and r(x) ~ r0 are valid only for monotonically increasing deformations. Another point to consider, is that actual tracks are not perfectly straight, but have small geometric imperfections. With minor modifications the formulations presented above are suitable also for the study of this problem. However, the resulting analyses are cumbersome and are also complicated by the uncertainty of the multitude of imperfection shapes encountered in an actual track. The results obtained in references [3] (p. 36) and [5] suggest, however, that the effect of the relatively small lateral imperfections encountered in an actual track will be to decrease the value of Tcr without affecting noticeably the corresponding T L-value. This in turn indicates the possibility that if the temperature increase in the rails of a track could be maintained (technically and economically) such that T O < Tz, where T~ is the value for the perfectly straight track discussed previously, then for engineering purposes there may be no need to determine the effect of lateral track imperfections on the safe temperature increase.

Acknowledgments The author wishes to thank Dr. Allan Zarembski, formerly Research Associate at Princeton University, for performing the numerical evaluations presented in this paper. Thanks are also due to Dr. Andrew Kish TSC/DOT and to Dr. i%. Michael McCafferty FI%A/DOT for reading and commenting on the content of this paper. 4

Acta Mech. 30/I--2

50

A.D. Kerr: Analysis of Thermal Track Buckling in the Lateral Plane References

[1] Kerr, A. D.: Lateral buckling of railroad tracks due to constrained thermal expansions. -- A Critical Survey, in: Railroad Track Mechanics and Technology, Proceedings of a Symposium held at Princeton University, April 21--23, 1975 (Kerr, A. D., ed.). Pergamon Press. (In print.) [2] Birmann, F. : Neuere Messungen an Gleisen mit verschiedenen Unterschwellungen (New measurements on tracks with different ties, ballast and subgrade). (In German.) Eisenbatmtechnische Rundsehau 6 (1957). [3] Kerr, A. D.: The effect of lateral resistance on track buckling analyses. Rail International, No. I (1976). [4] Kerr, A. D., E1-Aini, Y. : Determination of admissible temperature increases to prevent vertical track buckling. Princeton University Research Report 75-SM-11; December 1975. [5] Kerr, A. D. : Model study of vertical track buckling. I-figh Speed Ground Transportation Journal 7 (1973). [6] Kerr, A. D.: On the derivation of well posed boundary value problems in structural mechanics. International Journal of Solids and Structures 12, No. 1 (1976). [7] Novozhilov, V.V.: Teoria Uprugosti. Leningrad: Gos. Soyuz. Izd. Sudostroitelnoi Promyshlennosti. 1958. Translated into English as "Theory of Elasticity" by the Israel Program for Scientific Translations, Jerusalem, 1961, OTS 1641401. [8] ~artinet, A. : Flambement des voles sans joints sur ballast et rails de grande longueur (Buckling of the jointless track on ballast and very long rails). (In French.) Revue G6n@rale des Chemins de Fer, No. i0 (1936). [9] Mishehenko, K. N. : Besstykovyi Relsovyi Put (The jointless railroad track). (In Russian.) 5'[oscow: Gos. Transp. Zh/D Izd. 1950. [10] Numata, M. : Buckling strength of continuous welded rail. Bulletin International Railway Congress Association, English Edition, January 1960. [11] Kerr, A. D. : On the stability of the railroad track in the verticalplane. Rail International, 5[0.2 (1974). [12] Pershin, S. P.: Metody rascheta ustoichivosti besstykogo puti (Methods for analyzing the buckling of a jointless railroad track). (In Russian. ) Vyp. 147. Moscow: Trudy MIIT. 1962.

Dr. A. D. Kerr Department o/Civil Engineering Princeton University Princeton, NJ 08540, U.S.A.