Track Buckling

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Int. Z Meck Sc/.Vol.23,No. 10,pp.577-587,1981

0020-7403/81/100577-11502.0010

PrintedinGreatBritain.

PergamonPressLtd.

ON LOCALIZED

THERMAL

TRACK

BUCKLING

VIGGO TVERGAARD
The Technical University of Denmark, Lyngby, Denmark
and
ALAN NEEDLEMAN
Division of Engineering, Brown University, Providence, RI 02912, U.S.A.
(Received

I November

1980)

Summary--The thermal buckling of a railroad track in the lateral plane is analyzed with the
track modelled as an elastic beam resting on an elastic-plastic foundation representing the
ballast. The nonlinearity of the resistance forces exerted by the ballast on the track is
accounted for, both in the lateral and axial directions. For a perfectly straight track the critical
bifurcation mode is a periodic one and the effect of periodic imperfections on the instability
temperature is analyzed numerically. The transition to the localized buckling pattern observed
in practice takes place by a bifurcation from the periodic deflection pattern. The transition to
this localized mode can occur with only little growth of the periodic deflections. The
instability temperatures for some tracks with various nonperiodic initial imperfections are also
determined. It is shown that the instability temperature depends on both the magnitude and
the form of the initial imperfections.
1. INTRODUCTION

A railroad track consists of two parallel metal rails attached by closely spaced
cross-ties, which are embedded in a crushed stone layer termed the ballast. The
constrained thermal expansion of the rails can give rise to compressive forces
sufficiently great to buckle the track. Field observations and tests indicate that track
buckling often occurs in the lateral plane, as depicted in Fig. 1, although when the
track is subject to sufficient lateral restraint buckling occurs in the vertical direction
(out of the plane of Fig. 1) or involves a combination of lateral and vertical track
displacements. In this paper we focus attention on the lateral mode of track buckling.
The extensive literature on lateral track buckling, both experimental and analytical, has been reviewed by Kerr[l]. A number of early track buckling analyses, e.g.
[2], modelled the track as a beam, the rail cross-tie structure, resting on an elastic
foundation, representing the ballast, and obtained the bifurcation load (or,
equivalently, the bifurcation temperature) of the track. The bifurcation temperature
given by such an analysis was found to be greatly in excess of the observed buckling
temperature. It should also be noted that the critical bifurcation mode given by such
an analysis is a periodic one in contrast to the observed localized track buckling mode
indicated in Fig. 1.
Other investigations have emphasized the importance of post-bifurcation considerations in the track buckling phenomenon. Investigations of simple discrete track
models, e.g. [3], have indicated that small initial geometric imperfections could, as is
often the case in structural buckling, significantly effect the point at which loss of
stability occurs. However, such discrete model analyses are inherently qualitative in
nature. Another line of attack[4-6] has focussed on the far post-buckling behaviour
and associates the critical temperature with the minimum temperature for which an
equilibrium post-buckling state exists for a perfect track. This approach to post-

FIG. 1. Sketch illustrating the localized track buckling mode typically observed (top view).
(IJ)MS VoL 23, No. 10--A

577

578

V. TVERGAARDand A. NEEDLEMAN

b u c k l i n g b e h a v i o u r has b e e n e m p l o y e d b y v o n Kfirmfin and T s i e n [7] in the c o n t e x t of
elastic s t r u c t u r e s . Due to the p i o n e e r i n g w o r k of K o i t e r [ 8 ] it has for s o m e time b e e n
a p p r e c i a t e d that there is no direct c o n n e c t i o n b e t w e e n the m i n i m u m p o s t - b u c k l i n g
s u p p o r t load of a p e r f e c t elastic s t r u c t u r e and the m a x i m u m s u p p o r t load of its
i m p e r f e c t r e a l i z a t i o n . A n a l o g o u s l y , as will be s h o w n here, the m i n i m u m e q u i l i b r i u m
p o s t - b u c k l i n g t e m p e r a t u r e of a track is n o t directly related to its i n s t a b i l i t y temp e r a t u r e . S u c h a n a l y s e s have, h o w e v e r , highlighted the i m p o r t a n c e of the n o n l i n e a r
b e h a v i o u r of the f o u n d a t i o n o n w h i c h the track rests.
I n the p r e s e n t p a p e r we shall show how the localized b u c k l i n g m o d e of r a i l w a y
t r a c k s is related to the m e c h a n i s m of b u c k l i n g l o c a l i z a t i o n d i s c u s s e d p r e v i o u s l y in a
m o r e g e n e r a l p e r s p e c t i v e b y the a u t h o r s [ 9 , 10]. The t r a c k is m o d e l l e d as an elastic
b e a m r e s t i n g o n a n e l a s t i c - p l a s t i c f o u n d a t i o n r e p r e s e n t i n g the ballast. T h e b e a m is
s u b j e c t to a u n i f o r m t e m p e r a t u r e i n c r e a s e a n d c o n s t r a i n e d a g a i n s t t h e r m a l e x p a n s i o n ,
t h e r e b y giving rise to c o m p r e s s i v e stresses. T h e critical b i f u r c a t i o n m o d e is a periodic
o n e a n d the effect of initial p e r i o d i c i m p e r f e c t i o n s of v a r i o u s w a v e l e n g t h s is a n a l y z e d .
It is f o u n d that a m a x i m u m axial c o m p r e s s i v e f o r c e is a t t a i n e d and, after r a t h e r little
f u r t h e r deflection in the p e r i o d i c m o d e , the m a x i m u m t e m p e r a t u r e for w h i c h a stable
e q u i l i b r i u m state exists is r e a c h e d . T h e t r a n s i t i o n f r o m the p e r i o d i c m o d e to the
localized b u c k l i n g m o d e o b s e r v e d in p r a c t i c e a n d d e p i c t e d in Fig. 1 takes place b y the
m e c h a n i s m d i s c u s s e d in [9, 10], n a m e l y b y a b i f u r c a t i o n f r o m the p e r i o d i c deflection
p a t t e r n to a localized one. As s h o w n in [9], this b i f u r c a t i o n n e c e s s a r i l y takes place
after the m a x i m u m axial c o m p r e s s i v e force has b e e n a t t a i n e d ; but, as will be s h o w n
here, the b i f u r c a t i o n c a n o c c u r b e f o r e the i n s t a b i l i t y t e m p e r a t u r e is r e a c h e d . The
effect of a d d i t i o n a l localized g e o m e t r i c i m p e r f e c t i o n s o n the i n s t a b i l i t y t e m p e r a t u r e is
also illustrated.

2. PROBLEM FORMULATION
Although thermal track buckling may involve vertical as well as horizontal track displacements,
observations show that the continuously welded tracks presently in use tend to buckle in the horizontal
plane. In the present investigation both rails are assumed to remain in their initial horizontal plane (without
twisting) throughout the buckling process, and thus the eccentricity of the support loads exerted by the
ballast on the ties does not affect the solution.
For each rail the cross-sectional area and the area moment of inertia with respect to the vertical axis of
symmetry are denoted by Ar and L, respectively, Young's modulus is E and the linear thermal expansion
coefficient is ,~. The rails are assumed to remain elastic and are uniformly heated to a temperature T, with
the temperature increase denoted by AT = T - To. Here, the neutral temperature To, at which the axial rail
force vanishes, is chosen at installation as a compromise so as to avoid track buckling on hot sunny days,
and to avoid tensile fracture on very cold days,
The resulting axial force Nr and the resulting bending moment Mr in a rail are given by
Nr = E A r ( e - aAT),

Mr = Elm.

(2.1)

Here, the strain of the centre line • and the bending strain K, in terms of the axial displacements u and the
total lateral displacements w, are taken to be
= u.~ +a(w.~)'-x(g'.~)- K = w.,, - ¢'~,

(2,2)

where x is the coordinate along the centre line, ( ).~ denotes differentiation with respect to x, and ~ is the
initial lateral displacements of the unstressed rail.
The resistance against rotations of the rail relative to the cross-ties depends on the type of fastener
used. Some fasteners, such as cut-spike fasteners, give very little resistance, and in the present
investigation we shall neglect this resistance. Then, all the cross-ties remain parallel during buckling, and
the displacements and strains are identical in neighbouring points of the two rails. Consequently, the track
responds as a column with the axial force N and the bending moment M given by
N=EA(~-aAT),

M=EI~

(2.3)

where A = 2A, and I = 21,.
The restoring forces provided by the track foundation result from the forces exerted by the ballast on
the cross-ties. These forces consist of frictional forces on the bottom and sides of the ties and of normal
pressure against the vertical surfaces of the ties. Test results showing total resistance forces versus track
displacement have been given by Birmann [11] for several different types of rail-tie structures. Both in the
lateral direction and in the axial direction the resistance forces depend nonlinearly on the displacements.

579

On localized thermal track buckling

The particular track resistances to be used here represent some of the test results shown by Birmann[l 1],
which were also referred to in the analysis of Kerr[4]. These resistances were approximated in [4] as
constant restoring forces, independent of the track displacements; but here we account for the full
nonlinear response. The test results for the restoring lateral force F per unit length of track are reasonably
well represented by the relationship

co + ct

c°ffl

for ]F]-< F1

-

for ± F > F,

with Ft = 1961(N/m), co = 4 x 10-5 m, c, = 5 × l0 -4 m and c2 = 2.813 x l0 -5 m. The displacement w* is measured
relative to the initial unstressed geometry of the track at the neutral temperature. Thus, for a track with a
geometrical imperfection in the form of an initial lateral displacement ~, the displacement to be used in (2.4) is
w* = w - ft. Similarly, the test results for the restoring axial force K per unit length of track can be represented
by the relationship
do~

for IK]-< Ki

u= +=[do+dl(K~KT_l]+d2(~K _ i f

for ± K > K ,

(2.5)

with K~ = 1961(N/m), do = 4 x l0 -~ m, d~ = 5 x l0 -4 m and d2 = 3.345 x 10-4 m. The track resistance curves
given by (2.4) and (2.5) are shown in Fig. 2.
The track resistances given above refer to situations in which the displacements grow monotonically.
However, due to the frictional nature of the resistance, the same curves are not followed backwards if the
displacement rates are reversed. Thus, the foundation of the track column cannot realistically be modelled
as nonlinear elastic springs. Here, the foundation shall be modelled as "elastic-plastic" springs and for this
purpose the equations for the support forces are written in the incremental form
b" = Cw,

/( = O~i

(2.6)

where (') denotes an increment and C and D are incremental stiffnesses. For IFI -> IFI . . . . C is taken to be
given by dFIdw* according to (2.4); but for )FI < IF)~ax (elastic unloading) C = Fdco is assumed, corresponding to the linear part of (2.4). The definition of D relative to (2.5) is analogous.
Now, consider a track of length l, with zero axial displacements and simple support conditions
prescribed at the ends of each rail. The equilibrium solutions for increasing temperature of the rails are
obtained by the principle of virtual work

/ {NSE + M~K + F~w + K~u} dx = O.

(2.7)

Since the effect of the foundation is modelled in terms of elastic-plastic springs, the numerical solution is
based on the incremental form of (2.7), which can be written as

[' {EAiad + EI/cSK + Cfvaw + Dft6fi + Nw.x~Sw.x}dx = ( ' EaaAT"ai: dx
do

(2.8)

.10

using (2.3) and (2.6).
10000
F
(N/m)

10000
K
(N/m}

8000

8000

6000

6000

4000

4000

2000

2000

Oo

0.0

'
0.008

;
0.02

0
0.,6

0.02

0

y
.
. 0.008
.
. 0.0 2
0.004

w

Ca)

(m)

0.0,6

0.02
o

(b)

(m)

FlG. 2. Track resistance curves used in the present analysis. (a) Lateral resistance curve of
the form (2.4). (b) Axial resistance curve of the form (2.5). The parameter values characterizing these resistance curves are given in the text and are chosen to represent some of the test
results of Birmann [11].

580

V. TVERGAARD and A. NEEDLEMAN

The lateral buckling modes of tracks are often observed to be either symmetrical, as indicated in Fig. I,
or anti-symmetrical. Previous track analyses (see, e.g. [4]) show very little difference between the buckling
behaviour for the two types of modes, and here we chose to consider only the symmetric case. Therefore,
symmetry conditions are prescribed at x = 1/2 and only one half of the track is considered in the numerical
solution.
For a perfectly straight track on the foundation specified by (2.4) and (2.5) buckling is governed by the
initial lateral spring stiffness k = Fdc,. The buckling mode for an infinite perfect track is sinusoidal with the
critical temperature increase AT, and half wavelength a, given by
AT,

,A\ 3 } '

a, = 7 r ~ - )

.

(2.9)

In the present investigation, however, interest is focussed on tracks with initial geometrical imperfections
specified by an initial lateral deflection ~, of the form
g' = (~j + 5: e ~-'' l~,,,~:) sin 7rx.
a

(2.10)

Here, g~ is the amplitude of a periodic imperfection, 5., is the amplitude of a localized imperfection, and a is
the half wavelength of the imperfection. Thus, to consider a track with n halfwaves the length has to be
chosen as
(2.11)

I=na.

It should be emphasized that other types of imperfections than an initial waviness of the rails may be
quite significant for railway tracks. For example, variations of the foundation stiffnesses along the track can
occur. Also the reduced frictional forces between ties and ballast due to a slight lift-off of the rail-tie
structure in front of or behind a wheel can act as an imperfection. Analyses for a moving load show that
the lift-off increases as the speed of the train approaches a critical value[12], and due to damping the
greatest lift-off occurs in front of the wheel, both for a linear foundation[13] and for a foundation modelled
as nonlinearly elastic[14]. In the present investigation consideration is restricted to circumstances in which
buckling occurs solely due to heating, without the influence of wheel loads.
That initial geometrical imperfections may play a significant role in track buckling is, in fact, substantiated by the observations of Birmann and Raab[15], who carried out tests on 21 tracks with various
imperfection amplitudes. For nearly straight tracks (imperfections < 0.003 m) critical temperature increases
of 108-144°C were found, whereas for initial lateral deflections around 0.025 m the measured buckling
temperatures were 53-85°C. The length of the test tracks considered by Birmann and Raab was 46.5 m, and
the half wavelength of the imperfections was around 5-6 m.
3. NUMERICAL METHOD AND RESULTS
In the numerical computations to be presented here we model the track with 57 kg/m rails fastened to
wooden cross-ties by but-spike fasteners that was also considered by Kerr[4]. For this track the
cross-sectional area is A = 2A, = 1.45 × 10_2 m 2, the area moment of inertia of the cross-section is I = 2L =
8.99x 10-6m 4, Young's modulus is E = 2.06× 10tt N/m-' and the linear thermal expansion coefficient is
a = 1.05 × 10-5 (°C)-~. The track resistance curves of Birmann[11], represented here by (2.4) and 12.5), are
the same resistances approximated as constant in [4].
For the track specified above the critical temperature increase (2.9) in the absence of imperfections is
ATt = 608°C, with the corresponding half wavelength of the buckling mode a, = 1.38 m. This is far beyond
the level of critical temperature increases measured for tracks, and no realistic adjustment of the initial
lateral foundation stiffness k = Fdco specified by (2.4) brings AT, in accord with measured values.
Therefore, in the numerical investigations the imperfection-sensitivity of the critical temperature is
investigated.
3.1. Numerical method
The numerical determination of equilibrium solutions for a heated track is obtained by a linear
incremental procedure. At each temperature level the incremental equilibrium equation (2.8) is solved
approximately by the finite element method. Within an element the axial and lateral displacements are
approximated by Hermitian cubics, and integrals along the track are evaluated by 4 point Gaussian
quadrature. In all cases to be presented six elements are used over each half wavelength a.
As long as the equilibrium temperature grows, the temperature increment A'P can be prescribed directly
in the numerical solutions based on (2.8). However, the attainment of a maximum temperature leads to
numerical difficulties, if AT is prescribed. These numerical difficulties are avoided, without loosing the
symmetry of the finite element stiffness matrix, by using a special additional Rayleigh-Ritz procedure to
prescribe a lateral displacement increment instead of AT (see Tvergaard [16]).
3.2. Periodic solutions
Due to the periodic shape of the critical buckling mode for the perfect track, this type of imperfection is
first considered by taking g2 = 0 in (2.10). In this phase of our study the deflection pattern is required to
remain periodic throughout the deformation history, which is done by taking n = 1 in 12.11). The effects of
various half wavelengths a and imperfection amplitudes 6~ are shown in Figs. 3-6. These figures show plots
of the temperature increase AT versus the maximum deflection Wm and of the average axial force N,, vs w,,,
where Na is defined as

N,,=~f'N

dx.

13.1)

On localized thermal track buckling

140

-N(

AT

(MN

(%)

581

n=l

120
/ 61 = 0.005

/ 01 = 0.005 m

m

100

/ ~)1 = 0.01 m

80

/~)1
60

0.015 m

~1 = o.o2_~
f ~ ~ / _

. . f " ~ " _ b1=0.025 m

40

20
× maximum

II ; x'7,
0.02

0.04

0.06
(Q)

0.02

0.08 wm 0.1
(m)

0;4

0.;6

o.~8 Wm 01
(m)

(b)

FIG. 3. Temperature rise AT and average axial force N, vs lateral deflection amplitude w,, for
various imperfection amplitudes g~. The imperfections and the subsequent deflections are
required to remain periodic, with half wavelength a = 6 m.

120
aT
(°C)
100

80

3
-N a
n=l

IMI'

bl= 0.0

f~61

~ _

= 0.005 m

\ b 1=0.015m

60
\ bl= 0.01 m

40

0.015 m

20
x maximum

0.;2

0.64

0.66
(a)

I I,

0.68,,,m 0.1
(m)

0

0

0.02

x maxim,um
0.04

0.06
(b)

0.08 wm 0.1
(m)

FIG. 4. Temperature rise AT and average axial force N, vs lateral deflection amplitude w,, for
various imperfection amplitudes g~. The imperfections and the subsequent deflections are
required to remain periodic, with half wavelength a = 4 m.

For a half wavelength a = 6 m the behaviour corresponding to five different imperfection amplitudes is
shown in Fig. 3. For the three smallest imperfections g~ = 0.005 m, g~ = 0.01 m and 8~ = 0.015 m a maximum
temperature is reached. The branches of these curves, on which the temperature decays, represent unstable
equilibrium solutions. Thus, after the temperature maxima the tracks will snap dynamically to equilibrium
solutions at considerably larger deflections, and the present quasi static solutions after the maxima are only
included to give an indication of the order of deflections to be expected in the final buckled solution. No
temperature maxima are reached for the two larger imperfections g~ = 0.02 m and g~ = 0.025 m, so the
solutions shown are stable for increasing temperature. However, in all five cases considered in Fig. 3 a
maximum compressive force is reached in the track, and for the three smaller imperfections these force
maxima occur slightly before the temperature maxima.
Fig. 4. shows results similar to those of Fig. 3, but for a smaller half wavelength a = 4 m. Here, only the
smallest imperfection g~ = 0.005 m leads to a temperature maximum. For the larger imperfections the
solutions shown remain stable, and only rather small deflections are reached in the temperature range of
interest. The same tendency continues in Fig. 5 for the still smaller half wavelength a = 2 m. Here, even for

582

V. TVERGAARDand A. NEEDLEMAN

207Or

,///

5 [-

~

-No /

r





bl = 0.001 m

-

b 1 = 0.005 m

80

I /

/ ~ _ ~ 1

= 0.015 m

&O

o~

0.02

o.6~

o.6~

0.68 ~m 01

0.02

0.~

(ml

(a}

0.06

O,08~m 0.~
(m)

(b)

FIG. 5. Temperature rise AT and average axial force N, vs lateral deflection amplitude w~ for
various imperfection amplitudes g~. The imperfections and the subsequent deflections are
required to remain periodic, with half wavelength a = 2 m.

140
~T
{°C)
120

100

_Na - (MN)

n=l

= 0.01 m

/

$1 = 0.Ol m
, / ' b l = 0.015 m
_//bl = 0.015 m

i

80

--~//bl

= &O2m

60

z.O

20

Oo

o.o~

x

maximum

o.G~.

o.&
{o)

x

0.68 "'m o.1
(ml

oo

0.o2

maximum

0.6~

066
(b)

068Wm 0~
(m)

FIG. 6. Temperature rise AT and average axial force N, vs lateral deflection amplitude wm for
various imperfection amplitudes gl. The imperfections and the subsequent deflections are
required to remain periodic, with half wavelength a = 8 m.
the very small imperfection g~ = 0.001 m no temperature maximum is reached, and none of the tracks attain
a maximum compressive force in the range considered.
In Fig. 6 a longer half wavelength, a = 8 m, than that of Fi~. 3 is considered. A temperature maximum is
reached for all imperfections in the range from g~= 0.01 m to ~1 = 0.025 m, and in each case the compressive
force maximum occurs slighly before the maximum temperature. Comparison with Figs. 3 and 4 shows that
for a given imperfection level the maximum temperature, at which instability occurs, decreases for
decreasing half wavelength a. For each particular imperfection magnitude gt the maximum temperature
reaches a minimum for a certain half wavelength a. For half wavelengths slightly smaller than this value the
imperfection does not give rise to a temperature maximum. The figures indicate that for g~ = 0.02 m the
value of this critical half wavelength is very close to 6 m, and for most imperfection amplitudes of practical
interest it is in the range from 4 m to 8 m. It is noteworthy that these values of the half wavelength a are
considerably larger than the critical half wavelength ac = 1.38 m for the perfect track.
Idealizations of the foundation stiffnesses, similar to those used in [4], have also been tried here, by
taking Fi = 5884 (N/m), Co = 6 x 10:6 m, cl = 6 x 10~ m, Kt = 9807 (N/m), do = 10 -5 m, dl = 106 m and c2 = d2 = 0
in (2.4) and (2.5,). For a = 6 m and n = 1 this increases the maximum temperatures found in Fig. 3 by 25% or
more, and for 8~ equal to 0.02 m and 0.025 m maximum temperatures of 49°C and 42°C, respectively, result.
Thus, although the idealization of using constant resistances gives qualitatively reasonable results, the fully
nonlinear description used in this paper is necessary for a more quantitative description.

On localized thermal track buckling

583

3.3. Localization of the buckling patterns
The results presented in Figs. 3-6 presume that the initially periodic deflection pattern remains periodic.
However, it has been shown by Tvergaard and Needleman[9] for a wide class of structures, in which a
maximum axial compressive force is reached, that the initial periodic buckling pattern can bifurcate into a
localized pattern shortly after the force maximum. A simple one dimensional model of such structures
predicts bifurcation at the maximum force point[9], whereas a delay of this bifurcation beyond the
maximum force point is exhibited by more realistic models, such as that of a column on a bilinear elastic
foundation or a long, elastic-plastic, axially compressed plate strip supported at the edges. Thus, for
uniformly heated railway tracks such a bifurcation into a localized mode is expected slightly beyond the
maximum compressive force points shown in Figs. 3(b), 4(b) and 6(b), which occur before the temperature
maxima.
The possibility of bifurcation into a localized mode is first analysed for the case g~ = 0.02 m and a = 6 m
in Fig. 3, where a compressive force maximum but no temperature maximum occurs for the periodic
pattern. For a length of track with n = 9 half waves, and thus from (2.11), I = 54m, a bifurcation point is
found slightly after the maximum compressive force, Fig. 7. The post-bifurcation behaviour is obtained
numerically as the behaviour of a track with a slight deviation from the periodic imperfection g~ = 0.02 m
and 82 = 8~ x 10-*. This solution remains practically periodic up to the bifurcation point, at which the central
buckles continue to grow, while the remaining buckles start to decay, with elastic unloading in the lateral
support springs. On the post-bifurcation path a maximum temperature is reached slightly after the
bifurcation point, and the same temperature is reached again at wm= 0.06 m. The influence of track length is
studied by repeating the same computation for n = 17 (I = 102 m), which has little effect on the location of
the bifurcation point and on the initial part of the post-bifurcation curve. However, for larger deflections
the curve is somewhat below that of the shorter track, and it is expected that the post-buckling curve will
be still a bit lower, as the track length becomes infinite. The length necessary to represent a very long track
corresponds to the distance from the localized buckles, at which no noticeable restoring force in the axial
direction develops in the foundation.
Figure 8 shows a similar investigation of the bifurcation behaviour corresponding to the smaller
imperfection, g~ = 0.015 m, in Fig. 3. Here, a temperature maximum is reached for the periodic solution, so
closely after the compressive force maximum that the two points are nearly coincident in Fig. 8, For both
n = 9 and n = 17 bifurcation occurs between these two maxima, The initial portions of the two postbifurcation curves coincide, whereas for larger deflections the curve for n = 17 is below that for n = 9, in
agreement with the trend found in Fig. 7. Although the details cannot be seen in Fig. 8, it should be
mentioned that the post-bifurcation curve is initially below the periodic solution, until the two curves cross
on the decaying branch.
Two plots of lateral deflection patterns are shown in Fig. 9 corresponding to the curve for n = 17 in Fig.
8. Figure 9(a) shows the periodic pattern just before the bifurcation point. Figure 9(b) shows a later stage on
the post-bifurcation path, where growth of the wave amplitudes has localized in the central buckles (only
one half of the track is shown). It is noted that the wavelength of the buckles increases very little during the
localized growth, which is consistent with the comparison of initial imperfections and final buckled
configurations shown by Birmann and Raab[15] for some of their tests.
In Fig. 10 an intermediate case between those of Figs. 7 and 8 is considered, with g~ = 0.0175 m and
a = 6 m. The bifurcation behaviour for n = 17, given by the solid curve, is in good agreement with the two
previous figures. This computation has been repeated with two alternative descriptions of the foundation
resistances. First the foundation response is taken to be elastic (unloading is neglected). As shown by the
dotted curve in Fig. I0 this change has little influence on the result, the main difference being that the wave
amplitudes outside the localized region reapproach their initial value, whereas with eleastic unloading these
amplitudes remain at the level reached at bifurcation into the localized mode. The other alternative
considered is to neglect the axial foundation resistance (K ~ 0), keeping the description of the lateral
resistance unchanged. The little influence of this change on the initial part of the post-bifurcation behaviour
is of interest, because it has been claimed [4], in connexion with the search for post-buckling equilibrium
lOO
~T
(°C)
80
60

~=o

bl = 0.02 m, b2 = b1'10-4\

. o ~
n=9/

20

0

x
o

maximum
bifurcation
i

0

0.02

n=17/

0.04

i

0.06

,

0.08 wm 0.1
(m)

FIG. 7. Temperature rise AT vs maximum lateral deflection w,, for a = 6 m and ~ = 0,02 m,
when the deflection pattern is allowed to localize. The dashed curve shows the periodic
solution.

584

V. TVERGAARD and A. NEEDLEMAN
100
~T
(°C)
80
~ = o.o15 m. b2 = ~ ' 164 \

_

60

z,0

oo17/

20

0

x

maximum

o

bifurcation

~

0.02

o.o~

o.o6

o.oa w m o.1
(m}

FIG. 8. Temperature rise AT vs maximum lateral deflection wm for a = 6 m and g~ = 0.015 m,
when the deflection pattern is allowed to localize. The dashed curve shows the periodic
solution.

w

0.05

(al

(b)
Fro. 9. Lateral deflection patterns for a = 6m, n = 17, g~ = 0.015 m and g2= g~ x 10 4 . (a)
Deflections at wm= 0.0182 m, just before localization. (b) Deflections at wm = 0.093 m, after
localization.

states, that analyses neglecting the axial resistance are not realistic. In fact, according to the results shown
in Fig. 10, the critical part of the track buckling process, around bifurcation, is entirely controlled by the
lateral resistance. The need to account for axial resistance relates mainly to the determination of a
post-buckling equilibrium configuration at a given temperature, because in an infinite track without axial
resistance the relaxation of compressive forces would result in an unlimited amplitude of the localized
buckles.
In certain circumstances, the stiffening effect of the axial resistance can effect the development of a
localized buckling pattern. For the periodic imperfection with the larger amplitude, gt = 0.025 m in Fig. 3, a
bifurcation point is found rather late, at w,, = 0.06 m; but a computation with g~ = 0.025 m, g2 = g~ x 10-%
a = 6 m and n = 17 shows no tendency towards localization of the deformation pattern. Thus, for a
sufficiently large periodic imperfection in this particular wavelength no instability of the track occurs. This
behaviour is strongly dependent on the axial support stiffness, in contrast to the result found in Fig. 10. If
the axial resistance is neglected in this case (K ~- 0), bifurcation and subsequent localization is predicted
just after the maximum of the axial compressive force, as also found in [9, 10]. For the larger half
wavelength a = 8 m (Fig. 6) the same amplitude of a periodic imperfection does lead to bifurcation into the
localized mode shortly before the temperature maximum. Here, the rate of decay of the axial force No
beyond it's maximum is large enough to overcome the delaying effect of the axial support stiffness.
The pattern of imperfections found in practice is usually less regular than the periodic or nearly periodic
imperfections considered in Figs. 3-10. Therefore, in Figs. 11-13 the influence of more localized imperfections is investigated.
In Fig. 11 the half wavelength and wavenumber are taken to be a = 6 m and n = 17, respectively, as in
the previous four figures, and the maximum initial deflection is taken to be 0.02 m. A moderately localized
imperfection is specified by ~ = g~=0.0i m and a highly localized imperfection by g~ =0.002 m, g2 =
0.018 m. In both cases a critical temperature is reached somewhat above that for a periodic imperfection of
the same maximum amplitude (Fig. 7) and buckling takes place in a localized mode very similar to those
found in the previous cases. For the highly localized imperfection Fig. 12 shows a plot of the lateral
deflection pattern considerably after the temperature maximum. Here, the small initial wave amplitude
outside the region of localization has grown less than 1%.
For the same imperfection amplitudes the influence of a longer half wavelength, a = 8 m, is investigated

On localized thermal track buckling

585

100
aT
(°C)
80

bl= 0.0175 m.n:l "x~/
/

60

.-. /

/ elastk

40

20

00

0.02

0.0~

0.06

0.08 wm 0.1
(m)

FIG. 10. Temperature rise AT vs maximum lateral deflection w,~ for a = 6 m and 8, =
0.0175 m, when the deflection pattern is allowed to localize. The effect of alternative
foundation resistances is illustrated for n = 17. The dashed curve shows the periodic solution

(n

=

1).

100
aT

~°c)
8O

60

bl =0.002 m, b2:0.018 m, n=17

b1:0'02 r e ' n = 1 %

z,O

20

i
00

0.02

x , maximu m
0.04

0.06

,
0.08 wm 0.1
(rn)

FIG. 11. Temperature rise AT vs maximum lateral deflection w,, for a = 6 m , n = 17 and
8~ + 82 = 0.02 m. The effect of localized imperfections on instability is illustrated. The dashed
curve shows the periodic solution.

wm 0,0[

FIG. 12, Lateral deflection pattern considerably after the temperature maximum for a = 6 m,
n = 17, 81 = 0.1)t)2 m and 82 = 0.018 m.
in Fig. 13. Here, n = 17 corresponds to a track length of l = 136 m. In this case the critical temperatures for
the tracks with localized imperfections are below the temperature maximum for g~ = 0.02 m and n = l, at
which bifurcation from a periodic pattern takes place. Furthermore, the track with the most localized
imperfection has the lowest critical temperature, in contrast to the result displayed in Fig. 11. However, for
both localized imperfections and for the periodic imperfection with a maximum aplitude 0.02 m the critical
temperature corresponding to the smaller half wavelength a -- 6 m (Fig. 11) is below that corresponding to
a = 8 m (Fig. 13).
4. D I S C U S S I O N
The results presented here show that the overall behaviour in track buckling is influenced by the
presence of two bifurcation points. One bifurcation point is associated with a periodic mode. The other
involves a bifurcation from a periodic deflection pattern subsequent to the attainment of an axial

586

V. TVERGAARD and A. NEEDLEMAN
100
,',T
I°C)
80 !
bl= ~2= 0.01 re,n:17

60

20

x maximum

002

00~

0.06

0.08 wm 01
(m)

FIG. 13. Temperature rise AT vs maximum lateral deflection wm for a = 8 m, n = 17 and
61 + 82 = 0.02 m. The effect of localized imperfections on instability is illustrated. The dashed
curve shows the periodic solution.
compressive force maximum and gives rise to a localized final collapse mode, as is always observed in
thermal track buckling. This bifurcation into the localized mode plays a crucial role in thermal track
buckling. As illustrated in Fig. 7, there are situations in which an analysis that restricts attention to periodic
deflections predicts no instability temperature, whereas when the localized mode is allowed to develop an
instability does occur.
Bifurcation into the localized mode is predicted at realistic temperature rises for tracks with periodic
initial imperfections of certain critical magnitudes and shapes. In practice, of course, a completely periodic
imperfection will hardly ever be found, but the results for various track lengths in Figs. 7 and 8 indicate that
a near periodicity over a relatively short stretch of a very long track will have a similar effect. Furthermore,
even when track buckling is triggered by predominantly non-periodic imperfections, Figs. I I and 13, the
behaviour is influenced by the existence of the bifurcation point, as in most other structural buckling
problems.
The early track buckling analyses [2], which do not account for imperfections, result in periodic
bifurcation modes and give critical temperatures far in excess of observed values. Furthermore, the
periodic bifurcation mode has a half wavelength, for the track analyzed here, of 1.38 m, whereas accounting
for realistic imperfections gives a critical half wavelength closer to 6 m.
More recent analyses of post-buckling equilibrium states for perfect tracks based on assuming an
idealized elastic foundation (rigid-perfectly plastic without unloading) does result in a localized deflection
pattern [4]. These analyses determine the smallest temperature rise for which a post-buckling equilibrium
solution exists and term this the safe temperature rise; but no attempt is made to investigate how this
equilibrium can be reached without applying much higher temperatures. The instability temperatures
obtained in the present paper for imperfect tracks occur at much smaller lateral deflections than does the
minimum equilibrium post-buckling temperature for the perfect track. Thus, the effects of various track
parameters, such as the axial resistance, on these two temperatures can differ substantially. Furthermore,
quantitatively realistic buckling predictions require a full nonlinear description of the track resistances.
The overall picture of track buckling that emerges from the analysis carried out here is as follows.
Suppose that a certain maximum initial lateral deflection amplitude is allowed in the track, such as the
maximum of 0.025 m suggested by Birmann and Raab[15], or perhaps a smaller imperfection level. In
practice, in a very long track imperfections with many different wavelengths will be available, some of
these at or near the maximum permitted amplitude. Thermal buckling will occur in a track segment, which
has a relatively large imperfection in a certain critical wavelength. This critical wavelength, which for a
given track depends on the imperfection amplitude, is the one for which instability occurs in the localized
mode for the smallest temperature increase. For the track considered here the critical half wavelength is in
the vicinity of 6 m. For longer wavelength imperfections, the instability temperature rise is increased
whereas for somewhat shorter wavelength imperfections no instability occurs over the temperature range
of interest.
In this study, attention has been focussed on imperfections in the form of initial lateral deflections. A
region of relatively weak ballast, with a corresponding reduction of the lateral foundation resistance, may
further reduce the critical temperature increase for the track, particularly if it coincides with a region, in
which the geometrical track imperfections are near critical. Other factors affecting track buckling, which
have been neglected in the present development, include the lift-off of the rail-tie structure in front of or
behind a wheel, which would also tend to lower the critical temperature, and the resistance, due to the
fasteners, against rotations of the rail relative to the cross-ties. This resistance acts to increase the
instability temperature. A quantitative study of the effects of these factors or instabilities could be carried
out within the theoretical framework employed here.
Here, we have presumed that the track is subject to a prescribed temperature increase prior to buckling.
However, during the summer, an actual track is usually exposed to a succession of hot days and relatively
cool nights. It has been suggested by Bromberg[17] that accumulation of lateral track deflections during
these temperature cycles could result in track buckling temperatures lower than those given by an analysis
that presumes a monotonic temperature rise to instability. Our model of the track as a beam on an

On localized thermal track buckling

587

elastic-plastic foundation, perhaps with a different rule for plastic reloading subsequent to elastic unloading, would be suitable for investigating the effects of temperature oscillation induced permanent deflections
on instability.
1. A. D. KERR, Lateral buckling of railroad tracks due to constrained thermal expansion. Railroad Track
Mechanics and Technology (Edited by A. D. Kerr), Pergamon Press, Oxford (1978).
2. M. T. HUBER, Uber die Stabilit~it gerader liickenloser Gleise. Gleistechnik und Fahrbahnbau, H. 18
(1936).
3. V. K. VAISH, Lateral buckling of track. Rail International 5, 437 (1974).
4. A. D. KERR, Analysis of thermal track buckling in the lateral plane. Acta Mechanica 30, 17 (1978).
5. A. MARTINET,Flambement des voies sons joints sur ballast et rails de grande longeur. Revue G~n~rale
des Chemins de Fer, No. 10 (1936).
6. M. NUMATA, Buckling strength of continuous welded rail. Bull. Int. Railway Congress Association,
English Edition (Jan. 1960).
7. T. VON KARMANand H. S. TSIEN, The buckling of spherical shells by external pressure. J. Aero. Sci. 7,
43 (1939).
8. W. T. KOITER, Over de stabiliteit van het elastisch evenwicht. Thesis, Delft, H. J. Paris, Amsterdam
(1945). English transl.: N A S A TT F-10, 833 (1967); and AFFDL-TR-70-25 (1970).
9. V. TVERCAARDand A. NEEDLEMAN,On the localization of buckling patterns. J. Appl. Mech. 47, 613
(1980).
10. A. NEEDLEMANand V. TVERGAARD,Aspects of plastic post-buckling behaviour. Mechanics of Solids,
The Rodney Hill 60th Anniversary volume (Edited by H. G. Hopkins and M. J. Sewell), Pergamon
Press, to appear in 1981.
11. F. BIRMANN,Neuere Messungen an Gleisen mit verschiedenen Unterschwellungen. Eisenbahntechnische Rundschau 6, 229 (1957).
12. S. TIMOSHENKO,Method of analysis of statical and dynamical stresses in rail. Proc. 2nd Int. Congr.
Appl. Mech. Ziirich, 407 (1926).
13. J. T. KENNEY,Steady-state vibrations of beam on elastic foundation for moving load. J. Appl. Mech. 21,
359 (1954).
14. S. SCnOLER, Instability of railway tracks (in Danish). M.Sc. project, Dept. of Solid Mech., Techn.
University of Denmark (1977).
15. F. BIRMANNand F. RAAB, Zur Entwicklung durchgehend verschweisster Gleise. Eisenbahntechnische
Rundschau 9, 321 (1960).
16. V. TVER~AARD,Effect of thickness inhomogeneities in internally pressurized elastic-plastic spherical
shells. J. Mech. Phys. Solids 24, 291 (1976).
17. E. M. BROr~BE~G,The stability of the jointless track (in Russian). Izd. Transport (1966).

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