The continuously supported rail subjected to an axial force and a moving load

Int. J. mech. Sci. Pergamon Press. 1972. Vol. 14, pp. 71-78. Printed in Great Britain

THE

CONTINUOUSLY AN AXIAL

SUPPORTED

FORCE

RAIL

SUBJECTED

TO

AND A MOVING LOAD*

ARNOLD D. KERR Department of Aeronautics and Astronautics, New York University New York, N.Y., U.S.A. (Received 23 October 1970, and in revised form 12 March 1971)

Summary--The recent practice of welding railroad rails to each other suggests that considerable axial compression forces may be induced in the rails because of a rise in temperature. This in turn may reduce the critical velocity for the track to the range of operational velocities of modern high-speed trains. The purpose of the paper is to demonstrate that this is indeed a possibility. INTRODUCTION THE RESPONSE of a continuously supported beam subjected to moving loads was conducted first, in connection with the determination of stresses in railroad tracks, b y Timoshenko. 1 The obtained results indicated t h a t there exists a critical velocity, Ver, at which the deflections become very large. For assumed values of a rail and the foundation parameter, Timoshenko found t h a t Vet is about 1200 m.p.h., about ten times larger t ha n the highest speed of a locomotive at the time, and concluded t h a t the static equations are sufficient for the analysis of stresses in railroad rails. New technological problems, connected with the construction of high-speed rocket test tracks 2 and the use of floating ice plates on water as a pavem ent for moving vehicles, a accelerated the research activities in this area. A large number of these results for beams, as well as plates, are discussed in a recent survey b y Kerr. 4 The recent introduction of high-speed trains generated new interest in the response of continuously supported beams subjected to moving loads. In order to increase passenger comfort and at the same time decrease the wear of the rail and base the individual rails are presently joined by welding the rail ends to each other. 5 Because of the lack of expansion joints, falling or rising temperatures m ay cause considerable axial compression or tension forces in the rail. I t is reasonable to expect t h a t an axial compression force m a y reduce vet; possibly to the range of operational velocities of modern high-speed trains. The purpose of the present paper is to demonstrate t h a t this is indeed a possibility. I t appears t h a t there is no analysis which takes into consideration the effect of axial forces upon Vet of continuously supported beams (see ref. 4). t * Research sponsored under National Aeronautics and Space Administration grant No.

NGL-33-016-067.

1"One exception is a recent paper by Newland. 8 71

72

ARI~OLD D. KERR

T o s t u d y t h i s effect, i n t h e following, we a n a l y s e a n i n f i n i t e b e a m w h i c h r e s t s o n a W i n k l e r f o u n d a t i o n a n d is s u b j e c t e d t o a m o v i n g c o n c e n t r a t e d t r a n s v e r s e l o a d P a n d a c o n s t a n t a x i a l force N , as s h o w n i n F i g . 1. T h e i n v e s t i g a t i o n is

N

N

Fio. 1. based on the differential equation T~4w

. ~2w

a2w

.

El-~x4 + lv -~x2 + m T ~ - + lcw = PS(x, t),

(1)

w h e r e w is t h e l a t e r a l deflection, E1 is t h e f l e x u r a l r i g i d i t y o f t h e b e a m , m is t h e m a s s o f t h e b e a m p e r u n i t l e n g t h o f a x i s a n d k is t h e f o u n d a t i o n parameter. THE PROPAGATION

OF FREE

WAVES

Before discussing the solution of equation (1), we study first the propagation of free waves in the infinite beam, the results being needed later. For this purpose we substitute the wave type expression

w(x, t) = wo sin [ ~ (x-ct)],

(2)

into equation (1) with P = 0, and obtain

+ ~ ] wo sin [2~ (x-ct)] = O. The above equation is satisfied for any t and x when the term inside the first square brackets is zero, i.e. when

c=

i-K/ - m

m l t ~ l J"

(3)

Thus, expression (2) is a solution of the homogeneous equation (1) when c satisfies equation (3). I t represents an infinite wave train with amplitude w0, wavelength ~ and propagation velocity c. When _N = 0, equation (3) reduces to the one obtained by D6rr. e The corresponding graph is shown schematically in Fig. 2. The value CminlN-0is obtained from the condition [~c/8(2~/,~)]~ o = 0. I t is easily found that it is located at ~

(4)

and that

4/4kEI mid=-o =

m=

When c -- 0, equation (3) reduces to

N = EI

k

"

(5)

The continuously s u p p o r t e d rail s u b j e c t e d to an axial force a n d a m o v i n g load

73

C

CmIx.o n i .

~I I

2_._7

Hm i ~ Fro. 2. T h e graphical p r e s e n t a t i o n of t h e a b o v e e q u a t i o n is also shown in Fig. 2. The m i n i m u m v a l u e of N is o b t a i n e d from t h e condition [~N/~(2~r/,~)]¢=o = 0. I t is found to t a k e place for 2~r

.]~ k

-

(6)

~

and that Nmtn]~= o = 2 ~/(kEI) = N c r

(7)

is t h e critical buckling load of t h e infinite beam. 7 C o m p a r i n g e q u a t i o n s (4) a n d (6), it follows t h a t b o t h m i n i m a are located at t h e same v a l u e of (21r/A). I t m a y be easily shown t h a t for a n y fixed N ~
Cm~ =

--~

--N .

(S)

The a b o v e results suggest the rewriting of e q u a t i o n (3) in t h e following non-dimensional form e

Cmlnl,v=o

=

I

r 2 + ~I - N o r

,

(9)

where 2~./k = T ~/~7"

(10)

The graphical p r e s e n t a t i o n of e q u a t i o n (9) is shown in Fig. 3. I t should be n o t e d t h a t , for each N ~ Ncr, t h e p r o p a g a t i o n velocity c depends also u p o n t h e w a v e l e n g t h A a n d t h a t the w a v e trains of t y p e (2) do exist only for c >i Cmin. W h e n the w a v e l e n g t h A -~ 0 or ~ -~ ~ , t h e corresponding c -~ ~ . F r o m Fig. 3 it m a y also be seen t h a t for a fixed N ~ Ncr, to each c > Cmin, there always corresponds two w a v e s w i t h different w a v e l e n g t h )~. T h a t is, for a fixed N < 1Vcr, two waves, each w i t h a different w a v e l e n g t h )t, m a y p r o p a g a t e w i t h t h e s a m e velocity. To t h e p r o p a g a t i o n v e l o c i t y Cmin t h e r e corresponds only one w a v e (2) w i t h t h e w a v e l e n g t h g i v e n in e q u a t i o n (4). A n i m p o r t a n t feature of t h e results presented a b o v e

74

ARNGLD D. KERR ¢ CminlN,o -O50

Nor 7

~c=o

FIG. 3.

is t h a t Cmin decreases w i t h i n c r e a s i n g N (i.e. w i t h i n c r e a s i n g c o m p r e s s i o n force) a n d t h a t Cmin b e c o m e s zero a t Nor. W i t h d e c r e a s i n g N (i.e. w i t h i n c r e a s i n g t e n s i o n force), Vmin increases. F r o m e q u a t i o n (8) i t follows t h a t for a n y N > Nc~ t h e p r o p a g a t i o n v e l o c i t y Cram is a c o m p l e x n u m b e r . T h i s is also t h e case for a n N > N c r a n d a n y c v a l u e b e y o n d t h e c = 0 c u r v e s h o w n in Figs. 2 a n d 3.

BEAM

SUBJECTED

TO A MOVING

LOAD

P

W e c o n s i d e r n o w t h e r e s p o n s e of a b e a m w h e n it is s u b j e c t e d t o a l o a d P t h a t m o v e s a t a c o n s t a n t v e l o c i t y v0, as s h o w n i n Fig. 1. T h e differential e q u a t i o n w h i c h describes t h e b e a m r e s p o n s e is g i v e n in e q u a t i o n (1). B e c a u s e of t h e infinite e x t e n t of t h e b e a m a n d b a s e a n d its c o n s t a n t p r o p e r t i e s , as well as b e c a u s e of t h e a s s u m p t i o n t h a t v 0 = c o n s t , it a p p e a r s r e a s o n a b l e t o a s s u m e t h a t a f t e r a t i m e p e r i o d t h e t r a n s i e n t m o t i o n s will b e c o m e negligibly s m a l l a n d t h a t t h e b e a m d i s p l a c e m e n t s will a p p r o a c h a s t e a d y s t a t e . 4 T h u s , f r o m t h e p o i n t o f v i e w of a n o b s e r v e r w h o m o v e s w i t h t h e load, t h e deflections o f t h e b e a m will a p p e a r static. T h i s o b s e r v a t i o n suggests t h e possibility, o f t e n utilized before, of t r a n s f o r m i n g t h e p a r t i a l differential e q u a t i o n (1) i n t o a n o r d i n a r y differential e q u a t i o n in t h e m o v i n g r e f e r e n c e f r a m e (~, ~, ~) as s h o w n i n Fig. 1. W i t h t h e n e w co-ordinates =x-vet; ~? = y; ~ = z, (11) t h e differential e q u a t i o n (1) b e c o m e s d~w

~ d 2w

EI--s~+a~_ (N +mvo) ~ + k w

.

= P8(~¢).

(12)

The continuously s u p p o r t e d rail subjected to a n axial force a n d a m o v i n g load

75

Because of t h e s t e a d y - s t a t e assumption, t h e load P m o v e s at a c o n s t a n t a l t i t u d e a n d hence does n o t experience an acceleration in t h e z-direction. Thus, P in e q u a t i o n (12) represents o n l y t h e static i n t e n s i t y of t h e load. N o t e t h a t e q u a t i o n (12) is identical to t h e e q u a t i o n of a b e a m which rests on a W i n k l e r base, is compressed b y an axial force (N+mv~) and is s u b j e c t e d to a lateral load at ~ = 0. The simplest m e t h o d of solving e q u a t i o n (12) is to set t h e r i g h t - h a n d side equal to zero and to incorporate t h e c o n c e n t r a t e d load P t h r o u g h the m a t c h i n g conditions at ---- 0. Setting

4c~S

=

N" + mv~ EX

;

]c

(13)

w = 0.

(14)

4fl'=E-i'

e q u a t i o n (12) becomes 44W

d~¢4

+~

2d 2 w

~-~(~ + 4 #

~4

A s s u m i n g w = Aes~, and s u b s t i t u t i n g it into e q u a t i o n (14), we o b t a i n s 4 + 4c~2 s 2 + 4fl4 = 0.

(15)

The four roots of the a b o v e e q u a t i o n are sl,s,a,, = -+ 4{ - 2[ (x~ ~ 4( ~4 _ f~4)]}.

(16)

N o t i n g t h a t a~-f~ a = (c~s-f~s)(as +~s), we h a v e to distinguish three eases: ~' > fl'

which corresponds

to

v0 <

212C

--~v

-

•

,,,,

N o t e t h a t t h e velocities v 0 are real only w h e n

N < 2 4(kEZ) = Net.

(18)

I t is well k n o w n t h a t w h e n N > Ner the stable b e a m shape is not straight and hence t h e differential e q u a t i o n (1) is n o t applicable. F o r this case t h e necessary analysis is v e r y i n v o l v e d and is b e y o n d the scope of t h e present paper. I n t h e following we analyze t h e case w h e n N < N c r in order to d e t e r m i n e t h e rail response w h e n / V approaches t h e buckling load Ner ; a simpler p r o b l e m of more i m m e d i a t e practical significance, for which e q u a t i o n (1) is applicable. W e consider t h e case w h e n

v.<212( ) This corresponds to a~< f~2. The roots of e q u a t i o n (16) are complex numbers, n a m e l y sl,s,a,4 = _+~+i¢o

(19)

where

= 4(3s-~s);

~ =

4@+o,~).

(20)

The corresponding solution for t h e region a h e a d of t h e load P is w.(~:) = e+a~[A 1 cos (¢~) + A s sin (¢o~)] + e - a t [ A s cos (eel) + A t sin (¢o~)],

~> 0

(21)

a n d for t h e region b e h i n d t h e load

wa(~) = e+~[Aacos(w~)+Atsin(a)~)J+e-M[ATcos(co~)+Assin(¢o~)],

~<.0.

(22)

I t is reasonable to e x p e c t t h a t as ~ -> ~ , w~ = 0 a n d as ~ - ~ - 0% wa --- 0. T h u s A 1---A s=

0;

AT--As=

0.

(23)

76

ARNOLD D. KERR

The r e m a i n i n g four constants are d e t e r m i n e d f r o m the four conditions at ~ = 0

~o(o) = wb(0),

d 2w~ d~ 2 0 =

~dwa Io = ~dwb I0'

1

(24)

d 2wb (d 3w~ d awb~ P d~ * 0' \ d ~ ~ d~ 3]0 = ~ "

I t should be n o t e d t h a t in the last b o u n d a r y condition t h e t e r m i n v o l v i n g (dwa/d~ - dwb/d~) o is missing, since according to t h e b o u n d a r y condition a b o v e it, this t e r m vanishes at ~ = 0. The d e t e r m i n e d constants are

A~ = - A 6 - 4Ei~o(,~ 2 +co~ ) ,

(25)

P A 3 = + A 5 = 4EL~(1~+co~) • S u b s t i t u t i n g the constants into equations (21) a n d (22) we obtain, n o t i n g t h a t ~--- X - - V

0 t,

wa(x,t) = P e x p [ - ) q x - v ° t ) ] { e o c o s [ o ~ ( x - v o t ) ] + 2 s i n [ c o ( X - V o t ) ] } 4EI~oJ(~ 2 + ¢o~) wb(x,t)=

Pexp[+'~(x-v°t)]"

s

~E/~)--(¢oco

[¢o(x-vot)]-~sin[eo(x-vot)]}

H e n c e the w a v e caused b y P , and which m o v e s w i t h P

for

x>~vot,

(26)

for

x<~vot.

(27)

at a constant

velocity

% < ~/[~](4kEI/m 2) - N / m ] , is s y m m e t r i c a l w i t h respect to P for a n y N < Ncr. I t m a y be easily shown t h a t w h e n v 0 approaches t h e v a l u e ~/[~[(4kEI/m ~) - N / m ] , t h e deflections become infinite. Thus, t h e critical v e l o c i t y of the b e a m subjected to the axial compression force N is

vet =

~r-

-~

•

W h e n N = 0, e q u a t i o n (28) reduces to

,/[4kEI]

VcrlN=0 = ~ / ~ m ~ /"

(29)

N o t i n g equations (7) and {29), e q u a t i o n (28) m a y be r e w r i t t e n in the following f o r m :

V~r N V~rlN=o~ - ~ = 1.

(30)

Vcr = 4( 1 - N / N c r ) Vcrl~=o.

(31)

or

H e n c e w h e n N - ~ Nor the critical speed Vcr-~ 0. W h e n N is a tensile force, Vcr> VcrlN=O. The graphical p r e s e n t a t i o n of e q u a t i o n (31) is shown in Fig. 4. Comparing equations (8) and (28) it follows t h a t for a n y N < N ~ r , Vcr = Cmtn. Thus for a fixed N < Nor, the corresponding vcr is t h e lowest speed a t which a free w a v e of t h e f o r m (2) can propagate. Therefore, Fig. 3 m a y also be used to visualize the effect of N u p o n Vcr. To show t h e effect of N u p o n t h e t r a v e l i n g deflection profile, e q u a t i o n (26) was e v a l u a t e d for vo/(Vcrl~=o) = 0"2 and different values of N/Nor. These results are shown in Fig. 5. I t m a y be seen t h a t as t h e axial compression force N approaches t h e value (N/Ncr)o.a shown in Fig. 4, the a m p l i t u d e increases rapidly and the w a v e l e n g t h decreases, whereas an increasing tensile force has an opposite effect.

T h e c o n t i n u o u s l y s u p p o r t e d rail s u b j e c t e d t o a n axial force a n d a m o v i n g load

J

1'6

77

Vcr VcrlN.O

1"4

1'2 O@ oe!

04 (>2 |

-I.0

,

,

,

.

I

-0.5

=

i

I

J

0

0-5

TENSION

1.0

COMPRESSION

FzG. 4.

25.0 --~ x lOS(in/Ib) 200

-• '~-¢r, O.8

15"0--

%

v'~'c"0'2 r k-IO00 Ib/int E-30 x 106 Ib/in2

\

1 - 9 6 ' 8 in 4

\

1"°4""~

I0"0~ 0 ~ , ~ ~

o

- 5O-

\ ioc)~ \

~

~ _ __...2oo ~---_

/

_

/

//

-

-KNI -~IG. 5.

- 3oo

j

~¢(in)

78

ARNOLD D. KERR

CONCLUSIONS I t was f o u n d t h a t vcr increases w i t h an increasing tension force N , whereas Vcr decreases w i t h an increasing compression force, vcr a p p r o a c h i n g zero w h e n N--> Nor. H e n c e in the absence of e x p a n s i o n joints in the rails t h e critical velocity vcr m a y be reduced, b y a rise in t e m p e r a t u r e , to within the o p e r a t i o n a l velocities of trains. I n view of this finding it is e x p e c t e d t h a t a n analysis, b a s e d on a f o r m u l a t i o n which does include the inertia a n d d a m p i n g o f t h e base, will also exhibit a similar effect of N u p o n vor. REFERENCES 1. S. P. TIMOSHEI~KO,Proc. 2nd International Congross for Applied Mechanics, Zi~rich, pp. 1-12 (1927). 2. Collection of Papers, Technical Report, Holloman Air Force Base, New Mexico (April 1959). 3. A. Assu~, U.S. Army SIPRE Report No. 36 (1956). 4. A. D. KERR, U.S. Army, Cold Regions Research and Engineering Laboratory (CRREL) Report, in preparation. 5. E. L. COR~WELL, Modern Railways, Part I (January 1969); Part I I (February 1969). 6. J. DSRR, Ingen. Archiv. 162-192 (1943). 7. M. HET~YI, Beams on Elastic Foundation. University of Michigan Press, Ann Arbor (1955).

8. D. E. NEWL~D, J. mech. Engng Sei. 12, 373 (1970).