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Masters Theses Student Theses and Dissertations

1965

Infinite beams on an elastic foundation Infinite beams on an elastic foundation

Shi-peing Chang

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INFINITE BEAMS ON AN ELASTIC FOUNDATTON

BY

SHI--PEING CHANG1 1Cf31 31f

A

THESIS

submitted to the £aculty of the

UNIVERSITY OF MISSOURI AT ROLLA

in partial £ul£illment o£ the requirements £or the

Degree o£

MASTER OF SCIENCE IN CIVIL ENGINEERING

Rolla., Missouri

1965

1.1.5238

Approved by

(advisor) ~~ _/ L .>

/ ( ·, ,•I .

. J

•)

' .> ~~

ii

ABSTRACT

In this paper a partial differential equation for

the analysis of beams resting on an elastic foundation under

dynamic load is presented. A ~olution for this e~1ua:1Jion is

derived, and solutions for various boundary conditions o£

inf'ini te beams resting on an el·astic foundation are dis­

cussed. The application of the equation is demonstrated by

an example of a moving load traveling along an infinite

beam.

ACKNOWLEDGEMENT

·The author wishes to express his sincere apprecia­

tion to Professor Jerry R. Bayless for his gui~nce and

counsel in this study.

Appreciation is likewise extended to Dr. Peter G.

Hansen of the Mechanics Department for his criticism on

this topic.

The author feels deeply indebted to Professor

Sylvester J. Pagano of the Mathematics Department £or

his invaluable help in solving some difficult problems

in this study.

iii

Thanks are also extended to the staff of the Depart~

ment of Civil Engineering.£or their encouragement.

iv

TABLE OF CONTENTS

ABSTRACT • ••••••••••••••••.••••••••••••••.•••••••.••••••• ii

ACKNOWLEDGEMENT .••• • ••••••••••••••••••••••••.••••••••• iii

TABLE OF CONTENTS •••••••••••••••••••••••••••••••••••• iv

LIST OF FIGURES ....................................... v

LIST OF TABLES •••••••••••••••••••••••••••••••••••••••• vi

NOTATION ·• ••• ••••••••••••••••••••••••••••••••••••••••• vii

INTRODUCTION ••••••••••••••••••••••••••••••••••••••••• 1

REVIEW OF LITERATURE •••••••••••••••••••••••••••.••••• 3

DERIVATION OF EQUATION •••••••••·••••••••••••••••••••• ~

DEt1IVATION/

DISCUSSION ••

SOLUTIONS ............................... • • • ••• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

CONCLUSIOri ••••••••••••••••••••••••••••••• ............ ._, I

14

19

3l

BIBLIOGRAPHY •••••••••••••••••••••••••••••••••••••••• .., , 5

VITA ••••••••••••••••••••••••••••••••••••••••••••••• 37

v

LIST OF FIGURES

Figure

1 Deflection of a Beam on Rigid Supports •••••••••• 6

2 The Coordinate Geometry of the Elastic Curve ••••• 8

3 Deflection of a Beam Resting on an Elastic Foun-

dation ...•........•.........•.........••......• . 10

4 Behavior of Forces Applied to Beam Resting on an

Elastic Foundation ............................... 12

5 Load Conditions of the Infinite Beam ••.•.••••••• 16

6 Residue Curve ••••••••.••••• ~ .•••••.••••••• · ••• -••• 22

7.

8

9

Diagram

Diagram

A Rail.

for the Variation

:for the Variation

Resting .. on Ballast

-of Y/Y0 with n •••••••• 24

of (I) with n .•. • •••••• 27

••.• ·~i ~ .. ~ ••••• :.: ••.• -~ •.•••• 29

vi

LIST OF TABLES

Table

1 Modulus ·of Subgrade Reaction •.••••••• · ••••••••..• 28

2 Program for Solution of Equation 40 ••••••••••••• 33

3 Output Data of.Equation 40 ....................... 34

'l 6

E

b

D

E

g

I

(I)

i

k

/. M

n

p, q

P, y

t

v

w

X

NOTATIONS

parameter of integration

deformation per unit length

r2.dius of curvature

vii

reduced coordinate in x-direction of beam defined

as ?<- v-1::_

~ tranformation o:f t defined as ~-=- t

stress

strain

width of beam

denominator of the solution defined as D=o<+-2n2o<: 2+ 1

modulus of elasticity

gravitational force acceleration

moment of inertia

integr~

imaginary unit·

constant of foundation

characteristic length defined as ~4 = Eki

bending moment per unit length of berun 2 v-.1 .....;-"L

defined by n - g .2.~ ~-a. <. I

applied load

Fourier coefficients

time

velocity

unit weight of beam

horizontaJ. a.Xis of beam

viii

y vertical ~~is of beam or deflection

I. INTRODUCTION

The problem of beams resting on an elas·cic foundatio:l.

is often found in the design· of stru~tural members or

buildings, ·:railroads, airports,. highways, and othe::.~ ~-:;t:c·uc­

tures.

l

Due to the advancement of engineering, many thirut

today tl:lat the solution of this problem by the clascic

equation, which was derived nearly a century ago (18), (ll)

and is still beirig used, is not satisfactory because it

does not consider the existing dynamic action.

In recent decades the development of mathematics 2illd

the theory of elasticity have made unprecedented strides

that make possible the solution of some difficult dyn~fiic

problerns.

In this paper an· equation of motion for beams rEstinc

on ela.~tic £oundations and influenced by dynamic loads is

obtai~ed. The derivation of this eauation is modeled after . . the mo:t-e eJCact plate theory of Reissner · (ll). It ·.is aSfJl1:1ted

throughout this paper that the depth of the beam. m8.y be

neglected, the deflection is very small compared to the

dimensions o.f the beam, and the beam is loaded within the

elast:ic lirni t. Other assumptions axe that the founC··_,-~ __ o:_

has no mass and the properties of the .foundation obey

Winkler's assumption (14), ·that at every ?Oii:..·c ·-~he r€ac"';:;io·(~.

of the .foundation ~s proportional to the de:fle(.;·:. '.on at tl;.e

corresponding point. These assu.-nptions have been i'o,· ~,a

2

(6) to give usable results. With these assumptions, a

general partial di££erential equation o£ motion will be

derived.

·.

3

II. REVIEW OF LITERATURE

The problem of the elastic beam on an elastic foun­

dation has been popular in scientif'ic and engineering

literature ever since Zimmerman (18) presented his solution

for the analysis of the railroad track in 1888. This ,

solution was based on VfinkJ.er' s (14) · ass'U!nption·· that the

deflection at any point is proportional to the foundation

pressure at that point, and does not depend on the pressure

at any other point of the foundation.

S. P. Timoshenko (1~) was: the .first·· to use the r;olution

in this country when he found the strength of rails.

Westergaard(l6) used the solution to explain cracking in

concrete pavements.

Hetenyi ('6) has done a great job in his famous book,

Beams on Elastic Foundation, on the .theoretical develop­

ment of equations for various boundary conditions.

To explain the fact that the rate of change of the

deflection is a function of time for a beam resting on a . -

viscoelastic medium, which is defined as a material whose

force-deflection relatione are functions of time,·Freu­

denthaJ. and Lorsch ('5) put a velocity term-r~~into the

classical equation

~ EI ~x1- + ky =P

eo that the equation becomes

EI ~""!l +1: -~ + ky= P ~X.._ ~k •

To deal. wi. th tha particular case of transverse vi-.

4

bration of a beam which lies on an elastic foundation and

is subjected to a periodic longitudina~: load, p0

+ ptcos ·c,

Bolotin (l) has derived a partial differential equation . ti rl"Z.~ ::>'-!} EI C1x"'" 4- (p0 +- ptcoeGt) d)t.').. + ky ~ m 'dt4 = o.

\'I here y( x, t) = fk ( t) sin ~~ -x , ( k 1, 2, 3, •••••• )

and .fk(t) are as yet undetermined functions of time.

The equations presented by either Freudent4al ru1d

Lorsch or Bolotin are satisfactory for individual boundary

.conditions, but are not satisfactory to solve the general

problem of a moving load traveling along a beam which is

resting on an elastic foundation. The reason is because

a moving load applied to a beam will subject·the berun to

a vibration .force and a static force.

In order to solve this problem, this paper presents

a general partial differential equation and its solution,

which are found by means o.f a Fourier integral •

. .

-

III. DERIVATION OF THE GENERAL PARTIAL

DIFFERENTIAL EQUATION

In deriving the equations based on the dynamics of

beams, the assumptions of ordinary beam theory are used:

5

1. The material is isotro'~ic, that is, any part of

the material that is.large enough to contain a considerable

number of grains will displaJy the same properties of over-. .

all stress and strain regardless of the directions in which

the part has been cut or is loaded (9).

2. Stresses are· below the proportional limit which .....

means that the ratio of stress to strain is a constru1t.

3. Deflections are very small compared to the dimen- .

sions of.the beam.

4. A plane cross-section before bending remains plane

after bending.

5. The beam is initially straight.

6. Shearing deflections are neglected.

7. The neutral axis of a·beam in bending is the locus

of the centroids of the cross-sectional area.

Accepting the above-mentioned assumptions (17), the

deformation of the neutral axis, y, can be represented by

Figure 1.

Cd9 A

~ =-

-~ E -

l: ·=

6

P. . '

m I'

i ---:;;1 ""' I / .

------X.

,.....-= JB :;::> _c.._

,..l dx ~- L-------cEi;;;ti c curve :r (a),

(b) M

n

F:i.g .. J. Deflection of Beam, :OncRigi.d:.:Supports~ ,_. ··.

' , '

' , ...

then

~dx 2E

CdG 6dx ~= 2E

MC I

Cd6 MCdx ~= 2EI

d9= Mdx EI

Furthermore, since

or d,c

d9 = p

1 M

7

(1) y::-n The coordinate geometry of the neutral axis of this beam

is represented by Figure 2, from which

fd8 =ds

For small slopes de is approximately equal to dx, hence

p d = dx

From Equation 1 substituted into Equation 2 M d9:::- dx EI

As dx approaches zero

tanG= :~ or in radians,

l)_Q:y. o- dx

( 2)

( 3)

( 4-)

Differentiating both sides with respect to x, the equatio;::

will be

8

..

.,.. y

Fig. 2 The Coordinate Geometry o;f t·he Elastic Curve

d 2y _ M dx2. --,rr

9

( 5)

In order to discuss the behavior o£ dynamic forces

applied to the infinite beam resting on an elastic £oun­

dation, one must assume that k, the modulus of subgrade

reaction is a constant, which may be estimated from Table

1, the foundation has no mass, and the infinite beam.is

supported along its entire·1ength by the .foundation and

subjected to a vertical force, p, acting in the principal

plane o.f the symmetrical· cross section (Figure 3).

If y is the deflection at a point, by Winkler's

assumption, it can be found that

p=-bky

where b is width o.f the beam. Assuming a unit width beam,

i •. e. that b equals one, the equation becomes,

k=- ~ An infinitely small element enclosed between two

vertical cross-sections a distance dx apart in the beam

is under consideration and a load, p lbs/in., is applied

as in Figure 4. The sign conyention is that the upward

acting .forces are positive. Therefore,

I.F =o V - (V + dV) + kydx

Whence

dV + kydx

that is dV d~ :.ky-p

pdx= 0

pdx=O

/

I I

y T

y

I

l :P

"'-Beam

II

Fig. 3 Deflection of a.. Beam Resting on an :E:..c: .. ::>tic

Foundation. -··. ,,, . . \ ~ . ··. (

J.O

X ,

By the relation

v = drv! cJx

then

• Substituting Equation 5 into Equation 6,

d' dZ.Ij d~ EI d x_'Z-~ = - o\ X.l.

· or

that is d4!1

EI ot x4 =- - ky + p •

11

( 6)

(7)

This is the equation £or the deflection curve o£ a

beam supported on an elastic £oundation and subjected to

static loads. I£ a beam is loaded by vibration only, then

-- vJ "d'~ · P- T c;>t-a.

~ is the mass o£ a unit length ·o£ the beam, so that Eq. 7

becomes

EI d4-~ + ~ ~'_'j -1" K'/ = 0. (P) ~)(41-- 0 ~*"\.. \....1

. '

Nov1, a moving load, p, traveling along a beam is to

be considered. The de·sign of a beam which is restins on

an e~astic foundation and which is subjected to a moving

load must be based on the.consideration of the action of

both vibration and static forces ('8) so that the load

term must be added into Equation 8, which becomes

EI d4 !')4 .-+ ¥/ ":;;).,_'!} -\- k. '-'\ -= -o ( 9) ~')( g ~t.... .J ' •

This is the equation for beams on elastic found:;.tion con-

sidering dynamic ~oads. This equation satisfies the bour...­

dary condi tiona o::f not only infinite beams restinr; on c.n

.•

v 4~

pdx.

-

M +dM dx

.. 'kid~ ,

v+dV

Figure 4 Behavior o:f Forces Applied to Beam

Resting on an EJ.asti'c Foundation

12

. ,

elastic foundation but also finite beams resting on ·an

elastic foundation •

13

..

1~-

IV. DERIVATION OF SOLUTION

In order to solve the generaJ. equation f'or heams on

an elastic f'oundat.:i..on, a. load, p, moving along on iniini te

beam resting on an elastic .foundation at a velocity,. v, io ...

considered. At time t, the distance along the beam to the I

original axis is (Figure 5)

~- x- .vt , - ' J.. ' 'l = t

whe~e ~ is characteristic length de~ined by t=~E;: By chang:l,.ng coordinates f'rom x to ~ of' the equation,

~4.1 , 1 ~~!;) EI d X-4- + ~ ct~ .,. ky = :p

it may be observed:

~:::.~a!+ tl a'\ .::: Tl ~ +o a.){ d~ ox "~ d-~ a~ .

d L~ ~ I at d ~ _ l d~La-o-X~= ~Qat~-~~~~"'

'd4 '?- 'd· l d\~-~)_· l d4 'd­Jx4- ~"? d-~ ~ -:r c).~i-

(9)

(a)

(b)

d.'t-~~ + ~ a"l - -'V i>tr ~ (c) ~ - a-~ C>i:"" <1~ Tf - A. ~ + _;;~

Assuming the stress and deflection patterns set up wove

with the lOad ·at a constant veloc~ ty·, .y is a function o:f $ Only. Consequentl.y, ~i.s zero and ·· :-

d'\.. "1..

. d. 'L'I- ·6 v ~ ~ ~ ld" - ..; '2- ';) l( ';;;*"\.. = ~) (-(~) ~- tt a-1~

( <l)

n.nd by substituting Equation b end Equation d into· Ecruc..tion '

' 9, the differential equation will be

EI ~4-1: + U 'V.,_ o2. L 04- ""\ .... ~ + ky = p . A Cl' 1' . ?".) Al. d ~ -r...

Dividing Equation 10 by k the equation ?ecomes

d 4 J + w v2 d-"1."(- _ _I?_ 0 14 kg ,l ~ o-~ ~ + y - . k •

Let 2 vJ --v,_

n =~act~~. and Equation 11 becomes

4 . L .

;;;. ·tt + 2n2 ;} ~ + y =....L 0 r" C>-!1- . . k •

Both p _and y. are :functions of' ~ •

If', then, p and y are absolutely integrable, y(3)

15

(11)

(12)

(13)

and p(~) may be represented by a Fourier transform (13)

consisting of' integration with respect to a parameter ~ so

that o.0 ·

. y(i) = ;ll '(()() e:k>(! do<. (H) )_~~ and· . ~

. p(~).= ~{;(t~-)ekt)d..-. (15)

Where o<. is indePendent of 'f and

P(c:.t.) -[£<3) e-jpt."§dol.. (lG)

Substituting Equation 15 and the second and foi.lr-th deri­

vati yes of'_ Equation 14 into Equation 13, then t:he e1_;_uatio~1

becomes

((){4_ 2n2o£ 2+ l)Y(ol.) =-tP(o()

so that ~ _L p(.,.()

Y(tJ....) - I< o(-4- :Ltn~o(~+ I ( t 7)

Now a load o:f magnitU:de q(x, t) unii'ormly eli.,

over a length of' 2d moving with a constant velocity, -...;,

16

' . . . : )

--v-

, Figure 5~ Load Conditions on the In£inite Beam

traveling on the beam is considered (Figure 5). The

applied load is

p(x, t)= + where -d < x-vt <d.

Trans£orming the coordinate this load becomes

p($) = -&- •

where - t· ~ ~ ~ t · From Equat~on 16 ·~ . .. ~ ·

P(o<.): l~dr + ( 4t e-iolfd} + (' Odf

-fP l.y~ _J~ :. . ~ ( eio(dA, - e-:lo(d/.f ) .

2cti"d .

_ Ck ( eivtd/l_ .9 -icld/J. ) • d2io< \ .l~ ' th.

. ~~7( -A.~ 7J.. Sine e sino(d,.1 _. e - e.

"' 2iv

17

p(oi.) = * St~ o/t (18)

Subst'i tuting Equation 18 into Equation J. 7, on:e £inds .

• .J a//.. cJ. = i S,,., V' /,l (1 a)

y~ ) . /<otJ lo<4-2n~~+l) -:;

From Equation 14

. . 4 -6 ,:, oi r.c. .· ·. .iofJ d

y(_3) = ~ -o- Q((o<if-2/n";l.'),..+t)

or the real. £orm may be used

(2o)

Let

elld

Let

EI 0

•k

01 {of..) ::

18

(21)

(22)

(23)

( 24-)

These equations_ 0

are- the partic\Ll.,ar solutions oo£ Eq-. 9,

and are the 0 general. _solutions o~o Equation 10;

19 • I

·v. DISCUSSION '\

The two important elements to ~e discussed in this

section are deflection and moment.

A. Deflection:

When the case o£ a concentrated load applied to · .. the beam is conside~ed, d in.·Equation 23 app~oaches

Thus in Equation 22, 4> (of.) app:rroaches f ·and.·

<~>- _j_ r u>~o< ~ . . tJI,../. . Y J - 7fk/J. - o(4-2.n'-,l-...-l-/ """ ·

. 0 is evolved.

It may .be desirable to investigat•e )-

D =ot..4- 2n2o<2+ 1;

zero.

(2'))

the denominator, £or it is necessary to understand every · ' '

e~bo1 thoroug~y.

(1) When n:: 1 then '

D = o{ 4_ 2n~2+ I = (o<2-: J.) 2

Ifot.:'tl in this equation y will become infinite,

" so that it does not ex;tst.

( 2) ·when n -,1 then D has reaJ. roots therefore the de­

flection cannot be defined.

( 3) The only cas·e . .for which the defl~ction is defined

everywhere occurs . when the denominator does not have a

reaJ. root, that is .when n.c l. This in·ea.ns that '2.k8 l2..

w In the above discussion it can be seen that

_ zk!, Ll--·. w

w~l1 be·the critical. condit;Lon .for the solution. Whenever

20

the veloci iiy is larger than or equaJ.s to v c the deflection

of the beam-becomes ·infinitely·large.

· For an example, a rai1 with a cross sc:-.:tion as sho~-:n

in Figure 9, rests on baJ.last hav:Lng a modulus k = 230x6. 5

lb/in.. The mod~us o! eiasticity of. the rail.. is. 30~·o6psi,

IN.A~112 in1, and weight of the rail is 3.78 lb/in ••

Find the maximum velocity of the train that the rail can

·endure.

For

~ =- 1~~~~. 4 . =0 .00973 lb-sec~/in.

J).= (Er =/ ~30x10~x112 ~~ . 230x6.5 ·

= i ,497 in~ v. _ "1 2xl, 500xl ,497 c-r 0.00973

= 1461 ,ooo ,ooo - 21,500 iri/sec.

- 1,22~ mph.

It is obvious that no train will, as yet, move· at this

speed. But if the combined action of the velocity and

vertical loading are taken into consideration the influence

o:f velocity on the de:flec'tion and moment. of beams may well

.be large.

The wave produced by the traveling load may be rep­

resented by the equation '

y=y0sin~ ( ::z.6)

where y 0 is ·the deflection of the beam· subjected to a st2tic

21

load.

If' the beam is· subjected to· a··_p.ni:f'orm l~s.d, and the

load .is. stopped somewhere ·.on the beam, ·-for .whicb "i'= o, . Equation 23 becomes

. tf ~~ fP(oC) • y = clo<. 7C k. ()("4 -:l., ... oi~+l .

D

Oc::: n~ 1 (27)

Tho argument ndght be extended to the maximum deflec­

tion under a concentrated ioad. It enters the situation

when both d and ~approach zero. From equa~ion 23 then

Y- ,9 r dOl. lL - 0 c: n c! 1 ( 28) - 7CK J.. o<t 2n"~gl.*~-+ I

()

.T:bis il'ltegration involves a known Tshebysche:f'f polynomial

('1 ) that may be e:\f"alulit$0. by the residue calculus which is

based on the theory·o£ analytic :function, in partcular,

the residue theorem.

The complex function

1 z4:...2n2z 2+ 1

is integrated·over the conto~ Figure 6 and R approaches

0¢.• It can be shown that the integral over CR approaches

o. By the residue theorem,

1 4 dZ

2 2 = 2/[i~ Res.idues in 0

Z - 2n Z + 1 · ~

and

~+ ·/::: 2~l.~Res. )R ,JcR• . .

Letting R approach infinity, then

-~

. ol~ . + 0: 27ti~Res. o< .,._ ~, ~ ~-~--,

The so~ution ·of the ~ntegral is

22

-R + R

I

Fig. 6 ;.-·Residue Curve

23

j .o o/ol- n ( l ) =· a.n I 0 ..,{4,-;:ll')~iL+I - 2.~. y (- 't\ ").. . 4 ( f/- 11~ )

and the deflection equation for concentrated load is

Y _ nFi 't: - 7[ /<)42/J-n"'- 0~ n<: 1 (30)

When the velocity is zero, ·.

Yo= ( 31)

A graph o£ the vaxiation of y/y0

with n is given in

Figure 7. From this figure it can be seen that the ratio

y/y0 approaches infinity as n reaches a value very close I

to unity. This is a.main point which should be discussed

here. · The actual deflection will never become in.fini te ·.

in any case, but it must be borne in mind, however, that

the beam is assumed to be an undamped elastic system which

neglects the resistance to the velocity. In most physical

cases the e.f.fect o£ damping.would certainly tend to reduce

the deflection. ·Also it can be seen in the numerical

ex~ple stated be.fore that the critical velocity will never

be reached. In our calculation v0= 1220 mph. This is

greater than ten times the maximum speed o£ a normal loco­

motive, consequently, in the problem where n equals 0.1,

the influence on the ratio o£ de.flection is 0.01. This is

so small that it may be neglected here.

B. Bending moment:

When a uniformly distributed moving load is applied.

.. ;

24

. .. .

'·

3.0

2.8

2.6

2.4

2.2 .,.

'l. 2.0 Yo

1..8

1..6

1.. 0 1-.__,..--:---:­

o~o o.1. 0.2 0.3 0.4 o.s o.6 0.7 o.s o.g 1..0 n--.-

Fi~. a Diagram for the V a.riation of y /y 0 with n.

to beams the bending moment under such load can be f0L1c:

from Equation 24. It is

~0"'

M ( 0) = f £ .-¢;Col.) o< '"2. 1 7[. o<f-2n~'&.--t I

0

do( 0 ~ n~-l (32)

This integral converges to a finite value.

As a. concentrat·ed load is te considered, then d appro­

aches zero, and Equation 32 becomes _ _ur c(2-

M(O)- 7! J ~~;J.n%<'a....f-J o/o( 0 .

.0~ n' 1 (33)

or oO 1ij ~ 2. tic< 7C

0 I -2 n7o<-a. -r/.L a

M(O) ==

From this equation it can be seen that the integral \Vill

not remain defined, so that a solution does not c-:~ist.

It will be shown later that by ch2nging the integral into

another form a divergent solution will be obtaincc1_.·

The static condition of Equation 32 for 8, concentrated

load and n:: 0 is considered and-

Mo = ~Rr~=t~~ ()

To find the bending moment under the concentrated

moving load, one may go back to Equation 32, reduced by

Equation 34, and

!v~( 0) M = 0

?6

:::::: COL'- ( .... ,,/~[;;6 ~/;, o< tjz r.-< 't-o/- ro< t£. 2 n ~ ...._.IJ do<. ( 35 ) -n: ~ (ol "4-t.) ( c< ll_ ~rJ ~ z...-1-/..)

provided 0 ~ n~ 1

Equation 35 remains convergent.

Now, let d approach 0; that is,~ sinot..-j approaches Dl. f Since ¢;(o<) ~ 1/i when the loaded area is .finite, Equation

35 can be simplif~~d to give an upper bound for the increase

in MJ(O), this bound .being approached as the lo~ded area

decreases. Thus

- i/ r:.2...(o<f£1-l) :...o( L.('o( ~ ~ n":l~ 7.....+1.) do( M(O) M0 <:. JZL ·~ (_o(4-+l) (C>l4-::zn'2c:K7-+I_)

0 .

Dividing both sides of Equation 36 by M0 it becomeo

M(O) L + -:z1.;/771o0 . o<4- cJo( M;- I 7CMo o ·(o<4-+!}'(o<4-;2..n~~-+-J_)

0< n~ 1 (37)

( 38)

O~n<:~

27

2.00;· ,-I

I i

1.go;

1.8o! I

1.70~ I

l. 601 l.50t

I 1.401

1.301

1.20

1.10

(I) l.OO

o.go

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

o.oo / I

0.0 0.1 0.2· 0.3 0.4 0.5 0.6 0.7 0.8 o.g

I n I

I FIGURE 8

Diagram for the Variation of (I) with n .

. ---~~-----~-------·-·-----":~---- -~----. ----- ---

J_.;;t (I) ~(pt.) do<.

then

M(O) M

z.. L. I -r 4/2 n ( Z )

7C

28

(41)

(I) can be .found from Figu:r·e 8 whi-ch is plotted by the

autoplotter £rom the output data of' approximate solutions

o.f' Eq_uation 40 (Table 3). The solu~ions are f'ound by Simp­

son's law with dif'.f'erent n_values and six various upper

bound values which are f'rom zero to ten, twenty, .forty,

eighty, one hundred sixty, and three hundred twenty. It

is to be noted .from Table 3 that the increa::.:>e. in (I) j_s

very small as the value o.f the upper bound o.f the integr8l

increases .from f'orty to three hundred twenty. It shows

that the integral may be written

160 "' loO (I)= f'(o/..)doi..= r.f'(o<)d"' + .f(~)do(

oO • /o - " and 1 :f(o<) do( :Ln the integral :Ls ve~y small when N is

larger than .forty and it may be neglected.

To explain the use of' the graphs and equations o.f the \ .

solution, a :problem utilizing the rail previously treated

is considered (Figure 9)• It is assumed:

(1) A concentrated load of 40,000 lbs. traveling

with a velocity o.f 122 mph is applied.

(2) ~':2.=1,4.97 in? "

( 3) k =.k' xb = 230x6. 5 = 1, 500 lb/in?

(k' 'is evaluated .from Table 1.)

(4) vc• 1,220 mp~

::fl. ·-

::: 0 ·r-1 ..p

0 <1>

U2

,;

U2 C/l 0

s

Find

a. The maximum rail deflection.

b. The maximum bending moment.

From the given data it is.calculated that

. - v 122 n- v c = 12 20 = 0 •1

n 2= 0.01

30

Substituting n value into Equation 30, the deflection under

the load is

y _ 1.4l4x40 1000 . · - 4xl, 500x::Y ,, ~C?> '?.'xo. 99 5

::. 0. 252 in.

From Equation 38,

.M0 = ~x40,0~0x 11497

= 546~000 in-lbs.

By means of Figure 8 and Equation 41,

M~- 546,000x{1 + 0.0018x0.49)

so that

M~=- 551,000 in-lbs.

When the velocity is disregarded, n= o. Using n:: 0

in Equation. 30 and 41, the deflection is 0. 250 in.· 8l1d the

moment is 546,000 in-lbs •• Comparison of these V8lues

to the values for v= 122 mph. of y= 0.252 in. and

M== 551,000 in-lbs. shows ~hat the influence of velocity

is not large and may be neglected· in this case. I

From the above calculations it may be seen that ob~

taining the solutions by use o:f the graphs is s:L:.lpl e.

•

31

VI. CONCLUSION

From the theoretical point of view Equation 9 shows

that it may be possible to obtain-solutions for various

boundary conditions of not only infinite beams resting on

an elastic foundation subjected to dynamic load but also

finite beams. So -far as infinte beams resting on an elas­

tic foundation subjected to an uniformly distributed

moving load and concentrated moving load are concerned,

the particular solutions of Equation 9 presented in this

study show that in ordinary cases the influence of the

velocity of the loads on the deflection and moment of the

beam is small and may be neglected.

It should be pointed out, however, that the mass of

foundation will reduce the cri tic·al velocity, if it is

considered-in this study, and due to the_ uncertainty of

foundation pro:p.erties it is necessary to verify the solu-' tions in this paper by experiment. ·

. .

* Table 1. Monulus of Subgra~e Reaction

' Modulus.j_k'in lb/sq in./in.

lpo 1150 ! 2so 5b0 ~00

l -~ . -, I

Genera soil rating as subgrade, subbase or base

poor Fair to good Excellen1 Good ~ood Very poor sub grade 5ub,ro.de sub grade ~~~byo.cle subbase bo.~e

I

G-Gravel P-Poorly graded I GW · GC S-Sand L-Low to med. compressibility

GP M-"Mo'~Very fine sand, silt '

C-Clay ·H- High compressibility GF "f F-Fines; Material less I

sw than 0.1 rom sc

0-0rganic SP Vf-Vf ell . grade SF ~ CH ML

OH CL 1 OL

It MH

*Based on Casagrande's soil Classification. From S. P. Timoshenko ·"Theory of Plates and Shell" p-•• 259 1959.

8 00

~.,)t

b~

• ..

.

•

.... _ -- ..... - - --·. . ------ .... _ -·- ---- ~---- ... ___ . ._ -------- .......

Table 2. Proeram for Solution of Equation (40) *LIST PRINTER

33

*ALL STAT~E-M~E~N~T~M~A~P~----------~-------------------------------

C C***62975CSX02l S P CHANG 07/08/65 FORTRAN II ~---__ AP_pROXIMATE SOLUTION OF THE. MOMENT INTEGRATION

S=O.O PRINT 101

----~DO 5 K= l 10 B=10. S2=S**2

. ___ DQ ft_ __ J·_::=~l......t.S6!..._ ______________________ _ B4=B**4 A=B4/((B4+l.)*(B4-2.*(S2)*(B**2)+1.)) N-B Y=O X= 1.

_________ DO 1 I =2, N 2 X4=X**4 Y=Y+X4/((X4+l.)*(X4-2.*(S2)*(X**2)+1.0))

1 X-X+l. Z=O X=2.

--..-...-- DO _ 2 _I_= 3, N~..t.J;2;__ __________ -:------------X4=X**4 Z=Z+X4/((X4+1.0)*(X4-2.*(S2)*(X**2)+1.0))

----~2~X~=X~+~l~·~--~~~~~-----------------------------------SUM=(4.*Y+2.*Z+A)/3. PRINT 100,S 7 B,SUM

____ PUNCH 100 7 S, B, SUM 4 B=B*2• 5 S=S+0.1

101 FORMAT(3X 1 1HM 7 5X,1HB,lOX,3HSUM) . --~1-0~0 FORMAT(3X 1 F3.1 7 3X,FS.0,3X,El4.7)

CALL EXIT

END~------------------------------~----------~--

-

-- ------------=---.......----~-:-----

!·- . . .•. - • ··r··., . .. ,_ .. ,. ... ' -- ..... ····--·- · ..... - ... -~.-~. -- ----· ~· -------.--, - ....... ·- -- -. ........... -- ---- ----:-<-·•

34

·--~- .... -:----------------------· -·- ---. -· ···---·-

'lable ) • Output Data of Equation(40)

'' G o.o 10. o.o .26. o.o 40.

' o.o so. o.o 160.

--0.6 320. .1 10. • 1 20. ----.1 40. • 1 so. • 1 160. • 1 320. • 2 10. • 2 20. ----·-.2 40. • 2 so. • 2 160 • • 2 320 • • 3 10. • 3 20 • .3 40. .3 80. • 3 160 • • 3 320 • • 4 10 • • 4 20 • • 4 40 • • 4 so • • 4 160 • • 4 320 • • s 10 • • 5 20 • -- .

• 5 40 • • 5 80 • • 5 160 • • 5 '320 • • 6 10 • • 6 20 • • 6 40 • • 6 so • • 6 160 • --• 6 320 • • 7 0 10 • • 7 20 • ----• 7 40 • • 7 ao •

... - - --

SlJ :-~ 4. 79127 63 E-O 1

0

____ ----

4. S 2 46 2 2 6 E-0 1 4. S29 5 616E-O 1 4. S 302323E-O 1 4.a303166E-O 1 · 4.8303230E-Ol: 4.8308340E-01:

I

4.8641943E-01 i 4.8691343E-Ol i 4.8698046E-01: 4. 8698890 E-O 1 4.8698960E-Ol I 4.9540323E-O 1·

1 4.9874366E-Ol 4. 99 23783 E-O 1 : 4.9930490E-Ol :-4.9931333E-01 i 4.99 31400 E-O 1 I s • 11 sa a 2 3 E -a 1 . 5.2093600E-Ol 5. 2143046E-O 1 \ 5.2149753E-01: 5.2150600E-01 ~ 5.2150663E-Ol; 5.5271156E-Ol i 5.5606960E-01 i 5. 56 56450 E-O 1 I . 5. 5663156E-O 1 1

5. 5664003E-O 1 5.5664066E-Ol 6.0671540E-01 6 .lOOS 6S6E-O 1 I

6.105a230E-O 1 6. 10649 33 E-O 1 6. 10657 SO E-O 1 6. 106 5846E-O 1 6 .9173786E-O 1 6.9512513E...:01 6.95621S6E-O 1 6 .9568893E-O 1 ; 6.9 569740 E-O 1 6.9569a06E-01 8. 36 11090 E-O 1 i 0

8.3951S50E-01 !

S .400 1540E-O 1 8.4008253E-Ol

.. - --...--~ ...

--

.7 • 7

-~ • a .a • a • 8 .s • _fJ .9 .9 .9 .9 .9

--

160. 320 •

8·.4009096E-O 1 S.4009163E-Ol

~--10 • 1_2_4_2_2_2 2 E+_O 0_~ ~

1_0. 20 • 40.

1.1276529E+OO l.l281507E+OO

so • _:._ _ _,1::..:•:...1"""""2_i? 2 1 7 8 E +0 0,__ 160 • 320 •

1.1282261E+OO 1.12S2266E+OO

10 • _:::_:_ __ 1._9_7_4_~ 8 ~_2_E +.O o __ 20. 40 • so •

160. 320 •

1.9781420E+OO 1.9786401E+OO 1. 97 870 58 E+OO

.1.9787119E+OO 1. 9787120 E+OO

1.

2.

3.

35

BIBLIOGRAPHY

Bolotin, V •. v., The Dynamic Stability of Elastic Sys-. tern. Moscow USSR. Translated by '.'!eigp_rten,

V. I., Holden-Day Inc., 1964, p. 12)-126.

Carpenter, s .. T .. , Structural Mechanics. John Wiley & Sons, Inc., 1960, p.l2-15. ·

Churchill, R. V.,.Complex var.ibles and Annlications. McGraw-Hill Book Company,-In~.;~i960, p~ 162-167.

4. · ·Den Hartog, J. P., Advanced Strength of Matc,<.nl. McGraw-Hill Book Company, Inc., 1952, l'J· 141-142.

5.

6.

. 8.

9.

10 ·-

11.

12.

13.

Freudenthal, A; M. and Lorsch, H. G., The Infinite Elastic Beam on a Linear Viscoels.stic Founda-- · tion. ASCE Mech. Pro c. Vol. 83, No •. EM 1, Jan. 1957. · · ·

Hetenyi, M. Beams on Elastic Founda!;ion. University of Michigan Press, 1958,· . ·

Jahnke, E., Erode, F.· and Losch, F., Table of Higher Functions. McGraw-Hill Book Company, Inc., 1960, p. 96-97 •

Livesley, R. K., Some Notes on The ·Mathematical Theory . · · of a Loaded Elastic Plate Resting on An El2.S-

tic Foundation. Q. Jour. M~chanics and Applied Math. Vol. 6, 1953, p. 39-43.

Mansfield, E. H., The Bending and Stretching of Pls:tes. The Macmillan Company, 1964, p. 3-20.

Miller, K. s., Partial Differential Eq~ations in En­gineEring Problems. Prentice-Hall Book Comp8_ny 1953, p. 144-166.

. .

Reissner, E., On Bendin~ of Elastic Plates. Q. co~;~)l. Maths. V (194 7), p. 55 .•

Timoshenko, s. P., Strength of Rail. Transactions of t:1c Institute of Ways of Communication (St. Pc-ccr­sburg, 1915).

Ti.moshenko, s. P., Theory of Plates and Shells. I:'IcGrr:sl­. Hill. Book Company, Inc., 1959, p.259-2.-l.

• 36

14. .Winkler, E., Die Lehre von der Elastizist und Festie­keit (Praga Dominicus, 1::367), p. 182.

15. Westergaard, H. M., Mechanics of :Progressive Cr,clcing in Concrete .Pavements. Public Roads Vo:i.. 10, No. 4, .June 1929.

16. Westergaard, H. M., Theory of Elasticity and Pla~rci-· city. Harvard University :Press,. p. 2·~,1-235.

17. Younger, J. E., Advanced Dynamics. The Ronald :~ ress Company, 1958, p. 88-135, p. 2.1-235.

18. Zimmermann, H., Die Berechnung des Eisenbahnoberbanes. Berlin, 1888; 2nd ed., Berlin, 1930 .•

• 37

VITA

Sh~-peing Chang was born on June 26, 1937, in PW~ien,

China, the son o£ Mr. and Mrs. Shih-juh Chang. He received

his primary education ln FooChow, Fukien, China, secondc:.r:v

education in Taiwan, China, and was granted a :0. s. degree

in Civil Engineering £rom Taiwan Christian College in 1960.

After graduation, he served one and one-hal.f years as a

second lieutenant with the Chinese Air Force Engineering

Corps in Taiwan, China.

In October 1961 he worked as a junior engineer in the

Taiv1an Provincial. Water Conservancy Bureau until he C3Jne

to the United States.

In September 1964 he was enrolled as a graduate stu­

dent at the Uni veTsi ty o£ :Missouri at Rolla , Rolla .. ,

Missouri, majoring in Civil Engineering.

1.:15238