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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