Beams on a two-dimensional pasternak base subjected to loads that cause lift-off

BEAMS ON A TWO-DIMENSIONAL SUBJECTED TO LOADS THAT

PASTERNAK BASE CAUSE LIFT-OFF

ARSOLD D. KERR1 and DOUGLAS 1%‘. COFFINS +Departmcnt

.\ktract-Fir’;t.

of Civil Engineering and ZDepartment of hlcchamcal University of Dclawarc. Nw;Lrk. DE IV716 U.S.A.

Engineering.

1~0 clowl~ rclatcd prohlcms. shoun in Fig. I. published by Chcrnigovskuya

In Ivslrdwun~~uperDm~nrik~~ Soord~mii

i Rawhem

(1961.

Konsrrukr.sii

ncr Cprugom O.mownii (Edited Moscow) and Ting (1973. J. Frunklin Insf. 296(Z).

by B. G. Korenev. pp. Il3-IJI. Gosstroiizdat. 77-89) are discussed. It is shown that neither of their formulations is correct. The aim of this paper is to show how to correctly formulntc and solve problems of this type. Utilizing the variational appro~lch for \;trl,lblc matching points dcribcd by Kerr ( 1976. Irrr. J. S~liclr Sw~c.~wc~.v12. I I I ). ;L formulation for the problem ;1naly7cd hy Tine is prcscnlcd. that is mcchanlcally rcwonahle and mathematically ucll posed. The ;unalytwul solutwn ohtaincd is cvaluarcd numerically and then compared with rclalcd test results by Durclli (‘I trl. (IVW. ./. Sfnrcr. Dr. :tS<‘E 95. 1713-1725). This paper concludes with ;I discuGm of the results ohtaincd.

The analytical aspects of unhondcti contact problems for continuously

supported structures

have hccn discussed by Kerr

( 1976. 1979). when the b;~sc rcsponsc is rcprcscntcd by ;I I’astcrn;lk ti~undation model. For prohlcms of this type the contact region is not known (I /~iori. C’onscqucntly. the matching conditions at the point 01’ separation must include ;III adtlition;il cqu;Ition li)r the clctcrniination ot’ this unknown. As an cuniplc, Kerr ( 1976) considcrcd ;I linitc beam resting on ;I two-dimensional Pastcrnak foundation and subjcctcd to vsrtical loads such that lil’t-oil’ 01’ the beam is possible, ;IS shown in Fig. I. Ikvcloping variational c;~lculus for varinblc matching points. hc sho\vcd that this problem was incorrectly formulated by Chcrnigovskaya

the GISC when q(.~) = const and i’ = 0, she heuristic;illy conditions at the separation point. .V = I: -E/w”(l)

= --q(L-lyp

- Eh”‘(I)

= q( L - 1)

h(l)

- Gw”(l)

prcscribcd

1

= 0

(1961). Solving

the following

matching

(1)

whcrc 1tj.v) is the vertical dctlcction. 11.’= d~/d.\-, G is the paramctcr of the shear layer. k is the paramctcr of the spring Inycr. q is the uniform

load. If is the length ofcontact of base

and beam and 2L is the Icngth of the beam. For the GISC of ;I platr strip bending with II’ = W(S). El is rcplaccd by D = E/I’/[ I?( I -IO’)]. For the two-dimensional

Pastcrnak

foundation

in cylindrical

model the contact prcssurc is (Kerr,

1964) p(s)

= h(.r)

- Gd’(s).

(3

Thus. the third condition in (I) prcscribcs that the prcssurc is zero at the point of separation. This is not correct. The proper condition. which results from the vari;ltional formulation, is that at s = I the slope of the shear layer is continuous. Thcrcforc. the contact pressure at the separation point may have :I non-zero value. Kerr (1979) pointed out that because of the reduced order of the differential equation that governs the Pasternak foundation response (as compared to the elastic continuum) only one of the two anticipated conditions 413

A. D. KERR and D. W. COFFIN

Fig.

I. Bum problem under consideration.

may be satislicd. For the problem under consideration hc rctaincd.

and not the 0x0-prcssurc

the tangency

condition

is the one to

listed in (I).

condition

A similar prohlcm \V;IS also incorrectly formulated Iatcr by Ting (1972). t Ic solvctl the pmhlcm shown in Fig.

I for the cast

of q(.v) = 0, by prescribing

Ihc following conditions

aL .Y = I:

,I$/) = 0 w”(l) = 0 w”‘(I) = 0 This

formtkltion

incorrectly

stipulates

Whcrcas this is true for the Winklcr model. As ;I wnsqtluu

of the first

1 .

(3)

that the dcllcction at the separation point is xro.

found:ltion,

it does not apply to the Pastcrnak base

two conditions

in (3). Ting’s

formulation

implies ;I

zero contact prcssurc at the scpatxtion point .Y = 1. This

is not correct for fhc reason given

in discussing the Chcrnigovskaya

major error in Ting’s

is his omission of the dilfcrcntial In rhc following The closed-form

formulation.

the correct formulation

solulion

Another

formulaCon

ccluation for the base layer beyond the point of separation. of Ting’s

problem is skltcd and then solved.

obtained is then numerically

eval~~;~tu.J and fhc rcsulls arc com-

pared bvith those of the photoclnstic results rcportcd by Durclli

I-OKMULATION

1’1u/. (1969).

OF PROBLEM

The problem IO bc anulyzcd is shown in Fig. I with q(x) = 0. It consists of an elastic bc;lm resting on ;I two-dimcnsiorxll Pastcrnak foundation that is subjcctcd at the ccntcr to ;I vertical conccntratcd load. I’. According to the derivation by Kerr (1976). utilizing symwstry. assuming the validity of beam bending theory, and denoting Hi, as the vertical dckction

of the barn axis in 0 c .Y < 1. Nap

I < .Y < L. and II.,

as the vertical

and G = const. the following

dcllcction

as the vertical dcflcction of the beam mis of the shear layer in I < s c

z,

for EI

in

= const

difTcrcntial equations rcsull:

EIw’~ -Gw'; +hc~, = 0

my =o

0 < s < I

I
Gw:'-X-w, = 0

I < .r <

(3) 00

The corresponding boundary and matching conditions are :

P

II

-J.

5

416

A. D. KERRand 0. W.

d-1 j = 0; .d7 =

.-I

(,

=

COFFIN

0

-.- --,‘---...Xf(p-+ri$C/’

(1’) !pcos(#jt)sinh(k.l)--ssin(,~(fcosh(k.l)t

‘_I, = __._--~;p--_‘Ef(p- f/P)‘JI

i&p

cos (ply cash (elf-(x2-p“)

sin ($1 sinh (~2)

H.hCK

The corresponding the prcssurc distribution

csprcssions

Ibr the bcncling nwmcnts.

-(K’-p2)sin

and

bcani shearing Ibrccs and

in the contact rcgiori arc :

(p.u)sinh (KY)-7~pcos

(p.~)cosh (KY)

(lb)

Beamson tuo-dlmensionai p(x)

Pasternak base

417

= kw,(.Y)-GG,v;(.v)

PG

=-

1 -

z, sinh’ (~f)-_=~ sin’ (y/)-i-:! (~~~.)I

i + ([

4,‘liEr

+

sin’ (p/l+

i:,

-II sinh’ (K/)+ z__

Ih-pp

h.z+-;T

-“$

2

sin (pr) sinh (X.X)

1

cos (p-u) cash (KX)

sin (p.r) cash (KX)-

:

)

cos (us) sinh (KY)

.

(17)

where

Next. the case G > Z-is

analyzed. The solutions of the three differential equations

in (4) are

whcrc

The integration constants 11,. . , . . II,,, dctcnuincd are

B,

=

__~_._

_.

I’

____

-

2E//l’(a- -/F)(J B s=O;

sinh (II) sinh (/l/)-a’] (21)

= ,E,,~r~f, .__iT__

&

P = _ _,-,_ .i_.._ ({I’cosh XIp-I-r#.J

B ,” =o.

cash (\U)-r/l

B,,=O

B,

B,

where

[/I? cash Irl)

[/I sinh (zl)-

z sinh (/U)j

(I/)-r’cosh

cash (xl)-z’cosh

A. D. KERR and D. W. COFFIN

418

# = flcosh (xf)sinh

and the condition for the determination

r/l[x sinh (fll)-p

The corresponding distribution

w\-) =

(xl)

cash (/U)--/I’

cash (xl)] = 0.

(23)

bending moments. shearin g forces in the beam and contact pressure

are

-P ‘($

z sinh (xx)-Psinh

_p’,

I

+[/I’

+

(/Iv)

([I/)--r/i

!~(I[b’cosh (d)cosh

9

L{[(p/r)’ -

sinh (xl) sinh (/il)-/1’]cosh

cash (xl) cash (/I!)-

l][r’

r/l sinh (xl) sinh (PI)--r”]

(/I!)-$sinh

+/l[/~2cosh (rl)cosh

4

cash (/#)sinh

ofl is

sinh (zl)]+/l[r’

+ - ([r’ cash (rl)cosh #

f

f/Q)--z

(r/)sinh

(/I/)--sc/isinh

cash (xl) cash (/l/)--r/j

([M--/l’]

(r/)sinh

(xv)

cash ([Lv))

(73)

sinh (xv)

(/il)-u~Jsillh(ll.r)j

sinh (zl) sinh (/U)-P’]

(25)

cash (xv)

(xl) sinh (li()-5’jcos11(I~x))

.

(26)

I

This completes the solution to the prohlcm under consideration.

The case G = I,/‘kK/

is not of interest for this study. Nest. the solution obtained is compnrcd with cspcrimcntal results.

COMI’ARhTlVE

STt’D-t

The only results located related to the problem under consideration are those of photoelastic tests conducted by Durelli c*f a/. (1969), which were used by Ting (1973). Durelli PI (11.tested beams ofdifTerent lengths resting on an elastic foundation and subjected

Bums

Fig. 2. Test

on tao-dimensional

set-up b) Durclli CI ul. (IY69)

hard transparent plastic (CR-39)

ruhhcr (HYSDL

load.

(I

419

base

in = 2.54 cm)

7 It consisted of a beam set-up is shown in Fig. _.

to various loadings. Their experimental madcofa

Pwtemak

resting on ;I slabofsoft

transparent polyurcthanc

44X5) of the same width as the beam. The beam was sub.jccted to ;I vertical

f’ = I40 N (3 I .J6 lb). placed

in the center.

The elastic propcrtics for thcsc materials are

CR-39

r1 = 2.22 x IO” N cm

’ (2.X x IO” psi)

I’ = 0.4’ IIYSOL-44X5

E=

W.ONcm

‘(525psi)

(‘7)

\’ = 0.47

Lvhcrc E is Young’s modulus and v is Poisson’s ratio. The bending moments M(X) were calculated directly from the rccordcd fringe numbers. Then, noting that

the shearing forces in the beam I’(.r) and contact prcssurc distribution by successive numerical diKcrenti;ltion Durelli

of M(s)

Ed trl. prcssntcd the bending moment.

tributions in the non-dimcnsionnl

normalized

ill*(r)

beam shear and contact prcssurc dis-

form

= 4i.M(.r),‘P

c’*(s) = 2C’(s),‘P /I*

p(s) wcrc clctcrmined

with respect to s.

= Zp(.r),!PL

(28) 1

where i. = ,‘/ki4EI. They determined the k value needed by using the expression derived by Biot (1937). who matched the results for an infinite beam attached to a two-dimensional elastic continuum. According to their calculations. k = 27.58 N cm ’ (40 lb in _‘) and i. = 0. IO83 cm ’ (0.275 in ‘). This non-dimcnsionalization

was found necessary by Durclli EI al. since they compared

the photoclastic results to a Winkler

model. Because i. is a parameter that appears in the

A. D. KERR and D. W. CLIFFIS

IO

5 x

tiiJ

15

b-4

J

---_ I

15

x b-4

Winklcr foundarion analysis and not in the one which utilizes ;I Pastcrnak hasc. as considered in the prcscnt paper. the non-dimcnsionolized graphs wcrc converted back to their original form. The results are shown in Figs 3, 4 and 5, as dashed lines. In a subsequent discussion Pandit (1970) pointed out that the contact prcssurc distribution, as determined by Dureili PI nl. (shown in Fig. 5). contains the inaccuracies inhcrcnt in the process of double dilferentiation

of experimental results. and that this is

&ams on two-dimensional

Pasternak

421

base

most likel) to be the cause of the sharp increase of the contact pressure distribution the load. This

possibility

was acknowledged and accepted by Durelli

LVhen considering the contact pressure distribution vertical equilibrium

under

et uf. (1970).

in Fig. 5 it should be noted that

must be satisfied. Thus,

(29)

must hold true. IVhereas in the test P 2 = 70 N (31.46 lb). the integration of the area enclosed by the experimental contact pressure results in 92 N ; this is 3 I Y6 larger than P 2. Another

problem ivith

the experimental

pressure distribution

is that the point of

separation (beyond which the contact pressures are zero) corresponds beam. IS.24 cm (6”). This

to the end of the

contradicts their statement that when subjected to P = 140 N.

the beam separated from the foundation

at both ends. Furthermore.

moment diagram shown in Fig. 3. the point of lift-off from the center of the beam. Therefore.

according to the test

appears to be at .Y = I I.7 cm (4.5”)

the contact pressure distribution

calculated from

the bending moments recorded appears to be inaccurate near the load and should not exist past s = I I .2 cm.

In order to compnrc the analytic results based on the Pasternak prcscntcd previously

with the photoclastic

results. the I’astcrnak

must hc dctcrmincd tirst. The methods for dctcrmining rcccntly discussed by Kerr

(1085). The criterion

f~~undation model

foundation

ths foundation

parameters

paramctcrs wcrc

suggcstcd for the determination

of thcsc

paramctcrs is that the analytical results agree as closely IIS possihlc with the actual situation. for the range of loads under consideration.

It was utilized by Kncifati

(198.5).

‘I‘hcrcli~rc. in the follo\ving the two I’astcrnak paramctcrs arc dctcrmincd by collocating the test results with the corresponding points to hc usctl is t’qt1:11 ti\o thta

analytical cxprsssions.

The number of collocation

to the number of unknown foundation

paramctcrs. Thcrcfore.

points arc ncctlctl. f:or the problem unclcr consideration

the separation length I is

obtained from the test. fsccause of the small size of the test beam, it is reasonable to assume th:lt the clrect of the weight ol’thc beam is ncgligiblc on the test response. Thisjustilies the USCofthc presented analysts in the following compar;ltivc study. Since the foundation paramctcrs arc not known (I priori. the analytical results for both cases G < 2,/1?Ei conhirlcrcd.

and G > 2,/kEI

From the test curves in Figs 3,4 and 5. the moment distribution it ~V;ISobtained directly from the test data. Thercforc,

must be

is most accurate. since

it will be used for collocation purposes.

From this curve two data points arc chosen. They arc : ilt

.v, = Ocm

M, = 272 N cm (24.1 Ibin)

at

.vl = II.2cm(4.4in)

iz/? = Olbin

(30)

Substitution ofthssc values into the moment expressions given in (15) and (24). noting that according to f:icv C’ 3 the separation distance I = s1 = I I.2 cm. results in two non-linear equations for the determination of k and G for each of the two casts G 5 2Jkl. Using the fMSL

routine NEQNF

the foundation

and a Fortran

program for the problem under consideration

parameters are found to satisfy G > 2&c. X-= 9.65Ncm~‘(l4psi).

They are :

G = 1864.5N(4IC)Ib).

(31)

It is of interc>J_to note that this solution program when used on the two non-lincarequations for G > 2,/kEI did not converge. It is also noteworthy that I = .v? = I I.2 cm obtained from the test curkc in Fig. 3 and the above two parameters k and G do satisfy eqn (23); the analytical condition for the determination of 1.

A. 0.

P__

KERR and D. W. COFFIN

The bending moments. shear forces and contact pressure distribution cally determined using eyns (ZJt(26).

are then numeri-

They are plotted as solid lines in Figs 2.1rtnd

5.

According to Fig. 3 the photoelastic results and the analytical results agree very closrly. throughout.

for the beam bendine moments. although only the two extreme points (.Y = 0

and .r = I) \s‘ercmatched for the determination

of k and G. The diqreement

of the I’(.\-)

cur\es und the pf.u) curves seem to bc caused mainly by the inaccuracies introduced by the numerical JiITerentiations

of the .\I-test curves. as discussed previously. However. it should

he noted that the slope dis~ontinlIit~

at the concentrated load Pof the ~~n~tI~tic~lf~-obt~lin~d

p(.v) curve (Fig. 5) is another basic shortcoming+

of the Pasternsk

model and is caused by

the term Gu,” in eqn (2). It is reasoniible to expect that in an actual situation pressure ~ti~tribLiti~~n I’(.\‘) will have a horizontal

the wnlxt

tangent at P; the point of s~nirnetr~.

It mily be shown that the vertical equilibrium

equation

is satisfied for the ;~nalytical {t(s) expression. iis anticipated.

Utilizing

the vari;ltional

:I f~~rt~i~~l;~ti~~~~ for thf rcasonahlc :lIld

i\ppro;tch for variable matching points dcrivcd by Kerr ( 1976).

problem under ~~~(l~i~~~r~lti(~n was prcscntud that is me~h;~ni~;illy

\VCll p0SCd.ThC ;tiialylicul solution

lll;lltl~~~~;~lically

~lhlili!lCd

W;IS

numcric;~lly itnd then comp;trctl with rclatcd photocl;tstic test results by Durclli It is rtotcwrwthy

that

mined hy colloc;ltin~ qywnrcnt

~~lt~~~~l~~l~the two

beam-shrtriring force distrihulion It

~~~ul~~~;iti~~~~pnrumctcrs were dctcr-

I’astcrmk

two cxtrcmc points on the test cur\c ft)r bending moments.

is very close tltrouphout

the lo;ttl I”).

this curve. 7‘11~agrccmcnt is only satisfactory

I ‘( v} met the conlact-prcssurc

~iis~ribuli~~n p (cxccpl

is pointed out lhal this tlisagrccmcnl seems to hc

inaccuracies ;tss&ttcr!

with thu numcrial

dill&unti;rtion

p~~sc~~t~tl

when formltl:~ting

solution of the uorrcct formulation probfcrms of structures

t In addition to the occurrcncc bnsc, as discussod by Kerr ( I’M-iL

nf concentrated

CilllSlXl

the

for the fiwr

mainly by the

01’the M-test curve.

The disa~ssion of the incorrect analyses by Chcrnigovskuyrt atld the

CV~llll~llCd

CI d. ( IW~).

(1961) and Ting

(1973).

show that caution has to bc exercised

which rest on ;L Pasternak-type found:ilion.

reactions at the fret beam

endsthat

do

not scparnlefrom