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Transactions of the Engineering Institute of Canada, 2, 3, pp. 123-130,
1958-11-01
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Deflections of plates on elastic foundation
Gold, L. W.; Black, L. D.; Trofimenkoff, F. N.; Matz, D.
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CANADA
DIVISION OF BUILDING RESEARCH
DEFLECTIONS O F PLATES ON
ELASTIC FOUNDATION
by
L. W. Gold, L. D. Black, F. Trofirnenkof and D. Matz
REPRINTED FROM
TRANSACTIONS OF THE ENGINEERING INSTITUTE OF CANADA VOL. 2, NO. 3, SEPTEMBER 1958, p. 123
RESEARCH PAPER NO.
71
OF
THE
out permission of the original pr~blisher. T h e Dicision woultl be gla
to be of c~ssistc~t~ce it1 obtaining such permission.
Money Order or (I cheqrle made pcrycrble at par i n Ottarba, to the
General of Cancrda, credit National Rcsec~rch Corrncil) to the
y z ; ,
.*\-
'as-,
-
. .s i, > ? - -*:; "< :.!&;**s$ C :Deflections of
Plates
on
Elastic Foundation
8,.
W-.
Gold*, L. D. Rlacli,"
17.Trofimenkof,t anil
D.
MatzJy
E.1 [J(JI r r u p t ~ l o / obse1~uci11o1r.s ice/(> ccz~rlerl oltl or1 llrc c/qflPclro~i of cr givon an :l~)l)roxilllatc. solution to thc plObl(~lll of thc half-
larqe p/n(r ~ I V / / I 11 C O I ~ C C I I ~ I ~ L ( R ( / loczd czppl~ed (11 tI~r C F I , ( I ~ ; qf tr 1,latc. Wilson cxt :11> rc1)ortrd no progress 111 finding :]I, c.snct
large ha([-plcilc, i ~ ~ l l ~ (1 ~ O I L C ~ I ~ ~ I ~ I P ~ /o(rO oppl~ed a / /lie jwe edge; solution to this prol~lcm. S h : ~ p ~ r o " ~)ublishcrl thc: ~ c s u l t s of crnd (1 l r r ~ g e seclot~czl plofe uith a cot~~e~~f/ulr~cl load uppl?ed (11 (he n solution In the form of integrals that are tlifficult to
lrp. [I'he resctlls of l h ~ s slr~dy 7cere used lo rleler11~7ne lt~e rrqiliclzlr, c\r:lluate b u t tlitl not givr thc c.onlplcte tlevelopnlcllt of the ~~~oclttltts o j I c e j ~ o r t r ob.setuuteo~~s 11lczc1c 711 (lie fic~ld. Conc/usions s01uti01l. L i v ( , ~ \ ~ y i rt:ltcs ill his conclusions t h a t tllc proll-
trrc, dln1~11 cotre*er~rr~~g ll~e ~elrrl~ur 1~ecr11ng slrc.nqtlt oC lhr trrrlrc- Icm of tlrc tlcflcbctio~l of tlw loatl(~tl cornor \\.itti frcc bou~lclaries
loc~tlrel cl~rel ctlqr-lorcclerl plnlc~.~. IS still to b r s o l ~ ~ c t l .
Solutiol~h to tll(.sc ~ ) ~ o l ) l t - ~ l l s ;ale of furt11c.r illtc~rcst as they
,<],;I<;s
()I),e rv(, tllnt I()atlb arc> allplte(l f u r : ~ r c rc l c ~ . f i ~ ~ t t o l~osril)l(t r i t u a t i o t l ~ 111ct in tllc ficltl. It isI
slic~rt timc to lalge icr plater, tllr r c s u l t ~ ~ i g cIeiorm:~t~otl norlllally assulnctl. \\it11 justific:itiot~, that a c i m k 111 t11c is ,)rirn:lrily rl:lhtie :Lll(l thr c[cficetiOn cnrv(, iq li 3r ico call tri~llsrnit slrcb:t~ but not l~e~ltlilig illomc~lt. This is: i l 'I'llc 1)o:lritlg str(11lgth of ~ :I pl:~tt. \vh~rli I)ch:~vc~s ill trur :IS lol~g :IS the crack l h (.loset1 but :IS soon a s :L crsck
t,~,is I)(, Llic.tc,d from a kllon lctlgc, of tll(, thiCkllCSs O I ) C ~ I S . the c ( l ~ ( , involv~(l nil1 I)cha\.c as a free boundary. I t
tllc, i,l:ltc, :Llld tll(, flPSUrnl riaitlit\. ,lltlll,:ltc, strcllgth is cl(~sira11Ir to knon- \\-li:it is t h c lo:~d that t h r icc shret (8:an ot tllc. 111:1t(~ri:tl fro111 \\-IIIc'II it is I~I:I(IC. For 111ost ~ll:~t('l.i:~ls. e ' L r r ~ l t l l ( l ~ ' r s " ~ "
tl~osc. 1)rol)ertics ?:in 1nc:isurc~cl 1)y n.cll-c.stnl1lishct1 testing Sitlrc, it is tliffieult to ol)t:lin :t thco~ctical solution to thcsc mc.tllotls,
A
I
of~
tllrsc, tllct~,oc~s n o t s:ltisfaetOr~-~
~
~
for ~)rol)l(~ms, it \\-as tl(,c.idctl to ~n:ikc mcnsutc~~ncnts o n ~ ~ l n t c sicci ll(lc:IUsC tllC \v:ly
it
l)el,:Lves Utl(~Cr loud tl(,l,etlds sucll 011 a11 (,lxstie f o u ~ l d : ~ t i o ~ ~ itt or(ler to tlcrivc. eml~irical solu-suetors s ~ ~ a l ~ c ~ :Llld 5iZls O f t(,,t I)i(.cC, trllll)(,ratllrC. ratc tions. This 11:~l)('r 1.q)orts t l ~ e s c studies : L I I ~ the a p l ~ l i c a t ~ o n
loncling, allcl tluration of loatl, F,Irt]lcrmorc~, i t is :llnlost of thc. rc~sults t o t l c t c m ~ i ~ ~ c the rigiclity nlodulus of icc 11lntes
illil)~)s'ii)j)~~ to rCj)r(Idllc(l :,rtifieinl\y tI,(. f:,(.torS tIl:,t m:,) from o l ~ s c ~ r ~ ~ : ~ t i c ~ n s mad(* ill thc' ficltl. il~fluc~nc~c the cI:~sti(' tno(luli :111(1 ulti~ll:~tc str(311gth of i(v> ,
1 1 t I I I . For thcsc r c ~ ~ s o ~ l s , tllr v:~lucs of thcscx I he or^
p r o l ) ~ ' ~ tics nlc~:~s~trotl in tho I:~bor:~tot>- or n.itli sm:~11 s:unl,l(,s
Tliv
t l ~ f l ( ~ . t l o n ot :I t h i ~ ~ , unifornl, ~sotl.opic pl:atc, floitting in th(\ fic,ltl. callnot 1x1 ithc~l \\-it11 conficlet~ec~ nt thts t1111c. it1,,,,
,,,t,, :,lltl sul,lcet to conccntr:ttctl loncl must s:Ltisfyl)(>:~ril~g str(,rigtll e:~lc~~I:ltiol~s. I t is (lcsir:~t)lc to hnvc~ somc tllc follon ing tlift.erc,nti:ll c,quatioll sul,Jc,ct to given ljolllltlar\- nlc%llotl of ~nc~:~sl~ring tl~oso ~ ~ r o r ~ ( ~ t ~ r s 111lc10r 10:tdillg ~011- (,ollditions ( ~ l n l O s ~ l C l l ~ o ~ ) :
ditiolls t l ~ t :t1)11roxinl:1tc I ~ I O T ( ~ elos~l> t l l o ~ e tllot in ~~r:ic.tic~c~.
0bscrv:~tions l l a ~ . ~ been ~lintlr on thc tlcflcction of vc%r> DV'/I,
+
pgrr =P
. . . (1)I:t~gc ice shrcxts unclrr tlistributccl loads : ~ ~ ~ p l ~ c ~ d :it :I I:~rge tlistancc frotn frcci l)ou~~tl:trics.' T l ~ c ol)ser\~c~cl cl[~flcctions llavc3 llccLll lound to :tglccS 11 cllI \\
platc on :ul cxlastic t'out~tl:~tiol~
\\'J I ~ I ~ I I I . Such tcbsts I I : I ~ ( ~ the (lis:~(l\.:~~lt:~ge t l ~ : ~ t , for ic(, of (~o-or(~i1l:lto~ t11c tlli(~L~l(~ss 11or111:ilI\. t t r ( ~ l ~ I I 1)1:1(tic(>, t11c 1o:lcIs reqttir(,(l
t o g~\rc. ~ ~ ~ c a ~ ~ ~ r o l , l c tlcHc~ctio~~s : ~ r c very I:rrgc~. F~~rtl~c~runorcs, :ultl
(
--- I :+
-'
-"
+
- --'7
'
in pu,:~r u)-or(li~,:,tes. ~t is cliffic~~lt to 1)I:ico :I lo:~tl O I I thcl iccx : ~ r l t l rci~~lovcx it cluirkly I- 81, 1-2 '01c , ~ ~ o u g l ~ so tllut ( * r ( ~ 1 ) tloc,s not c~ontributc. signific:l~ltly to
thc tlcflc,ctio~~ p = t l c t ~ s i t ~ . of n-atcr
g = : ~ c c c l c ~ : t t i o ~ ~ tluo to gr:~vity
Somctinlc,:, it is possibl(~ to c,arlj out lo:rtlil~g
tc,\t<
on largcx I , = :ll,l,lic.tl c,orlc.c.rltr:,tc,(~ ~ o : l r ~i ~ c s p1:ltc.s \\ it11 a t1c.c str:~igl~t (~lg(5 01 (111 V O I . ~ l:~rg(l s ~ e t o r ~ : ~ l = (lc,fl(,ctiOll tll(, r)l:ltc
ice, 1)1:1tc~s 17r~fortu~~atc,ly. to us(, sucll lo:~tl tcssts ~ I I (I(>t(*r- I ) = &/13/12(1 - a') IS thc r ~ g i c l ~ t ~ ~llo(lulus . .c2) I t i f l l i i i t or I 1 1 t i 1 t S ~ I I I ~ I I I = t t l i c ~ l l e s h o1 tilC I)l:Lt(,
ire. r ( s q u ~ ~ ( % > :I kl)o\\ lt~tlgc~ ot tlit. tlcfl(~c.tio~r. tluc. to :I lo:itl g , yollllg~, rrlotlulllh for tllc, ,,late t l l ~ ~ t ( , l l ~ I ~ : ~ l ) p l ~ c ~ l :tt tl~cb tlc,c% c~clgc,, 01 :I I~nlt 1)1:1tc. otl a11 cl:~sttc ioulltl:~- = ~ ~ O ~ ~ ~ o l l ~ s r:lti,) for tll(, l)l:ltc lll:lt(,ri~l~
ti011 :111tl tllc. tlrHc.ctiot~, tluc. to a loatl : ~ l ) l , l ~ c ~ l : ~ t thcb tip, of :I
t i I t I I t i I I I I I : I ~ ~ ~ I I . I2x:~c.t so l u t i o ~ ~ s t o T t ~ e s o l u t ~ o ~ l lor ~r ~ i v ( b r ~ by \ \ ' ~ I I I : I I I for :I cor~c(b~~tr:ited tt~c~sc. 1)1ol,l(~11ls : ( I ( ' [lot r c , : t t l i l > ol~t:iiuc~tl \\.(~\t(>rg:i:ir(l II:I< lo:~tl :tt tllc. cca~~t~cs ol :I v ( y I:~rg(t 1)1:1tcb is
'S,,,,,, .tnd I c r S c c i ~ o n , l ) t \ t o l o ~ r uf h ~ ~ l d l r r g ILeacarcIt, X:ltror~,rl Ile.c.ua11
c o u n t i l , O t t n \ \ : ~ (1% = - --
'
I
kci I / I . . . ( 3 )where I. is the radi:\l,digttlll~c fro111 the point
of the lo:ld
lssel function'.
Fig. 1-.ipparatus used in the tnodel stud).
A. Foam rubber aheet. B. I'las~er-of-paris block. C. Screw 11setl for applyiug loads. 1). Frarne of loading apparatus. E. D > ~ ~ a m o ~ l ~ e L c r . F. I)eflec~olneter.
Therefore. clrterii~iiiing the, shape of the deflected surfacc deflections, a value for the
I
of tlir 11l:~tc and rubber com- experimrntally, in terms of tlic dlmcnsionless parnnletcrs bination \\.as ol~tainctl.It
\ \ a s founcl t h a t the distmcc for any ollc value of 1, elctc~r~iiiiic~s it for :my raluc of I . from the point of :~l)plir:~tiori of tlic load to t h e ne:i~cst edge\\.ah nl\\:~gs grc.:~ter than 61. Earlier obscr\~ations I)y J3lack
Apparatus (ulipublislird) of tlic tlcflrctions of :i lucitc platc floating on
Thc :~ppuratus usctl 111 the ~notlrl stuclg l h slio\\ 11 ill Plg. 1. lucrc.uy hat1 h110\\ 11 th:it if t l i ~ dist:~1ic(: fro111 the 110int of
toall,
.I, 4kt.
scluarc nllcl 1 ill. thick, :ll)plicsatio~~ of t l l c ~ loud to t h e l ~ c n ~ e s t eclgc \ \ a s only abouteI,l~,edt~(,el ill a ol l,aris kllock, 13,
cast
directly onto
3.51. the. funct~o~l:ll tlrgc~lclcncc ot the obsc~vccl clcflections or, ~l~~ surf:Lcc of the sllcet \ras nl:Ltlc 011 th" r:~tliua \\:is -;till gircn v c ~ y closcly b). kcxi I / l up to to better tllaii 1/100 in. Loads ~ r c r c applied by thc' ' / I
= 2 .u:llly ol~c~ratcd scrc\\-, C, nlountccl on the frame. I).
namomctc,r ring, I<, \\.itll a lirlc,:lr dificrcntial trnnsiorillcr "late
e tlcflcction-scl~siiig elcliic~lt. \\.ah usc~l to mc:lsurr tlic, l-birlg thc. sllcctb lor \ \ l l i r . l l 1 li:~el 1)cc11 clctc~l~ri~iic~tl LJ
cs Boor, I\ :is usc~cl to ~iic~:lsul c. thci cleflcctions of thcl H(Y~IOLI e)k tlic s11r1ac1, \ \ l l ~ c . l l 1.(3>111tc~l \ \ I I O I I :l co111+1,11t1:ltrcI r 1
r ring anci deficctonicter \\ere recorded 011 a Ill? I(~1igt1i oi the eclge \r ;I< : ~ t Icitht 201. Tllc sl1ec.t \ \ i ~ s :[gain
-2, a-J r c ~ o r d c ~ . I t \\:li 1)ossil)lt. to nicnsure prrlo:~tl(~l \ \ ~ t l ~ :I ~1illlo1111 1:1y(~r o! I(X:IC~ .hot to the1
Edge-loaded I'late
In Fig. 3, tlir noriii:~liectl tlcflcction inc~asurcel along tliv ctlge ant1 pc~rpclirlieula1. to tllv cbtlgr for the half-plate :uc plotted ~~#:li~lst ~ / 1 .
It
is illt(~rc~sti,tilig to not(& th a t thc tle- flection along thc c~lgcl c:u~ 1Je fittetl vory csloscly by the kci functioii, tlie solitl lint> in Fig. 3. Pcl~l~cntlic.ular to thc edge. the dcflec.tion goc,s t o zcs~.o n t ,./I = 2.8s as coml)nrccl to1./1 = 3.!13 :1lo11g tllc ctlgc. 111
FI:
4. liiic~i ot c+onst:unt deflection :u (, ~)lottotl :igoillst I , / / .Fro111 li'ig. 4 it IS I)ossiblc to I ~ ~ ~ ~ : L > I I I . ~ ' L J , I . : I ~ ) ~ ~ I C : L ~ ~ ~ ~ I I C
1 ~ 0 l ~ l l l c Of 111l)b~l' clls1)l:rcetl b!' the' tl(qfl?~ic't\ I~I!lt('. 'I'llC'
:~ppliccl lu:~cl, Put ~ c c ~ u i r e d to girc uilit t l e f l r c t ~ o ~ ~ . i b rel:~ted
to tliia vollmlc' hy
deptll as for thc centrr-loadcd plate and the loarls al~plied
~ ~ c r t i e : ~ l I y tliroug11 a ($0' ronc. 0.0
8
0 2Sectorial Plate c
U
W
Sectorial plates of various :inglcs 11-crc prcparrtl by ninkiiig 0 4
two-sc~ctor cuts in tlic tlulz~l plates. Thc scctorial platcs n.crc8
"
prelonclcd as for thc ctlgc,- ancl cc~itrc-lo:lclcrl platcs. -1 con- "w 6ccntratcd load was applic,tl through a GO0 colic to a point
011 t8hc% bisector : ~ t L: tlistancc of (1/20)1 from the tip of the
sector and thc tleflc~ctioli of tlie s u r f : ~ ~ c 11-liic11 r t ~ s i ~ l t ~ t l , \\-:IS
~uc~asurctl. Tho two-scrtor cuts wrrc.
: ~ t
1c:lst 101 long. 1 0 Tllc nlcasurctl tlcflcctions of' tlic etlgr,-1o:~clcd and st~rtor1:11plates wcrc plotted :~gailist dist:~iirc from the poilit of :~pplication of the load aiid :I sillootll cur\-c was drawn
Fig. 2-Korr~~alizecl tleflectio~~ plottecl againsL r/1 fo
through tlicm. From this curve, tllc dcflcction a t tllc point centre-loatlecl plate.
of application of loatl n a s cstimntctl. 7'hc tlcflections norc
then iiorin:~lizcd to unity a t the point of load npplicntion. I I I I
I',, = pg1- l l . ( Y / L ) dil
,,
K
Thcs(> norn~alizcd obsc~rratiolis wcrr plottrd :lgninst r / l . 0 0 -
~vhcre d,l is thca clcn~cnt ot surl:~cc :u.c,a rlcflcctcd 113- a n ~llllou1lt 1 d r / l ) .
C:lrving the glapliical il~tc.gl:~tion uut tu tllc c.011tour given by t o ( r / l ) = 0 : ~ n d lrtting 1'1 Ilc thc 1o:ltl corrcspontl-
i11g to this volumP of rul)l)cr tlisl)l:~cetl girts:: :
PERPENDICULAR TO a-~-m:-.a-D-- -
I,(''. ""lt cil.llc~ctlol1 :rt tlw yullli t r i :r~,l~ilr:ltlull ol t l l c . lo:1cl. I->~lip tlic \*aluc for pg :untl
1
obtninctl ~ I . ~ I I I tlic ol~sc~r\.ntionson thc centre-lontlcd pl:~te, the ratio ot I-'1 /IJo \v:~s foulid
3
- THE F R E E EDGE
,/"
/"' -'
FREE EDGE (DEFLECTIONOHSEIZVATIOSS PROPORTIONAL T O Kel (;I )
Cen tre-loadetl I'la tc L - 1 4 4 '
s t': 2 5 2 "
. .:
101"111 Fig. 2 arc plotted the norni:~lizotl tlcflcctioiis for thc -
plate along wit11 the kei 1./1 curvc. T\vo facts were :~cc~eptctl
:IS indicating t h a t
t11~
luodcl n-:li; I-nlitl : ANGLE SECTORIAL P L A T E -1. -1 kci function coultl t ~ c founcl n11ic.h fitted t,:~ch sclt -
of ohsrrvatioi~s. I I
2. The values obt>~ilied for tlic tlivnl satisfirtl the csprcs- - r 3 4 5
sioii n-llicli can bc. derived from rqu:~tions (2) :lilt1 (4) : L
I:ig. :$-Jol.~ll;tlizecl cleflec~io~ls for the free edge autl
[ I / / ? = ( / I , / / , * ) : ' perpc~lclieul;~r LO the eclge f o ~ . ~ l l c eclge-loaclecl plate.
r
1:
c nornlnlizcd deflection of the ccntrc line for all sectorial formal nlctllod or by trial :lntl cwor, :inti
Obscrvations of this naturc wc,rc, carried o u t o
o ANGLE 6 0 ° 1; 2 . 4 0 " ANGLE 9 0 ° L= 2 . 4 0 "
LINES OF CONSTANT DEFLECTION FOR NORMALIZED DEFLECTION O F CENTRE LINE FOR SECTORIAL
SECTORIAL PLATES OF ANGLE 6 0 ' OR LESS PLATES OF ANGLE 9 0 ° OR LESS
Fig. 5-Uormalieetl tleflccliou4 f o r 4erLorial pla~es. 01) v a l l ( > ~ s o1ld ridges, tllc reiltrc-linc deflection coin- lvas cut in the iec,
il loatl al)l)llCd th(, free edg(, LLlld
with t h a t for tllc sllla11 angle sector rxccpt near the resulting surfacc shal)c lnr:lsurcd, -it
a later date a r h ( > normalized deflection along the linc perpendicular sectorial platr with an anglc of 00" n a s cut in thc icc and free edge of the half-plat(, is almost t h e sanlc as along thc tleflcc~tion observed. F o r rach of the cxpcrimcnts, the
clcflection \yas lllcasurcc~ b?, lITith all
level the of :1 sc:hl(, on a \voo,jen post
located a t a known clistancc from the point of application of the lontl. T h r load w:is placed on the ice only long enough to one lncasurclllcnt, tllcll lelllo\rcd nllilc tile level IITas sigllt(,tl 011 tll(, liest lllc:lsrllillg pobition, 'rllc lrvcl mas
loc~atecl on thc rock surrou~itlilig tlic, pontl.
Obscrvations ~ c r c :llso carrictl out : ~ t :L log tlulnp of the Canadian 1ntern:ltion:~l I'nl~cr Conip:~ii~r, nortli of hIani~vaki,
P.Q.
A
level w:ts pl:~ec>cl on the ice. a t n distrtllcc a t 1c:lst 41from thc point of application of the loads. With thc levcl, thc tlcflcction a t various tlistnliccs froni the point of y~plication of the lontls ws~s observed. F o r these obscrv:ltions, the loads ncLrcb tr l ~ c k s c:rrrying logs. 'I'llc C>an:ldiun 1ntcrii:~tion:tl 1':~p~'r C~II~I~):LI~!. very kindly ~ o - o ~ ) ~ ) r a t e ~ l LJ. Ilaving tllr Lrucks stop long cnougl~ for a re:hcling to bc t:tkcll. The
results of the obscrvntions in the colt1 roo111 :111(1 in the
rc g i w n in Tahlc 11. 111 Fig. 7 log I is plottrd a p i n s t
. ,A line of slopc 3/4 is tlrann through the obscrv:~tionp c to tempcsraturc, to rrrors ill mr:~suro~nc~nt of tllo thick- ss and. in thc (.:LSC of th(l oI)s(~r\~:ltions in thc~ cold room, to
2
e finite size of t h r tnnk. I t n:is oi)sc.r\-c~l tiuriilg the tests the cold room t h a t tho \v:ltcr lc~rcl cllangrtl (luring loading le t o the disl~lacctl ~ ~ a t c ~ r . I t was assmnc~tl t h a t the c>hangc
\\r:~tcr Ic~rcl coiitributctl :I constant a ~ n o u ~ i t to t l i ~ de- FIGURE 7
ction
st
each point. 'l'hc. tlcflcctions mcasurod ill the fic,ltlVig. 7-'l'he depelltlence of he characteris~ic length on ice Lhicliness.
Ta1)le 1. Nornlalized 1)eflcc~ions of Lhe Centre Line of the S~nall Angle Sec~orial P l a ~ e
on the continuous shect, c:dgc-loaded shcct ant1 scctorial
Sorm:~lizctl Sorn~alixed
/ 1 I>eflection f , 1 Deflection sheet \vcrc normalized :untl plottccl in Figs. 2, 3, and 5
11) 0 /4r1 rcspectivrly.
0 1.000 1.75 0.107 From rquations ('2) ant1 (4)
0.23 0.505 2.00 0.064 0.50 0.610 2.2.; 0.032 1 = cll
"
'
0.75 0.460 2.50 0.010 1 .OO 0.340 2.75 -0.001 1.23 0.240 3.00 -0.008. = [
"
1"
0.163 for B in psi. 1.50 3.25 -0.010 12(1 - a2)p . (9)Prom Fig. 7 . (. = 40.0. F r o ~ n clat:~ given by XssurlO a ~ i d
'I'ahle 11. ~ ~ l o t t c c l in Fig. 7. 1. = 3S.O. F o r \vatcLr, p = O.O:3Ol lh./in.j.
Average air Substituting this I-aluv for p in c~quntion (9) :uncl solving for
Iec 'I'en1per:~ture for 'I'ypc ol Ol~hcrvc~l 1 B
thicla~css previous 2 days Loading inches - = 1.11
x
10' (lb./in.')inches "C. (I - u 2 ) ' 1 -
0.25 LII cold roo111 8ector1r~1 Plate 13
<
'
15 P r o n ~ cyuatiol~ (2) :0.44
-no
c.
1 1 \ " I (i
0.30 21
D
= 0.92;ix
10"hi (Ib.-in.)0.7.; - 1 ( ' I J I I ~ I I I I I I ) L I ~ Pl:~tr 30
0.80 0 37 tpplicatio~l of L I I ~ \.iodel S ~ ~ ~ c l i e s Lo
2.88 - 7 < 8s Bearing CapaciLy Consitlera~ions
3.25 -,I 1 98
I':dge-lo:~clcd Plate 10 1 I n cartcsian co-orcli~iatos, th r bcnclillg nloments : ~ r c glren 3.38 - ,I
4.88 - 10 Srcxtorial PIatc, 146 by 'L'imosl~enko~~ : 5.25 - 10
',
I ' 13732.,i - 1 .?I ('e~~t~r-lo:~dcd Plate 340
( 10) Fig. 6-Li11e:s of e~r~st:tnI elefleclio~l for 90" seelorial plate.
FIGURE 6
LINES OF CONSTANT DEFLECTION FOR 9 0 ' S E C T Q U PLATF
For tlle ccntrc-loatlrd plate, t h r m:lsimu~n bellcling m o ~ n r n t occurs nt thc paint of :lpl)li~:ltio~l of t11r 10x1, ~ v h c r c
011sorv:ition~ 011 icr ~ I i c ~ t ~ :111(1 glass 11Ii~tcs ~ l ~ o \ \ t ~ c l t11t1t lor the 11all'-l~lntc., c.l,:lcking st:~rtc,(l : ~ t ur vcr\, 11(':1r tlie point of application of ttlr loatl. 1,rt tllv f r c b c cclgc lit, alo~lg the
.I. auis. If it i~ absn~rlcxtl t l ~ t t l ~ c rl(~flcctio~l of tlw frcc cdgc of
tlie 1)lutc. is prol)o~tio~lul to t l ~ c suu~c. fu~lction 5 s the
(I('-
flc~c~tioli of tllc, cc~~~trc-Io:ltled ~ ) l u t c , tllct~ for tllc. s:llrir (I(,- Ilcctio~l :it the 1)u111t ol :~l)plic:iilui~ ul t l ~ c luacl, i)2ic-/i).c2 tor the edge-loncletl plntr n ould be ccluul to d21c / a m 2 for the
0 . L -
From equations (12) and (13), tlie ratio of the maximum bending nlo~nent of tllc edge-loaded platc t o t h a t of the centre-loaded platr for t h e same deflrction a t the point of application of t h e load is
ilssunling Poisson's ratio for ice is 0.33
a = 0.67 . (14)
It \\-as obscrvcd cspcrin~cntall!- t h a t for thc same load the ratio of the deflection of the rdge-loaded platc t o tllr centre-loaded plate
: ~ t
the point of application of the load is 3.43.If
it is assumed t h a t thc bending momcilt for failure is the same for the edge-loaded plate as the centre-loadccl plate, tlleil the load required t o cause the first crack in thc cclge-loacled platc is 1/(3.43 X 0.67) = 0.43 of the load forthe centre-loadccl plate of the same thickness.
During small-scale tests on edge-loaded ice platcs in the cold room, i t \\-as observed t h a t tlie plate clcvelopccl raclial cracks before failing completely. This has been obscrvcd for the contiiluous sheet and described by Assur.10 UTlien com- plete failure did occur, tlle piece broken out of the edgc liad a sha11c si~llilai to t h a t silo\\ 11 for tlic 0 tlcflectioll contour
in Fig. 4
From the present cxperi1nc~nt:~l obser.i~ations, it is not possible t o estimate the brar~iig capacity of t h e large sectorial ice platc rrl:itive to t h a t for tllc continuous ice sheet. Assuming a?tc/dy? = 0, it is possible to calculate
bcnding nlon~ents fronl equation (8) and the calculated inonlrnts agree vrry \\ ell I\ it11 those obtained graphically.
Unfortunately, :~ccording to equation (8) the maxinlunl nlomc~lt occurs a t the tip, :~ncl it is kno\\-11 cxperimcntally t h a t scctorial plates under thcsc conditions of loading fail a t sorllc clista~icae from the tip ("
'
(discussion)).It
must be concl~~cled, thcreforc, t h a t a t least for snlall values of r/1, a2w/ayz docs not eclual zero. Accurate mrasurrinrnt of cleflectiorls witliin the tlistancc r/1 = 0 to 1.5 for the snlallangle sectorial plat(, is quite d~fficult.
It
is possiblc t h a t the lstnlit clcflectio~i she\\-11 in Fig 5 arc curved 111T o est:~blish a liypotlicsis c~o~irc~rniiig the relative bcari~ig capacity of the continuous cclge-loacled :tncl sectorial ice shcct, Black11 c~:irriecl out obsrrvations on glass plates on rubber. Ilis obscrvations support the ronrlusion clecluc~rtl from the model studies \\.it11 clural platcs of tlic rc1:ttivr l ~ c n r i ~ i g capacity of thc, ctlgr-loatled :mtl c~riitrc-loatled
CONCLUSIOS
Tlw invcstigat~on of t h r tlcflrctiun m i r l ~ r c~oiiccntratctl load of plates on elastic lounclations \\as ~uitlcrtaken t o clevc~lop possiblr mcthocls for measurilig tlir rigidit! motlulus :ind ultiniotct strcngtl~ of ice ill th(\ fic.1~1. It 11:~s :ilso rclcog- nizcd t h a t solutions to the problein5 uf :L c.onrc~ntratcd loatl on the free edgc of :t 1:~rgc half-plat(% or on tlic tip of :i Iargc,
scctol.ia1 platc 'ivoultl I)(. rclrvaiit t o cc.rt:ii~~ l)rol)l(~~n..: ~ i i ( > t
in tlic ficld
Fur prcloaclctl tlmal platcs 011 fo~inl ~ u l ) l ~ c ~ ~t \\:is Iounrl
[0.55 cos r / 1 4 - 0.277 sill r / 1 4 $
+
0.431 1lcc:~usc the values for tlir cli:rractcristic leiigth 1, obtained by :~pplying these results to field observations o n ice, agreed very closely .ivith the values given by .lssur10 : ~ n d exhibited tlic tlcpc~idrncc on icr thickness csprctrcl froin theory, it is concluclcd t h a t thc observed dependence of t h e ~lornlalized deflections on ~ / 1 for clural on foam rubbcr is also valid for ice.r 7
l h c drpcndcncc of the characteristic length o n thickness for ice \\-as founcl t o be 1 = 4Oh3I1 (in.). This gives for
tlic rigidity moclulus of ic?
1) = 0.923 X 10jh"ll1.-in.)
Fro111 tllc ol~ser\recl tlcl~cndcncc, of the deflection on mclius for the cclgc-loaclccl plate, i t is concluclecl t h a t the edge- loadcd platc can carry : ~ h o u t 43 per cent of t h e nlaxinlum load for the centre-lonclecl pl:lte.
~CIihOWLE~DGEAIENT
Tlic :l~lthol.a gl.ate1ull.i- ackno\\ lctlgc tlict technical nssis- ta11cc3 givc.11 by L. . J o ~ ~ c s in preparing the, :tpparatus and ~ i l a k i ~ i g tlic m c ~ : i ~ u r c ~ i ~ r ~ n t s . 'I'licy :~lso wish to rsprrss their :ipprcci:ition to Dr. -1. hssur for his helpful c o n l ~ ~ ~ c i ~ t s and
p:irticularly for pointing out : ~ n error in the t e s t \\l~ich has since 1)cc.n corrected.
1. "Aircraft Operations 011 I'lotlting Ice Sheets" by S. R. Stearns, Proc. Amel. Soc. Civil Eng., Jour. of the .4ir Trans. Div., vol. 83, AT1, 1957.
2. "Uher clas Gleichge~richt sclln-in~mcnclcr clastischer Plat- ten" by
R.
Hertz, lViec1emnn11's Annalen cler Physilc ur~d Chemic, vol. 22, 1884.3. "Deflections of an Infinite Sheet" by M. Wym:~n, Can. Jozir. IZes., A, vol. 28, 1950, pp. 293-302.
4. "Sew Formulas for Stressc:s in Concrete Pk~vemcnts of Airfields" by H. M. TVestergaartl, 'fl/.rtns. Amer. Soc. Civil Eng., vol 113, 1948, pp. 125-444.
3. "Coupling 13etn.een Moving Loads and Flesu~.:tl \Vaves in Floating Ice Sheets" by ,I. 7:. \Vilso~~, et al. Snow, Ice and Permafrost Res. Est., Corps of TCng., G.S. Army, SIPRE Report 34, 1955.
6. "Deflections of a Semi-infinite Plate on an Elastic Founda- tion" by G. S. Shapiro, Institut mekhaniki Altademii S a u k SSSR Prik1adn:~ia ~nathenlatil~a i ~ n e k l ~ a ~ ~ i l t n , 1913, 'I'o~n 1'11, Snow, Ice and Permafrost Res. Est., Corps of Eng., U.S. Army, SIPRE Trans. 48, 1955.
7. "Some Sotes on the Mathematical 'l'lieory of a Loaded Elastic Plate Resting on an Elastic Foundation" by R. K.
I,ivesley, Q~icirt. J O I I ~ . illech. cind i l p p . ~llrtlh., 6, pt. 1, 1953, pp. 32-44.
8. "Hessel Functions for Engineers" by S. TT'. McLachlan, Oxford University Press, 1946.
9. "Theory of Plates and Shells" hy S. T i ~ n o s h e ~ ~ k o , McGra\rr- Hill, S e w York, 1940.
10. "Airfields 011 F l o a t i ~ ~ g Ice" hy A. dssur, St~oir., Ice and Permafrost Res. Est., C:orps of E I I ~ . , 17.S. 4rnly, SIPRE Iieport 36, 1956.
11. '"The Itelativc: Stic!ngtlls of Ccr~trcb-loatlctl, JGclge-loacled
: L I I ~ Sectoriill P1:~tcs CJII :111 Elfistic F U I I I ~ ~ ~ L ~ ~ U I I as Deter-
111i11cd by Model St~~clies" by I,. I). Black, 'f'~~tr~tsrccliorrs, F:.I.C., vol. 2, no. 3, 1958, p. 120.
r uutlrors havc ~ n s d c a n important colrtribution to exist- kiion.leclgc of the nction of plates on elastic foundations, cci:llly as pertaining to scmi-infinite and sectorial plates. results of thcsc nlodcl tests which provide a solution t o thc cdgc a ~ i d point load prol~leins will be apprcciatctl by thoso n-it11 a n intrrrst iir riigincering mechanics.
l'lic tlicor!, is 11 cll kliolvn for the centre-loaded infinite plat(% (I-Icrtz,' \T*j.~~iari,~ \T?estcrgaarcl I)':, but complete
succcssl'ul :~~i:~ljrtic:~l solutioiis to the cclgc-loatled half plate, :inti the point-loaclrd srctorial platc, arc not uvailal~le ( S l ~ a p i r o , ~ i v e s l c ~ ~ ~ ) : " . Thcreforc, tliis empirical solutio~i rcprcsents a \velcomc acl\~ancc toward complctc solution.
The tj.pc of icc plate loading simulatccl by tlirsc motlcls is oftcrr c~ic~ouiitcrcd in practicc for cracks ant1 leads are c~onimo~rplace rather t h a ~ i unusual. I t is truc, a s tlie autliors stat(,, tli:it 1:lrgc~ tost loatls are required to deflect ice sheets thirk c~iior~gl~ to support the Ileal-jr aircraft and vchic~lcs ~ron. rising sucli ict> airfields and roads. I t is also truc t h a t licltl tc,sting of tliin, ~iitrrginal ice sheets, although possible is clangerous and difFici~lt. Iloncvcr, such tcsti~ig, using .mallcr more convenient loads, must sooner or latrr provide the nc,ccssary information to gap the void I)etn.cc~i snfca :~ntl tillsafe loadilig on thin ice plntc~s.
.\~rothcr n.ay to obtain this knonlcdgc 1s t o usc ~iiotl(~l tcsts in the lal~orntory. Then tlie d a ~ i g c r and iriconvenic~~rcc
\I ould be c1imiii:~tcd. The authors have clioseii t h ~ s iilctliocl
of att:rck on tlic problem, ancl liavr I~rcli able to show t h a t tlir modcl represents the prototype by clc~~eloping :L tleflect~on
curvc lor tlic ccntre-londccl infinite platc \ ~ l i i c h satisfies tlic theorrtical equations dcvelopcd hy Ilcrtz :inti others. Tliis i i i ~ l ~ o r t n ~ i t corrclatio~l is provrd by inatcliiiig the 13essel kri tuiictio~i t o tlre ~~ormnlizcd cmpirrcnl curvc, and by sliowiuq t h a t tlic clcflcctioii of the model plate 011 a rubbcr foi~ntlatioi~
-xtisfic,s tl~cl l>asic theoretical relationslrip, lahl' I.
l<asc.tl oil these results, aiid also on tlic corrclatio~i \\lit11 1ciion.n tcsts of ice supportccl l ~ y \v:~tcr, one must concluclc t h a t tlir authors have developed nn apparatus ancl pro- ccclurc which simulates tlie field loaclirlg, foundation support, ancl bc~iiding conditions. From usc of this :~ppnratus ant1 procctlurc they have obtainecl valuablc cnll~irical results for tlie cases of cclge and sector poilit loncling \vliich call also 1)c fittcd vcry closely t o the kei function.
I t is oi interest to note t h a t LL rubl>cr f o ~ l ~ r c I : ~ t ~ o ~ i sinlillat-
i ~ r g a n elastic subgrndc has brrn usrcl successfully 1)y YIellingcr, 13chrmani1, and Carltoii to stuclj- concrc,tc pave- ments on elastic fo~nclations.'~'
'''
"i
The authors havc beell careful t o cliiniiiatc~ edge cft'ects us niuch as possible by using a platc largo enough to l~ro.iriclc
:i radius of 61 or grcatcr. Also, tlrrj hal-c climinatrcl time
cft'rct. or creep, I)y :L 1':lst application of load n liich not onlj sh:~rpc~rs tlic foc'i~s 011 elastic l)ropcrtics, but :~lso providrs
info1.1ilation for the clynainic cusc
I t is i1lrport:~nt to rcrnembcr t h a t s t i ~ t i ( Io:~(Ii~rg :LII(I
resulting creel), nlthough l)c!.ond the lailgc of tlrrsc invc\ti-
gatioris, arc, air c>clunlly i ~ i i p o r t a ~ l t casr. Tlic1 rrrajority of critical aircraft loadings will bc of tlir static type aiid occur during parking, loacling, or \\:~riii-up 01 c~ngi11c.s Thcrc>forc, this 1)rol)lcm illi~st bc kept ill 11ri11d
"'l'rolesaul- oI ( l i v i l 1111~il1r.!l.ilr~, l'ia:~yet. S c l r u u l I > [ L < ~ i ~ i l l ~ . e l ' i l l ~ I.):~rt loout11 Cullege. H:lnov?r, 5 e \ \ , Iiitli~pshire.
Nu~rlbers refer 10 biblioyrnyhy in paper by Gold a l ~ d o ~ h e r s . (p.123-128)
; ;\ddilioit:~I I . c ~ < ! ~ ~ > L I I . I ~ J listed a t t , l ~ e end 111 this ~liacl~nsioti.
st:rncIing of tlic i~iflucncing factors, a ~ i d validates \\.ell ~vitli :L l'(w. fiill-scalc tcsts, a truc p i r t u ~ c of
c : ~ ~ i I)e obt:~ir~ctl.
111 this study, slrlall-scale tcsts \\csrc carrietl o u t u
forcigli 1iiatcri:ils. I t is suggcstccl t l ~ : ~ t sucli 1113opcrties of d11r:~l : ~ n d r u b l ~ e r as tlrnsitp, Young's ~iiotlulus, ant1 Poisson's ratio 1)c 01)t:rinctl directly for coiiipnrisoii \\-it11 thc valucs thcsc propcrtics ol)tainecl from tllc plate clcflcctions.
111 siiiiulat~ncz tlie action of :L plat? 011
the :iutliors n-(,re ca~c~ful to ~ i i : i i ~ i t : ~ ~ ~ i (-0i1ti1iu0iis ro ~ i t a p1~~10:1(111i,g t l ~ r (Iural sllcet ~ v ~ t l r shot Tlic c0rrcI:~ti t\vc~rr tliv resulting tlrflcctloii c11rvc :illel the t h c rtirvr tcst~lic.; to tlre succoss of this procedure. I1
tliis writer is (~oi~rer~iocl a11o~1t tllc o
o l ~ t a i ~ i s froni tlrc L I ~ C ot a senli-i~ifin~tc
floatiiig icr s l i ( ~ t n ~ t l r :I load :~pplicd to its frre eclgc
-
1)c~c~oiiir progicssivc~ljr subliiergctl along this edge si~icc \\:itrr 11:rs no slrc~ir htr~ngtli. 'rlic lontlccl cclgcl 01 tho cl
rc~sist:riic~r t o deflrctroii I~c~yoirtl tlir rclgc. f ~ o m t h e rubber lounrl:~tioii :t \voulcl I)(, ot i~itcrcst to
of tliis lark of subiiic~rgc~~rcr : ~ i i ( 1 tlie extra rcsisti~nc
Tlic iiloclel tcsts o ~ i srctorial plates provide us ~ v i t h son1 ~iiost intrrcsti~i:, tl(~flcctioi1 plots. The lines of constan deflccstioii for sectorial : ~ ~ i g l r s of 60' or less arc: straigh :~ncl poil~ciidicular to tlrv centre, linc indicating thc action o :i tri2~ngul:rr. cnrrtilelrcr I)ralli ratlrcr tlia~l :L plate Tlii
1.c1sult corrc~latcxs \I rll n itli Mr. l3lac1cJs conclusion bs~setl o his trsts of gl:~ss sartorial plates
,
t1i:~t tlic strength of t h srctorial platc a t angles less than 60" is dirt,ctly proport1011 to th(> angle. This noi~lcl I>(> truc for 11 t ~ i z i ~ ~ g i i l a r cantilelI~caiil \I liorc. tlir secttion ~noclulus is proportional t o t h e \vid :uid thus :~lso to tlrc angle.
Tlic locatioii nii(l tlirc>rtion of the cracks i l l tlic gl
sectorial plates causc this writcr soiilc, colicern. Always cutting glass, ~ ~ i i t l c~specially in ol)taining slrnrp corners, must b r cstrn c~arrful to void sinall (.racks a n d s t co11ccntr:itions. I t appears likely t h a t minutc cracks on edges of the glass plates usrtl voii!tl prcclctc~rrniiir locsation of the failuic fracturrs. -1 vcry special treatmcii
tl~csc edges, both rluriiig cutting u~itl for stress relief, appears necessary.
r .
Ilic ovrr-:ill co~rclus~on rc~:lcllccl 1)y tlic tlisrussor is tha tlrcsc testi alicl thc, reports tlcscribing t l r r n ~ forin a valuabl s~cldition to t l i ~ sciclitific kno~vlcdgr ill this fic.1~1, c~spcciall- In tlie sc.nrch for :I soliltion to tlic, c.clgc-loaded plate. TI1
:luthors arc, t o bc corrgr:~trrlntotl for li~ic pcrformnnrcs a n rcl)orts
; i t l t l i t i o ~ ~ ; i l Bibliography
12. MEI,L~X(;>:~I, I?. &I. l ~ t 1 ( 1 Cz\~t~,,ros, P. I:., "Applicntio~i iVodels to Illcsig~r Stutlics of Conclete Airfield Pavement Proc., Ilighn-ay Itcsc:~l.cll Board, vol. 3-1, 1955, p. 57-
13. CAI~LTOX, P. F. ant1 B~r~nnr.ki~s, Itutli M., "A h l
Study of ltigitl Pavemelit Ueh:iviouu under Corner I':tlge Loadil~gs", Proc., I-Iigl~~vuy ltescarch Bou~cl, vol. 1956, 11. 139-1-16.
1 4 . CARLTOY, 1'. 1'. i l l l ~ l ~ ~ E ~ I I ~ \ I A S S , It
of Prestressccl lligitl Pnve~nelits Rc~scarrli 13onrtl, Rullrti~i 179, 195
rlcflcctions :~ritl this n a y ucc~cll~tcrl 34 c.vidcr1c.c that thc 72 11) /111.~ rffect of tlie $heal s t l e s ~ c s a t the ctlgc could be nrglected. With regard t o the co~ilrnents on Mr. 131stck7s paper con-
beginning of these cupc~rirnents. 'rlic edges of some nedgc.s wert> c:~rcfully polishccl aiid strcss laisers were artificially
Z'roperties o f the rlurnl lnduccd in others by deliberate nicking anrl nlarring of the
Specinlens n erc cut kroiii thr dural sliect having various cdgcs No clifft~rt11ict~ in strcngth or nlotle of fracture was orientations with rcspcct t o th(> clilrctio~i of rollirig and tlic dctcctetl I t is possible, honever, t h a t such ail effect, Young's n~odulus determined usin, :L tcstiiig machine. No :~lthough 5111311, :LCCOUII~S for tllc fact t h a t the strength
depelidclicc of Young's inoclulus on oricntatioii n.as observed. index for the complete plate lies above t h a t predicted by extrapolation from the d a t a for smaller angle sectorial
Young's modulus. 1.03 X 10' psi
. . .
Assuincd Poisson's ratio. 0.33 in. I n coriclusiori the tiutliors \\rould like t o express their
Plate thickness. . . 0.062 ill. a,ppreciation t o Professor Stearns for his comments and for Calculated value of 1. . . 1 . 4 I tlie interest \vhicli he tlisplayccl in the observations.
A list of all publications o f tlze Dioision of Btrilcling Research is ncailahle (2nd may be obtnirzsd frorrz the Pzrblicntior~ Section, Dioision o f Builtling Research, National Research Council, Ottrr~tin, Cnnclda.
