Bazant, Z.P. (1965). "Electric analogues for creep of concrete structures." Proc., Cant. on Exper. Methods of Investigating Stress and Strain in Structures, held at Techn. Univ. Prague, Building Res. Inst., 207-218.

P r o c e edi n g s

of the

Conference on Experimental Methods of Investigating

Stress and Strain in Structures

Praha, October 5 • 8, 1 q b 5 International Hotel

Reprint

Edited by the

Building Research Institute of the TechniCal University

in Prague,

Praha b Dejvice, Solinova 7,Czechoslovakia

August lqb5

207

B I!Il ! ant Zdenmc P., IDg., CSc.

Building Research Institute of the Technical University in Prague

IStavebm \ts�v C!eekQ\o 'VYsokQ\o �en! techni.�o v Prase/

.Ilectric �logues_!� Creep of Concrete Structur!!

/Elekt:ri� 8IDI!Ilogie pro dot.varovin:! betonovYch konstrukc:!/

Jsou moW dva �y analogie, v nich� k na�t:Ca a rychl.o­

stem deformace odpQv!daj! elektrick4 na��! a proud /AI ne­bo �cenl lSI. Model rdzn;fch dkond dotvarovmu betonu.Jro­

del pro hAem n sta.ticky neurl!itYch velil!in nehomogenm koJ:to strukce je tvoren obvodem /AI 0 n navzajem propojenlch am;yl!­

!mch a D zcfrojich napi!t.:!. �esp. IBI 0 n uzlovych p4i-ech a n zdroj:tch proudu, pH�eU jedltot.liri prvky obvodll jsou

tvofeoy eleaaen� obvody pro dotvarovdn!. Obvod sa slcl'­� ·se samoindukc! a odpoI'd IA/, resp. z kapecit a odport1 /8/, jet jsou obec� �asoyE! proml!�.

Modern concret.e structures, such ae concrete bridge.s cast by the c8lltileYel' method, concrete suspension beams, etc. , are fw the most part. clear� nonhomogeneous because of co­operation between concrete and ateeI, comerate parts of dif­fereDt ages , of various thicknesses, etc. An analysis of the effects of creep leads in such cases to relatively involved �t.e.s of ordinary differential equatioos or, more generally yet, t.o Volterra 's int.egal equatioos· which must be solnet on digital or analague comp�ters. Although their electric mo­delling is essentially known lsee [2] -by conversion to one differentia1 equation of higher orderl, one can arriYe at. an analogue circuit far more simply in a direct way, namely � finding an analogy for the l.aw of creep of the material and

an electric model corresponding to the given statically }n­

determinate structure. The present paper will first indicate

208

that Dischinger-Whitney s or more accurate�, Aru�an-lla­slov's laws of concrete creep can be �e8Cribed by 4imple rhe­ologic models consisting of springs and dashpots with time variable constants,i.e. by Maxwell'S or Boltzmann's models. Thell8:f"ter it will determine the electric circuit aualogous to the mechanical model aDd hence also to the law of creep, which can be built up of resistances, self-inductions and ca­paci ties. Finally it will show how to set up an analogue cir­cuit far solving the statically indet�rminate quantities of the entire structure by coupling together- such elementary circuits.

1. ADalosue. for creep of time invariable I118terial.. The li­near creep of I118terials with time in'l8riabl.e parameters, e.g.

viscoelasticity of plastics at constant temperature, can be represented by mechanical rheologic models consisting of spriags and !dashpots. As it is well known [41 • [5] • their

electric analogue can be realized by the following four types: IA! Ui= adf)df , Ii'"' b6( • R= a../bl ' L= aJbt lSI Ui .. b 6"i , �. a.dEi/df , 11= b"{/a • C= a..Jbfi tel Ui= a. fj , Ii= b G'i • 11= a. JbE • C= b'7. fa Ie/ Ui" 'b6'i • Ii" a. £, , R= 1P f Ja.. , L;:: 'b1()a. where ( fir f, denote the stress and defoct'lDation in a spring­or daebpot, E - the spring constant I �::. E E. i I, 1'[ -the viscosity of the dashpotl 6,- = 71 dE-ddt I, Ui-the voltage and Ii-tqe current in the correspoming electric element, R-the ohaic resistance I U..:. =- R Ii I, C-the capacity 1 Ii = C df{/dt /. L-the self-induction 1 (/i::: L dI,:Jdt I, a, b-the selectable parameters: of the analogue, t-the time.

2. ADal.oBBe for creep of time variable material.The creep of concrete is complicated by its dependence on coacrete age. Its representation can also be effected - 8S wil1 be shawn 1&­ter

_on - qy rheologic models assembled of springs and dash­

pot4, the parameters E, and "1.: of which are, however, ti­me Variabl.e. For a spring Qf such a model. it no l.oager holds that �:, E(t) E..: but rather that drl,.Jtit=E(ij't;.jJt, and siaUar-

209 13' for a dashpot (= 1[(i) df,/dt • Since i.t holds for a time va­riable resistance that Vi :: P.{ij Ii rather than dU.ltI'=Il.{fJj{: �_ loguea leI and IDI can evidently be not applied to creep of conerete. On the other haDd. cOlIIParing the equations for VB­riablle self-induction Vi = L(f} '1,-I,[t and -capacity Ii '" crt} d{fjrit with those far a spring, we note that'analoguea tAl and I§l aZ'e applicable to materia1s with tillle variable para!!et.ers and can represent the creep of concrete as an aging aterial..

SiDe. in a mechanical model stresses are additive in the paralle1 and deformations in the series., coupling, whereas in an electric circuit currents are additive in the parallel and voltages in the series coup:l.ing, series coupling corresponds to aeries, and parallel coupling to parallel in analogue IAI land /e/I, and parallel coupling to series coupling and vice versa in anal.ogue IB/ 18Jld /DI I.

Far variable parameters analogues A and B can be formulated ,yet .,re general�. Let us require that real 1;iIIIe i corres­ponds to time J, in the electric model, the correspondence '" = �(t) being aDiY arbitrary contiuuous increasing function with fiDite limit for t -1' GO i thus infinite age of concrete is associated with finite time in the model. FUrthermore, in aua10gues A and B aDiY arbitrary continuous function a.ft) Ey be chosen in lieu of cODBtant a ; in this way one of the paramet.ers Il(t} , L(t) , or ({t)c�become" conatant. It can be proved that the ana10gies are then given by relations lA' I U.{.IJ z 41t) «a,lt} 1.(.1) = 1,6,· (t) BtI)-..ili)...:. T/JI) = 1(t} .!ftL

_, 1: i , 1: � , • "[It} , ...... t 1:IE(t)· fa I U�):: 1, et1t} , Iit6}=tllt} �}, B6'-)= � , qJ).�"""fJ ...!J!L .t 4U) ---zr- "'EIf} . J. Creep law of concrete and its model. To obtain general relations for stress 6 and strain (: in uniaxial. state of stress' at l.inear creep' of concrete ('which we can assume for stresses 10wer than about 0.5 of strength. except for the case of a l.arge decrease in stre.as/, we can use the principle of superpositi�n, and by integration by parts derive Volterra's integral equation [l] • [6] , [9] as follows:

/1/ _ !fl!l _ It !.. (_1 17 £(t) - Nt) 6('[') �d E(r; + C(t,T'lj d-r' t.

210 wbere Eft} is the tiJlle Yariable modulus of elasticity of' con­U.U. t - the age of concrete, C(t/C')- the creep :f'unctien ex­preae:iDg creep strain at t:iJae t pa-oduced by th

e action of con­staat unit stress applied in time r • to - the instant of' tbe application of the first stress.

l&! The II08tsimpl.e and f'requently used, yet onlJr roughly ClescriptiYe expression for crt /t) is Diseb.i.Ager-Wbitpe;y' s l!!! [31 • [8J t £9]"

C(t,-[) '" tt(fj - rtr:'} /2/ Eo where f{t} is the creep f'ac.tor fan increasing continuous f'unct­ion of' ODe variable with f'inite limit tf(OD} I. Introducing /2/ in /1/ and differentiating with respect to f' ' we obtain di�ereatial e quation

'#E 1 'db' 6' /3/ - :o - -r-1ft ' E(t) 1f Eo with

the initial condition of' if '" E E. for t=to. The equation of' creep of' so-cal1.ed' llEawell' s lIOdel /Fig.la/ is as follows: dE. 1 di1 G /3a/ dt = Eft} 7t + 1( On cODIjiISrixig 131 and /3aI it becomes evident that the crHp of CODCl'ete according t.o law /'2./ can be represente d by JIax­welI.1 s mood /Pig.1a/ in wIlicht the

spring constant. is equal. t.o [(t) am dashpot viscosit.y 7l(tJ= d t7d.t • On introduciDg the­se expressions in /AI/'and IS II we 06tai� an electric analogue of this creep law. In analogue /A' / it is circuit Aa according to

Fig.2 f'or which it hoMs that /48/ L("") = d�j.} a.{t) /).[�) or dftt} a(t/ dt b£(t} , dt 'bED

:ru analogue IB' / it is circuit Sa according to Fig.2 where /4b/ ((..,J.; = '-Nt) aM R(I-} _ b Eo dt 'bEft} I tt.(t; fir/dt 'If we select;. parameter a. in the form of aft}:: «" • we con-• tT... b . . dff/dt Yemen � 0 tS1n reSlstance 12 constant in both analogues.Since

for coucrete we can approximately consider £ =: Ctn1�t., we ar:rive at co�stant self'-incluction and capacity if we choose -}(t):oIof'ffJf�. Hence t.ime invariable models are appr�imately satisfactory for tbis

creep law. (Constant self-induction or capacity would

211

be attained for wriabl.e £ by choosing �:;; Iii dy • This �an­

not be done, however, if' the structure is nonhomogeneous hE'cau­se,-J. would be different for different parts of the struct-are/'. . . by Arut.YG-� Amore exact 1.&w of concrete creep 1S g1ven man:=JlasIov s function [1] ,[6] ,(9J _ (i-Tn 15/ C(i,1'j-= 1f(r')(1-eT / where it is to advantage to choose p{'C) in the form of '1'= AD + A1(r-( (9). On introducing eq .. 151 to III we can.

that the same inteural appears in the first and the S8-prove

,0-. , • •

COM derivatives of eq./ll with respect to t • Ell.llU.nat1ng 1t

f'ro� the two' equations we get the foll.awing differential equa-

tion d'-£ _ dt(i' + ft+ £ -.1. d£) dff /6/ f �i:2. + ErE. - de· {{'I r Er tit dt with the initial conditions £" f.. "" 6' anci £ dF. 116' + r£,tI for -t =--to dl: =. til: • �ied by eq./ll and its fir�t derivat1ve.

A diffe rential. equation of ao-call.ed Boltzmann 5 lor stan­

dardl �el according t.o Fig.lb with time variabie paramet.ers

E � is obtained f'r0fll the def'ormation equations of' t V. l drf (dE. dEy) do" _ E dEv

the model eI.ements it "'" E at - dt / I dol; - Y dt , 6'- 6'y = ?[ �? by el.iminating Ey and 0; in.

the f'orm of' [9] : U d dE) dl d'l.F.. E(. t!:!J} 4.E _ !!;. + .i.1£+f + � _l. - .£ ./6a/ £ dt1- + ?{{f"yf tit! 'ilf - dt� ?[ (I: r dt Ji' dt dt with the initial conditions E �! '" �:+f6' and E£. '" 6' for t = tD•

U' (i) _1_ £ If;! = .L - 11ff) , equations 161 and 16al '{ .. TIf'(t}' Y 'J rp(t! t . . ' .th their initial condit�ons are xdent1cal. Hence Bolt� s Wl. I . model represents the creep law according to eq.15 • On wtro-

ducing E , Ey and?[ to /A' / or /B' / we obtain as a model of.

this creep law in analogues A' or B 'circuits Ab and Bb,respect1-

vely according to Fig.2 for which it hold2. that alt) • d1Yt} art) d.tJ>(t) � rr ,t} R(�}= -rrft) l7at L (.J.) == tit "b E(t} I L /�) '" di: b rlf'{tj+d"(t}/di ( l? or (t) iNi) a.{t} .r r2.(t} RrJ.} = _b __

/7b/ c(y.} G ��tj p4E(t) , (r(� '" at b ,r(f:}+tiJfffj//f I edt! T'-f'(t)

212 A. choice of eualo.gue parameter a(t} = t.lD/,,{tj reaw.ts in couetant reaiataace R. • SiDee we CaR approxiJlatel¥ coueide:r £=eoDBt., we arrive at couetant L IX" C by chooeiDg rtf) >= �. Ef'lt) tit ,_ i'or wariable £ we- could a1so erri ve at a couetant L or C by choos� lit) == 71o"jErtlt • This is, howeVer, not possible ill the case of DODilCllOPDeoua stru�tUres because � thus Obtai­ned would be different for difi'erent perts of the etructure). lleDce constant L" aDd C" . are not aChievahle' for this law. ltl BoltZlllarm' s model /Fig.lb/ and circuits All and Bb in Fig.2 al.so represent a lIIOil"e Pnera1 creep law, Dalllely ((fIr) '" .., 9(r)[Ir.{t}-I..('r)j [9). Barpr's rheologic lIlOdel accordiJrg to l'ig.le IIOdel.led � circuits A.c and Be describes the case of Con­crete still better. It is arrived at trom the following law oZ CClDDZ'ef.e: C(t,'() = 1ft) -j(Tj + !{r'J[It{tJ- It (r'Jj. file corresponding di:f'1'erential equation is also of the .. cond crGJer but cOllp8red with eqa./61 and: /68/, it contains � ,too. J& 'lhe el.ec:t.ri.c analogue s outlined in the fcresoing could be evolved more rapidXy by crdculating IA'I voltaae or /8'1 c:url'eal; response in tiDle I: per unit step /A' I of c:arrent or IS' I voltap in tiale -# in circuits Aa and Ba, or aIternate-:q 4b aDI2 Bb. respectivel,J, et.c. accordi� to Fig.2. Siuce the time wariation of the r�e of this voltase or current respoDse is in the form of :f'unction /2/ or 15/, etc" the respectiVe cir­cuits l"'8prese nt the creep according to la_l2/ and /5/, ete:. t. Models of �!J); of statically indete�te structures. 4.1 Let us :first determiDe an electric circuit Ja:ldelling the relatioa between 1Od._ lW aDd def01'l8atiao 'r(f} produced by it in the directiClll of foree % applied to a 8tatically in­determinate CODCrete structure. So far as coacrete is conc:� ed, ODe can 'Wery well accept the a8'sWllption that the functions o!' CODI1:rete creep ill various parts of the structure are affine [9J .[10] .[11] ,. i.e. that the creep function in all parts of the IIItructure � be written ill the fora o!' :Ie C (f, 1:') wIlere c( t (1:') is the chosen basic creep :function for- the entire atl"Uctare, and 1e is the coe t':ficient of creep at:finity of the 1'e8pe�iYe part. /e.g. � a 0 for steel, ,,= 1 for a h�PDeCIWI

213 structure). Defor.stitm tJ;(f, t'} produced in u.e t by eaa.­sturt. IDad I {r:} applied in time '[' is given by the principle of virtual. work in the f"0l'/& G' 6'

rt -/ 16'tI f.-'- + 1t C(f 't/ldJl =j-dY+C(t,7J �6'�uIY /8/ t; (i ,1'/ = l E{'t) , J r E('(j y

1iIae.re 6' 1111 th: t� invariable stress pl'oduced by couet� load 1 ('t) , 6' - the , stress due to virtual load X = 1 111 the X--direetion, Y - the volullle of the �ttlre. As the

last -tara of expresei tm /8/ immediately iDdicate.. loDg-tillla CCIIIPOnmt of d'x(t /r) is proportiODal. to i'unc:tion crt IT} las­am:ing creep afiinityt.

Dds is the reason why the elactric aaal.ogue for relati on between Ix aJJd i {fj wder an arbitn:P,Y load is apin giTell b.J circuits Aa aDd Sa. aDd Ab aDd Bb, reepeetiftly-, etc. ill 1'ig.2 for the Nepective laws of creep. !be pal"BIIIeters ef the­se circuits are asc:ertaiMd: in a silllilar wa:,y t:raa eq./81 tin­terchaJrgi.Dg 1'unc:tiODe E(f}, r(� or '1ft) , etc. in eqa./4a.bI .. na,bI with t:uDct1ons CorreSPOtIlJing to fIq./8/ ).

.!.ti lbe deformatiCID of a ataticaI1J indetermiDate �� ._oaa etructUl"e is bOUDd by the coDditioaa of compat1bilit;r requiriDg zero re'sulta."t deformation. af the equilibr� basic 1IIJat.ea statically determinate in the sense of the Yar1.OIU sta­tically inde.termiaats quantities X,; Ii. I, 2, .. .n/ proc!uced by ezt.eraal load 1 (t) aDd by various statically iD:ieterai­nats quantities

SiJlce in accordaDce with para. 4.1. these deformations are spin expresllled: at coastant atres8 � equations in the fGll'lll at /8/ - assumiltg creep affinity - circuits in the fora � Aa. aDi Be ar aI.teraate� .&b and Bb denoted a. Ii Z 101" I ik.. I wboae �ters are dete rmiDec! bola an equation in the fora at /8/ describing the respec tift atns8 aDd % �orDIStion. again i�et the relation for defarmation in the X -direction va. ezterual.

'load � li:jor vs. statically iDdetermiDate quantities

X • i'be electric .ode! of a 8tatical� indeterminate atruct-� is then. obta ined: b.J iDtercoDlleeting circuits / -(,;l. / aad /-ik.. / li,k. - 1 . ... aI'aocardiJlll,y 1Idth th.ecoDtitioas tsL eqq!_ libriua and compatibility.

214 In cir cui ts thas iz:terconne�d ,Kircl:lh.of:f' ' s . laws are valid irrespe<:;tive of the ' t l ' L,. ��er.na.

an-angeme·nt of the circuits (i.e. ""t-.etr:!'J" we consi1er Cl.r�lllt Aa or Ba, etc.) S.imilarl;y 8Sin �tructlll"e� conditions of eq�librium and cODipstibility apply �;respectlve Oi', t�e deformation law �f the I118tet'iaI. it is,

-enf'orp., suffICIent t to set up a Cl.rcuit·Jiiodelli 'th '. .. • ' . ng e so-.L.utlon of the statically indeterminate qUantities of an el� $tic structure consi sting only of elements of'one k1tid l1.eo c� �e � t'-inductions in analogue A �or of' capacities in aualogtie B !� 1nterchanging' these elements with elelllentary circuits / 1 Z / and I -, k. I of the form of As or Ab,- etc. we get a cir­cu.it lllodellitig the solution of the statl:cally' indeterminate :tl.3ntities 0: ,a stru�ture undergoing creep. Alter:aatively. us_ I�� only resl:"st.:mces,; we can first set upa circuitlllOd 1"" h - . + , . .. ' . e _l.Dg � e 801�c1.on of an elastic structure in analdgues C or D, and ln�erchllllging resistances with self-inductions or capacities. w,: obt�lin a IlOOdel of t.he elastic structure in analogues. A.'or

'

B •

• 1\8. th� definition _ of the analogues clearly indicates, sts­tl�a:ly lndeteI"lnillBt.e quantities are modelled 'in analogue' A.' /SlmlIarly 115 inC/ by currerit, arid in analogU.e a'/Or D/ by vOltage;::tU:rthermore, that the COOditions of equilibriUm "cor-respond in an.9logue A'to the first KirChhoff's 1 / ' '- .' . , ' . aw zero sum ofcllrrents meeting at a' node/, whi''te in analogUe a " : to the se�olld �r�hho.ff' slaw Izero sum of voltages in a closed loopl. 1'h: coruhtlons are a priori satis'f:ied [J 1 if wesolire the cir­Clil. t on the basis of loop currents in an!itogcie A. .'and on the oasis of voltilgesin nodal pail's-'in'analogue B:r.�The c6odit_ i o�s �f

. the structure cOmPa-e:fbthty' ih'-stailrltHy:

ind'eteFznina� te connections then'c:orrestiond tdC'tnesecond" 1G!�dhtiO:t't"'.5 law in �n:a16igueA'lOr:c/;'and"t6-t11e' f:t�Sf'KirC1fum�s�::r�'.' ' .. _ '1 .. - ... ,.; . l,n s na ogue &1 C I or '0/.

'T!1e:p!"'obl"em ·<:rf theC'it<cu.lt st�uctifi.e'fiasth'd$ be'c" ' ;-"d - . d' '.' ..•. , . '" . " . - . c> ' -. . ' . . . '. en re uce to Ul�t: of finding'g cirCuit lliod.eningth'g"':s81athmor'� sy-stemaf '-iI lih-ear nlge5ra-{c equa'tions Yihli·i"s'.YMMet":H·' �t-;," rn anal'o�-e'A?orC/; the circtf:tt:lilUkt:",·tolrtain':n ' lc:ro�swi;�X'

n elements, 0:1' wtlich each two looPsniastgen�r'ally have a

215

�el.8laentand e�. tWQ '�-P&irIl JllU;8t generaUybe in­tercontJected tlli.-ougtr ;orieel�#t,.'Each :lOop IIIlst colit .. i� a

SOUl'ce of vOl.tagemocte��iqg')�¥ .·det'ormation·or 'the �f3ic sy�

st_ in the X ...dtrecti�n;:tWoouc-ed by ¢ernaJ. l.oad li.e. by �hecurr$Jlt inthlf:r�ective l.oop if' the remainiDgl:OO� US UDder no current '- op.encireuit" siIid.l.arly in anal;c;gue athi. loadiug, t .. is ateack IiDIiiI.al 'pairaadelled: by the source of

CUl"1"eDt land the loa'd corresponds to.voltageittheremainiDg

IIOd,al pair. are .withoutvoltage -snort «::ircuit/. nm- deformat­

ion of the basic system produced by . load {= 1 in the -XJc -di­rection, i.e. the i'Ie:noility coefficient;' corresponds in ana­logue A.'to vOl.tage. on the terminals of l.oop iiDiuced. by unit.

CUJI'l"ent in l.oop:" Usll the remaiDiogloop. are without S01.irCe lopen circuit!, and in amlogue B' to - current in nodal pair iwithout voltage Ishort circuitl 'iDdu�ed by unit source

voltage in nodal pair k if al.l the reJilaining nodal pairs are

wi thoUt voltage· IshOrt cireuit/. In this'way -weeotain in,rmlogli.e A�as _daIs of 2-t.iJaes,

j-tilles, 4-times staticallYiDdete1"lllinat.e struct"il1"es, circuits

A2, Al,; A,C-c8ccO.rdingtoPig.)/the;- are so-calJ.ed N-poles/,

aDd· in aualogae B' generally' for n'-tiDles atatical.ly iDdeter­

DliDate Structure, circuit ·Btl 8ccordiDg to Fig.)d. Elementary

circuitaAa and A.b, and Ba 'and' Sb, respectively, accord:iDg to

fig.2 atlOuld be vieuallud illS the various eIeaente 'of these circuit.s. 1'hecoeffi'cient.'of the elelllents' 01' "this circUit can

be dedileed :fro. eq,i/8/wi�bOU:t the necessity of settizig up a

s;)l'steaof differential equations, ch8Il8ing it to a sing1e e­quatiollof the first order ·8rld.earehiDg for its arm1.ogile.­

/Note: whendl'awiJig the . c'ireui t . 'in tbe analogue.' th� gl'oups' 'cd' three self-inductions in T-conne·etion oe.curriIig therein. can be

replaced equivalently by two parallel self-inductions with q� mon self-iDduction, i.e. by, transformersl.

roillustr�t� � diSCl,1s�iqn,weshal+-write a sys"t em of

equations.eti�uirig from the,solut},6n Of: 8 ,circuit for the spe..;

cial. case. 'of ,�ri e18s:\::-ic' .s-truC;:tUr� in which . the vari�8 dements repreSe�t;1A/8elf;"iniiuctionS �or(3:1 c�p��,ities /while in ana­logd�;;'C, 'D they represent resistances/.· Using the first and

:lIB second Kirchhoff's :taws, respectively. we obtain IVa ayatea of alge�ic equations 1Ji. -= - "fi LI"J:. �. for �;� where L'"i� -I:. � ... !'Gig.3, circuits 42, AJ. 1.4/. or IBI a �tea of "'''' Q(�.) ( " ... dtJ. p{1/, .... equatl.cms Ii = -1: CiJc di'l< for ---r;! where eli = -z:::.. ell' . ":: f ) at" "�(J{r�l) /Fig.3, circuit wv. As the equations suggest, all the ncm.;.cti-agoaal coefficients of the systea matrix are negative for c.er­umdirections of the loop currents or voltages of the nodal peirs. Consequmdly t the circuits can be eapIoJ'ed directl,J only faJ!" solution of such structures in which the statically indeterminate qlllUItitiea can be introduced in a mam:ter that. renders all the non-diagoDal coefficients of the :nexibi1i� _tri:r uegative; this can be done for the majority of struct.­ures in practice. In other cases it would be necea&ar7 to in­troduce electric elements which would act as negative self­-iDiuctions, ne�tive capacities or nept!" remteDcea at f!IZq arbitrary stress variation. 5. Now. Various DOnIinear effect. oDsel"Yable in creep of con­crete can also be modelled electrically. T!lus e.g. irreyersib­Ie creep deformations can be represented in a mechaDical model wi th the aid of a ratchet pawl /Fig.2d/ [91 • In the: corresp­onding ele<:tric models Ad, ad in l'ig.2 the ratchet pawl is re­preeented by a diode. The nonJinear relation between creep sud sUe_ at high &tresses can be repz'e_nt&d. by noDliDear self-indUctions, capacities aDd relistaDces. 6. N"ote. The same models can aIao be used far repreaentatiClll of the creep of structures of plastics at variable tempent.u­re because in such CBses the I18terial pars.eters are a180 ti­_ variab:te. In the special instance of tille inwariable para­_tel's. such aodels represent the bebaYiOUl" f14 atruct.lJl"ea of plastics at conetant temperat\ll'e.

Ref'erences: 1 ArutyuDyan, N.X:.: SolIe· Problems of the Theory of Creep

lin Russianl.Tekhteorizelat., IIoecow 1952 2 Be:Iash, P.M.: Fund8lllenta1s of. Callput.er Technique IUl RI,1ssianl. Chapf..l., 1Jedra, IIoeCCM 1964 J Dischinger, F.: EIastische UDd pIastische Ved�er E:iseabetantrsgwerke UDd iuabesOl'JlliePe Bogen.­brtlcken. Bauingenieur 1939, 53-6l.286-94.et.c.

6

7

217

L Tbeor,y and jpplications, Eirich, F.R. et. al.: Rheo ogy. .." v k 1956 Vol.I. Aced. Press, .,.ew ... ar

lIoIzmftl1er, •• �t�}��d!ie e�e�i;� :;;1! �;�1.1CunSt.-

of La - �ime Processes on the ProkopoYit.ch, J.E.: Effe� · ng- • of structures States of stress and stra1n lin Russian/.- Gosstroyizdao:t 1963

. ring lin cruelx' 1'rDka oz.: The ThWry of Elec:trical Engl.nee , Vol.I, SNTL El!ague 1956

• d 11i 1m aussian/. Fiz-Ia Tet.elbaUIII, I.I.: Electr1c Mo e ng matgiz 1959

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1 . ADalysia of Reinforced COI!-crete Ulickiy, I.I. et a • • ..,",�ect to Loog-Tuae PrO-Structur�s Wl. .... . GoBS�zdBt,lCiyeY ceases I� BusS •

1960 . th Analysis of struc1i-Ba�ant, Z.P.: Creep of Concrete l.n e

1965 lin urea lin Czec:hl. SlfiL, Prague (print/ � Shrinkage of Con-Bafaat., Z.P.: The 'l'he:�"::�ous St.ructore8 aDd . cret� J in aaChl stavebniclcY �asopl.s �!!:-Aead. of sciencesll962, No.9, 552-576 . c:b.mmg des Kriechens uno. schwiDd­

Babllt, Z.P.: Die �rch

e:t �gener BetoDkonatruktioaen. ens 111 .·a Bridge Struct.

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