Modeling of the cables of prestressing

Transcription

1 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 1/25 Modelig of the cables of prestressig Summary To improve resistace of certai structures of civil egieer, prestressed cocrete is used: for that, the cocrete is compressed usig cables of prestressed out of steel. I Code_Aster, it is possible to do calculatios of such structures: the cables of prestressig are modelled, that is to say by elemets BAR with two odes is by elemets CABLE_GAINE with 3 odes, which are the kiematically related to the elemets of volume or plate which costitute structural the cocrete part. To carry out this calculatio, there exist three orders specific to these cables of prestressig, DEFI_CABLE_BP who allows geometrically to defie the cable ad the coditios of settig i tesio, AFFE_CHAR_MECA, operad RELA_CINE_BP, which makes it possible to trasform the iformatio calculated by DEFI_CABLE_BP i loadig for the structure, ad CALC_PRECONT who allows the applicatio of prestressig o the structure. Pricipal specificities of the modelig based o the elemets BAR or CABLE_GAINE i the adheret case are the followig oes: The profile of tesio alog a cable ca be calculated (I) either accordig to regulatio BPEL 91 [bib1] by takig accout of the retreat of achorig, of the loss by rectiliear ad curviliear frictio, of the relievig of the cables, the creep ad the shrikig of the cocrete, (II) or accordig to regulatio ETC- C while holdig of the retreat of achorig, of the loss by frictio ad relievig of the cables. I these two cases, the coectio cables/cocrete is supposed to be perfect, with the image of the sheaths ijected by a coulis. It is possible to defie a zoe of achorig (istead of a poit of achorig) i order to atteuate the sigularities of costraits due to the applicatio of the tesio o oly oe ode of the cable (effect of modelig). The behavior of the cables is elastoplastic, thermal dilatio beig able to be take ito accout. Thaks to the operator CALC_PRECONT, oe ca simulate the phasage of the settig i tesio of the cables ad the settig i tesio ca be doe i several steps of time i the evet of appearace of o-liearities. Lastly, the fial tesio i the cable is strictly equal to the tesio prescribed by the BPEL. The cables beig modelled by fiite elemets, their rigidity remais active throughout the aalyses. Pricipal specificities of the modelig based o the elemets CABLE_GAINE, i the cases ot-members are the followig oes: It is possible to model adheret coectios cable-cocrete, slippig or rubbig. I the adheret case, it is the profile of tesio calculated accordig to the BPEL91 which is imposed; i the rubbig ad slippig cases, the profile is obtaied by simulatig the settig i tesio of the cable ad possibly the retreat of achorig. The other types of loss caot be take ito accout.

2 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 2/25 Operators DEFI_CABLE_BP ad CALC_PRECONT are compatible with all the types of mechaical fiite elemets volumial ad the elemets of plate (DKT, Q4GG) for the descriptio of the cocrete medium crossed by the cables of prestressed ad the settig i tesio.

3 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 3/25 Cotets 1 Prelimiaries The operator DEFI_CABLE_BP Evaluatio of the characteristics of the layout of the cables Cubic iterpolatio by splie Method without iterpolatio Cotrol of the iterpolatio by splie Determiatio of the profile of tesio i the cable accordig to BPEL Geeral formula Loss of tesio by frictio Loss of tesio by retreat of achorig Deformatios differed from steel Loss of tesio by istataeous strais of the cocrete Determiatio of the profile of tesio i the cable accordig to the ETC-C Geeral formula Losses of tesio by frictio Losses of tesio by retreat of achorig Losses due to the relievig of steel Determiatio of the relatios kiematics betwee steel ad cocrete Defiitio of the close odes Calculatio of the coefficiets of the relatios kiematics Case where the cocrete is modelled by massive fiite elemets Case where the cocrete is modelled by fiite elemets of plate Case where the ode of the cable is projected o a ode of the grid cocrete Treatmet of the zoes of ed of the cable Note: calculatio of the tesio of the cable as a mechaical loadig The macro-order CALC_PRECONT Adheret case: why a macro-order for the settig i tesio? Stage 1: calculatio of the equivalet odal forces Stage 2: applicatio of prestressig to the cocrete Stage 3: swig of the exteral efforts i iterior efforts Noadheret case Procedure of modelig for a cable modelled i BAR Various stages: stadard case Typical case Precautios of use ad remarks Procedure of modelig for a cable modelled i CABLE_GAINE Features ad checkig Bibliography... 25

4 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 4/25 8 Descriptio of the versios of the documet...25

5 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 5/25 1 Prelimiaries Certai structures of civil egieer are made up ot oly of cocrete ad passive steel reiforcemets, but also of cables of prestressigs. The aalysis of these structures by the fiite elemet method the requires to itegrate ot oly the geometrical ad material characteristics of these cables but also their iitial tesio. The operator DEFI_CABLE_BP was coceived accordig to the regulatios of the regulatio BPEL 91 which makes it possible to defie the cotractual tesio of way. The mechaisms take ito accout by this operator are the the followig: the settig i tesio of a cable by oe or two eds, the loss of tesio due to the frictios developed alog the rectiliear ad curviliear ways, the loss of tesio due to the retreat of achorig, the loss of tesio due to the relievig of the cable, the loss of tesio due to the shrikig ad the creep of the cocrete. It is also possible to use the regulatios of the ETC-C to defie the tesio. I this case, the mechaisms take ito accout are the followig: the settig i tesio of a cable by oe or two eds, the loss of tesio due to frictios ad o-lie losses, the loss of tesio due to the retreat of achorig, the loss of tesio due to the relievig of the cable. Lastly, oe ca obtai the tesio i the cable by a mechaical calculatio by usig modelig CABLE_GAINE ad the law CABLE_GAINE_FROT who takes as a startig poit the BPEL 91. I this case, the mechaisms take ito accout are the followig: the settig i tesio of a cable by oe or two eds, the loss of tesio due to the frictios developed alog the rectiliear ad curviliear ways, the loss of tesio due to the retreat of achorig. The cables are modelled by elemets BAR with two odes or by elemets CABLE_GAINE with 3 odes, which implies to adopt a layout approached i the case of the layouts i curve. This ca be made with more close to reality without major restrictio (the odes of cables must be iside the volume of the cocrete elemets) takig ito cosideratio grid to the elemets to the cocrete. Structural the cocrete part ca be modelled thaks to ay type of volumial elemet 2D ad 3D or with the elemets plates DKT. The operator DEFI_CABLE_BP the possibility has of creatig coditios kiematics betwee the odes of elemets BAR ad elemets 2D or 3D who do ot coicide i space. This has the advatage of simplifyig the creatio of the grid ad of leavig free choice to the user i terms of provisio of the elemets ad their umber. For the elemets BAR ad elemets CABLE_GAINE i the adheret case, the coectio cables of prestressed/cocrete is of perfect type, without possibility of relative slip. I the other cases, the cable ca slip ito its sheath. The operator also allows to defie a coe of diffusio of the costraits aroud achorigs (oly if the cable is adheret) i order to limit to it the stress cocetratios much higher tha reality ad which are due to modelig. The secod pricipal fuctio of the operator DEFI_CABLE_BP is to evaluate the profile of the tesio alog the cables of prestressed by cosiderig the techological aspects of their implemetatio. At the time of the istallatio of the cables, prestressig is obtaied thaks to the hydraulic actuatig cyliders placed at oe or two eds of the cables. The profile of tesio alog a cable is affected by frictio (rectiliear ad/or curviliear), by the deformatio of the surroudig cocrete, the retreat of achorigs at the eds of the cables ad by the relievig of steels. This tesio ca the be take ito accout like a iitial state of stress at the time of the resolutio of the problem complete fiite elemet. The problem, it is that i this case, uder the effect of the tesio of the cable, the cocrete uit ad cable are compressed ivolvig a reductio i the tesio of the cable. To avoid this problem ad to have exactly the tesio prescribed by the BPEL or the ETC-C i the structure i balace, the tesio must be applied by the meas of the macro-order

6 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 6/25 CALC_PRECONT. I more thaks to this method, it is possible to make phasage o the settig i tesio of the cables or to impose the loadig i several steps of time, which ca be iterestig if the behavior of the cocrete becomes o-liear as of the phase of settig i tesio of the cables. I the cases slippig ad rubbig elemet CABLE_GAINE, profile of tesio calculated by DEFI_CABLE_BP is ot used. The settig i tesio is obtaied by a calculatio fiite elemets by always usig the macro-order CALC_PRECONT.

7 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 7/25 2 The operator DEFI_CABLE_BP 2.1 Evaluatio of the characteristics of the layout of the cables Cubic iterpolatio by splie We preset here the method used to obtai a geometrical iterpolatio of the cables, which is essetial to precisely calculate the curviliear X-coordiate ad the agle used i the formulas of loss of prestressig. Oe starts by buildig a iterpolatio of the trajectory of the cable (i fact a iterpolatio of two projectios of the trajectory i the two plas Oxy ad Oxz ), the startig from these iterpolatios, oe cosiders the X-coordiate curviliear, ad the agular deviatio cumulated, accordig to the formulas: x s x = 1 y ' 2 x z ' 2 x dx éq x x = y ' ' 2 x z ' ' 2 x [ y ' ' x z ' x y ' x z ' ' x ] 2 dx éq y ' 2 x z ' 2 x I order to preserve the topology of the cable (ad i particular the schedulig of the odes which composes it) the operator DEFI_CABLE_BP work startig from meshs ad of groups of meshs, (rather tha of odes ad groups of odes), i order to be able to calculate the sizes while followig the sequece of the odes alog the cable. The iterpolatio used for the calculatio of prestressig i the cocrete will be a cubic Splie iterpolatio carried out i parallel o the three space coordiates accordig to the curviliear X- coordiate. The coordiates of the odes of the cable are the real coordiates, i.e. the coordiates defied by the grid of the cable. All the calculatios preseted withi the framework of the operator DEFI_CABLE_BP are defied startig from the real geometry of the structures ad the real positios of the odes. Calculatios of tesio to the odes will be carried out odes i odes, i the order give by the topology of the grid, startig from the formulas quoted above [éq 2.1-1] ad [éq 2.1-2]. The calculatio of the cumulated agular deviatio ad the curviliear X-coordiate requires the precise calculatio of the derivative of the trajectory of the cable defied i the operator i a discrete way by the positio of the odes of the grid of cable. The polyomials of Lagrage have istabilities, i particular for irregular grids. Moreover, oe sigificat umber of poits of discretizatio will lead to polyomials of high degrees. I additio a small ucertaity o the coefficiets of iterpolatio will have as a cosequece a importat error o the results, i term of derivative. By choosig a polyomial iterpolatio of small degree, oe will obtai derivative secod worthless or ot cotiuous (accordig to the degree). The iterest of a cubic iterpolatio of Splie type is to obtai drifts secod cotiuous ad costs of calculatios of order, if is the umber of poits of the fuctio tabulée to iterpolate, with polyomials of small degree. The priciple of this method of iterpolatio is described exclusively i the case of a fuctio of the form x f x. Oe supposes that oe carries out a iterpolatio of the tabulée fuctio, startig from the values of the fuctio at the poits of discretizatio x 1, x 2,..., x, ad of its derivative secod. Oe ca thus build a polyomial of order 3, o each iterval x i, x i 1, of which the polyomial expressio is the followig oe: y= x x j 1 y x j 1 x j x x j ' y j x j 1 x j 1 Cy ' ' ' j Dy j 1 j

8 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 8/25 with: Oe ca check easily that: C= 1 [ x j 1 x 3 x j 1 x ] 6 x j 1 x j x j 1 x x j 1 x j 2 3 j D= 1 [ x x j x x j ] 6 x j 1 x j x j 1 x x j 1 x j 2 j y x j = y j et y ' ' x j = y j ' ' y x j 1 = y j 1 et y ' ' x j 1 = y j 1 ' ' It is the ecessary to estimate the values of the derivative secod with the poits of iterpolatio. By writig the equality of the iterpolatios o the itervals [ x i 1, x i ], ad [ x i, x i 1 ] derivative of order oe, at the poit x i, the followig expressio is obtaied: x j x j 1 6 y j 1 ' ' x j 1 x j 1 3 y j ' ' x j 1 x j 6 y j 1 ' ' = y j 1 y j x j 1 x j y j y j 1 x j x j 1 Oe obtais thus 2 equatios coectig the values of the derivative secod to the poits of discretizatio x 1, x 2,..., x. By writig the boudary coditios i x 1 ad x o the values of the derivative secod, oe obtais a system, which oe ca determie i a sigle way the value of all the derivative, ad thus obtai the fuctio of iterpolatio. Two solutios arise the for the establishmet of the boudary coditios: to arbitrarily fix the value of the derivative secod at the poits x 1, ad x, to zero for example, to allot the actual values of the derivative secod i these poits, if this data is accessible. Oe obtais a system of equatios havig for ukow factors them derived secods from the fuctio tabulée to iterpolate. This liear system has the characteristic to be tri-diagoal, which meas that the resolutio is about O. I practice the iterpolatio breaks up ito two stages: the first cosists i calculatig the values estimated of the derivative secod with the poits, operatio which is carried out oly oce, the secod cosists i calculatig, for a value give of x, the value of the iterpolated fuctio, operatio which ca be repeated as may times as oe wish it. Tests carried out o the fuctio sie, three periods, show that the results are strogly depedet amogst poits, as well as distributio of the poits of the curve to be iterpolated, (expected result), but that eve i delicate situatios (few poits ad very irregular curve) the iterpolatio does ot diverge. I other words, eve if the correlatio cocerig the trajectory of the cable is ot the very good (iterpolatio with very few poits) iterpolatio is roughly located i a fork close to the real trajectory. This case will ot arise i practice, but makes it possible to check the stability of the method of iterpolatio. For the problem that we cosider here, oe caot always write the trajectory of the cable i the form [ y x ],[ z x ], wheever this curve is ot bijective, i particular whe projectio of the trajectory i oe of the two plas Oxy or Oxz cyclic or is closed (case of a circular cocrete structure).

9 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 9/25 By takig a itermediate variable of the type u= x ', parameter always growig ad of icrease idetical i absolute value to that i variable x, oe ca brig back oeself to expressios [ y u ] bijective fuctios of the variable u. The cubic iterpolatio Splie described above is the applicable to the fuctio y u (like with the fuctio z u ). I practice, that led however to problems of coectios of taget (agular poits) at the poits where the variable x chage directio of variatio, ad with specific irregularities. Oe describes the trajectory of the cable like a parametric curve. Kowig a set of poits of the curve, the parameter most easily accessible is the the curviliear X-coordiate. Oe writes the trajectory of the cable i the form [ x p, y p ], i the pla Oxy, (respectively [ x p, y p, z p ] i a space with three dimesios). The cumulated cord p discretized at the tabulés poits of the fuctio which oe iterpolates P 1, P 2,..., P is calculated i the followig way: p 1 =0 at the poit P 1, p k = p k 1 + distace P k 1 P k at the poit P k Oe thus has two curves defied by a set of couple [X (I), p (I)] ad [there (I), p (I)] which oe ca directly apply the cubic Splie iterpolatio preseted before, ad which makes it possible to be freed from the difficulties ecoutered previously. The iterpolatio is made for the two coordiates, (or three coordiates, i dimesio 3), idepedetly oe of the other Method without iterpolatio It is possible to simply calculate the curviliear X-coordiate ad the agle without makig iterpolatio. This method is obviously less precise but it is very robust. Moreover, more the grid is fie, plus its precisio icreases. However the problems ivolved i the iterpolatio by splie precisely occur whe the grid is too fie compared to the irregularities which it cotais. This method without iterpolatio is used whe the iterpolatio by splie failed (see ). The calculatio of the curviliear X-coordiates is very simple. It cosists i summoig the legth of the meshs of cables. Calculatio of the cumulated agular deviatio: The followig example is eough to describe the calculatio of the agular deviatio by this method. Meshs M 1, M 2 ad M 3 costitute a cable. N 1 is the first ode of the cable, the value of its cumulated agular deviatio is =0. The agle theta is the agular deviatio betwee the meshs M 1 ad M 2. I ay poit of ] N 1 N 2 [ the cumulated agular deviatio is always worthless because the taget vector with the curve i these poits is N 1 N 2. I ay poit of ] N 2 N 3 [, the cumulated agular deviatio is equal to theta because the taget vector each oe of these poits is N 2 N 3 ( theta is the agle eters N 1 N 2 ad N 2 N 3 ). It was decided, that the taget vector with the curve i N 2

10 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 10/25 is the average of N 1 N 2 ad N 2 N 3. What gives that the agular deviatio cumulated i N 2 is = theta. 2 With same logic, the agular deviatio cumulated i N 3 is =theta beta, ad theta beta i 2 N Cotrol of the iterpolatio by splie I order to cotrol if the iterpolatios by splie for the three coordiates of space are correct, oe calculates the umber of chages of variatio of the derivative first ad the umber of chages of sig of the derivative secod. If the umber of chages of sig is smaller tha the umber of chages of variatio (+ a whole costat fixed at 10), it is cosidered that the iterpolatio is of good quality. I the cotrary case, oe passes to the method without iterpolatio. 2.2 Determiatio of the profile of tesio i the cable accordig to BPEL Geeral formula The operator DEFI_CABLE_BP allows to calculate the tesio F s alog the curviliear X- coordiate s cable. This oe is give startig from the rules of the BPEL 91 [bib1]. All i all, oe leads to the followig formulatio: F s = F s { x F x F 5 flu 0 ret 0 r j [ F s S a y 0] F éq s } where s idicate the curviliear X-coordiate alog the cable. The parameters itroduced ito this expressio are: F 0 iitial tesio, x flu stadard rate of loss of tesio by creep of the cocrete, compared to the iitial tesio, x ret stadard rate of loss of tesio by shrikig of the cocrete, compared to the iitial tesio, 1000 relievig of steel at 1000 hours, expressed i %, S a surface of the cross-sectio of the cable, y elastic ultimate stress of steel, 0 adimesioal coefficiet of relievig of prestressed steel. I this formula, F 0 idicate the iitial tesio with achorigs (before retreat), F s represet the tesio after the takig ito accout of the losses by frictio ad retreat of achorig, x flu F 0 represet the loss of tesio by creep of the cocrete, x ret F 0 the loss of tesio by shrikig of the cocrete, r j [ ρ F s m S a s y 0] F s losses by relievig of steels. Note: The itroductio ito these elemets of losses of tesio is optioal. Thus, if oe plas to do a calculatio of creep ad/or shrikig of the cocrete by usig a suitable law with STAT_NON_LINE, oe should ot itroduce these elemets ito the losses calculated by DEFI_CABLE_BP.

11 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 11/25 The evaluatio of the losses requires the kowledge of the curviliear X-coordiate s ad of the cumulated agular deviatio calculated as from the derivative first ad secod of the trajectory of the cable. The precise calculatio of these derivative requires a iterpolatio betwee the poits of passage of the cable. This iterpolatio is carried out usig Splies, better tha the polyomials of Lagrage which have istabilities, i particular for irregular grids (cf precedig paragraph). I what follows each mechaism iterveig i the calculatio of the tesio is detailed Loss of tesio by frictio We start by calculatig the tesio alog the cable by takig accout as of losses by cotact betwee the cable ad the cocrete: F c ( s)=f 0 exp ( f α φ s) where idicate the cumulated agular deviatio ad the itroduced parameters are: f coefficiet of frictio of the cable o the partly curved cocrete, i rad 1, coefficiet of frictio per uit of legth, i m 1, F 0 tesio applied to oe or the two eds of the cable Loss of tesio by retreat of achorig To take ito accout the retreat of achorig, the followig reasoig is made: the tesio alog the cable is affected by the retreat of achorig at a distace d that oe calculates by solvig a problem with two ukow factors: the fuctio F s who represets the force after retreat of achorig ad the scalar d : d [ F s F s ] ds, E a S a 0 fα s F s is worth F 0 e is the value of the retreat of achorig (it is a data) = 1 F s, the force after retreat of achorig, is give startig from the formula [bib1]: [ F s. F s ]=[ F d ] 2, The legth d will be give i a iterative way thaks to the precedig itegral. Other authors use differet relatios such as: [ F s F d ]=[ F d F s ] For the calculatio of d, three typical cases ca arise: 1) This loss by retreat of achorig is localised i the zoe of achorig. If the cable is curved, ad the sufficietly short legth of the cable, it ca happe that d that is to say larger tha the legth of the cable. I this case, the loss of prestressig due to the retreat of achorig applies everywhere. It is ecessary to calculate the surface ragig betwee the two curves F s ad F s, which must be equal to E a S a, ad which thus makes it possible to calculate F s.

12 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 12/25 2) If a tesio is applied to each of the two eds of the cable, let us call F 1 s the distributio of iitial tesio calculated as if the tesio were applied oly to the first achorig, ad F 2 s the distributio of iitial tesio calculated as if the tesio were applied oly to the secod achorig. The value which must be retaied i ay poit of the cable as iitial tesio is F s =Max F 1 s, F 2 s. 3) Lastly, if D is larger tha the legth of the cable, ad whe a tesio is applied to each of the two eds of the cable (superpositio of the two precedig cases), oe must apply the followig procedure: calculatio of F 1 s calculated iitial tesio as if the tesio were applied oly to the first achorig ad by takig accout of the retreat of achorig (as i typical case 1), calculatio of F 2 s calculated iitial tesio as if the tesio were applied oly to the secod achorig ad by takig accout of the retreat of achorig (as i typical case 1), calculatio of F s = Mi F 1 s, F 2 s Deformatios differed from steel The loss by relievig of steel, for a ifiite time, is expressed i the followig way: r j ρ 1000[ F s S a σ y μ 0] F s ( 1000 relievig with 1000 hour i %; 0 the coefficiet of relievig of prestressed steel ad σ y the guarateed value of the maximum loadig to the rupture of the cable). This relatio expresses the loss by relievig of the cables for a ifiite time. The BPEL 91 proposes j the followig formula: r ( j )= 0 0 where j idicate the age of the work i days ad r j+ 9.r m a ray m characteristic obtaied by submittig the report of the sectio of the cocrete structure, i m 2, by the perimeter of the sectio (i meters) of cocrete Loss of tesio by istataeous strais of the cocrete The istataeous losses are ot take ito accout i the formula [éq ] used i Code_Aster. What the BPEL calls loss of istataeous tesio is i fact the loss of tesio iduced i cables already posed by the istallatio of a ew group of cables. To model this pheomeo, it is ecessary to represet the phasage of settig i prestressed i Code_Aster calculatio, i.e. ot to tighte the whole of the cables at the same time but i a successive way by coectig them CALC_PRECONT (see test SSNV164). 2.3 Determiatio of the profile of tesio i the cable accordig to the ETC-C Geeral formula The operator DEFI_CABLE_BP allows to calculate the tesio F s alog the curviliear X- coordiate s cable accordig to the rules of the ETC-C [bib4]. The theoretical formula is the followig oe: F (s)=f 0 Δ F μ Δ F ac Δ F el Δ F r Δ ϵ cs Δ ϵ cc where: F 0 is the iitial tesio applied to the cable

13 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 13/25 Δ F μ are the losses of tesio by frictio, Δ F ac are the losses of tesio due to the retreat of achorig, Δ F el are the losses of tesio due to the elastic strai of the cocrete, Δ F r are the losses of tesio due to the relievig of steels, Δ F cs are the losses of tesio due to the shrikig of the cocrete, Δ F cc are the losses of tesio due to the creep of the cocrete. The losses due to the elastic strai are estimated accordig to the ETC-C at: Δ F el (s)= A p E p Δ σ c (x) 2E with E Youg modulus of the cocrete, A p ad E p the sectio ad Youg the modulus of steel, ad Δ σ c (x) the costrait iduced i the cocrete by prestressig. They ca be estimated by simulatig the phasage of the settig i prestressed thaks to the operator CALC_PRECONT. These losses are ot take ito accout i the operator DEFI_CABLE_BP. Losses of tesio due to the shrikig of the cocrete Δ F cc ad with the creep of the cocrete Δ F cs, ca be obtaied by imposig a equivalet field of deformatio after the settig i tesio of the cables. Still, they are thus ot take ito accout i the operator DEFI_CABLE_BP. With fial, the formula established i DEFI_CABLE_BP is the followig oe: F (s)=f 0 Δ F μ Δ F ac Δ F r éq The 3 types of loss are detailed i the paragraphs below Losses of tesio by frictio I accordace with the ETC-C, the losses by frictio are estimated by the followig formula: F c ( s)=f 0 ( 1 e μ(α+ k s) ) [éq ] where idicate the cumulated agular deviatio ad the itroduced parameters are: μ coefficiet of frictio of the cable o the cocrete E k the loss ratio o lie [m 1 ] F 0 tesio applied to oe or the two eds of the cable Losses of tesio by retreat of achorig The formula is idetical to the BPEL. To refer to the Losses due to the relievig of steel The formula give by the ETC-C is the followig oe: 0,75 1 F s / F F r s =0,8 0,66 ρ 1000.exp 9,1 F s / F prg. h prg F s where: éq s idicate the curviliear X-coordiate alog the cable relievig of steel at 1000 hours, expressed i %, F prg costrait with rupture i steel,

14 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 14/25 h the umber of hours after the settig ito prestressed correspodig to the date or the losses by relievig of steel are calculated. I this formula, F s represet the tesio after the takig ito accout of the losses by frictio ad retreat of achorig like ormally after takig ito accout of the elastic losses. Two optios of calculatio are proposed correspodig to the choice TYPE_RELAX=' ETCC_DIRECT' or TYPE_RELAX=' ETCC_DIRECT'. If the user chooses the optio TYPE_RELAX=' ETCC_DIRECT', the the tesio used to calculate the loss due to the relievig of steels does ot take ito accout the elastic losses but oly the losses by frictio ad retreat of achorig. If the user chooses the optio TYPE_RELAX=' ETCC_REPRISE', the the tesio used to calculate the loss due to the relievig of steels takes the 3 types of losses of prestressed ito accout. This tesio must be provided by the user (keyword TENSION_CT uder DEFI_CABLE). It will have bee obtaied durig the first calculatio that oe ca qualify state with short-term by modellig the losses, by frictio, retreat of achorig ad the elastic losses per modelig of the phasage (cf test SSNV229B for example of implemetatio). 2.4 Determiatio of the relatios kiematics betwee steel ad cocrete Sice the odes of the grid of cable do ot coicide ievitably with the odes of the cocrete grid, it is ecessary to defie relatios kiematics modellig perfect adhesio betwee the cables ad the cocrete. The followig paragraphs describe i the order the space geometrical cosideratios makig it possible to defie the cocept of viciity betwee the odes of elemets of cable ad cocrete, the the method of calculatig of the coefficiets of the relatios kiematics Defiitio of the close odes The calculatio of the coefficiets of the relatios kiematics requires to determie the odes close to each ode of the grid of the cable. The diagram which follows symbolizes a ode cables ad a mesh cocrete: Noeuds voisis 1 Noeud câble Elémet béto 2 4 Noeuds béto 3 The mesh defied by the odes 1,2, 3,4 the ode cotais cables. The close odes are thus the tops 1,2, 3,4. If the ode cable is located iside a elemet at p odes P 1, P 2,..., P, the odes P 1, P 2,..., P odes close to the ode are called cables. Oe treats i the same way, the elemets of plate without offsettig, ad the solid elemets. The calculatio of the offsettig of each ode of the grid cable is ecessary for the calculatio of the coefficiets of the relatios kiematics.

15 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 15/25 I the case of elemets of plate, whe the ode cable is characterized by a offsettig ot o oe, oe defies the odes close as the uit to the odes top of the elemet which cotais the projectio of the ode cables i the taget pla with the grid cocrete. If the ode cables (or well its projectio i the taget pla with the grid cocrete) belogs to a border of a elemet, i fact the tops of this border form the whole of the close odes Calculatio of the coefficiets of the relatios kiematics I the whole of descriptios which follow the sizes are systematically expressed i the total referece mark of the grid. The coectios kiematics are thus expressed accordig to the degrees of freedom expressed i this base. The ormals ad vectors rotatio are expressed i the total referece mark, except explicit cotrary metio. I modelig fiite elemets of the structure cable-cocrete, the displacemet of a material structural cocrete poit ca be expressed easily usig the fuctios of form of the elemet or mesh cocrete whose tops form the close odes, accordig to displacemets of the odes close to the discretizatio cocrete. I the same way, a size or a displacemet of a poit of the cable, (or of its projectio o the taget level of the grid cocrete) is idetical to the value of this size at the material structural cocrete poit which occupies this same positio (perfect coectio betwee the cocrete ad steel), ad is thus expressed accordig to the value of this same size at the tops of the elemet, usig the fuctios of form. If x, y, z are the coordiates of the ode cables, or those of its projectio, ad N 1,N 2,..., N fuctios of forms associated with the odes cocrete P 1, P 2,..., P tops of a elemet of the grid cocrete (or tops of a border of a elemet of the grid cocrete), ad x i, y i,z i coordiates of the ode i, the the iterpolatio of a variable u o the elemet is writte: u x, y, z = i =1 x, y, z. u x i, y i, z i = i=1 x, y, z. u i u beig able to be a coordiate, or ay other odal data. The coectios kiematics make it possible to express the idetity of displacemet betwee the ode of the grid cables, ad the material poit cocrete which occupies the same positio. This correspods to the assumptio of a perfect coectio betwee the cocrete ad the cable Case where the cocrete is modelled by massive fiite elemets By takig agai the precedig otatios ad while cosiderig dx c, dy c, dz c displacemets of the ode cables, ad dx b j, dy b b j,dz j displacemets of the odes j ( j=1, ) structure cocrete eighbors of the ode of the cable we obtai the followig relatios: c = {dx x c, y c, z c dx b i i =1 dy c = x c, y c, z c dy b i i =1 dz c = x c, y c, z c dz b i i=1 beig the umber of odes of the elemet cocrete eighbors of the ode of the cable, or that of oe of its borders. For each ode of the cable, oe obtais 3 relatios kiematics betwee displacemets of the odes of the two grids cables ad cocrete Case where the cocrete is modelled by fiite elemets of plate

16 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 16/25 P b 3 P c P P 1 P 2 That is to say P 0 c the iitial positio of a poit of cable i the ot deformed geometry ad is P c the positio of this same poit after deformatio. Let us call P 0 p the projectio of P 0 c o the surface of the average layer of the cocrete hull ot deformed ad P p the projectio of P c o the surface of the average layer of the cocrete hull deformed. That is to say 0 the ormal with the average pla of the cocrete hull i P p 0 ad that i P p. { }={ } [ p c x 0 { x0 x0 c b x 0 P p p c O is give by: y 0 y 0 y c b 0 y 0 0y 0y p c b} {0x z 0 z 0 z p 0 y 0 0z}] {0x } { }={ p x x c} [ c { x c x b b} { x x P p is give by: y p y c y c y b y y z p z z p y z }] { z} The poit P O p belogs to a mesh of cocrete plate whose odes are oted P 1 b,p 2 b et P 3 b. Oe defies the offsettig of the cable compared to the cocrete hull as the distace e= P 0 p P 0 c ad the assumptio is made that this offsettig does ot vary whe the structure becomes deformed: e= P 0 p P 0 c = P p P c Oe itroduces displacemets of the poits of the cable ad his projectio: P 0 c P c ={d x c d y c d z c P0 p P p ={dx p = i =1 x 0 p, y 0 p, z 0 p dx i b dy p = x p 0, y p 0, z p 0 dy b i i =1 dz p = i=1 x 0 p, y 0 p, z 0 p dz i b

17 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 17/25 Oe itroduces the vector rotatio plate at the poit P p ad degrees of freedom of rotatio of the odes of the plate: θ={drx b = i=1 x 0 p, y 0 p, z 0 p drx i b dry b = x p 0, y p 0, z p 0 dry b i i=1 drz b = i=1 x 0 p, y 0 p, z 0 p drz i b By defiitio of, oe a: 0 = 0 Oe ca the write: P0 p P 0 c =e 0 P p P c =e {dx dy c dz c By withdrawig these two equatios, by takig accout of the defiitio of oe fids: {dx c dx p =e. dry p. 0z drz p. 0y dy c dy p =e. drz p. 0x drx p. 0z dz c dz p =e. drx p. 0y dry p. 0x By ijectig ito this last equatio the fuctios of form, oe has fially: b c x p, y p, z p dx =e. b i x p, y p, z p dry. b i 0z x p, y p, z p drz i 0y. i=1 i=1 i =1 i=1 i=1 x p, y p, z p b dx =e. b i x p, y p, z p drz i i=1 x p, y p, z p b dx =e. b i x p, y p, z p drx i i=1 0x. i =1. 0y i=1 x p, y p,z p drx i b x p, y p, z p dry i b Case where the ode of the cable is projected o a ode of the grid cocrete. 0z. 0x The distace eters projectio P 0 p ode cables P 0 c ad a ode cocrete P i b is give by: x d = P P 0 P b i = [ c b x i x c b x 0] 0 i y c b y i y c b y. i. z c b z i z c b z i If it happes that this distace is worthless (i practice lower tha 10-5 ), it is that the ode cable is projected at the top of a cocrete mesh, ad the the relatios kiematics are simplified: {dx c dx p i =e. dry p i. 0z drz p i. 0y dy c dy p i =e. drz p i. 0x drx p i. 0z dz c dz p i =e. drx p i. 0y dry p i. 0x

18 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 18/25 These relatios are the geeral relatios i which: N j x p, y p, z p =0 if j i. 2.5 Treatmet of the zoes of ed of the cable The modelig of a cable of prestressed such as it is made i Code_Aster cosist i represetig the uit cables, sheath of passage, ad all the parts of achorig, oly thaks to oe cotiuatio of elemets of bar. The lik betwee the elemets of cables ad the cocrete medium is esured by coditios kiematics o the degrees of freedom of each ode of the cable, ad those of the crossed elemets cocrete. Whe the settig i tesio of the cable is applied, it is observed that the reactios geerated at the eds of the cables o the cocrete create levels of costraits much higher tha reality, ad cause the damage of the cocrete. As example, i certai studies, oe could observe compressive stresses of more tha 200 MPa, which largely exceeds the experimetal value observed ( 40 MPa ). I reality, this pheomeo is ot observed thaks to the istallatio of a coe of diffusio of costrait (see drawig below) which distributes the force of prestressed o a large surface of the cocrete. I the case of the model with the fiite elemets, this surface does ot exist, sice the force is directly take agai by a ode. Real situatio Model EF without coe This way of modelig has several disadvatages: the cocetratio of this effort crushes the cocrete, the space discretizatio of the model chages the results. To cure this problem, the keyword CONE of the operator DEFI_CABLE_BP allows to distribute this force of prestressed either o a ode, but o all the odes cotaied i a volume (all the odes of this volume are depedet betwee them to form a rigid solid) delimited by a cylider of ray R ad legth L, represetig the equivalet of the zoe of ifluece of the coe of bloomig of a achorig (see figure below). rayo logueur The idetificatio ad the creatio of the relatios kiematics betwee the odes of the cocrete ad the cable are made i a automatic way by the order DEFI_CABLE_BP, where ew data R ad L will be to provide by the user.

19 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 19/ Note: calculatio of the tesio of the cable as a mechaical loadig We made the choice to leave the elemets of cable i the mechaical model support of calculatio by fiite elemets (liear or ot). So there is o calculatio of equivalet force to defer to the odes of the grid. Oe is simply satisfied to say that the cables of prestressig have a state of iitial stress ot o oe. This state of stress is that deduced from the tesio as calculated by DEFI_CABLE_BP. For reasos of simplicity, the data-processig object created by the operator DEFI_CABLE_BP is a table memorizig of the values to the odes of the cable. The let us cosider two related elemets of the cable: e1 tops N1 ad N2, ad e2 of top N2 ad N3. We suppose that l 1 ad S 1 are the legth ad the sectio of a elemet e1 ad that l 2 ad S 2 are the legth ad the sectio of the elemet e2. N2 N1 e1 e2 N3 DEFI_CABLE_BP will calculate with the ode N2 a tesio T N 2 defied by: T N 2 = 1 e 1 2 T s ds l 1 e 2 T s ds l 2 Coversely, for calculatio fiite elemet, the operator STAT_NON_LINE will cosider that the iitial costrait i the elemet e1 is 0 e1 = T N 1 T N 2 2S 1 Note: It will be always cosidered that the law of behavior of the cable is of icremetal type. 3 The macro-order CALC_PRECONT The macro-order CALC_PRECONT is to carry out the settig i tesio of the cables i the cocrete startig from the data cotaied i the cocept cable_precot resultig from DEFI_CABLE_BP. Two procedures differet of settig i tesio of the cables are preset i this macro-order. Oe or the other of these procedures is automatically selected accordig to whether the cocept cable_precot was created with the keyword MEMBER = YES or NOT. If MEMBER = YES, profile of tesio calculated i DEFI_CABLE_BP is trasformed ito iitial loadig by AFFE_CHAR_MECA.

20 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 20/25 If ot, the settig i tesio is simulated completely by a mechaical calculatio by imposig forces o the degrees of freedom of slip of the elemets CABLE_GAINE startig from the cotaied iformatio i the cocept cable_precot. 3.1 Adheret case: why a macro-order for the settig i tesio? It is possible to trasform the tesio i the cables, calculated by DEFI_CABLE_BP, i a loadig directly take ito accout by STAT_NON_LINE thaks to the order AFFE_CHAR_MECA operad RELA_CINE_BP (SIGM_BPEL=' OUI'). I this case, the tesio is take ito accout as a iitial state of stress at the time of the resolutio of the complete problem fiite elemets. Iitialemet f 0 f 0 A l'équilibre f f The resolutio of the problem makes it possible to reach a state of balace betwee the cable of prestressed ad the rest of the structure after istataeous strai. Ideed, uder the actio of the tesio of the cable, the uit cables (S) ad cocrete will be compressed compared to the iitial positio (cable i tesio, grid ot deformed). The legth of the cable will thus decrease, ad the iitial tesio also, cosequetly, will decrease. Oe thus obtais a fial state with a tesio i the cable differet from the tesio calculated iitially. It is the essetial proportioally to icrease the tesio applied i situ o the level them achorigs to take accout of this loss. The use of the macro-order CALC_PRECONT allows to avoid this phase of correctio, by obtaiig the state of balace of the structure with a tesio i the cables equal to the lawful tesio. I additio because of adopted method, it allows besides applyig the tesio i several steps of time, which ca be iterestig i the evet of plasticizatio or of damage of the cocrete. It makes it possible moreover to tighte the cables i a osimultaeous way ad thus i a way closer to reality of the buildig sites. To profit from these advatages, the loadig is applied i the form of a exteral loadig ad ot like a iitial state, which allows the progressive loadig of the structure. I additio, to avoid the loss of tesio i the cable, the idea is ot to make act the rigidity of the cables durig the phase of settig i tesio (cf [bib3]). The various stages carried out by the macro-order are here detailed Stage 1: calculatio of the equivalet odal forces This stage cosists i trasformig the iteral tesios of the cables calculated by DEFI_CABLE_BP i a exteral loadig. For that, a first is carried out STAT_NON_LINE oly o the cables which oe wishes to put i prestressig, with the followig loadig: embedded cable the tesio give by DEFI_CABLE_BP

21 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 21/25 t t t Figure a: Loadig at stage 1 Oe calculates the odal efforts o the cable. Oe recovers these efforts thaks to CREA_CHAMP. Ad oe builds the vector associated loadig F Stage 2: applicatio of prestressig to the cocrete The followig stage cosists i applyig prestressig to the cocrete structure, without makig take part the rigidity of the cable. For that, oe supposes for this calculatio that the Youg modulus of steel is ull. Oe ca choose to apply the loadig of prestressed i oly oe step of time or several steps of time if the cocrete is damaged. The loadig is thus the followig: blockig of the rigid movemets of body for the cocrete, odal efforts resultig from the first calculatio o the cable, the coectios kiematics betwee the cable ad the cocrete. Ecable = 0 F Figure a: Loadig at stage Stage 3: swig of the exteral efforts i iterior efforts Before cotiuig calculatio i a traditioal way, it is ecessary of retrasformer the exteral efforts which made it possible to deform the cocrete structure i iterior efforts. This operatio is doe without modificatio o displacemets ad the costraits of the whole of the structure, sice balace was reached at stage 2: it is about a simple artifice to be able to cotiue calculatio. The loadig is thus the followig: blockig of the rigid movemets of body for the cocrete, the coectios kiematics betwee the cable ad the cocrete, tesio i the cables.

22 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 22/25 t t t Figure a: Loadig at stage Noadheret case The value give to the keyword TENSION_INIT of DEFI_CABLE_BP is imposed i force o the degree of freedom GLIS active odes of achorig of the cable whereas with the passive odes this degree of freedom is blocked to zero. I the case of a achorig ACTIF/ACTIF, it is ecessary to proceed i two times: 1. A force is imposed o the first active achorig while the degree of freedom GLIS is blocked o the secod. 2. Oe takes agai calculatio by imposig the force o the secod achorig, the degree of freedom GLIS first is the blocked to zero with a load of the type DIDI. If a retreat of achorig is specified i DEFI_CABLE_BP, aother calculatio is lauched while imposig o the degree of freedom GLIS a displacemet equal to the retreat of achorig give. This load beig also of type DIDI. To cotiue calculatio afterwards CALC_PRECONT, it is eough to block the degree of freedom GLIS odes of achorig with a load DIDI. 4 Procedure of modelig for a cable modelled i BAR 4.1 Various stages: stadard case To maage to model a prestressed cocrete structure the procedure to be followed is the followig oe: to model the cocrete elemets (DKT, Q4GG, 2D or 3D), to model the cables of prestressed by elemets bars with two odes (BAR), to allot to the elemets bars the mechaical characteristics of the cables of prestressig, to defie the parameters materials for steel like BPEL_ACIER ad BPEL_BETON or ETCC_ACIER ad ETCC_BETON, thaks to the operator DEFI_CABLE_BP to calculate the data kiematics (relatios kiematics betwee the odes of the cable ad those of the cocrete elemets) ad statics (profile of tesio alog the cables), to defie the data kiematics like mechaical loadig, to call upo the operator CALC_PRECONT, to solve the problem with the operator STAT_NON_LINE by itegratig oly the data kiematics ad the loadigs other tha prestressig. For more practical iformatio, to refer to the documet [U ].

23 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 23/ Typical case So for a reaso where the other, the user does ot wish to use the macro-order CALC_PRECONT it is possible to adopt the followig procedure: to model the cocrete elemets, to model the cables of prestressed by elemets bars with two odes (BAR), to allot to the elemets bars the mechaical characteristics of the cables of prestressig, thaks to the operator DEFI_CABLE_BP to calculate the data kiematics (relatios kiematics betwee the odes of the cable ad those of the cocrete elemets) ad statics (profile of tesio alog the cables), to apply these data kiematics ad statics like a mechaical loadig, to solve the problem with the operator STAT_NON_LINE by itegratig all the loadigs. At the coclusio of this calculatio, it is ecessary to determie the coefficiets of correctio to apply to the iitial tesios applied to the cables (o the level of the declaratio of the operator DEFI_CABLE_BP) allowig to compesate for the loss by istataeous strai of the structure. Oce the commad file modified by these coefficiets of correctio, the modelig of the cables of prestressig is accomplished. Attetio, i the case of sequece of STAT_NON_LINE, it is appropriate startig from the secod call, to iclude i the loadig oly the relatios kiematics ad ot the tesio i the cables, uder pealty of addig this tesio, with each calculatio. 4.3 Precautios of use ad remarks It is recommeded to limit the recourse to a large umber of relatios kiematics uder pealty of weighig dow the computig time. However, whe a ode of the elemets of bar costitutig the cables coicides topologically with a ode cocrete, there is o kiematic additio of relatio. If a first is carried out STAT_NON_LINE before puttig i tesio i the cables, it is preferable to disable the cables, either by ot takig them ito accout i the model, or i their affectig a tesio costatly worthless (law of behavior WITHOUT), ad by icludig i the loadig the relatios kiematics bidig the cable to the cocrete. If oe carries out a phasage of the settig i prestressig, it is ecessary to thik of icludig the relatios kiematics i the loadig for the cables already teded at the precedig stages. 5 Procedure of modelig for a cable modelled i CABLE_GAINE To maage to model a prestressed cocrete structure the procedure to be followed is the followig oe: to model the cocrete elemets (3D), to model the cables of prestressed by elemets 1D with three odes ( CABLE_GAINE), while takig care that the elemets are ot liear uder pealty of ot beig able to evaluate the losses by frictio due to the curve, to allot to the elemets CABLE_GAINE mechaical characteristics of the cables of prestressig, to defie the parameters materials for the steel ad the law of frictio (CABLE_GAINE_FROT), but too BPEL_ACIER ad BPEL_BETON (eve if i practice, the parameters are ot used i the rubbig cases), thaks to the operator DEFI_CABLE_BP to calculate the data kiematics (relatios kiematics betwee the odes of the cable ad those of the cocrete elemets) ad statics (profile of tesio alog the cables),

24 Titre : Modélisatio des câbles de précotraite Date : 10/06/2014 Page : 24/25 to defie the data kiematics like mechaical loadig, to call upo the operator CALC_PRECONT, to solve the problem with the operator STAT_NON_LINE by itegratig oly the data kiematics ad the loadigs other tha prestressig, ad by blockig the degree of freedom GLIS odes of achorig with a loadig of TYPE=' DIDI'. For CALC_PRECONT ad STAT_NON_LINE, the covergece criteria should be used preferably RESI_REFE_RELA to be sure to have reached covergece. 6 Features ad checkig The orders evoked i this documet are checked by the cases followig tests: CALC_PRECONT SSLV115 [V ] Prestressed cocrete elemet i compressio ad gravity SSNV164 [V ] Settig i tesio of cables of prestressed i a beam 3D SSLS137 [V ] Prestressed cocrete plate with excetré cable i iflectio ZZZZ347 [V ] Validatio of the rubbig cables CABLE_GAINE DEFI_CABLE_BP SSLV115 [V ] Prestressed cocrete elemet i compressio ad gravity SSNV164 [V ] Settig i tesio of cables of prestressed i a beam 3D SSNP108 [V ] Prestressed cocrete elemet i compressio SSNP109 [V ] Cable of prestressig excetré i a right cocrete beam SSNV137 [V ] Cable of prestressed i a right cocrete beam SSNV229 [V ] Validatio of formulas ETCC i DEFI_CABLE_BP ZZZZ111 [V ] Validatio of the operator DEFI_CABLE_BP