Exercise 1: Prestressed cross-section; pretensioned beam with bonded strands

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1 RK Preast and restresses onrete strutures, sring 2016 xerise 1: Prestressed ross-setion; retensioned bea with bonded strands Calulate stresses due to restressing and self weight at transfer for the bea in the figure Bea: deth h580 width b280 Strands: 8 φ 12,5 strands ( 93 2 /strand) grade 1600/1800 lasti odulus of the strands MPa restress just before the transfer σ o 1317 MPa Distane of the strands fro the botto fibre 50 Conrete: C40/50; ean onrete strengtn at transfer f i 0,75 f lasti odulus of onrete at transfer i MPa Length (san) of the bea L10

2 DIMNSIONING OF TH PRSTRSSD STRUCTUR 1) Choosing the ross-setion diensions and the required aount of reinforeent (ultiate liit state, t ) 2) Choosing the required restress fore (servie liit state, t ) 3) Prestress loss alulation 4) Stress and raking analysis (servie liit state, t ) 5) Ultiate liit state, t flexural resistane shear resistane torsion + obined ations 6) Lifting for reast unit 7) Defletion (servie liit state, assebling and final stage t ) 8) nhorage and salling at the end of the struture 9) Fire resistane

3 CHOOSING TH INITIL PRSTRSS 1) Choosing the degree of restress Fully restressed struture The struture is fully oressed, no tension stresses water tight strutures exlosure lass Partially restressed struture Bridges ; no tension Buildings: usually tension stresses are allowed under quasi-eranent load raking is allowed, the rak width is liited C 2 table 7.1 N(FI): Requireents at the raking liit state deending on the exosure lass 2) Choosing the initial restress - Code liitations: C2: σ 0 0,8 f uk or 0,9 f,0,1k - The required balaned fores - Craking (or tension stress) in the tension side during restressing stage, transfer (release) stage or lifting stage before any iosed loads - Resistane of oressed area during restressing stage, transfer (release) stage or lifting stage before any iosed loads - Tyially the initial restress is σ MPa - Preast restressed struture with bonded strands: usually the whole ain tensile reinforent is restressed strands

4 Low restress - disadvantages: - raking - roble with strain about flexure - restressing steel is not otiially used Yield fore of φ 12,7 (99 2 ) orresonds reinforing bar T 20 (500HW) High restress disadvantages: - uward defletion (aber) is great - uh debonding is required - uh salling reinforeent at the end of the struture - too high restress ay ause restressing work ore diffiult stiation of the restress losses Influening fator Influene High losses Sall losses M quasi-er /M total ree M quasi-er /M total sall M quasi-er /M total high Conrete stress σ at the ree σ high σ sall oint of restress tendons Cross-setion shrinkage light, slender, thik, assive ree thin Initial restress σ o relaxation σ o high σ o sall Tyially the restress losses are: re-tensioned struture: %, norally about 20 % ost-tensioned struture: %, norally about 15 %

5 NLYSIS OF PRSTRSSD CROSS-SCTION Presstressing eans that the restressing reinforeent has stresses before loading of the struture. 1. PRTNSIOND STRUCTUR WITH BONDD STRNDS - Strand are tensioned against the anhors of the bed before asting the onrete Usually strands are straight - When the strength of onrete is develoed to the required strength (transfer strength, norally about % of noinal strength of onrete) the strands are released fro the anhors by utting (sudden release) or by hydrauli jak (gradual release) - Beause of the transfer strength there is bond between the strands and onrete; so the deforation (strain) of the strand is the sae as the deforation of onrete at the sae oint. - When the strand is released it tends to shorten but the shortening is ostly eliinated by the surrounding onrete due to bond between the stand and onrete this eans that the ounterfore of the re-tensioned fore of the strand transits fro the anhor to the onrete as a oressive fore.

6 - Due to bond between the strand and onrete the struture funtion for a noral iosed oressive fore like a reinfored onrete setion where the strand funtion as a reinforeent. The oressive ounterfore of the restress fore is distributed between onrete ross setion and the strand so that onrete (at the level of the strand) and strand get the sae oressive strain due to the noral fore. The fore of the strand derease due to this elasti shortening strain. Initial before the transfer: The ounterfore of the restressing fore of the strands ats as a oressive noral fore to the ross-setion oosed by onrete and the strands (transfored ross-setion) Prestressed strand has tensile restressing fore P and the stress P σ o Counterfore of the strand is affeted to the anhor bloks. fter the transfer: Conrete ross setion gets oressive fore whih is the ounterfore of the restressing fore of the strand Coressive fore transfers to onrete by bond between the strand and onrete Due to bond the strand gets the sae oressive strain as onrete at the level of the strand > Prestressed strand gets oressive fore whih redue the initial fore of the strand quilibriu: Resultant of the onrete stresses + redued fore of strands 0 For iosed loads: Struture funtion as an unraked reinfored ross setion. Stresses are distributed between onrete ross setion and the strands so that they have the sae hange of strain at the sae level. Strands at like a noral reinforeent getting tensile stresses whih eliinated the effet of the elasti shortening at the transfer. Tensile stress of the strand will redue the tensile stresses of onrete. So the effet of the elasti shortening is artly reversible. Due to the bond between strand and onrete the stress analysis is obtained using so alled transfored ross-setion where the strands and other reinforeent are taken into aount with reset of the elasti odulies; it eans the area of the strands are ultilied by the fator /-1, where is the elasti odulus of the strands and is the elasti odulus of onrete.

7 For the transfer situation the elasti odule of onrete i reresents the transfer strength f i > i, I i, W ai, W yi P is the restressing fore just before the transfer Prestressing load is treated in say anner as the noral oressive iosed load -P Stresses: Conrete: P P y P M σ y (initial onrte stress before the transfer is 0) I i W i i i Prestressing strand: P P y σ,p σo + n e + y (initial stress is σ o ) i I i y is the distane of the restressing strand fro the entroid The negative bending oent due to the eentriity y of the restressing fore M P y

8 For iosed loads: The elasti odule of onrete reresents the final strength of onrete f > the ross-setion values should be alulated newly Change of the stresses due to the iosed load N i, M i : Conrete: N i M i P M σ + y + (initial onrete stress before the transfer is 0) I W Total stress σ σ + σ + Prestressing strand: N i M σ n e + I i y Total stress in the strand σ σ,p + σ

9 BSIC SSUMPTIONS 1. Coatibility ondition > sae strain in the reinforeent and onrete at the sae level (ε s ε ) > no sli between reinforeent and onrete > lane setions reain lain setion under the loading > Bernoulli s assution > strain ε at the ertain distane y fro the entroid an be obtainde by linear equation ε ε + ψ y 0 where ε o is the strain at the entroid due to a noral fore ψ is the urvature due to a bending oent y is the distane fro the entroid 2. Constitutive equations of onrete and reinforeent steel > stress-strain-relation > Hooke s law: σ ε Conrete: σ ε Prestressing reinforeent: σ ε (ε s ε ) 3. quilibriu equations > Stress resultant of onrete + stress resultant of reinforeent iosed load Conrete stress resultant N σ d Moent of stress resultant about the entroid M σ y d Reinforeent stress resultant N s σ Moent of the reinforeent fores about the entroid M σ y is the area of one restressing strand y s is the distane of a strand fro the entroid s s quilibriu: N + N s N i M + M s M i

10 Ultiate liit state. Due to bond between onrete and strands the strands get due to loading the sae strain ε π as onrete at the level of the strands. The total strain int the strands is ε tot ε + ε. The strain due to restressing ε P / where P is the effetive restress after losses (loss due to elasti shortening exluded). The stress of the strands is alulated for the stress-strain diagra of the strand with the totla strain ε ε tot

11 2. POST-TNSIOND STRUCTUR - Duts and anhors are assebled in the ould; - Tendons oosed by several strands are assebled is the duts - Tendons have usually araboli urve for - Conrete struture with the ordinary reinforeent is asted - Tendons are tensioned when the strength of the onrete has been develoed to the required value > ost-tensioning; In this situation the duts are not yet injeted > there is no bond between the tendons and onrete. Tendons are tensioned against the anhors > Prestressing fore goes to the onrete at the ends of the struture by the anhors - Duts are injeted by ortar > when the ortar has hardened there exist bond between the tendon and onrete.

12 Tensioning: No bond between the tendon and onrete The oressive ounterfore of the restressing fore goes to the onrete only by the anhors Cross-setion is the net onrete ross setion where the duts fors the holes Stresses: net n dut Where net is the net onrete ross-setion dut is the area on one dut n is the nuber of the duts Coressive fore auses the elasti shortening of the struture and the distane between the anhors > the strain of the tendons derease > fore of the tendon derease > restressing loss due to the elasti shortening Conrete: P P y P M σ y (initial onrete stress before the transfer is 0) I W net net net net Prestressing tendon: σ σ + σ (initial stress is σ o ),P o,el y is the distane of the restressing strand fro the entroid σ,el is the loss of elasti shortening of the distane between the anhors σ,el L n 1 σ 2n 0 i L dl The negative bending oent due to the eentriity y of the restressing fore n is the nuber of the tendons M P y The average value an be used beause there is no bond between the tendons and onrete at the tie of restressing.

13 For iosed loads: Duts are injeted > bond between the tendon and onrete > onrete gets tensile strain due to iosed loads > due to the bond the tendons gets the sae tensile strain as onrete at the level of the tendon > transfored ross-setion an be used for analyzing stresses fro the loads affeting after the injeting the duts. The elasti odule of onrete reresents the final strength of onrete f > the ross-setion values should be alulated newly without dut holes (duts have been injeted with ortar) Change of the stresses due to the iosed load N i, M i : Conrete: N i M i P M σ + y + (initial onrete stress before the transfer is 0) I W Total stress σ σ + σ + Prestressing strand: N i M σ n e + I i y Total stress in the strand σ σ,p + σ

14 Ultiate liit state. Due to bond between onrete and tendons after grouting the duts the strands get due to loading the sae strain ε π as onrete at the level of the strands. The total strain int the strands is ε tot ε + ε. The strain due to restressing ε P / where P is the effetive restress after losses (loss due to elasti shortening inluded beause elasti shortening haens before grouting the duts). The stress of the strands is alulated for the stress-strain diagra of the strand with the totla strain ε ε tot

15 3. PRMNNTLY UNBONDD TNDONS - Tendon is in the dut whih is filled with grease. The dut does not injeted with ortar. > there is no bond between the tendon and onrete Duts and anhors are assebled in the ould; - Tendons oosed by several strands are assebled is the duts - Tendons have usually araboli urve for - Conrete struture with the ordinary reinforeent is asted - Tendons are tensioned when the strength of the onrete has been develoed to the required value > ost-tensioning; Tendons are tensioned against the anhors > Prestressing fore goes to the onrete at the ends of the struture by the anhors Tensioning: No bond between the tendon and onrete The oressive ounterfore of the restressing fore goes to the onrete only by the anhors Cross-setion is the net onrete ross setion where the duts fors the holes net n dut Where net is the net onrete ross-setion dut is the area on one dut n is the nuber of the duts Coressive fore auses the elasti shortening of the struture and the distane between the anhors > the strain of the tendons derease > fore of the tendon derease > restressing loss due to the elasti shortening For iosed loads: Duts are not injeted > no bond between the tendon and onrete The net onrete area is used Ultiate liit state: There is no bond between onrete and the tendons. Length of the tendon inreased due to urvature of the struture under the load in the ultiate liit state. The inrease of the stress fro thje effetive restress to the stress in the ultiate liit state is σ.uls 100 MPa (aording to the Finnish National nnex σ.uls 50 MPa).

16 Prestressed onrete setion (with bonded retensioned strands) Transfer stage elasti shortening Bea with entri restressing, ross-setion h*b The aount of the strands, restressing just before the transfer σ i Strain in the strands just before the transfer ε i σ i / Prestressing fore (tension fore) just before the transfer P i σ i ε i When the oressive ounter fore P i of this restressing fore ating to the struture oosed by the onrete and the strands, the onrete gets the oressive strain ε (< 0) Conrete stress σ ε (<0) The stress resultant of onrete N σ ε (<0) The net ross setion of onrete - Bond between the onrete and reinforeent (strands) > Strands gets the sae strain as the onrete at the loation of the strands, so ε ε Strain of strands just after the transfer ε ε i + ε ε i + ε Stress in the strands just after the transfer σ ε (ε i +ε ) The stress resultant just after the transfer P σ (ε i + ε ) quilibriu: P + N 0 (ε i + ε ) + ε 0 ε ( + ) -ε i εi ε + Pi Pi + ( ) + 1 Pi The transfored ross-setion oosed by the onrete and the strands

17 1 + Stress in the onrete i P ε σ The stress resultant of the onrete N σ i P lasti shortening of the strands i P ε ε The hange of the stress of the strands due to the elasti shortening i e P ε σ The stress in the strands just after the transfer + + σ σ σ i i e i 1 1 P 1 P

18 Prestressed onrete struture Transfer stage - elasti shortening xale Bea b*h with entri restessing; area of the restressing strands b : 500 h : 500 Bruttoarea : b h rea of restressing strands : Initial stress just before the transfer σ i : 1250 MPa lasi odulus of the strands : MPa σ i Strain of the strand just before the transfer ε i : ε i 6.41 %o Prestressing fore just before the transfer P i : σ i P i 1250 kn Conrete C35/45 (harateristi ylinder /harateristi ubi oressive strength) Charateristi strength f k : 35MPa Mean strength f : f k + 8 MPa f 43 MPa 0.3 f lasti odulus of onrete : MPa 10 MPa MPa Transfer strength usually about 70 % of the final strength Develoent of onrete strengh aording to C2 Ceent tye araeter s : 0.2 (raid-eent) Develoent strength of strength s1 β ( t) : e 28 t C2, equation( 3.4)

19 Mean strength, when age of onrete is t f t ( t) : β ( t) f t transfer: f i : 0.7 f f i 30.1 MPa > β i : 0.7 Develoent of elasti odulus as a funtion of tie f t ( t ) f : β ( t ) f i 0.7 f t ( t) 0.3 f ( t) : f C2 equation (3.5) 0.3 f i lasti odulus of onrete at transfer i : f i MPa Ratio of elasti odulies n e : n e i rea of the transfored ross-setion : + n e ( ) Netto area of onrete : xial rigidity of the bea ross-setion : i + ( ) : i MN ( ) : i + i : i + 1 i : i P i P i Conrete strain ε : ε : ε i 0.16 %o Conrete stress σ : ε i σ MPa Netto area of onrete : Conrete stress resultant N : σ N kn Change of strain of the strands ε : ε ε 0.16 %o Change of stress of the strands due to the elasti shortening σ e : ε σ e MPa

20 Stress of the strands just after the transfer σ : σ i + σ e σ MPa Fore of the strands just after the transfer P : σ P kn quilibriu P + N 0 N

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS 006 The Graw-Hill Copanies, n. ll rights reserved. Fourth E CHTER ure ECHNCS OF TERLS Ferdinand. Beer E. Russell Johnston, Jr. John T. DeWolf Bending Leture Notes: J. Walt Oler Teas Teh Universit ECHNCS