1.054/1.541 Mechanics and Design of Concrete Structures (3-0-9) Outline 5 Creep and Shrinkage Deformation

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1 1.54/1.541 Mehanis and Design of Conree ruures pring 24 Prof. Oral Buyukozurk Massahuses Insiue of Tehnology Ouline /1.541 Mehanis and Design of Conree ruures (3--9 Ouline 5 and hrinkage Deformaion of onree o Conree under sress undergoes a gradual inrease of srain wih ime. The final reep srain may be several imes as large as he iniial elasi srain. o is he propery of maerials by whih hey oninue deforming over onsiderable lengh of ime under susained sress. o Relaxaion is he loss of sress by ime wih onsan srain. o In onree, reep deformaions are generally larger han elasi deformaion and hus reep represens an imporan faor affeing he deformaion behavior. o Conree under onsan axial ompressive sress loaded onsanly load removed (unloading ins (elasi reovery reep reep reovery elasi deformaion = ins permanen or residual deformaion Time Experimens indiae ha for working sress range i.e. sresses no exeeding.5 reep srains are direly proporional o f sress ha is a linear relaionship beween σ and r. 1 / 11

2 1.54/1.541 Mehanis and Design of Conree ruures pring 24 Miroraking effe on reep in high sresses. Mehanisms of onree reep o Two phenomena are involved: 1. Time dependen deformaion ha ours when onree is loaded in a sealed ondiion so ha moisure anno esape. basi reep 2. of he maerial when moisure exhange is permied. drying reep o Basi reep is primarily influened by he maerial properies only, while drying reep and shrinkage also depend on he environmen and he size of he speimen. o The real siuaion migh be he ombinaion of he wo phenomena, someimes, one being he dominaing faor. o deformaion onains hree regions: 1. Primary reep iniial inrease in deformaion 2. eondary reep relaively a seady deformaion region 3. Teriary reep leads o reep fraure peifi reep o σ sp =, σ <.5f where = a funion of ime. ress levels above.8f reep produes failure in ime. o Relaionship beween f and sp : 6 f (psi sp ( 1 per psi Final srain ( 1 6 per psi / 11

3 1.54/1.541 Mehanis and Design of Conree ruures pring 24 oeffiien o C = ins where ins = insananeous (iniial srain. Faors influening reep: o Inernal faors (omposiion Aggregae (onenraion + siffness Waer/emen raio Aggregae permeabiliy Aggregae reep Aggregae siffness Aggregae grading and disribuion Cemen o Exernal faors (environmen, ime hisory ize hape Cross-seion Environmenal faors (ambien humidiy, emperaure ress inensiy Time (age of loading he loading hisory is imporan o he oal deformaion (srain. age of loading Mahemaial modeling of reep: o rain deomposiion The oal srain of onree may be deomposed as " = σ + = E + C + = E + C + + T = E + 3 / 11

4 1.54/1.541 Mehanis and Design of Conree ruures pring 24 where σ = sress-indued srain, E = reversible srain, C = reep srain, = sress-independen inelasi srain, = shrinkage srain, T = hermal expansion, and " = inelasi srain. o Inremenal formulaion of sress-srain relaion { dσ} = [ D]{ d} where { dσ } = hange in sress ensor, [ D ] = onsiuive law of he maerial, and { d } = hange in srain ensor. o Firs-order and higher orders formulaions of reep and shrinkage deformaion: Firs-order formulaions: The inremenal elasi siffness marix hanges from one ime sep o he nex as proporional o he elasi modulus. Appliable o homogeneous maerials. Higher order formulaions: The inremenal elasi siffness marix is differen and no proporional o he elasi modulus from one ime sep o anoher. Appliable o non-homogeneous maerials. o Inremenal quasi-elasi sress-srain law = J σ + where = a olumn marix onsising of he srain inremens, J = ompliane (square marix (funion, σ = a olumn marix onsising of he sress inremens, and 4 / 11

5 1.54/1.541 Mehanis and Design of Conree ruures pring 24 = a olumn marix onsising of he inelasi srain inremens. Linear mehods for he alulaion of reep srain o Effeive modulus mehod Toal sress-srain relaion: 1 ( = D σ + E where eff (, ( ( = urren ime, = he ime a whih he insananeous elasi modulus is haraerized, 1 ν ν 1 ν 1 D = for he ase of isoropy, 1+ ν 1+ ν sym. 1+ ν E eff (, ( ( 1 E = = = effeive modulus of elasiiy, J, 1 + φ, = sress-independen inelasi srain, 1 1 +φ, J(, = + C(, = = ompliane funion (he reep E E funion represening he srain (elasi + reep a ime, E ( = modulus of elasiiy haraerizing he insananeous deformaion a ime, (, E( J( φ =, 1 = reep oeffiien represening he raio of he reep deformaion o he iniial elasi deformaion, and 5 / 11

6 1.54/1.541 Mehanis and Design of Conree ruures pring 24 (, 1 (, = φ E( = ins E( = σ = speifi reep funion ( (, C sp. C = Inremenal srain The mehod is applied inremenally from sysem is hanged afer. o if he sruural ( 1 σ ( = D σ ( + ( φ, + Eeff, E where = inremenal form of he sress a ime, σ ( ( = inremenal form of he sress-independen inelasi srain, and = ime of firs loading. Effeive modulus mehod is exa when he sress is onsan from o. o Linear variaion mehod Assuming he mehanial srain is zero up o ime and jumps o he value of a, apply linear variaion of srain from and obain he following expression: ( ( = a+ (, φ The sress for > is ( ( a R(, E( σ = + where a, = onsans. o One an hen obain by algebrai manipulaion he basi inremenal sress-srain relaion of he age-adjused effeive modulus. 6 / 11

7 1.54/1.541 Mehanis and Design of Conree ruures pring 24 o Age-adjused effeive modulus mehod Effeive elasi modulus wih a orreion of he reep oeffiien: E (, E = + χφ 1, where χ = he relaxaion oeffiien (aging oeffiien, age-adjused effeive modulus =.5~1.. The quasi-elasi inremenal sress-srain relaion of he ageadjused effeive modulus: 1 ( = D E σ + (, (, (, ( ( φ ( = Dσ + E ( where =. ( ( We assume isoropi maerial wih a onsan Poisson s raio, whih is approximaely rue for onree. Praial onsideraions of reep prediion o Considering he speifi srain funion expressed as 1 φ (, τ = + C, τ E ( τ ( Tha is srain per uni sress a ime for he sress applied a age τ. o The speifi reep funion C(, τ = F( τ f ( τ o Forms of he aging funion F ( τ and f ( τ F τ : 7 / 11

8 1.54/1.541 Mehanis and Design of Conree ruures pring 24 F τ = a+ b τ Power law: F τ = a+ be τ Exponenial law: f ( τ Power law: m f τ = a τ Logarihmi expression: f ( τ = a+ log 1 + ( τ where ( -τ is measured in days. Experimenal daa is used o obain oeffiiens., σ.4 < f o Prediion of reep deformaion: Perform shor-erm experimenal resuls Develop shor-erm reep urves o be used in long-erm prediion. When experimenal daa is no available, he design has o rely on a relevan ode. Various ode reommendaions have been quie onroversial. ( ACI 29 Commiee defines he reep oeffiien ϕ, as.6 ( + ( ϕ, = ϕ, where.6 1 ( = ime sine appliaion of load, ϕ, = αk K K K K K = ulimae reep, K and relae o uring proess, K relaes o he K2 3 K4 K5 6 hikness of member,,, and K relae o onree omposiion. (α = 2.35 Priniple of superposiion For analysis and design purposes he ime dependen linear relaion beween sress and reep srain an be wrien in he following way: 8 / 11

9 1.54/1.541 Mehanis and Design of Conree ruures pring 24 ( = f ( + F ( r sp, τ τi sp, τi i= τ where f = iniial sress in onree a he ime τ of firs loading, sp (, τ = speifi reep srain a ime for onree loaded a ime τ, F ( τ i = addiional sress inremen or deremen applied a sp (, i ime τ < τ i <, τ = speifi reep srain a ime for onree a age τ >. i Effe of sress on reep is predied by he superposiion mehod. srains produed in onree a any ime by an inremen of sress applied a any ime τ are independen of he effes of any oher sress inremen applied earlier or laer han ime τ. This mehod of analysis predis reep reovery and generaes sress-srain urves of a shape similar o experimenal resuls. I is assumed ha he onree reeps in ension a he same rae as i reeps in ompression. under variable sress an be obained by superimposing appropriae reep urves inrodued for orresponding hanges in sress a differen ime inervals. This is rue if reep is proporional o applied sress. under differen saes of sress: o akes plae in ension in he same manner as in ompression. The magniude in ension is muh higher. o ours under orsional loading. o Under unaxial ompression, reep ours no only in he axial bu also in he normal direions. 9 / 11

10 1.54/1.541 Mehanis and Design of Conree ruures pring 24 o Under muliaxial loading reep ours in all direions and affeed by sresses in oher normal sresses. Non-linear faors in he evaluaion of reep srain o The influenes of Temperaure dependene in ombined sress hrinkage o Humidiy and emperaure variaion Cross oupling of reep wih shrinkage or hermal expansion. o Craking or srain-sofening o Cyli loading o High sress and muliaxial visoplasiiy Examples of omplex sruures wih signifian effe of reep on deformaion and failure behavior o Thermal effe (pressure vessels, proess vessels o egmenal onree bridge deformaion (onsruion and servie sages hrinkage deformaion o hrinkage is basially a volume hange of he elemen whih is onsidered o our independenly of exernally imposed sresses. This volume hange an also be negaive and is hen alled swelling. Thin seions are pariularly susepible o drying shrinkage and herefore mus onain a erain minimum quaniy of seel. In resrained onree shrinkage resuls in raking, before any loading. The minimum reinforemen is provided o onrol his raking. o Causes of shrinkage: Loss of waer on drying 1 / 11

11 1.54/1.541 Mehanis and Design of Conree ruures pring 24 Volume hange on arbonaion o On average ulimae shrinkage 4 sh, u = 8 1. o ACI 29 Commiee suggess: = sh sh, u h h s f e where sh, u= ulimae faor of shrinkage, = faor of ime, h = faor of humidiy, h = faor of hikness, s = faor of slump, f = faor of fineness, e = faor of air onen, and = faor of emen propery. 11 / 11