Advanced Analysis of Concrete Structures Lectures
Advanced Analysis of Concrete Structures Prof.Ing. Vladimír Křístek, DrSc. Doc.Ing. Alena Kohoutková, CSc. Ing. Helena Včelová Department of Concrete Structuresand Bridges
Web site: http://concrete.fsv.cvut.cz~kristek
Textbook Z. Šmerda, V. Křístek : Creepand Shrinkageof Concrete Elements andstructures, Elsevier, Amsterdam, Oxford, New York, Tokyo, 1988
Deformationof concrete in terms ofloadduration range of linearity (approximately)
Strainindependent on stress Short-term Long-term Reversible Causedby temperature ε t = αδt Irreversible Causedby shrinkage ε s Straincaused by stress Short-term Long-term - creep Reversible elastic Hooks s law delayedelastic Irreversible plastic delayedinelastic
Measurement ofshrinkageand creep development
Strain due to shrinkage and constant stress acting from time t o = 7 days until t = 500 days creep afterunloading shrinkage ageof concrete
Factors affecting creep and shrinkageof concrete 1. Compositionof concrete : type andquantity ofcement: highercontainof cement largercreepandshrinkage grinding ofcement: finergrinding largercreep andshrinkage water-cement ratio: highercontainofwater-larger creepandshrinkage grainsizeofaggregate: fineraggregate larger creeoandshrinkage
2. Density of concrete Higherstrengthconcreteandconcretewithclosed structure: lowercreepandshrinkage 3. Hygrometric conditions Drierconditions: largercreepandshrinkage 4. Cross-sectionaldimensionandshape Thinelements: more intensivecreepandshrinkage causedby rapid drying up characterizedusually by cross-sectional areaandcircumference
Significant timefactors Ageofstress application (to, t ort): concrete loadedatearly age exhibits higher creep rate Investigatedage (t) Stress duration-(t-t or t -τ):long term loading results in high creep Curing of concrete wet conditions -> beneficial effect
To express creep and shrinkage effects, two approaches are available : Point models Cross-sectional models
Application of the point model drying : a wall constant surface humidity 50% distribution of humidity for various ages free deformation of individual layers stresses and cracking
Appliacationof the cross-sectional model: Variation of internal forces due to changeof structural system older part younger part
Time development of internal forces
Roof spatial shell structure
Load carrying capacity according to various assumptions
1. Point models Creep and shrinkage of concrete Free shrinkage Free thermal dilatation Pickettefect(stress inducedshrinkage, stress inducedthermal dilatation) Basic creep Drying creep Only this approach allows to obtain real stresses
2. For different thickness of slabs and webs Differential shrinkage Differential drying creep 3. Cross-sectional models Commonapproachindesignpractice Only internal forcesm, N, Q and deflections canconfidentially be obtained however,not stresses and thier distributions over cross sections
Expression of concrete strain the first method: by the creep coefficient Strain due to stress s acting from age t 0 to age t e(t,t 0 ) = s (1 + φ(t,t 0 )) /E(t 0 ) φ(t,t 0 ) is the creep coefficient
Creep coefficient φ(t,t 0 ) Multiplier of instanteneous strains Range: from 0 to 6 (depending ofconcreteage, durationofloading, concretecomposition, external humidity, etc.) Expressed on the base of experimental investigations
Measurement ofshrinkageand creep development
Expression of concrete strain the second method: by thecompliancefunction Strain due to stress s acting from age t 0 to age t e(t,t 0 ) = s J(t,t 0 ) J(t,t 0 ) is the compliance function
Compliance function J(t,t 0 ) Strain due to unit stress acting from age t 0 to age t (pro instantaneous load, J is the reverse value of Young s modulus) Depends on age of concrete at loading, load duration, concrete composition, environmental conditions, etc. Expressed on the base of experimental investigations
Relationbetween thecreep coefficientand thecompliance function j(t,t 0 ) = J(t,t 0 ) E(t 0 ) -1
Linearity of thecreep effects Two constantstresses s 1 and s 2 acting fromage t o, produce the creep strain at age t (σ 1 + σ 2 ) φ(t,t o ) / E(t o ) Thetotal strainatage t is e = (s 1 + s 2 ) (1 +j(t,t o ))/E(t o ) (only for s 1 + s 2 less thancca 40% of strength)
Strain variation due to stress history e t 1 t 2 t 3 t s 1 s 2 s 3 Stress increments σ 1, σ 2, σ 3 acting from t 1, t 2, t 3,., the corresponding strain at age t is e = s 1 (1 +j(t,t 1 ))/E(t 1 ) + s 2 (1 +j(t,t 2 ))/E(t 2 ) + + s 3 (1 +j(t,t 3 ))/E(t 3 ) +. (σ 1 + σ 2 +.) must be less than 40% of strength
Strain variation due to stress history wrong method e t 1 t 2 t 3 t s 1 s 2 s 3 Stress increments σ 1, σ 2, σ 3 acting from t 1, t 2, t 3,., the corresponding strain at age t is not e = s 1 j(t 2,t 1 )/E(t 1 ) + + (s 1 +s 2 ) j(t 3,t 2 )/E(t 2 ) + + (s 1 +s 2 + s 3 ) j(t 4,t 3 )/E(t 3 ) +.
Loading and unloading e (t)= s(1 +j(t,t 1 ))/E(t 1 ) - s(1 +j(t,t 2 ))/E(t 2 ) After unloading: Strains are varying without stress action! Modeledas two equal stresses of opposite signs acting from ages t 1 a t 2