Advanced Analysis of. Lectures. PDF vytvořeno zkušební verzí pdffactory

Advanced Analysis of Concrete Structures

Lectures PDF vytvořeno zkušební verzí pdfFactory www.fineprint.cz

Advanced Analysis of Concrete Structures Prof.Ing. Vladimír Křístek, DrSc. Doc.Ing. Alena Kohoutková, CSc. Ing. Helena Včelová Department of Concrete Structures and Bridges

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Web site:

http://concrete.fsv.cvut.cz~kristek


Textbook

Z. Šmerda, V. Křístek :

Creep and Shrinkage of Concrete Elements and Structures, Elsevier, Amsterdam, Oxford, New York, Tokyo, 1988


Deformation of concrete in terms of load duration

range of linearity (approximately)


Strain independent on stress Reversible

Irreversible

Caused by temperature εt = α Δt

Short-term

Caused by shrinkage εs

Long-term

Strain caused by stress Reversible

Irreversible

Short-term

elastic – Hooks’s law

plastic

Long-term creep

delayed elastic

delayed inelastic


Measurement of shrinkage and creep development


Strain due to shrinkage and constant stress acting from time to = 7 days until t = 500 days

creep after unloading shrinkage age of concrete


Factors affecting creep and shrinkage of concrete 1. Composition of concrete : ● type and quantity of cement: higher contain of cement – larger creep and shrinkage ● grinding of cement: finer grinding – larger creep and shrinkage ● water-cement ratio: higher contain of water - larger creep and shrinkage ● grain size of aggregate: finer aggregate – larger creeo and shrinkage PDF vytvořeno zkušební verzí pdfFactory www.fineprint.cz

2. Density of concrete Higher strength concrete and concrete with closed structure: lower creep and shrinkage

3. Hygrometric conditions Drier conditions: larger creep and shrinkage

4. Cross-sectional dimension and shape Thin elements: more intensive creep and shrinkage caused by rapid drying up – characterized usually by cross-sectional area and circumference


Significant time factors ● Age of stress application (to, t´ orτ): concrete loaded at early age exhibits higher creep rate ● Investigated age (t) ● Stress duration - (t - t´ or t -τ): long term loading results in high creep ● Curing of concrete – wet conditions -> beneficial effect PDF vytvořeno zkušební verzí pdfFactory www.fineprint.cz

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


Appliacation of the cross-sectional model: Variation of internal forces due to change of structural system older part


younger part

Time development of internal forces


Roof– spatial shell structure


Load carrying capacity according to various assumptions


Creep and shrinkage of concrete 1. Point models ● Free shrinkage ● Free thermal dilatation ● Pickett efect (stress induced shrinkage, stress induced thermal dilatation) ● Basic creep ● Drying creep Only this approach allows to obtain real stresses PDF vytvořeno zkušební verzí pdfFactory www.fineprint.cz

2. For different thickness of slabs and webs ● Differential shrinkage ● Differential drying creep 3. Cross-sectional models ● Common approach in design practice ● Only internal forces M, N, Q and deflections can confidentially be obtained ● however, not stresses and thier distributions over cross sections PDF vytvořeno zkušební verzí pdfFactory www.fineprint.cz

Expression of concrete strain – the first method:

by the creep coefficient

Strain due to stress σ acting from age t0 to age t

ε(t,t0) = σ (1 + φ(t,t0)) /E(t0) φ(t,t0) is the creep coefficient PDF vytvořeno zkušební verzí pdfFactory www.fineprint.cz

Creep coefficient φ(t,t0) ● Multiplier of instanteneous strains ● Range: from 0 to 6 (depending of concrete age, duration of loading, concrete composition, external humidity, etc.) ● Expressed on the base of experimental investigations


Measurement of shrinkage and creep development


Expression of concrete strain – the second method:

by the compliance function Strain due to stress σ acting from age t0 to age t

ε(t,t0) = σ J(t,t0) J(t,t0) is the compliance function


Compliance function

J(t,t0)

● Strain due to unit stress acting from age t0 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


Relation between the creep coefficient and the compliance function

ϕ(t,t0) = J(t,t0) E(t0 ) - 1


Linearity of the creep effects Two constant stresses σ1 and σ2 acting from age to, produce the creep strain at age t

(σ1 + σ2) φ(t,to ) / E(to) The total strain at age t is

ε = (σ1 + σ2) (1 +ϕ(t,to )) / E(to) (only for σ1 + σ2 less than cca 40% of strength) PDF vytvořeno zkušební verzí pdfFactory www.fineprint.cz

Strain variation due to stress history ε

t1 t2

σ1

σ2

t3

t

σ3

Stress increments σ1, σ2 , σ3 … acting from t1, t2, t3,…. , the corresponding strain at age t is

ε = σ1(1 +ϕ(t,t1 )) / E(t1) + σ2(1 +ϕ(t,t2)) / E(t2) + + σ3(1 +ϕ(t,t3)) / E(t3) + ….

(σ1 + σ2 + ….) must be less than 40% of strength


Strain variation due to stress history – wrong method

ε

t1

σ1

t2

σ2

t3

t

σ3

Stress increments σ1, σ2 , σ3 … acting from t1, t2, t3,…. , the corresponding strain at age t is not

ε = σ1ϕ(t2,t1 ) / E(t1) + + (σ1 +σ2) ϕ(t3,t2) / E(t2) + + (σ1 +σ2+ σ3) ϕ(t4,t3) / E(t3) + …. PDF vytvořeno zkušební verzí pdfFactory www.fineprint.cz

Loading and unloading ε (t) = σ(1 +ϕ(t,t1 )) / E(t1) σ(1 +ϕ(t,t2)) / E(t2)

After unloading: Strains are varying without stress action! Modeled as two equal stresses of opposite signs acting from ages t1 a t2 PDF vytvořeno zkušební verzí pdfFactory www.fineprint.cz

Advanced Analysis of. Lectures. PDF vytvořeno zkušební verzí pdffactory

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