Creep of Concrete - Seoul National University · 2019. 3. 15. · Creep of Concrete Creep of...

Creep of Concrete

Creep of Concrete

• Relationship between creep and shrinkage

• Both originate from the hydrated cement paste

• The strain-time curves are similar

• The factors that influence the drying shrinkage also influence the creep and are generally in the same way

• In concrete the micro-strain of each, 400 to 1000x10-6, islarge and can’t be ignored in structural design

• Both are partially reversible

• Origins of creep are believed to reside in the response of C-S-Hto stress

Definition of Terms

Creep of Concrete

• Creep

• is the phenomenonof a gradual increasein strain with timeunder a given level ofsustained stress

• Stress relaxation

• is the phenomenonof gradual decreasein stress with timeunder a given level ofsustained strain

(Jason Weiss)

Creep

Creep of Concrete

• When a concrete element is restrained

• There is progressive decrease of stresses with time due to the visco-elasticity of concrete

• Under restrained conditions, deformation and cracking ofconcrete depends on

• stress induced by shrinkage and creep strains

• stress relief due to the visco-elasticity of concrete

(Kelly 1963)

Elastic and Creep Strains

Creep of Concrete

• Elastic recovery - approximately the same as the instantaneousstrain on the first application of load

• Creep recovery - gradual decrease in strain after the instantaneous recovery

• Irreversible creep

Definition of Terms

Creep of Concrete

• Under typical service conditions, concrete is more likely to bedrying while under load. It has been found that under suchconditions creep deformations are greater than if the concrete is dried prior to loading

• Free shrinkage εsh

• Basic creep εbc

• Total strain duringloading and drying εtot

• Drying creep εdc

• Total creep εcr= εbc + εdc

• Creep coefficient

e

crC

(Mindess et al 2003)

Column Shortening in a Tall Building

Creep of Concrete

Summation of strains in a

reinforced concrete column

during construction of a tall

building (Russell and Corley

1977)

Factors Influencing Creep

Creep of Concrete

• Applied stress

• Creep strain increases withapplied stress.

• Relationship is not linear

• Approximately linear in the stressrange generally used

• For practical reasons, a linearrelationships is often used tocompare different concretespecimens loaded at differentstress levels

• Specific creep

• typical value 150X10-6/MPa

cr


Factors Influencing Creep (cont’d)

Creep of Concrete

• W/C ratio

• Conflicting data (not possibleto change one parameterindependent of others)

• w/c ↓, specific creep ↓

• Compressive strength ↑,specific creep ↓

• Curing condition

• The time of moist curing ofconcrete at loading affect themagnitude of creep

• An increase in thetemperature of curing reduceboth basic & drying creep

(Neville 1970)



Creep of Concrete

• Temperature (concrete maintained at high T while under load)

• < ~80 °C, temperature ↑, creep ↑, > ~80 °C, uncertain

• Moisture

• Presence of free moisture in concrete is necessary conditionfor creep

• Creep is a function evaporable water, falls to zero when noevaporable water is present. The greatest ↓ in evaporable water,hence in creep, occurs on drying to 40% RH while water is lostfrom capillary pores.

• The lower the RH of environment, the higher the creep

(Troxell et al 1958)


Creep of Concrete

• Aggregate

εcon= εp(1-Va)n

• Amount

• Modulus of elasticity of aggregate

(Troxell et al 1958)


Creep of Concrete

• Specimen geometry

• Volume-to-surface ratio andspecimen thickness affect the totalcreep in much the same way asdrying shrinkage is affected

• Creep Recovery

• Only a small portion of total creepstrain is recoverable whenconcrete is unloaded

• Proportion of irreversible creep ↑with time under load

• After ~30 days under load,additional creep is largelyirreversible

• Drying creep is irreversible

(Hanson & Mattock 1966)

Mechanisms of Creep

Creep of Concrete

• Thermally activated creep

• Creep strain originates through deformation of microvolume ofpaste, called “creep center”

• “Creep center” undergoes deformation to a low energy stateunder the influence of energy by external sources

• Deformation can only occur by going through an energy barrier inthe form of intermediate, high-energy state

• The ability of a creep center to cross the barrier depends on

• Height of energy barrier

• Input of energy fromexternal sources

• Stress, strain

• Change in moisture

• Change in temperature


Mechanisms of Creep (cont’d)

Creep of Concrete

• Role of absorbed water

• Nature of “creep center”?

Operating process?

• Slip between adjacentparticles of C-S-H under ashear stress; the ease andextent of slip depend onthe force of attractionbetween particles

• Measurable slip occursonly when sufficientthickness of water existsbetween the particles

Mechanisms of Creep (cont’d)

Creep of Concrete

• Role of absorbed water (cont’d)

• Creep can also results from diffusion of water inmicropores under stress

• When external stress is applied, the stress exerted onwater in micropores is ↑

• Thickness of absorbed water ↓

• Water diffuses from micropores to capillary poreswhere no stress exists, causing bulk deformation

• Reordering of C-S-H can lead to increased bondingbetween them, leading to elimination of micropores,“aging”

Creep of Concrete

(Mehta & Monteiro 2006)

Creep of Concrete

(Mehta & Monteiro 2006)

Modeling and Code

Contents:

Viscoelastic behavior (Relaxation vs Creep)

Classical linear viscoelasticity

Constitutive equation for modeling of concrete

ACI method

1. Material constants

2. Shrinkage

3. Creep

CEB-FIP

1. Material constants

2. Shrinkage

3. Creep

Examples

Modeling and Code

Viscoelastic behavior

Viscous fluid (Newtonian)

Elastic solid (Hookean)

Viscoelastic material

𝜎

𝜎

𝜎

𝑡

𝑡

𝑡

Viscoelastic solid

Viscoelastic fluid

𝜀

𝜀

𝜎

𝜎

𝜎0

𝜎0

𝜎0

𝜎0

𝜀0

𝐸

𝜂

Modeling and Code


Creep response < >

Relaxation response < >

𝜎

𝑡

𝜎0

𝜀

𝑡

𝜀(𝑡)

𝜀 𝑡 = 𝐽 𝑡 ⋅ 𝜎0 ⋅ 𝐻 𝑡

𝜎 𝑡 = 𝜎0 ⋅ 𝐻 𝑡

𝑡

𝜀0

𝑡

𝜎(𝑡)

𝜀 𝑡 = 𝜀0 ⋅ 𝐻 𝑡

𝜎 𝑡 = 𝐸 𝑡 ⋅ 𝜀0 ⋅ 𝐻 𝑡

𝜀 𝜎

Modeling and Code


Where Heaviside step function

𝑡

1.0

𝑡

𝐻 𝑡 =

𝐻 𝑡 𝐻 𝑡 − 𝑡0

𝑡0

Modeling and Code

Classical linear viscoelasticity (linear superposition or Boltzmann’s superposition)

i) Infinitesimal deformation (neglect second-order term)ii) Instantaneous stress is proportional to input strainiii) Relaxation rate is proportional to instantaneous stressiv) No aging effect is assumed

Creep response

𝜎

𝑡

𝜀

𝑡

𝜀(𝑡)

𝜎 𝑡

Assumptions:

𝑡0 𝑡0

𝛥𝜎0 𝛥𝜀0

𝛥𝜀1𝛥𝜎1

𝜎 𝑡 =△ 𝜎0 ⋅ 𝐻 𝑡 +△ 𝜎1 ⋅ 𝐻 𝑡 − 𝑡0𝜀 𝑡 =△ 𝜀0 𝑡 +△ 𝜀1 𝑡 − 𝑡0

= 𝐽 𝑡 ⋅△ 𝜎0 ⋅ 𝐻 𝑡 +𝐽 𝑡 − 𝑡0 ⋅△ 𝜎1 ⋅ 𝐻 𝑡 − 𝑡0

Modeling and Code

Classical linear viscoelasticity (linear superposition or Boltzmann’s superposition)

i) Infinitesimal deformation (neglect second-order term)ii) Instantaneous stress is proportional to input strainiii) Relaxation rate is proportional to instantaneous stressiv) No aging effect is assumed

Relaxation response

𝜎

𝑡

𝜀

𝑡

𝜀(𝑡)

𝜎 𝑡

Assumptions:

𝑡0 𝑡0

𝛥𝜎0𝛥𝜀0

𝛥𝜀1 𝛥𝜎1

𝜀 𝑡 =△ 𝜀0 ⋅ 𝐻 𝑡 +△ 𝜀1 ⋅ 𝐻 𝑡 − 𝑡0𝜎 𝑡 =△ 𝜎0 𝑡 +△ 𝜎1 𝑡 − 𝑡0

= 𝐸 𝑡 ⋅△ 𝜀0 ⋅ 𝐻 𝑡 +𝐸 𝑡 − 𝑡0 ⋅△ 𝜀1 ⋅ 𝐻 𝑡 − 𝑡0

Modeling and Code

Constitutive equations

1) Maxwell model (Reuss)

2) Kelvin model (Voigt)

3) Standard linear solid model

4) Another standard linear solid model

Spring

Dashpot

𝜎 𝑡 = 𝐸𝜀 𝑡𝐸

𝜎

𝑡

𝜎0/𝐸

𝑡

𝜀

𝐸𝜀0

𝜂 𝜎 𝑡 = 𝜂 𝜀 𝑡

𝜎

𝑡𝑡

𝜀

Modeling and Code



𝐸

𝜂

Equilibrium eq.

Compatibility eq.

Constitutive rel. (spring)

Constitutive rel. (dashpot)

Constitutive rel. (Maxwell)

𝜎𝐸 𝑡 = 𝜎𝜂 𝑡 = 𝜎 𝑡

𝜀 𝑡 = 𝜀𝐸 𝑡 + 𝜀𝜂 𝑡

𝜎𝐸 𝑡 = 𝐸𝜀𝐸 𝑡

𝜎𝜂 𝑡 = 𝜂 𝜀𝜂 𝑡

𝜀𝜂 𝑡 = 𝜎(𝑡)

𝐸+𝜎(𝑡)

𝜂

Modeling and Code



𝐸

𝜂

Constitutive rel. (Maxwell)

Creep test

Relaxation test

𝜀 𝑡 =𝜎0𝐸+𝜎0𝜂𝑡

𝜎 𝑡 = 𝐸𝜀0𝑒−𝐸 𝑡 𝜂

𝑡

𝜀

𝑡

𝜎

𝜀𝜂 𝑡 = 𝜎(𝑡)

𝐸+𝜎(𝑡)

𝜂

Modeling and Code



𝐸 𝜂

Equilibrium eq.

Compatibility eq.

Constitutive rel. (spring)

Constitutive rel. (dashpot)

Constitutive rel. (Kelvin)

𝜎 𝑡 = 𝜎𝐸 𝑡 + 𝜎𝜂 𝑡

𝜀 𝑡 = 𝜀𝐸 𝑡 = 𝜀𝜂 𝑡

𝜎𝐸 𝑡 = 𝐸𝜀𝐸 𝑡

𝜎𝜂 𝑡 = 𝜂 𝜀𝜂 𝑡

𝜎 𝑡 = 𝐸𝜀 𝑡 + 𝜂 𝜀 𝑡

Modeling and Code


Constitutive rel. (Kelvin)

Creep test

Relaxation test

𝑡

𝜀

𝑡

𝜎


𝐸 𝜂

𝜎 𝑡 = 𝐸𝜀 𝑡 + 𝜂 𝜀 𝑡

?

𝜀 𝑡 =𝜎0𝐸

1 − 𝑒−𝐸𝑡𝜂

Modeling and Code


Creep test

Relaxation test

𝑡

𝜀

𝑡

𝜎

3) Standard (linear) solid model

𝜂𝑣

𝐸𝑒

𝐸𝑣

Modeling and Code


Creep test

Relaxation test

𝑡

𝜀

𝑡

𝜎

4) Another standard (linear) solid model

𝜂𝑣

𝐸𝑣𝐸𝑒

Modeling and Code

Modeling for Concrete

Early age behavior or Time dependent properties

for creep

for shrinkage

𝐿0

𝛥𝐿

𝐿0= 𝐽 𝑡 𝜎0

𝛥𝐿

𝐿0= 𝜀𝑠ℎ𝑟𝑖𝑛𝑘𝑎𝑔𝑒 𝑡

Modeling and Code

ACI model (ACI 209, Prediction of creep, shrinkage, and temp. effects in

concrete structure)

1) Material constants

𝑓𝑐′ 𝑡 =

𝑡

𝑎 + 𝑏𝑡𝑓𝑐,28′

𝐸𝑐 =0.4𝑓𝑐

′ − 𝑓𝑐,50⋅10−6

𝜀 𝑐 40% − 50 ⋅ 10−6

𝐸𝑐 = 0.043 ⋅ 𝑊1.5 𝑓𝑐′ 𝑡

𝑡

Norm

aliz

ed

𝑓𝑐′ 𝑡

𝐸𝑐 𝑡

𝑡28

Cement type I

IIIII

IVV

Moisture:Steam:

Moisture:Steam:

4 0.851 0.95

2.3 0.920.7 0.98

a b

Modeling and Code

ACI model (ACI 209R-92)

2) Shrinkage

Drying 90~95% 60~70%Autogenous 5~10% 30~40%

Normal High-strength

𝜀𝑠ℎ(𝑡, 𝑡0) =𝑡 − 𝑡0

𝛼

𝑓 + 𝑡 − 𝑡0𝛼 𝜀𝑠ℎ,𝑢

𝛼 = 1.0

𝑓 = 26.0 ⋅ exp 0.0142𝑣

𝑠

𝛾𝑠ℎ = 𝛾𝑐𝑝 ⋅ 𝛾𝜆 ⋅ 𝛾ℎ ⋅ 𝛾𝑠 ⋅ 𝛾𝜓 ⋅ 𝛾𝑐 ⋅ 𝛾𝛼

𝜀𝑠ℎ,𝑢 = −780 ⋅ 10−6𝛾𝑠ℎ

1 2 3 4 5 6 7

Modeling and Code


2) Shrinkage

𝛾𝑐𝑝

𝛾𝜆

𝛾ℎ

(Ref = 7 day)

(Ref = 40%)

(Ref = 150mm)

𝛾𝜆=1.40 − 0.010 ⋅ 𝜆 40~60%

3 − 0.030 ⋅ 𝜆 80~100%

𝛾ℎ =1.23 − 0.00015 ⋅ ℎ 𝑡 − 𝑡0 ≤ 1𝑦𝑟

1.17 − 0.00114 ⋅ ℎ 𝑡 − 𝑡0 > 1𝑦𝑟

ℎ(𝑚𝑚) 𝛾ℎ

50

100

150

1.35

1.17

1.00

or

𝛾ℎ = 1.2exp −0.00472𝑣

𝑠

Initial moisture curing periodday 1 3 7 14 28 90

𝛾𝑐𝑝 1.2 1.1 1 0.93 0.86 0.75

1

2

3

Modeling and Code


2) Shrinkage

𝛾𝑐

𝛾𝑠

𝛾𝜓

𝛾𝛼

(Ref = 68mm)

(Ref = 50%)

(Ref = 410kg/m3)

(Ref = 6.25%)

𝛾𝛼 = 0.95 + 0.008 ⋅ 𝛼

𝛾𝑐 = 0.75 + 0.00061 ⋅ 𝑐

𝛾𝜓=0.30 + 0.014 ⋅ 𝜓 𝜓 ≤ 50%

0.90 + 0.002 ⋅ 𝜓 𝜓 > 50%

𝛾𝑠 = 0.89 + 0.00161 ⋅ 𝑠

4

5

6

7

Modeling and Code


3) Creep

𝐽 𝑡, 𝑡′ =1

𝐸𝑐 𝑡′+𝜙 𝑡, 𝑡′

𝐸𝑐 𝑡

𝜙 𝑡, 𝑡′ =𝑡 − 𝑡′ 𝜓

𝑑 + 𝑡 − 𝑡′ 𝜓𝜙𝑢

𝑑 = 10, 𝜓 = 0.6

or

𝑑 = 𝑓 = 26.0 ⋅ exp 0.0142𝑣

𝑠, 𝜓 = 1.0

𝜙𝑢 = 2.35 ⋅ 𝛾𝑡𝑎 ⋅ 𝛾𝜆 ⋅ 𝛾ℎ ⋅ 𝛾𝑠 ⋅ 𝛾𝜓 ⋅ 𝛾𝑐 ⋅ 𝛾𝛼

1 2 3 4 5 6 7

Modeling and Code


3) Creep

𝛾𝑡𝑎

𝛾𝜆

𝛾ℎ

𝛾𝑠

𝛾ℎ =1.14 − 0.00363 ⋅

𝑣

𝑠𝑡 − 𝑡0 ≤ 1𝑦𝑟

1.10 − 0.00268 ⋅𝑣

𝑠𝑡 − 𝑡0 > 1𝑦𝑟

1

2

3

4

𝛾𝑡𝑎 =1.25𝑡′−0.118 𝑚𝑜𝑖𝑠𝑡, 𝑡′ ≥ 7𝑑

1.13𝑡′−0.094 𝑠𝑡𝑒𝑎𝑚, 𝑡′ ≥ 3𝑑

𝛾𝜆 =1.0 𝜆 < 40%

1.27 − 0.0067𝜆 𝜆 ≥ 40%

𝛾𝑠 = 0.82 + 0.00264 ⋅ 𝑠

Modeling and Code


3) Creep

𝛾𝜓

𝛾𝑐

𝛾𝛼

5

6

7

𝛾𝜓= 0.88 + 0.0024 ⋅ 𝜓

𝛾𝛼 =

𝛾𝑐 = 0.75 + 0.00061 ⋅ 𝑐

0.46 + 0.09 ⋅ 𝛼 (𝛼 ≥1)

1

Modeling and Code

FIB code (CEB-FIP MC90)


𝑓𝑐𝑚 = 𝑓𝑐𝑘 + 𝛥𝑓

𝛥𝑓 = 8𝑀𝑃𝑎

𝑓𝑐𝑚 𝑡 = 𝛽𝑐𝑐 𝑡 𝑓𝑐𝑚

𝛽𝑐𝑐 𝑡 = exp 𝑠 1 −28

𝑡 𝑡1

𝑡1 = 1d s =

0.35 (type 1 moist. curing)

0.15 (type 1 steam curing)



0.12 (type 3 steam curing)

Modeling and Code



𝐸𝑐𝑖 = 𝐸𝑐𝑜𝑓𝑐𝑚𝑓𝑐𝑚𝑜

13

𝐸𝑐𝑜 = 2.15 × 104𝑀𝑃𝑎 = 21.5 𝐺𝑃𝑎

𝑓𝑐𝑚𝑜 = 10𝑀𝑃𝑎

𝐸𝑐𝑖 𝑡 = 𝛽𝐸 𝑡 𝐸𝑐𝑖

𝛽𝐸 𝑡 = 𝛽𝑐𝑐 𝑡

𝑡

𝜀

creep

Elastic or instantaneous

𝐸𝑐 = 0.85𝐸𝑐𝑖

Note: chord vs initial tangent modulus (in ACI)

Modeling and Code


2) Drying shrinkage

𝜀𝑠ℎ 𝑡, 𝑡0 = 𝜀𝑠ℎ𝑜𝛽𝑠 𝑡 − 𝑡0

𝜀𝑠ℎ𝑜 =

𝜀𝑠ℎ𝑜 = 𝜀𝑠 𝑓𝑐𝑚 𝛽𝑅𝐻

𝜀𝑠 𝑓𝑐𝑚 = 160 + 10𝛽𝑠𝑐 9 −𝑓𝑐𝑚𝑓𝑐𝑚𝑜

× 10−6

𝛽𝑠𝑐 =

𝑓𝑐𝑚 =

𝛽𝑅𝐻 =−1.55 1 −

𝑅𝐻

100%

3

0.25

Notional shrinkage coefficient

Cement type

5 (N) or (R)

4 (SL)

6 (RS)

Comp. strength

40% ≤ 𝑅𝐻 ≤ 99%

𝑅𝐻 ≥ 99%

Modeling and Code


2) Drying shrinkage

𝛽𝑠 𝑡 − 𝑡0 = 𝑡 − 𝑡0 𝑡1

𝛽𝑠ℎ + 𝑡 − 𝑡0 𝑡1

0.5

𝜀𝑠ℎ 𝑡, 𝑡0 = 𝜀𝑠ℎ𝑜𝛽𝑠 𝑡 − 𝑡0

𝛽𝑠ℎ = 350 ×ℎ

ℎ0

2

ℎ0 =

ℎ =2𝐴𝑐𝑢

(Notional size of member, Ac=section area, u=perimeter length)

100 mm

Modeling and Code


3) Creep

𝜀𝑐𝑐 𝑡, 𝑡′ =𝜎𝑐 𝑡′

𝐸𝑐𝑖𝜙 𝑡, 𝑡′

𝜀𝑐 𝑡, 𝑡′ = 𝜎𝑐 𝑡′1

𝐸𝑐𝑖 𝑡′+𝜙 𝑡, 𝑡′

𝐸𝑐𝑖

𝐽 𝑡, 𝑡′

𝜙 𝑡, 𝑡′ = 𝜙0𝛽𝑐 𝑡, 𝑡′ (notational creep coefficient)

Modeling and Code


3) Creep

𝜙0 = 𝜙𝑅𝐻𝛽 𝑓𝑐𝑚 𝛽 𝑡′

𝜙𝑅𝐻 = 1 +1 − 𝑅𝐻/𝑅𝐻0

0.46ℎℎ0

1 3

𝛽 𝑓𝑐𝑚 =5.3

𝑓𝑐𝑚 𝑓𝑐𝑚𝑜

𝛽 𝑡′ =1

0.1 + 𝑡 ′ 𝑡10.2

𝛽𝑐 𝑡 − 𝑡′ = 𝑡 − 𝑡′ 𝑡1

𝛽𝐻 + 𝑡 − 𝑡′ 𝑡1

0.3

𝜙 𝑡, 𝑡′ = 𝜙0𝛽𝑐 𝑡, 𝑡′ (notational creep coefficient)

𝛽𝐻 = 150 1 + 1.2𝑅𝐻

𝑅𝐻0

18ℎ

ℎ0+ 250 ≤ 1,500

Modeling and Code

Example #1

A concrete slab is exposed to drying at 75% RH seven days after casting. Compute the drying shrinkage strain after (a) 60 days; (b) 180 days.

Modeling and Code

Example #2

Steam-cured precast beams are prestressed after 24 hours when the compressive strengthReaches 25 MPa. The level of prestress is 7 MPa. Determine the potential free strain that willOccur over the first year if the beams are exposed to 70% RH.

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