PDF Creep of Concrete - Seoul National University

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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, is large 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-H to stress

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Definition of Terms

• Creep

• is the phenomenon of a gradual increase in strain with time under a given level of sustained stress

• Stress relaxation

• is the phenomenon of gradual decrease in stress with time under a given level of sustained strain

(Jason Weiss)

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Creep

• 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 of concrete depends on

• stress induced by shrinkage and creep strains

• stress relief due to the visco-elasticity of concrete

(Kelly 1963)

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Elastic and Creep Strains

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

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

• Irreversible creep

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Definition of Terms

• Under typical service conditions, concrete is more likely to be drying while under load. It has been found that under such

conditions creep deformations are greater than if the concrete is dried prior to loading

• Free shrinkage ε_sh

• Basic creep ε_bc

• Total strain during

loading and drying ε_tot

• Drying creep ε_dc

• Total creep ε_cr= ε_bc + ε_dc

• Creep coefficient



e

C 

cr

(Mindess et al 2003)

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Column Shortening in a Tall Building

Summation of strains in a reinforced concrete column during construction of a tall building (Russell and Corley 1977)

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Factors Influencing Creep

• Applied stress

• Creep strain increases with applied stress.

• Relationship is not linear

• Approximately linear in the stress range generally used

• For practical reasons, a linear relationships is often used to compare different concrete specimens loaded at different stress levels

• Specific creep

• typical value 150X10

   

^cr ^-6/MPa

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Factors Influencing Creep (cont’d)

• W/C ratio

• Conflicting data (not possible to change one parameter independent of others)

• w/c ↓, specific creep ↓

• Compressive strength ↑, specific creep ↓

• Curing condition

• The time of moist curing of concrete at loading affect the magnitude of creep

• An increase in the

temperature of curing reduce both basic & drying creep

(Neville 1970)

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• 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 condition for creep

• Creep is a function evaporable water, falls to zero when no

evaporable water is present. The greatest ↓ in evaporable water, hence in creep, occurs on drying to 40% RH while water is lost from capillary pores.

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

(Troxell et al 1958)

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• Aggregate

ε_con= ε_p(1-V_a)ⁿ

• Amount

• Modulus of elasticity of aggregate

(Troxell et al 1958)

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• Specimen geometry

• Volume-to-surface ratio and

specimen thickness affect the total creep in much the same way as drying shrinkage is affected

• Creep Recovery

• Only a small portion of total creep strain is recoverable when

concrete is unloaded

• Proportion of irreversible creep ↑ with time under load

• After ~30 days under load, additional creep is largely irreversible

• Drying creep is irreversible

(Hanson & Mattock 1966)

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Mechanisms of Creep

• Thermally activated creep

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

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

• Deformation can only occur by going through an energy barrier in the 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 from external sources

• Stress, strain

• Change in moisture

• Change in temperature

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Mechanisms of Creep (cont’d)

• Role of absorbed water

• Nature of “creep center”?

Operating process?

• Slip between adjacent

particles of C-S-H under a shear stress; the ease and extent of slip depend on the force of attraction between particles

• Measurable slip occurs only when sufficient

thickness of water exists between the particles

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Mechanisms of Creep (cont’d)

• Role of absorbed water (cont’d)

• Creep can also results from diffusion of water in micropores under stress

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

• Thickness of absorbed water ↓

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

• Reordering of C-S-H can lead to increased bonding between them, leading to elimination of micropores,

“aging”

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(Mehta & Monteiro 2006)

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(Mehta & Monteiro 2006)

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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

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 Viscoelastic behavior Viscous fluid (Newtonian)

Elastic solid (Hookean)

Viscoelastic material

𝜎

𝑡

Viscoelastic solid Viscoelastic fluid

𝜀

𝜎

𝜎 𝜎₀

𝜎₀

𝜀₀

𝐸 𝜂

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 Viscoelastic behavior

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

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 Viscoelastic behavior

Where Heaviside step function

𝑡 1.0

𝑡 𝐻 𝑡 =

𝐻 𝑡 𝐻 𝑡 − 𝑡₀

𝑡₀

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 Classical linear viscoelasticity (linear superposition or Boltzmann’s superposition) i) Infinitesimal deformation (neglect second-order term)

ii) Instantaneous stress is proportional to input strain iii) Relaxation rate is proportional to instantaneous stress iv) No aging effect is assumed

Creep response

𝜎

𝑡

𝜀

𝑡 𝜀(𝑡) 𝜎 𝑡

Assumptions:

𝑡₀ 𝑡₀

𝛥𝜎₀ 𝛥𝜀₀

𝛥𝜀₁ 𝛥𝜎₁

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

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

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 Classical linear viscoelasticity (linear superposition or Boltzmann’s superposition) i) Infinitesimal deformation (neglect second-order term)

ii) Instantaneous stress is proportional to input strain iii) Relaxation rate is proportional to instantaneous stress iv) No aging effect is assumed

Relaxation response

𝜎

𝑡 𝜀

𝑡 𝜀(𝑡)

𝜎 𝑡

Assumptions:

𝑡₀ 𝑡₀

𝛥𝜎₀ 𝛥𝜀₀

𝛥𝜀₁ 𝛥𝜎₁

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

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

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 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 𝑡 𝐸

𝜎

𝑡 𝜎₀/𝐸

𝑡 𝜀

𝐸𝜀₀

𝜂 𝜎 𝑡 = 𝜂 𝜀 𝑡

𝜎

𝑡 𝑡

𝜀

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 Constitutive equations 1) Maxwell model (Reuss)

𝐸 𝜂

Equilibrium eq.

Compatibility eq.

Constitutive rel. (spring)

Constitutive rel. (dashpot)

Constitutive rel. (Maxwell)

𝜎_𝐸 𝑡 = 𝜎_𝜂 𝑡 = 𝜎 𝑡

𝜀 𝑡 = 𝜀_𝐸 𝑡 + 𝜀_𝜂 𝑡

𝜎_𝐸 𝑡 = 𝐸𝜀_𝐸 𝑡

𝜎_𝜂 𝑡 = 𝜂 𝜀_𝜂 𝑡

𝜀𝜂 𝑡 = 𝜎(𝑡)

𝐸 + 𝜎(𝑡) 𝜂

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 Constitutive equations 1) Maxwell model (Reuss)

𝐸 𝜂

Constitutive rel. (Maxwell)

Creep test

Relaxation test

𝜀 𝑡 = 𝜎₀

𝐸 + 𝜎₀ 𝜂 𝑡

𝜎 𝑡 = 𝐸𝜀₀𝑒^−𝐸^{𝑡 𝜂}

𝑡 𝜀

𝑡 𝜎

𝜀𝜂 𝑡 = 𝜎(𝑡)

𝐸 + 𝜎(𝑡) 𝜂

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 Constitutive equations 2) Kelvin model (Voigt)

𝐸 𝜂

Equilibrium eq.

Compatibility eq.

Constitutive rel. (spring)

Constitutive rel. (dashpot)

Constitutive rel. (Kelvin)

𝜎 𝑡 = 𝜎_𝐸 𝑡 + 𝜎_𝜂 𝑡

𝜀 𝑡 = 𝜀_𝐸 𝑡 = 𝜀_𝜂 𝑡

𝜎_𝐸 𝑡 = 𝐸𝜀_𝐸 𝑡

𝜎_𝜂 𝑡 = 𝜂 𝜀_𝜂 𝑡

𝜎 𝑡 = 𝐸𝜀 𝑡 + 𝜂 𝜀 𝑡

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Constitutive rel. (Kelvin)

Creep test

Relaxation test

𝑡 𝜀

𝑡 𝜎

2) Kelvin model (Voigt)

𝐸 𝜂

𝜎 𝑡 = 𝐸𝜀 𝑡 + 𝜂 𝜀 𝑡

?

𝜀 𝑡 = 𝜎₀

𝐸 1 − 𝑒⁻

𝐸𝑡 𝜂

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Creep test

Relaxation test

𝑡 𝜀

𝑡 𝜎

3) Standard (linear) solid model

𝜂_𝑣 𝐸_𝑒

𝐸_𝑣

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Creep test

Relaxation test

𝑡 𝜀

𝑡 𝜎

4) Another standard (linear) solid model

𝜂_𝑣 𝐸_𝑣 𝐸_𝑒

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 Modeling for Concrete

Early age behavior or Time dependent properties

for creep

for shrinkage

𝐿₀

𝛥𝐿

𝐿₀ = 𝐽 𝑡 𝜎₀

𝛥𝐿

𝐿₀ = 𝜀_{𝑠ℎ𝑟𝑖𝑛𝑘𝑎𝑔𝑒} 𝑡

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 ACI model (ACI 209, Prediction of creep, shrinkage, and temp. effects in concrete structure)

1) Material constants

𝑓_𝑐^′ 𝑡 = 𝑡

𝑎 + 𝑏𝑡𝑓_𝑐,28^′

𝐸_𝑐 = 0.4𝑓_𝑐^′ − 𝑓_𝑐,50⋅10⁻⁶ 𝜀_{𝑐 40%} − 50 ⋅ 10⁻⁶

𝐸_𝑐 = 0.043 ⋅ 𝑊^1.5 𝑓_𝑐^′ 𝑡

𝑡

Normalized 𝑓_𝑐^′ 𝑡

𝐸_𝑐 𝑡

𝑡₂₈

Cement type I IIIII

IVV

Moisture:

Steam:

Moisture:

Steam:

4 0.85 1 0.95

2.3 0.92 0.7 0.98 a b

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 ACI model (ACI 209R-92) 2) Shrinkage

Drying 90~95% 60~70%

Autogenous 5~10% 30~40%

Normal High-strength

𝜀_𝑠ℎ(𝑡, 𝑡₀) = 𝑡 − 𝑡₀ ^𝛼

𝑓 + 𝑡 − 𝑡₀ ^𝛼 𝜀_𝑠ℎ,𝑢 𝛼 = 1.0

𝑓 = 26.0 ⋅ exp 0.0142𝑣 𝑠

𝛾_𝑠ℎ = 𝛾_𝑐𝑝 ⋅ 𝛾_𝜆 ⋅ 𝛾_ℎ ⋅ 𝛾_𝑠 ⋅ 𝛾_𝜓 ⋅ 𝛾_𝑐 ⋅ 𝛾_𝛼 𝜀_𝑠ℎ,𝑢 = −780 ⋅ 10⁻⁶𝛾_𝑠ℎ

1 2 3 4 5 6 7

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 ACI model (ACI 209R-92)

2) Shrinkage

𝛾_𝑐𝑝

𝛾_𝜆

𝛾_ℎ

(Ref = 7 day)

(Ref = 40%)

(Ref = 150mm)

𝛾_𝜆= 1.40 − 0.010 ⋅ 𝜆 40~60%

3 − 0.030 ⋅ 𝜆 80~100%

𝛾_ℎ = 1.23 − 0.00015 ⋅ ℎ 𝑡 − 𝑡₀ ≤ 1𝑦𝑟 1.17 − 0.00114 ⋅ ℎ 𝑡 − 𝑡₀ > 1𝑦𝑟

ℎ(𝑚𝑚) 𝛾_ℎ

50 100 150

1.35 1.17 1.00 or

𝛾_ℎ = 1.2exp −0.00472𝑣 𝑠

Initial moisture curing period ^day ¹ ³ ⁷ ¹⁴ ²⁸ ⁹⁰

𝛾_𝑐𝑝 1.2 1.1 1 0.93 0.86 0.75

1

2

3

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 ACI model (ACI 209R-92) 2) Shrinkage

𝛾_𝑐 𝛾_𝑠

𝛾_𝜓

𝛾_𝛼

(Ref = 68mm)

(Ref = 50%)

(Ref = 410kg/m3)

(Ref = 6.25%)

𝛾_𝛼 = 0.95 + 0.008 ⋅ 𝛼 𝛾_𝑐 = 0.75 + 0.00061 ⋅ 𝑐

𝛾_𝜓= 0.30 + 0.014 ⋅ 𝜓 𝜓 ≤ 50%

0.90 + 0.002 ⋅ 𝜓 𝜓 > 50%

𝛾_𝑠 = 0.89 + 0.00161 ⋅ 𝑠

4

5

6

7

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 ACI model (ACI 209R-92) 3) Creep

𝐽 𝑡, 𝑡′ = 1

𝐸_𝑐 𝑡′ + 𝜙 𝑡, 𝑡′

𝐸_𝑐 𝑡

𝜙 𝑡, 𝑡′ = 𝑡 − 𝑡′ ^𝜓

𝑑 + 𝑡 − 𝑡′ ^𝜓 𝜙_𝑢 𝑑 = 10, 𝜓 = 0.6

or

𝑑 = 𝑓 = 26.0 ⋅ exp 0.0142𝑣

𝑠 , 𝜓 = 1.0 𝜙_𝑢 = 2.35 ⋅ 𝛾_𝑡𝑎 ⋅ 𝛾_𝜆 ⋅ 𝛾_ℎ ⋅ 𝛾_𝑠 ⋅ 𝛾_𝜓 ⋅ 𝛾_𝑐 ⋅ 𝛾_𝛼

1 2 3 4 5 6 7

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𝛾_𝑡𝑎

𝛾_𝜆

𝛾_ℎ

𝛾_𝑠

𝛾_ℎ =

1.14 − 0.00363 ⋅ 𝑣

𝑠 𝑡 − 𝑡₀ ≤ 1𝑦𝑟 1.10 − 0.00268 ⋅ 𝑣

𝑠 𝑡 − 𝑡₀ > 1𝑦𝑟

1

2

3

4

𝛾_𝑡𝑎 = 1.25𝑡′^−0.118 𝑚𝑜𝑖𝑠𝑡, 𝑡′ ≥ 7𝑑 1.13𝑡′^−0.094 𝑠𝑡𝑒𝑎𝑚, 𝑡′ ≥ 3𝑑

𝛾_𝜆 = 1.0 𝜆 < 40%

1.27 − 0.0067𝜆 𝜆 ≥ 40%

𝛾_𝑠 = 0.82 + 0.00264 ⋅ 𝑠

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𝛾_𝜓

𝛾_𝑐

𝛾_𝛼

5

6

7

𝛾_𝜓= 0.88 + 0.0024 ⋅ 𝜓

𝛾_𝛼 =

𝛾_𝑐 = 0.75 + 0.00061 ⋅ 𝑐

0.46 + 0.09 ⋅ 𝛼 (𝛼 ≥1) 1

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 FIB code (CEB-FIP MC90) 1) Material constants

𝑓_𝑐𝑚 = 𝑓_𝑐𝑘 + 𝛥𝑓 𝛥𝑓 = 8 𝑀𝑃𝑎

𝑓_𝑐𝑚 𝑡 = 𝛽_𝑐𝑐 𝑡 𝑓_𝑐𝑚

𝛽_𝑐𝑐 𝑡 = exp 𝑠 1 − 28 𝑡 𝑡₁

𝑡₁ = 1d s =

0.35 (type 1 moist. curing) 0.15 (type 1 steam curing) 0.40 (type 2 moist. curing) 0.25 (type 3 moist. curing) 0.12 (type 3 steam curing)

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 FIB code (CEB-FIP MC90) 1) Material constants

𝐸_𝑐𝑖 = 𝐸_𝑐𝑜 𝑓_𝑐𝑚 𝑓_𝑐𝑚𝑜

1 3

𝐸_𝑐𝑜 = 2.15 × 10⁴𝑀𝑃𝑎 = 21.5 𝐺𝑃𝑎 𝑓_𝑐𝑚𝑜 = 10𝑀𝑃𝑎

𝐸_𝑐𝑖 𝑡 = 𝛽_𝐸 𝑡 𝐸_𝑐𝑖 𝛽_𝐸 𝑡 = 𝛽_𝑐𝑐 𝑡

𝑡 𝜀

creep

Elastic or instantaneous

𝐸_𝑐 = 0.85𝐸_𝑐𝑖

Note: chord vs initial tangent modulus (in ACI)

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 FIB code (CEB-FIP MC90) 2) Drying shrinkage

𝜀_𝑠ℎ 𝑡, 𝑡₀ = 𝜀_𝑠ℎ𝑜𝛽_𝑠 𝑡 − 𝑡₀ 𝜀_𝑠ℎ𝑜 =

𝜀_𝑠ℎ𝑜 = 𝜀_𝑠 𝑓_𝑐𝑚 𝛽_𝑅𝐻

𝜀_𝑠 𝑓_𝑐𝑚 = 160 + 10𝛽_𝑠𝑐 9 − 𝑓_𝑐𝑚

𝑓_𝑐𝑚𝑜 × 10⁻⁶

𝛽_𝑠𝑐 = 𝑓_𝑐𝑚 = 𝛽_𝑅𝐻 =

−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%

(42)

 FIB code (CEB-FIP MC90) 2) Drying shrinkage

𝛽_𝑠 𝑡 − 𝑡₀ = 𝑡 − 𝑡₀ 𝑡₁ 𝛽_𝑠ℎ + 𝑡 − 𝑡₀ 𝑡₁

0.5

𝜀_𝑠ℎ 𝑡, 𝑡₀ = 𝜀_𝑠ℎ𝑜𝛽_𝑠 𝑡 − 𝑡₀

𝛽_𝑠ℎ = 350 × ℎ ℎ₀

2

ℎ₀ =

ℎ = 2𝐴_𝑐 𝑢

(Notional size of member,

Ac=section area, u=perimeter length) 100 mm

(43)

 FIB code (CEB-FIP MC90) 3) Creep

𝜀_𝑐𝑐 𝑡, 𝑡′ = 𝜎_𝑐 𝑡′

𝐸_𝑐𝑖 𝜙 𝑡, 𝑡′

𝜀_𝑐 𝑡, 𝑡′ = 𝜎_𝑐 𝑡′ 1

𝐸_𝑐𝑖 𝑡′ + 𝜙 𝑡, 𝑡′

𝐸_𝑐𝑖 𝐽 𝑡, 𝑡′

𝜙 𝑡, 𝑡′ = 𝜙₀𝛽_𝑐 𝑡, 𝑡′ (notational creep coefficient)

(44)

 FIB code (CEB-FIP MC90) 3) Creep

𝜙₀ = 𝜙_𝑅𝐻𝛽 𝑓_𝑐𝑚 𝛽 𝑡′

𝜙_𝑅𝐻 = 1 + 1 − 𝑅𝐻/𝑅𝐻₀ 0.46 ℎ

ℎ₀

1 3

𝛽 𝑓_𝑐𝑚 = 5.3 𝑓_𝑐𝑚 𝑓_𝑐𝑚𝑜

𝛽 𝑡′ = 1

0.1 + 𝑡 ′ 𝑡₁ ^0.2 𝛽_𝑐 𝑡 − 𝑡′ = 𝑡 − 𝑡′ 𝑡₁

𝛽_𝐻 + 𝑡 − 𝑡′ 𝑡₁

0.3

𝜙 𝑡, 𝑡′ = 𝜙₀𝛽_𝑐 𝑡, 𝑡′ (notational creep coefficient)

𝛽_𝐻 = 150 1 + 1.2 𝑅𝐻 𝑅𝐻₀

18 ℎ

ℎ₀ + 250 ≤ 1,500

(45)

 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.

(46)

 Example #2

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