Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev

Mechanics and Design of Concrete Structures

Civil Engineering (CE) : Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev

The document Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Mechanics and Design of Concrete Structures.
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Creep and Shrinkage Deformation
→ Creep of concrete
o Concrete under stress undergoes a gradual increase of strain with time. The final creep strain may be several times as large as the initial elastic strain.
o Creep is the property of materials by which they continue deforming over considerable length of time under sustained stress.
o Relaxation is the loss of stress by time with constant strain.
o In concrete, creep deformations are generally larger than elastic deformation and thus creep represents an important factor affecting the deformation behavior.
o Concrete under constant axial compressive stress

Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev

→ Experiments indicate that for working stress range – i.e. stresses not exceeding 0.5 f'c– creep strains are directly proportional to stress that is a linear relationship between σ and εcr .
→ Microcracking effect on creep in high stresses.

→ Mechanisms of concrete creep
o Two phenomena are involved:
1. Time dependent deformation that occurs when concrete is loaded in a sealed condition so that moisture cannot escape. → basic creep
2. Creep of the material when moisture exchange is permitted. → drying creep

o Basic creep is primarily influenced by the material properties only, while drying creep and shrinkage also depend on the environment and the size of the specimen.
o The real situation might be the combination of the two phenomena, sometimes, one being the dominating factor.
o Creep deformation contains three regions:
1. Primary creep → initial increase in deformation
2. Secondary creep → relatively a steady deformation region
3. Tertiary creep → leads to creep fracture

→ Specific creep
o Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev

where ε= a function of time. → Stress levels above 0.8f'c creep produces failure in time.
o Relationship between f'c and εsp :
Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev

→ Creep coefficient
Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
where εinst = instantaneous (initial) strain.
 

Factors influencing creep:
o Internal factors (composition)
→ Aggregate (concentration + stiffness)           Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
→ Water/cement ratio                                      Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev                                         
→ Aggregate permeability                                Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
→ Aggregate creep Creep                                 Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
→ Aggregate stiffness Creep                              Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
→ Aggregate grading and distribution      
→ Cement
o External factors (environment, time history)
→ Size
→ Shape
→ Cross-section                                                    Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
→ Environmental factors (ambient humidity, temperature)
→ Stress intensity                                               Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
→ Time (age of loading) → the loading history is important to the total deformation (strain).
                                    → age of loading Creep  Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev


→ Mathematical modeling of creep:
o Strain decomposition
The total strain of concrete may be decomposed as
Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
where
εσ = stress-induced strain,
εE = reversible strain,
εC = creep strain,
ε​0 = stress-independent inelastic strain,
εS = shrinkage strain,
εT = thermal expansion, and
ε" = inelastic strain.

o Incremental formulation of stress-strain relation
{dσ} = [D]{dε}

where {dσ} = change in stress tensor,
[D] = constitutive law of the material, and
{dε} = change in strain tensor.

o First-order and higher orders formulations of creep and shrinkage deformation:
→ First-order formulations: The incremental elastic stiffness matrix changes from one time step to the next as proportional to the elastic modulus. Applicable to homogeneous materials.
→ Higher order formulations: The incremental elastic stiffness matrix is different and not proportional to the elastic modulus from one time step to another. Applicable to non-homogeneous materials.

o Incremental quasi-elastic stress-strain law

∆ε = J∆σ + ∆ε0 

where
∆ε = a column matrix consisting of the strain increments,
J = compliance (square) matrix (function),
∆σ = a column matrix consisting of the stress increments, and
∆ε0 = a column matrix consisting of the inelastic strain increments.

→ Linear methods for the calculation of creep strain
o Effective modulus method
Total stress-strain relation:
Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
where t = current time,
t' = the time at which the instantaneous elastic modulus is characterized,
Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev   = effective modulus of elasticity,
ε(t) = stress-independent inelastic strain,

Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev compliance function (the creep function) representing the strain (elastic + creep) at time t

E (t')= modulus of elasticity characterizing the instantaneous deformation at time , t'

Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev = creep coefficient representing the ratio of the creep deformation to the initial elastic deformation, and

Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev specific creep function Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev εsp).

→ Incremental strain The method is applied incrementally from tto  t if the structural system is changed after .t0

Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev

where ∆σ(t) = incremental form of the stress at time t,
∆ε0(t) = incremental form of the stress-independent inelastic strain, and
t0 = time of first loading.

→ Effective modulus method is exact when the stress is constant from t' to t. 

o Linear variation method
→ Assuming the mechanical strain ε −ε0 is zero up to time tand jumps to the value of a, apply linear variation of strain from tto  t and obtain the following expression:

Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
where a, c = constants.

One can then obtain by algebraic manipulation the basic incremental stress-strain relation of the age-adjusted effective modulus.

o Age-adjusted effective modulus method
Effective elastic modulus with a correction of the creep coefficient:

Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev

where χ = the relaxation coefficient (aging coefficient, age-adjusted effective modulus) = 0.5~1.0.

The quasi-elastic incremental stress-strain relation of the ageadjusted effective modulus:

Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev

We assume isotropic material with a constant Poisson’s ratio, which is approximately true for concrete.

→ Practical considerations of creep prediction
o Considering the specific strain function expressed as

Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev

That is strain per unit stress at time t for the stress applied at age τ .

o The specific creep function

Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
o Forms of the aging function F (τ ) and f (t −τ ) :

→ F (τ )
• Power law: Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
• Exponential law: Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
f (t −τ )
• Power law: Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
• Logarithmic expression: Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev  where (t -τ ) is measured in days.

→ Experimental data is used to obtain coefficients.

o Prediction of creep deformation:
→ Perform short-term experimental results → Develop short-term creep curves to be used in long-term prediction.
→ When experimental data is not available, the design has to rely on a relevant code. Various code recommendations have been quite controversial.
→ ACI 209 Committee defines the creep coefficient as  ϕ (t t0
Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
where (t − t0) = time since application of load,

Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev = ultimate creep,

K1 and K2 relate to curing process, K3 relates to the thickness of member, K4,K5 , K6 and relate to concrete composition. (α = 2.35)

→ Principle of superposition
For analysis and design purposes the time dependent linear relation between stress and creep strain can be written in the following way:
Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev

where f0 = initial stress in concrete at the time τ of first loading,
Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev = additional stress increment or decrement applied at time  Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev  = specific creep strain at time t for concrete at age Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev

→ Effect of stress on creep is predicted by the superposition method.
→ Creep strains produced in concrete at any time t by an increment of stress applied at any time τ are independent of the effects of any other stress increment applied earlier or later than time τ.
→ This method of analysis predicts creep recovery and generates stress-strain curves of a shape similar to experimental results. It is assumed that the concrete creeps in tension at the same rate as it creeps in compression.
→ Creep under variable stress can be obtained by superimposing appropriate creep curves introduced for corresponding changes in stress at different time intervals. This is true if creep is proportional to applied stress.

→ Creep under different states of stress:
o Creep takes place in tension in the same manner as in compression. The magnitude in tension is much higher.
o Creep occurs under torsional loading.
o Under unaxial compression, creep occurs not only in the axial but also in the normal directions.
o Under multiaxial loading creep occurs in all directions and affected by stresses in other normal stresses.

→ Non-linear factors in the evaluation of creep strain
o The influences of
– Temperature dependence
– Creep in combined stress
– Shrinkage o Humidity and temperature variation

→ Cross coupling of creep with shrinkage or thermal expansion.
o Cracking or strain-softening
o Cyclic loading
o High stress and multiaxial viscoplasticity

→ Examples of complex structures with significant effect of creep on deformation and failure behavior
o Thermal effect (pressure vessels, process vessels)
o Segmental concrete bridge deformation (construction and service stages)

→ Shrinkage deformation
o Shrinkage is basically a volume change of the element which is considered to occur independently of externally imposed stresses. This volume change can also be negative and is then called swelling. Thin sections are particularly susceptible to drying shrinkage and therefore must contain a certain minimum quantity of steel. In restrained concrete shrinkage results in cracking, before any loading. The minimum reinforcement is provided to control this cracking.

o Causes of shrinkage:
→ Loss of water on drying
→ Volume change on carbonation

o On average ultimate shrinkage Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
o ACI 209 Committee suggests:
Creep and Shrinkage Deformation Civil Engineering (CE) Notes | EduRev
where
εsh,u = ultimate factor of shrinkage,
St = factor of time,
Sh = factor of humidity,
Sth = factor of thickness,
Ss = factor of slump,
Sf = factor of fineness,
Se = factor of air content, and
Sc = factor of cement property

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