Z.C. Grasley, D.A. Lange, A.J. Brinks, M.D. D’Ambrosia University of Illinois at Urbana-Champaign
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Z.C. Grasley, D.A. Lange, A.J. Brinks, M.D. D’Ambrosia
University of Illinois at Urbana-Champaign
MODELING AUTOGENOUS SHRINKAGE OF CONCRETE ACCOUNTING FOR CREEP CAUSED BY AGGREGATE RESTRAINT
Sponsors: PCA, NHI/FHWA, IDOT, CEAT
Why a composite model? Models that allow the prediction of concrete shrinkage
as f(Sp, mech. properties) are valuable modeling tools Predict the effect of segregation on shrinkage of SCC layers Input for FEM model that considers differential drying
shrinkage with depth Bridge deck or pavement Curling or cracking
While our model will be validated using autogenous shrinkage, should apply to drying also
Many models have already been developed, but… Existing models based on theory of elasticity An example: Pickett’s model uses elasticity theory to
predict concrete shrinkage S=S(E,Eg,, g,Sp,g)
Problem: cement paste is viscoelastic, so Pickett’s model tends to over-predict shrinkage as time increases because creep relaxes restraining stress
Solution: rework Pickett’s model using a viscoelastic constitutive theory rather than elastic
Pickett, G., Effect of aggregate on shrinkage of concrete and hypothesis concerning shrinkage. American Concrete Institute -- Journal, 1956. 27(5): p. 581-590.
(1 )pS S g
Evidence of Pickett Problem
-200
-150
-100
-50
0
0 20 40 60 80 100
ViscoelasticElasticAlpha = 1.7Mix-1
Stra
in x
10-6
Age (d)
Creep
Visualizing the effect of aggregate restraint
Shrinkage predicted by elastic modelShrinkage of viscoelastic materialShrinkage considering dilution only
Aggregate
Paste
2(1 )pS S g
> Sviscoelastic
(1 )pS S g
Sdilution
1(1 )pS S g
> Selastic
Physical model representation
qagg
qconc
qagg = qconc
Conversion of Pickett’s model
0
3(1 )( )(1 2 )
1 2( , ') ( ') '
gt
g
f t
E J t t f t dt
(1 )pS S g gg EE /)21(21
)1(3
where
where
(1 )pS S g Viscoelastic
Elastic
f(t) = loading function= Poisson ratio of concreteg = Poisson ratio of aggregateE = Young’s modulus of concreteEg = Young’s modulus of aggregateJ(t,t’) = viscoelastic compliance of concreteSp = paste shrinkageg = aggregate volume fraction
Accounting for aging
')'()',()(0
dttttJtt
0 '
( ')1 1( , ')( ') ( )
th
t
J tJ t t d
E a t a
Solidified gel
da()a(t)
Pore water
Gel solidifying at time
g(a,)
Constitutive equation for aging viscoelastic material
Solidification theory
Bazant, Z.P., Viscoelasticity of Solidifying Porous Material - Concrete. J. of the Eng. Mech. Div., ASCE, 1977. 103(EM6): p. 1049-1067.
Materials modeledSG Unit Mix-1 Mix-2 Mix-3
Cement (Type I) 3.15 kg/m3 392 357 403
Fly Ash (Class C) 2.65 kg/m3 93 193 90
Coarse Aggregate, 3/4" (20 mm) 2.70 kg/m3 218 810 343
Coarse Aggregate, 3/8" (10 mm) 2.70 kg/m3 638 0 604
Fine Aggregate (FM = 2.57) 2.64 kg/m3 832 792 824
Water 1.00 kg/m3 185 179 158
Superplasticizer (CAE) 1.06 l/m3 2.44 1.12 1.38w/cm 0.38 0.33 0.32
Required model parameters
Elastic modulus Paste autogenous shrinkage Concrete autogenous shrinkage Concrete creep Aging function (elastic and creep) Aggregate elastic properties
Measuring shrinkage and creep
Measured paste shrinkage
-1300
-1100
-900
-700
-500
-300
-100
0 20 40 60 80 100 120 140 160
Age (d)
Stra
in x
10-6
Mix-1Mix-2Mix-2 repeatMix-3
w/cm = 0.38
w/cm = 0.33w/cm =0.32
Measured concrete shrinkage
-200
-150
-100
-50
0
0 20 40 60 80 100
Mix-1Mix-2
Mix-3
Stra
in x
10-6
Age (d)
High paste content
w/cm = 0.38
w/cm = 0.33
w/cm =0.32
Determining creep function
100
150
200
250
300
350
400
0 50 100 150 200 250
Cre
ep S
train
x 1
06
Age (d)
Mix-1
Kelvin Chain
Measuring elastic response
Determination of Aging Function
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30
Agi
ng fu
nctio
n (n
orm
aliz
ed a
t 28
d)
Age (d)
New model improves fit
-200
-150
-100
-50
0
0 20 40 60 80 100
ViscoelasticElasticAlpha = 1.7Mix-1
Stra
in x
10-6
Age (d)
Model prediction of Mix-1 shrinkage
(1 )pS S g
Improvement again
-400
-350
-300
-250
-200
-150
-100
-50
0
0 20 40 60 80 100
ViscoelasticElasticAlpha = 1.7Measured
Stra
in x
10-6
Age (d)
Model prediction of Mix-3 shrinkage
Even better
-400
-350
-300
-250
-200
-150
-100
-50
0
0 20 40 60 80 100
ViscoelasticElasticAlpha = 1.7Measured
Stra
in x
10-6
Age (d)
Model prediction of Mix-2 shrinkage
Does high paste content better fit? Why? Less damage?
Tangential stress is function of b/c
Higher g
Higher likelihood of damage, nonlinearity of creep
Reduction in shrinkage
Damage/nonlinearity
Measured shrinkage
Predicted shrinkage – viscoelastic model
Time
b
c
PasteAggregate
Why not perfect fit? Linear viscoelasticity is assumed No damage such as microcracking is considered
around aggregates Dependence of J(t,t’) on g is ignored Aging function determined from elastic tests A time-independent, stress history independent
Poisson’s ratio was assumed
Current work Importance of aggregate dependence
Solve model equations with J(t,t’) as f(g) Use paste creep and elastic properties
Assumption of constant Poisson ratio Solve model in terms of E(t,t’) and K(t,t’) (substitute for Poisson
ratio) Use new experimental methods to measure K
Compare to existing model predictions Combine model with paste shrinkage prediction model Account for nonlinearity and/or damage effects
Summary New model has been developed for predicting concrete
shrinkage Model is extension of Pickett’s model Includes creep Improves on Pickett’s elastic model
Creep is present as result of aggregate restraint Model still over-predicts concrete autogenous shrinkage
Nonlinearity and damage Increasing g in mixture design may reduce shrinkage not
only by reducing paste content, but also by inducing stress-relaxing damage ~ additional creep
Effect of creep on alpha
(1 )pS S g
Larger alpha = lower predicted shrinkage better fit
1
1.2
1.4
1.6
1.8
2
0 20 40 60 80 100
Mix-1 Viscoelastic AlphaMix-1 Elastic Alpha Mix-2 Viscoelastic AlphaMix-2 Elastic Alpha Mix-3 Viscoelastic AlphaMix-3 Elastic Alpha
Alp
ha V
alue
s
Age (d)
Evidence of tangential cracks around aggregates
Bisschop, J., Drying shrinkage microcracking in cement-based materials. 2002, Delft University: Delft, The Netherlands.
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