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1
Lecture 5Drying shrinkage
Pietro Lura
Concrete & Construction Chemistry
Shrinkage and Cracking of Concrete: Mechanisms and Impact on Durability, ETHZ, 19.10.11
2
Contents
� Drying conditions
� Drying of cement pastes at equilibrium
� Drying gradients
� Self-induced stresses and degree of restraint
2
3
Drying shrinkage cracking
Photo: Loser 2008
4
Stimulus -response
Weiss 2008
Ivan Pavlov (1849 –1936)Nobel prize 1904
3
5
Volume change due to mechanical loading
Weiss 2008
� Stimulus – application of a force
� Response – length change and compensation in the transverse direction
F = K ∆
Ten
sio
n
Co
mp
ress
ionFo
rce
Deformation
K
6
Volume change due to temperature change
δTEMP
Temperature Deformations
• Stimulus is change in temperature
• Response is the change in length (volume)
• This happens in nearly all materials
δTEMP=α · L · ∆TεTEMP=α · ∆T -1
000
-800
-600
-400
-200 0
200
400
600
800
1000
Longitudinal Strain (µε)
-20
0
20
40
60
80
100
Tem
per
atu
re C
han
ge
(C)
1α
Weiss 2008
4
7
Shrinkage due to moisture loss
-70 -60 -50 -40 -30 -20 -10 0Change in Relative Humidity (%)
-5000
-4000
-3000
-2000
-1000
0
1000
Sh
rin
kag
e S
trai
n (
µεµε µεµε)
30 40 50 60 70 80 90 100Relative Humidity (%)
0% SRA
-70 -60 -50 -40 -30 -20 -10 0Change in Relative Humidity (%)
-5000
-4000
-3000
-2000
-1000
0
1000
Sh
rin
kag
e S
trai
n (
µεµε µεµε)
30 40 50 60 70 80 90 100Relative Humidity (%)
0% SRA
( )31 RHNSH −=∞− βε
• Stimulus is change in moisture (drying external or internal) and chemical reaction
• Response is the change in length (volume)
Weiss 2008
8
Drying conditions (1)
Moisture
RH T v
RH
5
9
Drying conditions (2)
Pore Volume
after Bentz and Jensen CCC 2004
r1r2
( )W
cap V
RTp
⋅= RHln
( ) RT
Vcosr W
⋅⋅⋅−=
RHln
2 ϑγPore Radius
r3
r3 r2 r1
10
Drying shrinkage of old cement pastes (1)
Loser and Lura 2009-2011
w/c 0.3 cement pastes, water cured for 6 months prior to dryingSmall samples, 10x10x80 mm3
5-6 weeks before equilibrium at each RH step
-1000
-800
-600
-400
-200
0
200
70 75 80 85 90 95 100
Relative humidity [%]
Stra
in [
m/m
]
CEM III/B w/c 0.3
6
11
Drying shrinkage of old cement pastes (2)
Drying to 94⇒86⇒75%
0.1
1
10
100
1000
10000
405060708090100
Relative humidity (%)
Cap
illar
y p
ress
ure
(M
Pa)
-140
-120
-100
-80
-60
-40
-20
0
Kel
vin
rad
ius
(nm
)
p cap
Kelvin radius
rK=17 nm
rK=7 nm
94% 86%
pcap= -20.4 MPa
pcap= -8.4 MPa
−⋅
⋅=ε
S
capcap K
1
K
1
3
pS
� Capillary tension calculated directly from RH
� Degree of saturation S from water loss at given RH, if water loss at complete drying is knownRough estimate: S=0.9 at 94%, 0.8 at 86%, 0.7 at 75%
rK=4 nm
pcap= -38.9 MPa
Bentz et al. 1998
75%
Loser and Lura 2009-2011
12
Drying shrinkage of old cement pastes (3)
−⋅
⋅=ε
S
capcap K
1
K
1
3
pS
)21(3
EK
ν−⋅=
� Bulk modulus of paste K calculated from measured E modulus and ν=0.15 (assumed)Measured E modulus at 182 d E=24.6 GPaCalculated K=11.7 GPa
� Bulk modulus of solids Ks=45 GPaKs influences results less than K
Bentz et al. 1998
Loser and Lura 2009-2011
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13
Drying shrinkage of old cement pastes (3)
Relatively good agreement up to 86% RH, difference at 75%Input could be more refined (e.g. calculation of S)Other models could be used (Lecture 4A)
-1000
-800
-600
-400
-200
0
200
70 75 80 85 90 95 100
Relative humidity [%]
Stra
in [
m/m
]
CEM III/B w/c 0.3
Simulation (Bentz et al.
Loser and Lura 2009-2011
14
Drying shrinkage of young concrete
Note: this is an approximation, drying affects also hydration Age of Specimen (Days)
Drying = Sealed - Unsealed
Unsealed
Sealed
Mea
sure
d S
hri
nka
ge
( µεµε µεµε)
0 28 56 84
0
400
800
1200
1600
Sealed Specimen
Unsealed Specimen
Drying
0 28 56 84
0
400
800
1200
1600
Sealed Specimen
Unsealed Specimen
Drying
Weiss 2008
8
15
Samples with different exposure (1)
Khelidj et al. MS 1998
Sealed, drying from 2 sides, drying from all sides
16
Samples with different exposure (2)
Khelidj et al. MS 1998
Gradient in RH in samples sealed from 2 sides
Different shrinkage depending on exposure and drying rate
9
17
Moisture diffusion in a sample
Weiss 2008
40 60 80 100
Relative Humidity (%)
1.0
0.8
0.6
0.4
0.2
0.0
No
rmal
ized
Sp
ecim
en
Th
ickn
ess
(X/D
)
HAMBIENT HINTERNAL
Surface
As the Specimen
Dries
( )CgradHdivdt
dH =
18
Linear diffusion
Weiss 2008
erfc
IRH),( txRH
x
tD
: relative humidity
: distance from the surface (m)
: time (sec)
: complementary error function
: diffusion coefficient (m2/s)
⋅−−=Dt2
xerfc)H(RHHt)RH(x, SII RR
: Internal RH
SRH : drying surface RH
(100%)
(50%)
Dt2=γ
10
19
Humidity profile
Weiss 2008
50 60 70 80 90 100
Relative Humidity (RH%)
60
40
20
0
Dep
th f
rom
th
e su
rfac
e (m
m)
=0.001 =0.004 =0.008 =0.02 =0.05
=0.1 =0.2 =1.0 =100
RHIRHS
γ (m)
20
Humidity profile
Weiss 2008
40 60 80 100
Relative Humidity (%)
1.0
0.8
0.6
0.4
0.2
0.0
No
rmal
ized
Sp
ecim
en
Th
ickn
ess
(X/D
)
HAMBIENT HINTERNALSurface
As the Specimen
Dries
40 60 80 100
Relative Humidity (%)
1.0
0.8
0.6
0.4
0.2
0.0
No
rmal
ized
Sp
ecim
en
Th
ickn
ess
(X/D
)
HAMBIENT HINTERNALSurface
Drying
Self-Desc
Higher W/C Lower W/C( )dt
dHCgradHdiv
dt
dH SelfDesc+=Weiss 2008
11
21
Moisture gradients and curling
MOISTURE GRADIENT
INITIAL
INCREASING
AGE
CONCRETE SLAB
SHRINKAGE STRAIN GRADIENT
DRYING
INITIAL SLAB CURLING IN A SLABWeiss 2008
22
Drying shrinkage cracking
Loser and Leemann MS 2009
12
23
Cracking frame (1)
Cracking frame at EmpaConcrete / Construction Chemistry Laboratory
� Concrete is poured into the mould and sealed with plastic sheets
� Demoulding after 1 day or less, concrete normally exposed to drying
� Can be used for sealed specimens made of high performance concrete (autogenous shrinkage only)
24
Cracking frame (2)
Tension
Compression
� Passive system, shrinkage of concrete is restrained by steel columns
� There is always some deformation of the steel columns, shrinkage of concrete is never totally restrained
� Stress in concrete is calculated from strain measured on steel columns
150 mm
100 mm
sc FF =
csssc A/AE ⋅⋅ε=σsscc AA ⋅σ=⋅σ0.5xFs0.5xFs Fc
13
25
Cracking frame (3)
0
0.5
1
1.5
2
0 7 14 21 28Time (days)
Str
ess
(MP
a)
CEM ICEM III/B
� Stress increases due to shrinkage (both due to drying and to hydration)
� The higher the stiffness, the higher the stress
� Stress is reduced by creep/relaxation in the cement paste
� Stress is proportional to the degree of restraint
� Stress can lead to cracking of the concrete (not in this specific case)
csred,cc Ek ε⋅⋅=σDegree of restraint
Reduced elastic modulus
26
Cracking frame (4)
� Creep/relaxation reduces the elastic stresses
� The more concrete creeps, the more the stresses are reduced, the lower the probability of cracking
� Here, concrete made with blast furnace slag cement (CEM III/B) shows less stress relaxation
0
2
4
6
8
0 7 14 21 28Time (d)
Str
ess
(MP
a)
Elastic stress CEM I
Measured stress CEM I
Elastic stress CEM III/B
Measured stress CEM III/B
csred,cc Ek ε⋅⋅=σcsel,cel,c Ek ε⋅⋅=σ
14
27
Calculation of degree of restraint (1)
� If restrained specimen has no strain, k=1 or 100%
� Degree of restraint of cracking frame depends only on geometry and modulus of elasticity)AE/()AE(/ sscccs ⋅⋅=εε
FreestrainedReFree /)(k εε−ε=
)/1/(1)/()(k csscssc εε+=ε+εε−ε+ε=0.5xFsFc
sc FF =
sssccc AEAE ⋅⋅ε=⋅⋅ε
)]AE/()AE(1/[1)/1/(1k sscccs ⋅⋅+=εε+=
0.5xFs
scFree ε+ε=εsstrainedRe ε=ε
28
Calculation of degree of restraint (2)
Degree of restraint decreases with concrete stiffness
0
10
20
30
40
50
0 7 14 21 28
Time (days)
Co
ncr
ete
elas
tic m
od
ulu
s (G
Pa)
0.5
0.6
0.7
0.8
0.9
1
Deg
ree
of r
estr
ain
t k (-
)
15
29
Summary
� Drying shrinkage is the response to a relative humidity change in the concrete
� Development in time of drying shrinkage depends on specimen size, final value similar
� In HPC, self-desiccation combines with drying
� Moisture gradients and shrinkage gradients
� Cracking frames and degree of restraint
30
Acknowledgements
� J. Weiss
� R. Loser