On a three-dimensional lattice approach for modelling corrosion induced cracking and its influence on bond between reinforcement and concrete

0.25 MB·English

by  Peter Grassl

On a three-dimensional lattice approach for modelling corrosion induced
cracking and its influence on bond between reinforcement and concrete
Peter Grassl and Trevor Davies
Departmentof Civil Engineering,UniversityofGlasgow,Glasgow,UnitedKingdom
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ABSTRACT: Thepresentwork involvesthediscretemodellingofcorrosioninducedcrackingand itsinfluence
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aon the bond between reinforcement and concrete. A lattice approach is used to describe the mechanical inter-
Jactionofacorrodingreinforcementbar,thesurroundingconcreteandtheinterfacebetweensteelreinforcement

1and concrete. The cross-section of the ribbed reinforcement bar is taken to be circular, assuming that the inter-
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actionoftheribsofthedeformedreinforcementbarandthesurroundingconcreteisincludedinacap-plasticity

]interfacemodel. Theexpansionofthecorrosionproductisrepresented byan eigenstraininthelatticeelements
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cforming the interface. The lattice modelling approach is applied to the analysis of corrosion induced cracking
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and its influence of the bond strength. The model capabilities are assessed by comparing results of analyses
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lwith those from unconfined pull-out tests reported in the literature. Future work will investigate the influence
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mtofthestiffnessofinterfaceelementsand theeffect oflateralconfinement oncorrosion inducedcracking.
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aKeywords: lattice,concrete, cracking, reinforcement, corrosion
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d1 INTRODUCTION bond strength is determined empirically. Thus, these
n modelsareoflimitedvalidityforthepredictionofthe
Thepresentworkinvolvesthemodellingofcorrosion
o
influenceofcorrosionon bond.
cinduced cracking of reinforced concrete by means of
[athree-dimensionallatticeapproach.Corrosionofre- Only very few models describe the expansion of

1inforcement involves the transformation of steel into therust,theradialpressureandthetransversestresses
vexpansiverust.Iftheexpansionofrustisrestrained,it on the concrete explicitly (Lundgren 2005b). These
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resultsinradialpressureintheconfiningmaterial.For models have the potential to establish an analytical
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4reinforced concrete, this radial pressure and accom- relationship between the expansion of rust, crack-
1panying transverse tensile stresses may cause crack- ing and spalling. They can be combined with realis-
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1ing (Andrade etal. 1993). This cracking is not desir- tic bond models (Lundgren2005a), so that the influ-
0able,sinceitreducestheanchoragecapacityofthere- ence of corrosion induced cracking on the bond ca-
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inforcement (Leeet al. 2002). Most of the anchorage pacity can be predicted without theneed of empirical
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:capacity of deformed reinforcement is provided by relationships. However, this modelling framework is
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ribsonthesurfaceofthebar,whichresisttheslipbe- computationally demanding, since it requires three-
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tween concrete and reinforcement by transferring in- dimensionalmodellingofthemechanicalresponseof
rclined radial forces into the concrete (Tepfers 1979). theconcrete,thebondbetweenreinforcementbarand
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The capacity of the concrete to resist these forces concrete, andthereinforcementbaritselfas shownin
can besignificantlyreduced bythecorrosioninduced Figure1.
cracking. Consequently, there is a considerable inter-
In earlier work, three-dimensional continuum
estindevelopingmodelswhichcanpredictthemech-
based models were used to describe the cracking
anismofcorrosioninducedcrackinganditsinfluence
of concrete surrounding the reinforcement. However,
onthebondcapacity ofreinforced concrete.
three-dimensional continuum modelling of cracking
Many of the present models include the effect of in concrete is challenging, since it is not straight-
corrosion induced cracking on bond by reducing the forward to include the localised deformations into
bond strength of the interface between concrete and the continuum description. Combined with the mod-
steel (Leeet al. 2002). In these models, the relation- elling of the bond between concrete and reinforce-
ship between the amount of rust and the reduction of ment, it can be exceedingly difficult. This might ex-
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tessellation based on the randomly placed nodes.
The cross-sectional properties of these elements are
obtained from the dual Voronoi tessellation of the
same set of random nodes. For the interface be-
tween reinforcement and concrete, the lattice nodes
are not placed randomly but at special locations,
such that the middle cross-sections of the lattice el-
ements form the boundaries between the two phases
(Bolanderand Berton 2004). The nodes of the lattice
elements have six degrees of freedom, namely three
Figure 1: Three-dimensional modelling of reinforced con- translationsandthreerotations.Thesedegreesoffree-
crete:Concrete, reinforcement andbondbetweenconcrete dom are related to three displacement and three ro-
and reinforcement are considered as individual phases. tation discontinuities at the centroid of the middle
Left: Concrete cube (light grey) with reinforcement bar cross-section of the elements. The three rotation dis-
(dark grey); right: detail of interface between reinforce- continuitiesarerelatedtomomentsbyelasticrelation-
mentandconcerte. ships.Thethreedisplacementdiscontinuitiesareused
in interface constitutivemodels to determine the cor-
plainthesmallnumberofmodelswhichareavailable responding tractions. In the present study, an elastic
based on this three-dimensional approach. Discrete interface model is used for the reinforcement. One
approaches,suchaslatticeandparticlemodels,might limitation of the present lattice approach is that it
be a favourable alternative to describe this discontin- only predicts Poisson’s ratios of less than 1/4 for
uous problem. Their potential to model corrosion in- 3D and less than 1/3 for 2D. This is restrictive for
duced cracking and its influence on bond is assessed the 3D modelling of the steel reinforcement, which
inthepresentstudy. has aPoisson’s ratio greater than 1/4. Thislimitation
Lattice approaches have been used successfully could be overcome by combining the lattice model
in the past to model the failure of concrete, with continuum tetrahedra as discussed for the 2D
as reported by Schlangenand van Mier(1992) and case by Grassl etal. (2006). However, this approach
Bolanderand Saito(1998). The model by Bolander is beyond the scope of the present study. The inter-
has been shown to accurately reproduce analytical face model for concrete is based on a combination
solutions for elasticity and potential flow problems of plasticity and damage, which describes the soften-
(Yipet al.2005;Bolanderand Berton 2004).Further- ing and reduction of the unloading stiffness in ten-
more, it allows for the use of constitutive models, sion as well as the nonlinear hardening response in
which are formulated by means of tractions and dis- high confined compression (Grassl 2009). For the in-
placement jumps, as commonly used in interface ap- terface between concrete and reinforcement, a new
proachesforconcretefracture(Caballero et al.2006). cap-plasticity model is proposed which is described
These have been shown to result in an element-size inmoredetailin thenextsection.
independent description of crack-openings. This lat-
tice approach is used in the present study to describe 2.1 Elasto-plastic cap model for the bond between
themechanicalresponseofthreephases,namelyrein- concreteand reinforcement
forcement, concreteand bond between reinforcement In thelatticemodel,thenodaldegreesoffreedomare
and concrete. In this approach, the lattice elements related to displacement jumps at the middle cross-
do not represent the meso-structure of the materi- section of the lattice element. This three-dimensional
T
als (Zubelewiczand Bazˇant 1987). Instead, they are displacement jump u = (u ,u ,u) is transformed
c n s t
used to discretise the continuum. Thus, constitutive into strains ε = (ε ,ε ,ε)T by means of the interface
n s t
models are required for all three phases, which are thicknesshas
u
describedin moredetail inthefollowingsections. ε = c (1)
h
2 MODELLINGAPPROACH Thethreesubscriptsn, s and tdenotethenormaland
two tangential directions in the local coordinate sys-
Thepresentlatticeapproachforthemodellingofcor-
tem of the interface (Figure 2a). The thickness of the
rosion induced cracking follows the framework de-
interface h (Figure 2b) is chosen to be equal to the
veloped by Bolander and his co-workers. The nodes
lattice elements crossing the interface between rein-
of the lattice are randomly located in the domain
forcement steel and concrete. The strains are related
to be analysed, subject to the constraint of a min- to the nominal stress σ = (σ ,σ ,σ)T by the elasto-
n s t
imum distance (Zubelewiczand Bazˇant 1987). The
plasticstress-strainrelationship
arrangement of the lattice elements is determined
from the edges of the tetrahedra of the Delaunay σ =D (ε−ε −ε ) (2)
e c p
whereα isthefrictionalangleandf isthecompres-
c
sivestrength.Furthermore,
βαf
c
a= (4)
αβ+ 1+β2α2
p
where β is the ratio of the short and long radii of the
cap ellipse (Figure 3). At the point where the two
parts of the yield surface meet, the normal stress is
a
σ =− .Therateoftheplasticstrains
n0
βα 1+β2α2
(a) inEquationp2 is
∂g
ε˙ = λ˙ (5)
p ∂σ¯
where g is the plastic potential and λ is the plastic
multiplier.In thepresentstudy,g ischosen tobevery
similarto theyield function f. The only difference is
that α is replaced by thedilatancy angleψ so that the
(b) magnitude of the normal plasticity strain generated
during shear loading can be controlled. The plastic-
Figure 2: Interface. (a) Tangential plane of interface with ity model is completed by the loading and unloading
thelocalcoordinate systemn,sandt.(b)Cross-section of conditions:
interfaceofthickeness h.
f ≤0, λ˙ ≥0, λ˙f = 0 (6)
Thisplasticitybondmodelissimilartotheonedevel-
oped by Lundgren (2005a). However, in the present
work, theresponse is assumed to be perfectly plastic,
i.e. the shape of the yield surface is independent of
the plastic strains. The bond model will be extended
tohardeningandsofteninginfuturework.Theimple-
mentationofthepresent modelissimplifiedby intro-
ducing a smooth transition between the cap and the
frictional law. Thus, a special vertex stress return in
thetransitionregionisnotrequired.
2.2 Modelforcorrosionbetween concreteandrein-
forcement
Figure 3: Yield surface: Mohr-Coulomb friction law com-
The effect of corrosion is idealised as an eigenstrain
binedwithacap.
ε , which is determined from the free expansion of
c
whereD istheelasticstiffness,ε =(ε ,0,0)T isthe thecorrosionproductuc asεc =uc/h(Figure4).This
e c c expansionisrelatedtothecorrosionpenetrationdepth
eigenstrain describing the expansion of the corrosion
x as
productand ε =(ε ,ε ,ε )T istheplasticstrain. c
p pn ps pt
The yield surface of the plasticity model consists u = r2+(2rx −x2)(λ −1)−r (7)
c c c c
of a Mohr-Coulomb friction law combined with an p
where λ is the ratio of the volume ratio of rust and
ellipticalcap.Theshapeofthecapsurfaceisadjusted c
steel. The corrosion penetration x is related to the
so that a smooth transition between the two surfaces c
corrosionpercentage ρ as
isobtained (Figure3). Thiscombinationwas initially c
proposed by Swan andSeo (2000) and further devel-
ρ
oped by Dolarevicand Ibrahimbegovic(2007). The x = r 1− 1− c (8)
c
yield function f depends on the normal stress σ and (cid:18) r 100(cid:19)
n
theshearstress normσ = σ2+σ2 as
q s t 3 COMPARISON WITH EXPERIMENTAL RE-
p
σ +ασ ifσ ≤σ SULTS
q n n0 n
 Thelatticeapproachisusedtomodeltheexperiments
f = σ2+ (σn+fc−a)2 − a2 ifσ ≤σ (3) reportedbyLee etal. (2002).Thegeometryandload-
 q β2 β2 n n0 ingsetupoftheexperimentsisshowninFigure5.Re-
inforcementbars((cid:31)=13mm)embeddedinconcrete





Figure4:Representation ofthecorrosionprocessasanex-
pansivelayerofrust.
cubeswereinitiallysubjectedtocorrosionandsubse-
Figure6:Meshforthelatticeanalysis.
quentlypulledout.
approach described earlier. The lattice for the analy-
ses is shown in Figure 6. For the reinforcement and
the interface between reinforcement and corrosion,
the mesh is structured. For the concrete the lattice is
random.
Three analyses were performed. In the first anal-
ysis the reinforcement was pulled out without initial
corrosion. In the other two analyses, corrosion per-
centages of ρ = 3.2 and 16.8 % were considered
c
before the pullout. Assuming uniform corrosion, the
corrosion percentages were transformed according to
Equation 8 to corrosion penetrations of x = 105 and
c
(a)
571 µm. With an expansion ratio of λ = 1.4, this
c
gives,accordingtoEquation7,freecorrosionproduct
expansions of u = 41.6 and 214.8 µm, respectively.
c
With an interface element thickness of 1 mm, this
gives eigenstrains of ε = 0.0416 and 0.2148. In all
c
three analyses, the load F was controlled by the end
slip in the form of relative horizontal displacements
of nodes A and B as shown in Figure 5a. The results
of the analyses are compared to the experimental re-
sults in the form of average bond stress-slip curves
shown in Figure 7. Here, the average bond stress was
(b)
determinedasτ =F/(π(cid:31)ℓ),whereℓ=6(cid:31)istheem-
bedded length.
Figure5:Geometryandloadingset-upforthebycorrosion
pull-out test reported by Leeetal.(2002). The reinforce- The pre-peak regime of the load-split curves ob-
ment bar with a diameter (cid:31) = 13 mm is placed eccentri- tained in the analyses is in very good agreement with
cally in y-direction in the concrete specimen. No lateral the experiments. However, the post-peak responses
reinforcement isprovided. obtained in the analyses deviate considerably from
thosereported intheliterature.Inparticular,theanal-
The concrete used in the experiments is charac- ysis of the corrosion free case exhibits a more brit-
terised by a Young’s modulus of E = 22.6 GPa, tle response than reported in the experiments. On the
c
Poisson’s ratio of ν = 0.17, a tensile strength of other hand, the load-slip curves with initial corrosion
c
f = 2.7 MPa and a compressive strength of f = exhibit a more ductile response than reported in the
t c
24.7 MPa. The Young’s modulus of the reinforce- experiments. This is surprising, since it is expected
ment is E = 183 GPa. In the present study ,the re- that cracking in the concrete cover should reduce the
s
sponse of concrete, reinforcement and bond between pressure at the interface and, thus, also the tangential
concrete and reinforcement is modelled by the lattice stresses. Consequently, a more consistent pattern of
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a] analysis
P 7 experiments
M
s [ 6
s
e 5
r
st 0 %
d 4
n
o 3
b
e
g 2 3.2 %
a
er 1
v 16.8 %
a
0
0 0.2 0.4 0.6 0.8 1
slip [mm] (a)
Figure7:Comparisonofpredictedaveragebondstress-slip
curvesandexperimental datareported byLeeetal.(2002)
forthreecorrosion percentages ρ =0,3.2and16.8%.
c
post-peakresponsemightbeexpectedacrossallthree
cases.Morestudiesarerequiredtoexplorethisobser-
vation.
For the analyses without corrosion, the crack pat-
terns for the peak bond stress and the maximum slip
(presented in Figure 7) are shown in Figure 8. Crack
patterns are visualised as those middle cross-sections
of lattice elements in which the norm of the crack
(b)
opening vector is greater than 10 µm and increasing.
Thus,onlyactivecracks are presented.
Figure 8: Crack patterns for the pullout analysis for the
At the peak of the average bond stress-slip curve,
corrosion-free case at (a) peak and (b) end of the aver-
the concrete cover is cracked at its thinnest section
age bond stress slip curve. Cracks initiate at the interface
(Figure 8a). With further slip, additional cracks initi-
between reinforcement and concrete and propgate to the
atefromthereinforcementandpropagateradiallyinto
specimensurface.
thespecimen asshowninFigure8b.
In Figure 9 the crack patterns are shown, for the
the decrease of the bond strength if the concrete is
two corrosion cases, at the end of the corrosion pro-
pre-cracked. Very good agreement with experimental
cess. For both corrosion cases, cracking of the con-
results in the pre-peak regime of the bond stress-slip
crete cover occurs before the pullout, which corre-
curves was obtained. More studies are required to in-
sponds to the observations reported in the literature
vestigate the post-peak response of the bond stress-
(Leeet al. 2002).
slipcurves.Also,furtherstudieswillbeperformedto
investigate the influence of the element length of the
4 CONCLUSIONS interface between reinforcement and concrete on the
In the present work a lattice approach is used to de- analyses results. Also, we will study the influence of
scribe the mechanical interaction of a corroding rein- the stiffness of the lattice elements on corrosion in-
forcementbar,thesurroundingconcreteandtheinter- duced cracking and its interplay with lateral confine-
face between steel reinforcement and concrete. The ment.
cross-sectionof theribbed reinforcement bar is taken
tobecircular,assumingthattheinteractionoftheribs ACKNOWLEDGEMENTS
of the deformed reinforcement bar and the surround- The simulations were performed with the
ing concrete is included in a cap-plasticity interface object-oriented finite element package OOFEM
model. This lattice approach is capable of represent- (Patza´k 1999; Patza´k and Bittnar2001), extended by
ingmanyoftheimportantcharacteristicsofcorrosion thepresentauthors.
induced cracking and its influence on bond. The ide-
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