FE modeling of expansive oxide induced fracture of rebar reinforced concrete

Pergamon

Engineering Fracture Mechanics Vol. ~6, No. 6, pp. 797-812, 1997

PIh

S0013-7944(96)00132-4

© 1997 Elsevier Science Ltd. All fights reserved Printed in Great Britain 0013-7944/97 $17.00 + 0.00

FE M O D E L I N G OF EXPANSIVE O X I D E I N D U C E D F R A C T U R E OF R E B A R R E I N F O R C E D C O N C R E T E JOE PADOVAN Department of Mechanical and Polymer Engineering, College of Engineering, University of Akron, Akron, Ohio 44325-3903, U.S.A. JIN JAE Department of Mechanical Engineering, College of Engineering, University of Akron, Akron, Ohio 44325-3903, U.S.A. Abstract--This paper develops a finite element based simulation strategy which can model the time dependent expansive oxide induced fracturing of corroding rebar reinforced concrete. The oxide growth history is modeled by empirical relations which can be correlated to represent the expansive process for a given in situ condition. Due to the generality of the procedure developed, problems with multiple long progressively growing cracks can be handled in an entirely automated manner. To ascertain crack tip physics, moving templates are attached to each crack. These individually move with the cracks as they progress under a given expansive oxide growth history. To quantify the procedure several benchmark example problems are presented. © 1997 Elsevier Science Ltd

1. INTRODUCTION WHILE REBARreinforced concrete represents an economical building material[l], in environments which are high in moisture, corrosion can often lead to significant structural degradation [2]. This follows from the expansive character of iron oxide [3] which, when embedded as in the case of corroded rebar[4], induces time dependent progressively larger scales of fracturing. Such attributes tend to significantly reduce service life particularly in environments laden with electrolyte rich moisture. While the use of coatings may mitigate such behavior, in chemically active environments, even protective measures may fail in the short term. Because of this contingent on environmental factors, proper design may need to account for the progressive oxide expansion induced fracturing process. In context of the foregoing, this paper will develop a finite element (FE) modeling scheme which can simulate the time dependent expansive oxide induced fracturing of reinforced concrete. Here, the oxide growth rate will be modeled by empirical relations which can be correlated to experiments, so as to represent the expansive history for a given in situ behavior. The FE simulation scheme incorporates the capability of handling the propagation from multiple fracture sites. Additionally, oftentimes crack trajectories may extend over large distances relative to the product dimension. To accomplish this, the procedure will employ the concept of multiple moving FE templates [5, 6] to evaluate crack tip physics. This includes the determination of stepwise crack increment growth size and orientation. Applied in a history dependent incremental format, complete component straddling crack trajectories can be handled. In the sections which follow, detailed discussions will be given on the time dependent empirical simulation of oxide growth history, the FE modeling of the growth of propagating multiple cracks, the associated solver strategy and extensive benchmark applications. This includes simulating multiple simultaneously acting oxide induced crack propagation events in columns and slabs. To quantify modeling integrity, the results are correlated with empirical data. As will be seen, such comparisons illustrate the accuracy of the FE oxide induced crack growth simulation scheme.

2. CORROSION MODELING

The corrosion of steel rebar is a complex electromechanochemical interaction. Overall it includes: (1) the diffusion of electrolyte-rich saline solution through the concrete, and subEFM 56/6~C

797

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J. PADOVAN and J. JAE

sequently around and through the rebar oxide coating; (2) the progressive oxide growth under the influence of electrochemical conditioning, temperature and back pressure from the constraining concrete. Generally, FeO2 is an expansive product increasing some 200 + % in volume over the original nonoxided rebar metal[3]. Contingent on the near field compressive stress state, the layer of oxide surrounding the rebar can be somewhat compacted leading to moderately less expansion. This is highly dependent on the rate of oxidation. A secondary mechanism leading to a reduction of the oxide layer thickness follows from the fact that FeO2 is somewhat water soluble. In very wet situations, significant amounts can be leached out of the near field boundary layer to the surrounding concrete. Regardless of such mechanisms, significant expansion usually occurs in wet environments. This is greatly accelerated in the presence of electrolyte rich saline solutions. Generally, the growth of the oxide layer can be modeled via the empirical relation [3] d d-t (30) =/z(t~o)t~,

(1)

where t is time, tSo is the thickness of the oxide layer and (#, fl) are empirical curve fitting parameters. As the layer grows in thickness, the rate of growth slows down somewhat as free oxygen radicals become stagnated in the outer layers. In this context,/3 < 0, thus 1

/~o ~ (1 - fl)(/zt) 1 - 13.

(2)

Most typically/3 ranges O(-1/2). The parameter # is highly dependent on the mechanochemical interactions. In this context, in dry regions, a 6o ~ 0(1 mm) growth per decade would be a generous upper bound in oxide thickness. In contrast, in a wet region high in salinity, i.e. electrolytes, 60 ~ 0(1 mm) per year would be a lower bound on growth. For FE modeling purposes, noting Fig. 1, the empirical simulation of the oxide growth process requires the definition of two nodal kinematical issues namely: (1) the thickness of rebar metal converted to oxide yto,k; and (2) the interference fit ytmk at the kth interfacial nodal pair. Here Yos~ t and ytmk are measured perpendicular to the original interfacial surface. Since the rate of compaction of the oxide may evolve in time, the noted kinematical measures may not remain in a constant proportion to each other. In this context the nodal growth rates can be empirically modeled by the relation Concrete

Fig. 1. Free growth of oxide layer: the interference fit.

FE modeling of fracture of reinforced concrete

799

d

~(lYNckl) ~ f N G k ~ {/ZIYNGklt~}NGk

(3)

d dt (lYoskl) "~fos, ~ {UlYos, l~}osk ,

(4)

where lysckl, lyo.,,I, f u n k and fosk respectively represent the magnitude of the net oxide layer thickness, the amount of rebar corroded/oxided and the empirical, i.e. approximating, functions defining growth of the noted kinematical quantities. Given that the growth is measured perpendicular to the original interface, Ysck and Yo~k take the form

Yuak ~ g u a k FUk ~

[ 1/ (1 -- fl) (/xt) 1

fl

(5)

FNK

NGk

Yosk ~ gosk l"Nk "~

/

(1 -- fl) (#t) 1

/

fl

I'.k,

(6)

osk

such that Fsk is a vector defining the perpendicular orientation. Noting Fig. 1, the relationship between the net oxide thickness, the amount of rebar reverted and the interference kinematics takes the form (7)

YNGk = YlGk d- Yosk"

Based on eqs (5) and (6), we see that the nodal interference is given by the expression YIGk = { g N G K - gosk} ]'Nk.

(8)

In situations where gNGk and gosk are directly proportional, namely (9)

gosk ~ ~ gNGk,

then

YX6K~ (1 - () gNGk ~$Nk ~ (1 -- 0

I 11 (1 --/3)(~t) 1

fl

FNK,

(10)

NGk

where here ( defines the expansion ratio. Equations (7) and (8) can be used to establish the nodal interference fit at all oxide-concrete interfacial node points. This, of course, requires the distributional character of the #, fl activation parameters. Such properties are directly dependent on the environmentally induced moisture penetration and associated electrolyte concentration. In the context of the foregoing, for any concrete-rust interface nodal pair, the inter-relationship between the concrete and rust motions (y¢, YR) and the interference fit Ym takes the form Yc = Y/6 + YR.

(1 1)

Viewed from a global point of view, the constraint defined by eq. (11) converts the nodal vector Y to the form Y---->Y/~ + Y,

(12)

YxI6 = ( 0 . . . . Y16,, ..... Ym,2, ..),

(13)

such that

where nl, n2,.. define concrete-rust interfacial node numbers.

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J. P A D O V A N and J. JAE

Given that significant growth is possible, very high stresses can be induced at the rebaroxide-concrete interface. Due to the flaky character of the oxide layer, it has significantly reduced tensile strength compared to the rebar's low carbon steel. In this context, due to the circular shape of the reinforcement, the diametral fields are highly compressive while the circumferential are tensile. Note, the tensile levels are such that circumferential fracturing of the oxide layer is an ongoing event. Hence, the compressive diametral and shear fields provide the primary load path into the rebar. To model the foregoing oxide layer's mechanical properties, a bimodular formulation is employed. Specifically, in the compressive state the constitutive relation takes the form ass

=

[0]

[Cs]J

Ls

'

(14)

where (aN; a S ) t :

(O'rr , aO0, a Z Z , O'rO, O'OZ, O'Zr )

(Lrv; L s ) ' = (Lrr, Loo, Lzz, 2Lro, 2Loz, 2Lzr),

05) (16)

such that [CN] and [Cs] take the usual isotropic form. In tension, the properties are cut off in the perpendicular direction. This requires a material orientational transformation, thus yielding an orthotropic response. Note, compaction effects are dealt with by calibrating the empirical growth rules defined by eqs (1)-(3). Contingent on the phase of growth, a variety of stress field fluctuations are possible around the rebar. In this context, a nonlinear algorithm will be required to handle the bimodularity and potential large deformation (small strain) movement for field from crack sites. This will be discussed later.

3. CRACK MODELING Most modern day concrete mixtures behave in a brittle manner with low tensile strengths 0(400 psi/2.76 MPa) and order of magnitude higher compressive properties, namely 0(4000 psi/ 27.6 MPa)[1]. As oxide induced diametral growth continues, the concomitant increase in circumferential stress in the surrounding concrete ultimately leads to progressive brittle tensile fracture. The time scale of the process is gauged by the oxide growth rate O((t)l/0-~)). For the current purposes, this will be simulated by a compressive-tension cutoff model. In the vicinity of the tension rupture site, the crack tip will be modeled to establish 1. the proper stress intensity effects; 2. the next direction of cracking; and 3. the progressive ongoing crack growth rate. As a base assumption, we shall model the concrete as a homogeneous brittle medium wherein the ultimate crack length is several orders of magnitude larger than the aggregate material size (a very long crack). This mitigates the effects of aggregate shape. Under this assumption, the oxide induced crack growth can be likened to extension of fracture in a brittle medium [7] undergoing monotonic loading. In a 2-D context, the state of stress in the tip region takes the form[8] ax - ~

K~

cos(0/2)(1 - sin(0/2) sin(30/2)) - ~

Kn

sin(0/2)(2 + cos(0/2) cos(30/2))

KI ~rY- 2424 ~ cos(0/2)(l+ sin(0/2) sin(30/2))+ ~ r _

~xy - -

KI

(17)

sin(0/2)cos(0/2) cos(30/2)

(18)

sin(0/2) cos(0/2) cos(30/2) + KlI cOS(0/2)(1 -- sin(0/2) sin(30/2)).

(19)

The parameters KI and Kn are the mode I and II intensity factors. To determine the direction

FE modeling of fracture of reinforced concrete

801

of cracking, several procedures can be employed, i.e. maximum tangential stress [9], minimum strain energy density[10], maximum energy release rate[l l] and so on. Each of these criteria has been reported by many researchers to give fairly good estimates on both the crack growth direction and the onset of crack propagation in metal components undergoing monotonic loading conditions[12, 13]. While this situation is not as clear for concrete, nonetheless, as with the tension cutoff fracture criterion, the foregoing directionality scheme will provide a qualitative evaluation of the state of the intensity factors. For the current problem, the crack/cracks may migrate large distances sometimes bridging the gap between successive rebars. In fact, they may lead to a complete cross-sectional failure or cleave off significant fascia material. Such events will be described in the following sections. What is of primary importance is a way to evaluate the stress intensity factors of several cracks as they migrate along a variety of paths. Here we shall employ FE analysis wherein the state of the crack tip is monitored and its local physics is gauged via a moving template. Recently Guo and Padovan [6] employed such a scheme to handle fatigue crack growth problems. The wake of the crack is simulated by element birth/death. Specifically, once the tension cut off is reached at a given element integration point, then the properties are killed in the direction perpendicular to the crack direction defined by the local tip physics. If the surrounding kinematics induces closure, then the birth option can be activated. The template [6] employs 1/4 node elements [14] to simulate the tip singularity. Overall the template follows the crack tip along its route of growth. In this context, it is detached from the global mesh. To establish the proper stress state in the template, load/deflection data are obtained along its boundaries via interpolation of global fields. This is done incrementally along the route of cracking. The number of increments being dependent on convergence requirements and incremental crack lengths. Once such data are received, the FE model used to define the template is solved to obtain (1) Global mesh

Movable mesh

Crack trajectory

;uccessive template positions

I I

Crack orientation

Crack tip Global :oordinates Fig. 2. Moving template strategy defining propagation of multiple cracks.

802

J. PADOVAN and J. JAE Template boundarynodes

Fig. 3. Templatemesh associatedboundaryjuxtaposed global mesh and crack wake elements.

the intensity factors, (2) the direction of cracking; and (3) each cracks new incremental growth. Figures 2 and 3 illustrate the overall process. For linear elastic brittle problems, there are several kinds of relationships available which may be employed to relate the stress intensity factor to FE results[13]: (1) displacement based, (2) strain energy release rate based, or (3) via the J integral. While the latter two schemes can be employed with relatively coarse meshes, their major deficiency is the inability to separate out mixed mode effects. While a more sophisticated J integral scheme is available [13], it is more cumbersome to employ. In this context, the analytical expression for the displacement variation along rays emanating from the crack in the vicinity of the singularity is employed namely[6,13]: u = (KI/4#) rv/-r-~ {(2t¢- 1) cos(0/2)- cos(30/2)}- (Kii/4#)v/-~/2~r){(2K + 3) sin(0/2) + sin(30/2)}

(20) v = (KI/4#) ~

{(2x + 1) sin(0/2) - (30/2)} + (Kii/4/z) v/~-/2rc {(2K - 3) cos(0/2) + cos(30/2)}, (21)

such that 3 - 4v; K = { (3 - v)/(1 + v);

plane strain plane stress

Now based on the 1/4 node element defined in Fig. 4, stress intensity factors can be evaluated by direct substitution of the crack opening displacements into eqs (20) and (21). Here the values of 0 are the polar angle of the element (1-2) edge. This yields u

"~ U2 +

(4u5 - ul - 3u2) x / ~

+ (2ul + 2u,. - 4us)r/L

(22)

803

FE modeling of fracture of reinforced concrete Quad

I. II.

4

7

3

Initial global analysis Incremental oxide growth for T +AT time step

III.

Global reanalysis

IV.

Transfer boundary data to templates

V.

Template solutions

VI.

Define local crack tip physics * Orientation of incremental growth

6

• Incremental growth length • Reposition/orient templates 1

5

2

VII.

Wake formation at global level • Element death or remesh • Check for crack closure

Triangular

VIII.

Next time increment: repeat II-VII

Fig. 5. Flow of control of crack propagation analysis.

1

5

(2,3,6)

Fig. 4. 1/4 node element.

v ~

v2 + (4v5

-

vl -

3v2) v/rv/rv/~+ (2Vl + 2v2 -

4vs)r/L.

(23)

It is common practice that the stress intensity factors are evaluated from the two crack face sides, namely where 0 = + re. Hence, by averaging we obtain KI

Kn

=

-

~r(1 + K)

-

-

zr(1 + x)

(4v2

-

v3 -

3vl)

(24)

(4u2

-

u3 -

3Ul).

(25)

The operation of the various templates involves a two-way flow of information. In the forward phase, they receive global mesh displacement data at all associated peripheral nodes. These form the boundaries of the templates when centered at the current location of crack tips (Fig. 3). Such data is continuously updated via subsequent recursive analysis performed on the coarse global model as the cracks incrementally progress across the part (Fig. 2). Based on such boundary information, the template equations are themselves incrementally solved to establish the crack tip physics. On the backward phase, the newly calculated crack orientations and lengths are returned to the global mesh to enable proper wake formation and template positioning. After updating the oxide layer thickness to the next time step increment, the solution to the global field equations is obtained. The process is then repeated. The total flow of control is given in Fig. 5. Note, due to material or kinematical nonlinearity, the global reanalysis may require an iterative solver.

4. NONLINEAR SOLVER STRATEGY

While prototypically the strains in concrete structures are quite small, once cracking proceeds, fairly large deformations may occur as detachment progresses. Such events require the use of the appropriate stress and strain measures, i.e. the 2nd Piola-Kirchhoff stress and Green-Lagrange strain tensor combination. This together with the bimodular behavior of the

804

J. PADOVAN and J. JAE

oxide and crack during tension-compression transitions leads to opening/closure effects, hence generating nonlinear response characteristics. To simulate crack propagation, the combination of (l) the recursive re-evaluation of the global fields, (2) the subsequent template calculations and (3) resulting incremental crack wake formulation, all lead to a natural environment for the use of a Newton-Raphson [15] solver. In FE format, the governing equations take the form[15] I

[B*] t S dv

F,

(26)

R

where S is a vector of 2nd Piola-Kirchhoff stress components and dL = [a*] dY,

(27)

such that [B*] defines the mapping matrix between nodal (dY) and Lagrangian strain (dL) increments. As the oxide growth proceeds, we have that

YtG(t)----+YIG(t + At).

(28)

To setup the requisite solution algorithm, we introduce the formulation y(Y(t)) = [ [B*]T S dv - F.

(29)

dR

For the interference fit problem, 7 takes the form y = y[Y(t) + Ym(t)].

(30)

After an oxide growth spurt of,

YIG(t)------>YIG(t+ At), it follows that y : y[Y(t -t- A) + Ym(t + At)]

(31)

must be solved. To start the process, use of Taylor series yields 0

y(t+At)-.~y[Y(t)+Yta(t+At)]+-~{y[Y(t)+Ym(t+At)]}AY+O{(IIAYII)2}.

(32)

where here 0{} defines the order of the follow-up terms and the Jacobian matrix is given by the expression [KT(Y(t) + Vm(t + At))] = 0-~ {y(V(t) +

Ym(t +

At))}.

(33)

Recast in an algorithmic format, eq. (32) takes the form ~vt+z~t z_ yt+m~lAyt+Zxt = Ft+z~t - - f [B*]T S dv v,+~,.v,+~, iv -JXTI.Xe - 1 IG ]J ~ --~-- - - - - [ G

(34)

yt+At g = vt+At --gl + Aye+At.

(35)

"

JR

such that

Equation (34) must be used in conjunction with the requisite bi-modular update of the oxide layer properties and any wake elements undergoing closure. Convergence at any time step is achieved when the usual norm check is satisfied, namely IIAY~+Atll

IIY~+Atll

<< tol.

(36)

Note that if complete detachment occurs, then the nonfunctional section is killed off along with any associated external load sites. This is achieved by stripping all the appropriate rows and columns from [KT], Y and F.

FE modeling of fracture of reinforcedconcrete

805

Columns

000 0 0 000

•

Corrodingrebar

000 • • 000

(D Cleanrebar

Slabs Fig. 6. Distribution of corroding rebar in column and slab configurations. 5. B E N C H M A R K S I M U L A T I O N S To benchmark the foregoing development, several simulations were considered. Each of these were backed up by data taken from corrosion failed concrete structures reinforced by rebar. Since it is hard to simulate the exact time dependency of the electrolyte chemistry leading to the oxide growth, the prime goal was to verify the modes of fracture in each of the various problems selected. These consisted of two major structural types, i.e. columns (Fig. 6) and walls/slabs (Fig. 6). In each, the rebar is undergoing various regionally dependent forms of oxidation. Overall, the following problems are considered: (1) column--all rebar are uniformly corroding; (2) column--corner rebars are uniformly corroding; (3) column--central rebars are uniformly corroding; (4) slab--all rebars are uniformly corroding; (5) slab--all rebars are nonuniformly corroding; and (6) slab with isolated corroding rebar. Figure 6 also illustrates the pertinent problem configurations. The material properties associated with the concrete, rebar and rust are given in Fig. 7 and Table 1. This includes the compression-tension modeling of the concrete. To finish off the simulation, Fig. 8 illustrates the global and template level FE models crpj O"t IYc _--~ O'pi O"t

Compression failure envelope J

O'¢

Fig. 7. Properties of concrete.

Tensile failure envelope

806

J. PADOVAN and J. JAE Table 1. Properties of concrete, rebar and rust Rebar

Young's modulus Poisson's ratio

2.07 x 105 MPa

30000 ksi 0.3

Concrete Initial tangent modulus Poisson's ratio Uniaxial cut-off tensile strength Uniaxial maximum compressivestress

4.2 x 104 MPa

6100 ksi 0.02 0.458 ksi

3.16 MPa

Principal stress ratio

25.8 MPa 0.002 cm/cm 22.2 Mpascals 0.003 cm/cm Triaxial compressivefailure curve Curve number

SPl(/) SP3(LI) at SP2 = SP1 SP3(L2) at SP2 = flx SP3 SP3(L3) at SP2 = SP3

I=1 0.00 1.00 1.25 1.20

O"c

Compressive strain at ac Uniaxial ultimate compressivestress Uniaxial ultimate compressivestrain

2 0.25 1.35 1.70 1.60

3 0.50 1.75 2.10 2.00

4 0.75 2.15 2.55 2.40

-3.74 ksi -0.002 in/in -3.225 ksi -0.003 in/in

5 1.00 2.50 2.95 2.80

6 1.20 2.80 3.30 3.10

employed. By adjusting the boundary conditions with the appropriate assortment of rollers, fixed nodes and free surfaces, all six problems can be defined. Since the benchmark examples are taken from post-mortems of on site failed components, our primary purpose here will be to match the final crack trajectory. The actual oxide buildup was simulated by a monotonic growth rate. For the current purposes, the time scale was chosen to induce major fracturing within a decade with initiation in O(1 year), i.e. wherein several sites exceed the 0(400 psi/2.76 MPa) tensile fracture strength. Figures 9-20 present in a graphical context the stages of growth of the various tension-compression induced cracks. As can be seen, various cracks started early but progressively stopped growing as shading reduced tip stress levels. This can be established by following the incremental trajectory histories given in Figs 9, 11, 13, 15, 17 and 19. In most situations, the main crack progressively shaded secondary slow growing ones. Figures 10, 12, 14, 16, 18 and 20 depict the growth histories of the primary cracks for the case of monotonic oxide buildup. In the simulation depicted in Figs 17 and 18, the rust deposition was modeled as a nonuniform growth process which varied linearly around the circumference. The m a x i m u m growth rate was set at the point closest to the surface and at 20% of that value at the farthest position (Fig. 18). Comparing Figs 15 and 16 and Figs 17 and 18, one sees an interchange in which a crack dominates the early stages of fracture. This follows from the fact that the nonuniform oxide buildup tends to concentrate the stress in the near surface region; hence, leading to the early dominance of crack 2. In both cases, crack 1 leads to a complete detachment of the surface concrete. Crack 2 tends to lead to detachment in chunks. Such processes are usually present in upright walls where electrolyte rich water can work its way down vertically placed rebar. Here gravity tends to promote both a radial and axial corrosion front. This follows from the fact that while water tends to pool at the lowest axial section of oxide penetration, the top of the bar will be exposed to fresh electrolyte leading to corrosion along the entire rebar. Note rust is highly porous, hence vertical or inclined rebar can promote pathways leading to whole structure failure. For the current work, such effects can be modeled through time evolving empirical definitions of the site specific corrosion rate. Figure 21 also depicts the post-mortem experimentally obtained crack trajectories. As can be seen, good correlation is achieved for the primary cracks which cause total component failure. Several interesting differences were noted, namely: (1) the actual cracks appeared to be more jagged; and (2) less secondary cracks were noted in the experiment. The jaggedness is a direct result of two major issues: (i) the local nonhomogeneous aggregate to cement material property transitions; and (ii) while the model must employ the weakest concrete properties homogeneously, the actual specimen may have significantly stronger local properties. Items (i) and (ii) noted above obviously have a distributional, i.e. probabilistic nature, which depends on aggregate size and positioning. This can be modeled analytically by providing the simulation with probabilistic properties. This will be the objective of the follow-on study.

FE modeling of fracture of reinforced concrete

I I I I I 1 ~ 1 1

.........

807

Column

IIIII

¢ l l l l l l l l l l l l l l I I I t l l l l l l l [ l l l I I I I 1 1 1 1 1 1 ] 1 1 1 1

l l I I I I I l

l l I I l l I l

I I I I l l I I

I I I I l l I I

l I I I I I I I

l l I l I I I I

l l I l l l I I

l I l l l l I I

l I l l I I I I

I l I I I l I I

I l l I I l I I

I l l I I l I I

l I I I I I I I

l l l I l l I l

Singular elements

II II Template

Fig. 8. FE simulation of column, slab and templates.

Slab

808

J. PADOVAN and J. JAE Time = 1.8 Y

Time

=4.~

V

Time = 2.4 Y

Time = 4.8

V

Time = 6.0 years / Crack 3 ,

~, Crack 2 Fig. 9. Incremental trajectory history for case 1: column-all rebars are uniformlycorroding• Crack propagates from center bar to corner bar \

3

• Crack 1 i Crack 2 _ Crack 3

' f /" //

~0

~2

. ~

~.-. .

2

.

.

.

.

3

4 5 6 Year Fig. 10, Growth rate of primary cracks case 1. Time

=2.4~

Time = 4.0 Y V

Time = 8.0 years /

Crack2 ~ Crack 1¢7.,A-/ Fig. 11. Incremental trajectory history for case 2: column--corner rebars are uniformlycorroding•

FE modeling of fracture of reinforced concrete

809

2.0 • Crack 1 ~Crack 2 Crack 3

~ 1.5 ¢-

~ 1.0

1 / ]

/'~'/ ,

,/S

r..) 0.5

- -

,

,

r

2

,

4 Year

2.5

i

6

8

Fig. 12. Growth rate of primary cracks case 2.

Time

=1.2V

Time

Time

=6 ~

=2 . ~

Time

=

Time = 8 . 0 / ~

Crack 1 Fig. 13. Incremental trajectory history for case 3: column--central rebars are uniformly corroding.

2.5 2.0

• Crack 1 • Crack 2

-

/ /

" 1.5

2 1.0 0.5

/

#" /

/

Crack through surface

t

! Year Fig. 14. Growth rate of primary cracks case 3.

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J. PADOVAN and J. JAE

Time = 1 year

Time = 2

Time = 2.4

C 2.0

'

C'rack peeled o'ff ~ c°ncrete~

t

~ 1.5

Time = 3 years

Time = 3.4

1.0

Time = 4.6 (..)

2

3

4

Year Fig. 16. Growth rate of primary cracks case 4.

Fig. 15. Incremental trajectory history for case 4: slab--all rebars are uniformly corroding.

Time = 1 year

Time = 2

Time = 2.4

/

2.0

Crack peeled off " ~ t concrete s u f f a ~ .

~" 1.5

,.o Time = 3 years

Time = 3.4

Time = 4.6

.f

" 0.5

/ 1.5

t

, f ' J "

3.0

4.5 6.0 Year

7.5

9.0

Fig. 18. Growth rate of primary cracks case 5.

Fig. 17. Incremental trajectory history for case 5: slab--all rebars are nonuniformly corroding.

FE modeling of fracture of reinforced concrete

Time = 1 year

Time =

Time = 2.4

Time = 3.4

Time = 4.6

I

Time = 3 years ~

~

~Crackl Crack 2

Fig. 19. Incremental trajectory history for case 6: slab--isolated corroding rebar.

t00 ~ i

/ .

25

~" 0.75 e-

0.50 ~Crack 1 inner I " Crack 1 outer I ° Crack 2 0 ~ ~ Year Fig. 20. Growth rate of primary cracks case 6.

U 0.25

811

812

J. PADOVAN and J. JAE Case 1

Case 2

Case 4

Insite

Model

Fig. 21. Major crack trajectories of on-site experimental post-mortem components and model results.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

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