Multi-Mechanical Approach to Structural Performance ...steel w/o corrosion Confined expansion of RC system where D: Initial steel diameter, γ: Mass fraction loss, α: Rust expansion

$Page 1: Multi-Mechanical Approach to Structural Performance ...steel w/o corrosion Confined expansion of RC system where D: Initial steel diameter, γ: Mass fraction loss, α: Rust expansion$
Journal of Advanced Concrete Technology Vol. 3, No. 1, 107-122 February 2005 / Copyright © 2005 Japan Concrete Institute 107

Scientific paper

Multi-Mechanical Approach to Structural Performance Assessment of Corroded RC Members in Shear Kukrit Toongoenthong1 and Koichi Maekawa2

Received 15 April 2004, accepted 30 September 2004

Abstract A multi-mechanical model that explicitly takes into account corrosion cracks in structural safety performance is pre-sented to deal with materialized corrosive substances around steel bars and equilibrated damage in structural concrete. The multi-mechanics of corrosive product and cracked concrete are integrated with nonlinear multi-directional fixed crack modeling so that corrosion cracks in structural concrete can be simulated in a unified manner. Structural analysis of corroded RC beams is carried out for experimental verification of the multi-mechanical model in terms of shear ca-pacity and ductility. RC beams, which primarily fail in shear or flexure, are discussed and special attention is addressed to the conversion of failure modes and the absolute capacity. Consideration of inherent cracking on corroded RC mem-bers is proven to be crucial for structural performance assessment and the anchorage failure of longitudinal reinforce-ment is found to cause considerable decay of member capacity.

1. Introduction

Over the life cycle of any reinforced concrete (RC) structure, environmental actions are more or less un-avoidable. Although alkalinity of concrete can effec-tively prevent reinforcement corrosion, it has been rec-ognized as one of the primary sources of durability problems. Questions on safety and remaining service-ability of such corroded reinforced concrete are raised as a critical engineering issue of importance (Okada et al. 1988; Takewaka et al. 1984; Enright et al. 1998; Li 2003). Here, in order to quantitatively assess the service life, ambient conditions causing steel corrosion need to be evaluated with a reliable approach in both material and structural stages.

At present, nonlinear mechanics based simulation is successful for initially non-damaged RC structures in which inherent defects are not necessarily considered in general. In order to effectively utilize a computational approach to performance assessment, possible defects or damages should be properly included in the analysis as initial conditions. Possible damages are attributed to loss of cross-sectional areas of reinforcing bars and vol-ume expansion of corroded substances (Lee et al. 1998; Yoon et al. 2000; Coronelli et al. 2004). Once the corro-sion starts, the risk of splitting cracks that directly affect the macroscopic bond is also raised. Without consider-ing these possible damages, reliable evaluation of cor-roded RC structures will not be achieved. In the use of simulation taking into account inherent damages, struc-

tural performance assessment throughout the life of an RC structure and appropriate maintenance planning can be achieved as schematically shown in Fig. 1, where damage brought on by ambient actions is an input to the nonlinear behavioral simulation of structural concrete (Okamura et al. 1991; Maekawa et al. 2003). The dam-age information can be obtained either by means of simulations based on thermo-dynamic multi chemo-physics (Maekawa et al. 1999, 2003) or non-destructive testing and inspection of an existing structure.

What is required here is an analytical approach that can take into account the mechanical damages caused by corrosion cracks (Yoon et al. 2000; Coronelli et al. 2004). In this study, a multi-mechanical model to deal with formed corrosion substances and cracking is pro-posed so that corrosion cracking can successfully be treated in a unified manner by integrating it into nonlin-ear finite element analysis. As the crack-to-crack inter-action becomes significant, the authors mainly focus on this aspect and employ multi-directional non-orthogonal fixed crack modeling (Okamura et al. 1991; Maekawa et al. 2003), because self-equilibrated stress cracking caused by corrosion and/or volume change of concrete does not necessarily intersect normally with cracks in-duced by external forces, unlike in the case of seismic dynamic analysis under one-way reversed cyclic loads.

2. Multi-mechanics of corrosive substances and concrete

In order to consider cracking caused by an expansive corrosive product or silica-gel from an alkali-silicate reaction, an idealized material model having the me-chanical properties of formed substances is placed in-side an integrated RC element to reproduce the multi-mechanics of steel, corroded substances and concrete. Due to the group effect of reinforcement, the resultant

1Post Doctoral Fellow, Department of Civil EngineeringThe University of Tokyo, Tokyo, Japan. Email: [email protected] 2Professor, Department of Civil Engineering, The University of Tokyo, Tokyo, Japan.

108 K.Toongoenthong and K. Maekawa / Journal of Advanced Concrete Technology Vol. 3, No. 1, 107-122, 2005

force induced by expansive material around steel bars is assumed to be equivalent to the 1-D stress field for criti-cal corrosion cracks, which penetrate the whole sections. Consequently, structural safety related to corrosion cracking can be computed under external forces.

The authors assume a two-phase problem composed of corrosion product with non-corroded steel as one integrated phase and surrounding concrete as another (See Fig. 1 and Fig. 2). When corrosion takes place, the expansion of the rust is constrained by the surrounding concrete. Accordingly, the compressive stress is intro-duced to the corroded reinforcement system, while the tensile stress is induced to the surrounding concrete (Lundgren 2002). Internal equilibrium and deforma-tional compatibility may lead to pre- and post-cracking deformation and stresses. In a pre-crack condition, in-duced tensile stress to concrete may cause cracking if its value exceeds the tensile strength. In a post-cracking state of corrosion, stress transfer across corrosion crack planes may be derived from this multi-mechanical mod-eling.

Figure 2 shows a schematic diagram for the ex-panded geometry of corroded steel and concrete. If there is no restraint on the corrosion product from the sur-rounding concrete, the corrosion product can expand freely with no stress. First, volume loss of the corroded

steel (denoted by Vloss) from the mother steel is calcu-lated by,

42DVloss πγ= (1)

where, D is the steel diameter (m) and γ denotes the volume fraction loss (= mass fraction loss: non-dimension) of steel per unit length ranging from 0 to 1; 0 means no corrosion, while 1 means complete loss of the reinforcement section. The value of γ can be con-verted from the accumulated mass loss per unit surface area of steel (Ω: kg/m2) obtained from the simulation (Maekawa et al. 1999, 2003) as,

Dsteelργ /4Ω= (2)

where, ρsteel is defined as the steel density (~7.65x103 kg/m3) and the mother steel exhibits smaller diameter due to corrosion loss. The remaining diameter of un-corroded steel (denoted by Dcorroded-steel) yields,

γ−=− 1DD steelcorroded (3)

As the volume loss is converted to expansive corroded product, we have the final volume of corroded product as,

dr

rro

δmaxδm

dr

rro

δmaxδm

DuCOMthermo-dynamic oriented system

Material Information

(by Maekawa, Chaube, Kishi, Ishida)

Material Status considering durability

-corrosion -ASR-carbonation, etc.

Mechanical Behavior

Continuum MechanicsDeformation CompatibilityMomentum Conservation

WCOMD(2D) & COM3(3D)By Okamura and Maekawa

Non-destructive Testing

•Surface Crack Width•Half-cell potential•Resistivity measure•Infrared thermograph•Radiography

Mutual Linkage

Possible cross-check Steel

Concrete Concrete

Mechanical Corrosion ModelRational & unified evaluation

of structural performance

Num

eric

al

corro

sive

cra

ckin

g

Hor. crack

Ver. crackσ

εTension stiffening model

Fig. 1 Rational structural performance assessment of corroded RC.

K.Toongoenthong and K. Maekawa / Journal of Advanced Concrete Technology Vol. 3, No. 1, 107-122, 2005 109

42DVcorr αγπ= (4)

where, α is defined as the coefficient of expansion of corroded substance and reported in literature ranging from 2 to 6 depending on the composition of oxides (Liu and Weyers 1998; Martin-Perez 2000). The cor-roded product consequently forms around the mother steel. The diameter of the corroded reinforcement sys-tem, which consists of mother steel and its corrosive product, is obtained in the case of stress-free and geo-metrically uniform expansion as,

( )11 −+= αγDD yercorrodedla (5)

Then, the reinforcement free-expansion strain (εs, free) of isotropy yields,

( ) 111, −−+= αγε frees (6)

The value stated above is the mean strain of the cor-roded system at the specific level of corrosion ‘γ’. The stress induced by restraint of the surrounding concrete is computed by multiplying the total strain of the corroded system with its average stiffness representing multi-mechanical properties. The average stiffness of the cor-roded system, which is denoted by Es,eq is computed based on the steel and corroded layers. As the corroded system is assumed to be uniform characteristic, the av-erage stiffness of the corroded system can be expressed based upon the volume fraction as,

( )( )GE

E

s

eqsγαγ

αγ

+

−

−+=

111

, (7)

where, G is the stiffness of the corroded substance (~7 GPa) and Es is the stiffness of steel (~200 GPa). Note that the linear relationship of corroded substance is as-sumed in this formulation.

Let us first consider the post-cracking state of corro-sion in a finite element including reinforcement. In fact, the post-cracking model is indispensable to verify struc-tural performance under loads as well as prediction of crack initiation. In this study, the authors direct their attention primarily to the post-crack & multi-mechanics aspects rather than crack initiation. The crack plane is assumed in the transverse direction of the Y-axis as shown in Fig. 2. Here, the normal stress is transferred just through the multi-material system of corroded and non-corroded steel, and shear transfer is chiefly carried along the concrete crack by interlocking. Under the condition of crack separation, incremental normal strain of the control volume can be described as the sum of the concrete and multi-mechanical steel. The stress develop-ing in the corroded steel is nearly inversely proportional to the reinforcement ratio, and the strain field of the multi-mechanical system of corroded steel is a part of the total strain expressed as,

eqs

yy

c

yysyycyy E

dE

dd

,, )1()1(

ψσ

ψψσ

ψψεεψε +−≅+−= (8)

D

Free expansionof rust

steel w/o corrosion

Confined expansionof RC system

where D: Initial steel diameter, γ: Mass fraction loss, α: Rust expansion coefficient

L

L

D L+δ

concrete

( )11 −+ αγD

D+δ

D+δ

steel

σyy

σyys

x

y

Fig. 2 Schematic representation for expansion of corroded steel in RC.


where, ψ denotes the ratio of local corroded steel strain over the mean strain in the y-direction. When rein-forcement is assumed to be uniformly suspended in the control volume or a finite element as shown in Fig. 2, we have,

2/1p≅ψ (9)

where, p is defined as the reinforcement ratio by the control volume or a finite element. Then, the element based stiffness of coupled cracked concrete and multi-mechanical corroded steel yields,

yycyyceqs

cyy dEdE

EE

d εψεψψψσ ≅

+−=

−1

,1 (10)

Here, let us consider the free expansion as the control volume after cracking. The transverse stress is equal to zero, the continuum strain of cracked concrete can be assumed to be null, and the corroded steel strain coin-cides with εs, free. In considering the volume ratio of cor-roded steel over the control volume or the finite element, we have p1/2εs, free as the normal expansion free strain of the multi-mechanical system, which consists of cracked concrete, corroded product layers and the non-corroded mother steel. In considering the stress-free strain of the element and the derivative of total stress-strain ex-pressed by Eq. (8), we have,

( )

freesyys

yy

yycfreess

yy

freesyyfreesyycs

yy

E

E

,

2,

,,

0

0

0

εψεσ

εψεσ

εψεεψεψσ

⋅>=

<⋅⋅−=

⋅<≤⋅−⋅= … (11)

where, the total transverse stress becomes null when the total strain exceeds ψεs, free When the transverse strain turns negative, cracks will close and start to transfer compressive normal stress. In this case, cracked con-crete primarily bears the stress transfer, and this stress can be computed based on the smeared crack constitu-tive modeling. Then, the additional stress transferred by the corroded steel in Eq. (11) can be overlaid on the original multi-directional fixed crack approach.

As far as shear transfer across a crack is concerned, corroded steel may play a negligible role compared with that of concrete rough crack interlock. Normal stress transfer in the longitudinal direction is carried by the reinforcement and can be modeled as previously done. Thus, the transverse component of reinforcement stress is added to the conventional framework of multi-directional fixed crack modeling of RC elements (Maekawa et al. 2003). This constitutes the newly pro-posed approach in the spatial averaging scheme of smeared cracks. In this approach, creep of concrete con-tinuum and the non-uniformity of concrete stress field, which are of importance to the pre-crack state, need not be explicitly taken into account since crack kinematics may absorb creep deformation, thus making continuum

deformation of concrete insignificant in relation to the overall deformation. This modeling shall be verified in the post-cracked RC structural behaviors as discussed in the following chapter.

3. Verification in terms of post-cracking member level

In this chapter, the multi-mechanical model for corroded steel system is integrated into the nonlinear finite ele-ment analysis (Okamura and Maekawa 1991; Maekawa et al. 2003). Verification of multi-mechanics is aimed for through the analysis of RC members subjected to combined corrosion and external loads. Numerical simulation of corroded RC members is conducted for flexural and shear modes of failure.

When reinforcement corrosion takes place inside a member, loss of effective reinforcement area is observed. When corrosion advances further, the tensile stress in-side the surrounding concrete is raised due to self-equilibrated force to restrain the expansion of corrosive product. After a critical stress, cracking occurs along main reinforcing bars and macroscopic bond deteriora-tion is initiated since the layer of steel and the concrete beam web are mechanically divided. Here, the loss of corroded steel sectional area is simply represented as a reduced reinforcement ratio in numerical analysis. The bond loss (Auyeung et al. 2000; Coronelli 2002; Stanish et al. 1999) associated with longitudinal cracking along the main reinforcement is represented by a degradation in tension stiffening, which represents the ability to carry a certain amount of tension transferred from the reinforcement to concrete between cracks.

In this study, macroscopic bond deterioration is me-chanically considered through the section-penetrating corrosion cracking as previously stated, and the micro-scopic bond deterioration around the main reinforce-ment is represented by an altered tension stiffening pa-rameter (Okamura and Maekawa 1991; Maekawa et al. 2003). Finally, the effect of cracking damage on the resultant member capacity is considered by employing the multi-mechanical model introduced in the nonlinear analysis as,

=

=

+=

0),(

)()(

),()(

sxy

yys

yy

xxs

xx

kls

ij

kls

ijklc

ijij

σγεσ

εσεσ

εσεσσ

(12)

where, σijc is the concrete stress with cracking and for-mulated by the multi-directional fixed crack modeling (Okamura and Maekawa 1991; Maekawa et al. 2003) and σijs is the mean stress component carried by the reinforcement steel. When the longitudinal direction is specified as the X-axis, σxx

sdenotes the longitudinal steel stress and is computed in terms of the mean normal strain of the element along the reinforcement. Stress


component σyys is the multi-mechanical transverse stress

in terms of corrosion degree and the transverse strain as given by Eq. (11), and illustrated in Fig. 3.

Figure 4 summarizes the analysis approach for struc-tural performance evaluation of corroded members. Here, careful consideration should be given to the cor-rosion crack plane. A longitudinal crack plane over the thickness of members is crucial since crack-to-crack interaction is mobilized in the structural analysis.

On the other hand, a vertical splitting crack along re-inforcement has less influence on crack-to-crack inter-action, but the free elongation of reinforcement is hardly restricted due to the local bond deterioration caused by corrosion. Thus, bond deterioration due to splitting cor-rosion cracks is considered in terms of tension-stiffness (Okamura and Maekawa 1991; Maekawa et al. 2003).

It is known that the tension-stiffness characteristic or

bond performance depends on the magnitude of corro-sion (Auyeung et al. 2000, Coronelli 2002; Stanish et al. 1999). As a matter of fact, local bond is enhanced under light corrosion due to corrosion-induced pressure, but it undoubtedly becomes degraded after bond splitting cracking. As this paper focuses on the ultimate limit state that accompanies bond cracking, tension softening characteristics corresponding to plain concrete are solely applied to the bond deteriorated finite elements. This means that the tensile stress transfer by bond is simply neglected. This assumption will be evaluated in the following sections of experimental verification. 3.1 Analysis of corroded RC beam dominated by shear The focus here is on RC beams, which are originally designed to have greater flexural capacity than shear

Constitutive Law of Steel in Transverse

γ>>

Increa

se m

ass l

oss

( )( ) 111, −−+= αγε frees

RC Concrete Reinforcement

Mean steel stress

Mean steel strain

Average steel response in concrete

Local response

Constitutive Law for Steel in Longitudinal

Corroded steel stress

Element strain

“Newly Added”

Concrete Constitutive Law (Tension)

Concrete Constitutive Law (Shear)

Concrete Constitutive Law (Comp.)

Average tensile stress

Average tensile strain

She

ar s

tress

trans

ferr

ed

Mean shear strain

Com

p. s

tress

Comp. strain

DuCOMthermo-dynamic oriented system

(by Maekawa, Chaube, Kishi, Ishida)

γAccumulated corrosion mass loss

is computed

( ) ( )klsteelijkl

concreteij

RCij εσεσσ +=

( )( )

=0

γ,yysyy

xxsxx

steelij εσ

εσσ

Superposition of concrete and reinforcement stresses

Fe→Fe2++2e-

Ionic current

Electric currentAnode Cathode

1/2O2+H2O+2e-→2OH-

Fe→Fe2++2e-

Ionic current

Electric currentAnode Cathode

1/2O2+H2O+2e-→2OH-

O2O2 Cl‐Cl‐CO2CO2

freescpE ,ε

freesp ,ε⋅

p fy

Fig. 3 Assumed nonlinear stress-strain relationship of corrosion product.

Steel

Concrete Concrete•Corrosion amount•Material properties•Structural details

Num

eric

al in

put

2-D numerical cracking

Mechanical Corrosion Model

Real damaged RC member

Check the cracking condition ( width, direction)

Hor

izon

tal P

lane

corr

osio

n cr

ack

Crackplane

Crack!

Vert

ical

Pla

neco

rros

ion

crac

k

Bond Deteriorationσ

εTension stiffening model (Okamura)

Ctu

tf

= εεσ

Bond lossIncreased ‘c’ value

Effective reinforcement ratio

ρρ ⋅

=′

ss

AA' As

A’s

OKNG

Non-linear FEM; WCOMD(2D)&COM3(3D)•Path dependency Residual stress, strain•Cracking Initial anisotropy

Structural response

Load

Deformation

Structural PerformanceEvaluation of Corroded

RC Member

Fig. 4 Analysis approach for structural performance evaluation of corroded RC.


strength. Thus the shear resistance mechanism is of sig-nificant importance and subjected to longitudinal crack-ing by corrosion. Actually, it was confirmed by Piman-mas and Maekawa (2001) that there exists a strong in-teraction of non-orthogonal cracks in the shear behav-iors of RC members. The shear capacity of corroded RC beams with longitudinal deep cracking in space was also experimentally investigated and the significance of the location of these partial local damages was clarified (Toongoenthong and Maekawa 2004). The effect of lo-cal longitudinal cracking on shear capacity of RC dam-aged members can be adverse, insignificant or even favorable, depending upon the specific damage location in space and member structural detail (Toongoenthong et al. 2004).

Consequently, non-linear analyses that can take into account corrosion cracking may carry greater weight. Following are the authors’ analyses of damaged RC beams with partial corrosion and artificial defects for magnifying the mechanical effect of corrosion cracks. In order to verify the applicability of the developed nonlin-ear FEM chiefly with regard to crack-to-crack interac-

tion, RC beams with different patterns of local cracking and defects are analyzed by employing the multi-mechanical corrosion model.

3.1.1 Analysis with local damage induced in-side the shear span An experiment on RC beams with longitudinal cracking limited to a small area was conducted (Toongoenthong and Maekawa 2004), i.e., compression fiber damaged beam (case 1), tension fiber damaged beam (case 2), corroded cracking confined to a beam support inside a shear span (case 3) and a combination thereof (case 4). Figure 5 shows the specimen dimensions and rein-forcement arrangement together with the FEM mesh used in the analysis. The analytical result of the refer-ence with no corrosion is shown in Fig. 6. The com-puted shear capacity is 162 kN, which favorably con-forms to the experimental value.

An RC beam with corrosion cracking around com-pression fiber is shown in Fig. 7. The damaged beam with accelerated corrosion around the compression fiber shows evidence of a crack width of approximately 0.3 to

1000200

2400

1000250

350

24006D16

fy= 343 MPa(unit: mm)

38CL

Fig. 5 Specimen dimensions, reinforcement arrangement and FEM mesh for the analysis.

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5Mid span deflection (mm)

Load

(kN

)

Experiment Analysis

CL.

CL.Experiment

Analysis a) Load-deflection relation b) Failure crack pattern

Fig. 6 Experiment and analysis results of referential non-damaged beam.


0.5 mm on one side and zero on the other, and the mass loss of the corroded reinforcement of 1.89% as tabu-lated in Table 1. Reduction in shear capacity is ap-proximately 8% compared to the non-damaged case. Nonlinear analysis was performed with the remaining reinforcement ratio after corrosion. The local bond dete-rioration by corrosion cracks was considered by apply-ing tension stiffening equivalent to plain concrete (Okamura and Maekawa 1991; Maekawa et al. 2003). To take into account the pre-existing corrosion cracking, numerical cracking was introduced into the target body in the first step followed by subsequent shear loading. Experimentally obtained mass loss of reinforcing bars was directly input into the multi-mechanical model and numerical corrosion cracks were computationally repro-duced as shown in Fig. 7. The computed corrosion crack width was 0.155mm and the experimental crack width

taken as the average of both web sides was 0.15 to 0.25 mm (see Table 1).

Figure 7 shows the load-deflection responses of the compression fiber damaged and non-damaged reference beams. A small reduction in shear capacity and stiffness similar to the experimental one were computed. The crack pattern of the first step at the production of corro-sion cracks is shown along with the failure crack pattern in Fig. 7. The merging of diagonal shear and corrosion cracks around compression fiber was observed in the analysis to be similar to the experimental observation.

Next, the damaged beam with pre-cracking around tension fibers was analyzed. The experiment shows the crack width to be 0.2 to 0.7mm after accelerated corro-sion. The percentage of mass loss of the corroded rein-forcement was reported as 8.76% and the computed crack width was 0.714mm, a little larger than in reality.

Table 1 Crack width and mass loss of steel from the accelerated corrosion method.

Compression mid-spanCase 1

Tension mid-span Case 2

Tension near support Case 3

Crack width (mm) 0*)~0.3~0.5 0.2~0.7 0.3~0.5 Mass loss (%) 1.89 8.76 6.7

Computed crack width (mm) 0.155 0.714 0.551 *) Visible cracking (0.3-0.5mm) in one side but cracking of the other side was hardly detected.

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4

Mid Deflection(mm)

Load

(kN)

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4mid span deflection(mm)

Load

(kN

)

Analysis Experiment

Non-damaged

Compression-fiberdamaged

Non-damaged

Compression-fiberdamaged

Small reduction of shear capacity

Mid span deflection (mm) Mid span deflection (mm)

Load

(kN

)

a) Load-deflection relation

Induced cracking damage (1st step)

Final failure crack pattern (2nd step)

Experimental failure crack pattern

b) Crack pattern in the analysis (left) and the experiment (right)

Fig. 7 Analysis and experimental results of compression damaged beam.


A 13.6% decrease in shear capacity was reported com-pared with the non-damaged beam. The analysis proce-dure was the same as in the case of the compression damaged one. Figure 8 shows the analytical result and the small reduction in shear capacity, which favorably conforms to the experimental facts. The induced crack pattern and failure mode are shown along with the ex-perimental ones. The crack-to-crack interaction in these cases is small enough not to influence the safety per-formance although some geometrical merging of corro-sion and diagonal shear cracks occurs.

An RC beam with corrosion cracking near the beam support inside the shear span was targeted. The experi-mental shear capacity was 154.5 kN. The corrosion crack width was observed to be approximately 0.3 to 0.5 mm on both webs and the computed crack width was 0.551 mm, as shown in Table 1. The multi-mechanical model yields a slightly larger crack width through cor-rosion in simulation. This is attributable to the seepage of corrosive gel, which is not explicitly considered in modeling. The major corrosion cracks were found lying on the bottom face as shown in Fig. 9. This figure also shows the computed load-deflection and crack patterns. The computed shear capacity is 148 kN or a reduction of approximately 11.3% compared with the reference case. The merging of diagonal shear with existing corro-

sion cracks around the support can be observed in both analysis and experiment. Here, reduction in shear capac-ity of just 4.6% was seen in the experiment. As men-tioned earlier, the major actual corrosion cracks were found on the bottom face of this beam. As a sensitivity analysis, only tension softening equivalent to plain con-crete and a reduced reinforcement ratio were assumed without cracking damage, as shown in Fig. 9. The trial analysis result is closer to the experimental one, but no shear stiffness degradation was found until failure unlike in reality. The experimental facts that were ob-served lie somewhere between these two extreme cases. This implies that the microscopic bond deterioration by corrosion is not absolutely critical. 3.1.2 Analysis with local damage extended to anchorage zone The experimental verification regarding crack-to-crack interaction was further carried out by use of artificial damage devices extended to the anchorage zone. The main aim was to ensure the ability of simulating an-chorage performance by the multi-mechanical and multi-directional fixed crack approach before the dis-cussion in section 3.1.3 and section 3.2 below. Figure 10 shows the detail of the specimen including an artifi-cial crack-like defect (Pimanmas et al. 2001) tested in

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4

Mid Deflection(mm)

Load

(kN)

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4mid span deflection(mm)

Load

(kN

)

Analysis Experiment


Non-damagedNon-damaged

Tension midspandamaged

Tension midspandamaged

Load

(kN

)

Mid span deflection (mm) Mid span deflection (mm)

a) Load-deflection relation


Final failure crack pattern (2nd step) Experimental failure crack pattern b) Crack pattern in the analysis (left) and the experiment (right)

Fig. 8 Analysis and experimental results of tension-fiber damaged beam.


the experiment. Instead of corrosive rough cracking, an artificial inherent crack-like defect produced by a flat paper plate was introduced during reinforcement fabri-cation. As this defect has a smooth surface with negligi-bly small shear transfer and a sharp interaction with diagonal shear cracking, the anchorage performance is thought to be much deteriorated.

The specimen has a shear-span to depth ratio of 3.2 and a tensile reinforcement ratio of 1.29%. The dam-aged anchorage length at the beam end is 0.2 m and the compressive strength of the concrete is 34 MPa. Yield strength of reinforcement is 343 MPa. The experimental result shows a small shear capacity of 60 kN compared with the reference case of 150 kN accompanying the sliding along the defect outside the shear span. In the

analysis, the pre-cracking damage was first introduced by using multi-mechanical modeling as shown in Fig. 11. Here, the shear transfer of this fictitious cracking is assumed to be zero for representing the flat plane ge-ometry. Monotonic loading was subsequently applied at mid span. Figure 11 shows the analytically obtained load-deflection relation and crack patterns compared to the experiment. The analysis gives the maximum resul-tant shear capacity of 59 kN for the beam with embed-ded damage, and the drastic reduction in member shear capacity matches the reality. The crack pattern exhibits merging of the initial defect and externally force-induced cracking as the experiment did. The computed post-peak residual force is sustained to a greater extent than in reality. This difference may be attributed to the

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4

Mid Deflection(mm)Lo

ad(k

N)

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5mid span deflection(mm)

Load

(kN

)

Analysis with horizontal crack

Analysis with vertical crack


Non-damaged

Analysis Experiment

Non-damaged

Tension near support damaged

Load

(kN

)

Mid span deflection (mm) Mid span deflection (mm) a) Load-deflection relation



C L. Side face

Bottom face

Corrosive cracks

b) Crack pattern in the analysis (left) and the experiment (right)

Fig. 9 Analysis and experimental results of tension-near-support damaged beam.

5D16fy=343 MPa

Paper plate

350

250 1000

Artificial damage Artificial damage

(Unit : mm)CL

200

38

Fig. 10 Specimen detail in damaged beam with cracking until anchorage.


small deviation of the direction of diagonal crack propagation from the tip of the artificial defect since a rather large finite element was applied even in the crack tip area. According to this verification, it can be said that the analysis is applicable to capacity estimation at least up to the peak strength.

3.1.3 Analysis with local damage in a whole shear span Here, discussion and verification will be extended to defects over the whole shear span of beams. There are two key issues. One is the loss of macroscopic bond. The web concrete is mechanically detached from the main reinforcement layer by corrosion cracking over the thickness. Ikeda and Uji (1980) experimentally demon-strated that the shear capacity of RC beams is elevated when the bond between tension main reinforcement and concrete disappears due to retarded formation of diago-nal shear crack and an arch action. Another issue is the increased risk of failure at the beam anchorage, which also leads to degradation in capacity. Thus, two possible factors co-exist with this problem where the multi-mechanical model is to be verified. Figure 12 shows the details of the specimen, reinforcement arrangement and location of longitudinal cracks in the experiment (Toon-

goenthong et al. 2004). In this case, a premature failure around the anchorage

zone was experimentally observed unlike the cases pre-viously discussed. The resultant shear capacity is 146 kN, which is slightly lower than the capacity of the non-damaged beam (167 kN) as shown in Fig. 13. The an-chorage capacity of the beam details might not be enough to sustain the arch action capacity. Nonlinear finite element analysis was performed to verify this phenomenon with the multi-mechanical model in a similar way to the previous cases. The computed load-deflection response was compared with the experimen-tal results as shown in Fig. 13. The analysis showed fair agreement with the experimental results. The premature failure around the anchorage zone was well captured in the analysis too.

As previously stated for the longitudinal cracking in the whole shear span, the shear strength can be pre-served when sufficient anchorage of longitudinal rein-forcement is provided. In order to confirm this fact, sig-nificant web reinforcement of 3% was numerically placed to prevent premature failure of the anchorage zone. Similarly, induced pre-cracking damage in the whole shear span was performed and followed by monotonic loading until failure. Figure 14 shows the

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Experimental failure crack pattern a) Load-deflection relation b) Crack patterns

Fig. 11 Analysis result of beam with inherent crack-like defect until the anchorage zone.

5D16fy=343 MPa

Paper plate

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Artificial damage Artificial damage

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Fig. 12 Specimen dimension and reinforcement arrangement with whole span damage.


expected enhanced shear capacity when an arch action effectively develops with adequate anchorage capacity. The computed crack pattern is also shown in Fig. 13. In this pattern, no premature failure is seen at the beam end. Here, it is obvious that anchorage performance is crucial, and special attention is required when a complete shear span is subjected to cracking where higher anchorage capacity is sustained, to satisfactorily form an arch ac-tion.

Verification was conducted on the RC beam with lon-

gitudinal corrosion cracks over the whole shear span reported by Satoh et al. (2003). Figure 15 shows the specimen details and reinforcement arrangement. These RC specimens are composed of three beams. One is the reference non-damaged beam. The other two were sub-jected to corrosion by submerging each specimen into a sodium pond for accelerated corrosion, resulting in uni-formly distributed reinforcement corrosion along the entirety of the longitudinal steel. The two corroded beams were reported to have different levels of corro-

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Experimental failure crack pattern

Pre-mature failure causes reduced ultimate capacity

in analysis and experiment

a) Load-deflection relation b) Crack patterns

Fig. 13 Analysis result of the whole span damaged beam with insufficient anchorage.

020406080

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Non-damaged beam

Whole span-damaged beamwith enough anchorage

Reduced capacity(Insufficient anchorage)

Reference

Expected improved shear capacitywith sufficient anchorage provided

(no premature failure)

a) Load-deflection relation b) Crack patterns

Fig. 14 Analysis result of the whole span damaged beam with sufficient anchorage.


sion along the main reinforcement. The lightly corroded beam had a mass loss of 2.1% and heavily corroded beam had a mass loss of 3.3%. In this experiment, the main reinforcing bars were bent up 90 degree inside the anchorage zone as shown in Fig. 15. Thus, a satisfactory anchorage capacity is expected is spite of the existence of corrosion.

Figure 16 shows the experimental load-displacement relation and crack pattern of all RC specimens. Both corroded beams show a favorable shear capacity com-pared with the non-damaged beam. As illustrated also in Fig. 16, there are clear differences in failure crack pat-terns between the two corroded damaged beams and the non-damaged reference beam. Although the diagonal shear failure was observed in the non-damaged beam, no localized diagonal shear crack was formed in the case of the lightly corroded beam.

Similar to the previous cases of analysis, the target specimens were analyzed by introducing numerical pre-cracking at the location similar to the experiment in the first step of analysis. Then the monotonic load was ap-plied until failure. Figure 17 and Fig. 18 show the load-displacement relations and failure crack patterns ob-

tained for the cases of the lightly and heavily corroded beams, respectively. It can be seen that the analytical results of both cases are in good agreement with the experimental facts in both the load-displacement curves and failure crack patterns. Furthermore, analysis of both corroded beams without pre-corrosive-cracking was also performed to ensure the significance of the analyti-cal approach by considering only the reduced rein-forcement ratio and tension stiffening. As illustrated in Fig. 17 and Fig. 18, the analysis result without consider-ing pre-cracking is clearly far off the experimental fact.

3.2 Analysis of corroded RC beam in flexure It is numerically confirmed that longitudinal corrosion cracking can be insignificant, adverse or even favorable on resultant shear capacity and that it depends upon the damage location and structural details. It is also known that flexural capacity is closely associated with the re-maining sectional areas of steel (Rodriguez et al. 1997; Lee et al. 1998; Mangat et al. 1999). In other words, corrosion cracking has small influence on the flexural capacity.

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Fig. 15 Specimen detail and reinforcement arrangement by Satoh et al. (2003).

Sound beamSmall corrosion beamLarge corrosion beam

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Sound beam

a) Experimental load-deflection relation b) Failure crack patterns

Fig. 16 Behaviors of corroded beams by Sato et al. (2003).


However, it should be noted that there is also an al-ternative view of ductility and post-peak behaviors. Past studies (Tachibana et al. 1990; Coronelli and Gam-barova 2004) have shown that the transition of the fail-ure mode from steel yielding to bond-shear appears in the beam subjected to corrosion. Figure 19 shows the specimen details and reinforcement arrangement. The concrete compressive strength is 35.6 MPa and the yield strength of reinforcement is 353 MPa with a tensile re-inforcement ratio of 1.3%. The shear span to depth ratio is 4.2. The observed corrosion cracking is also shown in

Fig. 19. Corrosion cracks developed up to the bottom surface as well as the side face, with a maximum crack width of 0.75mm (Tachibana et al. 1990).

The corrosion induced pre-cracking was numerically introduced first followed by conventional loading. Fig-ure 20 shows the analysis result with different magni-tudes of numerical cracking damage. The transition of the failure mode from flexure to bond-shear has also been successfully captured as shown in Fig. 20 with crack patterns and the post-peak degradation in load-displacement relation, which differs from the typical

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Analysis with corrosion cracking Analysis w/o corrosion crackingReserved shear capacity can be observed only if the cracking damage is

included!

Light corrosion beam

Experiment Failure Crack Pattern

Analysis Failure Crack Pattern

a) Load-deflection relation b) Failure crack patterns

Fig. 17 Simulated shear capacity and cracking of small corrosion beam.

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included!

Large corrosion beam

Experiment Failure Crack Pattern

Analysis Failure Crack Pattern

a) Load-deflection relation b) Failure crack patterns

Fig. 18 Simulated shear capacity and cracking of large corrosion beam.


ductile behavior of the non-damaged beam. In order to refocus on the importance of pre-cracks

penetrating the whole section, pre-corrosive cracking was intentionally excluded in the sensitivity analysis. Here, reduced reinforcement ratio is defined according to the lost sectional area, and bond deterioration is rep-resented by reduced tension-stiffness equivalent to plain concrete (no local bond assumed) in the analysis. Fig-ure 21 shows the analytical result and the failure crack pattern. The flexural failure mode accompanying the yield of the main reinforcement was obtained without considering pre-corrosive cracking. The failure crack patterns in the fictitious analysis and experiment do not

match. Thus, it is clearly seen that the consideration of corrosion cracking is substantial in the structural per-formance assessment of corroded RC members and that the macroscopic bond deterioration associated with cor-rosive penetrating cracks dominates the overall feature of ultimate limit states.

4. Conclusions

A multi-mechanical approach to structural concrete with corrosive substance, non-corroded steel and sheared cracked concrete was proposed in 2D for post-cracking states of stresses and strains, and applied to the 2-

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R u s t s lud g e A rea

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0.1 0 0 .3 5 0 .2 0 0 .10 0 .2 5 0 .4 0 0 .2 0 0 .2 5 0 .35

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Fig. 19 Experimentally observed corrosion cracking by Tachibana et al. (1990).

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Non-d amaged b eamcapac ity (exp eriment)

15-d ay damaged b eamcap ac ity (experiment)

Non-damaged beam

15-day corroded beam

Non-damaged beam

15-day corroded beam a) Numerical failure crack patterns b) Experimental failure crack patterns

Fig. 20 Numerical simulation and experimental shear failure with corrosion induced pre-cracking by multi-mechanical model (standard analysis).


dimensional structural analysis with multi-directional fixed crack modeling. As the result of this investigation, the following conclusions have been reached. (1) Sectional penetrating cracks and their widths were simulated using a multi-mechanical model and were experimentally verified using RC beams subjected to accelerated corrosion. The measured mass loss ratio of reinforcement was used for input information and the corresponding crack strain in the steel layers was calcu-lated. Comparison with the experimental facts reveals that interaction of corrosion and diagonal shear cracks can be treated but that initial crack width was a little overestimated. The seepage of corroded gel products through cracks shall be considered in future for dealing with RC in a real time scale and under actual environ-ments. (2) The computational approach was verified to capture the shear capacity, which depends upon the locations of corrosion cracks, and the crack-to-crack interaction was found to be well simulated. The loss of bond caused by corrosion crack may change the failure mode and capac-ity of anchorage performance as well as the strength in shear spans. The corrosion cracking in the shear span results in an elevated shear capacity brought due to the arch action formed but simultaneously this may cause an increase in the risk of failure at anchorage zones. This feature can be satisfactorily captured by the com-putational approach. (3) Pre-existing defects at anchorage zones were con-firmed as potential sources of trouble. Capacity deterio-ration is predictable using the multi-mechanical model where the transverse normal stress to reinforcement is carried by corroded steel under a single or multi-directional cracking state, and shear-compression stress transfer across cracking is achieved by the cracked con-crete. (4) Verification was extended to wholly corroded beams

in both shear spans and anchorage. Numerical simula-tion was found to yield satisfactory results.

As the proposed simple multi-mechanical model is based upon a cracked state of concrete stress and strain, applicability of this modeling to pre-cracking states is not necessarily acceptable. For computing crack initia-tion, non-uniform stress and strain fields may be con-sidered and time-dependent creep and fracture charac-teristics are of substantial importance (Aoki and Shino-mura 2003). However, when we deal with larger mass loss (heavy damage) and its structural safety, these fac-tors can be simply treated. Acknowledgments The authors appreciate fruitful and valuable discussion with Dr. T. Ishida and Dr. T. Kishi of The University of Tokyo. This study was financially supported by Grant-in-Aid for Scientific Research (S) No.15106008. References Aoki, Y. and Shimomura, T. (2003). “Experimental

study on tensile deformation and cracking criteria of concrete for the evaluation of resistance of concrete to drying shrinkage crack.” Journal of Materials, Concrete Structures and Pavements, JSCE, 59 (732), 135-148.

Auyeung, Y. B., Balaguru, P. and Chung, L. (2000). “Bond behavior of corroded reinforcement bars.” ACI Materials Journal, 97(2), 214-220.

Berra, B. (2003) “Steel-concrete bond deterioration due to corrosion: finite-element analysis for different confinement levels.” Magazine of Concrete Research, No.3, June.

Coronelli, D. (2002). “Corrosion cracking and bond strength modeling for corroded bars in reinforced concrete.” ACI Structural Journal, 99(3), 267-276.

Coronelli, D. and Gambarova, P. (2004). “Structural

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Analysis reference

Analysis with bond-loss andreduced reinforcement ratio

Non-damaged beam capacity (experiment)

15-day damaged beam capacity (experiment)

a) Load-displacement relation

Experiment

Analysis

15-day corroded beam15-day corroded beam

b) Fictitious crack pattern unlike the real-ity when no pre-cracking is considered.

Fig. 21 Sensitivity analysis of 15-day corroded beam without pre-cracking.


assessment of corroded reinforced concrete beams: modeling guidelines.” Journal of Structural Engineering, ASCE, 130(8), 1214-1224.

Enright, M. P. and Frangopol, D. M. (1998). “Service-life prediction of deteriorating concrete bridges.” Journal of Structural Engineering, ASCE, 124(3), 309-317.

Ikeda, S. and Uji, K. (1980). “Studies on the effect of bond on the shear behavior of reinforced concrete beams.” Journal of JSCE, 293, 101-109.

Lee, H. S., Noguchi, T. and Tomosawa, F. (1998). “Fundamental study on evaluation of structural performance of reinforced concrete beam damaged by corrosion of longitudinal tensile main rebar by finite element method.” Journal of Structural Concrete Engineering, AIJ, 506, 43-50.

Li, C. Q. (2003). “Life-cycle modeling of corrosion-affected concrete structures: Propagation.” Journal of Structural Engineering, ASCE, 129(6), 753-761.

Liu, Y. and Weyers, R. E. (1998). “Modeling the time-to-corrosion cracking in chloride contaminated reinforce concrete structures.” ACI Material Journal, 95(6), 675-681.

Lundgren, K. (2002). “Modeling the effect of corrosion on bond in reinforced concrete.” Magazine of Concrete Research, No.3, June.

Maekawa, K. Kishi, T. and Ishida, T. (2003). “Multi-scale modeling of concrete performance -Integrated material and structural mechanics (invited).” Journal of Advanced Concrete Technology, 1(2), 91-126.

Maekawa, K., Pimanmas, A. and Okamura, H. (2003). “Nonlinear Mechanics of Reinforced Concrete.” Spon Press, London.

Maekawa, K., Chaube, R. and Kishi, T. (1999). “Modeling of Concrete Performance.” SPON Press, London.

Mangat, P. S. and Elgarf, M. S. (1999). “Flexural strength of concrete beams with corroding reinforcement.” ACI Structural Journal, 96(1), 149-158.

Martin-Perez, B. (2000). “A study of the effect of chloride binding on service life predictions.” Cement and Concrete Research, 30, 1215-1223.

Okada, K., Kobayashi, K. and Miyagawa, T. (1988).

“Influence of longitudinal cracking due to reinforcement corrosion on characteristics of reinforced concrete members.” ACI Structural Journal, 85(2), 134-140.

Okamura, H. and Maekawa, K. (1991). “Nonlinear Analysis and Constitutive Models of Reinforced Concrete.” Gihodo Press, Tokyo.

Pimanmas, A. and Maekawa, K. (2001). “Control of crack localization and formation of failure path in RC members containing artificial crack device.” Journal of Materials, Concrete Structures and Pavements, JSCE, 52 (683), 173-186.

Pimanmas, A. and Maekawa, K. (2001). “Multi-directional fixed crack approach for highly anisotropic shear behavior in pre-cracked RC members.” Journal Materials, Concrete Structures and Pavements, JSCE, 50 (669), 293-307.

Rodriguez, J., Ortega, L. M. and Casal, J. (1997). “Load carrying capacity of concrete structures with corroded reinforcement.” Journal of Construction and Building Materials, ASCE, 11(4), 239-248.

Satoh, Y. et al. (2003). “Shear behavior of RC member with corroded shear and longitudinal reinforcing steels.” Proceedings of the JCI, 25(1), 821-826.

Stanish, K., Hooton, R. D. and Pantazopoulou, S. J. (1999). “Corrosion effects on bond strength in reinforced concrete.” ACI Structural Journal, 96(6), 915-921.

Tachibana, Y., Kajikawa, Y. and Kawamura. M. (1990). “The mechanical behavior of RC beams damaged by corrosion of reinforcement.” Concrete Library of JSCE, 14, March, 177-188.

Takewaka, K. and Matsumoto, S. (1984) “Behaviors of reinforced concrete members deteriorated by corrosion of reinforcement.” Proceedings of JCI, 6, 177-180.

Toongoenthong, K. and Maekawa, K. (2004). “Shear capacity of damaged RC beam with partial longitudinal cracks in space.” Proceedings of JCI, 26, 385-390.

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Multi-Mechanical Approach to Structural Performance ...steel w/o corrosion Confined expansion of RC system where D: Initial steel diameter, γ: Mass fraction loss, α: Rust expansion

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