2005 SLOVAK UNIVERSITY OF TECHNOLOGY18

1 INTRODUCTION

The process of a concrete cover’s cracking and delamination in corroding reinforced concrete structures results from the creation of corrosion products in the rebars area. The amount of ferrous ions and corrosion products strictly depends on the electric current’s density function. From an electrochemical point of view, the creation of corrosion products can be defined in the form of a summary reaction (the sum of anode and cathode reactions), cf. (Zybura, 1990).

(1)

The process of a concrete cover’s degradation is a two-stage process. The first part describes the process of the mass transport of and O2 into the rebar surface area and the initiation of the electrode process. The second part describes the cracking of the concrete cover as the result of the accumulation of corrosion

products. In this work an analysis of a concrete cover’s degradation has been presented. The analysis was performed through the use of a Willam – Warnke elastic - brittle model.

2 THEORETICAL BACKGROUND OF THE PROBLEM

A multicomponent media model with a dominant phase (skeleton) will be considered. On the basis of works

(Krykowski, 2006), (Krykowski 2005), (Bowen, 1980) it will be assumed that the problem is described by a differential equation of mass and momentum transfer. Additionally, the geometrically linear motion of the skeleton and diffusion mass transport without convection is assumed. The mass flux is assumed in accordance with Fick’s law. However, the constitutive relations from using the relationship resulting from the residual inequality of the problem,

T. KRYKOWSKI

THE SIMULATION OF CONCRETE FRACTURING PROCESSES AS THE RESULT OF REINFORCEMENT CORROSION

KEY WORDS

• FEM, • concrete cracking, • nonlinear material, • corrosion.

ABSTRACT

This work presentsa simulation of a concrete cover’s cracking process as a result of corrosion. An elastic - brittle material model with a distortion effect which depends on an electric current was defined. Finally, an exemplary analysis was presented.

Dr. T. Krykowski, ul. Akademicka 5 44-100 Gliwice, Poland, +48 32-237-15-42, Tomasz.Krykowski@polsl.pl.Professional career: 1996 – MSc, Silesian University of Technology, Gliwice; 2001 – PhD, Silesian University of Technology, Gliwice. Main research interests: Mechanics of continuous media, finite element method, nonlinear mechanics, degradation processes of concrete.

2006 SLOVAK UNIVERSITY OF TECHNOLOGY

2006/3 – 4 PAGES 18 – 22 RECEIVED 18. 9. 2006 ACCEPTED 27. 11. 2006

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19PROBABILITY AND SENSITIVITY ANALYSIS OF SOIL-STRUCTURE INTERACTION ...

cf. (Krykowski, 2006), is in the form:

σ = C : (ε – εin – εd) (2)

In equation (2), εin - is the nonelastic strain tensor which describes the propagation of cracking in concrete; εd - is the distortional strain tensor resulting from the corrosion products’ accumulation on the reinforced surface area; ε - is the strain tensor. It will be assumed that the distortional strain tensor is described by the following relation, cf. (Krykowski, 2006):

εd = (cp – c0p)γ (3)

In equation (3), γ is the tensor of the diffusion’s volumetric expansion; describes the increase in the corrosion product’s concentration on the reinforced area.

3 SUBSTITUTE CORROSION EXPANSION COEFFICIENT

The corrosion product’s accumulation process does not depend directly on the amount of chemical substances which are transported from the environment into the rebar surface. First, as previously mentioned, this process depends on the density of the electric current that flows in the rebars. Taking into consideration the approach presented in the work (Krykowski, 2006), it is necessary to introduce the so-called substitute distortional corrosion coefficients. Such coefficients which will join the volumetric expansion of the rebars with the density of the electric current can be presented in the following form:

(4)

In equation number (4): m – is the coefficient which depends on the type of corrosion, cf. (Roberts et al., 2000), φ0 - is the initial reinforcement’s diameter, N – the proportion coefficient, corrosion products to the mass of steel that is removed from the reinforcement, icorr- density of the electric current, λ - the conversion coefficient

.

4 CONCRETE MODEL

In the analysis a model based on the theory of elastic – brittle materials will be used. It will be assumed that the failure surface of the model is the Willam– Warnke failure surface. This surface can be described cf. (Chen, 1994), (Majewski, 2003) on the basis of the

following equations:

, (5)

In equation (5), the following denotation has been used: , , , θ - lode parameter. The expression

can be defined cf. (Chen, 1994), (Majewski, 2003) by the use of the following equations:

(6)

(7)

(8)

(9)

(10)

The parameters a0, a1, a2, b0, b1, b2 are the material constants.

5 THE COMPUTATIONAL EXAMPLE

An analysis of a concrete cover’s fracturing process in a concrete beam subjected to chloride ions has been performed. The analyzed cross section and the finite element grid were presented in Fig. 1. It is assumed that there are no external forces in the element. The state of the stresses is determined by the distortional state produced by the corrosion products. It is assumed that the concentration of chloride ions on the boundary of the cross section is C0 = 0.75 [%] of the cement mass, and the critical concentration is Ck = 0.4 [%] (the electrode process starts when the concentration of chlorides on the surface of the area exceeds the critical concentration C ≥ Ck). The following data was accepted for the analysis: density of the corrosion of electric current icorr = 0.2 [µA/cm2] , proportionality coefficient N = 3, initial reinforced diameter φ0 = 20 . 10–3 [m], conversion coefficient [µA/cm2] → [mm/rok] . λ = 0,0115, the corrosion type’s parameter m = 2. The analysis was performed using the ANSYS system. In particular, a module of the transient analysis of the heat flow for modeling the transport of the chlorides ions and the module to a nonlinear analysis of the material with

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thermal distortions for simulation of the concrete cover’s cracking has been used. Material parameters, cf. (Majewski, 2003), (Comite Euro-international du beton, 1993), (PKN, 2002).a) Concrete B20- elastic - fracturing Willam – Warnke material

model, characteristic compression strength fck = 16 [MPa], medium compressive strength fcm = fck +8 = 24 [MPa], characteristic strength for tension fctk = 1.3 [MPa], medium strength for tension fctm = 1.9 [MPa], Poisson coefficient for concrete ν = 0.2, elastic module Ecm = 29 [GPa], initial elastic module for concrete

, diffusion coefficient of chloride ions Cl–, D = 2 . 10–8 [cm2/s] .b) Steel 34GS, linearly elastic model, elasticity module E = 200

GPa, Poisson coefficient ν = 0.3. In the analysis the simplified algorithm, cf. (Krykowski, 2006), allows for an analysis of the problem through the use of commercial programs. The reduced algorithm of the analysis can be presented in the following form:• The analysis of the transport of chloride ions in concrete

– determination of critical time .

• The initiation of the electrochemical process (it is assumed that the density of the corrosion causing electric current is constant until the concrete cover’s cracking).

REPEAT UNTIL, (concrete cover cracks, time increase ∆t = 1 [month])

• Initiation of diffusion strain tensor γ and the increase in the corrosion’s products concentration ∆cp in the form:

,

. (11)

• The elastic-brittle analysis for the concentration of corrosion products increases.

ENDWe will assume that the degradation time is equal to the sum of the critical time that is necessary to initiate the corrosion process and to the time of the concrete cover’s cracking.

TDEGRADATION=TTRANSPORT+TCRACKING (12)

Fig. 1 The element’s cross section and the computational model of the concrete cover

γ

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The cracking time will be defined, cf. Figs. 2, 3, 4, 5, as the sum of the times necessary to obtain the critical Cl– ions’ concentration on the rebar surface and the creation time of the fully cracked zone in the cover.

TDEGRADATION = 20 [Years] + 9 [Months] ≅ 20 years and 9 months (13)

This paper was created with the help of grant: FP6 Marie Curie Transfer of Knowledge MTKD-CT-2004-509775.

Fig. 2 The grid of changes in the concentrated chloride ions at points A and B.

Fig. 3 The map of cracks in the concrete after time T = 5 [months]

Fig. 4 The map of cracks in the concrete after time T = 9 [months]

Fig. 5 The map of the fracturing propagation front in the concrete after T = 9 [months]

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REFERENCES

• Bowen, R. M. (1980) Incompressible Porous Media Models by use of the Theory of Mixtures. Int. J. Eng. Sci., 18, 1980, pp. 1129-1148.

• Krykowski, T. (2006) Computational Model of Corrosion Damage in Reinforced Concrete. Archives of Civil Engineering. Polish Academy of Science. Institute of Fundamental Technological Research. Volume L2. Issue 2, pp. 319-337, Warsaw 2006.

• Roberts, N.B., Atkins, C., Hogg, V., Middleton, C. (2000) A Proposed Empirical Corrosion Model for Reinforced Concrete. Proceedings of Institution of Civil Engineers: Structures & Buildings 2000, No. 140, pp. 1-11. ISSN: 0965-0911

• Krykowski, T. (2005) Zastosowanie mes do wyznaczania czasu degradacji otuliny w konstrukcjach żelbetowych (The Application of FEM in the Simulation of Reinforced Concrete Cover Degradation). Proceedings of the 4th International Conference on New Trends in the Statics and Dynamics of Buildings, Faculty of Civil Engineering STU, Slovak Society of Mechanics SAS, Bratislava 2005, Slovakia (in Polish).

• Chen, W-F. (1994) Constitutive Equations for Engineering Materials, Amsterdam –London – New York – Tokyo: Elsevier Science, ISBN: 0-444-88408-4.

• Majewski, S. (2003) Mechanika betonu konstrukcyjnego w ujęciu sprężysto – plastycznym. (Mechanics of Structural Concrete in Terms of Elasto-Plasticity) Gliwice: Wydawnictwo Politechniki Śląskiej, ISBN 83-7335-129-9.

• Zybura, A. (1990) Degradacja żelbetu w warunkach korozyjnych (The Degradation of Reinforced Concrete in the Corrosive Conditions). Gliwice: Zeszyty Naukowe Politechniki Śląskiej, Praca habilitacyjna, Z. 72, ISSN 0434-0779.

• Comite Euro-international du beton (1993) CEB-FIP Model Code 1990.

• PKN Polski Komitet Normalizacyjny (2002) PN-B-03264: Konstrukcje żelbetowe i sprzężone. Obliczenia statyczne i projektowanie (Plain Reinforced and Prestressed Concrete Structures. Analysis and Structural Design). Polish National Code.

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