Measuring crack width and spacing in reinforced concrete members

Measuring crack width and spacing in reinforced concrete members Syed Yasir Alam, Thibault Lenormand, Ahmed Loukili, J.P. Regoin

To cite this version: Syed Yasir Alam, Thibault Lenormand, Ahmed Loukili, J.P. Regoin. Measuring crack width and spacing in reinforced concrete members. B. H. Oh, O. C. Choi, L. Chung. 7th International conference on Fracture Mechanics of Concrete and Concrete Structures (FraMCoS-7), May 2010, Jeju, North Korea. pp.377-382, 2010.

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Fracture Mechanics of Concrete and Concrete Structures Recent Advances in Fracture Mechanics of Concrete - B. H. Oh, et al.(eds) ⓒ 2010 Korea Concrete Institute, Seoul, ISBN 978-89-5708-180-8

Measuring crack width and spacing in reinforced concrete members S. Yasir Alam, T. Lenormand, A. Loukili & J.P. Regoin Ecole Centrale de Nantes, France

ABSTRACT: Cracking behavior of reinforced concrete is usually understood by the cracking of a concrete prism reinforced with a central bar subjected to tension. Bending, which is majority of the real cases, is dealt in the Eurocode by an empirical adjustment of the coefficients. In this paper an experimental program is devised to study the structural size effect of reinforced concrete members on crack width and spacing. Bending tests are performed on three different sizes of beams which are geometrically similar in two dimensions. The main reinforcement ratio is constant in all the beams keeping the same number of bars. The cover to the main reinforcement is also scaled with the beam size. Crack width and cracks spacing are measured using digital image correlation technique. Strain in the main reinforcement is measured using embedded electric strain gauges. It is found that Eurocode underestimates the crack width and crack spacing. Measured values of crack widths show an important structural size effect, which is not accounted by Eurocode crack width formula. 1 INTRODUCTION Serviceability limit states (SLS) for reinforced concrete (RC) structures are usually applied to ensure their functionality and structural integrity under service loading conditions. In current design codes (EC2 EN 1992), the serviceability limit states are defined by providing three control parameters (1) Limitation of stresses in the material (2) Control of crack width and spacing (3) Deflection (short term and long term) checks. The maximum stresses, deflections and crack sizes are computed from critical combinations of actual, applied loads (called service loads), in conjunction with the structure’s geometry (type and location of reinforcement) and boundary conditions. However, it will be seen in this paper that the crack size also depends on material resistance to crack growth, which is not same as the material strength and which varies with the size of the structure. In this paper an experimental study is presented, in which three geometrically sized beams were tested in three point bending to obtain the experimental crack width and crack spacing, when the structure is subjected to service loadings. 2 CRACK CONTROL IN RC STRUCTURES It is recognized that cracking in reinforced concrete members may be of two forms: (1) Cracking due to restraint provided by structure to volume change, and (2) cracking due to applied loads. In this paper only the cracking due to applied loads is discussed.

There are three principal elements in the provisions of crack control (EC2 EN 1992): (1) The provision of minimum reinforcement area (2) A method for calculating design crack width (3) Simplified rules which will avoid the necessity for explicit calculation of crack width in most normal situations. 2.1 Design Crack width The approach almost universally used to explain the basic cracking behavior of reinforced concrete is to consider the cracking of a concrete prism reinforced with a central bar which is subjected to pure tension. Bending does influence the phenomena but this is dealt with in the Eurocode by an empirical adjustment of the coefficients. According to Eurocode (EC2 EN 1992), the design crack width can be determined using the following expression wk

(1)

= s r ,max (ε sm − ε cm )

where, wk = design crack width ; sr,max = maximum crack spacing ; εsm = mean strain in the reinforcement ; εcm = mean strain in concrete between cracks. In the expression (1) εsm - εcm may be calculated as: σ s − kt ε sm − ε cm =

fct ,eff

ρ p ,eff

(1 + α e ρ p ,eff )

Es

≥

0 .6

σs Es

(2)

J = − D ( h , T ) ∇h where (1) σ = stress in the tension reinforcement assuming cracked section ; ρ = Acoefficient /A , which is theisratio of The proportionality D(h,T) called area of tension reinforcement ( A ) to the effective moisture permeability and it is a nonlinear function tension area ofhumidity concreteh and around the steelT (Bažant (A ). of the relative temperature However, A depends implicitly upon the concrete & Najjar 1972). The moisture mass balance requires cover in that theprovided variationoninthe timetension of the face. waterSee massdetails per unit EC2 EN 1992. volume of concrete (water content w) be equal to the divergence of the moisture flux J 2.2 Maximum final crack spacing (2) − ∂ = reinforcement ∇•J When is fixed at reasonably closed ∂ spacing (spacing ≤ 5(c+Φ/2)), according to EurocodeThe (EC2 ENcontent 1992) the maximum final crack water w can be expressed as thespacsum ing can be calculated as: of the evaporable water we (capillary water, water vapor, and adsorbed water) and the non-evaporable φ (chemically (3) s , max = c + bound)k 1 k 2water wn (Mills 1966, ρ , It is reasonable to Pantazopoulo & Mills 1995). assume that the evaporable water is a function of relative humidity, h, degree of hydration, αc, and where, we=w; ek(h,, kαc,are αs) degree fume;reaction, αs, i.e. Φ = of barsilica diameter c = concrete cover = age-dependent sorption/desorption isotherm coefficients depending upon steel-concrete bond (Norling Mjonell 1997).type Under this assumption and properties and loading respectively. See EC2 by 1992 substituting Equation 1 into Equation 2 one EN for details. obtains In equation (3) concrete cover c is explicitly introduced into the expression of the crack width as ∂w the influence of ∂w ∂h by A. Beeby.∂w suggested studied e α& e α& + w& e + ∇ • ( D ∇h) = He + − (3) c s n (Beeby concrete α transfer ∂α length ∂h ∂t cover hover ∂the c s 2001). The transfer length increases monotonically with the increase of concrete cover. This effect is inthe slope ofofthecrack sorption/desorption where ∂wine/∂htheis calculation troduced spacing. Stress isotherm (also called moisture capacity). The in the concrete around the reinforcement is directly governing equation (Equation 3) must be completed dependant on the stress transfer length between conby appropriate and initial conditions. crete and steel.boundary Since cracking occurs in concrete at relation the amount of evaporable the The points where between stress exceeds the tensile resistance water and relative humidity is called ‘‘adsorption of concrete, therefore, crack spacing is a function of isotherm”cover. if measured relativity concrete In this with study,increasing concrete cover to humidity and ‘‘desorption isotherm” in the opposite height of the beam ratio is kept constant to take into case. Neglecting (Xi et al. in account the effecttheir of difference overall structure size1994), on the the following, ‘‘sorption isotherm” will be used with crack opening and crack spacing. reference to both sorption and desorption conditions. By the way, if the hysteresis of the moisture 2.3 Analysis of test data into account, two different isotherm would be taken evaporable water relative humidity, Itrelation, has been suggested thatvswhen analyzing themust test be used according to the sign of the variation of the data of crack width and crack spacing in reinforced relativity tensile humidity. The following shape of considerations the sorption concrete members, isotherm HPC is influencedtobyENmany parameters, should befor made (Commentary 1992): especially those that influence extent and ratetheofmathe - The materials used should be similar to chemical reactions and, in turn, determine pore terials use today in the building structures. The low2 structure and pore size qualities distribution bound rebars, concrete less(water-to-cement than 20 N/mm ratio,steel cement chemical composition, content, and qualities less than 400 N/mm2SFshould not curing time and method, temperature, mix additives, be used. etc.). In stress the literature various be - The range should be formulations serviceability can range. found to describe the sorption isotherm of normal For this purpose, results 2in the stress range in steel concrete et al.N/mm 1994).should However, in the present from 150 (Xi to 350 be considered for paper the semi-empirical expression proposed by tests involving direct actions. For indirect actions, Norling Mjornell (1997) is adopted because it s

p,eff

s

c,eff

s

c,eff

c,eff

w t

r

3.4

0 . 425

p eff

1

Proceedings of FraMCoS-7, May 23-28, 2010

2

explicitly evolutionstress of hydration steel stressaccounts range upforto the yielding of steel reactionbe and SF content. This sorption isotherm should considered. reads - To determine the crack spacing the number of cracks present at the last phase of the test is always considered since it is the ⎡ closest to the stabilized ⎤ cracking, as given by equation (3). ⎢ ⎥ (h α α ) = G (α α ) − weIn + ⎥ for the s s ⎢ scarce(gdata thec literature ac very is found ∞ α − α )h ⎥ ⎢ c c ⎦and (4) direct comparison between the ethe calculated ⎣ experimental crack widths. ∞ ⎡ ⎤ 1

,

,

,

1

1

10

g α c − α c )h −

10(

K (α c α s )⎢e ⎢ ,

1

1

1

⎣ 3 SIZE EFFECT IN CONCRETE

⎥ ⎥ ⎦

1

where the size firsteffect term is(gel represents the Structural theisotherm) central problem in prephysically bound (adsorbed) waterareand the second dictions of fracture. Fracture tests normally conterm (capillary isotherm) represents and the then capillary ducted on relatively small specimens this water. This expression is valid only for low content information is extrapolated to large structures. the amount of of The SF. The coefficient question of size Geffect has become a crucial 1 represents water per unitinvolume held toin design the gelconcrete pores at struc100% consideration the efforts relativeforhumidity, and inevitably it can be expressed tures, which there is a large (Norling gap beMjornell 1997) as tween the scales of large structures (dams, reactor containments, bridges) and of laboratory tests. The cstructural size is normally underG (αeffect α ) =on k vg α c + k s αstrength (5) c of the vg characteristic ss stoodc as sthe effect structure size (dimension) on the nominal strength of the structure when similar structures are compared material parameters. From the wheregeometrically kcvg and ksvg are (Bazant 2002). maximum amount of water per unit volume that can quasi-brittle materials fillInallall pores (both capillary poreslike and concrete, gel pores),fracone ture is preceded byone a gradual obtainsdispersed microcrackcan calculate K1 as ing that occurs within a relatively large fracture process zone ahead of the crack⎡ tip ⎛⎜of ⎞ ⎤ − α ⎟h ⎥ g αac∞ continuous ⎢ c ⎝ ⎠ strain crack. When a concrete structure is loaded, the w − α s + α s −G ⎢ −e ⎥ c s ⎥ energy produced by the applied⎢⎣ load is converted to ⎦ (6) K (αenergy α ) = consumed to create new fracture surfaces the c s ⎛ ⎞ ∞ − α ⎟h ⎜ g α in and the energy absorbed the c c ⎝ ⎠ − fracture process e zone. For large-size structure the latter is negligible compared to theparameters former, whereas small-size ksvg aand g1 can The material kcvg and for structure these can be comparable (Bazant 2002). be calibrated by fitting experimental data relevant to Therefore, the larger the structure size, the lower the free (evaporable) water content in concrete nominal strength. However, the 2009b). concrete strengthat various ages (Di Luzio & Cusatis approaches a constant when the size of the concrete structure becomes very large. 2.2 Temperature evolution NoteSize that,effect at early age, since 3.1 on crack width the chemical reactions associated with cement hydration and isSFalsoreaction Itareis exothermic, obvious thatthe thetemperature cracking behavior influfield is not uniform enced by structural size. A recent study has shown for non-adiabatic even if the environmental that the cracking insystems plain concrete is size dependant temperature is constant. Heat conduction can be mechanism (Alam & Loukili 2009). It was observed described in concrete, at least for temperature not that when geometrically similar specimens are tested exceeding 100°C (Bažant & Kaplan 1996), by under same loading and boundary conditions, the Fourier’s law,ofwhich propagation crack reads is not the same in large beams asq =it −isλ∇inT small beams. A numerical study (Ouyang (7) & Shah 1994) of reinforced concrete based on fracture mechanics rules has also shown the cracking where qto isbe size the dependant. heat flux,They T iscalculated the absolute behavior crack temperature, and λ is the heat conductivity; this width in tensile members with constant centralinreinforcement and member length but varying cross sec,

1

10

0

0.188

0.22

1

1

,

1

10

1

1

1

− D (hwith , T )∇6mm h in diameter. The tional areas. They found that crack width decreases vided by deformedJ =bars as the width of the member increases. Results were steel bars in both cases are type fyk ≥ 500 MPa. not compared with experimental data. They found The proportionality coefficient D(h,T) that previously obtained experimental data is very 4.2 Beams moisture permeability and it is a nonlinea scattered because crack width depends, not only on of the relative humidity h and temperature are tested. of balanc the geometry and loading condition of the structure, Three sizes of beams & Najjar 1972). The dimensions moisture mass in Table 2. The beams have a mas but also on the size of the structure. In the current the beams are given that the variation in time of the water = 100 ofmm), however, the length design approach (Equations (1) & (2)), crack width constant width (bvolume concrete (water content w) be eq Concrete cover at theflux tension is evaluated primarily using empirical methods that and height are scaled. divergence of the moisture J require some empirical constants. The size effect on face of the beam is also scaled with same proportion of beams. The beams are rethe crack width cannot be accounted for by the con- between different sizes − ∂ = ∇reinforcement •J ratio by varyventional analysis unless additional empirical con- inforced with a constant ∂ ing the diameter of the bars; however, the number of stants are used (Shah et al. 1995). the A limited amount of experimental data is avail- bars is kept constant. TheSteel waterstirrups contentarew used can beinexpressed a hanger bars of 6mm diameable in the literature concerning the study of size ef- area of high shear.of Two the evaporable water we (capillary wa stirrups. These fect on the crack width and crack spacing. Most of ter are used to support vapor, steel and adsorbed water)bars andare the non-e are not provided 1). wn (Mil the results deal with the reinforced concrete mem- cut where the stirrups (chemically bound) (Fig. water bers subjected to tension or having small dimenPantazopoulo & Mills 1995). It is reas sions(Commentary to EN 1992). In this paper an exassume that the evaporable water is a fu perimental program is devised to study the effect of Table 2. Specimens geometry. relative humidity, h, degree of hydration structural size on the crack width and crack spacing. Beam ρ cover degree of silica fume reaction, αs, i.e. we=w Three sizes of beams with geometrically similar = age-dependent sorption/desorption mm mm mm mm % mm length and height, but with constant thickness are (Norling Mjonell 1997). Under this assum tested in three-point bending. The tensile reinforce- Small D1 100by 100 substituting 1 into Equati 400 300Equation 1.99 10 ment ratio is kept constant; however, the concrete obtains cover to the reinforcement was scaled according to Medium D2 100 200 800 600 1.84 20 the overall size of the beam. The aim of this study is ∂w ∂w we ∂h1600 1200 1.83 e α40 e α& + w to evaluate the concrete crack width and crack spac- Large D3 100− ∂400 & + ∇ • ( D ∇h ) = c + s h ∂ α ∂ α ∂ ∂ h t ing expressions present in the Eurocode (equations c s 1-3) with the experimentally measured crack width D1 and spacing. where ∂we/∂h is the slopeA of the sorption/ isotherm (also called moisture capac governing equation (Equation 3) must be 4 EXPERIMENTAL PROGRAM by appropriate boundary and initial conditi The relation between Athe amount of e 4.1 Materials water and relative humidity is called ‘‘ Section A-A isotherm” if measured with increasing The concrete used to prepare the beams is made D2 humidity and ‘‘desorption isotherm” in th from Portland cement type 52.5, sand, aggregate & case. Neglecting their difference (Xi et al. water using the mix proportions in Table (1). A the following, ‘‘sorption isotherm” will be Table 1. Concrete mix design details. reference to both sorption and desorption c Cement (Portland 52.5) 312 kg/m3 By the way, if the hysteresis of the Sand 818 kg/m3 isotherm would be taken into account, two A vs relative humi relation, evaporable water Coarse aggregate (5 - 12.5 mm) 936 kg/m3 Section A-A be used according to the sign of the varia Water 219 kg/m3 relativity humidity. The shape of the D3 isotherm for HPC is influenced by many p A The compressive strength of concrete is tested at especially those that influence extent and 28 days using cylindrical specimens with diameter2 chemical reactions and, in turn, determ 10cm and height 20cm and is found to be 40 N/mm . structure and pore size distribution (waterSince the beams are reinforced with rebars, a super ratio, cement chemical composition, SF plasticizer with a dosage of 0.25% by weight of cecuring time and method, temperature, mix ment is added to increase the workability of fresh A etc.). In the literature various formulatio concrete. Section A-A found to describe the sorption isotherm The longitudinal reinforcement of beams is con(Xi et details. al. 1994). However, in th 1*. Geometryconcrete and reinforcement stituted of deformed bars with 10, 14 & 20 mm in Figure paper the semi-empirical expression pro *The figure is not scaled. diameter, while, the transverse reinforcement is proNorling Mjornell (1997) is adopted b w t

b

h

L

l


J = −Test D (h,procedure T ) ∇h 4.3 (1) All the experiments are performed using the same The machine proportionality coefficientdisplacement. D(h,T) is called testing with controlled The moisture permeability and it is a nonlinear test setup is shown in Figure 2. The load is function applied the mid relative h and temperature T (Bažant atof the spanhumidity with a cylindrical jack, ensuring the & Najjar The ismoisture masstobalance requires point load.1972). The load transferred the beam using the variation in timetheof concrete the waterdamage mass perat unit athat rubber pad, to avoid the volume of concrete (water content w ) be to the load point. The test is carried out usingequal a constant divergence of the moisture flux J mm/min. The disٛ0.5 vertical displacement rate of placement is measured using a laser sensor, which measures the − ∂ = ∇ • the J displacement at the mid section of (2) ∂ under the load. The sensor is attached to a beam steel hanger, which is supported at the supports to content w can be expressed the supsum takeThe intowater account the settlement of beam atas the (capillary water, water of the evaporable water w ports. e vapor, and adsorbed water) and the non-evaporable (chemically bound) water wn (Mills System 1966, Pantazopoulo & Mills 1995). It is of reasonable to assume that the evaporable water is sition a acquifunction of of relative humidity, h, degree of hydration, αc, and steel dedegree of silica fume reaction, αs, i.e. wformae=we(h,αc,αs) = age-dependent sorption/desorption isotherm (Norling Mjonell 1997). Under this assumption and by substituting Equation 1 into Equation 2 one Light obtains projector w t

−

∂w ∂h e + ∇ • ( D ∇h) = ∂we h ∂α ∂h ∂t

∂w α&c + e α&s + w&n ∂α c s

(3)

where ∂we/∂h is the slope of the sorption/desorption called moisture capacity). The governing equation (Equation 3) must be completed each beam, two electric strain gauges are atby For appropriate boundary and initial conditions. tached to the lower face of tension steel mid span The relation between the amount ofatevaporable to measure longitudinal in steel‘‘adsorption during the water and the relative humiditystrain is called test (Fig. 3).if measured with increasing relativity isotherm” Resultsand are ‘‘desorption discussed here for threeinpoint bending humidity isotherm” the opposite tests on three sizes of beams. After the fabrication, case. Neglecting their difference (Xi et al. 1994), in these beams were stored atisotherm” 20°C andwill relative humidthe following, ‘‘sorption be used with ity of 50%toforboth 24 sorption hours. After removalconditions. of molds, reference and the desorption these beams in 100% relative By the way,were if stored the hysteresis of the humidity moisture for 28 days isotherm would be taken into account, two different relation, evaporable water vs relative humidity, must be used according to the sign of the variation of the relativity humidity. The shape of the sorption isotherm for HPC is influenced by many parameters, especially those that influence extent and rate of the chemical reactions and, in turn, determine pore structure and pore size distribution (water-to-cement ratio, cement chemical composition, SF content, curing time and method, temperature, mix additives, etc.). In the literature various formulations can be found3*.to Strain describe sorption normal Figure gaugesthe attached to theisotherm steel bars atofmid span. concrete al. 1994). However, in the present *view before(Xi the et pouring of concrete. paper the semi-empirical expression proposed by Norling Mjornell (1997) is adopted because it Figure 2. Test(also setup. isotherm


explicitly for the evolution of hydration 4.4 Digitalaccounts image correlation reaction and SF of content. This and sorption The measurement crack width crackisotherm spacing reads is carried out using a digital image correlation system. The digital image correlation is an optical ⎤ method to measure the ⎡⎢surface displacement field ⎥ between al) = time (α instants. ) − This technique has we (h α c αtwo G α + ⎢ ⎥ s c s ∞ (g α(Alam h ⎥ Loukili − α c )& ready been applied on ⎢concrete e etcal. 2001, ⎦ Corr(4) 2009, Choi & Shah 1997,⎣ Lawler et ∞ al. 2007). It is based on ⎡the (comparison of two im⎤ g α c − α c )h ⎥ after ages based on Kgrayscale before− and (α α )⎢erecorded c s ⎢ ⎥ deformation. The first image is called the reference ⎣ ⎦ and the second is the deformed image. The displacement field term is then through the where the offirst (gel determined isotherm) represents the movement the subimage in an area defined as the physically bound (adsorbed) water and the second correlation zone. isotherm) The correlation consists considterm a(capillary represents theofcapillary ering segment (subimage) in the reference image water.locating This expression is valid only for low content and that segment (subimage) in the of of SF. The coefficient G1 represents the amountdeformed image, where the maximum likelihood is water per unit volume held in the gel pores at 100% achieved. In this and study, a be commercial software relativeishumidity, it can expressed (Norling Vic2D used to perform the image correlation. The Mjornell 1997) as size of subimage is taken as 29x29 pixels, which is regularly spaced in the reference image (of size c s ( ) G α α = k α c + k α s of a grid. Each subi(5) 1392x1040 in vg the sform c s pixels) vg c mage in the reference image is correlated to the one in deformed image by calculating a correlation coefandcorrelation ksvg are material parameters. From the where kincvgthe ficient area around the subimage. maximum amount of system water per unit volume The resolution of the depends directlythat on can the fill all pores (both capillary pores and gel pores), one distribution of gray levels which depends on the texcan calculate K1 as one ture of the material. Toobtains obtain a random pattern, a speckle pattern of black and white paint is sprayed ⎡ ⎛ ⎞ ⎤ ∞ − α ⎟h ⎥ on a onto the surface of the specimen. The ⎜ g α images ⎢ c c ⎝ ⎠ s + α s s − are G ⎢ −captured e ⎥ two single facew of− theα cspecimen using ⎥ ⎢ digital which give 256⎣ levels of gray inten⎦ (6) )= K (α c α scameras, ⎛ ⎞ ∞ sity. Cameras are placed in such a way to film an ⎜ g α − α ⎟h c c at ⎠ −either side of the area 15x10 cm by each e ⎝ camera mid span. The mid span portion is filmed by both ksvg as andthe greferThe material kcvg isand cameras. The midparameters span section taken 1 can be calibrated by fitting experimental data relevant to ence to relate the measurement values of the two free (evaporable) in concrete at cameras as shown inwater Figure content 4. various ages (Di Luzio & Cusatis 2009b). 1

,

,

,

1

1

10

10

,

1

1

1

1

,

1

10

0

0.188

0.22

1

1

1

,

1

10

1

1

mm Mid span, reference 2.2 Temperature150evolution line for two cameras Note that, at early age, since the chemical reactions associated with cement hydration and SF reaction are 100 exothermic, the temperature field is not uniform mm non-adiabatic systems even if the environmental for temperature is constant. Heat conduction can be described in concrete, at least for temperature not exceeding 100°C (Bažant & Kaplan 1996), by 1 CAM 2 Fourier’s law,CAM which reads 15 mm

Figure 4.∇Camera zones for digital image correlation. q = −λ T

(7)

where q is the heat flux, T is the absolute temperature, and λ is the heat conductivity; in this

J

= − D ( h , T ) ∇h

The proportionality coefficient D(h,T) moisture permeability and it is a nonlinea of the relative humidity h and temperature & Najjar 1972). The moisture mass balanc that the variation in time of the water mas volume of concrete (water content w) be eq divergence of the moisture flux J − ∂ = ∇•J ∂ w t

The water content w can be expressed a of the evaporable water we (capillary wa vapor, and adsorbed water) and the non-e (chemically bound) water wn (Mil Pantazopoulo & Mills 1995). It is reas assume that the evaporable water is a fu relative humidity, h, degree of hydration degree of silica fume reaction, αs, i.e. we=w = age-dependent sorption/desorption (Norling Mjonell 1997). Under this assum by substituting Equation 1 into Equati obtains −

∂w ∂h e + ∇ • ( D ∇h) = ∂we h ∂α ∂h ∂t

∂w α&c + e α&s + w ∂α c s

Figure 5. Measurement of crack opening displacement (COD) and crack spacing in small (D1) beam.

5 RESULTS AND DISCUSSION The measurement of crack opening displacement (COD) and crack spacing by digital image correlation is presented in Figure 5. COD is measured as displacement jump on the surface of the beam. Multiple cracks are found, but only the maximum COD is considered in the further analysis The crack width measured using digital image correlation technique is plotted against the crack width obtained using eurocode crack width formula (eq. 1) as a function of steel strain εsm. The crack width is measured at the mouth of the crack i.e. at the lowest fiber of the beam, where the crack width is maximum. The results are shown in Figure 6 for comparison. It can be seen that eurocode formula underestimates the crack width for all the three sizes. The difference between measured and calculated value exceeds when the strain is higher. Figure 4 shows the comparison for a stress range under yield limit of steel.

where ∂we/∂h is the slope of the sorption/ isotherm (also called moisture capac governing equation (Equation 3) must be by appropriate boundary and initial conditi The relation between the amount of e ) water and relative humidity is called ‘‘ m (m isotherm” if measured with increasing W humidity and ‘‘desorption isotherm” in th case. Neglecting their difference (Xi et al. the following, ‘‘sorption isotherm” will be reference to both sorption and desorption c By the way, if the hysteresis of the (mm) isothermW would be taken into account, two relation, evaporable water(W vsk,exp relative Figure 6. Comparison of experimental crack width ) and humi be used according to the sign of the varia calculated maximum crack width (Wk,max) during crack formarelativity humidity. The shape of the tion stage (εsm< yield limit). isotherm for HPC is influenced by many p thateffect influence A more detail especially observationthose of the of sizeextent on and chemical reactions in inturn, the eurocode calculated crack opening and, is made the determ structure pore sizecrack distribution Figure 7. It is clear that theand calculated width is (waterratio, cement chemical composition, SF more or less comparable for small size beam but the time and method, temperature, mix error exceeds withcuring size increases. etc.). In the literature various formulatio found to describe the sorption isotherm concrete (Xi et al. 1994). However, in th paper the semi-empirical expression pro Norling Mjornell (1997) is adopted b 0,5

0,4

D3

0,3

D2

ax ,m k

D1

0,2

Calc~test

0,1

0

0

0,1

0,2

0,3

0,4

0,5

k,exp


= − D ( h , T ) ∇h

J

(1)

0,16

Eurocode 2 EN 1992

The proportionality coefficient D(h,T)Experiment is called moisture permeability and it is a nonlinear 0,12 ε = 0,4 x function 10 of the relative humidity h and temperature T (Bažant &)m(m0,08 Najjar 1972). The moisture mass balance requires ε = 0,6 x 10 that W the variation in time of the water mass per unit = 0,9 x 10 to the volume of concrete (water content w) beε equal 0,04 divergence of the moisture flux J -3

-3

k

-3

− ∂0,00 0= ∇ 50• J ∂

1

,

,

,

1

1

10

100

150

200

250

Specimen size (mm)

300

350

400

450

(2)

Figure 7. water Size effect on experimental and calculated The content w can be expressed asmaximum the sum crack width. of the evaporable water w (capillary water, water e

vapor, water) and non-evaporable The and crackadsorbed width obtained fromwthe Eurocode formula (Mills (chemically bound) water n does not take into account1995). size effect. However,1966, the Pantazopoulo & Mills It is reasonable to values obtained are different for three sizes of assume that the evaporable water is a function of beams. But, this should not be ofconsidered as αsize efrelative humidity, h, degree hydration, c, and fect as the Eurocode formula depends only on the i.e. bars, we=we(h,αc,αs) degree ofcover silica and fumediameter reaction,ofαs,the concrete however, = age-dependent sorption/desorption isotherm the true size effect is the effect of the dimensions of (Norling Mjonell 1997). Under this assumption and the structure on the overall fracture mechanism. As by structure substituting Equation 1 into large Equation 2 one the becomes sufficiently the size of obtains the heterogeneity becomes negligible compared to the∂wsize of the structure ∂and material follows a brittle ∂w ∂hHowever, when w failure. the size of the e e e α& + α& structure + w&n be+ ∇ • ( D ∇h ) = − (3) h small ∂α thec size ∂α of sheterogeneity ∂h ∂sufficiently t comes c s becomes comparable to the size of the structure and the material the slope material follows a ductile failis the of the sorption/desorption where ∂we/∂hand ure. isotherm (also called moisture capacity). The governing equation (Equation 3) must be completed 200 by appropriate boundary and initial conditions. The relation between the amount of evaporable water 150 and relative humidity is called ‘‘adsorption isotherm” if measured with increasing relativity ) humidity and ‘‘desorption isotherm” in the opposite m m ( case. Neglecting their difference (Xi et al. 1994), in x 100 thesar,m following, ‘‘sorption isotherm” will be used with reference to both sorption and desorption Eurocode 2conditions. EN 1992 By 50the way, if the hysteresis of the moisture Experiment isotherm would be taken into account, two different relation, evaporable water vs relative humidity, must 0 according to the sign of the variation of the be used 0 humidity. 100 200 shape 300 of the 400 sorption 500 relativity The Specimen size (mm) isotherm for HPC is influenced by many parameters, Figure 8. Maximum spacing of cracks, comparison especially those that influence extent and rateof ofmeasthe ured values with the calculated chemical reactions and, values. in turn, determine pore structure and pore size distribution (water-to-cement The cement maximum spacingcomposition, between cracks is also2 ratio, chemical SF content, measured when steel stress was equal to 300 N/mm curing time and method, temperature, mix additives,. This is compared (Fig. formulations 8) with the spacing etc.). spacing In the literature various can be obtained the the Eurocode (eq.of3).normal It is found to from describe sorptionformula isotherm seen that is underestimated for the medium (D2) and concrete (Xi et al. 1994). However, in the present large (D3) sized beam. The reason may be that paper the semi-empirical expression proposed the by concrete brittle the because size of theit Norling becomes Mjornell more (1997) is when adopted beam increases. Thus the crack propagation is much Proceedings of FraMCoS-7, May 23-28, 2010

1

6 CONCLUSION

1

g α c − α c )h −

10(

K (α c α s )⎢e ⎢ ,

w t

explicitly accounts for the evolution of hydration faster (Alam & Loukili 2009) and energy release reaction and SF This sorption rate is higher in content. large structures (Kaplanisotherm 1961). reads for small structures, crack propagation is However much slower and strain energy is mainly released due to microcracking in⎡ the fracture process ⎤ zone. Since the presence of reinforcement hinders ⎢ ⎥ the dewe (h α c α sof) =fracture G (α c αprocess + are ⎥ s )⎢ − zone, velopment more cracks ∞ (g α − α c )h ⎥ cenergy being formed to release⎢⎣theestrain stored in ⎦ (4) the material. ∞ ⎡ ⎤ 1

⎣

⎥ ⎥ ⎦

1

where the first term (gel isotherm) represents the An experimental investigation is performed to study physically (adsorbed) and the the effect ofbound structural size on water crack width andsecond crack term (capillary isotherm) represents theunder capillary spacing in reinforced concrete structures serwater. This expression is valid only for low content vice loadings. The results show that the measured represents the amount of of SF. more The coefficient G1with values or less agree the calculated values unit volume heldbeam in thesize. gel But poresthere at 100% atwater lowper strains and small is a relative humidity, and itoncanthebe experimental expressed (Norling significant size effect crack Mjornell 1997) as width and crack spacing, which is not incorporated in the current Eurocode design formulas. c α c+ ks α s G (α c α s ) = k vg (5) c vg s REFERNCES where kcvg and ksvg are material parameters. From the Alam, S.Y. &amount Loukili, of A. water 2009. Etude des effets d’échelle sur maximum per unit volume that can la propagation des fissures dans le béton par la technique de fillcorrélation all pores (both capillary pores and gel pores), one d’images, Rencontres AUGC. Saint-Malo. canFrance. calculate K1 as one obtains ,

1

Bazant, Z.P. 2002. Scaling of structural strength. London: ⎡ Hermes Penton. ⎛ ⎞ ⎤ ∞ ⎜ g α − α ⎟h ⎥ ⎢ c c Beeby A.W.w2001. Calculation of Crack Width , PrEN ⎝ ⎠ 1992-1 − α s + α s −G ⎢ −e ⎥ c 7.3.4.s (Final draft). Chapter ⎥ ⎢ ⎦ (6) ⎣ Beeby ) = 2001. Crack control provisions in the new euroK (α c αA.W. s ⎛ ⎞ ∞ code for the design of concrete structures. ACI Special pub⎜ g α − α ⎟h lication 204: 57-84. e ⎝ c c ⎠ − Corr, D.; Accardi, M., Grahan-Brady, L. & Shah, S.P. 2007. Gigital image correlation analysis interfacial debonding and ksvg and g1 can The material parameters kcvg inof properties and fracture behaviour concrete. Engineering be Fracture calibrated by fitting experimental data relevant to Mechanics 74: 109-121. free S.(evaporable) waterMeasurement content in concrete onat Choi, & Shah, S.P. 1997. of deformation concrete subjected to compression image correlation. various ages (Di Luzio & Cusatisusing 2009b). Experimental Mechanics 37(3): 307-313. EN 1992-1-1, December 2004. Eurocode 2: Design of concrete – Part 1-1: General rules and rules for buildings. 2.2structures Temperature evolution Brussels. European Committee for Standardization (CEN). Note that, at early since the chemical reactions Jacobs, J-P. June 2008. age, Commentary Eurocode 2. Brussels, European Concrete associated with Platform cement ASBL. hydration and SF reaction Kaplan, M. F. 1961.theCrack Propagation andisthenotFracture of are exothermic, temperature field uniform Concrete. ACI Journal proceedings 58(11): 591-608. for non-adiabatic systems even if2001. the environmental Lawler, J.S.; Keane, D.T. & Shah, S.P. conduction Measuring temperature is constant. Heat canthree be dimensional damage in concrete under compression. ACI described in concrete, at least for temperature not Materials journal 98(6): 465-475 exceeding (Bažant & Kaplan 1996), for by Ouyang, C. &100°C Shah, S.P. 1994. Fracture energy approach predicting cracking of reinforced concrete tensile members. Fourier’s law, which reads ACI Structural Journal 91(1): 69-78. Shah, q = −S.P.; λ ∇T Swartz, S.E. & Ouyang, C. 1995. Fracture Me10

0

0.188

0.22

1

1

1

,

1

10

1

1

chanics of concrete: Applications of fracture mechanics(7) to concrete, rock and other quasi-brittle materials. Newyork: John Wiley where q is& sons. the heat flux, T is the absolute

temperature, and λ is the heat conductivity; in this