VIII International Conferene on Frature Mehanis of Conrete and Conrete Strutures FraMCoS-8 J.G.M. Van Mier, G. Ruiz, C. Andrade, R.C. Yu and X.X. Zhang Eds) MODELLING THE POSTPEAK STRESS DISPLACEMENT RELATIONSHIP OF CONCRETE IN UNIAXIAL COMPRESSION TAKEAKI KOSHIKAWA Hokkaido University Faulty of Engineering Kita 13, Nishi 8, Kita-ku, Sapporo 6-8628, Japan e-mail: takeaki@eng.hokudai.a.jp, www.hokudai.a.jp Key words: Uniaxial Compression, Postpeak, Loalization, Frature, Dissipated Energy, Conrete Abstrat. This paper presents a onstitutive equation for the postpeak stress displaement relationship of onrete in uniaxial ompression. The relationship is modelled in relation to the proess of postpeak energy dissipation inreasing the postpeak displaement of the loalized damage zone. The proposed equation inludes three material parameters: ompressive frature energy, ompressive strength, and ritial postpeak displaement. The nondimensional form of the equation shows that a parameter determined from these three material parameters ontrols the shape of the postpeak stress displaement urve. The preditions produed by the proposed equation are fit to uniaxial ompression test results from onrete speimens with different slenderness ratios and strengths. The fitting results and experimental responses agree quite well. 1 INTRODUCTION It is generally aepted that during the strain softening behaviour of onrete in its postpeak region under uniaxial ompression, the damage is loalized in ertain zones and strain loalization ours [1]. Due to the loalization phenomena, the postpeak portion of the stress strain relationship is not a true material property, but is dependent on speimen size, with longer speimens exhibiting more brittle postpeak behaviour. Development of an appropriate postpeak relationship based on strain loalization is important in aurately estimating the dutile apaity of onrete strutures. Published tests [1 3] have shown that the postpeak displaement of the loalized damage zone is independent of speimen size. Some theoretial approahes have been proposed for analyzing the ompressive behaviour of onrete based on this physial evidene. They inorporate a stress strain relationship to desribe the prepeak response and a stress displaement relationship to desribe postpeak behaviour as a material property [4 6]. These approahes an adequately predit onrete speimens uniaxial ompression test results, but the postpeak stress displaement relationship is approximated by an overly simple equation. A more realisti equation for the postpeak stress displaement relationship may be needed to further improve preditions. The purpose of this paper is to develop a postpeak stress displaement relationship for onrete in uniaxial ompression. We propose a method to derive a realisti desription based on the orrelation between the postpeak stress displaement relationship and the postpeak dissipated energy displaement relationship. We present the nondimensional form of the proposed equation desribing the postpeak stress 1
a) b) ) Figure 1: Correlation between a) σ ε relationship, b) σ δ p relationship and ) G δ p relationship. displaement relationship to show how the parameters influene the shape of the relationship. We use least squares fitting to uniaxial ompression test results from onrete speimens with different slenderness ratios and strengths to estimate the material parameters used in the proposed equation, then ompare the fitted results and experimental responses. 2 POSTPEAK STRESS DISPLACEMENT RELATIONSHIP 2.1 Derivation of fundamental expression We adopted the approah for the loalized ompression model proposed by Hillerborg [4] to model the postpeak stress displaement relationship of onrete. Consider a onedimensional onrete speimen under uniaxial ompression. The speimen has a height of H, and a uniform ross-setional area along the height. A typial stress strain σ ε) relationship for a onrete speimen in ompression as observed experimentally is illustrated in Figure 1a). The relationship of the prepeak region up to the ompressive strength f an be onsidered to be approximately the same throughout the speimen. At the ompressive strength, strain loalization takes plae, and strain softening begins in a loalized damage zone. In the postpeak region, the ompressive behaviour of the whole speimen an be desribed as a ombination of the behaviour inside and outside the damage zone. The postpeak displaement δ p in the loalized damage zone ontinues to inrease, while the remainder of the speimen unloads. Assuming that a linear unloading path from the ompressive strength f with slope E o during unloading ours outside the damage zone, the postpeak displaement δ p shown in Figure 1a) an be given by δ p = ε ε o + f σ E o ) H 1) where ε and σ are the strain and stress, and ε o is the strain orresponding to the ompressive strength f. The progress of the postpeak displaement in the loalized damage zone is aompanied by energy dissipation. The area under the urve of the postpeak stress displaement σ δ p ) relationship shown in Figure 1b) orresponds to the amount of postpeak energy dissipated per unit speimen area [2, 7]. In this investigation, we assume that stress is a funtion of the postpeak displaement, i.e. σ = σδ p ) as shown in Figure 1b), and the postpeak energy dissipated G is defined as G = δp σdδ p 2) where δ p and σ are the postpeak displaement and stress at a given point on the urve of the relationship in Figure 1b), respetively. The total postpeak energy dissipated by the time the postpeak displaement reahes the ritial value, orresponding to the stress ondition σ =, an be defined as the ompressive frature energy. Aording to Equation 2), the postpeak dissipated energy is also assumed to be a funtion 2
of the postpeak displaement, i.e. G = Gδ p ), whih satisfies the following equation. σ = dg dδ p 3) Equation 3) implies that the postpeak stress displaement relationship an be defined by determining the postpeak dissipated energy displaement G δ p ) relationship with an appropriate equation and then differentiating that equation with respet to the postpeak displaement. The essential features of the postpeak dissipated energy displaement relationship shown in Figure 1) are the following: 1. The postpeak dissipated energy is an inreasing funtion of the postpeak displaement, starting from G = at δ p = and inreasing up to G = at δ p =. 2. The slope of the postpeak dissipated energy displaement urve orresponds to the stress σ = f at δ p = and σ = at δ p =. Taking these features into aount, the postpeak dissipated energy displaement relationship is assumed to be defined by the following proposed equation. G = a 1δ p + a 2 δ 2 p 1 + a 3 δ p + a 4 δ 2 p 4) where a 1, a 2, a 3 and a 4 are onstants determined from the boundary onditions of the relationship. Considering an additional boundary ondition, the horizontal tangent at the point of ompressive strength on the postpeak stress displaement urve, d 2 G/dδp 2 = dσ/dδ p = at δ p =, together with the previously desribed boundary onditions of G = and σ = f at δ p =, G = and σ = at δ p =, the onstants are given by a 1 = f a 2 = f a 3 = a 4 = f 2 ) f 2 f 1 ) 2 5) Thus, using Equations 3) to 5), the following expression an be derived. { ) } f 1+2 f 2 )δ p 2f 3 δ 2 δpu σ = 2 p { ) } 1+ f 2 )δ p + f 2δ 2 1 2 p 6) As Equation 6) learly shows, the shape of the postpeak stress displaement urve in pratial appliation depends on three material parameters: ompressive frature energy, ompressive strength f, and ritial postpeak displaement. 2.2 Nondimensional form expression To demonstrate the shape of the postpeak stress displaement relationship obtained using Equation 6), we introdue the following new parameter A p, determined using those three material parameters. A p = f 7) As an be seen from Figure 1b) and Equation 7), the parameter A p represents the ratio between the area under the urve of the postpeak stress displaement relationship and the area of a retangle given by f times. Considering softening behaviour, the parameter A p must be a positive value less than 1. Using the parameter A p, Equation 6) an be rewritten in the following nondimensional form. σ = f 1 1+2 A p 2 1 A p 2 { 1+ ) δp ) δp + 2 ) ) 2 δp A p 3 ) 2 ) } 2 2 8) δp 1 A p 1 3
In this expression, stress and postpeak displaement are redued to nondimensional forms by dividing them by ompressive strength f and ritial postpeak displaement, respetively. The parameter A p thus beomes a single parameter that determines the relationship. Figure 2 shows the nondimensional form of the postpeak stress displaement relationship with various values of A p. The value of A p influenes the shape of the relationship, with larger values exhibiting more dutile postpeak behaviour. Based on Figure 2, it seems appropriate for parameter A p to be less than.5 for realisti representation of the behaviour of plain onrete. Another possible expression of the nondimensional form of Equation 6) that makes use of parameter A p is given by σ 1+2 1 2A p ) f δ p 2 3A p ) A f δ p p = f { ) } 2 2 1+ 1 2A p ) f δ p +1 A p ) 2 f δ p ) 2 9) In this expression, /f is used to redue the postpeak displaement to a nondimensional form. As a speial ase, if A p = is used and the value of the ritial postpeak displaement is assumed to be infinity, Equation 9) an be rewritten in the following simple form σ f = 1 + 2 f δ p G { F ) } 1 + f δ p + f 2 2 1) δ p If A p = 1 is used, Equation 9) or 8)) beomes σ f = 1 11) These two speial ases are illustrated in Figure 3. 3 FITTING TO EXPERIMENTAL RE- SULTS In the postpeak stress displaement relationship proposed in the previous setion there are relative stress relative postpeak displaement Figure 2: Nondimensional form of postpeak stress displaement relationship with various parameters A p. relative stress postpeak displaement Figure 3: Postpeak stress displaement relationship with parameters A p = and 1. three material parameters,, f and or A p ), that need to be determined for pratial appliations. While the ompressive strength f is easy to determine experimentally, the ompressive frature energy and ritial postpeak displaement are hard to measure using uniaxial ompression tests beause the tests generally annot ontinue to the point where the stress drops to zero. However by fitting the proposed relationship to experimental data obtained through the point where the stress drops to a ertain postpeak level, these material parameters an be estimated. In this investigation, we onduted uniaxial ompression tests on onrete speimens, and then fitted the postpeak stress displaement relationship using least squares to estimate the material parameters of eah onrete speimen. 4
Table 1: Mix proportions Mix W/C s/a Unit weight kg/m 3 ) Ad %) %) W C S G C %) I 56.2 44.5 179 318 784 998.6 II 32. 47.2 175 547 747 853.6 1.1 * air entraining agent ** air entraining and high-range water reduing agent Figure 4: Conrete speimens with different heights. 3.1 Uniaxial ompression test In the uniaxial ompression test, speimens with different slenderness ratios and strengths were tested. As shown in Figure 4 and Table 1, we used two sets of ylindrial speimens with the same diameters D = 1 mm and heights H = 2 mm and 4 mm, prepared with two onrete mixes. To prevent the effet of bleeding in onrete, the speimens were ast 2 mm taller than their proper heights, and 1 mm was ut from the top and bottom of eah speimen. At the time of loading, frition reduing pads onsisting of two.5 mm thik Teflon sheets with silion grease were plaed between the speimen and the loading platens. The uniaxial ompression test results showing ompressive strength f, strain at ompressive strength ε o and the modulus of elastiity E are given in Table 2. The ompressive strengths were about 25 N/mm 2 and 5 N/mm 2 for the Mix I and II onretes, respetively. 3.2 Fitting result The postpeak stress displaement relationship was fitted to the experimental results using Equation 9). The postpeak displaement δ p of the test results was alulated by Equation 1), assuming that the slope E o is equal to the modulus of elastiity E. The ompressive frature energy was assumed to be the oeffiient estimated from the least squares fitting proedure. By setting parameter A p to various values less than.5, orresponding values of ompressive frature energy were obtained Table 2: Speimens and test results Mix H/D H f ε o E mm) N/mm 2 ) 1 6 ) kn/mm 2 ) I 2 2 25.5 1874 24.3 I 4 4 25.3 185 22.5 II 2 2 51.2 2875 26.4 II 4 4 49.8 267 25.2 Compressive Frature Energy N/mm) Critial Postpeak Displaement mm) 3 2 1 Mix I, H/D= 2 Mix I, H/D= 4 Mix II,H/D= 2 Mix II,H/D= 4.1.2.3.4.5 Parameter 1 5 a) Compressive frature energy Mix I, H/D= 2 Mix I, H/D= 4 Mix II,H/D= 2 Mix II,H/D= 4.1.2.3.4.5 Parameter b) Critial postpeak displaement Figure 5: Fitting results for various parameters A p. 5
Stress N/mm 2 ) 3 2 1 1 2 Postpeak Displaement mm) a) Mix I Postpeak Dissipated Energy N/mm) 2 1 1 2 Postpeak Displaement mm) a) Mix I Stress N/mm 2 ) 6 4 2 1 2 Postpeak Displaement mm) b) Mix II Postpeak Dissipated Energy N/mm) 2 1 1 2 Postpeak Displaement mm) b) Mix II Figure 6: Experimental and proposed σ δ p relationships using A p =.1). Figure 7: Experimental and proposed G δ p relationships using A p =.1). for all the speimens as shown in Figure 5a). Also, the values of ritial postpeak displaement alulated using Equation 7) are shown in Figure 5b). As an be readily seen from Figure 5, for all speimens, the variation of estimated values of or ompared to the values of A p show the same tendeny although eah speimen s values are different. Sine the values of derease only slightly as the values of A p inrease, they an be thought to be generally onsistent in this range of A p. On the ontrary, the orresponding values of are signifiantly affeted by the values of A p. We ould not measure the atual values of in this test, but aording to the fitting results, we assumed them to be at least greater than 1 mm. As an example, we present a omparison between the resulting postpeak stress displaement relationships using A p =.1 and the experimental results in Figure 6. Overall, the proposed relationship is quite onsistent with the the experimental results. Another set of omparisons of the fitting results using A p =.1 against the experimental results are shown in Figures 7 and 8 for the postpeak dissipated energy displaement relationship and the stress strain relationship, respetively. The postpeak dissipated energy G of the test results was obtained from the area under the urve of the postpeak stress displaement relationship alulated by the trapezoidal rule. The strain ε of the postpeak region based on the proposed relationship was alulated using Equation 1). For the prepeak region, we used the following equation [8] with the parameter values listed in Table 2. 6
Stress N/mm 2 ) Stress N/mm 2 ) 3 2 1.4.8 Strain 6 4 2 a) Mix I.4.8 Strain b) Mix II Figure 8: Experimental and proposed σ ε relationships using A p =.1). σ f = E ε o ε f ε o 1 + E ε o 2 f ) 2 ε ε o ) 12) ε ε o The results of these figures, inluding Figure 6 indiate that the proposed approah for modelling the postpeak behaviour of onrete based on the orrelation of these relationships is valid. 4 CONCLUSIONS We have presented a onstitutive equation for the postpeak stress displaement relationship of onrete in uniaxial ompression. The derivation of the equation is based on the orrelation between this relationship and the postpeak dissipated energy displaement relationship. The proposed equation inludes three material parameters: ompressive frature energy, ompressive strength, and ritial postpeak displaement at the point when the stress drops to zero. The nondimensional form of the equation shows that the parameter A p, determined from those three material parameters, ontrols the shape of the postpeak stress displaement urve. For realisti representation of postpeak behaviour in plain onrete, values of parameter A p that are less than.5 seem appropriate. The results of least squares fitting of the equation to uniaxial ompression tests of onrete speimens with different slenderness ratios and strengths show that the value of parameter A p has more effet on the estimated value of the ritial postpeak displaement than does the ompressive frature energy. For any speimen, the fitted results using A p =.1 and the experimental responses agree quite well with respet to the stress strain relationship, the postpeak stress displaement relationship and the postpeak dissipated energy displaement relationship. REFERENCES [1] Van Mier, J.G.M., 1986. Multiaxial strainsoftening of onrete, Part I: Frature, Part II: Load-Histories. Mater. Strut. 19:179 2. [2] Jansen, D.C. and Shah, S.P., 1997. Effet of length on ompressive strain softening of onrete. J. Engrg. Meh. 123:25 35. [3] Van Mier, J.G.M., Shah, S.P., Arnaud, M. et al., 1997. Strain-softening of onrete in uniaxial ompression. Mater. Strut. 3:195 29. [4] Hillerborg, A., 199. Frature mehanis onepts applied to moment apaity and rotational apaity of reinfored onrete beams. Engrg. Frat. Meh. 35:233 24. [5] Fantilli, A.P., Mihashi, H. and Vallini, P., 27. Post-peak behavior of ementbased materials in ompression. ACI Mat. J. 14:51 51. 7
[6] Carpinteri, A., Corrado, M., Manini, G. and Paggi, M., 29. The overlapping rak model for uniaxial and eentri onrete ompression tests. Mag. Con. Res. 61:745 757. [7] Vonk, R.A., 1993. A miromehanial investigation of softening of onrete loaded in ompression. Heron, 38:1 94. [8] CEB Comité Euro-International du Béton), 199. CEB-FIP Model Code 199. Bulletin d Information, 195, CEB 8