Experimental study and analytical formulation of mechanical behavior of concrete

Construction and Building Materials 47 (2013) 662–670

Contents lists available at SciVerse ScienceDirect

Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

Experimental study and analytical formulation of mechanical behavior of concrete Xudong Chen, Shengxing Wu ⇑, Jikai Zhou College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China

h i g h l i g h t s  Mechanical behavior of normal concrete was tested and analyzed.  Strain at peak stress increases with an increase in concrete strength.  Existing expressions relating elastic modulus and strength is discussed.  A mathematical model was developed for predicting stress–strain curves.  Suitability of existing models for stress–strain curves is assessed.

a r t i c l e

i n f o

Article history: Received 15 February 2013 Received in revised form 22 April 2013 Accepted 4 May 2013 Available online 10 June 2013 Keywords: Concrete Mechanical behavior Experimental study Modeling

a b s t r a c t An experimental investigation was carried out to generate the mechanical behavior of normal concrete cores with a strength range of 10–50 MPa, including the compressive strength, elastic modulus, strain at peak stress and stress–strain relationships. From several formulations for concrete in this study, it was observed that a conservative estimation of the elastic modulus and strain at peak stress can be obtained from the value of compressive strength. The accuracy of predictions of a number of analytical models available in the literature is discussed. This paper shows the development of a statistical damage mechanics model for concrete at uniaxial loading in compression to ultimate failure. This model is formulated by using Weibull’s statistical theory of the strength of materials. The body of heterogeneous concrete material is simulated as a continuum comprising a large population of microscopic ‘‘weakestlink’’ elements. This model provides a good prediction of experimental results in this study. When compared other existing models, it gave better prediction. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Reinforced concrete is widely used to build infrastructures in many countries. The main constituent materials of reinforced concrete structures are plain concrete and steel bars. The concrete is essential to carry compressive stresses, however, the steel is essential to transmit tensile stresses. The discussion of instantaneous deformations of concrete under load is timed from a theoretical viewpoint because deformations provide indirect information concerning the internal structure as well as the failure mechanism of concrete [1]. From a practical standpoint, the ultimate strength design of reinforced concrete elements brought the stress–strain relationship into focus. Also, a knowledge of the deformability of concrete is necessary to compute deflections of structures, to compute stresses from observed strains, to design sections of highway

⇑ Corresponding author. Tel.: +86 25 83786551; fax: +86 26 83786986. E-mail address: [email protected] (S. Wu). 0950-0618/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2013.05.041

slabs, and to compute loss of pre-stress in pre-stressed members [2,3]. The stress–strain curves of concrete are dependent on two major parameters; testing conditions and concrete characteristics. Testing conditions include variables such as stiffness of testing machine [4–6], shape and size of the specimen [7,8], strain rate [9– 11], type of strain gauge and gauge length [6,12]. Concrete characteristics depend on many interrelated variables such as water–cement ratio [13,14], the mechanical and physical properties of the cement [15,16] and aggregate [17,18], and the age of the specimen when tested [19,20]. The evaluation of such parameters based on one series of test results may not be accurate for another series of experiments under different conditions. The nonlinear behavior of the stress–strain relation of concrete is well known. Many investigators have tried to represent the relationship by standard mathematical curves, e.g., a parabola, hyperbola, ellipse, cubic parabola, or combinations like parabola with a straight line or a sine wave with a cubic parabola and so on [21,22]. Some researchers have approximated the stress–strain curve into a triangle, a rectangle

X. Chen et al. / Construction and Building Materials 47 (2013) 662–670

or a trapezium [23]. The above relationships may have the advantage of simplifying the computation of ultimate moment of reinforced concrete sections. However, they can be classified only as empirical methods since the assumed stress distribution does not represent an observed phenomenon [24]. Many theoretical attempts have also been devoted to the modeling of the constitutive behavior of concrete, such as the fracture mechanism [25] and damage mechanics theory [26]. In particular, the fracture mechanisms is more practical for existing concentrated cracks while damage mechanics theory is more suitable for crack initiation, growth, and coalescence in the case of distributed microcracks. Moreover, composite theory [27], micromechanics [28], and probabilistic characterizations [29] can be incorporated into both the fracture and damage mechanics approaches. The damage mechanics theory was utilized in this paper. The core specimens used for this investigation have several distinct advantages over the cast specimens which are conventionally used in concrete technology. When concrete is cast a weak zone near the top surface is formed due to migration of paste and water during the compaction and setting of concrete. Although this weakness in concrete test specimens was noted as early as 1963 [30], however this physical phenomenon has not always been appreciated when interpreting observed modes of fracture in test specimens [31]. In any investigation into the study of concrete behavior the presence of this weak layer will lead to an unknown extent obscuring the resulting conclusions. The uncertainty arising from such a layer was, for this study, eliminated by cutting the thin layer of mortar placing near the top surface. This thin layer of mortar exhibits different properties than the interior concrete. Thus strain gauge performance on such surfaces could be misleading, especially when strains are measured because of the widely differing specific values of mortar and concrete. Davies [30] has concluded that such differences are insignificant. However, the authors find this investigation conclusive and are of the opinion that further experimentation is necessary before a satisfactory conclusion on this subject can be formulated. Another advantage of such specimens is that the development of cracking can be continuously observed in the course of testing under constant strain rate. In this paper, a statistical damage constitutive model for the compressive behavior of cored concrete is presented. To evaluate the adequacy of the proposed model equation, it was analyzed and compared with experimental results obtained from this study. The accuracy of predictions of a number of analytical models available in the literature is also discussed.

in six cases and 10 mm in the two remaining cases. 200  200  550 mm concrete beams were cast. To ensure adequate curing, the beam specimens after demoulding were wrapped under a wet hessian cloth, wetted continuously by sprinkling water. The cores, which are presented in this study, with diameter d of 74 mm were drilled from 200  200  550 mm concrete beams, cast and wet air-cured under laboratory conditions until being tested at the age of 60 days. Testing samples were cored at the 60 days of curing and the compressive strength tests were performed on all cores at the age of 60 days. 2.2. Compression test Compressive strength test was performed on all the cores with diameter of 74 mm. The cores were cut to obtain a length/diameter (l/d) of 2. The maximum parallel error between the two end-faces was always less than 0.1°. The load was applied through steel plates, one of them pivoting. Longitudinal strains were measured by means of three strain gauges parallel to the direction of the applied load and centered at mid-height of the specimens. The uniaxial compressive load was applied using a universal testing machine (UTM, Closed-Loop Servo-Hydraulic Testing machine) with a capacity of 1000 kN. Loading was applied at a rate of 0.003 m per second. Failure occurred between 3 and 5 min after initial loading. 2.3. Test results and discussion The strength test results are summarized in Table 2. Fig. 1 shows the stress– strain diagrams of concrete cores. The stress–strain curve of a concrete core deviates gradually from the straight line mainly because of the progressive propagation of internal cracking in the specimen. For the concrete with higher strength, the ascending part of the stress–strain curve is more linear than for the lower strength concrete. The unstable descending parts of the curves were not measurable with the experimental setup used. Tables 3 and 4 show the expressions of some codes given by several authors to predict the values of elastic modulus and strain at peak stress. The elastic modulus in this paper was measured as the chord modulus according to ASTM C469 [46]. Such predictions are plotted together with the mean experimental results in Figs. 2 and 3, for elastic modulus and strain at peak stress, respectively. It can be seen from Fig. 2 that the experimental relationship between elastic modulus and compressive strength is beneath the majority of the code provisions, but above the ACI 318 [35]. Fig. 3 shows the relationship between strain at peak stress and compressive strength. From the results of this limited study, the formulations by Nicolo et al. [45] seems to give the best estimation for strain at peak stress.

3. Assessment of existing models for stress–strain curves The compressive stress–strain behavior of concrete is a significant issue in the flexural analysis of reinforced concrete beams and columns. Some researchers have attempted to represent the stress–strain relationship of concrete in compression. The work of the researchers is presented in the following sections. Barnard [47] proposed the following second order parabola to represent the stress–strain relationship of concrete in compression.

"     # e e 2  r ¼ rp 2

2. Experimental study and results

ep

2.1. Mix proportions and specimens The tests covered eight types of concrete. Mixture proportions, as listed in Table 1, were selected so as to obtain concretes of nominal classes ranging from 10 MPa to 50 MPa. All concrete mixes were batched with siliceous river aggregate, characterized by continuous size curves, with a maximum aggregate size of 20 mm

Mix

Mix proportions (C:W:S:A)

Type of Portland cement

Maximum aggregate size (mm)

Mix1 Mix2 Mix3 Mix4 Mix5 Mix6 Mix7 Mix8

1:0.7:2.7:4.6 1:0.65:2.6:3.9 1:0.7:2.7:4.6 1:0.48:1.5:3.1 1:0.6:2.2:3.7 1:0.4:1.1:2.5 1:0.42:1.8:5.2 1:0.36:1.4:5.2

325 325 425 425 425 425 425 425

20 20 20 20 20 20 10 10

ep

ð1Þ

where r is the stress at any strain e, and rp is the compressive strain or peak stress at a strain ep.Desayi and Krishnan [48] proposed the following quotient of two polynomials to represent the stress– strain relationship of concrete in compression.

r¼ Table 1 Mix proportions and some properties of concrete mixtures.

663

Eit e  2 1 þ eep

ð2Þ

where Eit is the initial tangent modulus such that Eit = 2rp/ep. Baldwin and North [49] proposed the following quotient of twosecond order polynomials to represent the stress–strain relationship of concrete in compression.

2

   2 3 e e 6 A ep þ B ep 7 r ¼ rp 4    2 5 e e 1 þ C ep þ D ep

ð3Þ

664

X. Chen et al. / Construction and Building Materials 47 (2013) 662–670

Table 2 Summary of test results.

Table 3 Formulations to predict elastic modulus from compressive strength.

Mix type

Specimen No.

Test (MPa)

Mean (MPa)

S.D. (MPa)

References

Mix1

1 2 3 4 5

10.69 10.67 9.82 9.43 9.44

10.27

0.84

EHE [32]

6 7 8 9 10

17.29 16.85 16.63 15.62 15.78

16.44

11 12 13 14 15

19.73 18.76 18.53 18.52 18.28

18.67

16 17 18 19 20

30.37 27.82 26.05 25.30 30.22

27.48

21 22 23 24 25

33.27 25.87 25.96 25.02 27.33

32.50

26 27 28 29 30

41.16 35.07 37.97 32.18 32.14

35.71

31 32 33 34 35

43.48 42.61 41.52 41.09 40.94

41.93

36 37 38 39 40

49.93 48.12 48.71 49.05 45.76

48.32

Mix2

Mix3

Mix4

Mix5

Mix6

Mix7

Mix8

NBR 6118 [33] CEB [34] ACI 318 [35] 0.72

Hueste et al. [36] Norwegian code [37] Gardner and Zao [38]

0.56

Elastic modulus pffiffiffiffi E ¼ 10000  3 fc pffiffiffiffi E ¼ 5600  fc qffiffiffiffi 3 fc E ¼ 21:5  10 pffiffiffiffi E ¼ 43  q1:5 fc  106a c  pffiffiffiffi E ¼ 5230  fc E = 9.5  (fc)0.3 pffiffiffiffi E ¼ 9  3 fc

Note: fc is the compressive strength; qc is the concrete density; and a is the air content.

2

2.37

3.33

   2 3 e þ ðD  1Þ e A e ep p 7 r ¼ rp 6 4    2 5 e e 1 þ ðA  1Þ ep þ D ep

Popovics [50] shown that most of the stress–strain functions have a similar shape regardless of concrete strength. Such an equation is shown as following:

e m ep m  1 þ ðe=ep Þm

ð5Þ

m ¼ 2:76  105 rp þ 1:0

ð6Þ

r ¼ rp 3.88

1.08

Cook and Chindaprasirt [51] proposed the following mathematical model to describe the compressive stress–strain behavior of concrete.

"  k1 # E0 e 1 e  k ð1  WÞ þ E0 eW 1  k ep 1 e

r¼

1 þ k1 1.57

ð4Þ

ð7Þ

ep

where E0 is initial elastic modulus, k is curvature of the stress–strain curve, W is a factor in the stress–strain curve approximation. The above parameters could be obtained as follows:

"

1 W ¼ exp  k

 k #

e ep

ep ¼ 0:001755 þ 8:74  106 rp k ¼ 1:0 þ 0:009r1:551 p

ð8Þ

ð9Þ ð10Þ

Carreira and Chu [41] proposed the following mathematical expression to represent the stress–strain relationship for concrete in compression.

  b eep rp r¼  b b  1 þ eep b¼

h

r p i3 32:4

þ 1:55

ð11Þ

ð12Þ

Almusallam and Alsayed [52] proposed a simple mathematical model that can represent the stress–strain spectrum of concrete. Fig. 1. Stress–strain curves of various concrete.

Boundary conditions yield three independent equations with four unknown constants A, B, C and D. Thus, the number of unknowns can be reduced to two constants,

r¼n

ðK  K p Þe h in o1=n þ K p e ðKK p Þe 1þ r0

ð13Þ

where K is the initial slope of the stress–strain curve, r0 is a reference stress and n is a curve-shape parameter. The parameters can be expressed as follows:

665

X. Chen et al. / Construction and Building Materials 47 (2013) 662–670 Table 4 Formulations to predict strain at peak stress from compressive strength. References

Strain at peak stress

Liebenberg [39] Tadros [40] Carreira and Chu [41] Ahmad and Shah [42] Saenz [43] Lee [44] Nicolo et al. [45]

ep = (0.0546 + 0.003713  fc)  102 ep = (1.6 + 0.01  fc)  103 ep = 0.71  105  fc + 0.00168 ep = 1.65  105  fc + 0.001648 ep = 1.491  105  fc + 0.00195 ep = fc/(46.886 + 2.6  fc) ep = 0.00076 + [(0.626  fc - 4.33)  107]0.5

Tasnimi [53] presents a series of compressive tests on concrete cylinders in order to develop a stress–strain model concrete under axial compressive load. A mathematical representing the entire range of concrete under uniaxial stress is developed as follows:

 4

r e ¼ ð2c  3Þ rp ep

ep

ð21Þ

ep

where c is a constant related to the tangential modulus of elasticity.

c¼

where ep is the strain at peak stress.

 3   e e þ ð4  3cÞ þc

Eitm ep

ð22Þ

rp

!

r2:8 p þ 0:05rp q0:2 c

Eitm ¼ 2:25 ln 

ð23Þ



5 ep ¼ 26:73r0:5 p þ 114:78  10

ð24Þ

Xiao et al. [54] proposed the following analytical expression for the uniaxial compression behavior of normal concrete.

8    2  3 > e þ ð3  2aÞ e þ ða  2Þ e ;

r :

ep

ep

ð e= ep Þ 2

bðe=ep 1Þ þðe=ep Þ

Fig. 2. Relationship between the elastic modulus and compressive strength for concrete.

Fig. 3. Relationship between the strain at peak stress and compressive strength for concrete.

n¼



ln 2

K ln rr10  KKp p



ð14Þ

where

r1

"     # e1 e1 2  ¼ rp 2

e1 ¼

e0

e0

r0

ð15Þ

ð16Þ

K  Kp r0 ¼ 5:6 þ 1:02rp  K p e0 K p ¼ 5470  375rp pffiffiffiffiffiffi K ¼ 3320 rp þ 6900

ð17Þ ð18Þ ð19Þ

e0 ¼ ð0:2rp þ 13:06Þ  104

ð20Þ

;

for

ep

e  ep

for

e < ep ð25Þ

where a and b are constants to be determined. The smaller the a value is, the smaller is the proportion of the plastic deformation at the peak stress with respect to the total deformation. The parameter b is related to the area under the descending portion of the stress–strain curve. A cross comparison between the experimental curve and those obtained by using the different theoretical models analyzed was completed. The comparison highlights that every model agrees well with their respective experimental data, and less well with the experimental data obtained by other authors [55], because each proposed equation was obtained by using a regression analysis to interpolate their own experimental data. The experimental stress–strain curves obtained from this study are compared with the various model predictions in Figs. 4–12. These results shows that the theoretical curves of Tasnimi [53] have a totally different shape from those experimental observed. The stress–strain relationships proposed by Barnard [47], Baldwin and North [49], Desayi and Krishnan [48] and Cook and Chindaprasirt [51] overestimate the region near to the peak stress, but show a somewhat similar slope of experimental curves. Carreira and Chu [41] model give an adequate interpretation of the phenomenon, but only for limited ranges of strengths. The model proposed by Xiao et al. [54], on the other hand, generally agrees well with the experimental curves. 4. Development of the stress–strain curve Continuous damage model defines damage as the density of defects/discontinuities on a cross section in a given orientation, amplified by their stress-raising effects [56]. In general, damage is represented by tensors due to its directional nature [57]. When the weighted fractional loss of an area of a cross section is the same regardless of the orientation of the cross section, then damage is isotropic and is described by a scalar variable D taking values between 0 and 1. Damage is considered to be isotropic in this paper. The concept of effective stress, along with the principle of strain equivalence [56], may be used to derive the constitutive law for a damaged material. In the framework of small deformation, total strain e can be divided as

e ¼ ee þ ev

ð26Þ

where ee and ev is elastic and visco-plastic strains, respectively.

666

X. Chen et al. / Construction and Building Materials 47 (2013) 662–670

Fig. 4. Comparison of Barnard model with test data.

Fig. 7. Comparison of Popovics model with test data.

Fig. 5. Comparison of Desayi and Krishnan model with test data. Fig. 8. Comparison of Cook and Chindaprasirt model with test data.

Fig. 6. Comparison of Baldwin and North model with test data. Fig. 9. Comparison of Carreira and Chu model with test data.

For the iso-thermal case, if the elastic deformation is assumed to be uncoupled with strain hardening, the Helmholtz free potential energy W can be represented as follows:

Wðe; q; DÞ ¼ we ðee ; DÞ þ wp ðg; ev Þ

ð27Þ

where we is the elastic part of Helmholtz specific free energy; wp is the plastic part of Helmholtz specific free energy; D is internal state variable representing damage, which is a scalar variable; and g is internal variable representing ductility.

X. Chen et al. / Construction and Building Materials 47 (2013) 662–670

Fig. 10. Comparison of Almusallam and Alsayed model with test data.

Fig. 12. Comparison of Xiao et al. model with test data.

Fig. 13. Comparison of proposed model with test data.

Fig. 11. Comparison of Tasnimi model with test data.

we can be defined by the strain equivalent assumption as follows: we ðee ; DÞ ¼

1 ð1  DÞEe2e 2

ð28Þ

Based on the second law of thermodynamics, the damage and plastic deformation of materials are irreversible thermodynamic processes. Therefore, the inequality of Clausius-Duheim must be satisfied, as follows:

re  W  0

ð29Þ

So, the following equation can be deduced:

r¼

@we ¼ ð1  DÞEðe  ev Þ @ ee

ð30Þ

Eq. (30) is an elastic–plastic damage constitutive model. When D = 0, ev obeys the plastic mechanics law and Eq. (30) changes into the classic damage model [58]

r ¼ Eð1  DÞe

667

ð31Þ

where E is the elastic modulus, e is the total strain, and r the stress of the material. Concrete is assumed to be composed of numerous elements, which is called the mesoscopic elements. As for these elements themselves, suppose that they are relatively large enough to contain many defects and, on the other hand, they are adequately

small in dimension compared with the whole structure of the concrete. Hence a distinct influence of individual defects may be ignored in such a case, and then the mesoscopic element can be considered as a particle within a framework of continuous mechanics theory. To proceed in this way, we have the possibility of exploring the damaging (or failure) behavior of a concrete on the basis of the properties of those mesoscopic elements involved [59– 61]. Therefore, if the defects existing in a concrete are considered to be randomly induced during the loading phase, then the damage or failure with respect to individual mesoscopic elements is also viewed to be random, more precisely, the strength level of mesoscopic elements may be stochastically distributed. Next, before presenting the equation describing an evolutionary condition of the damage, the major assumptions are given as follows [62]. (1) In terms of each mescoscopic element prior to failure, it exhibits linear-elasticity, whose stress–strain relationship obeys Hooke’s law. (2) The strength level F of the mescoscopic elements satisfies the Weibull distribution function [63], whose probability density P(F) can be formulated by

PðFÞ ¼

 m1   m  m F F exp  F0 F0 F0

ð32Þ

668

X. Chen et al. / Construction and Building Materials 47 (2013) 662–670

in which m is the shape parameter denoting the degree of material homogeneity, and F0 is the scale parameter associated with the strength of concrete elements. It is noteworthy that because of the intrinsic complexity with respect to deformation mechanics of concrete, difficulties arise in attempting to apply any simple theory to accounting flexibly for all statistical aspects of concrete deformation [64]. Here, a Weibull distribution given by Eq. (32) is, by convention, adopted to describe the strength distribution due to its widespread acceptance regarding concrete properties [65–67]. It is timely to explore now the presentation of damage evolution equation incorporating statistical considerations. The damage process of a concrete in loading conditions can be considered to be continuously evolving, progressively accumulating failure behavior of the mesoscopic elements. Assuming that Nf represents the quantities of failed mesoscopic elements and N represents the total quantities of mesoscopic elements, the extent of damage may be assessed by the ratio of Nf to N. This implies that the damage variable D mentioned previously can also be measured in the form of such ratio, which is expressed by Eq. (33) presented below [68,69]:

D¼

8 h   i < 1  exp  F m F  0 F0 D¼ :0 F < 0

ð36Þ

As shown in Eq. (36), F < 0 and F P 0 correspond to intact (undamaged) states and damaged states, respectively, in which F = 0 is the exact damage threshold. The Mohr–Coulomb failure criterion may be expressed as:

~ 1 ð1 þ sin uÞ  r ~ 3 ð1  sin uÞ ¼ 2c cos u F¼r

ð37Þ

~ 1 and r ~ 3 are effective stress; c and / are cohesion and interwhere r frg ~ g ¼ ð1DÞ nal friction angle.Substituting fr into Eq. (37) leads to:

F¼

r1 1D

ð1 þ sin uÞ 

r3 1D

ð1  sin uÞ

ð38Þ

Meanwhile, there exists:

Nf N

ð33Þ

1 E

e1 ¼ ðr~ 1  2lr~ 3 Þ ¼

To employ Eq. (33), the right-hand side of it was determined explicitly. In the case where the value of the strength of mesoscopic elements changes from F to F + dF, the number of failed mesoscopic elements can be obtained as NP(F)dF. As a consequence, the mathematical denotation of the number of failed mesoscopic elements Nf can be derived when the strength value ranges between 0 and F:

Nf ¼

statistical variations in mechanical properties of concrete elements, Eq. (35) can also be regarded as a statistical evolution of the damage. We then take the effects of the damage threshold on the damage evolution, and Eq. (35) becomes:

Z 0

F

  m 

F NPðyÞdy ¼ N 1  exp  F0

ð34Þ

From Eqs. (33) and (34), it follows that

  m  F D ¼ 1  exp  F0

ð35Þ

Eq. (35) manifests that a correlation is found between the damage variable D (describing the state of concrete damage) and the element strength F (satisfying a statistical distribution P(F) of the Weibull form with two parameters m and F0). Thus, by introducing

1 ðr1  2lr3 Þ ð1  DÞE

ð39Þ

where l is the Poisson’s ratio. Combing Eqs. (38) and (39) yields the following expression:

F¼

Ee1 ½r1 ð1 þ sin uÞ  r3 ð1  sin uÞ r1  2lr3

ð40Þ

For the uniaxial tests, the above expression can be rewritten as:

F ¼ Ee1 ð1 þ sin uÞ

ð41Þ

Eq. (39), should satisfy the following boundary conditions: (1) When e1 = 0, r1 = 0. (2) When e1 = 0, ddre11 ¼ E. (3) When e1 = ep, r1 = rp. (4) When e1 = ep, ddre11 ¼ 0. where ep is strain at which the stress is equal to a peak stress rp. Differentiating Eq. (39), we can obtain the following expression:

Table 5 Corrlation coefficient (R2) and standard error (S) for concrete specimens. Model

Mix1

Mix2

Mix3

Mix4

Mix5

Mix6

Mix7

Mix8

Barnard [47]

R2 S

0.995 0.219

0.999 0.155

0.996 0.670

0.990 0.453

0.999 0.451

0.980 1.211

0.997 1.747

0.996 2.257

Desayi and Krishnan [48]

R2 S

0.997 0.385

0.998 0.558

0.999 0.407

0.996 1.121

0.994 1.247

0.998 2.082

0.988 3.097

0.989 3.550

Baldwin and North [49]

R2 S

0.986 0.381

0.980 0.738

0.979 0.798

0.973 1.387

0.971 1.366

0.987 1.089

0.971 2.264

0.971 2.882

Popovics [50]

R2 S

0.945 1.269

0.992 0.677

0.930 3.340

0.947 3.583

0.954 4.090

0.954 4.318

0.968 5.447

0.955 6.467

Cook and Chindaprasirt [51]

R2 S

0.999 1.399

0.989 0.489

0.992 1.080

0.999 3.078

0.979 3.965

0.998 4.323

0.998 9.066

0.998 9.911

Carreira and Chu [41]

R2 S

0.989 0.950

0.989 1.242

0.996 0.666

0.996 1.084

0.997 0.923

0.999 0.797

0.997 1.183

0.989 2.636

Almusallam and Alsayed [52]

R2 S

0.991 0.461

0.999 0.293

0.997 0.619

0.999 0.565

0.998 0.840

0.999 0.719

0.998 1.238

0.999 0.618

Tasnimi [53]

R2 S

0.767 3.267

0.789 5.249

0.767 6.532

0.814 7.437

0.822 8.457

0.780 9.649

0.844 11.730

0.843 12.996

Xiao et al. [54]

R2 S

0.997 0.264

0.973 0.306

0.969 0.446

0.982 0.774

0.968 0.824

0.989 1.653

0.993 2.447

0.995 2.512

This paper

R2 S

0.998 0.184

0.997 0.133

0.987 0.163

0.999 0.209

0.998 0.362

0.998 0.527

0.985 0.240

0.999 0.694

X. Chen et al. / Construction and Building Materials 47 (2013) 662–670

@ r1 @ e1

" )   m  (  m1 # F F 1 @F 1 þ e1 m ¼ E exp  F0 F0 F 0 @ e1

ð42Þ

Obviously, Eqs. (42) satisfy the conditions of (1) and (2).When the boundaries conditions (3) and (4) are substituted into Eq. (42), the following formulate can be obtained.

  m  F F0

rp ¼ Eep exp 

Acknowledgement

ð44Þ

where Fp = Eep(1 + sin /). Solving Eqs. (43) and (44), the expressions of m and F0 can be written as:

m¼

1 r ln epp 1

F 0 ¼ ðmF p Þm

lowed by that proposed by Carreira and Chu model. However, the latter is relatively complex; the strain at peak stress was assumed to be a function of concrete strength. A further publication is in preparation for further developing this statistical damage mechanics model to simulate the microcrack growth process within a body of concrete material at multiple cyclic loading in uniaxial compression.

ð43Þ

" 1 þ ep

 m1 # Fp 1 m Eð1 þ sin uÞ ¼ 0 F0 F0

669

ð45Þ ð46Þ

and F0 are calculated by experimental data, the modified statistical constitutive model for concrete can be determined by Eqs. (35), (41), (45), and (46). A comparison of stress–strain curves generated by the present analytical model with those obtained in the present experimental study is shown in Fig. 13, which show a very close match between the analytical and experimental curves. The correlation coefficient (R2) and standard errors (S) of the existing models and the model proposed in this paper are shown in Table 5. The standard errors (S) between the experimental and predicted results of the model proposed in this paper are lower than that of the existing models, indicating very good fits. The correlation coefficient (R2) is above 0.98 also indicates good correlation between the model in this paper and corresponding experimental data. Through this study, we can find that, it is more definite to see the strength of concrete element as random variable instead of the axial strain. Meanwhile, the statistical mechanics damage model includes the Mohr–Coulomb failure criterion, which is widely applied in classic plastic theory. In addition, the new model is derived from three-dimensional general stress state, so the three-dimensional form of the model could be deduced and obtained. One limitation of this model is that it does not take into account the effect of rate of stressing (or rate of straining) on the stress–strain relation. However, most of the well-known proposed equations ignore it and probably there is no ultimate load theory which has taken it into account. Experimental evidence [70,71] indicates this effect on the failure load as well as on the stress– strain curve. Hence, for an accurate investigation of the ultimate moment of sections, the time element should also be considered. 5. Conclusions In this study, the mechanical property of concrete cores with compressive strength ranging from 10 MPa to 50 MPa was investigated. Strain at peak stress increases with an increase in concrete strength. Experimental investigation and analytical study were performed to develop a mathematical model for the prediction of stress–strain curves of concrete cores under compressive load. The main model parameters were determined analytically and only a few parameters from both statistical damage mechanics and experimental data were obtained. The model was checked against experimental results and provided good agreement with measured values. It also gave better predictions than some of the models available in the literature. Among the existing nine analytical models tested in light of the present test data, Xiao et al. model produced the best outcome, fol-

The authors are grateful to the National Natural Science Foundation of China (Grant No. 51178162) and the Fundamental Research Funds for the Central Universities (Grant No. 2011B11047) for the financial support.

References [1] Youssef MA, Moftah M. General stress–strain relationship for concrete at elevated temperatures. Eng Struct 2007;29:2618–34. [2] Taerwe L, Van Gysel A. Influence of steel fibers on design stress–strain curve for high-strength concrete. J Eng Mech-ASCE 1996;122(8):695–704. [3] Ohno K, Ohtsu M. Crack classification in concrete based on acoustic emission. Constr Build Mater 2010;24:2339–46. [4] Nemati KM, Monteiro PJM, Cook NGW. A new method for studying stressinduced microcracks in concrete. J Mater Civ Eng-ASCE 1997;10(3):128–34. [5] Zisopoulos PM, Kotsovos MD, Pavlovic MN. Deformational behavior of concrete specimens in uniaxial compression under different boundary conditions. Cem Concr Res 2000;30:153–9. [6] Richardson DN. Review of variables that influence measured concrete compressive strength. J Mater Civ Eng 1989;3(2):95–112. [7] Tokyay M, Ozdemir M. Specimen shape and size effect on the compressive strength of higher strength concrete. Cem Concr Res 1997;27(8):1281–9. [8] Chin MS, Mansur MA, Wee TH. Effect of shape, size, and casting direction of specimens on stress–strain curves of high-strength concrete. ACI Mater J 1997;94(3):209–18. [9] Bischoff PH, Perry SH. Compressive behavior of concrete at high strain rates. Mater Struct 1991;24:425–50. [10] Ficker T. Quasi-static compressive strength of cement-based materials. Cem Concr Res 2011;41:129–32. [11] Wu S, Chen X, Zhou J. Influence of strain rate and water content on mechanical behavior of dam concrete. Constr Build Mater 2012;36:448–57. [12] Chen PW, Chung DDL. Concrete as a new strain/stress sensor. Composites Part B 1996;27:11–23. [13] Pann KS, Yen T, Tang CW, Lin TD. New strength model based on water–cement ratio and capillary porosity. ACI Mater J 2003;100(4):311–8. [14] Vu XH, Malecot Y, Daudeville L, Buzaud E. Effect of water/cement ratio on concrete behavior under extreme loading. Int J Numer Anal Meth Geomech 2009;33:1867–88. [15] Popovics S. Another look at the relationship between strength and composition of concrete. ACI Mater J 2011;108(2):115–9. [16] Zhang YM, Napier-Munn TJ. Effects of particle size distribution, surface area and chemical composition on Portland cement strength. Powder Technol 1995;83:245–52. [17] Aitcin PC, Metha PK. Effect of coarse-aggregate characteristics on mechanical properties of high-strength concrete. ACI Mater J 1990;87(2):103–7. [18] Kuder K, Lehman D, Berman J, Hannesson G, Shogren R. Mechanical properties of self consolidating concrete blending with high volumes of fly ash and slag. Constr Build Mater 2012;34:285–95. [19] Yi ST, Kim JK, Oh TK. Effect of strength and age on the stress–strain curves of concrete specimens. Cem Concr Res 2003;33:1235–44. [20] Gutsch AW. Properties of early age concrete – experiments and modeling. Mater Struct 2002;35:76–9. [21] Simith GM, Young LE. Ultimate theory in flexure by exponential function. ACI J 1955;52(3):349–60. [22] Simith GM, Young LE. Ultimate flexural analysis based on stress–strain curves of cylinders. ACI J 1956;53(6):597–610. [23] Fall M, Belem T, Samb S, Benzaazoua M. Experimental characterization of the stress–strain behavior of cemented paste backfill in compression. J Mater Sci 2007;42:3914–22. [24] Popovics S. A review of stress–strain relationship for concrete. ACI J 1970;67:243–8. [25] Hiksdorf HK, Brameshuber W. Code-type formulation of fracture mechanics concepts of concrete. Int J Fract 1991;51:61–72. [26] Loland KE. Continuous damage model for load-response estimation of concrete. Cem Concr Res 1980;10:395–402. [27] Hansen TC. Influence of aggregate and voids on modulus of elasticity of concrete, cement mortar, and cement paste. ACI J 1965;62:193–216. [28] Yang CC, Huang R. A two-phase for prediction the compressive strength of concrete. Cem Concr Res 1996;26(10):1567–77.

670

X. Chen et al. / Construction and Building Materials 47 (2013) 662–670

[29] Miled K, Limam O, Sab K. A probabilistic mechanical model for prediction of aggregate’s size distribution effect on concrete compressive strength. Phys A 2012;391:3366–78. [30] Davies RD. The strains under constant stress of cast and sawn concrete specimens. Mag Concr Res 1963;15:31–2. [31] Celik AO, Kilinc K, Tuncan M, Tuncan A. Distributions of compressive strength obtained from various diameter cores. ACI Mater J 2012;109(6):597–606. [32] Spanish code for structural concrete EHE. Real Decreto 2661/1998, Madrid, December 11; 1998. [33] Brazilian association of technical standards NBR 6118: design of concrete structures. Rio de Janeiro; 2003. [34] Comité Euro-International du Béton. CEB-FIP model code 1990. London: Thomas Telford; 1993. [35] American concrete institute. ACI Committee 318: building code requirements for structural concrete. Farmington Hills, MI; 1999. [36] Hueste MBD, Chompreda P, Trejo D, Cline DBH, Keating PB. Mechanical properties of high-strength concrete for prestressed members. ACI Struct J 2004;101(4):457–65. [37] Norwegian council standardization. Design of concrete structures. Norwegian code, NS 3473, Oslo, Norway; 1992. [38] Gardner NJ, Zhao JW. Mechanical properties of concrete for calculation of long term deformations. In: Proceedings of the second Canadian on cement and concrete, Vancouver, Canada; 1991. p. 150–9. [39] Liebenberg AC. A stress strain function for concrete subjected to short-term loading. Mag Concr Res 1962;14:85–99. [40] Tadros GS. Plastic rotation of reinforced concrete members subjected to bending, axial load and shear. PhD thesis, University of Calgary; 1970. [41] Carreira DJ, Chu KH. Stress–strain relationship for plain concrete in compression. ACI J 1985;82:797–804. [42] Ahmad SH, Shah SP. Behavior of hoop confined concrete under high strain rates. ACI J 1982;82:634–47. [43] Saenz LP. Discussion of a paper by Desayi P and Krishnan, equation for the stress strain curve of concrete. Ibid 1964;61(9):1229–35. [44] Lee I. Complete stress–strain characteristics of high performance concrete. PhD thesis, New Jersey Institute of Technology; 2002. [45] De Nicolo B, Pani L, Pozzo E. Strain of concrete at peak compressive stress for a wide range of compressive strengths. Mater Struct 1994;27:206–10. [46] ASTM C469-02e1. Standard test method for static modulus of elasticity and Poisson ratio of concrete in compression. [47] Barnard PR. Researches into the complete stress–strain curve for concrete. Mag Concr Res 1964;16:203–10. [48] Desayi P, Krishnan S. Equation for the stress–strain curve of concrete. ACI J 1964;61(3):345–50. [49] Baldwin B, North MA. A stress–strain relationship for concrete at high temperatures. Mag Concr Res 1973;25:208–12. [50] Popovics S. A numerical approach to the complete stress–strain curve of concrete. Cem Concr Res 1973;3:583–99. [51] Cook DJ, Chindaprasirt P. A mathematical model for the prediction of damage in concrete. Cem Concr Res 1981;11:581–90.

[52] Almusallam TH, Alsayed SH. Stress–strain relationship of normal, highstrength and lightweight concrete. Mag Concr Res 1995;47:39–44. [53] Tasnimi LA. Mathematical model for complete stress–strain curve prediction of normal, light-weight and high-strength concretes. Mag Concr Res 2004;56:23–34. [54] Xiao J, Li J, Zhang C. Mechanical properties of recycled aggregate concrete under uniaxial loading. Cem Concr Res 2005;35:1187–94. [55] Paas MHJW, Oomens CWJ, Schreurs PJG, Janssen JD. The mechanical behavior of continuous media with stochastic damage. Eng Fract Mech 1990;36(2):255–66. [56] Shao JF, Rudnicki JW. A microcrack-based continuous damage model for brittle geomaterials. Mech Mater 2000;32:607–19. [57] Piechnik S, Pachla H. Law of continuous damage parameter for non-aging materials. Eng Fract Mech 1979;12:199–209. [58] Williams KV, Vaziri R. Application of a damage mechanics model for predicting the impact response of composite materials. Comput Struct 2001;79:997–1011. [59] Amaral PM, Fernandes JC, Rosa LG. Weibull statistical analysis of granite bending. Rock Mech Rock Eng 2008;41:917–28. [60] Momber AW. The fragmentation of standard concrete cylinders during the compressive testing. Int J Fract 1998;92:29–34. [61] Chen X, Wu S. Influence of water-to-cement ratio and curing period on pore structure of cement mortar. Constr Build Mater 2013;38:804–12. [62] Elgueta M, Diaz G, Zamorano S, Kittl P. On the use of the Weibull and normal cumulative probability models in structural design. Mater Des 2007;28:2496–9. [63] Weibull W. A statistical distribution function of wide applicability. J Appl Mech-T ASME 1951;18:293–7. [64] Neville DJ. Application of a new statistical function for fracture toughness to failures at microcracks in brittle materials. Int J Fract 1990;44:79–96. [65] Yip WK. New damage variable in failure analysis of concrete. J Mater Civ EngASCE 1996;8(4):184–8. [66] Sim J-II, Yang K-H, Kim H-Y, Choi B-J. Size and shape effect on compressive strength of lightweight concrete. Constr Build Mater 2013;38:854–64. [67] Camilleri J, Anastasi M, Torpiano A. The microstructure and physical properties of heavy oil fuel ash replaced Portland cement for use in flowable fill concrete and the production of concrete masonry units. Constr Build Mater 2013;38:970–9. [68] Huang C, Subhash G, Vitton SJ. A dynamic damage growth model for uniaxial compressive response of rock aggregates. Mech Mater 2002;34:267–77. [69] Lee G, Ling T-C, Wong Y-L, Poon C-S. Effects of crushed glass cullet sizes, casting methods and pozzolanic materials on ASR of concrete blocks. Constr Build Mater 2011;25(5):2611–8. [70] Liu R, Durham SA, Rens KL. Effects of post-mercury-control fly ash on fresh and hardened concrete properties. Constr Build Mater 2011;25(8):3283–90. [71] Wu S, Chen X, Zhou J. Tensile strength of concrete under static and intermediate strain rates: correlated results from different testing methods. Nucl Eng Des 2012;250:173–83.