Stress–strain model for normal- and light-weight concretes under uniaxial and triaxial compression

Construction and Building Materials 71 (2014) 492–509

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Stress–strain model for normal- and light-weight concretes under uniaxial and triaxial compression Jian C. Lim, Togay Ozbakkaloglu ⇑ School of Civil, Environmental and Mining Engineering, University of Adelaide, South Australia 5005, Australia

h i g h l i g h t s  Two experimental databases of confined and unconfined concretes are reported.  Parameters defining pre-peak, peak and post-peak conditions of concretes were established.  A unified stress–strain model for confined and unconfined concretes is presented.  The model is applicable to concrete specimens with various dimensions, densities and strengths.  The model provides improved predictions of stress–strain behavior over existing models.

a r t i c l e

i n f o

Article history: Received 29 May 2014 Received in revised form 1 August 2014 Accepted 23 August 2014

Keywords: Concrete High-strength concrete (HSC) Confinement Triaxial Stress–strain Water–cement ratio Density Slenderness Size effect Light-weight

a b s t r a c t Accurate prediction of stress–strain relationship of concrete is of vital importance to accurately predict the overall structural behavior of reinforced concrete members. The various types of concrete that are available in the construction industry today makes it essential that the models developed for the prediction of their behavior are of high versatility. Review of the existing literature revealed that existing stress–strain models for unconfined and confined concretes are limited in their application domains, defined by the parametric range of the experimental results considered in their development. The review also indicated that a unified model that is applicable to normal- and light-weight concretes is not yet available. The aim of the present study was to develop a unified confinement model that is applicable to various types of concrete, ranging from light-weight to high-strength. To this end, two large databases of experimental results of concrete specimens tested under uniaxial and triaxial compression were assembled through an extensive review of the literature. The databases covered a wide range of concrete properties, thereby allowing detailed observation of the important factors influencing the compressive behavior of concrete. The analysis of the unconfined concrete database resulted in the development of expressions for the prediction of elastic modulus, compressive strength and corresponding axial strain of various types of concrete. In addition, through a comprehensive analysis of the combined test database a unified stress–strain model was developed to predict the peak and residual conditions and the complete stress–strain behavior of unconfined and actively confined concretes. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction It is well established that lateral confinement of concrete enhances its compressive strength and axial deformation capacity [1–6]. A comprehensive review of the literature that was undertaken as part of the current study and those previously reported in Refs. [6,7] revealed that over 500 experimental studies have been conducted on the axial compressive behavior of unconfined, actively confined, and fiber reinforced polymer (FRP)-confined ⇑ Corresponding author. Tel.: +61 8 8313 6477; fax: +61 8 8313 4359. E-mail address: [email protected] (T. Ozbakkaloglu). http://dx.doi.org/10.1016/j.conbuildmat.2014.08.050 0950-0618/Ó 2014 Elsevier Ltd. All rights reserved.

concretes, resulting in the development of over 110 stress–strain models. However, due to the limitations in the parametric ranges of the experimental results considered in their development, the applicability of the existing models are often restricted to specific specimens subsets. The current availability of variety of concrete confinement techniques and reinforcing materials [4,8–21], and the abundance of concretes with different mechanical and material properties [9,22–26] poses a challenge for engineers in finding a suitable model given the possible composite combinations of these materials. The work presented in this paper was motivated by the need to develop a unified model applicable to various types of concrete

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2. Experimental test databases 2.1. Database of unconfined concrete The database of unconfined concrete was assembled from 209 experimental studies and consisted of 4353 datasets. 1167 datasets from 161 studies that reported the specimen axial strain at peak compressive stress of concrete (eco) are presented in Tables A1–A3 in Appendix, whereas the remaining datasets are presented in Tables A4–A7. The results in Tables A1–A3 and A4–A7 were sorted into seven groups according to the type of concrete (NWC or LWC) and the cross-sectional shape of specimen (circular or square). Out of the 4353 datasets presented in Tables A1–A7, 2279 of the datasets were NWC cylinders, 1167 were LWC cylinders, 864 were NWC prisms, 43 were LWC prisms. In Tables A1–A7, the following information was available for each dataset in the database: the number of identical specimen; the geometric properties (cross-sectional dimension B and height H); the specimen age; the water-to-cementitious binder ratio (w/c); the density of concrete (qc,f); the type and size of aggregates; the silica fume-to-cementitious binder percentage (sf/c); the mineral additive-to-cementitious binder percentage (ma/c); the elastic modulus of concrete (Ec); and the compressive strength of concrete (f0 co). In Tables A1–A3, the axial strain corresponding to the peak compressive stress (eco) and its measurement method is available in the last two columns. It should be noted that in some of the datasets, details of the aggregate type were not available from the source documents. Given the omission of such details, the aggregate types are noted as either normal-weight or light-weight aggregates in Tables A1– A7, according to the type of concrete (NWC or LWC). Regarding the percentages of mineral additives in concrete mixes of specimens presented in Tables A1–A7, except for silica fume that is presented in the 9th column, details of other mineral additives, such as fly-ash, slag, and hi-fi are presented in the same column in the 10th column. To distinguish their types in this column, these mineral additives are noted with superscripts ‘f’, ‘s’, ‘h’, respectively. In the database presented in Tables A1–A7 in Appendix, the specimen cross-sectional dimensions (B) varied from 50 to 406 mm, the specimen heights (H) varied from 25 to 1016 mm and the specimen aspect ratios (H/B) varied from 0.25 to 8, the water–cementitious binder ratios (w/c) varied from 0.16 to 1.27, the concrete densities (qc,f) varied from 666 to 2584 kg/m3, the concrete elastic moduli (Ec) varied from 9620 to 57,800 MPa, and the compressive strengths (f0 co) and the corresponding axial strains (eco) varied from 5.3 to 171.1 MPa and 0.07 to 0.53%, respectively.

observations on the influences of the cross-sectional shape on the observed behavior are also supplied. Thereafter, the specimen cross-sectional dimension (B) is referred to as the specimen diameter (D).

3. Elastic modulus and peak condition of unconfined concrete Based on the observed difference in their compressive behavior, concretes with a density (qc) greater than 2250 kg/m3 were categorized as NWC, whereas concretes with a density below the limit were categorized as LWC. A same transition boundary between NWC and LWC at concrete density of 2250 kg/m3 were previously reported in Tasdemir et al. [23] based on the observed difference in concrete heterogeneity and material properties. In the database results, details of fresh concrete density (qc,f) of specimens are commonly available from source documents, whereas the densities of air dried (qc,a) and oven dried hardened concretes (qc,o) are less commonly reported. Given the availability of information about the fresh densities of concrete (qc,f), this parameter was therefore used in the analysis of the database results. Fig. 1 shows the comparisons of the densities of air dried (qc,a) and oven dried concretes (qc,o) to fresh concrete (qc,f). The slight variations between the densities of fresh (qc,f), air dried (qc,a) and oven dried (qc,o) concretes can be accounted using the expressions given by the trendlines of Fig. 1, of which qc,f is in unit kg/m3. 3.1. Modelling of compressive strength of concrete Several studies have been reported to date on the modelling of concrete compressive strength (Refs. [28–33]). However, a unified expression to estimate the compressive strength of different types

Density of Dried Concrete ( ρc) (kg/m3 )

under unconfined and confined conditions. To this end, firstly two extensive databases of unconfined and actively confined concrete test results, which covered various concrete types, were assembled. The database results indicated significant differences in the stress–strain behavior of different types of concrete, ranging from light-weight (LWC) to normal-weight (NWC), and normal-strength (NSC) to high-strength (HSC). Based on these results, changes in the compressive behavior of concrete with various test parameters were then investigated, and the influential parameters were established. Finally, through a comprehensive examination of the results in the databases, a unified stress–strain model that it is applicable to: i) both LWC and NWC, ii) both NSC and HSC, and iii) both unconfined and actively confined concretes was developed.

3000

Normal and light weight concretes Air dried

2500

Oven dried

2000

ρc,a = 1.02ρc,f -76.1 Data = 349 R² = 0.971

1500 ρc,o = 1.10ρc,f -378.5 Data = 111 R² = 0.962

1000 500 0

0

500

1000

1500

2000

Density of Fresh Concrete ( ρc,f )

2500

3000

(kg/m3 )

Fig. 1. Variation of densities of air dried (qc,a) and oven dried concretes (qc,o) with fresh concrete density (qc,f). 2.2. Database of actively confined concrete

120

Concrete Strength ( f'co ) (MPa)

The database of actively confined concrete, presented in Ref. [7], was assembled from 25 experimental studies that consisted of 346 test datasets, and 31 additional datasets from tests recently undertaken at the University of Adelaide [27]. All of the specimens in the database had circular cross-sections, with cross-sectional dimensions (B) varying from 50 and 160 mm. The specimen heights (H) varied from 88 to 320 mm, the specimen aspect ratios (H/B) varied from 1 to 3, and the compressive strength (f0 co) and the corresponding axial strains (eco), obtained from unconfined concrete cylinder tests, varied from 7.2 to 132.0 MPa and 0.15% to 0.40%, respectively. Various instruments were used in existing studies to measure the axial strains (eco) of specimens, including in-built extensometers of compression machines, linear variable displacement transducers, and axial strain gauges. The unconfined concrete cylinders had the same geometric dimensions as the corresponding confined specimens. The active confinement ratio (f⁄l /f0 co), defined as the ratio of the hydrostatic confining pressure of the triaxial cell to the unconfined concrete strength, varied from 0.004 to 21.67. It is worth noting that, given the limitation of the actively confined concrete database only to specimens with circular cross-sections, for a consistent treatment of the test results, only the specimens with circular cross-sections from both unconfined and actively confined concrete databases were included in the development of the models that are presented later in the paper. However, wherever possible,

f'co = 21(w/c) -1.0 +32(sf/c)0.5 Data = 100 R² = 0.883

Normal-weight concrete Light-weight concrete

100

Concrete cylinders D = 152 mm H/D = 2 Age = 28 days

80 60

f'co = (21(w/c) -1.0 +32(sf/c)0.5)(ρc,f /2400)1.6 Data = 170 R² = 0.663

40 20 0

0

0.5

1

1.5

Water-Cementitous Binder Ratio (w/c) Fig. 2. Variation of concrete compressive strength (f0 co) with water-cementitious binder ratio (w/c).

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120 Normal-weight concrete

100 80 60

ρc,f < 2250 kg/m 3

ρc,f ≥ 2250 kg/m 3

Concrete Strength ( f'co ) (MPa)

Concrete Strength ( f'co ) (MPa)

120 Light-weight concrete Concrete cylinders D = 152 mm H/D = 2 Age = 28 days sf/c = 0% w/c = 0.30 -0.35

f'co = 21(w/c) -1.0 ( ρc,f /2400)1.6 Data = 66 R² = 0.461

40 20 0 500

1000

1500

2000

Density of Concrete ( ρc,f )

2500

3000

100 80

f'co = 21(w/c) -1.0 +32(sf/c) 0.5 Data = 14 R² = 0.287

60

Concrete cylinders D = 152 mm H/D = 2 Age = 28 days w/c = 0.27 ρc,f =2350 -2450 kg/m3

40 20

Normal-weight concrete

0

0

0.05

0.1

120 ρc,f ≥ 2250 kg/m 3

Concrete Strength ( f'co ) (MPa)

Concrete Strength ( f'co ) (MPa)

ρc,f < 2250 kg/m 3

Normal-weight concrete

60

Light-weight concrete Concrete cylinders D = 152 mm H/D = 2 Age = 28 days sf/c = 0% w/c = 0.45 -0.50

f'co = 21(w/c) -1.0 ( ρc,f /2400)1.6 Data = 79 R² = 0.676

40 20 0 500

80 60

1500

2000

2500

Concrete cylinders D = 152 mm H/D = 2 Age = 28 days w/c = 0.33 ρc,f =2350 -2450 kg/m3

40 20

Normal-weight concrete

0 1000

f'co = 21(w/c) -1.0 +32(sf/c) 0.5 Data = 18 R² = 0.230

100

3000

0

0.05

0.1

ρc,f < 2250 kg/m 3

Concrete Strength ( f'co ) (MPa)

Concrete Strength ( f'co ) (MPa)

Normal-weight concrete

ρc,f ≥ 2250 kg/m 3

Light-weight concrete Concrete cylinders D = 152 mm H/D = 2 Age = 28 days sf/c = 0% w/c = 0.45 -0.50

f'co = 21(w/c) -1.0 ( ρc,f /2400)1.6 Data = 79 R² = 0.676

40 20 0 500

0.25

60

120

60

0.2

(b)

(b)

80

0.15

Silica fume ratio (sf/c)

Density of Concrete ( ρc,f ) (kg/m3 )

100

0.25

(a)

120

80

0.2

Silica fume ratio (sf/c)

(kg/m 3 )

(a)

100

0.15

50 40 30

1500

2000

Density of Concrete ( ρc,f )

2500

3000

(kg/m3 )

(c) Fig. 3. Variation of concrete compressive strength (f0 co) with concrete density (qc,f): (a) w/c = 0.30–0.35; (b) w/c = 0.50–0.55; and (c) w/c = 0.70–0.75.

of concrete is not yet available. To establish such an expression, the mix designs of various types of concrete in the database were carefully studied, and important parameters identified to have prominent effect on the compressive strength of concrete were quantified. Fig. 2 shows the variation in the compressive strength of concrete (f0 co) with the w/c ratio, for both NWC and LWC. It is generally understood that the compressive strength (f0 co) of both NWC and LWC increase with a reduction in the w/c ratio, as shown in Fig. 2. However, the relative influences of the w/c ratio, concrete density (qc,f), and silica fume-to-cementitious binder ratio (sf/c) on the compressive strength (f0 co) are much less understood. Fig. 3 shows the change in the compressive strength of concrete (f0 co) with concrete density (qc,f) and Fig. 4 shows the change in the compressive strength of concrete (f0 co) with silica fume-to-cementitious binder ratio (sf/c), of several subgroups of specimens that fall within the selected ranges of w/c ratios for comparisons. As illustrated in the figures, the concrete compressive strength (f0 co) increases with

Concrete cylinders D = 152 mm H/D = 2 Age = 28 days w/c = 0.63 ρc,f =2000 -2100 kg/m3

20 10

Light-weight concrete

0 1000

f'co = (21(w/c) -1.0 +32(sf/c)0.5)(ρc,f /2400)1.6 Data = 11 R² = 0.761

0

0.05

0.1

0.15

0.2

0.25

Silica fume ratio (sf/c)

(c) Fig. 4. Variation of concrete compressive strength (f0 co) with silica fume ratio (sf/c): (a) w/c = 0.27, NWC; (b) w/c = 0.33; NWC; and (c) w/c = 0.63, LWC.

an increase in concrete density (qc,f) or silica fume ratio (sf/c) for a given w/c ratio. Based on the observed trendlines of Figs. 2–4, the relative influences of the w/c ratio, concrete density (qc,f), and silica fume ratio (sf/c) on the compressive strength of concrete (f0 co) were statistically quantified through multivariable regression analysis, which resulted in the expression given in Eq. (1). The new expression proposed in this study is applicable for the prediction of the 28-day compressive strength (f0 co) of 152  305 mm concrete cylinder up to 120 MPa, for NWC and LWC with water–cementitious binder ratios (w/c) ranging from 0.2 to 1.3, concrete densities (qc,f) ranging from 650 to 2550 kg/m3, and silica fume-to-cementitious binder ratios (sf/c) ranging from 0 to 0.2. In Eq. (1), the average density of NWC (qc,f) of 2400 kg/m3 is adopted as the reference value in establishing the change in concrete compressive strength (f0 co) with concrete density (qc,f). Fig. 5 shows that the predictions of the proposed expression (Eq. (1)) are in good agreement with the experimental results.

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J.C. Lim, T. Ozbakkaloglu / Construction and Building Materials 71 (2014) 492–509

f'co = (21(w/c) -1.0 +32(sf/c)0.5)( ρc,f /2400)1.6 Data = 270 R² = 0.891

100 80 60

Concrete cylinders D = 152 mm H/D = 2 Age = 28 days

40 20

Normal-weight concrete Light-weight concrete

0

0

20

40

60

80

100

120

Proposed Expression [(21(w/c)-1.0 +32(sf/c)0.5 )(ρc,f /2400) 1.6 ] (MPa) Fig. 5. Comparison of concrete compressive strengths (f0 co) with model predictions.

0



f co ¼

qffiffiffiffiffiffiffiffiffi q 1:6 21 c;f þ 32 sf =c w=c 2400

ð1Þ

where f0 co is in MPa and qc,f is in kg/m3. 3.2. Modelling of elastic modulus of concrete Extensive research has been conducted to study the relationship between compressive strength of concrete (f0 co), its density (qc,f) and elastic modulus (Ec) [34–48]. Table 1 presents the prediction statistics of the existing expressions proposed for the prediction of the elastic modulus of concrete (Ec) on the experimental results of the unconfined concrete database. It should be noted that not all the datasets included in the database contained all the relevant details required for model assessment. As a result, out of the 4353 datasets, 1471 test results of NWC cylinders and 739 test results of LWC cylinders from the experimental database were used in the assessment of the concrete elastic modulus (Ec). In the comparisons shown in Table 1, average absolute error (AAE) was used to establish overall model accuracy; standard deviation (SD) was used to establish the magnitude of the associated scatter for each model; and mean (M) was used to describe the associated average overestimation or underestimation of the model, where an overestimation was represented by a mean value greater than 1. Based on the prediction statistics in Table 1, it is clear that further improvement to prediction of concrete elastic modulus (Ec) is possible. Fig. 6 shows the typical stress–strain curves of unconfined NWC and LWC in compression. Fig. 7 shows the variation of concrete

elastic modulus (Ec) with the compressive strength (f0 co). As evident from Fig. 7, the elastic moduli (Ec) of both NWC and LWC increase with an increase in the compressive strength of concrete (f0 co). Fig. 8 shows the variation concrete elastic modulus (Ec) with the concrete density (qc,f), for several subgroups of specimens that fall within the selected ranges of compressive strengths (f0 co) for comparisons. As illustrated Fig. 8, the elastic modulus (Ec) of LWC is significantly lower than that of NWC, and varies with the concrete density (qc,f) at a given compressive strength (f0 co). The influence of the concrete age was also investigated through the analysis of the results from specimens with concrete ages ranging from 28 to 1975 days, but no significant influence of the age on the concrete elastic modulus (Ec) was found. Likewise, the specimen cross-sectional shape (i.e. circular and square), which was studied through the use of additional test results of the concrete prisms (Tables A3, A6 and A7), was found to have no significant on the elastic modulus (Ec). Based on the observed trendlines of Figs. 7 and 8, the relative influences of the compressive strength (f0 co) and concrete density (qc,f) on the elastic modulus of concrete (Ec) were statistically quantified using multivariable regression analysis, which resulted in the expression given in Eq. (2). The expression is applicable to concrete cylinders up to 120 MPa for NWC and LWC with concrete densities (qc,f) ranging from 650 to 2550 kg/m3. In Eq. (2), the average density of NWC (qc,f) of 2400 kg/m3 is treated as the reference value to establish the change in concrete elastic modulus (Ec) with concrete density (qc,f). Fig. 9 shows that the predictions of

Peak condition

f'co

Axial Stress ( f c)

Concrete Strength ( f'co ) (MPa)

120

Peak condition

Normalweight concrete

Ec

Light-weight concrete

Ec

Axial Strain (εc ) Fig. 6. Typical stress–strain curves of normal- and light-weight concretes.

Table 1 Statistics on performances of models in predictions of elastic modulus of concrete (Ec). Model

Jensen [34] Ahmad and Shah [35] Oluokun et al. [36] ACI-363 [58] NS-3473 [37] CEB-FIP [38] ACI-318 [62] CSA [39] Iravani [40] Wee et al. [41] TS-500 [42] Fam and Rizkalla [43] Persson [44] Gesoglu [45] Kim et al. [46] Mesbah et al. [47] Nassif [48]

Year of publication

1943 1982 1991 1992 1992 1993 1995 1995 1996 1996 2000 2001 2001 2002 2002 2002 2005

Prediction of Ec of NWC

Prediction of Ec of LWC

Average absolute error (%)

Mean (%)

Standard deviation (%)

Average absolute error (%)

Mean (%)

Standard deviation (%)

20.0 15.6 20.5 16.2 15.6 36.9 23.4 16.6 21.0 31.9 31.5 24.4 15.9 21.2 56.5 25.4 15.7

110.6 98.8 113.9 107.6 103.5 135.0 118.3 108.5 81.6 129.9 129.2 120.9 90.7 113.8 45.1 122.1 99.7

24.9 20.9 22.8 22.1 20.3 28.3 22.6 20.2 15.1 25.8 26.3 22.4 19.9 22.4 10.1 22.6 21.1

119.3 20.5 20.6 24.7 24.9 226.1 21.9 25.0 50.2 176.3 201.9 118.4 64.1 81.0 52.5 120.7 20.3

215.9 92.2 106.3 111.6 111.3 319.5 110.5 112.6 145.1 271.1 296.0 215.0 161.2 177.5 105.1 217.1 93.0

139.7 31.0 34.5 48.0 46.4 402.1 34.7 46.4 97.2 246.1 347.1 143.9 111.0 87.8 115.6 145.4 31.3

J.C. Lim, T. Ozbakkaloglu / Construction and Building Materials 71 (2014) 492–509

the proposed expression (Eq. (2)) are in good agreement with the experimental results.

60000 Ec = 4400( f'co ) 0.5 Data = 1471 R2 = 0.652

qffiffiffiffiffiffi q 1:4 0 c;f Ec ¼ 4400 f co 2400

40000

E c= 4400( f'co ) 0.5( ρc,f /2400)1.4) Data = 739 R2 = 0.311

20000

Concrete cylinders

3.3. Modelling of axial strain at peak compressive stress

Normal-weight concrete Light-weight concrete

0

0

50

100

150

Concrete Strength ( f'co ) (MPa)

Concrete Elastic Modulus (Ec ) (MPa)

Fig. 7. Variation of concrete elastic modulus (Ec) with concrete compressive strength (f0 co).

60000 Normal-weight concrete Light-weight concrete

40000

Concrete cylinders f'co = 15 - 20 MPa

Ec = 4400(f'co ) 0.5 ( c,f /2400)1.4 Data = 147 R² = 0.536

20000

0 1000

ρ ≥ 2250 kg/m 3

ρ < 2250 kg/m 3

1500

2000

Density of Concrete (

2500 c,f )

3000

(kg/m3 )

Concrete Elastic Modulus (Ec ) (MPa)

(a) 60000

Normal-weight concrete Light-weight concrete

40000

Concrete cylinders f'co = 35 - 40 MPa Ec = 4400(f'co ) 0.5 ( c,f /2400)1.4 Data = 205 R² = 0.402

20000

0 1000

ρ ≥ 2250 kg/m 3

ρ < 2250 kg/m 3

1500

2000

Density of Concrete (

2500

3000

3 c,f ) (kg/m )

(b) Concrete Elastic Modulus (Ec) (MPa)

ð2Þ

where Ec and f0 co are in MPa and qc,f is in kg/m3.

60000

Extensive research has been conducted to study the relationship between the compressive strength of concrete (f0 co) and the corresponding axial strain (eco) [22,23,41,49–68]. Table 2 presents the prediction statistics of the existing expressions proposed for the prediction of the axial strain at peak compressive stress (eco) on the experimental results of the unconfined concrete database. Out of the 1167 datasets that reported the axial strains at peak compressive stress (eco), 810 test results from the experimental database that contain relevant information required for model assessment of eco. 663 of these results came from NWC cylinders and 147 came from LWC cylinders. In the existing expressions, the concrete compressive strength (f0 co) is often considered the sole parameter influencing the axial strain at peak compressive stress (eco). Based on the comparison statistics in Table 2, it is clear that the modelling accuracy for the axial strain prediction can be further improved through the incorporation of other influential factors. Fig. 10 shows the variation of the axial strain at peak compressive stress (eco) with compressive strength of concrete (f0 co). As can be seen from the figure, the axial strain at peak compressive stress (eco) of LWC and NWC are significantly different. Apart from the influence of concrete density (qc,f), other parameters including specimen diameter (D) and aspect ratio (H/D) were also observed to have major influence on the axial strain at peak compressive stress (eco). To establish the relative influence of these parameters, 152  305-mm normal-weight concrete cylinders with concrete densities (qc,f) ranging from 2250 to 2550 kg/m3 (i.e., 2400 kg/ m3 ± 6%) were first selected as the reference specimens. The base expression (f0 co0.225/1000) established from the axial strains at peak compressive stress (eco) of these reference specimens, as illustrated in Fig. 11, were then used to quantify the relative changes in the axial strains at peak compressive stress (eco) in other specimens that have different geometric dimensions (H and D) and concrete densities (qc,f) than the reference specimens. Fig. 12 shows the variation of the axial strain at peak compressive stress (eco) with the concrete density (qc,f), for several subgroups of specimens that fall within the selected ranges of concrete compressive strengths (f0 co) for comparisons. As can be seen from the figure, for a given com-

Normal-weight concrete Light-weight concrete

40000

Concrete cylinders f'co = 50 - 60 MPa Ec = 4400(f'co ) 0.5 ( c,f /2400)1.4 Data = 171 R² = 0.300

20000

ρ ≥ 2250 kg/m 3

ρ < 2250 kg/m 3

0 1000

1500

2000

Density of Concrete (

2500 c,f )

3000

(kg/m3 )

(c) Fig. 8. Variation of concrete elastic modulus (Ec) with concrete density (qc,f): (a) low-strength concrete; (b) normal-strength concrete; and (c) high-strength concrete.

Concrete Elastic Modulus (Ec) (MPa)

Concrete Elastic Modulus (Ec) (MPa)

496

60000

Ec = 4400( f'co )0.5(ρc,f /2400)1.4

40000

20000

0

Normal-weight concrete NWC = 1471 AAE = 14.9% M = 100.8% SD = 18.0% Light-weight concrete LWC = 739 AAE = 18.7% M = 99.4% SD = 27.9%

0

20000

40000

60000

Proposed Expression [4400( f'co )0.5 ( ρc,f /2400) 1.4 ] (MPa) Fig. 9. Comparison of concrete elastic moduli (Ec) with model predictions.

497

J.C. Lim, T. Ozbakkaloglu / Construction and Building Materials 71 (2014) 492–509 Table 2 Statistics on performances of models in predictions of axial strain at peak compressive stress (eco) of unconfined concrete. Model

Year of publication

Ros [49] Saenz [50] Tadros [51] Popovics [52] Ahmad and Shah [53] Tomaszewicz [54] Carreira and Chu [55] Shah and Ahmad [56] Ali et al. [57] Taerwe [59] Collins et al. [60] De Nicolo et al. [22] Hsu and Hsu [61] Ariogˆlu [63] Attard and Setunge [64] Wee et al. [41] Tasdemir et al. [23] Mansur et al. [65] Lee [66] Wang et al. [67] Lu and Zhao [68] Chen et al. [83]

0.005 0.004

Mean (%)

Standard deviation (%)

Average absolute error (%)

Mean (%)

Standard deviation (%)

28.8 17.1 14.4 13.1 15.1 13.0 16.1 19.0 12.2 13.0 13.5 15.0 20.3 12.5 16.0 16.0 16.8 22.6 17.1 27.6 22.2 12.8

105.3 85.1 89.4 103.4 105.8 97.6 86.3 114.9 96.5 94.3 96.3 99.8 117.4 96.9 108.2 86.1 100.5 78.3 86.7 73.2 79.0 100.0

36.1 14.7 14.5 15.3 17.6 15.3 12.9 17.8 14.3 14.5 16.0 18.2 17.7 14.7 17.9 14.8 20.5 14.1 15.0 14.7 13.8 15.9

32.2 21.4 19.5 15.5 14.5 19.3 19.8 15.2 18.0 17.4 17.3 21.6 17.4 18.7 13.7 24.5 21.3 36.0 26.1 35.6 36.8 18.4

69.1 80.2 82.4 91.7 92.4 83.2 82.4 103.7 85.6 85.8 88.4 79.8 108.5 84.4 100.7 76.3 79.5 65.0 74.7 65.4 64.3 99.3

16.5 16.7 15.8 15.2 14.9 13.4 16.0 17.8 14.2 15.0 17.1 14.0 19.6 13.7 15.4 14.7 12.4 11.6 11.8 14.1 10.6 21.0

0.003 0.002 0.225 kd /1000) k k co = ( f'co s a kd = 1, ks = 1, ka = 1 Data = 663 R2 = 0.367

0.001

Concrete cylinders Normal-weight concrete Light-weight concrete

0.000

0

50

100

150

200

Concrete Strength ( f'co ) (MPa) Fig. 10. Variation of axial strain at peak compressive stress (eco) with compressive strength of concrete (f0 co).

0.005

Axial Strain at Peak Stress (

co )

Concrete cylinders D = 152 mm H/D = 2 ρc,f ≈ 2400 kg/m 3

0.004 0.003 0.002

0.225 /1000 co = f'co Data = 257 R2 = 0.380

0.001 Normal-weight concrete

0.000

0

50

100

Prediction of eco of LWC

Average absolute error (%)

0.225 kd /1000) k k co = ( f'co s a kd = (ρc,f /2400) 0.45 , ks = 1, ka = 1 Data = 147 R2 = 0.626

co )

Axial Strain at Peak Stress (

1950 1964 1970 1973 1979 1984 1985 1985 1990 1992 1993 1994 1994 1995 1996 1996 1998 1999 2002 2006 2008 2013

Prediction of eco of NWC

150

200

Concrete Strength ( f'co ) (MPa) Fig. 11. Variation of axial strain at peak compressive stress (eco) with compressive strength of concrete (f0 co) for 152  305-mm normal weight concrete cylinders.

pressive strength (f0 co), the axial strain at peak compressive stress (eco) decreases with an increase in concrete density (qc,f), and the reduction becomes more pronounced in the cases of higher

strength concretes. As illustrated in the trendlines of Fig. 12, the influence of concrete density (qc,f) on the axial strain at peak compressive stress (eco) were related to the power of the base expression as a function of (2400/qc,f)0.45. Apart from the concrete density (qc,f), Figs. 13 and 14 show that the specimen diameter (D) and aspect ratio (H/D) also slightly influence the axial strain at peak compressive stress (eco) of concrete. As illustrated in Figs. 13 and 14, for a given compressive strength (f0 co), the axial strain at peak compressive stress (eco) decreases with either an increase in specimen diameter (D) or aspect ratio (H/D). Based on the observed trendlines in Figs. 13 and 14, the relative influences of the specimen diameter (D), and aspect ratio (H/D) were incorporated into the base expression as multipliers (152/D)0.1 and (2D/H)0.13, respectively, which resulted in the final expression given in Eq. (3). The expression is applicable to concrete cylinders up to 120 MPa for NWC and LWC with concrete densities (qc,f) ranging from 650 to 2550 kg/m3. Fig. 15 shows that the predictions of the proposed expression (Eq. (3)) are in good agreement with the experimental results. It might be worth noting that the influence of the concrete age was also investigated through the analysis of the results from specimens with concrete ages ranging from 28 to 1975 days, but no significant influence of the age on the axial strain at peak compressive stress (eco) was found. Furthermore, investigation of the influence of specimen cross-sectional shape (i.e. circular and square) through the use of additional test results of the concrete prisms (Tables A3, A6 and A7) indicated no significant influence of specimen cross-sectional shape on the axial strain at peak (eco). 0 0:225k

eco ¼ kd ¼

ks ¼

d f co ks ka 1000

!0:45 2400

qc;f  0:1 152 D

ð3Þ

ð4Þ

ð5Þ

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J.C. Lim, T. Ozbakkaloglu / Construction and Building Materials 71 (2014) 492–509

0.004 co )

0.004

Normal-weight concrete Light-weight concrete

Axial Strain at Peak Stress (

Axial Strain at Peak Stress (

co )

0.005

= ( f'co 0.225 kd /1000) kd = (2400/ ρc) 0.45 Data = 41

co

0.003 0.002 0.001

Concrete cylinders f'co = 10 -15 MPa D = 152 mm H/D = 2

0.000 500

1000

1500

2000

2500

0.003

0.002 co

0.001

0.000

3000

Concrete cylinders f'co = 20 -25 MPa ρc ≈ 2400 kg/m 3 H/D = 2

50

100

150

350

co )

co )

Axial Strain at Peak Stress (

Axial Strain at Peak Stress (

Normal-weight concrete Light-weight concrete

0.004 0.003 0.002 = ( f'co 0.225 kd /1000) kd = (2400/ ρc) 0.45 Data = 74

co

Concrete cylinders f'co = 40 -50 MPa D = 152 mm H/D = 2

0.000 500

1000

0.003

0.002 co

0.001

0.000

1500

2000

2500

3000

50

100

150

co )

co )

Axial Strain at Peak Stress (

Normal-weight concrete Light-weight concrete

0.003

= ( f'co 0.225 kd /1000) kd = (2400/ ρc) 0.45 Data = 33

co

Concrete cylinders f'co = 70 -80 MPa D = 152 mm H/D = 2

0.000 500

1000

2000

Density of Concrete ( ρc,f )

2500

3000

kg/m3

350

300

350

0.003

0.002 co

0.001

= ( f'co 0.225 /1000) ks ks = (152/D) 0.1 Data = 43

Concrete cylinders f'co = 70 -80 MPa ρc ≈ 2400 kg/m 3 H/D = 2

50

100

150

200

250

Diameter (D)

(c)

(c) Fig. 12. Variation of axial strain at peak compressive stress (eco) with concrete density (qc,f): (a) low-strength concrete; (b) normal-strength concrete; and (c) highstrength concrete.

 0:13 2D H

300

0.004

0.000

1500

250

(b)

0.005

0.002

200

Diameter (D)

(b)

0.004

= ( f'co 0.225 /1000) ks ks = (152/D) 0.1 Data = 74

Concrete cylinders f'co = 40 -50 MPa ρc ≈ 2400 kg/m 3 H/D = 2

Density of Concrete ( ρc,f ) kg/m3

Axial Strain at Peak Stress (

300

0.004

0.005

ka ¼

250

(a)

(a)

0.001

200

Diameter (D)

Density of Concrete ( ρc,f ) kg/m3

0.001

= ( f'co 0.225 /1000) ks ks = (152/D) 0.1 Data = 47

ð6Þ

In Eqs. 3–6, f0 co is in MPa, qc,f is in kg/m3, and D and H are in mm, and kd, ks, and ka, respectively, are the coefficients to allow for concrete density, specimens size and specimen aspect ratio. Based on the geometric dimensions of the specimens used in the development of the model expressions, it is recommended that in Eqs. (5) and (6), the range of aspect ratios (H/D) be limited to 2–8, diameters (D) to 50–400 mm, and heights (H) to 100–850 mm.

4. Peak and residual conditions of actively confined concrete Since the 1920s [69,70], a significant research effort has been dedicated to understanding the behavior of concrete under triaxial

Fig. 13. Variation of axial strain at peak compressive stress (eco) with specimen diameter (D): (a) low-strength concrete; (b) normal-strength concrete; and (c) high-strength concrete.

compression, resulting in the development of a number of stress– strain model for actively confined concrete [3,24,64,70–79]. Fig. 16 shows the typical stress–strain curves of unconfined and actively confined concretes, together with the important coordinates along the curves. As can be seen from the figure, the stress–strain curve of actively confined concrete consists of a parabolic ascending branch that reaches the peak compressive stress (f⁄cc) and is followed by a gradually descending second branch. After the peak stress, interparticle cohesion in the concrete continues, and the remaining strength generated through frictional action that forms a stabilized plateau in the curve is known as the residual stress (fc,res) [75]. The accuracy of models in the predictions of these important coordinates significantly affects the predicted shape of the stress–strain curve. To compare the performances of the existing actively confined concrete models, the predictions of the

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Axial Strain at Peak Stress (

Axial Strain at Peak Stress (

co )

co )

0.004

0.003

0.002

co

0.001

0.000

Concrete cylinders f'co = 25 -30 MPa ρc ≈ 2400 kg/m 3 D = 152 mm

1

2

3

4

= (f'co 0.225 /1000) ka ka = (D/2H) 0.13 Data = 36

5

6

7

0.005 0.225 kd /1000) k k co = ( f'co s a kd= ( ρc,f /2400) 0.45 , ks = (152/D)0.1, ka = (2D/H)0.13

0.004 0.003 0.002 0.001

Light-weight concrete LWC = 147 AAE = 12.7% M = 101.2% SD = 13.9%

0.000 0.000

8

Normal-weight concrete NWC = 663 AAE = 11.3% M = 99.6% SD = 13.0%

Aspect Ratio (H/D)

0.001

0.002

0.003

0.004

0.005

Proposed Expression [( f'co 0.225 kd /1000) ks ka ]

(a) Fig. 15. Comparison of axial strains at peak compressive stress (eco) with model predictions.

0.003

4.1. Modelling of peak stress and strain of actively confined concrete 0.002 co

0.001

0.000

Concrete cylinders f'co = 40 -50 MPa ρc ≈ 2400 kg/m 3 D = 152 mm

1

2

3

4

= (f'co 0.225 /1000) ka ka = (D/2H) 0.13 Data = 30

5

6

7

8

7

8

Aspect Ratio (H/D)

(b)

Axial Strain at Peak Stress (

co )

0.004

0.003

0.002 co

0.001

0.000

= ( f'co 0.225 /1000) ka ka = (D/2H ) 0.13 Data = 47

Concrete cylinders f'co = 85 -95 MPa ρc ≈ 2400 kg/m 3 D = 152 mm

1

2

3

4

5

6

Aspect Ratio (H/D)

(c) Fig. 14. Variation of axial strain at peak compressive stress (eco) with specimen slenderness (H/D): (a) low-strength concrete; (b) normal-strength concrete; and (c) high-strength concrete.

strength enhancement ratio (f⁄cc/f0 co), peak strain ratio (e⁄cc/eco), and residual stress ratio (fc,res/f0 co) of the existing models were assessed using the test results of the actively confined concrete database. In the calculations of the strength enhancement ratios (f⁄cc/f0 co) and the peak strain ratios (e⁄cc/eco), the compressive strength (f0 co) and the corresponding axial strain (eco) of unconfined concrete were obtained from cylinder test results reported in the original studies. Out of the 377 datasets, 341, 243, and 173 test results from the experimental database were used in the assessment of the strength enhancement ratio (f⁄cc/f0 co), peak strain ratio (e⁄cc/eco), and residual stress ratio (fc,res/f0 co), respectively. Based on the prediction statistics in Table 3, a number of existing models were found to perform well in their predictions [24,64,75–77]. Nevertheless, the results suggest that, in the prediction of the peak stress and strain and residual stress of actively confined concrete, a further improvement in the modelling accuracy is possible.

Based on the results from the test database of actively confined concrete [7], the expressions established for the predictions of the compressive strength (f⁄cc) and the corresponding axial strain (e⁄cc) are given in Eqs. (7) and (8). In these equations, the active confinement ratio (f⁄l /f0 co) is expressed as the parameter influencing the compressive strength (f⁄cc) and the corresponding axial strain (e⁄cc), whereas the unconfined concrete strength (f0 co) is considered a subsidiary parameter affecting the compressive strength (f⁄cc) of actively confined concrete. As illustrated in Figs. 17 and 18, the trendlines of the proposed expressions (Eqs. (7) and (8)) are in good agreement with the experimental results. In the calculations of the strength enhancement ratios (f⁄cc/f0 co) and the peak strain ratios (e⁄cc/eco), the compressive strength (f0 co) and the corresponding axial strain (eco) of unconfined concrete were obtained from cylinder test results reported in the original studies. As can be seen from the comparison of the performance statistics (AAE, M, SD) of the proposed model in Figs. 17 and 18 with those of the existing models shown in Table 3, the proposed model outperformes the existing models in the predictions of the strength enhancement ratios (f⁄cc/f0 co) and the peak strain ratios (e⁄cc/eco). It might be worth noting that, in the prediction of the e⁄cc/eco ratios, the proposed model would have outperformed the existing models even more significantly, if in the model assessment eco values were established using the expressions proposed by the models as opposed to the experimental values.

Peak condition

f *cc

Inflection point

Axial Stress ( fc)

Axial Strain at Peak Stress (

co )

0.004

Peak condition

f'co

Confined concrete

fc,res

Residual stress Inflection point Ec Unconfined concrete co

c,i

*

cc

c,i

Axial Strain (εc) Fig. 16. Typical stress–strain curves of unconfined and actively confined concretes.

Strength Enhancement Ratio ( f *cc / f' co )

28.9 16.3 – – – 18.5 113.4 76.8 – – – 93.9

– – – – 105.1 – – – – – 18.3 –

38.3 – 50.9 – 25.0 31.5 – – 44.9 27.6 34.3 31.7 30.1

4 3 2

f *cc /f'co= 1+5.2 f'co-0.09 ( f *l/f'co ) a a = f'co -0.06 Data = 341 R2 = 0.913 AAE = 7.3% M = 100.2% SD = 8.7%

1 0

0

0.2

0.4

0.6

0.8

1

1.2

25

36.1 – 50.6 – 21.6 25.4 – – 45.1 22.8 28.6 25.3 24.7

124.9 – 137.1 – 89.6 101.0 – – 134.4 91.5 111.5 104.2 96.4

20.4 24.1 – – – 14.4

Fig. 17. Variation of strength enhancement ratio (f⁄cc/f0 co) with confinement ratio (fl⁄/f 0 co).

Peak Strain Ratio (ε * cc / ε co )

Mean (%)

20 15 10 5 0

ε*cc /εco= 1+0.045( f *l /f'co )1.15 /εco Data = 243 R2 = 0.794 AAE = 18.9% M = 99.7% SD = 24.8%

0

0.2

0.4

0.6

0.8

1

1.2

9.6

11.7 33.4 11.0 10.8 9.3 10.8 17.1 9.9 9.7 9.7

Fig. 18. Variation of peak strain ratio (e⁄cc/eco) with confinement ratio (f⁄l /f0 co).

9.9 94.9 29.7 83.8 9.2 95.4 9.4 105.3 7.4 97.2 12.2 89.3 13.5 107.7 7.7 97.7 8.0 96.3 13.1 88.5 Same as Jiang and Teng [77] 8.3 95.4 Same as Attard and Setunge [64]

Prediction of fc,res/f0 co

Average absolute error (%) Standard deviation (%) Mean (%) Average absolute error (%)



 f cc

¼

0 f co

þ

fl 0 f co

0 0:91 5:2f co



 cc

e

f ¼ eco þ 0:045 0l f co

!a 00:06

where a ¼ f co

ð7Þ

!1:15 ð8Þ

where f⁄l and f0 co are in MPa and eco is to be calculated using Eq. (3).

1929 1970 1988 1995 1996 1998 2001 2001 2005 2007 2007 2010 2012

4.2. Modelling of residual stress and inflection point of post-peak relationship

Richart et al. [70] Mills and Zimmerman [71] Mander et al. [3] Xie et al. [72] Attard and Setunge [64] Ansari and Li [73] Candappa [74] Imran and Pantazopoulou [75] Binici [76] Jiang and Teng [77] Teng et al. [78] Xiao et al. [79] Samani and Attard [24]

Average absolute error (%)

Mean (%)

Standard deviation (%)

Prediction of e⁄cc/eco

5

Confinement Ratio ( f *l / f'co )

Year of publication

Prediction of f⁄cc/f0 co

6

Confinement Ratio ( f *l / f'co )

Model

Table 3 Statistics on performances of models in predictions of strength enhancement ratios (f⁄cc/f0 co), peak strain ratios (e⁄cc/eco), and residual stress ratios (fc,res/f0 co) of actively-confined concrete.

– – – – 28.3 –

J.C. Lim, T. Ozbakkaloglu / Construction and Building Materials 71 (2014) 492–509

Standard deviation (%)

500

Accurate determination of the residual stress (fc,res) and the axial strain at the inflection point of descending branch (ec,i) is vital for the prediction of the post-peak stress–strain behavior of actively confined concrete. Based on the results from the test database of actively confined concrete [7], the expression established for the predictions of residual stress (fc,res) is given in Eq. (9). As illustrated in Fig. 19, the residual stress ratio (fc,res/f0 co) increases with the confinement ratio (f⁄l /f0 co) and nearly approaches the trendline of the peak stress ratio (f⁄cc/f0 co), as shown in dashed line in Fig. 19, when the confinement ration (f⁄l /f0 co) approaches 0.4. In such condition, the residual stress ratio (fc,res/f0 co) was observed to be slightly lower than the strength enhancement ratio at peak condition (i.e., f⁄cc/f0 co – 0.15), as evident from the comparison of trendlines in Fig. 19. This condition is represented through the constraint assigned to the residual stress (fc,res) in Eq. (9).

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J.C. Lim, T. Ozbakkaloglu / Construction and Building Materials 71 (2014) 492–509



Residual Stress Ratio ( fc,res / f'co )

6

ec;i ¼ 2:8ecc

5

ð10Þ

4

⁄

where f

Trendline of f *cc /f'co

3

cc, fc,res

fc,res /f'co = 1.6( f *cc /f'co)( f *l0.24 /f'co 0.32) ≤ f* cc /f'co -0.15 Data = 173 R2 = 0.937 AAE = 13.4% M = 100.0% SD = 19.1%

1 0

0.2

0.4

0.6

0.8

Confinement Ratio (

f*

l/

1

1.2

f'co )

Fig. 19. Variation of residual stress ratio (fc,res/f0 co) with confinement ratio (f⁄l /f0 co).



Axial Stress ( fc)

and f0 co are in MPa, and qc,f is in kg/m .

A model applicable to both unconfined and actively confined concretes is presented in this section. The ascending branch of the stress–strain curve is to be predicted using the expression developed by Popovics [52] (Eq. (11)), with the use of the concrete brittleness constant proposed by Carreira and Chu [55] (Eq. (12)). To allow for the change in shape of the stress–strain curve at ascending branch for various types of concrete, the concrete elastic modulus (Ec), compressive strength (f⁄cc) and strain (e⁄cc), are to be predicted using Eqs. (2), (7) and (8).

fc ¼

Eq. 11

*

f c /E c

* cc -f cc /E c

Eq. 13

Inelastic

Elastic

Ec

εd

H r = Hd

fc ,

w/H r

Ec

r¼

c

(a) f *cc ,





ec =ecc r  r r  1 þ ec =ecc f cc

* cc

ð11Þ

ð12Þ





f c ¼ f cc 

f cc  f c;res   e e 2 1 þ e c ecc c;i

ε d(H d/H) f c /E c

* -f * /E c cc cc

Ec

Ec

Hd

H

c,h

δc,h /H Eqs. 13 & 14

Axial Strain (ε c,h)

(b) Fig. 20. (a) Typical stress–strain curves of a reference specimen with height Hr; and (b) adjusted stress–strain curve of a specimen with height H.



fl

00:32

ec > ecc

ð13Þ

cc

For unconfined concrete, the f⁄cc and e⁄cc in Eqs. (11)–(13) become f co and eco, respectively. It should be noted that the proposed expression (Eq. (13)) was developed based on the results from the reference specimens with a 152 mm diameter and 305 mm height. To account for the influences of the specimen size and slenderness on post-peak axial strains, an expression is given in the following section (i.e., Eq. (14)) for the adjustment of axial strains of specimens with geometric dimensions that differ from those of the reference specimens. 5.1. Post-peak axial strain adjustment to allow for specimen size and slenderness

! 0:24

f co

if

0

w/H

fc ,



ec 6 ecc

Ec  Ec  f cc =ecc

Eq. 11

f c;res ¼ 1:6f cc

if 0 6

The descending branch of the stress–strain is to be predicted using a new expression developed in this study, which is given in Eq. (13) and was established using a large number of experimental stress–strain curves of unconfined and actively confined specimens in the databases covering various types of concrete. In Eq. (13), the residual stress of the confined concrete (fc,res) is to be calculated using Eq. (9), and the inflection point (ec,i) of the descending branch is to be calculated using Eq. (10).

δc /H r

Axial Strain (ε c)

Axial Stress ( fc)

3

5. Proposed axial stress–strain model

2

0

      qc;f 0:4 f c;res 00:12 f c;res 00:47 f co þ 10ecc 1   f co  2400 f cc f cc



0

and f c;res 6 f cc  0:15f co

ð9Þ

where f⁄cc, f⁄l , and f0 co are in MPa. The inflection point (ec,i) is important for accurate modelling of the location where the change in shape of the descending branches of stress–strain curves occur. As illustrated in Figs. 6 and 16, the inflection points (ec,i) mark to the locations of the change in the sign of curvature of the descending branches of axial stress–strain curves from negative to positive. Based on the descending branch results of unconfined and actively confined specimens in the databases that had complete axial stress–strain curves, the expression for predicting the inflection point (ec,i) was established as a function of the peak stress and strain (f⁄cc, e⁄cc) and residual stress (fc,res) and concrete density (qc,f), as given in Eq. (10).

As was reported in a number of studies, the total deformation of concretes consists of elastic and inelastic components [24,80– 83]. The inelastic component is a displacement occurs in a compression damage zone (Hd) within a segment of the specimen height, whereas the elastic component is a strain occurs along the entire specimen height (H) [80,81]. Due to the difference in the relative deformations of the elastic and inelastic components, the average axial strain (ec) varies with the change in heights (H), as evident from a number of test results [84–87]. To account for the relative contributions of the elastic and inelastic components in the average axial strain (ec), axial strain adjustments, which were based on the general concept established by Markeset and Hillerborg [81], were applied in a number of studies [24,82,83,86,87]. Based on the concept established by Markeset

502

J.C. Lim, T. Ozbakkaloglu / Construction and Building Materials 71 (2014) 492–509

100

80

Hsu and Hsu [61]

ø76 x 152 mm

Wischers [90] Proposed model

Proposed model

Axial Stress ( fc ) (MPa)

Axial Stress ( fc) (MPa)

ø150 x 300 mm

60

40

20

80 60 40 20 0

0 0

0.002

0.004

0.006

0.008

0.01

0

0.012

0.004

0.008

(a) 140

Axial Stress ( fc ) (MPa)

Proposed model

Axial Stress ( fc ) (MPa)

Wee et al. [41]

ø100 x 200 mm

Ahmad and Shah [91]

ø75 x 152 mm

80 60 40 20

Proposed model

120 100 80 60 40 20 0

0 0

0.002

0.004

0.006

0

0.008

0.001

0.002

0.003

0.004

0.005

0.006

Axial Strain (ε c,h)

Axial Strain (ε c,h)

(f)

(b) 50

120

80 60 40 20

Axial Stress ( fc ) (MPa)

Proposed model

100

Watanabe et al. [87]

ø100 x 400 mm

Dahl [92]

ø100 x 200 mm

Axial Stress ( fc) (MPa)

0.016

(e)

100

Proposed model

40 30 20 10 0

0 0

0.001

0.002

0.003

0.004

0.005

0

0.006

0.001

0.002

0.003

0.004

0.005

0.006

Axial Strain (ε c,h)

Axial Strain (ε c,h)

(g)

(c) 70

100

Desnerck et al. [93]

ø118 x 355 mm

Proposed model

80 60 40 20

Axial Stress ( fc) (MPa)

Taerwe [59]

ø155 x 380 mm

Axial Stress ( fc) (MPa)

0.012

Axial Strain (εc,h )

Axial Strain (ε c,h)

Proposed model

60 50 40 30 20 10 0

0 0

0.002

0.004

Axial Strain (ε c,h)

(d)

0.006

0.008

0

0.002

0.004

0.006

0.008

Axial Strain (ε c,h)

(h)

Fig. 21. Comparison of the proposed model predictions with unconfined specimen results of different concrete strengths from: (a) Wischers [90]; (b) Ahmad and Shah [91]; (c) Dahl [92]; (d) Taerwe [59]; (e) Hsu and Hsu [61]; (f) Wee et al. [41]; (g) Watanabe et al. [87]; and (h) Desnerck et al. [93].

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J.C. Lim, T. Ozbakkaloglu / Construction and Building Materials 71 (2014) 492–509

50

100

ø76 x 152 mm Light weight concrete ρc,f = 1954 -2046 kg/m3

Kaar et al. [94]

80

Axial Stress ( fc) (MPa)

Axial Stress ( fc) (MPa)

Proposed model ø152 x 305 mm Light weight concrete ρc,f = 1650 -1750 kg/m3

60 40 20

40

Shah et al. [95] Proposed model

30 20 10 0

0 0

0.001

0.002

0.003

0

0.004

0.001

0.002

0.003

0.007

Zhang and Gjorv [96]

Proposed model

60

Axial Stress ( fc) (MPa)

Axial Stress ( fc) (MPa)

Kaar et al. [94]

ø152 x 305 mm Light weight concrete ρc,f = 1650 -1750 kg/m3

50 40 30 20 10 0

0.001

0.002

0.003

Proposed model

80 60 40 20 0

0.004

ø100 x 280 mm Light weight concrete ρc,f = 1750 -1865 kg/m3

0

0.001

Axial Strain (ε c,h)

0.002

50

Shannag [97]

Proposed model

Axial Stress ( fc) (MPa)

Proposed model

20 10

0.001

0.002

0.003

0.005

50

Shah et al. [95]

30

0

0.004

(e)

ø76 x 152 mm Light weight concrete ρc,f = 1882 -1924 kg/m3

40

0.003

Axial Strain (ε c,h)

(b)

Axial Stress ( fc) (MPa)

0.006

100

70

0

0.005

(d)

(a)

0

0.004

Axial Strain (ε c,h)

Axial Strain (ε c,h )

0.004

0.005

0.006

40 30 20 10 0

0.007

ø100 x 200 mm Light weight concrete ρc,f = 2025 -2066 kg/m3

0

0.002

Axial Strain (ε c,h)

0.004

0.006

0.008

0.01

Axial Strain (ε c,h)

(f)

(c) 50 Shannag [97]

Axial Stress ( fc) (MPa)

Proposed model

40

ø100 x 200 mm Light weight concrete ρc,f = 2025 -2066 kg/m3

30 20 10 0

0

0.002

0.004

0.006

0.008

Axial Strain (ε c,h)

(g) Fig. 22. Comparison of the proposed model predictions with unconfined specimen results of light-weight concretes from: (a and b) Kaar et al. [94]; (c and d) Shah et al. [95]; (e) Zhang and Gjorv [96]; and (f and g) Shannag [97].

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350 ø100 x 250 mm f'co = 73.3 MPa

500

Axial Stress ( fc ) (MPa)

Axial Stress ( fc) (MPa)

600

f * l = 138.2 MPa

400

f * l = 69.2 MPa

300 f * l = 34.9 MPa

200

f * l = 22.6 MPa f * l = 13.7 MPa

100

f * l = 6.8 MPa f * l = 3.5 MPa f * l = 0 MPa

0 0

0.01

Newman [98] Proposed model

0.02

0.03

0.04

0.05

0.06

ø55.5 x 110 mm f'co = 92.3 MPa

300

f * l = 44.4 MPa

250

f * l = 35.5 MPa f * l = 26.3 MPa

200

f * l = 21.9 MPa f * l = 17.3 MPa f * l = 16.5 MPa f * l = 12.8 MPa f * l = 8.3 MPa

150 100 50 0 0

0.07

0.01

0.02

0.03

400

90 Hurlbut [99]

80

ø55.5 x 110 mm f'co = 119 MPa

f * l = 13.8 MPa

Proposed model

70

Axial Stress ( fc) (MPa)

Axial Stress ( fc ) (MPa)

0.05

(e)

(a)

ø54 x 108 mm f'co = 19 MPa

60

f * l = 6.9 MPa

50 40

f * l = 3.5 MPa

30 20

f * l = 0.7 MPa

f * l = 60.0 MPa f * l = 48.0 MPa

300

f * l = 36.1 MPa f * l = 30.0 MPa

200

f * l = 24.0 MPa f * l = 18.0 MPa

100

f * l = 12.0 MPa f *l = 6.1 MPa

Xie et al. [72]

f *l = 0 MPa

10

Proposed model

f * l = 0 MPa

0

0 0

0.005

0.01

0.015

0

0.02

0.01

0.02

0.03

250

250

ø160 x 320 mm f'co = 53.5 MPa

øl00 x 200 mm f'co = 123 MPa

f * l = 29.4 MPa

150

f * l = 34.3 MPa

f * l = 24.5 MPa f * l = 19.6 MPa f * l = 14.7 MPa f * l = 9.8 MPa

100

f * l = 4.9 MPa f * l = 0 MPa

Bellotti and Rossi [100]

Axial Stress ( fc) (MPa)

f * l = 39.2 MPa

200

200 f * l = 20 MPa

150 f * l = 15 MPa

100

f * l = 10 MPa f * l = 5 MPa

50

0 0.02

0.03

0.01

0.015

(c)

(g)

f* l = 23.3 MPa f * l = 20.3 MPa

120

f * l = 14.3 MPa f * l = 11.3 MPa

80

f * l = 8.3 MPa f * l = 2.3 MPa

40

f * l = 5.3 MPa

ø100 x 200 mm f'co = 132 MPa

200 f * l = 15 MPa

150 f * l = 10 MPa f * l = 5 MPa

100 50

Attard and Setunge [64]

Xie et al. [72] *

f l = 0.8 MPa

Proposed model

0 0.02

0.03

0.02

250

f* l = 29.3 MPa

160

0.01

0.005

Axial Strain (ε c,h)

ø55.5 x 110 mm f'co = 60.2 MPa

0

0

Axial Strain (ε c,h)

200

f * l = 0 MPa

Proposed model

0

0.04

Axial Stress ( fc) (MPa)

0.01

Attard and Setunge [64]

f * l = 1 MPa

Proposed model

0

0.05

(f)

(b)

50

0.04

Axial Strain (ε c,h)

Axial Strain (ε c,h)

Axial Stress ( fc) (MPa)

0.04

Axial Strain (ε c,h)

Axial Strain (ε c,h)

Axial Stress ( fc) (MPa)

Xie et al. [72] Proposed model

f * l = 3.8 MPa

f * l = 0 MPa

0.04

Proposed model

0 0

0.005

0.01

Axial Strain (ε c,h)

Axial Strain (εc,h)

(d)

(h)

0.015

0.02

Fig. 23. Comparison of the proposed model predictions with actively confined specimen results from: (a) Newman [98]; (b) Hurlbut [99]; (c) Bellotti and Rossi [100]; (d–f) Xie et al. [72]; (g and h) Attard and Setunge [64]; (i) Candappa et al. [74]; (j) Gabet et al. [101]; and (k and l) Lim and Ozbakkaloglu [27].

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J.C. Lim, T. Ozbakkaloglu / Construction and Building Materials 71 (2014) 492–509

180

180

140

f * l = 8 MPa

f * l = 4 MPa

120 100

f * l = 0 MPa

80 60 40 Candappa et al. [74]

20

120

0.004

0.006

0.008

60

0.02

0.04

0.06

(i)

(k) 250

f * l = 500 MPa

600

f * l = 200 MPa

400

f * l = 100 MPa

200

*

f l = 50 MPa

Gabet et al. [101]

f * l = 0 MPa

0.02

200 f * l = 25 MPa

150

f * l = 20 MPa

100

f * l = 12.5 MPa f * l = 10 MPa

f * l = 15 MPa

f * l = 5 MPa

50

0.06

0.08

0.1

f * l = 2.5 MPa

f * l = 0 MPa

Proposed model

0.04

0.12

0.08

ø63 x 127 mm f'co = 128 MPa

f * l = 650 MPa

Axial Stress ( fc) (MPa)

Axial Stress ( fc) (MPa)

0

Axial Strain (ε c,h )

800

0

Lim and Ozbakkaloglu [27] Proposed model

Axial Strain (ε c,h)

ø70 x 140 mm f'co = 30 MPa

0

f * l = 0 MPa

20

1200 1000

f * l = 10 MPa f * l = 5 MPa

40

0.012

f * l = 15 MPa

f * l = 12.5 MPa

80

0

0.01

f * l = 20 MPa

100

0 0.002

f * l = 25 MPa

140

Proposed model

0

ø63 x 127 mm f'co = 50.4 MPa

160

Axial Stress ( fc) (MPa)

Axial Stress ( fc) (MPa)

f * l = 12 MPa

ø98 x 200 mm f'co = 103.3 MPa

160

Lim and Ozbakkaloglu [27] Proposed model

0 0

Axial Strain (ε c,h )

0.01

0.02

0.03

0.04

Axial Strain (ε c,h)

(j)

(l) Fig. 23 (continued)

and Hillerborg [81], Eq. (14) was derived in the present study for the adjustment of the axial strains of specimens having diameters (D) and heights (H) different from those of the reference specimens of the proposed model (i.e., Dr = 152 mm, Hr = 305 mm). After the adjustment of the axial strain through Eq. (14), the axial strain (ec) of the reference specimen in Eq. (13) is to be replaced with the adjusted axial strain (ec,h) to establish the complete stress–strain relationship of a specimen with given geometric properties.

ec;h ¼ ec if 0 6 ec 6 ecc ec;h ¼

     Hr ðf c  f cc Þ Hr ðHd  Hr Þ þ ecc þ ec  ecc 1 þ ed if ec > ecc Ec H H H ð14Þ

where ec,h is the axial strain for a specimen of height H, Hr is the reference specimen height of 305 mm, Hd is the height of the compression damage zone, equal to twice the specimen diameter (i.e., Hd = 2D), and ed is the inelastic strain in the damage zone, to be calculated using a new expression (Eq. (15)) developed in this study based on analysis of the results from the test databases. 0

ed ¼

00:5 0:02f co

ed ¼ 0:02f 00:5 co

f co  f c 0 f co

!

 f cc  f c  f cc  f c;res

where f0 co is in MPa.

for unconfined concrete !

ð15Þ for confined concrete

To illustrate the strain adjustment process, Fig. 20(a) shows the stress–strain curve of a reference specimen (i.e., Dr = 152 mm, Hr = 305 mm), whereas Fig. 20(b) shows the adjusted stress–strain curve of a specimen having different geometric dimensions from those of the reference specimen. As illustrated in Fig. 20(a), the elastic component of the axial strain consists of a linear portion that follows a loading path equal to the concrete elastic modulus (Ec) and a parabolic portion that approaches the axial strain (e⁄cc) corresponding to the peak compressive stress (f⁄cc). After the peak condition is reached, a linear unloading path with a slope that is equal to the elastic modulus of concrete (Ec) is assumed for the elastic component of the axial strain, and the remaining strains are attributed to the inelastic displacement occurs in the damage zone (dc,h) [81]. As illustrated in Fig. 20(a), the inelastic displacement (dc,h) consists of an inelastic strain (ed) occurs within the damage zone (Hd), and a localized displacement (w) that results from the formation of macrocracks (i.e., dc,h = edHd + w) [81]. When the height of damage zone (Hd) is equal to the reference height (Hr), a typical stress–strain curve, as illustrated Fig. 20(a), can be found. In the case of a specimen having a height (H) greater than the height of the damage zone (Hd) as shown in Fig. 20(b), the axial strain adjustment to account for the relative contributions of each elastic and inelastic component becomes necessary. To this end, Eqs. (14) and (15), which were developed on the basis of the results from the test databases, are proposed in the present study. The proposed model is not recommended for specimens with a height (H) lower than 2D, where the effect of the frictional resistance supplied by the loading platens at specimen ends becomes evident on the compressive behavior of the specimen [88,89]. It

J.C. Lim, T. Ozbakkaloglu / Construction and Building Materials 71 (2014) 492–509

Axial Stress ( fc) (MPa)

50

D = 84 mm H = 170 -510 mm f'co = 38.8 -39.2 MPa

40

100

Rokugo and Koyanagi [84]

D = 100 mm H = 200 -550 mm f'co = 88.5 -90.8MPa

Proposed model

Axial Stress ( fc) (MPa)

506

30 H/D = 2

20 H/D = 4

10

80

Jansen and Shah [86] Proposed model

60 40 H/D = 5.5

H/D = 2

20

H/D = 2.5

H/D = 6

H/D = 4.5

0

H/D = 3.5

0 0

0.002

0.004

0.006

0.008

0

0.001

0.002

Axial Strain (ε c,h)

0.003

0.004

(a)

0.006

35

50

Watanabe et al. [87]

Jansen and Shah [86]

40

Axial Stress ( fc) (MPa)

Proposed model

Axial Stress ( fc) (MPa)

0.005

Axial Strain (ε c,h) (c)

D = 100 mm H = 200 -550 mm f'co = 42.8 -44.7 MPa

30 H/D = 2

20 H/D = 5.5

10

H/D = 4.5

H/D = 2.5 H/D = 3.5

Proposed model D = 100 mm H = 200 -800 mm f'co = 28.4 -30.3 MPa

30 25 20 15 10 H/D = 3

H/D = 4

5

H/D = 2

H/D = 6

H/D = 8

0

0 0

0.002

0.004

0.006

0.008

0

0.002

0.004

0.006

0.008

Axial Strain (ε c,h)

Axial Strain (ε c,h)

(d)

(b) 25 Nakamura and Higai [102]

Axial Stress ( fc) (MPa)

Proposed model

20

D = 150 mm H = 150 -600 mm f'co = 19.8 -20.0 MPa

15 10

H/D = 1 H/D = 4

5 0

H/D = 3

0

0.005

H/D = 2

0.01

0.015

Axial Strain (ε c)

(e) Fig. 24. Comparison of the proposed model predictions with unconfined specimen results of different aspect ratios from: (a) Rokugo and Koyanagi [84]; (b and c) Jansen and Shah [86]; (d) Watanabe et al. [87]; and (e) Nakamura and Higai [102].

is recommended that the specimen aspect ratio (H/D), diameter (D), height (H) and concrete compressive strength (f0 co) in Eqs. (14) and (15) be limited to the experimental validation ranges previously noted in Section 3.3.

6. Comparison with experimental results Fig. 21 compares the model predictions with experimental stress–strain curves of unconfined NWC of different compressive strengths (f0 co) [41,59,61,87,90–93]. Fig. 22 compares of the model predictions with experimental stress–strain curves of unconfined LWC of different concrete density (qc,f) [94–97]. Fig. 23 compares the model predictions with experimental stress–strain curves of actively confined NWC with different

confining pressures (f⁄l ) [27,64,72,74,98–101]. Fig. 24 compares of the model predictions with experimental stress–strain curves of unconfined NWC with different aspect ratio (H/D) [84,86,87,102]. As evident from the comparisons, the predictions of the proposed model are in good agreement with the experimental results that consist of a wide range of compressive strengths (f0 co), concrete densities (qc,f), specimen dimensions (H and D), and confining pressures (f⁄l ). To complete the comparisons, the predictions of the proposed model is also compared with those of the existing models that are capable of generating complete stress–strain curves of unconfined and confined concretes [3,24,64,76–79]. The results of this comparison are shown in Fig. 25, which illustrate the improved accuracy of the proposed model in predicting the behavior of concretes under various levels of confinement.

J.C. Lim, T. Ozbakkaloglu / Construction and Building Materials 71 (2014) 492–509

Axial Stress ( fc) (MPa)

80

Mander et al. [3] Attard and Setunge [64] Binici [76] Jiang and Teng [77] Teng et al. [78] Xiao et al. [79] Samani and Attard [24] Proposed model Test result [90]

D = 150 mm H = 300 mm f'co = 69 MPa

60

40

20

0 0.000

0.002

0.004

0.006

0.008

0.010

0.012

Axial Strain (ε c,h)

(a)

Axial Stress ( fc) (MPa)

unconfined and actively confined concretes was then developed and is presented. The model is applicable to normal and light-weight concretes with compressive strengths up to 120 MPa. The important features of the proposed stress–strain model include: (i) applicability to concretes with various densities and strengths, (ii) accurate prediction of the peak and residual stresses and strains of unconfined and confined concretes, (iii) consideration of the change in shape of stress–strain curve with the type of concrete, and (iv) consideration of specimen size and slenderness effects. The proposed model provides improved predictions of the peak stress and strain of unconfined and confined concretes compared to existing models. In addition, the model is capable of predicting the change in shapes of stress–strain curves of various types of concrete, including normal and light-weight concretes. Appendix A. Supplementary material

140

Mander et al. [3] Binici [76] Teng et al. [78] Samani and Attard [24] Test result [72]

120 100

Attard and Setunge [64] Jiang and Teng [77] Xiao et al. [79] Proposed model

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.conbuildmat. 2014.08.050.

80

References

60 40 20 0 0.000

D = 55 mm H = 110 mm f'co = 92 MPa f * l = 3.8 MPa

0.010

0.020

0.030

0.040

0.050

Axial Strain (ε c,h)

(b) 350

Axial Stress ( fc ) (MPa)

507

300

D = 55 mm H = 110 mm f'co = 92 MPa f * l = 44.4 MPa

250 200 150 Mander et al. [3] Binici [76] Teng et al. [78] Samani and Attard [24] Test result [72]

100 50 0 0.000

0.010

0.020

0.030

Attard and Setunge [64] Jiang and Teng [77] Xiao et al. [79] Proposed model

0.040

0.050

Axial Strain (ε c,h)

(c) Fig. 25. Comparison of stress–strain curves predicted by existing models: (a) unconfined concrete; (b) lightly-confined concrete; and (c) heavily-confined concrete.

7. Conclusions This paper has presented the results of an investigation into the stress–strain behavior of concrete in compression. Two large databases of experimental results of unconfined and actively confined specimens were assembled through an extensive review of the literature. Based on the unconfined concrete database, a wide range of parameters influencing the compressive behavior of various types of concrete ranging from light-weight to high-strength were carefully studied. This resulted in the development of unified expressions for the predictions of elastic modulus and compressive strength and the corresponding axial strain of various types of concrete. The database was then studied together with the companion actively confined concrete database, in order to capture the change in behavior of unconfined and confined concretes. A unified stress–strain model for the predictions of compressive behavior of

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