Empirical models and design codes in prediction of modulus of elasticity of concrete

doi:10.1007/s11709-018-0479-1

Front. Struct. Civ. Eng.

2019, Vol. 13

Issue (1) : 38-48 https://doi.org/10.1007/s11709-018-0479-1

REVIEW

Empirical models and design codes in prediction of modulus of elasticity of concrete

Behnam VAKHSHOURI(

), Shami NEJADI

Centre for Built Infrastructure Research, University of Technology Sydney, Sydney, Australia

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Abstract

Modulus of Elasticity (MOE) is a key parameter in reinforced concrete design. It represents the stress-strain relationship in the elastic range and is used in the prediction of concrete structures. Out of range estimation of MOE in the existing codes of practice strongly affect the design and performance of the concrete structures. This study includes: (a) evaluation and comparison of the existing analytical models to estimating the MOE in normal strength concrete, and (b) proposing and verifying a new model. In addition, a wide range of experimental databases and empirical models to estimate the MOE from compressive strength and density of concrete are evaluated to verification of the proposed model. The results show underestimation of MOE of conventional concrete in majority of the existing models. Also, considering the consistency between density and mechanical properties of concrete, the predicted MOE in the models including density effect, are more compatible with the experimental results.

Keywords modulus of elasticity normal strength normal weight concrete empirical models design codes compressive strength density

Corresponding Author(s): Behnam VAKHSHOURI

Online First Date: 08 May 2018 Issue Date: 04 January 2019

Cite this article:

Behnam VAKHSHOURI,Shami NEJADI. Empirical models and design codes in prediction of modulus of elasticity of concrete[J]. Front. Struct. Civ. Eng., 2019, 13(1): 38-48.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-018-0479-1
https://academic.hep.com.cn/fsce/EN/Y2019/V13/I1/38

Fig.1 Stress-strain diagram of concrete and its components

Fig.2 Classification of factors affecting the elastic modulus of concrete [⁸]

Fig.3 Predicted MOE values of Eq. (1) versus experimented values

Fig.4 Comparison between predicted MOE in codes of practice vs. predicted MOE by proposed model.

Fig.5 Comparison between predicted MOE in empirical models vs. predicted MOE by proposed model.

Fig.6 Comparison of experimental vs. predicted MOE in proposed model and best matching models in codes of practice

Fig.7 Comparison of experimental vs. predicted MOE in proposed model and best matching empirical models

Design code	Model	Limits and coefficients
ACI 318- 08 [³⁸]	$E c = 0.043 w 1.5 (f c') 0.5$ MPa, kg/m³ $E c = 4730 (f c') 0.5$ MPa	1440<w<2500 kg/m³
Modified ACI 318-95 [⁵⁴]	$E c = 30.16 w 1.5 (f c') 0.5 + 484200$ E in psi; $f c'$ in psi; w in pcf
ACI 363R-92, (1997)[⁵⁵]	$E c = 3320 (f c') 0.5 + 6890$	21< $f c'$ ?<83 MPa
ACI-209-2R-08[⁵⁶]	$E c = 0.043 w 1.5 (f c') 0.5$ MPa, kg/m³ $E c = 4730 (f c') 0.5$ MPa	for 1440<w<2500 kg/m³
ACI 312-92 [⁸]	E=14000+3250 $f c'$ ^0.5 E=9500( $f c'$ +8)^1/3 E=0.043w^1.5 $f c'$ ^0.5MPa
CSA A23.3-04[⁴⁸]	$E c = (3300 (f c') 0.5 + 6900) (w 2300) 1.5$ MPa $E c = 4500 (f c') 0.5$	20≤ $f c'$ ≤40 MPa 1500<w<2500 kg/m³
CAN A23.3- M94[³⁹]	$E c = 5 (f c') 0.5$ GPa
NZS-3101-95[⁵⁷]	$E c = (3320 (f c') 0.5 + 6900) (w 2300) 1.5$ MPa
CEB-FIP (1993) [⁴⁴]	$E c = 10000 (f c' + 8) 13$
CEB-FIP (1990) [⁵⁸]	$E c = 21500 α (f c' 10) 13$	α=1.2 basalt, dense limestone, 1= quartzite , 0.9 limestone, 0.7 sandstone aggregate
EC2-04 (2004) [⁵⁹]	$E c = 22000 (f c' 10) 0.3$ MPa	sc between 0 and 0.4f_cm
AASHTO-LRFD [⁴⁵]	$E c = 0.043 k 1 w c 1.5 (f c') 0.5$ kg/m³ and MPa K₁?= 1.0 unless determined by physical test	1280<w<2400 kg/m³ 14≤ $f c'$ ≤48 MPa
FHWA (2000) [⁴²]	$E c = 3837 (f c') 0.5$	28≤ $f c'$ ≤193 MPa
OHBDC-1983 [⁴⁶]	$E c = 5000 (f c') 0.5$ psi
NCHRP-2003 [⁴³]	$E c = 33000 k 1 k 2 (0.14 + f c' / 1000) 1.5 (f c') 0.5$ ksi	k₁ =1.0, k₂ =90th percentile upper bound and the 10th percentile lower bound 2320<w<2480 kg/m³
AS-3600(2009) [⁵¹]	$E c = 0.043 w c 1.5 (f c') 0.5$ $E c = 5050 (f c') 0.5$ normal weight concrete	<40 MPa, sc between 0 and 0.4f_cm
AS-3600(2009) [⁵¹]	$E c = w c 1.5 (0.024 (f c') 0.5 + 0.12)$	$f c'$ >40 MPa
JSCE (2007) [³⁶]	$E c = 4700 (f c') 0.5$ (ACI-318-08) E_c= 10.792ln( $f c'$ ) - 9.0675 best fit	18≤ $f c'$ ≤80 MPa
NS-3473(1992) [⁶⁰]	$E c = 9.5 (f c') 0.3 (W 2400) 1.5$ GPa, kg/m³ $E c = 9500 (f c') 0.3$
EHE (1998) [⁶¹]	$E c = 10, 000 f c' 3$
NBR-6118 (2003) [⁶²]	$E c = 5600 (f c') 0.5$
(AIJ- Japan) [⁴⁷]	$E c = 2.1 × 105 (w 2.3) 1.5 (f c' 200) 0.5$ E, f’c: kgf/cm^{2 ,}w= t/m³
TS-500 (2000) [⁶³]	$E c = 3.25 (f c') 0.5 + 14$
IS 456 (BIS, 2000) [⁵²]	$E c = 5000 (f c') 0.5$
GBJ 11-89 (1994) [¹]	$E c = 102 / [2.2 + (34.7 f c')]$
IDC 3274 [¹]	$E c = 5.7 (f c') 0.5$
GDC 2000 [¹]	$E c = 4.76 (f c') 0.5$
SABS-0100 (1992) Modified [⁴]	$E c (G P a) = K 0 + a f c u$	K₀ (GPa) = 17 (ferro quartzite); 20 ( Jukskei granite); 29 (Eikenhof andesit α (GPa/MPa) = 0.4 (ferro quartzite); 0.2 ( Jukskei granite and Eikenhof andesit
NTE E.060(2009) [⁴⁹]	$E c = 0.043 w 1.5 (f c') 0.5$ MPa, kg/m³ $E c = 4730 (f c') 0.5$ MPa
SP 52-101-2003 [¹⁴]	Ec= 11.652ln( $f c'$ ) - 7.4713	10≤ $f c'$ ≤60 MPa
BS 5400-4(1990) [³⁷]	$E c = 8.6475 (f c') 0.348$	20≤ $f c'$ ≤60 MPa
BS 8110 (1997)[⁵⁰]	$E c = 20 + 0.2 f c'$	20≤ $f c'$ ≤60 MPa
Dutch VBC-95 [⁴⁰]	$E c = 22250 + 250 f c'$ MPa
RakMK-D3-2012 [⁵³]	$E c = 5000 (w 2400) + (f c') 0.5$

Tab.1 Existing models in codes of practice to predict MOE of normal strength concrete

f’c (MPa)	10	15	20	25	30	35	40	45	50	55	60
E (GPa)	19	24	27.5	39	32.5	34.5	36	37	38	39	39.5
Best fit	E_c= 11.652ln( $f c'$ ) – 7.4713 R²= 0.9968 best fit

Tab.2 The best fitting equation with the CS-MOE data in SP 52-101(2003) [¹⁴]

f’c (MPa)	18	24	30	40	50	60	70	80
E (GPa)	22	25	28	31	33	35	37	38
Best fit	E_c = 4700( $f c' 0.5$ By ACI-318- R²=0.984
Best fit	E_c = 10.792ln( $f c'$ ) - 9.0675 Best fit R² = 0.9983

Tab.3 The best fitting equation with the CS-MOE data in JSCE (2007) [³⁶]

$f c'$ (MPa)	20	25	30	35	40	60
E (GPa)	25	26	28	31	34	36
Best fit	$E c = 8.6475 (f c') 0.348, R 2 = 0.992$ , best fit

Tab.4 The best fitting equation with the CS-MOE data in BS 5400-4(1990) [³⁷]

Tab.5 Classification of international design codes using the same MOE model

Researchers(s)	Model	Limits and coefficients
Carrasquillo, et al.(1981) [⁶⁴]	$E c = 3.320 (f c') 0.5 + 6.900 (w 2346)$ GPa , kg/m³ $E c = 3.320 (f c') 0.5 + 6.900$
Dinakar (2008) [⁶⁵]	$E c = 4.55 (f c') 0.5$ in fsp
Yanjun Liu (2006) [¹⁷]	$E c = α (f c') 0.5$ E_c , $f c'$ in psi	α=55949 Miami oolite limestone, 62721 for Georgia granite , 43777 for stalite lightweight aggregate
Rashid et al. (2002) [¹⁶]	$E c = 8900 (f c') 0.33$	20< $f c'$ <130 MPa
Kheder and Al-Windawi (2005) [⁶]	$E c = 5.323 (f c') 0.453$ MPa, GPa
Soleymani (2006) [⁶⁶]	$E c = (w c 2300) 1.5 (3000 (f c') 0.5 + 6900)$ $E c = (w c 2300) 1.5 (3000 (f c') 0.5 + 6900)$ $E c = 4500 (f c') 0.5$	-Best fit with eq. 8-6 in CSA-A23.3(1994) -Best fit with eq. 8-6 in CSA-23.3(1994) -worst fit with eq. 8-7 in CSA-A23.3(1994)
Ravindrarajah et al. (1985) [⁶⁷]	$E c = 4.630 (f c') 0.5$
San Luis Obispo (2011) [⁶⁸]	$E c = 6.59 (f c') 0.38$
Noguchi et al. (2009) [³]	MPa, kg/m³	40≤ $f c'$ ≤160 MPa
Haranki (2009) [⁵]	$E c = 31.92 × w 1.5 (f c') 0.5 + 345, 328$ psi , lb/ft³
Gardner and Zhao(1991) [⁶⁹]	$E c = 9 (f c') 13$	$f c'$ >27 MPa
Ahmad and Shah (1985) [⁷⁰]	$E c = 3.38 × 10 − 5 × λ 2.5 (f c') 0.65$
Jobse and Mustafa (1984) [⁷¹]	$E c = 0.103 w c 1.5 (f c') 0.5$
Cook (1989) [⁷²]	$E c = 3.22 × 10 − 5 × λ 2.5 (f c') 0.315$ kg/m^{3 ,}MPa
Gutierrez and Canovas (1995) [⁷³]	$E c = 8430 f c' 3$
Leemann and Hoffmann (2005)[²⁶]	$E c = 5480 (f c') 0.5$
Min and Gjorv (1991) [⁷]	$E c = 1.19 (f c') 2 / 3$ GPa , MPa	For light-weigth concrete
Levtchitch et el. (2004) [¹⁵]	$E c = 10000 (1.9 + 0.45 f c')$ MPa
Jensen (1943) [⁵²]	$E c = 6 × 106 / (1 + 2000 / f c')$ psi	1280<w<2400 kg/m³ 14≤ $f c'$ ≤48 MPa
Pauw (1960) [⁴¹]	$E c = 13.82 w 1.79 (f c') 0.44$

Tab.6 Empirical models to predict the modulus of elasticity of normal strength concrete

Tab.7 Coefficient of correlation factor for MOE (models in codes of practice)

Tab.8 Coefficient of correlation factor for MOE (Empirical models)

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