evaluating the dynamic elastic modulus of concrete using...

Research ArticleEvaluating the Dynamic Elastic Modulus of Concrete UsingShear-Wave Velocity Measurements

Byung Jae Lee,1 Seong-Hoon Kee,2 Taekeun Oh,3 and Yun-Yong Kim4

1R&D Center, JNTINC Co. Ltd., 9 Hyundaikia-ro, 830 Beon-gil, Bibong-Myeon, Hwaseong, Gyeonggi-do 18284, Republic of Korea2Department of Architectural Engineering, Dong-A University, 37 Nakdong-Daero, 550 Beon-gil, Saha-gu,Busan 49315, Republic of Korea3Department of Safety Engineering, Incheon National University, 119 Academy-ro, Yeonsu-gu, Incheon 22012, Republic of Korea4Department of Civil Engineering, Chungnam National University, 99 Daehak-ro, Yuseong-gu, Daejeon 34134, Republic of Korea

Correspondence should be addressed to Seong-Hoon Kee; [email protected]

Received 4 January 2017; Revised 27 April 2017; Accepted 2 May 2017; Published 24 July 2017

Academic Editor: Giorgio Pia

Copyright © 2017 Byung Jae Lee et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The objectives of this study are to investigate the relationship between static and dynamic elastic moduli determined usingshear-wave velocity measurements and to demonstrate the practical potential of the shear-wave velocity method for in situdynamic modulus evaluation. Three hundred 150 by 300mm concrete cylinders were prepared from three different mixtureswith target compressive strengths of 30, 35, and 40MPa. Static and dynamic tests were performed at 4, 7, 14, and 28 days toevaluate the compressive strength and the static and dynamic moduli of the cylinders. The results obtained from the shear-wavevelocity measurements were compared with dynamic moduli obtained from standard test methods (P-wave velocity measurementsaccording to ASTM C597/C597M-16 and fundamental longitudinal and transverse resonance tests according to ASTM C215-14).The shear-wave velocity measured from cylinders showed excellent repeatability with a coefficient of variation (COV) less than1%, which is as good as that of the standard test methods. The relationship between the dynamic elastic modulus based on shear-wave velocity and the chord elastic modulus according to ASTM C469/C469M was established. Furthermore, the best-fit line forthe shear-wave velocity was also demonstrated to be effective for estimating compressive strength using an empirical relationshipbetween compressive strength and static elastic modulus.

1. Introduction

In the design of structures, the elastic modulus of concrete(𝐸𝑐) is a fundamental parameter in estimating the defor-mation of a structural element under service conditions. Inpractice, 𝐸𝑐 has been estimated from compressive strengthbased on the design code rather than on direct measurement.This practice could underestimate 𝐸𝑐 and demand highercompressive strength to achieve a desired 𝐸𝑐 than is actuallyrequired in structural design [1]. Furthermore, 𝐸𝑐 has beendemonstrated to be an effective parameter for conditionassessment of concrete in existing structures [2, 3]. Destruc-tive tests such as core extraction have been used widely toacquire accurate information on elastic properties of con-crete. However, they are labor-intensive and time-consuming

and cannot be applied ubiquitously over the entire area of thestructure.

There are several nondestructive evaluation (NDE)meth-ods to estimate elastic properties of concrete includingultrasonic pulse velocity measurements according to ASTMC597/C597M-16 [4] and resonance frequency tests accordingto ASTM C215-14 [5]. The elastic modulus determined byNDE methods is typically called the dynamic elastic mod-ulus, which is generally greater than static elastic modulusmeasured in accordance with ASTM C469/C469M-14 [6].Dynamic modulus has been assumed to be initial tangentmodulus at zero stress determined in the standard testbecause only negligible stress is applied during ultrasonicpulse velocity (UPV) measurements and resonance fre-quency tests [7]. Philleo [8] also explained that the difference

HindawiAdvances in Materials Science and EngineeringVolume 2017, Article ID 1651753, 13 pageshttps://doi.org/10.1155/2017/1651753

https://doi.org/10.1155/2017/1651753

2 Advances in Materials Science and Engineering

between dynamic and static moduli is based on the factthat the nonhomogeneous characteristics of concrete affectthe two moduli in different ways. Moreover, the differencebetween the two moduli decreases as concrete strengthincreases: the dynamic elastic modulus is generally 20, 30,and 40% higher than the static elastic modulus for high-,medium-, and low-strength concrete, respectively [2]. Lydonand Balendran [9] proposed an empirical relationship asfollows:

𝐸𝑐 = 0.83𝐸𝑑 (GPa) . (1)

The British testing standard BS8100 Part 2 [10] providesanother empirical equation as follows:

𝐸𝑐 = 1.25𝐸𝑑 − 19 (GPa) . (2)

It is noteworthy that this equation does not apply to concretecontaining more than 500 kg of cement per cubic meter ofconcrete or to lightweight aggregate concrete [7]. A moregeneral relationship was proposed by Popovics [11] for bothlightweight and normal density concrete, taking into accountthe effect of concrete density as follows:

𝐸𝑐 = 446.09𝐸𝑑1.4𝜌𝑐 (GPa) , (3)

where 𝜌𝑐 is the density of the hardened concrete in a unit ofkg/m3.

The two standard NDE methods for evaluation ofdynamic elastic modulus have advantages and limitations.TheP-wave velocitymethod is convenient to use and has clearadvantages over the resonance method in that the testingis not confined to regularly shaped laboratory specimensand results are not sensitive to inelastic effect [8]. P-wavesmove particles parallel to the direction of propagation andcan propagate through media with enough lateral stiffness:they thus can propagate in solid, liquid, and gas media.Consequently, P-wave velocity in porous materials is affectedby the properties of three difference phases in the media.Dynamic elastic modulus determined by P-wave velocitymeasurements has been known to possess the characteristicsof concrete, heterogeneous, and porousmaterial, such as typeof aggregate, water content, air voids, and porosity comparedwith the static elastic modulus. In addition, P-wave velocityvalues depend on confinement conditions in solid media:the P-wave velocities in unbounded solid media (𝐶P), thinplates (𝐶𝑐,plate), and thin rods (𝐶𝑐,rod) are called constrained,partially constrained, and unconstrained compressive wavevelocities, respectively, and are expressed as the followingequations [12]:

𝐶P = √ 𝐸 (1 − V)𝜌 (1 + V) (1 − 2V) , (4)

𝐶𝑐,plate = √ 𝐸𝜌 (1 − V2) , (5)

𝐶𝑐,rod = √𝐸𝜌 . (6)

0 0.1 0.2 0.3 0.4 0.5Poisson’s ratio [u]

0

1

2

3

4

5

6

7

8

For constrained compressive waves

For unconstrained compressive wavesFor shear waves

For partially constrained compressive waves

Wav

e velo

city

nor

mal

ized

byVs

VP/VS

Vs/VsVc,plate/Vs

Vc,rod/Vs

Figure 1: The variations of constrained, partially constrained,and unconstrained compressive waves and shear waves in elastichomogeneous and isotropic media with Poisson’s ratio. The wavevelocities are normalized by the S-wave velocity.

Figure 1 shows the variations of P-waves with Poisson’s ratiofor elastic homogeneous and isotropicmedia. Given Poisson’sratio, P-wave velocity is in a range between unconstrainedand constrained velocities [13]. P-wave velocity increaseswithincreasing the degree of confinement, and the confinementeffect increases as Poisson’s ratio increases. In the field prac-tice, however, it is sometimes not easy to evaluate accuratedynamic elastic moduli from P-wave velocity measurementsbecause of difficulties in estimating the confinement effect inthe field and the absence of theoretical or practical formularelating P-wave velocity and dynamic elastic modulus thattakes into account the confinement effect. In this sense, someof previous researchers argue that the P-wave velocitymethodis nonreliable and not recommended for estimating thedynamic elasticmodulus of concrete [1, 7, 14]. In contrast, res-onance frequency testing has been usedwidely to evaluate thedynamic elastic modulus of concrete, which is less sensitivethan P-wave velocity. However, resonant frequency testingnecessarily needs regularly shaped concrete specimens; thus,in situ application of this method is labor-intensive and time-consuming and is inappropriate for general application overthe entire area of a structure.

There has been some theoretical and experimental evi-dence demonstrating that the shear-wave measurement haspotential to be an effective in situNDEmethod for estimatingsolid stiffness of various cementitious materials (e.g., cementpaste, mortar, and concrete).Theoretically, shear waves moveparticles perpendicular to the direction of propagation andthus can propagate through solid media with enough shearrigidity. Consequently, S-wave propagation is less sensitiveto liquids or gases in porous materials. Previous researchershave demonstrated by experiments in the laboratory that

Advances in Materials Science and Engineering 3

Table 1: Mixture proportions of concrete.

ID Cement type W/B S/A

Unit quantity (kg/m3)

W C S G SCMs Chemical admixture

FA SC AE(binder%)

SP(binder%)

Mix 1Type I

0.45 0.46 259 121 777 934 58 69 0.9 —Mix 2 0.35 0.47 308 166 761 886 81 85 — 1Mix 3 0.3 0.46 357 165 714 868 94 99 — 1Note. SCMs: Supplementary cementitious materials, W: water, C: cement, S: sand, G: gravel, FA: fly ash, SC: slag cement, AE: air-entraining agent, and SP:superplasticizer.

S-waves are less sensitive to water and/or air content and aneffective parameter for evaluating stiffness (or strength) gainof early-age cementitious materials. Voigt et al. [15] showedthat the reflection of the S-wave is closely related to the degreeof interparticle bonding regardless of water content in cementpaste. Zhu et al. [16] demonstrated experimentally that S-wave velocity is not affected by air content in a range of0.2% to 5.2% in cement paste, while P-wave velocity is toosensitive to air content. Liu et al. [17] successfully monitoredsetting and hardening processes ofmortar and concrete usingultrasonic S-wave velocity in the laboratory and verified thatS-wave velocity correlated well with penetration resistancein mortar regardless of test setup and water-to-cement ratio.Carette and Staquet [18] observed that S-wave velocity ismoresensitive to the setting process of cement-based materialsthan P-wave velocity. In addition, unlike P-waves, S-wavesare theoretically not sensitive to confinement conditions ofconcrete [12]. In a homogeneous and isotropic medium, theS-wave velocity is expressed as the following equationwithoutregard to the confinement conditions [12]:

𝐶S = √ 𝐸2𝜌 (1 + V) . (7)

Furthermore, some researchers have observed that S-wavevelocity is amore stable parameter for evaluating compressivestrength of hardened concrete than in P-wave velocity. Anet al. [19]. observed that S-wave velocity is closely correlatedwith compressive strength without regard to the types ofaggregate, curing age, and curing conditions. Ciancio andHelinski [20] demonstrated that the S-wave velocity mea-surement is effective in evaluating compressive strength offiber reinforced shotcrete in field tests owing to its decreasedsensitivity to the presence of pore water. In addition, Lenciset al. [21] observed that S-wave velocity is less sensitive toreinforcing steels in concrete than P-wave velocity. In theresearch, it was demonstrated that the influence of reinforcingsteels is insignificant for S-wave velocitymeasurements in theindirect transmission method when concrete cover is at least40mm, which is valid in most concrete structures in the fieldpractice. Samokrutov et al. [22] demonstrated that using S-waves compared to P-waves reduces the amount of backscat-tering and signal attenuation in the direction parallel to thepropagatingwave in hardened concrete. Recently, Lee andOh[23] measured S-wave velocities using MIRA equipment thatgenerates shear-wave tomography [24] and demonstrated

the effectiveness of shear-wave measurements for conditionassessment of reinforced and prestressed concrete slabs.In the study, it was demonstrated that S-wave velocity isstatisticallymore stable than P-wave velocity because of lowerenergy attenuation and less sensitivity of confinement effects.In summary, the S-wave velocity method has potential as anin situ NDE method for evaluating the elastic modulus andcompressive strength of concrete. However, it is still difficultto draw a general conclusion due to a scarcity of experimentalstudies on evaluating the dynamic elasticmodulus with the S-wave velocity measurement.

The objectives of this study are to investigate the relation-ship between static and dynamic elastic moduli determinedusing S-wave velocity measurements and to demonstrate thepractical potential of the S-wave velocity method for in situdynamic modulus evaluation. In this study, three hundred150 by 300mm concrete cylinders were prepared from threedifferent mixtures with target compressive strengths of 30,35, and 40MPa. Static and dynamic tests were performedat 4, 7, 14, and 28 days to evaluate the compressive strengthand the static and dynamic moduli of the cylinders. Theresults obtained from the S-wave velocity measurementswere compared with dynamic moduli obtained from stan-dard test methods (P-wave velocity measurements in ASTMC597/C597M-16 and fundamental longitudinal and trans-verse resonance tests in ASTM C215-14).

2. Experimental Program

2.1. Materials and Preparation of Specimens. The concrete inthis study consists of Type I Portland cement, river sand,crushed granite with a maximum size of 25mm, and sup-plementary cementitious materials (fly ash and slag cement).Three different water-to-binder ratios (W/B) of 0.3, 0.35,and 0.45 were used, which results in target compressivestrengths of 40, 35, and 30MPa, respectively. The threedifferent concretemixtures (hereafter referred to asmix 1,mix2, and mix 3) were prepared with ready-mixed concrete, andthe proportions are summarized in Table 1.

Three hundred concrete cylinders were cast in 150 by300mm plastic molds in accordance with ASTM C31/C31M[25].The cylinders were water-cured after being demolded onthe next day. A series of static and dynamic modulus tests for25 cylinders were conducted at different ages: 4, 7, 14, and 28days.


Figure 2: Test setup for measuring compressive strength and staticelastic modulus of a concrete cylinder.

2.2. Static Tests for Compressive Strength and Elastic Modulus.The cylinders were ground at both ends before testing toremove any surface irregularities as well as to ensure that theends were perpendicular to the sides of the specimen. Staticelastic modulus and compressive strength of the cylindersweremeasured using a universal testingmachine (UTM)witha capacity of 1000 kN according to ASTM C469/C469M-14[6] and ASTM C39/C39M-14 [26], respectively. Tests wereperformed at a loading rate of approximately 0.28MPa/s.Deformations were measured using three sets of linearvoltage differential transducers attached to two fixed rings(Figure 2). The apparatus consisted of two aluminum ringswith screws for attachment to the specimen. The spacingbetween screws on the top and bottom rings was 150mm.This served as a gauge length for calculating axial strain fromthe measured deformations. The static elastic modulus ofconcrete is defined as a chord modulus from the stress-straincurve, with a first point at a strain level of 0.00005 (𝜀1) andthe second point at 40% of maximum stress (𝑓𝑐) as follows:

𝐸𝑐 = 0.4𝑓𝑐 − 𝜎 (𝜀1)𝜀 (0.4𝑓𝑐) − 𝜀1 . (8)

It is of interest to compare dynamic elastic modulus andinitial tangent modulus, which corresponds to a small strainand is calculated from the tangent drawn at the origin of thestress-strain curve [27]. Unfortunately, the measured stress-strain curve of cylinders in this study always showed certainfluctuations at very low strain, which posed considerabledifficulties in consistent and reliable calculation of the initialtangent modulus. Instead, an initial chord modulus wasdefined from two points corresponding to strain levels of0.000001 (𝜀1) and 0.00005 (𝜀2) as follows:

𝐸𝑖 = 𝜎 (𝜀2) − 𝜎 (𝜀1)𝜀2 − 𝜀1 . (9)

Note that typical stress-strain curves of cylinders in this studyshow that stress increases almost linearly with strain at very

low values of strain, that is, approximately 0.00005.Therefore,the initial chord modulus is assumed to be equivalent to theinitial tangent modulus of cylinders in this study.

2.3. P- and S-Wave Measurements for Estimating DynamicElastic Modulus. The S-wave velocity of concrete, 𝐶S, wasmeasured using the P-wave velocity procedure described inASTMC597/C597M-16 [4] with a pair of S-wave transducers(ACS T1802) that were connected to a pulser-receiver (Ultra-con 170) (see Figure 3(a)). The source transducer was drivenby a 200V square pulse, generating a transverse ultrasonicpulse. The signals generated by the source transducer andmeasured by the receiving transducer were digitized at asampling frequency of 10MHz using an NI-PXIe 6366 oscil-loscope. Figure 4(a) shows typical impulse signal generatedby a source transducer and an ultrasonic wave measured bya receiving transducer for the S-wave velocity measurement.The arrival of the transient stress waves through cylinderswas determined from themeasured ultrasonic wave using themodified threshold method [28]. In this method, an approx-imate arrival time was first obtained using the conventionalthreshold method in the literature [29]. Next, an accuratearrival time was calculated by fitting a line to the single data.The intersection of the fitting line and the calculated zero-signal level defines the S-wave travel time. Finally, the S-wavevelocity was calculated by dividing the length of cylinders (𝐿)over the travel time (𝑡) (i.e., 𝐶s = L/t). In this study, dynamicelastic modulus based on𝐶S is determined by rearranging (7)as follows:

𝐸𝑑,S = 𝛼S𝜌𝐶S2, (10)

where 𝐸𝑑,S is dynamic elastic modulus of the concretespecimen using S-wave velocitymeasurements, 𝛼S is constantdependent on Poisson’s ratio v, that is, 2(1 + V), and 𝜌 ismass density of concrete (=𝑚/𝜋𝑟2𝐿;𝑚, 𝑟, and 𝐿 are the mass,radius, and length of the cylinder, respectively, measured attesting day).

For comparison purposes, dynamic elastic moduli ofconcrete cylinders were also determined using standardNDEmethods, including P-wave velocity and resonant frequencytests. In the P-wave velocity test, a longitudinal pulse of52 kHz is transmitted through the specimen, and the P-wavevelocity of concrete, 𝐶P, is measured in accordance withASTMC597/C597M-16 [4].The test setup for P-wave velocitymeasurements was the same as that for the S-wave velocitymeasurements except for using a pair of P-wave transducers(MK954). Figure 4(b) shows typical impulse signal generatedby a source transducer and an ultrasonic wave measured bya receiving transducer for the P-wave velocity measurement.The dynamic elastic modulus based on 𝐶P is determined byrearranging (4) as follows:

𝐸𝑑,P = 𝛼P𝜌𝐶P2, (11)

where 𝐸𝑑,P is dynamic elastic modulus of the cylinderusing the P-wave velocity measurements and 𝛼P is constantdependent on Poisson’s ratio 𝜐, that is, (1 + 𝜐)(1 − 2𝜐)/(1 − 𝜐).


(a) (b)

Figure 3: Testing configurations for (a) ultrasonic shear-wave velocity using a pair of S-wave transducers (ACS T1802) and (b) resonancefrequency of concrete specimens.

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006Time (s)

−1

−0.5

0

0.5

1

1.5

Am

plitu

de (V

)

First arrivalof S-wave

A square pulse generated bya source transducer

A signal measured bya receiving transducer

(a)

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006Time (s)

−2

−1

0

1

2

3

Am

plitu

de (V

)

First arrivalof P-wave

A square pulse generated bya source transducer

A signal measured bya receiver transducer

(b)

Figure 4: Typical impulse signals generated by a source transducer and ultrasonic waves obtained from ultrasonic pulse velocity tests for (a)S-wave and (b) P-wave measurements.

2.4. Resonance Frequency Measurements for Estimating Dy-namic ElasticModulus. Fundamental longitudinal and trans-verse resonant frequencies of the cylinders were measured toestimate dynamic elastic moduli in accordance with ASTMC215-14 [5] (see Figure 3(b)). A steel ball having a diameter of10mm was used as an impact source for generating incidentstress waves in concrete specimens; the steel ball was effectivefor generating wideband frequency signals from very low to20 kHz, covering the frequency range of resonant frequencytests in this study. The dynamic response of the concretecylinder was measured by an accelerometer (PCB 353B16),with ±5% frequency range of 1 to 10 kHz and resonantfrequency around 70 kHz, attached to the concrete specimenaccording to ASTM C215-14 [5]. The acquired signals usingthe accelerometer were stabilized using a signal conditioner(PCB 482C16) and digitized at a sampling frequency of 1MHzusing an NI-PXIe 6366 oscilloscope. Resulting time signalswere converted into the frequency domain using the FFT (fastFourier transform) algorithm. Figure 5 shows typical spectralamplitude from the mix 2 concrete cylinders with ages of4, 7, 14, and 28. The resonant frequencies of the cylindersare manifested as dominant amplitudes in the amplitudespectrum. The most dominant frequency was regarded asthe fundamental resonant frequency for longitudinal 𝑓L andtransverse 𝑓T modes, and the frequency values were used tocalculate dynamic elastic moduli of the cylinders as follows:

𝐸𝑑,LR = 𝛽L𝑚𝑓L2 (Pa) ,

𝐸𝑑,TR = 𝛽T𝑚𝑓T2 (Pa) ,(12)

where 𝐸𝑑,LR and 𝐸𝑑,TR are dynamic elastic moduli using lon-gitudinal and transverse fundamental resonant frequencies,respectively; 𝛽𝐿 is a constant depending on the dimensionsof the cylinder =5.093(𝐿/𝐷2); 𝐿 and 𝐷 are the lengthand diameter of a cylinder, respectively; 𝛽𝑇 is a constantdepending on dimensions and Poisson’s ratio of cylinder(=1.6067(𝐿3𝑇/𝐷4)); and 𝑇 is a correction factor obtainedfrom ASTM C215-14[5].

3. Result and Discussion

3.1. Experimental Variability of Static and Dynamic Tests.Experimental variability of the measured static and dynamicparameters is of interest when investigating the consistencyand reliability of the test methods. Uncertainty of the averagevalue of the in-place test results is a function of the standarddeviation of the results and the number of tests. In this study,the coefficient of variation (COV, the standard deviation, 𝜎,divided by the mean value, 𝜇, of a set of specimens) was usedas a means of evaluating the experimental variability of thestatic and dynamic measurements of the concrete cylinders;outliers were detected by the modified 𝑍-score method [30]and were removed in the statistical analysis. The statisticalparameters (𝜇 and COV) of the accepted static and dynamictest results are summarized in Table 2.


Table 2: Summary of statistical analysis.

Day 4 Day 7 Day 14 Day 28𝑁 𝜇 COV 𝑁 𝜇 COV 𝑁 𝜇 COV 𝑁 𝜇 COV

𝑓𝑐 [MPa]Mix 1 25 7.9 4.36 25 10.1 4.32 25 13.9 3.42 25 19.2 4.25Mix 2 24 25.1 2.94 25 30.0 3.06 24 37.1 4.23 25 43.1 4.87Mix 3 24 23.4 2.83 25 29.3 2.62 25 39.4 4.32 25 44.8 3.63

𝐸𝑐 [GPa]Mix 1 24 10.1 6.13 25 11.8 4.62 25 14.5 5.31 25 16.9 5.39Mix 2 25 16.5 5.35 24 17.7 4.34 25 20.4 3.86 25 23.5 4.57Mix 3 24 16.8 5.44 24 18.7 4.23 24 21.5 4.02 24 23.0 3.57

𝐶p [m/s]Mix 1 25 3225 2.49 25 3474 1.81 25 3740 1.68 25 3958 1.34Mix 2 25 3948 1.51 25 4064 1.41 25 4197 1.07 25 4283 1.03Mix 3 25 3894 1.45 25 4055 1.07 25 4195 0.97 25 4302 0.82

𝐶s [m/s)Mix 1 — — — 22 1664 1.15 25 1790 1.09 25 1936 1.32Mix 2 — — — 20 2015 1.00 25 2104 0.83 25 2165 0.76Mix 3 25 1942 0.93 25 2039 0.90 25 2138 0.90 25 2191 0.87

𝑓L1 [Hz]Mix 1 25 4596 1.56 25 4636 1.37 25 5248 1.21 25 5616 1.25Mix 2 25 5602 1.51 25 5752 0.98 25 5990 0.83 25 6178 0.87Mix 3 25 5580 1.29 25 5818 1.04 25 6064 1.22 25 6246 1.13

𝑓T1 [Hz]Mix 1 25 3346 3.28 25 3208 1.25 25 3232 1.25 25 3442 1.08Mix 2 25 3428 1.89 25 3490 1.49 25 3622 1.32 25 3778 1.32Mix 3 25 3412 1.35 25 3532 1.14 25 3678 1.52 25 3796 1.36

Note.𝑁 = the number of accepted samples used for the statistical analyses; 𝜇 = average of samples; and COV = coefficient of variable (=𝜎/𝜇 × 100 [%]).

0 4000 8000 12000 16000 20000Frequency (Hz)

Nor

mal

ized

spec

tral

ampl

itude Fundamental 2nd mode 3rd mode

4 days

7 days

14 days

28 days

mode

(a)

0 4000 8000 12000 16000 20000Frequency (Hz)

Nor

mal

ized

spec

tral

ampl

itude

Fundamental mode4 days

7 days

14 days

28 days

(b)

Figure 5: Typical spectral amplitude measured from the mix 2 con-crete cylinders with ages of 4, 7, 14, and 28 days using (a) longitudinaland (b) transverse resonance frequency tests.

The COV of the compressive strength of concrete 𝑓𝑐ranges from 2.62% to 4.97% for concrete specimens withdifferent mixture proportions and testing ages, which arebetween good (4.0 to 5.0) and excellent (<3.0) categories,according to ACI 214R [31]. The COV of the static elasticmodulus 𝐸𝑐 ranges from 3.57% to 6.13%. A slightly higherCOV and more rejected observations in 𝐸𝑐 are mainly dueto the fact that the measurement of 𝐸𝑐 involves more sourcesof variation than the measurement of 𝑓𝑐. The COVs of both𝑓𝑐 and 𝐸𝑐 are reasonably consistent at the different ages of 4,7, 14, and 28 days.

The COV of the P-wave velocity (𝐶P) ranges from 0.82%to 2.49%, which is consistent with observations by ACI com-mittee 228 [32]. Furthermore, the COV of the fundamentallongitudinal 𝑓L and transverse 𝑓T frequencies ranges from0.83% to 1.56% and from 1.08% to 3.28%, respectively. Nooutlier was detected by the modified 𝑍-score for the threedata sets 𝐶P, 𝑓L, and 𝑓T. The low COVs and lack of detectedoutliers show the excellent consistency of 𝐶P, 𝑓L, and 𝑓T.

Furthermore, the COV of the accepted S-wave velocity(𝐶S) ranges from 0.76% to 1.32%, which is comparable to thatof 𝐶P, 𝑓L, or 𝑓T. This statistical result clearly shows that S-wave velocity measurements have excellent consistency andcould be a consistent and reliable test method in practice.However, in early-age concrete, accurate detection of thefirst arrival time of S-waves is often difficult because of theinterference between direct P- and S-waves; even using S-wave transducers, P-wave components with low amplitudealways appear along with the S-wave components in the time


0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Dynamic elastic modulus Ed,LR (GPa)

Best-�t line (a = 0.34,

Stat

ic el

astic

mod

ulus

Ec

(GPa

)

Test results: mix 1Test results: mix 2Test results: mix 3Line of equality

Lydon and Balendran [1986]BS8110 part 2Popovics [1975]

b = 1.23, r2 = 0.93)

(a)

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40


Lydon and Balendran [1986]BS8110 part 2Popovics [1975]Best-�t line (a = 0.33,b = 1.24, r2 = 0.93)

Stat

ic el

astic

mod

ulus

Ec

(GPa

)

Dynamic elastic modulus Ed,TR (GPa)

(b)

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Stat

ic el

astic

mod

ulus

Ec

(GPa

)

Best-�t line (a = 0.16,b = 1.36, r2 = 0.9)



Dynamic elastic modulus Ed,P (GPa)

(c)

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Best-�t line (a = 0.49,b = 1.22, r2 = 0.9)

Stat

ic el

astic

mod

ulus

Ec

(GPa

)



Dynamic elastic modulus Ed,S (GPa)

(d)

Figure 6: Relationship between static and dynamic elastic moduli: (a) 𝐸𝑐 versus 𝐸𝑑,LR, (b) 𝐸𝑐 versus 𝐸𝑑,TR, (c) 𝐸𝑐 versus 𝐸𝑑,P, and (d) 𝐸𝑐 versus𝐸𝑑,S.

domain.This phenomenonwas substantial in observations ofcylinders with relatively low compressive strength (i.e., mixes1 and 2 at 4 days), and measured observations did not havea normal distribution. Therefore, those data sets were notincluded in the statistical analysis; special care is needed to

measure the S-wave velocity of concrete having relatively lowcompressive strength.

3.2. Relationship between Static and Dynamic Elastic Moduli.Figure 6 presents the relationship between static and dynamic


Table 3: Summary of constants 𝑎 and 𝑏 and 𝑅2 for the best-fit lines in Figure 4.

Relationship 𝑎 𝑏 𝑅2𝐸𝑑,LR − 𝐸𝑐 0.34 1.23 0.93𝐸𝑑,TR − 𝐸𝑐 0.33 1.24 0.93𝐸𝑑,P − 𝐸𝑐 0.16 1.36 0.90𝐸𝑑,S − 𝐸𝑐 0.49 1.22 0.90Note. The basic form of the equation is 𝐸𝑐 = 𝑎𝐸𝑑

𝑏.

elastic modulus determined using different NDE methods,𝐸𝑑,LR, 𝐸𝑑,TR, 𝐸𝑑,P, and 𝐸𝑑,S, with an assumed Poisson’s ratioof 0.2. The use of Poisson’s ratio of 0.2 is reasonable forcommon concrete in practice [7]. The expression relatingstatic to dynamic elastic modulus was obtained by nonlinearregression using the following power function relationship:

𝐸𝑐 = 𝑎𝐸𝑑𝑏, (13)

where 𝐸𝑐 is the chord elastic modulus in GPa in accor-dance with ASTM C469/C460M-14 [6], 𝐸𝑑 is the dynamicelastic modulus in GPa, and 𝑎 and 𝑏 are best-fit constantsaccording to the regression analysis. It has been observedthat the difference between static and dynamic elastic modulidecreases with increasing dynamic elastic modulus. There-fore, the nonlinear behavior was taken into account by usinga power function. Furthermore, (13) results in zero staticelastic modulus for zero dynamic elastic modulus. Table 3summarizes the constants 𝑎 and 𝑏, and 𝑅2 for the four best-fit curves, shown in Figure 6. For comparison, the samefigure presents previously mentioned expressions ((1)–(3))relating the static and dynamic elastic moduli of concrete.Table 4 summarizes mean absolute error (MAE) between themeasured and the predicted static elasticmodulus of concreteusing the four best-fit curves proposed in this study and thethree expressions ((1)–(3)).

The four dynamic moduli (𝐸𝑑,LR, 𝐸𝑑,TR, 𝐸𝑑,P, and 𝐸𝑑,S)obtained from the resonant frequency tests and UPVmethodare greater than the static elastic modulus 𝐸𝑐 (see Figure 6).However, the ratios between static anddynamic elasticmodu-lus vary according to themeasurementmethods. Figures 6(a)and 6(b) show that the 𝐸𝑐/𝐸𝑑,LR (or 𝐸𝑐/𝐸𝑑,TR) ratio, with anaverage of 0.72 and COV of 7.15%, was closer to the line ofequality than 𝐸𝑐/𝐸𝑑,P. In the figure, the equation proposedby Lydon and Balendran [9] slightly overestimates 𝐸𝑐 fromthe two resonance moduli (𝐸𝑑,LR or 𝐸𝑑,TR), with a meanabsolute error (MAE) of 2.63GPa. The equation proposedby the British Standard Institute [10] significantly underes-timates 𝐸𝑐, with a MAE of 10.95. In contrast, the equationproposed by Popovics [11] shows very good agreement withthe experimental results in this study and appears to beeffective for estimating a reasonably accurate 𝐸𝑐 from 𝐸𝑑,LRor 𝐸𝑑,TR. However, as shown in Figure 6(c), the 𝐸𝑐/𝐸𝑑,P ratio,with an average of 0.56 and COV of 9.2%, is too far awayfrom the line of equality as well as from the three proposedequations relating static and dynamic elastic modulus.

Figure 6(d) shows that the ratio 𝐸𝑐/𝐸𝑑,S, with an averageof 0.87 and COV of 6.2%, was even closer to the line of equal-ity than 𝐸𝑐/𝐸𝑑,LR (or 𝐸𝑐/𝐸𝑑,TR). The equations proposed by

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Chor

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Initial chord elastic modulus Ei (GPa)

Figure 7: Relationship between initial chord modulus and standardchord modulus.

British Standard Institute and Popovics resulted in relativelyhigh MAE of 6.36GPa and 3.76GPa, respectively. However,the best-fit curve obtained in this study and the equationsuggested by Lydon andBalendran [9] shows good agreementwith the experimental results in this study, with a relativelylowMAEof 0.82 and 1.19GPa, respectively.Therefore, the twoequations could be a good candidate as a practical formularelating 𝐸𝑑,S and 𝐸𝑐 with reasonable accuracy for normalconcrete with similar mixture proportions in this study.

3.3. Relationship between Initial Chord and Dynamic ElasticModuli. Figure 7 shows the relationship between the initialchord modulus and the standard chord modulus. The major-ity of the data points indicate that the initial chord modulusis greater than the standard chord modulus.This implies thatthe nonlinear stress-strain behavior of concrete is one of thepossible reasons for differences between static and dynamicelastic modulus. It has been believed that the dynamic elasticmodulus represents the tangent elastic modulus (or initialchord modulus in this study) at low strain [7]. Therefore, itis of some interest to compare dynamic elastic modulus toinitial chord modulus to better understand the nonlinearityeffect.


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Figure 8: Relationship between initial chord and dynamic elastic moduli: (a) 𝐸𝑖 versus 𝐸𝑑,LR, (b) 𝐸𝑖 versus 𝐸𝑑,TR, (c) 𝐸𝑖 versus 𝐸𝑑,P, and (d)𝐸𝑖 versus 𝐸𝑑,S.

Figures 8(a)–8(d) compare the initial chord and dynamicelastic moduli from different methods. As expected, thedifference between initial chord and dynamic elastic moduliis smaller than that between standard static and dynamicmoduli. Note thatmore scatter in Figure 8,𝐸𝑖 and𝐸𝑑 relation,

than in Figure 6, 𝐸𝑐 and 𝐸𝑑 relation, is attributable to somefluctuating data measured in a low strain level for calculating𝐸𝑖.

Figure 8(d) shows that 𝐸𝑑,S is close to the line of equality,with an average 𝐸𝑖/𝐸𝑑,S ratio of 1.04 and COV of 11.0%.


Table 4: Mean absolute error (MAE) of expressions relating static and dynamic elastic modulus.

Equation converting 𝐸𝑑 into 𝐸𝑐 MAE (GPa)𝐸𝑑,P 𝐸𝑑,S 𝐸𝑑,LR 𝐸𝑑,TRLine of equality 13.38 2.78 7.06 6.76(1) (Lydon and Balendran [9]) 8.11 1.19 2.87 2.63(2) (British Standard Institute [10]) 19.24 6.36 11.33 10.95(3) (Popovics [11]) 7.41 3.76 1.11 0.97Best-fit line 1.05 0.82 0.84 0.85Note. MAE is mean absolute error (=Σ|𝑌 − ��|/𝑁, where 𝑌 is the predicted values using the three proposed equations and �� is the measured values and𝑁 isthe number of specimens).

Therefore, the difference between static and dynamic elasticmodulus of 𝐸𝑑,S is mainly attributed to the nonlinear stress-strain curve of concrete up to 40% of compressive strength.In contrast, resonance dynamic moduli (𝐸𝑑,LR and 𝐸𝑑,TR),shown in Figures 8(a) and 8(b), still show considerabledifference from 𝐸𝑖. Furthermore, 𝐸𝑑,P shown in Figure 8(c)is considerably higher than 𝐸𝑖. It seems that there are otherinfluential factors causing differences between static anddynamic elastic modulus for 𝐸𝑑,P and the two resonancemoduli (𝐸𝑑,LR and 𝐸𝑑,TR).

It has been suggested that differences between static anddynamic elastic moduli are attributable to the fact that thenonhomogeneous characteristics of concrete affect the twomoduli in different ways [7, 8].The staticmodulus of concretehas been demonstrated to be more dependent on the elasticproperties of the paste, whereas, for P-wave velocity, theelastic modulus of the aggregates is the more influentialfactor. In normal-weight concrete, the elastic modulus ofaggregates is generally greater than the paste, providing areason for higher dynamic modulus compared to the staticmodulus. Similarly, the nonhomogeneity of concrete appearsto affect the fundamental resonance frequencies of concretecylinders. In contrast, the S-wave velocity method is likelythe least sensitive to the heterogeneity of concrete. However,additional experimental data are needed to confirm thishypothesis due to the limited quantity of experimental datain this study and scarcity of experimental studies exploringthe relationship between static modulus and dynamic elasticmodulus based on S-wave measurements.

3.4. Relationship between Compressive Strength and DynamicElasticModulus . Theplot in Figure 9 represents the relation-ship between static elastic modulus and compressive strengthmeasured in accordance with ASTM C469/C469M-14 [6]and ASTM C39/C39M-14 [18], respectively. Figure 9 alsocompares the experimental results with code equations [33–36] and a recently proposed equation by Noguchi et al. [37].TheCEB-FIP and Eurocode 2 link elasticmodulus𝐸𝑐 to com-pressive strength 𝑓𝑐 (𝐸 = 22000(𝑓𝑐/10)1/3 in MPa) for bothnormal-strength and high-strength concrete. ACI proposestwo different equations: 𝐸 = 0.043(𝑓𝑐)1/2(𝑤)1.5MPa for (𝑓𝑐 ≤38MPa) by the ACI 318 committee and𝐸 = (3320(𝑓𝑐/10)1/2+6900)(𝑤/2300)1.5MPa for (21MPa < 𝑓𝑐 < 83MPa) by the

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Static elastic modulus Ec (GPa)

Com

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thfc

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Figure 9: Relationship between compressive strength 𝑓𝑐 and staticelastic modulus 𝐸𝑐.

ACI 363 committee. Noguchi et al. [37] proposed a prac-tical equation based on an extensive experimental databaseincluding normal-strength and high-strength concrete: 𝐸 =𝑘1𝑘233500(𝑓𝑐/60)1/3(𝑤/2400)2, where 𝑘1 and 𝑘2 are correc-tion factors for coarse aggregates and SCMs. Figure 9 showsthat the three code equations significantly underestimatecompressive strength, which is mainly due to the fact thatsome critical factors (e.g., the types of coarse aggregates andSCMs) have not been properly taken into account in thecode equations. However, the Noguchi equation [37] predictsexperimental results with greatly improved accuracy by usingcorrection factors 𝑘1 = 𝑘2 = 0.95.

Figure 10 compares measured and predicted compressivestrength based on different equations relating static and


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Figure 10: Relationship between predicted andmeasured compressive strength determined from various dynamicmoduli: (a)𝐸𝑑,LR, (b)𝐸𝑑,TR,(c) 𝐸𝑑,P, and (d) 𝐸𝑑,S.

dynamic elastic modulus. The Noguchi equation [37] wasused to correlate compressive strength and dynamic elasticmodulus as follows:

𝑓𝑐 = 𝜂 (𝐸𝑐)3 = 𝜂 (𝑎𝐸𝑑𝑏)3 , (14)

where 𝑓𝑐 is estimated concrete strength, 𝜂 = 60(𝑤/2400)−3/2(𝑘1𝑘233500)−3, and 𝑎 and 𝑏 are constants dependingon equations relating static and dynamic elastic modulus(see Table 3). Figure 10 shows that the proposed equation

(14) slightly underestimates the compressive strength of con-crete when converting dynamic into static elastic modulususing the best-fit curves. The MAEs between measuredand estimated values using 𝐸𝑑,P, 𝐸𝑑,S, 𝐸𝑑,LR, and 𝐸𝑑,TR are4.9, 4.2, 4.3, and 4.3MPa, respectively. However, it wasobserved in Figure 10 that the predicted compressive strengthis linearly correlated with measured compressive strength,and the plots are consistently biased toward underestimationof the compressive strength. Consequently, it was found thatthe proposed equation (14) could improve the accuracy of


predicted compressive strength through a simple calibrationprocess of adding a certain correction factor: MAE valueswould be a good candidate for a simple calibration factor.Therefore, it can be concluded that the predetermined equa-tions relating dynamic to static elastic modulus, summarizedin Table 4, are effective for estimating compressive strengthwithin reasonable errors.

4. Conclusions

An experimental study was conducted to explore the rela-tionship between static and dynamic elastic moduli usingultrasonic pulse velocity methods and resonance tests andto demonstrate the potential of the S-wave velocity methodas an in situ dynamic elastic modulus evaluation method inpractice. Conclusions are drawn as follows:

(1) It was demonstrated that the coefficient of variation(COV) of the S-wave shows very good repeatability.It is as good as P-wave velocity measurements andcomparable to resonance tests. The average COVof the S-wave velocity (𝐶S) is 0.97%; the averageCOV of P-wave velocity (𝐶P) is 1.38%; the averageCOV of resonant frequency measurements are 1.19%and 1.35% for longitudinal and transverse resonancefrequencies, respectively.

(2) The four dynamicmoduli (𝐸𝑑,LR,𝐸𝑑,TR,𝐸𝑑,P, and𝐸𝑑,S)obtained from the resonance frequency and ultra-sonic pulse velocity (UPV) tests are greater than thestatic elastic modulus 𝐸𝑐. Test results from this studyand the literature show that the best-fit curve andthe equation suggested by Popovics [11] ((3) in thisstudy) are effective for predicting static elastic modulifrom resonance moduli of concrete, with MAEs of 1.0and 1.2 GPa, respectively. For velocity moduli from S-wave velocity measurements, the best-fit curve andthe equation suggested by Lydon and Balendran [9]((1) in this study) show good agreement with theexperimental results in this study, with MAEs of0.82GPa and 1.19GPa, respectively. However, velocitymodulus from P-wave measurements is excessivelygreater then static elastic modulus. The empiricalequations (see (1), (2), and (3)) result in significanterrors between the static and dynamic moduli, withMAE in a range of 7.41 GPa to 19.24GPa.

(3) 𝐸𝑑,S was observed to be almost linearly correlatedwith 𝐸𝑖, with a mean absolute error of 2GPa. There-fore, it can be concluded that the difference betweenthe initial elastic modulus of concrete and 𝐸𝑑,S ismainly attributable to the nonlinear stress-straincurve of concrete. In contrast, resonance dynamicmoduli (𝐸𝑑,LR and 𝐸𝑑,TR) are slightly higher thanthe initial tangent modulus of concrete 𝐸𝑖. Moreover,𝐸𝑑,P is considerably higher than 𝐸𝑖. There shouldbe other influential factors (such as heterogeneityof concrete) causing differences between static anddynamic elastic modulus for the P-wave modulus(𝐸𝑑,P) and the two resonance moduli (𝐸𝑑,LR and𝐸𝑑,TR).

(4) With an established equation between static and dy-namic elastic modulus, the dynamic elastic moduluscan be used to estimate the compressive strength ofcylinders. It is demonstrated that the predeterminedequations relating dynamic to static elastic modu-lus are effective for estimating compressive strengthwithin reasonable errors.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this manuscript.

Acknowledgments

This research was supported by Basic Science Research Pro-gram through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT & FuturePlanning (no. 2015R1A5A1037548).

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