prediction of the dynamic modulus of elasticity

7/28/2019 Prediction of the Dynamic Modulus of Elasticity

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nnb f = (2)

wherefn = the frequency of the n

th natural moden = the wavelength associated with the n

th natural frequencyn = the mode number.

The wavelength can be expressed in terms of the rod or bar length as:- for a free-free or fixed-fixed boundary condition

n

Ln

2= n=1,2,3,, (3)

- for a fixed-free boundary condition

n

Ln

4= n=1,3,5,, (4)

whereL = the length of rod or bar.

Hence the dynamic modulus can be back calculated by

2bE = (5)

The bar wave velocity, b , is determined by measuring the resonant frequency of the various modes and

substituting in Eq. (2) with the associated wavelength corresponding to the boundary condition that resemblesthe actual condition.

On the other hand, the equation of the lateral vibration of Euler type (i.e. neglecting rotary and shear effects)beams is given by[6]

02

2

4

42

=

+

t

w

x

wc (6)

where

AEIc

=

A = the beams cross-sectional areaI = the moment of inertia of the beams cross-section about its centroid.

It can be shown that the natural frequencies of the beam are [6]:

4

2

2

)(

AL

EILf nn

= (7)

From Eq. (7), the dynamic modulus of elasticity was determined as

I

AL

L

fE

n

n4

4

22

)(

4

= (8)

Values of(nL) for some common boundary conditions for the transverse vibration of beams are given in Table1.

Table 1. Values ofnL for common boundary conditions of beams

Boundary condition 1L 2L

Free-free or fixed-fixed 4.730041 7.853205

Fixed-Free 1.875104 4.694091

For axially loaded prismatic beams, one can estimate the flexural natural frequencies according to the followingequation[7]:

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2

1 )1(0

0nbn

n

P

P

f

f

P

P

+=

=

; n=1,2,3,, (9)


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where

0Pnf = natural frequency in the presence of an axial compressive load

0=Pnf = natural frequency in the absence of an axial compressive load

P = applied axial forceand

2

2 4

L

EIPb = = critical buckling load (10)

It should be noted that the first two natural frequencies of the beam are insignificantly influenced by the axialcritical load.

3 Experimental Program

3.1 Materials and mix proportions

The materials used for casting the test specimens comprised of Type I cement; crushed limestone coarseaggregate; and natural desert quartz fine aggregate. The water-cement ratio (w/c) considered in this study was

0.6. The compressive strength of the mix used was approximately 25 MPa. Details of the mix proportions aregiven in Table 2.

Table 2. Mix proportions per cubic meter of concrete

Material Quantity

Cement, kg 282

Water-cement ratio 0.6

3/4 Coarse aggregate, kg 699

3/8 Coarse aggregate, kg 376

Fine aggregate, kg 785

3.2 Casting and curing

For this work, standard and special cylinders were constructed. For compressive strength, three standard 150 x300 mm cylinders were prepared and tested. Four special cylinders (150 mm in diameter and 600 mm in length)were cast of which two were plain and the other two were reinforced. The reinforcement was provided by asingle 12 mm in diameter concentrically placed bar with 20 mm top and bottom covers. The concrete batchingand testing were conducted under laboratory conditions. The actual dimensions and weights of these specialcylinders were measured. The density of individual cylinders were calculated and presented in Table 3.

Table 3. Designations, dimensions, and densities of the special concrete cylinders

Cylinder Diameter,mm

Length,mm

Mass,kg

Density,kg/m3

NS-P-1/2 151.5 610 26.3 2390

NS-P-2/2 151.8 611 26.7 2393

NS-RC-1/2 151.2 610 26.7 2308

NS-RC-2/2 151.6 610 26.7 2425

Curing for all specimens started just after de-molding (24 hr after casting). All specimens were stored in thelaboratory under potable water (standard curing method) for 28 days. Specimens were always kept underidentical moisture conditions, since it is expected that moisture condition of concrete specimens would affect themeasurements of the dynamic modulus.

3.3 Dynamic testing setup

The experimental set-up used to conduct the measurement of the resonant frequencies of the specimens is shown

in Fig. 1. Accelerometers, with 250 mv/g sensitivity, are used to pick up the dynamic signals. The acquired timedomain analog signal is then amplified and passed to a low pass filter with a 10 KHz cut-off frequency. Thesignal is then fed to 12-bit analog to digital converter for digitisation at a sampling frequency of 5000 Hz.


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repeatability of the results. The fundamental frequency is employed in Eq. (8) to calculate the dynamic modulusof elasticity as reported in Table 4.

Table 4. Frequencies and moduli of concrete for free-free test, NS-P specimens

Cylinder Fundamental

Frequency,Hz

Dynamic

Modulus Ed,GPa

Static

Modulus Es,GPa

Ed/Es

NS-P-1/2 1278 29.71 28.1 1.06Prior toLoading NS-P-2/2 1278 29.83 29.1 1.03

NS-P-1/2 1200 26.2 28.1 0.93After Loading

NS-P-2/2 1200 26.3 29.1 0.90

4.1 Effect of stress level

Following the above tests, the special plain concrete cylinders NS-P-1/2 and NS-P-2/2 were tested to obtain

the stress-strain curve up to a stress `4.0 cf . This upper limit was used because it represents a limit which

most concrete structures are designed for at service loads. The stress-strain curve is shown in Fig. 2. It should be

noted that the stress-strain curve exhibits a linear behavior for the mix design used in this study. From thiscurve, the static modulus of elasticity was obtained and is shown in Table 4. It can be seen from Table 4 that thedynamic modulus for plain unloaded concrete is moderately higher than the static modulus, with a percentagedifference of less than 6%. This would be expected because of the almost linear stress-strain curve shown in Fig.2 which brings the tangent modulus to almost coincide with the secant modulus [8].

Figure 2. Stress-strain curves for the NS-P concrete cylinders (`4.0 cf )

The NS-P cylinders were then removed from the compression-testing machine and tested to obtain the afterloading dynamic modulus for free-free boundary conditions. The results are shown in Table 4 (after loadingcase). It can be seen that there is a decrease in the dynamic modulus, which may be attributed to micro crackingand creep of the concrete.The NS-P cylinders were placed again in the compression-testing machine and loaded at stress levels of 0.1 fc

`,0.2 fc

`, 0.3 fc`, and 0.4 fc

`. At each of these stress levels, the frequency spectrum was obtained and the naturalfrequencies were estimated, hence the fundamental flexural frequency was obtained. The dynamic modulus was

back calculated considering that we have a cylinder subjected to a compressive load with a fixed-fixed boundarycondition.It should be noted that the testing sequence in this study was considered. For instance, for one of the cylindersthe dynamic testing was first performed, and then the stress-strain curve was obtained. While for the othercylinder the testing sequence was reversed to see if it would affect significantly the results. This was done byobtaining the stress-strain curve, and then performing the dynamic testing. It was found that the testing sequencedid not significantly affect the measured results for the stress levels used in this study.The results are summarized in Fig. 3, which plots the variation of Ed/Es ratio in relation to the stress level. It can

be seen for a stress level not exceeding 0.4 fc` the Ed/Es ratio is in the range of 10% to 10%. This is again is

expected since an almost linear stress-strain curve for the stress levels considered was obtained.

0

24

6

8

10

12

0 100 200 300 400

Strain x E-6

Stress,

MPa

NS-P-1/2

NS-P-2/2


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0.6

0.8

1

1.2

1.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

fc / f 'c

Ed/

Es

NS-RC-1/2 NS-RC-2/2 load removed 1/2 load removed 2/2

Figure 5. Effect of stress level on Ed/Es ratio for the NS-RC concrete cylinders

5 Conclusions and Recommendations

This paper described the determination of the dynamic modulus of elasticity of concrete cylinders subjected toan axial compressive load from the fundamental flexural frequency for free-free boundary and fixed-fixedboundary conditions. The tests were performed on the cylinders prior to and after an axial loading. Based on theresults obtained in this study the following conclusions can be drawn:1. The designed data acquisition system employed in this study utilized the fundamental flexural frequency to

measure the dynamic modulus of concrete easily and accurately.2. For the mix design used in this study, it was found that the dynamic modulus of elasticity and the static

modulus of elasticity are almost close, when measurement are taken for similar stress levels below 0.4 fc`,

since the stress-strain relation exhibited a linear behavior in that range. This was contrary to the perceivedgeneral notion of having the dynamic modulus considerably higher than the static modulus.

3. The presence of micro cracks seemed to lower the dynamic modulus of the plain concrete specimens ascompared to reinforced specimens.

4. The effect of reinforcement on the fundamental flexural frequency needs further investigations considering

more and eccentric bars.5. Other parameters affecting the dynamic modulus like moisture content, temperature, and mix proportions

need to be investigated.

6 References

[1] ASTM C469, Standard Test Method for Static Modulus of Elasticity and Poissons Ratio of Concrete inCompression, Annual Book of ASTM Standards, American Society for Testing and Materials,1994.

[2] ASTM C215, Standard Test Method for Fundamental Transverse, Longitudinal, and TorsionalFrequencies of Concrete Specimens, Annual Book of ASTM Standards, American Society for Testingand Materials, 1991.

[3] Mindess, S., and Young, J., Concrete, Prentice-Hall, Englewood Cliffs, N.J., 1981, 671 pp.[4] Neville, A. M.,Properties of Concrete, Fourth edition, Addison Wesley Longman Limited, 1998, 844 pp.

[5] Richart, F. E., Jr., Hall, and J. R., Jr., Woods, R. D., Vibrations of Soils and Foundations, Prentice-Hall,Englewood Cliffs, N.J., 1970, 414 pp.

[6] Rao, S.,Mechanical Vibrations, Second edition, Addison-Wesley, 1990, 718 pp.[7] Robert Blevins,Formulasfor Natural Frequency and Mode Shape, Wiley, 1975.[8] Mesbah, H. A., Lachemi, M., and Aitcin, P.-C., Determination of Elastic Properties of High-Performance

Concrete at Early Ages,ACI Materials Journal, V. 99, No. 1, Jan.-Feb. 2002, pp. 37-41.

prediction of the dynamic modulus of elasticity

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