Bundesanstalt für Materialforschung und -prüfung

International Symposium (NDT-CE 2003)

Non-Destructive Testing in Civil Engineering 2003
Start > Contributions >Lectures > Materials 2: Print

Nonlinear nondestructive methods for evaluating strength of concrete

I. Shkolnik, Wayne State University, Detroit, Michigan, USA [Visiting Professor, skholnikj@juno.com]
H. Aktan, Wayne State University, Detroit, Michigan USA [Professor and Director of Center for Structural Durability (http:/webpages.eng.wayne.edu/durabilitycenter/ ) Haluk.Aktan@wayne.edu]
R. Birgul, Mugla University, Mugla, Turkey [Asst. Professor, rbirgul.mu.edu.tr]


Experimental data obtained using various methods shows that when concrete is subjected to increasing uniaxial compression, there are ranges of stress, reflecting the reaction of concrete material structure to external force and indicating the nonlinearity of deformation. It is also an established fact that the acoustic properties of solids depend on the properties of their flaws. Utilizing the nonlinear force deformation behavior of concrete, new nonlinear acoustical methods are described for evaluating the strength of concrete. There are two main steps involved in this research. The first step is the description of the physical process using nonlinear approach and known relations representing the fracture of concrete, and establishment of a relationship for the nonlinear strength parameter. The second step is development of new nondestructive test methods for evaluating the nonlinear parameter of concrete. The nonlinear parameter is defined as change in the static or dynamic modulus of elasticity under loading. It will be shown that analytical solution of strength is in substantial agreement with the corresponding formula in ACI Building Code 318. It is proven that the nonlinear parameter depends on structural defects present in concrete, moisture content, age, mineralogical content of cement, and integral energy losses. Three methods for the measurement of nonlinear parameters of concrete that will be presented are;

  1. spectral,
  2. frequency, and
  3. phase.

The testing methods do not require special equipment, may be used when conventional acoustic methods are not applicable, and point out new perspectives for the use of ultrasonic and impact-echo methods for evaluation of strength parameters of concrete in-situ. As important examples, the application of the three methods will be demonstrated for use in predicting both the strength of concrete at different moisture states, as well as the strength of high performance concrete.


In recent years, the emphasis in nonlinear ultrasonics has shifted towards the determination of properties that correlate strongly with mechanical properties of engineering materials such as adhesion and hardening of precipitates (Panraz and Arnold 1994, Cantrell and Yost 1996). However, to our knowledge, there are no nonlinear acoustical methods for predicting strength of concrete. At the present time, the conventional ultrasonic method for evaluating the strength of concrete does not consider the nonlinear behavior of concrete. Quick, simple, and widely acceptable procedures are needed for evaluation of the reaction of concrete's structure on the external loading, i.e. for measurement of the nonlinearity of concrete deformation.

Based on the nonlinear behavior of concrete, this article discusses the fundamentals as well as nondestructive methods for evaluation of the strength of concrete.

Basic Statements

When developing nonlinear methods for evaluating the strength of concrete, the most important condition is that within a wide range of strain (10-6- 10-4), the relationship between stress ( s) and strain (e) of brittle materials can be described as parabolic (Bell 1984):


It is then necessary to relate the fracture of brittle materials to their corresponding modulus of elasticity and the nonlinear parameter (E0 is the initial modulus of elasticity). Carrying the analogy over the stress-free region from the line crack model, it is assumed that the stress-free region is a volume, which is proportional and within the order of magnitude equal to the initial porosity of concrete.Thus total energy (U) for the system under consideration can be written as (Sedov 1973; Shkolnik 1993):


where "p" is the relative volume of defects (p= NVn/V=nVn<1); n is the number of defects per unit volume; Vn is the volume of a spherical defect with the diameter D; G is the density of surface energy; K=(62/3/4)(3.14)1/3= 1.2.

It is assumed that for the nonlinear model (1), the porosity under loading as well as the modulus of elasticity depends on strain (stress). Thus, the values of strength are found to be the roots of the condition U /s = 0 applied using Eq. (2). After simple manipulation and substitution of R for the root s 2, we have from Eq. (2):



Eqs. (2a and 2b) describe the relationship between compressive strength "R" of concrete, modulus of elasticity "E", (E0~ 50GPa for concrete), and the nonlinear parameter E / s (Shkolnik 1993).

From Eq. (2b), it follows that the modulus of elasticity should have a maximum value given by the product of strength and the nonlinear parameter. Such an approach explains existing experimental relationships between modulus of elasticity and strength of concrete in which modulus of elasticity decreases in spite of an increase in strength if the nonlinear parameter of concrete decreases simultaneously (Freudental, A.M.; Walker, S. 1959). In general, Eq. (2b) is in substantial agreement with the corresponding formula in ACI Building Code 318.

Methods and Results

Static Loading.
It is known that modulus of elasticity increases and strength decreases with increasing moisture content in concrete. Experiments show that saturated concrete has pseudo-elastic properties such that the elastic modulus and pulse velocities are higher than in dry specimens (Nilender et al.1969; Shkolnik 1993). As a consequence, the velocities of longitudinal and shear waves remain constant under loading of saturated specimens. After desorption and under a stress of 6MPa, static modulus as well as pulse velocities increase by 2-3 %. Therefore, E0 in Eq. (2a) may be defined as a pseudo-elastic modulus and equal to modulus of elasticity (E) of the saturated concrete:


Table 1 contains corresponding experimental data (Moskvin et al. 1974) for concrete of various water/cement ratios (w/c). Three mix proportions were prepared using the same fine and coarse aggregate and cement with mix proportions of 1:3.33:5.55 for w/c = 0.7; 1:2.17:4.50 for w/c = 0.5; 1:1.5:3.5 for w/c = 0.4. Various specimens were subjected to four methods of inundation and subsequent load testing as shown in Table 1.

W/C Method of Inundation 1 Moisture Content (%) E
Stress,sc (MPa) Strain
Modulus of Elasiticity Ec=sc / ec (MPa) / E s R (MPa) R (MPa)by Eq.(2c) Error
0.714.84343001355022900904 1719-13
0.726.243680011480231001234 15150
0.514.38350001879522100733 2324-6
0.525.06369001780521000 94121207
0.535.353750016775210001010 211910
0.414.05357002085024000574 2631-19
0.424.9372002083023600693 2527-8
0.435.11393001977024000826 24241
0.445.2402001976524300854 24241
Table 1: Data of static tests (Moskvin et al. 1974) and Eq. (2c).

Methods of Inundation are as detailed below:
  1. Inundation for two days at atmospheric pressure.
  2. Vacuum inundation under a pressure of 10-3 mm Hg for seven hours followed by inundation under atmospheric pressure for two days.
  3. Vacuum inundation under a pressure of 10-3 mm Hg for twelve hours followed by inundation under atmospheric pressure for four days.
    Vacuum inundation under a pressure of 10-3 mm Hg for twenty-four hours followed by inundation under atmospheric pressure for seven days.

Table 1 shows the critical stress values (sc), and strain values (ec), depicting the onset of a significant increase in bond cracking. For saturated specimens, the change in modulus at the critical stress can be approximated as E / s = E-Ec / sc. It is also seen in Table 1 that the nonlinear parameter, reflecting the response of moist concrete's structure to external load, increases with moisture content more significantly than the increase of the modulus of elasticity. Therefore, in spite of the increasing modulus of elasticity, the strength decreases, which displays agreement with Eq. (2c).

As was pointed out above, for the nonlinear model (1), the modulus of elasticity depends on strain. To investigate that relationship, mix proportions of concrete with five water-cement ratios (0.35, 0.40, 0.45, 0.5, 0.55) were batched. Young's Modulus, Poisson's ratio, and compressive strength tests were performed on designated specimens according to ASTM C 469 and ASTM 39. During the 1st cycle of loading, the ratio of normal stress to corresponding strain, i.e. modulus of elasticity, was measured within the strain range (10-1000) me . Then the nonlinear parameter dE/ds was been found for data with a correlation coefficient at least -0.9. Table 2 shows the comparison of experimental compressive strength with the corresponding prediction using Eq. (2 a) and Eq. (2c). Bold type represents the values of strength using Eq. (2a) or Eq. (2c) for dry and wet concrete respectively, leading to the minimal prediction error. It is seen in Table 2 that the average relative error is less than 15%. Excluding, with probability more than 90%, such errors as 75%, and then 71%, the average error can be reduced from 13% to 6.4%, as seen in the rightmost column.

W/C Strain Ranges(me) dE/ds E,(Mpa) fc',(Mpa) Rw',(Mpa)Eq.(2c) Rd',(Mpa)Eq.(2a) Minimal error, %
Average error, %13106
Table 2: Prediction of strength using Eq. (2a) and Eq. (2c).

The described nonlinear approach also allows investigation of the physical meaning of the coefficient recommended by ACI Building Code 318. Indeed, Table 3 illustrates the values of the coefficient between modulus of elasticity and strength for different water-cement ratios according to ACI Building Code and by use of Eq. (2a). Corresponding relative error is within 1%. Therefore, the empirical coefficient reflects the nonlinear behavior of concrete and can be calculated using the nonlinear parameter dE/ds while holding Eo=50GPa as a constant. It is possible (Table 2 and Table 3) simultaneously evaluate the strength of concrete, using only the 1st cycle during the test according to ASTM C 469, and to reduce the quantity of specimens for compressive strength.

dE/ds Coefficient
Eq. (2a)
  Average 5230 Average 5300
Table 3: Physical interpretation of the coefficients.

Dynamic Loading
A relationship is established between the strength of concrete specimens under static and dynamic (e.g. impact, cyclic) loading based on the nonlinear behavior of concrete (Kuksenko et al. 1993; Betekhtin et al. 1994; Shkolnik 1996).

It is logical to suppose that nonlinearity of deformation, reflecting a reaction of the structure to the external force, may be measured by increase the accuracy of measurement Therefore new testing methods were developed: spectral, resonant frequency shift, and phase shift (Shkolnik 1993).

In spectral method an ultrasonic wave of known frequency, f= 50KHz, was applied trough a concrete samples. The energy wasted by scattering when the wave "stumbles" upon inner defects is proportional the energy spend on forming the second harmonic, f=100KHz. Consequently, in spectral method the nonlinear parameter was defined by the ratio of the second harmonic amplitude to the first one. The ratio of the second harmonic to the first one, taking into account the transmission coefficients of measuring device, was within a few fractions to 1 percent (Shkolnik 1992).

From a practical point of view, it is important to measure the nonlinear parameter in the field. For this purpose, two nondestructive test methods were developed based on the measurement of resonant frequency shift and phase dependence to dynamic loading. The first procedure is based on measurement of the resonant frequency shift of the fundamental mode, and the second procedure is based on the phase shift dependence on the excitation amplitude (Shkolnik 1985; Shkolnik 1992).

The resonance frequency shift measurement allows the use of the well-known impact-echo method (Sansalone and Carino 1991) for measuring the nonlinear parameter of concrete in situ.

In the phase dependence measurement, the nonlinear parameter can be determined from the phase change due to the changes in the dynamic modulus of elasticity. The phase difference, " D f " between two amplitudes of excitation was measured on several high performance concrete specimens. A linear regression of static strength with phase difference D f produced R1 = 66.5-6.84D f , with a mean-square error of less than 10 % in the investigated range satisfying the practical requirements (Fig.1).

Fig 1: The relationship between phase shift and concrete strength.

The nondestructive test procedures can be used when conventional acoustic methods are not applicable, and significantly reduce the effort required for evaluating the static, impact, and fatigue strengths of concrete.


Nonlinear parameters are integral structure-sensitive characteristics that contain information on structural defects of the material, mineralogical content of Portland cement, and moisture content. Nonlinear parameters can be used for quantitative evaluation of concrete strength under quasi-static or dynamic loading. The nonlinear approach verifies the relationship between static elastic modulus of concrete and strength accepted by ACI 318. Although additional tests are needed for better statistical representation, data presented allows the evaluation of the strength of concrete specimens from standard compressive tests according to ASTM C 469. The findings also indicate that in situ non-destructive stress wave techniques can be applied for the prediction of strength and fracture of concrete.


  1. Bell, J.F. (1984). "Mechanics of solids." The experimental foundations of solid mechanics, Vol.1, Springer-Verlag Berlin Heidelberg New York Tokyo.
  2. Betekhtin, V.I., Kuksenko, V.S., Slutzker, A.I. and Shkolnik, I.E. (1994)."Fracture kinetics and the dynamic strength of concrete." Phys. Solid State - American Institute of Physics, pp. 1416-1421.
  3. Cantrell, J. H. and Yost, W. T. (1996). Review of Progress in Quantitative Nondestructive Evaluation, Vol. 15, New York: Plenum, pp. 1361.
  4. Freudental, A.M. and Roll, R. (1958). "Creep and creep recovery of concrete under high compressive stress." Journal of the American Concrete Institute, Vol. 54, No. 12.
  5. Kuksenko, V.S., Slutzker, A.I. and Shkolnik, I.E. (1993). "Estimating the dynamic strength of concrete." Tech. Phys. Letter American Institute of Physics, Vol. 19, No. 2, pp. 127-129.
  6. Moskvin et al. (1974). "Concrete for building under severe climatic conditions." Stroiizdat, Russia (in Russian).
  7. Nilender, Y.A., Pochtovik, G.Y. and Shkolnik, I.E. (1969). "The relation of the velocity of elastic wave propagation on internal energy losses in concrete under loading." Concrete and Ferroconcrete, Vol. 7, pp. 7-8, Russia (in Russian).
  8. Pangraz, S., and Arnold, W. (1994). Review of Progress in Quantitative Nondestructive Evaluation, Vol. 13, New York: Plenum, pp. 1995.
  9. Sansalone, M. and Carino, N.J. (1991). "Stress wave propagation methods." Handbook on Nondestructive Testing of Concrete, M. Malhotra and N.J. Carino, eds., CRC Press, Inc., pp. 275-304.
  10. Sedov, L.I. (1973). "Mechanics of continuous media." Vol. 2, Nauka, Moscow, (in Russian).
  11. Shkolnik, I.E. et al; Patent No. 1146593 (Russia), 1985.
  12. Shkolnik, I.E. et al; Patent No. 1770889 (Russia), 1992.
  13. Shkolnik, I.E. (1993). "Nondestructive testing of concrete: new aspects." Testing and Evaluation, Vol. 10, pp. 351-358.
  14. Shkolnik, I.E. (1993). "Diagnostics of concrete quality: new aspects." Technoproject, Moscow, (in Russian).
  15. Shkolnik, I.E. (1996). "Evaluation of dynamic strength of concrete from results of static tests." ASCE Journal of Engineering Mechanics, Vol. 122, No. 12, pp. 1133-1138.
  16. Shkolnik, I.E., Udegbunam, O.C. and Aktan, H.M. (1997). "Ultrasonic methods of evaluating concrete permeability." NDT-CE' 97, Vol. 1, pp. 111-120.
  17. Walker, S. (1959). "Modulus of elasticity of concrete." Proceedings of the ASTM, Vol. 19, No. 2.
  18. Yaman, I.O., Inci, G., Yesiller, N. and Aktan, H.M. (2001). "Ultrasonic pulse velocity in concrete using direct and indirect transmission." ACI Journal of Materials, Vol. 98, No. 6, pp. 450-457.
  19. Yaman, I.O., Hearn, N. and Aktan, H.M. (2002). "Active and non-active porosity in concrete - evaluation of existing models." Union of Testing and Research Laboratories for Materials and Structures (RILEM), Vol. 35, No. 246, pp. 110-116.
  20. Yaman, I.O., Hearn, N. and Aktan, H.M. (2002). "Active and non-active porosity in concrete - experimental evidence." Union of Testing and Research Laboratories for Materials and Structures (RILEM), Vol. 35, No. 246, pp. 102-109
STARTPublisher: DGfZPPrograming: NDT.net