Abstract:
Methods and apparatus using time domain reflectometry to determine electrical properties of concrete for calculating physical and mechanical properties of the concrete that are indicative of concrete quality and durability. The dielectric constant of a concrete is determined to calculate the gravimetric water content of the concrete for use in finding the water-cement ratio of the concrete. Electrical conductivity of curing concrete is determined to calculate compressive strength of the concrete for use in predicting strength development in the concrete.

Description:
CROSS-REFERENCE TO RELATED APPLICATION  
       [0001]     This application claims the benefit of U.S. Provisional Patent Application Ser. No. 60/547,846, filed Feb. 26, 2004, which application is hereby incorporated by reference along with all references cited therein. 
     
    
     TECHNICAL FIELD OF THE INVENTION  
       [0002]     The present invention relates generally to methods and apparatus for measuring properties of concrete and, more particularly, to methods and apparatus for measuring electrical properties of concrete.  
       BACKGROUND OF THE INVENTION  
       [0003]     Predicting behavior of concrete through preliminary analysis of concrete properties is important for construction operation decisions. Conventional approaches for evaluating properties of concrete in the filed are often inaccurate, destructive, time consuming, and costly. Two important ways of evaluating concrete are measuring the water-cement ratio of the concrete mixture prior to placement and measuring the compressive strength of the concrete at certain introductory time intervals to predict long term strength and durability.  
         [0004]     Water-cement ratio is an important property for concrete strength and durability so getting reasonably accurate measurements of it for freshly mixed concrete is of great practical importance. The current field practice to estimate water-cement ratio is from batch mixture quantities, i.e., water-cement ratio is calculated from the amount of water and the amount of cement added to the mixture. The estimated results are not generally found accurate, as the quantities do not accurately account for factors such as moisture contained in aggregate or water added at the job site.  
         [0005]     Given the practical importance of water-cement ratio, many techniques have been proposed to measure this property. These generally involve measurement of water content and measurement of cement content. Field experiences indicate that accurate measurement of the water-cement ratio of fresh concrete is very difficult to achieve and currently there are no fast and reliable technologies available. For example, the nuclear gauge, which is widely used for quality control of soil compaction, is found ill suited for field water-cement determination use due to its unsuitability for igneous aggregates, poor field performance with limestone aggregates, and extensive training and certification procedure for operators. Several ultrasonic research tests have also been proposed. However, although the research unveiled some potential application of ultrasonic technology in fresh concrete characterization, none of the approaches produced satisfactory solutions for the instantaneous determination of the water-cement ratio. Accordingly, there still remains a need for a technique that more accurately measures the water-cement ratio in a shorter period of time to improve quality control and quality assurance of concrete.  
         [0006]     Compressive strength of concrete is an important factor that controls the service life of concrete structures. Measurement of concrete strength development is important for quality assurance and quality control. The direct way to evaluate concrete strength is from laboratory compression tests performed on batch samples obtained in the field. The process is time consuming and the strengths obtained are generally not representative of the behavior of concrete in the actual structure due to difference in curing conditions. It is not uncommon to obtain significant scattering of measured concrete strength from compression tests on cylinders. Besides, the tests can only be performed to obtain concrete strength at fixed time since the tests are destructive.  
         [0007]     Alternative approaches exist which are based on correlating concrete strength to other properties that can be measured more easily and nondestructively. An important characteristic of concrete strength is that it increases with curing time, which is attributed to the continuing hydration process. Thus, characterization of the hydration process provides an alternative approach for determining concrete strength. Various indirect methods can be used for this purpose, including calorimetry, thermal techniques, and nuclear magnetic resonance spectroscopy, which are generally expensive and require sophisticated methods of data analysis. Alternative approaches, such as ultrasound, are based on measurement of ultrasonic velocity, which is more related to concrete modulus and was applied to study the initial setting of concrete. There are also other empirical approaches based on the relationship between concrete strength and porosity. All these approaches, while working under certain situations, have shortcomings with respect to field applications. Accordingly, there still remains a need for a technique that accurately predicts compressive strength of concrete within a short period of time after placing that is non-destructive and economical.  
       SUMMARY OF THE INVENTION  
       [0008]     The present invention relates to a method and apparatus for testing concrete using the principle of time domain reflectometry (TDR) associated with electromagnetic waves traveling in a medium. Two pieces of information may be obtained from the TDR signal in accordance with certain aspects of this invention. One aspect of the invention makes use of the relationship between apparent dielectric constant and gravimetric water content, and another aspect of the invention makes use of the relationship between electrical conductivity and compressive strength.  
         [0009]     According to one aspect of the present invention, a method of measuring water content of concrete applies an electrical signal suitable for time domain reflectometry to a plurality of spikes placed in the concrete and analyzes a reflected time domain reflectometry signal to determine an apparent dielectric constant. The gravimetric water content of the concrete is then calculated using a predetermined relationship between the apparent dielectric constant and a ratio including density of water and total density of the concrete.  
         [0010]     According to another aspect of the present invention, a method of measuring water-cement ratio of concrete applies an electrical signal suitable for time domain reflectometry to a plurality of spikes placed in the concrete and analyzes a reflected time domain reflectometry signal to determine an apparent dielectric constant. The gravimetric water content of the concrete is then calculated using a predetermined relationship between the apparent dielectric constant and a ratio including density of water and total density of the concrete. The cement content may be determined from batch records or by other methods. The water-cement ratio is then calculated.  
         [0011]     According to a further aspect of the present invention, a method of predicting compressive strength development of concrete applies an electrical signal suitable for time domain reflectometry to a plurality of spikes placed in the concrete and analyzes a reflected signal using time domain reflectometry to determine electrical conductivity of the concrete. The compressive strength is then calculated based on the electrical conductivity of the concrete.  
         [0012]     According to another aspect of the present invention, a method of determining behavior of concrete applies an electrical signal suitable for time domain reflectometry to a plurality of spikes placed in the concrete and analyzes a reflected time domain reflectometry signal to determine an electrical property of the concrete. A physical property is then calculated using a predetermined relationship between the electrical property and time.  
         [0013]     According to a further aspect of the present invention, an apparatus for measuring water content of concrete comprises a plurality of spikes adapted to be placed into the concrete, means for applying an electrical signal suitable for time domain reflectometry to the plurality of spikes, means for analyzing a reflected signal using time domain reflectometry to determine an apparent dielectric constant of the concrete, and means for calculating gravimetric water content of the concrete using a predetermined relationship between the dielectric constant and a ratio including density of water and total density of the concrete.  
         [0014]     According to a still further aspect of the present invention an apparatus for predicting compressive strength development of concrete comprises a plurality of spikes adapted to be placed into the concrete, means for applying an electrical signal suitable for time domain reflectometry to the spike, means for analyzing a reflected signal using time domain reflectometry to determine electrical conductivity of the concrete, and means for calculating compressive strength of the concrete using a predetermined relationship between the electrical conductivity and time.  
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0015]      FIG. 1  is a side elevational view of a TDR apparatus of the present invention placed in concrete.  
         [0016]      FIG. 2  shows the influence of material properties on a TDR waveform.  
         [0017]      FIG. 3  shows TDR monitored free water content in concrete mixture 1.  
         [0018]      FIG. 4  shows TDR monitored free water content in concrete mixture 2.  
         [0019]      FIG. 5  shows changing dielectric properties and temperature of the concrete mixtures over time.  
         [0020]      FIG. 6  shows changing electrical conductivity and temperature of the concrete mixtures over time.  
         [0021]      FIG. 7  shows the data of  FIG. 6  replotted with an arithmetic time scale.  
         [0022]      FIG. 8  shows the effect of temperature on TDR measured dielectric constants of concrete.  
         [0023]      FIG. 9  shows the effect of temperature on TDR measured electrical conductivity of concrete.  
         [0024]      FIG. 10  shows compressive strength results from Mixture 1 concrete cylinder testing.  
         [0025]      FIG. 11  shows compressive strength results from Mixture 2 concrete cylinder testing.  
         [0026]      FIG. 12  shows the relationship between TDR measured electrical conductivity and compressive strength for both concretes.  
         [0027]      FIG. 13  shows estimated compressive strength of Mixture 1 using a method of the present invention compared to actual measurements.  
         [0028]      FIG. 14  shows estimated compressive strength of Mixture 2 using a method of the present invention compared to actual measurements.  
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0029]     For the purpose of promoting an understanding of the principles of the invention, reference will now be made to the embodiments illustrated in the drawings and specific language will be used to describe the same. It will nevertheless be understood that no limitation of the scope of the invention is thereby intended, such alterations and further modifications in the illustrated device and such further applications of the principles of the invention as illustrated therein being contemplated as would normally occur to one skilled in the art to which the invention relates.  
         [0030]     A time domain reflectometry (TDR) apparatus  10  of the present invention, shown in  FIG. 1 , is used to measure electrical properties of the concrete  12 . The apparatus  10  generally comprises a probe  14 , a plurality of equally spaced spike sensors  16  (available at most local hardware stores), a coaxial cable  18 , and a TDR tester (such as TDR 100  tester available from Campbell Scientific, Inc., not shown). An apparatus of the type described above is described in more detail in U.S. Pat. No. 6,215,317 to Siddiqui et al., which patent is hereby incorporated by reference. Data collected by the TDR tester is analyzed in accordance with the present invention by a general purpose computer running a specially developed computer program that implements the equations described below.  
         [0031]     A typical TDR signal and information content for TDR measurement in a material is shown in  FIG. 2 . A “peak” and a “valley” are caused by reflections and are characteristic of TDR signals measured in geomaterials. The “peak” is caused by the first reflection, which occurs when the electromagnetic pulse crosses the air/material interface. The “valley” is caused by the second reflection, which occurs when the electromagnetic pulse arrives at the end of the measurement probe.  
         [0032]     Dielectric constant, and electrical conductivity are two pieces of important information that can be obtained from analysis of a TDR signal. Material dielectric constant is analogous to Young&#39;s modulus in that it determines the electromagnetic wave speed. It can be determined from travel time analysis and is generally called apparent dielectric constant, denoted K a . K a  represents the real part of the frequency dependent dielectric permittivity. Equation (1) gives the mathematic expression for computing dielectric constant from TDR measurement.  
               K   a     =       (       L   a       L   p       )     2             (   1   )             
 
 where L p  is the length of the probe in the material and L a  is the scaled horizontal distance between the two reflections, called apparent length. 
 
         [0033]     The electrical conductivity, EC b , causes attenuation of TDR signal and is another important piece of information that can be obtained from TDR waveforms. Different approaches can be used to obtain electrical conductivity from a TDR signal. Equation (2) uses an approach based on analysis of the long-term response of a TDR system to determine electrical conductivity.  
               EC   b     =       1   C     ⁢     (         V   s       V   f       -   1     )               (   2   )             
 
 where V s  is the source voltage, V f  is the long term voltage level, and C is a constant related to probe configuration, determined from Equation (3) for coaxially configured probes,  
             C   =       2   ⁢   π   ⁢           ⁢     L   p     ⁢     R   s         ln   ⁡     (       ⅆ   0       ⅆ   i       )                 (   3   )             
 
 in which L p  equals the length of the probe in the material, R s  the internal resistance of the pulse generator (typically 50 ohms), and d o  and d i  are the diameters of outer and inner coaxial conductors, respectively. 
 
         [0034]     Water plays an important role in concrete mixtures. It serves as a necessary agent for hydration reactions, in which free water molecules become chemically bound with cement particles. The amount of water involved in these reactions is typically only a fraction of the water added to the mixture. As the result, the major factor that controls the amount of water used in concrete is to provide a mixture that can be placed and is workable. Water in concrete mixtures comes from two main sources, i.e., water added during mixing process and the adsorbed water from aggregates. The moisture contents of coarse aggregates generally range from 0.5% to 2% and those of fine aggregates range from 2% to 6%, which can introduce significant amount of water into the concrete mixture. Currently, there is no effective approach for field measurement of water content in freshly placed concrete.  
         [0035]     The strong correlation between TDR-measured dielectric constant and the amount of water in concrete is attributed to the much larger dielectric constant of free water (around 81 at room temperature) as compared with the dielectric constant of air (around 1) or geomaterial solids (around 3 to 7).  
         [0036]     Gravimetric water content and concrete dry densities can be related to concrete dielectric constant using Eq. (4).  
                   K   a       ⁢       ρ   w       ρ   d         =     a   +   bw             (   4   )             
 
 where a and b are concrete specific constants obtained from calibration tests. 
 
         [0037]     For rapid determination of water content, a batch sample can be obtained and put into a cylindrical mold of known volume, from which total density of concrete in the mold, ρ t , can be determined. The relationship between total density and dry density in given by Eq. (5)  
               ρ   d     =       ρ   t       1   +   w               (   5   )             
 
 Substituting Eq. (5) into Eq. (4) and solving for the water content gives:  
             w   =             K   a       ⁢       ρ   w       ρ   t         -   a       b   -         K   a       ⁢       ρ   w       ρ   t                     (   6   )             
 
 Equation (6), with appropriate values of a and b for concrete, can be used to obtain the free water content of concrete. 
 
         [0038]     Two different concretes were studied using a method of the present invention. The mixture proportions of the two concretes are shown in Table 1. The samples were obtained from field and put into standard 6′×12″ plastic molds with volume of 6.107×10 −3  m 3 . Additional samples were obtained to determine oven dry water content.  
                                                                   TABLE 1                           Mixture proportions and water-cement ratio from batch mix records                Gravel   Sand   Cement   Water   Water-           (kg/m 3 )   (kg/m 3 )   (kg/m 3 )   (kg/m 3 )   Cement ratio                        Mixture 1   1089   916   306   162   0.52       Mixture 2   1101   916   336   161   0.48                  
 
         [0039]     Water contents in concrete are computed from TDR-measured dielectric constant by Eq. (6). A temperature compensation factor (Eq. (7)) was applied before computing water content to compensate the effects of temperature on TDR-measured dielectric constant, based on a linear relationship observed between the value of apparent dielectric constant K a  at a given temperature and the value of K a  at a standard temperature, e.g. 20° C. Specifically, the value of K a  obtained from the TDR signal at a given temperature is normalized to the standard temperature by multiplying the TDR-measured value of K a  by a temperature compensation factor (TCF), where, for the above two concrete mixtures,  
               TCF     K   ab       =     1       0.0019   ·   T     +   0.952               (   7   )             
 
 where T is temperature in ° C. 
 
         [0040]     Physical interpretation as well as typical range of constants a and b are predominantly dependent on dielectric properties of dry solid phases and b being mostly decided by pore fluid. The values of constants a and b used in Eq.(6) for the concrete tested were set to a=1.0 and b=14.5.  
         [0041]     A summary of measured water contents by TDR for the fresh concrete samples are shown in Table 2. There are several observations from this table. For both of these concretes, the oven dry water content is slightly larger than the water contents calculated from batch receipts (0.4% (for Mixture 2) and 0.3% (for Mixture 1)). These are equivalent to aggregates moisture content of 0.6% and 0.8% respectively, which are at the lower end of typical moisture range of aggregates discussed above. It is expected that the effects of aggregate moisture could be much more pronounced in other situations.  
         [0042]     Table 2 shows Eq. (6) provides better accuracy for water content measurement in concrete than from what can be determined from batch records because it accounts for moisture content contained in aggregate used in the concrete mixtures.  
                                     TABLE 2                           Water contents by TDR method                    Oven Dry   Water Content               Dry   Water   from Batch       Concrete   Density   Content   Weights   Eq. (6)       Source   (kg/m 3 )   (%)   (%)   (%)               Mixture 2   2041   7.4   7.0   7.5       Mixture 1   2078   7.1   6.8   7.0                  
 
         [0043]     Water in concrete exists in two different types, i.e., free water and chemically bound water. These two types of water show significantly different dielectric behavior. It has been established that free water has relaxation frequency of around 18 GHz while the relaxation frequency for bound water is within MHz range. (Relaxation frequency is a term that is analogous to resonant frequency for vibrating systems.) The TDR system used in the present invention has an effective frequency into the low gigahertz range and is more sensitive to the amount of free water than to bound water. Thus, it is a more direct indicator of amount of free water in concrete. This makes TDR-measured dielectric constant more instructive and easier to interpret compared with the system used in the past.  
         [0044]     The plots of TDR-measured dielectric constant with time are shown in  FIGS. 3 and 4  for both concretes. The dielectric constant consistently decreased with time. The TDR measured dielectric constant decreases at a high rate at the initial stage, which indicates the high intensity of hydration reactions. The rate of decrease becomes smaller with time, which reflects reduced intensity of hydration. The free water contents calculated using Siddiqui-Dmevich equation (Eq. (6)) are also plotted in this figure, which clearly shows the decreasing amount of free water in concrete with time. After 196 days, the free water content in concrete sample from Mixture 1 was around 3.0% and after 166 days that Mixture 2 was around 3.5%. The fact that TDR measurements can be easily automated makes it an attractive tool for monitoring the free water content in concrete.  
         [0045]     The TDR-measured water content can be combined with the information of cement content from batch receipts to make an estimate of water-cement ratio. The calculated water-cement ratio of concrete samples from Mixture 1 was 0.53 and that of Mixture 2 was 0.52, which are slightly higher than calculated from batch receipts (by 0.1 and 0.4), respectively. As mentioned before, the moisture contents of aggregates in these concretes are believed to be at the lower end of typical moisture content range. The resulting difference in water-cement ratio can be more significant for aggregates with higher water content or in situations where water is added at the job site.  
         [0046]     While hydration causes the change of concrete mechanical structure and corresponding increase of concrete strength, it simultaneously changes concrete physico-chemical and electrical properties. Thus, electrical properties of concrete and mortar, especially the electrical conductivity, are strongly related to the strength of concrete.  
         [0047]     While chemical reactions are the most important process occurring during concrete curing, the exact nature of the entire hydration process is complicated and not fully understood. Generally speaking, hydration reactions take place between cement powder and water upon mixing and theoretically, the process continues forever. Major products of the reactions include calcium silicate, calcium aluminate, ettringite, etc. Calcium silicate is the major component affecting concrete strength and calcium aluminates predominantly determine the time of initial setting. A significant amount of heat is generated during hydration process.  
         [0048]     A direct consequence of the hydration process is a change in the microscopic structure. This results in the increase of modulus and strength. There is a strong linear correlation between concrete strength and degree of hydration. The hydration process, which significantly changes the microstructure of concrete, changes the electrical behavior of concrete as well. Bulk water becomes chemically bound water, which shows significantly different dielectric behavior compared with free bulk water. The formation of solid structures by hydration reactions reduces the amount of free ions in pore solution, which results in a decrease of electrical conductivity. Thus, concrete electrical behavior can be a strong indicator of the progress of the hydration process. As the hydration process directly results in the increase of concrete strength, electrical properties can thus be used to monitor strength development.  
         [0049]     The dielectric properties of concrete are dependent on factors such as the amount of water in the pore system and the concrete microstructure. The measured results of dielectric properties are also influenced by electrode configuration and measurement frequency band. Optimized design, both in sensor geometry and measurement frequency range, is necessary to achieve the best measurement results.  
         [0050]     Accurate measurement of dielectric properties of concrete is critical to achieving the best measurement accuracy. Various technologies and system designs can be used for this purpose, including technologies based on measurement of frequency dependant behavior as obtained with an impedance analyzer or network analyzer. While these systems collect more information, measurements are generally expensive and data analysis is difficult. Such systems typically are not suitable for field applications.  
         [0051]     In addition to automatically monitoring dielectric properties of the two concrete mixtures with time, strengths at 1 day, 7 days, and 28 days also were measured using specimens collected at the time of concrete placing. The compressive tests were performed in certified laboratories.  
         [0052]      FIGS. 5 and 6  show how the dielectric properties, electric conductivity and dielectric constant, change with time. Dielectric constant consistently decreases with time, which is an indication of the decreasing amount of free water in concrete. Due to hydration reactions, free water becomes chemically bound water, which has much smaller dielectric constant than free water. The observed changes of dielectric constant shown in  FIG. 5  are different from previous study results that found the dielectric constant first increases and then decreases with time. Since the frequency used in previous studies were within the relaxation frequency range of chemically bound water, measurements were sensitive to the behaviors of both free water and chemically bound water. The previously observed increase of dielectric constant at the initial stage was the result of the dominant role played by the increasing amount of bound water. The decrease of dielectric constant in the longer term was dominated by the decreasing amount of free water. While previous studies helped to explain mechanisms of hydration in the fresh concrete mixture, they also caused difficulty in the interpretation of results since effects of bound water and free water could not be separated. The effective measurement frequency of the TDR tester used in the present invention is in the low Gigahertz range, which is beyond the relaxation frequency of bound water. Thus, the dielectric constant measured by TDR is predominantly influenced by the amount of free water in concrete. This makes the measurement much easier to interpret since the reduced amount of free water reflects the increased amount of bound water.  
         [0053]     The decrease of electrical conductivity as shown in  FIG. 6  is more significant, as it provides a strong indication of the structural changes and reduced amount of free ions in the concrete. The data for the initial part of the test in  FIG. 6  is replotted with an arithmetic time scale to show the behavior of freshly mixed concrete in  FIG. 7 . The electrical conductivity increases slightly after the mixing ( FIG. 7 ), which possibly is caused by effects of consolidation and particle rearrangement.  
         [0054]     Another important observation from  FIG. 6  is that for both of these concrete samples, after completion of the initial stages, the change of electrical conductivity decreases linearly with the logarithm of time. The slope of this line, which is believed to be related to rate of hydration, is similar for both concretes. At any given time after the initial stages, the electrical conductivity of the Mixture 2 concrete is smaller than that of Mixture 1 concrete.  
         [0055]     From  FIGS. 5-7 , we can see that the temperature curves of the two concretes are similar. A group of tests were performed to investigate the effects of temperature on TDR measured dielectric constant and electrical conductivity. The cured concrete specimens from both mixtures were sealed and stored for 24 hours in temperature controlled room of 4° C. and 40° C., respectively, and then allowed to return to ordinary laboratory room temperature. As the mixture temperature was being restored to room temperature, TDR and thermocouple readings were taken to monitor the change of dielectric constant and electrical conductivity with temperature.  
         [0056]     The measured values of dielectric constant and electrical conductivity for the two mixtures are normalized by those at room temperature (22° C.) and are plotted in  FIGS. 8 and 9 .  FIGS. 8 and 9  show that both the dielectric constant and electrical conductivity increase linearly with temperature within the temperature range of the study. The effect of temperature on electrical conductivity measurement is much more significant than that of dielectric constant as indicated by the steeper slope in  FIG. 9 .  
         [0057]     From these observations, the following temperature compensation factors (Eq.(8)) are recommended to compensate for the effects of temperature on TDR measured dielectric constant and electrical conductivity.  
                 TCF     K   a       =     1       0.0019   ·   T     +   0.952         ⁢     
     ⁢         TCF   EC               b       =     1       0.0247   ·   T     +   0.453                 (   8   )             
 
 where T is temperature in ° C. 
 
         [0058]     The results of compressive strength obtained from concrete cylinder testing are shown in  FIGS. 10 and 11 . There is significant (about 20%) although not unreasonable scatter of results, which is possibly due to effects of sampling disturbance and curing process. The phenomena are common when evaluating concrete strength from cylinder samples. Hyperbolic curves give good fit to the data and can be used to describe the evolution of compressive strength with time.  
         [0059]     The compressive strengths at different curing times predicted by the hyperbolic curves in  FIGS. 10 and 11  are plotted against the temperature compensated TDR measured electrical conductivity in  FIG. 12 .  FIG. 12  indicates that the compressive strengths show reverse relationships with electrical conductivity, which are similar for the two concretes tested.  
         [0060]     These curves show linear trend in the middle and are slightly curved at high electrical conductivities (initial stage) and at low electrical conductivities (long term). The reverse trend between concrete strength and electrical conductivity is believed to be valid since the hydration process reduces the amount of free ions in concrete (and thus reduces the electrical conductivity) and at the same time increases its compressive strength.  
         [0061]     It is observed that a curve in the form of Eq. (9) gives good fit to the data in  FIG. 12  and gives the reasonable strength values for extreme conditions.  
               f   c     =       α   ⁡     [       π   2     -       tan     -   1       ⁡     (     β   ⁡     (       EC   b     -     EC   0       )       )         ]       ⁢     P   a               (   9   )             
 
 where ƒ c  is compressive strength (same units as P a ); α is an empirical constant (no units), β is an empirical constant (units of m/mS), and EC 0  (units of mS/m) are obtained from calibration tests; the term EC b  (in units of mS/m) is TDR measured electrical conductivity after temperature compensation by Eq. (8); and P a  is the atmospheric pressure (P a =0.098 MPa for SI units and P a =14.7 lb/in 2  for U.S. Customary units). 
 
         [0062]     Using Eq. (9), three curves are plotted in  FIG. 12 , one for Mixture 2 concrete, one for Mixture 1 concrete, and one for both concretes combined. The equation for the curve for the combined data and corresponding error bars of ±10% are given in the figure by the darker solid line. The combined data were also used with equations recommended in the prior art, also shown in  FIG. 12 , having a concave upward shape.  
         [0063]     From  FIG. 12  it can be seen that fitted curve by Eq. (9) gives reasonable estimation of compressive strength from TDR measured electrical conductivity. The estimated strength generally falls within ±10% of the optimized compressive strengths from cylinder tests. Equations recommended in the prior art on the other hand cannot accurately describe the data trend, especially at low electrical conductivity (corresponding to long term strength). The measured electrical conductivity for Mixture 1 concrete was 11.96 mS/m after 196 days and that of Mixture 2 is 11.95 mS/m after 166 days. The estimated strengths by Eq. (9), using the parameters obtained from the combined data, are 47.6 MPa and 47.6 MPa, respectively. Equations recommended in the prior art on the other hand, gives unreasonable estimated strengths of 225.0 MPa and 224.7 MPa, respectively. Thus, Eq. (9) is believed to be more robust for estimating compressive strength from electrical conductivity.  
         [0064]     To apply Eq. (9), a group of calibration tests are needed to determine the calibration constants. The calibration involves making several cylinders for a given mix design. For one of the cylinders, the TDR probe and a temperature sensor are installed to monitor the dielectric constant, electrical conductivity, and temperature with time. Compression tests are performed on the other cylinders to determine compressive strength at different ages, typically one day, seven days, and twenty eight days. The compressive strength and temperature compensated electrical conductivity are then analyzed in a spreadsheet to obtain the calibration constants in Eq. (9). Once the calibrations are obtained, the measured electrical conductivity, either in the field or in the laboratory, can be applied to estimate the compressive strength.  
         [0065]      FIG. 6  shows that the electrical conductivity linearly decreases with time on a logarithmic scale after about one day, which is similar for both concretes. This can be described by Eq. (10).  
                 EC   b     ⁡     (   t   )       =         EC   b     ⁡     (     t   1     )       +       (       Δ   ⁢           ⁢     EC   b         log   ⁢           ⁢   cycle       )     ⁢   log   ⁢           ⁢     (   t   )                 (   10   )               
 where: t is the curing time in days, EC b (t 1 ) is the electrical conductivity measured at one day, ΔEC b /(log cycle) is the change in electrical conductivity over one log cycle, all in units of mS/m. For Mixture 1 concrete, the corresponding values of EC b (t 1 ) and ΔEC b /(log cycle) are 37.75 mS/m and −5.05 mS/m, respectively. The corresponding values for Mixture 2 concrete are 36.29 mS/m and −4.96 mS/m, respectively. Note that t in Eq. (12) may have decimal values, but that values of t must not be smaller than the linear portion of the curve on the log-time plot used to establish the coefficients. For example, for the Mixture 2 curve in  FIG. 6 , the value of t must be greater than approximately 0.5 days. 
 
         [0066]     Once its relationship with time is established, the electrical conductivity measured at curing times sufficiently long to establish the straight line on the semi-log plot could be used to estimate long-term electrical conductivity, and consequently, the long-term compressive strength by combining Eqs. (11) and (12) as shown in Eq. (11).  
               f   c     =       α   ⁡     [       π   2     -       tan     -   1       ⁢     {     β   ⁡     [         EC   b     ⁡     (     t   1     )       +       (       Δ   ⁢           ⁢     EC   b         log   ⁢           ⁢   cycle       )     ⁢     log   ⁡     (   t   )         -     EC   0       ]       }         ]       ⁢     P   a               (   11   )             
 
 where the parameters are defined above for Eqs. (9) and (10). 
 
         [0067]      FIGS. 13 and 14  show the predicted compressive strength versus time in days (log scale) by Eq. (11) for both Mixture 1 concrete and Mixture 2 concrete. The parameters used in Eq. (11) to predicted these curves are summarized in Table 3, where the values of αβ, and EC 0  are from the equation shown in  FIG. 12  (solid curve with the thick line) and the values of EC b (1 day) and ΔEC b (per log cycle) are from fitting the straight lines through the linear portion in  FIG. 6  for each of the concrete samples. The actual measured compressive strengths are also plotted for comparison. They generally fall within ±10% of predicted strength. Even though these curves are based on tests only up to 28 days,  FIGS. 13 and 14  indicates that the longer term strengths given by Eq. (11) appear quite reasonable.  
                                                   TABLE 3                       Parameters of Eq. (12) for curves in  FIGS. 13 and 14                                      α   155               β (m/mS)   0.3996           EC 0  (mS/m)   33.97                Concrete   Mixture 1   Mixture 2           EC b (t 1 ) (mS/m)   37.75   36.29           ΔEC b /log cycle   −5.05   −4.96           (mS/m)                        
         [0068]     While the invention has been illustrated and described in detail in the drawings and foregoing description, the same is to be considered as illustrative and not restrictive in character, it being understood that only the preferred embodiment has been shown and described and that all changes and modifications that come within the spirit of the invention are desired to be protected.