Abstract

The estimation of mechanical properties of concrete is one of the most important applications of the ultrasonic method. This technique often involves the measuring of the ultrasonic pulse propagation velocity. However, there exists some factors influencing the effective ultrasonic pulse propagation velocity in concrete. This paper is devoted to a discussion of the behavior of the effective ultrasonic pulse propagation velocity as a function of the free water contained in concrete. The travelling of an ultrasonic ray through a heterogeneous medium composed by cement and aggregates(solid phase), water(liquid phase) and air(gaseous phase) was empirically modelled. A relationship between the transition time of the ultrasonic ray in the testing piece and the free water quantity was considered. We assumed an empirical intermediate relationship between the ultrasonic path length in porous space in concrete filled with water and the free water quantity in the testing piece. From experimental data points we evaluated the parameters of the model and obtained a fitting function for describing the dependence between the effective ultrasonic pulse propagation velocity and the free water quantity in concrete. We also analyzed the error propagation through the algorithm used. It allowed to decide under which conditions the linear approach between the effective ultrasonic pulse propagation velocity and the free water quantity in concrete can be applied.

Keywords:
concrete, modelling, ultrasonic pulse velocity, moisture.

I. INTRODUCTION

The estimation of mechanical properties of concrete, in particular, that of compressive strength, is one of the most important applications of the ultrasonic method [1-6]. However, the effective application of this method is subject to the influence of several factors [5,7-10]. One of these factors is the water which penetrates in concrete through the pores [11-14]. Under the influence of this factor, the compressive strength and the effective ultrasonic pulse propagation velocity dierently behave as the free water content changes. In fact, the latter increases as the free water content in the specimen increases. On the other hand, the compressive strength practically behaves constant when the free water content is modified [2,15].

Under these circumstances, some considerations will be made. The free water in concrete is one of the most important building problems widely considered today. The concrete strength is generally considered its most important property among other characteristics for establishing its quality. This property of the concrete may be influenced by its free water content. Several authors consider that the durability of a building structure is determined by the rate at which water and chemical components that it contains infiltrate and move through the porous building materials [24].

On the other hand, the ultrasonic transmission method is often used for estimating, among other properties, the concrete strength. The application of the ultrasonic transmission technique to measuring the ultrasound propagation velocity through concrete allows to estimate the value of concrete strength [5]. The relationship between these two quantities is often established when the concrete in service is dried. When free water infiltrates and moves through the pores the measuring of the effective ultrasonic pulse propagation velocity can not be directly used for estimating the value of the concrete strength. It is still an open problem. However, the application of the ultrasonic transmission method for determining the ultrasound pulse velocity can allow to estimate the free water content in the porous structure but it can not allow to decide about the damage of the building. The free water content affects the ultrasonic pulse propagation velocity but this measure says nothing about the current strength of concrete. If the dried state of concrete can be obtained back again, then the measuring of the ultrasonic pulse propagation velocity made in that condition could indicate a possible change in the value of concrete strength. This problem has a high attracting social attention in a tropical country where several types of meteorological events often produce a high level of moisture in building during several days. M. Rodriguez [5] has studied the influence of moisture in building in tropical condition finding that this phenomenon produces a significatively high damage in building structure.

M. Kuntz and P. Lavallée [24] have studied the infiltration of liquid and the propagation of the moisture front in
non-saturated porous media using an one-dimensional diffusion equation for describing the phenomenon. They considered a generalized scaling law of the type xt^-a with a = 1/(n + 1) and being n Î Â The water transfer in partially saturated materials was assumed to follow a general nonlinear dillusion equation being water content the variable to determine. They found the values a = 0.58 in brick and a = 0.61 in the limestone diverging both from the value a = 0.5. The results mean that the volume of absorbed water in these materials may be underestimated, having a particular importance for evaluating the effect of water content in building structures. This result expresses that there exists a great theoretical interest about the complexity of this problem along with the practical approach.

E. Ohdaira and N. Masuzawa [16] have examined the possibility of the NDE of concrete from a knowledge of the behavior of the ultrasound propagation velocity with water content in concrete. They measured the ultrasound velocity and the frequency component on ultrasonic propagation as a function of water content in concrete. They made testing pieces of concrete being 10 cm in diameter and 20 cm in length. These testing pieces were embedded in water for 50 days. After this, the water in each test piece was gradually extracted and the ultrasound propagation velocity was conveniently measured for having 15 points in the relationship between water content and the ultrasound propagation velocity. They found that the ultrasound propagation velocity decreases approximately linearly in proportion to the decrease in water content. Similarly the same occurs for the frequency of the maximum transmission.

M. Rodriguez and R. Bonal [23] made a comment on the paper referred above. They considered that the relationship between the ultrasound propagation velocity and water content can better be described by using a statistical model. They fitted the experimental data points obtained by Ohdaira and Masuzawa [16] by applying this method and obtaining an empirical fitting equation. They concluded that the ultrasound propagation velocity exponentially depends on the free water quantity in concrete.

In this work an empirical nonlinear model for the relationship between the effective ultrasonic pulse propagation velocity v and the quantity of free water in concrete (m_w) was considered by us to better describe that behavior. The experimental data given in [16] were considered again. It was also discussed under which conditions the linear approach can be used.

II. MODELLING THE TRAVELLING OF AN ULTRASONIC RAY THROUGH A HETEROGENEOUS MEDIUM

We consider the experimental disposition scheme for applying the transmission technique in ultrasonic method to determine the ultrasonic pulse propagation velocity. An ultrasonic ray travelling through a concrete testing piece containing free water can be modelled by considering an equivalent scheme consisting of a heterogeneous medium composed by three phases of the matter: gaseous phase (air), liquid phase (free water), and solid phase (fine and coarse aggregate, cement). The total transmission time t of the ultrasonic ray in the acoustical system may be decomposed as

(1)

where we have separately considered the time interval spent by the ultrasonic ray for travelling through each phase in the heterogeneous medium: t_a corresponds to the air, t_w corresponds to the free water, and t_s corresponds to the solid phase of concrete. Finally, d represents the time interval spent by the ultrasonic ray for travelling outside the testing piece, and it will be considered a constant value during the measuring procedure on a given testing piece. Its numerical value in all these experiments was taken to be d = 0.6ms [16]. The equation (1) may be written as

(2)

where h_a, h_w, h_s are the ultrasonic ray path lengths in air, in free water, and in solid phase, respectively, h is the testing piece length (200 mm), v_a = 330 m/s, v_w = 1500 m/s, v_s is the ultrasonic propagation velocity in the concrete solid phase, and finally, v represents the effective ultrasonic propagation velocity in the heterogeneous medium (concrete).

When the testing piece does not exhibit moisture at all it can be considered that the term h_w/v_w in the equation (2) is approximately equal to zero and the transition time t of the ultrasonic ray through the testing piece takes its maximum value t_max. It corresponds to a minimum value of the effective ultrasonic propagation velocity v in the heterogeneous medium. Besides, the variable h_a takes its maximum value h_p which corresponds to the ultrasonic ray path length in porous space when only the air is present. In this case, the equation (2) can be written as

(3)

On the other hand, if the testing piece is quite saturated with the free water, we can consider that the term h_a/v_a in the equation (2) is approximately equal to zero, the variable h_w takes its maximum value h_p, the variable t takes its minimum value t_min corresponding to the maximum value of the variable v. Then, the equation (2) will become

(4)

We have assumed in the analysis above that the quantities h_s,v_s, and h are fixed. It is also considered that

(5)

(6)

From the equations (3) and (4) we can find that

(7)

where the constant k₁ = 2364ms/m. By using the equation above, we can estimate the ultrasonic ray path length in porous space, where t_max corresponds to the transition time of the ultrasonic ray in the testing piece when it is saturated with free water, and t_min is the value of the same variable when the testing piece is quite dried.

On the other hand, we can derive a relationship for the dependence of the variable t (transition time of the ultrasonic ray in the testing piece) with the variable h_w (ultrasonic ray path length in the free water in the porous space in concrete). In fact, from the equations (2), (3), and (5) it can be written that

(8)

It can be observed that the correction term d in equation (2) cancels itself and it does not appear in equation (8). Variables t and h_w in the equation (8) are well defined in the intervals t Î [t_max , t_min] and h_w Î [0, h_p]. The variable t can be directly measured, but the variable h_w can not. Here we considered that the variable h_w depends on the quantity of free water (m_w) in the testing piece, but such a dependence will not be a simple one. The way in which h_w depends on m_w is not a known matter for theoretical explaining, but we suppose that such a dependence is influenced by the morphology of pores and by the statistical distribution law of pores in the testing piece volume, among other factors. These aspects will be considered in further investigations. We know that the variable h_w and the variable m_w are well defined in the intervals h_w Î [0; h_p] and m_w Î [0; m_w;max]. We assume the following empirical relationship

(9)

being Q a positive dimensional constant and the exponent g should be in the interval g Î (0, 1). The particular values which the proportional constant Q and the exponent g take in a particular experiment will be determined by the factors indicated above. Introducing the equation (9) in the equation (8) the following relationship is obtained

(10)

where now both variables t and m_w can be directly measured. It is convenient to linearizing the equation (10) by applying natural log function to both sides of the equation,

(11)

Applying a linear regression method to a log-log plotting of the variables (t_max - t) and (m_w) the best straight line is fitted to the experimental data point (m_w, t_max - t), and the model parameters g and Q can be determined. In this plot, the point (0,0) can not be taken into account.

III. EXPERIMENTAL DATA POINTS AND FITTING FUNCTION

In order to evaluate the proposed empirical model, given by the equation (11), we considered the experimental data points obtained by Ohdaira and Masuzawa [16]. They measured the transition time t of an ultrasonic ray in a cylindrical testing piece of concrete being 200mm in length by applying the ultrasonic transmission technique. The corresponding value of the quantity of free water m_w in the testing concrete piece was measured by using a special device. The cotas of error for the variables t and m_w, and the parameter h, all of them directly measured, are given. They applied the procedure to three types of testing piece named A-1, B-1, and C-1. From the experimental data points obtained by these authors we only considered those that were subjects of our analysis. Besides, the experimental data points considered by us were organized as data vectors because it made easier data processing when certain software was used. Ohdaira and Masuzawa [16] measured the transition time t five times for each value of the free water m_w. Statistical validation allows us to average these five values by taking the mean value as one element (in ms) of the data vector called (time). The vector (water) contains the values of the variable m_w in kilogram. These vectors are separately given below for each testing piece.

Testing piece A-1:

(12)

(13)

Testing piece B-1:

(14)

(15)

Testing piece C-1:

(16)

(17)

We estimated the random cota of error for the vector (time) and composed it together with the instrumental cota of error whose value is equal to 0.2 ms [16] for having the vector (cota-time) in ms,

Testing piece A-1:

(18)

Testing piece B-1:

(19)

Testing piece C-1:

(20)

The vector (time) was corrected by the constant value d = 0.6 ms (equation (1)) for obtaining the vector (timec) in ms,

Testing piece A-1:

(21)

Testing piece B-1:

(22)

Testing piece C-1:

(23)

By using the vector (timec) the vector (velocity) was constructed by applying the relationship

(24)

where h = 200 mm is the length of each testing piece and t - d are the elements of the vector (timec). The elements of the vector (velocity) are given in m/s,

Testing piece A-1:

(25)

Testing piece B-1:

(26)

Testing piece C-1:

(27)

Finally, the cotas of error for the transition time and for the length of the testing piece (dh = 0.0005 m) were propagated through the formula (24) written in the form of finite increment,

(28)

The vector (dvelocity) in m/s represents the cota of error vector of the vector (velocity) propagated by the equation above. For each type of testing piece we obtained,

Testing piece A-1:

(29)

Testing piece B-1:

(30)

Testing piece C-1:

(31)

By considering the equation (11) the log-log plotting was applied to each of the three types of testing pieces. Figures 1, 2 and 3 show the results. The best straight line fitting the experimental data points produced the following values for the model parameters g and Q,

Fig 1: Linear regression, testing piece A-1.

Fig 2: Linear regression, testing piece B-1.

Fig 3: Linear regression, testing piece C-1.

(32)

(33)

Figure 4 depicts the plotting of the equation (9) using the values of the model parameters indicated above.

Fig 4: Curves of the empirical model: ultrasonic ray path length in the free water in the porous space in concrete versus free water content

On the other hand, by applying the equation (7), the parameter h_p in meter was estimated

(34)

From the equation (2) and (10), the effective ultrasonic pulse propagation velocity v as a function of m_w is given by

(35)

This equation is the fitting function for the experimental data points (m_w,v). Using the numerical values of the parameters h, k₂, Q, k₁, and g, the equation (35) for each of the testing pieces are respectively,

(36)

(37)

(38)

Figures 5, 6, and 7 separately depict the experimental data points (m_w,v) and the fitting function given in each case by the equations (36), (37), and (38).

Fig 5: Fitting function, testing piece A-1.

Fig 6: Fitting function, testing piece B-1.

Fig 7: Fitting function, testing piece C-1.

IV. DISCUSSION OF THE RESULTS

The influence of moisture in the testing piece of concrete on the effective ultrasonic pulse propagation velocity is highly significant. Experimental results confirm this fact. Though the ultrasonic transmission technique for measuring the effective ultrasonic pulse propagation velocity does not exhibit high accuracy, it does allow to determine the level of such an influence and the behavior of the effective ultrasonic pulse propagation velocity with water content can be well defined. The functional form by which such an influence is determined depends, among other factors, on the morphology of pores and on the statistical distribution law of pores in the testing piece volume. However, we have considered an empirical relationship between the variable h_w and the variable m_w that indirectly very well matches to experimental data points through the relationship (35) between the effective ultrasonic pulse propagation velocity v and the quantity of free water m_w in the testing piece. The finding of this fitting function has been the aim of this work. Figure 4 depicts the empirical relationship between h_w and m_w for each type of testing pieces. It is interesting to observe that the values of the parameters g and Q increases with the increase in cement content in the testing piece in the considered range. It is just a conjecture. This aspect will be considered in further investigations.

We observe that the relationship between v and m_w is not a linearly proportional function. Figures 5, 6 and 7 for testing pieces A-1, B-1, and C-1, show a suitable fitting of the experimental data points by the equation (35).

It is important to check whether the proposed model correctly describes the behavior of the experimental data
(v = f(m_w)). The equation (35) can not be linearized, but if m_w = 0, then the velocity coincides with the minimum velocity v₀ (velocity in dry state) and, consequently, equation (35) can be written in the form

(39)

where y = v is the effective ultrasonic pulse propagation velocity, a = v₀ represents the minimum velocity (in dry state), x = m_w is the free water content, b = Q(k1/k2) and c = g are parameters related with the parameters indicated in the equation(35). When x = 0, from equation (39) is obtained that y = a.

The results of the statistical analysis are presented in table no.1, for each one of the samples A₁, B₁, and C₁.

R R.squared R.squared adj Typical error of the estimate Durbin-Watson .955 .912 .905 .26315 1.187
	Non Standardized		standardized Coeffcients	t	Sig.	Confidence Interval 95 %
	Coeffcients
	B	Typ. Error	Beta			Lower bound	Upper bound
Ln b	-8.944	.197		-45.314	.000	-9.374	-8.514
C	.870	.078	.955	11.177	.000	.700	1.040
Table 1: Test Piece A-1

R R.squared R.squared adj Typical error of the estimate Durbin Watson .948 .898 .89 .31323 1.667
	Non Standardized		standardized Coeffcients	t	Sig.	Confidence Interval 95 %
	Coeffcients
	B	Typ. Error	Beta			Lower bound	Upper bound
Ln b	-8.828	.227		-38.867	.000	-9.323	-8.333
C	.921	.089	.948	10.294	.000	.726	1.116
Table 1: Test Piece B-1

R R.squared R.squared adj Typical error of the estimate Durbin Watson .948 .955 .955 .16848 1.265
	Non Standardized		standardized Coeffcients	t	Sig.	Confidence Interval 95 %
	Coeffcients
	B	Typ. Error	Beta			Lower bound	Upper bound
Ln b	-9.271	.122		-75.767	.000	-9.535	-9.006
C	.793	.048	.977	16.557	.000	.689	.896
Table 1: Test Piece C-1

The model very well fits the experimental data, and the values of parameter c do not significatively differ from those of g obtained in the equation (32). In all the cases they are within the deviation range proposed by the model. One can conclude that the model given in equations (36, 37, 38), appropriately describes the experimental data set.

Next, we analyze under which conditions the behavior of the variable v with the variable m_w can be considered a linearly proportional relationship. From the equation (35) we can obtain the following relationship by applying the Newtonian binomial formula. The quadratic approximation is

(40)

The linear approximation of the equation above is

(41)

and the contribution to the effective ultrasonic propagation velocity of the quadratic term is,

(42)

The vector (errorv) in m/s contains the contributions of the quadratic term in equation (42). For each testing piece we can write,

Testing piece A-1:

(43)

Testing piece B-1:

(44)

Testing piece C-1:

(45)

We can compare the vectors (dvelocity) and (errorv) for deciding when the equation (41) is suitable to apply. When m_w ³ (0.16) in kilogram it can be observed that the contribution of the quadratic term (equation 42) is greater than the propagated cota of error for the experimental value of the effective ultrasonic propagation velocity v given by the equation (28). The equation (41), considering that the exponent g @ 1, approximately well describes the linear behavior considered in [16], but this approach implies a greater error in velocity when m_w is approximately greater than 0.16 kg.

On the other hand, the equation (35) very well quantitatively describes the effect of moisture in concrete on v. In fact, the formula (35) can be conveniently written as

(46)

where 100 * h_p/h represents the ratio in percent between the ultrasonic ray path length in porous space and the testing piece length. Applying this equation to experimental data points for each testing piece of concrete it may be found that such a small value of the ratio 100 * h_p/h as @ 1.45 percent can yield a highly significative value of the fraction dv/v, approximately 14-16 percent. This result is well confirmed by the experimental data points (v, m_w).

V. CONCLUSIONS

The result of this analysis confirms that the empirical fitting function very well quantitatively describes the behavior of effective ultrasonic pulse propagation velocity with the free water content in concrete testing piece and that this empirical model well explains under which condition the linear approximation can be applied. It also allows to estimate the propagation of error for the indirectly-measured involved variables.

ACKNOWLEDGEMENT

We acknowledge Pedro A. Sánchez Fernández from the Department of Foreign Languages for the revision of the English version, and Yoelsy Leiva for the edition. We also thank the Ministry of Higher Education of Cuba for partially financial support of the research. Finally, we express our thankfulness to E. Ohdaira and N. Masuzawa for publishing their experimental data points.

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