Regression analysis of compressive strength and hardness

The rebound index vs. strength relationship can be determined if the experimental data are available. The usual practice is to consider the average values of the replicate compressive strength and NDT results as one data pair at each strength level. The data pairs are usually presented using the NDT value as the independent variable (along the X axis) and the compressive strength as the dependent variable (along the Y axis). Regression analysis is performed as a conventional least-squares analysis on the data pairs to obtain the best-fit estimate for the strength relationship. The technical literature calls the attention that the boundary conditions of the conventional least-squares analysis are violated in the case of rebound index vs.

strength relationships (Carino, 1993), therefore it is not recommended because the uncertainty in the strength relationship would be underestimated.

The two most important limitations of the conventional least-squares analysis are: 1) no error (variability) is considered to be existing in the X variable (here: the rebound index); 2) the error (i.e. standard deviation) is constant in the Y variable (here: the compressive strength) over all values of Y. The first assumption can be violated by the uncertainty of the NDT method – characterized by its within-test coefficient of variation (which may have a larger variability than that of the strength tests); and the second assumption can be violated because standard deviation may change by the compressive strength both for strength testing and NDT.

Mathematical statistics considers a data plot scatter to be heteroscedastic, when the error (i.e. standard deviation) is not constant in the Y variable; the variation in Y differs depending on the value of X (Tóth, 2007). Regression analysis of heteroscedastic data needs performing a Y variable transformation to achieve homoscedasticity (constant standard deviation in the Y variable). Conventional least-squares analysis regression can be used only if the data are homoscedastic. A suitable Y variable transformation is the Box-Cox Normality Plot (NIST, 2009) which is defined by a λ transformation parameter as:







( Y

^

1 ) / )

Y (

T Eq. (2.18)

For λ = 0, the natural logarithm of the data is taken; this is the most common estimation in the case of rebound index (R) vs. strength (f) relationships. If a linear relationship is used, it is formed as follows:

ln (f) = a + B·ln (R) → f = e

^a

·R

^B

= A·R

^B

Eq. (2.19)

In Eq. (2.19) the exponent B determines the degree of nonlinearity of the power function. If B = 1, the strength relationship is a straight line passing through the origin with a slope of A. If B ≠ 1, the relationship is nonlinear.

Regarding the problem of error in the X variable the regression procedure proposed by Mandel is

suggested instead of the conventional least-squares analysis regression (Carino, 1993; ACI, 2003b). The

most important difference to the conventional least-squares analysis is that Mandel’s method minimizes the

sum of squares of the deviations from the regression line in both X and Y directions, on the contrary to the

conventional least-squares analysis which minimizes only the deviations from the regression line in Y

direction.

Graphical representation of the surface hardness vs. compressive strength relationships usually indicates heteroscedastic behaviour; i.e. increasing standard deviation in strength (Y variable) for increasing rebound index (X variable). Even the manufacturer of the original rebound hammers suggests increasing standard deviations to be taken into account for increasing rebound indices (Pascale et al, 2003). Examples for the heteroscedastic behaviour are indicated in Fig. 2.17a, b and c (Greene, 1954; Zoldners, 1957; Schmidt, 1951).

It should be highlighted that researchers usually do not separate the experimental data of the corresponding rebound index vs. compressive strength results by different influencing parameters in the graphical representations – and the situation has not changed during the last 60 years. Therefore, exclusively the univariate regression curves are available in the technical literature.

0 20 40 60

15 20 25 30 35 40 45 50

rebound index, R28, – fcm

fcm+s

fcm–s 60

40

20

0

15 20 25

30 35 40 45 50 fc,28, ×10² psi

15 20 25 30 35 40

50 40

30 20

10

rebound index, R28, – fcm

fcm+15%

fcm–15%

fc,28, ×10² psi 50

40

30

20

10

15 20 25

30 35 40

0 10 20 30 40 50

10 20 30 40 50

rebound index, R28, – fcm

fcm+s

fcm–s fc,28,N/mm²

50 40 30 20 10 0

10 20 30 40 50

Fig. 2.17 Heteroscadastic behaviour of the rebound hardness vs. compressive strength relationship (1 psi = 6.894×10

^-3

N/mm

²

, a) Greene, 1954; b) Zoldners, 1957; c) Schmidt, 1951).

Surface hardness and compressive strength of concrete, however, are depending on several parameters

(e.g. type of cement, amount of cement, type of aggregate, amount of aggregate, compaction of

structural concrete, type of formwork, method of curing, quality of concrete surface, age of concrete,

carbonation depth in the concrete, moisture content of concrete, mass of the structural element,

temperature and state of stress) of which influences may be represented when a multivariate regression

analysis is carried out.

The most significant influencing parameters for the compressive strength of normal weight concretes are the water-cement ratio, the type of cement and the age of the concrete. The amount of cement, the amount of aggregate, the storing method and further concrete technology parameters have only secondary influences. The type and amount of aggregate can have significant influence in the case of lightweight aggregate concretes.

It is shown here as an example that non-separation of experimental data can lead to completely misleading trends of the analysis and the separation of experimental data can clearly uncover the real material behaviour and, therefore, gives the only way to understand the mechanisms of the rebound surface hardness testing of concrete. Two from the earliest publications are referred as example, i.e. papers by Schmidt (1951) and Herzig (1951). Both papers are based on detailed laboratory tests carried out at EMPA Laboratories, Switzerland.

Schmidt analysed in his paper the experimental results of 550 cube specimens tested both for rebound surface hardness and compressive strength. The non-separated results are adopted in Fig. 2.17c where the univariate regression curve power function is represented together with the lower and upper bound curves based on the reported deviations from the mean values. It can be realized that an apparent heteroscedastic behaviour appears when the compressive strength of concrete is represented as a dependent variable of the rebound index. Herzig was the only researcher who presented the experimental results from the same tests but the data were reasonably separated by the amount of cement, the storing method and the age of concrete at testing (water content or water-cement ratio is not given in his paper).

As a primary influence, the separation by the age of concrete provides high-contrast differences. Three typical representative curves are selected from his several separate curves (Fig. 2.18a).

0 10 20 30 40

10 20 30 40

rebound index, R, –

fcm, N/mm² a)

air, 200 kg/m³ wet, 300 kg/m³

air, 300 kg/m³

2d

3d 5d 28d

40

30

20

10

0

10 20 30 40

2d

2d 3d

3d

5d

5d

28d

28d

10 20 30 40

0 10 20 30

40 b)

40 30 20 10

0

10 20 30 40 40

30

20

10

0

rebound index, R, – fcm, N/mm²

Fig. 2.18 Influences of data separation on the rebound index vs. compressive strength relationship (after Herzig, 1951).

Herzig’s results are adopted in Fig. 2.18b also as one population of data to highlight the possibility to find a false empirical regression curve corresponding to rebound index vs. compressive strength responses as non-separated data.

It can be observed that a strong correlation of a power function can be resulted. Here, the heteroscedastic

behaviour is not pronounced as the data covers only 56 data pairs and not the complete test result of the

550 cubes. It can be realized from Herzig’s original, separated data analysis that further primary influences

could come into play besides the age of concrete (e.g. water-cement ratio) not mentioned in his analysis.

In document Rebound surface hardness and related properties of concrete (Pldal 34-37)