MATHEMATICAL MODEL TO ESTIMATE OF THE DETERIORATION OF WOODEN POLES IN CONTACT WITH SOIL USED IN RURAL AREAS

In São Paulo State, mainly in rural areas, the utilization of wooden poles is observed for different purposes. In this context, wood in contact with the ground presents faster deterioration, which is generally associated to environmental factors and, especially to the presence of fungi and insects. With the use of mathematical models, the useful life of wooden structures can be predicted by obtaining "climatic indexes" to indicate, comparatively among the areas studied, which have more or less tendency to fungi and insects attacks. In this work, by using climatological data of several cities at São Paulo State, a simplified mathematical model was obtained to measure the aggressiveness of the wood in contact with the soil.


INTRODUCTION
According to FIORELLI et al. (2008), treated logs have been increasingly used in rustic construction, composition of poles, as well as in agricultural sheds.The reforestation species most used as logs are pine and eucalyptus (MOLINA et al., 2009).In addition, natural logs undergo deterioration by rotting in the region that is in contact with the ground, an area called "upwelling region."It is worth mentioning that no material is inherently durable when subjected to environmental actions, and the biological nature of wood makes it susceptible to fungus and insect attacks.However, if the wood is treated, not only it becomes resistant to decomposing organisms, but also to fire, (FONTE & CALIL JUNIOR, 2007).In this context, it is also important to mention that rotting fungi and bacteria are probably one of the wood degradation forms that most commonly leads to extensive material destruction and consequent loss of resistance (HAYGREEN & BOWYER, 1985).However, wood degradation can be evaluated through mathematical models for which equations are proposed to verify the reduction of its properties.There are also kinetic models that consider wood decay properties due to oxidative processes in terrestrial and aquatic environments (APRILE et al., 1999).On the other hand, there are also models for assessing deterioration of wooden poles above the soil (SCHEFFER, 1971).This work aims to obtain a simplified mathematical model of aggression for wood in contact with the ground, using weather data provided by IAC -Instituto Agronômico de Campinas, in several cities of São Paulo State, Brazil.
Leicester's model LEICESTER et al. (2003) proposed a mathematical model to verify the decomposing of wood in contact with soil, based on monitoring small poles for over 30 years in Australia.The model considers the existence of a lag time to beginning of the rotting process, which, after its onset, evolves at a constant speed ("r" ratio); a new start time may occur, with the establishment of a constant decay with maintenance of performance (Figure 1).This model is used to determine the "Climatic Index" -I ig , which depends on the functions f(R mean ) and g(T mean ), which are dependent on "Average Annual Precipitation" -R mean and "Average Annual Temperature" -T mean , respectively, being N dm the number of dry months per year, defined as a dry month those which the average rainfall is less than or equal to 5 mm (Equations 1-5).
LEICESTER ( 2001) considers the influence of wood types present in a structural element, as well as the speed of decay for different types of wood (Figure 2).To determine the decomposing initiation time of the process (lag) and the speed of decay (r), mathematical equations (equations 5-9) are adopted along with tabulated parameters (Table 1 where, r un,core,stake -rate of decay of new, untreated wood in experimental poles; r un,heart,stake -decay speed rate of untreated core in experimental poles; r un,sap,stake -decay speed rate of untreated sapwood in experimental poles; r un,heart,dc4,stake -decay speed rate of untreated external core of species of class 4 durability; lag un,core,stake -initial time of decay of new, untreated wood in experimental poles; lag un,heart,stake -initial time of decay of untreated core in experimental poles; lag un,sap,,stake -initial time of decay of untreated sapwood in experimental poles, and lag un,heart,dc4,stake -initial time of decay of external core of species of class 4 durability.

MATERIAL AND METHODS
In the present study, a simplified mathematical model was obtained to describe the decaying wood stakes in contact with the ground.This study took into account a total of 113 municipalities in the State of São Paulo, Brazil seeking to encompass locations that were representative across the whole state.Thus, we considered coastal towns and also inner cities in the State.Climate data was provided by IAC -Instituto Agronômico de Campinas, and used to determine the values of I ig , according to the initial model proposed by LEICESTER (2001), and subsequently, to calculate I ig;simplified , according to the simplified model proposed in this work.It is worth mentioning that although data from 113 meteorological stations located in various São Paulo State municipalities, determination of I ig and I ig;simplified values was performed only for stations that had a minimum record period of five years until the collection date.For this reason 11 weather stations were excluded from the study.This decision avoided the analysis of values that were not very representative in terms of annual variations in climate.Moreover, in order to determine whether the cities classified by the regions of decay (Table 5) had populations of temperature, precipitation and different dry months, Ftest (Snedecor distribution) was applied to the variables, dividing them into two populations; I ig lower than 2.50 (I ig <2.50) and I ig higher than 2.50 (I ig > 2.50).F-test was carried out at a significance level of 5% (α = 0.05).

RESULTS AND DISCUSSION
The descriptive statistics of climate data of the municipalities studied is presented for temperature, annual rainfall and dry months (Table 2).The absolute frequency distributions are given as a function of temperature, annual rainfall and dry months from histograms with normal distribution curve (Figures 3-5).

Application of Leicester Model
Descriptive statistics (Table 3) was carried out for the values obtained with the application of climate data applied to the initial Leicester model, available in LEICESTER et al. (2003).4).The descriptive statistics for variables of the populations of I ig <2.50 and I ig > 2.50 were also obtained (Table 5).There is no statistical evidence that the populations of temperature and precipitation are the equivalent for I ig <2.50 and I ig > 2.50.However, there is statistical evidence that the populations of dry months for I ig <2.50 and I ig > 2.50 are comparable.Therefore, one can conclude that the variable N dm (dry months) is not for this data set in analysis a significant variable contributing to the determination of I ig .

Proposition of the simplified model
The simplification proposed herein is the removal of the variable " N dm -dry months", which is the most difficult to obtain.This proposition is justified, as when carrying out the F-test for this variable for the population of I ig <2.50 and I ig >2.50, no evidence has been obtained about the statistical difference between these data sets.Thus, the simplification of the model started with only the removal of the variable N dm .The first model to be used is given below: It was compared the descriptive statistics for th I ig and I ig; simplified (Table 6).The dispersion of I ig and I ig; simplified values was also compared (Figure 6).One can observe in this case a positive displacement (increased values) of the population of data as well as the average.To validate the simplification proposed, it was necessary to observe whether there was correlation between the values of the current model and the simplified model, considering the linear regression and the regression of residuals for I ig and I ig; simplified , respectively (Figures 7-8).It was observed that the r 2 value -coefficient of determination -for this regression was 83.3%.According to ANDERSON et al. (2002), the coefficient of determination r 2 can be expressed as a percentage of the sum of squares that can be explained by the regression proposed, ranging from 0 to 100%.It can be observed (Figure 7) that until about I ig; simplified of 2.50, there was no considerable dispersion of values around the regression line.However, above 2.5, data dispersion around the regression line was high.Until I ig; simplified values between 2.50, the values of I ig; simplified of 2.50 and I ig were more similar.It is observed that the correlation between I ig and I ig; simplified has no independence of errors due to the I ig; simplified function.This feature can be explained by considering that for the same value of the reduction factor (1 -N dm /6), the difference between the values of I ig and I ig; simplified increases proportionally to the second variable.Adopting the same aggressiveness classification as used by LEICESTER ( 2001), we obtained the distribution of I ig; simplified (Table 7).We also present descriptive statistics for the populations of I ig; simplified (Table 8).We carried out the F-test to identify whether it is possible to consider that the two populations I ig and I ig; simplified are similar.Thus, for α = 0.05 we have:  From the test above, we verify that there is statistical evidence that the populations I ig and I ig; simplified are equivalent.
According to the previous procedure, F-test was performed between the populations of temperature and precipitation, to verify whether the populations of these parameters I ig; simplified differ by dividing them again into two populations: municipalities with I ig; simplified lower than 2.50 (I ig; Considering a linear equation: For the values of the x-axis to be equal to the y-axis, it is necessary that the linear coefficient (b) to be zero and the value of the angular coefficient (a) of the line to equal to 1 (equation 15).In other words, if the I ig and I ig; simplified values are numerically equal to the slope, it should be equal to 1 and the linear coefficient be equal to zero.
Thus, the higher the value in module for the linear coefficient (b) and the slope (a), the greater the distance from the identity line (y = x).From the comparison (Table 9) of the values of linear and angular coefficients of the linear regression performed to observe the values on which passages were more similar, it was found that the coefficient values were very close to the values for the identity line only for regression low I ig values.This means that the difference between I ig and I ig; simplified was virtually constant in this case, and this correlation was closest to a straight line parallel to the identity, and thus it was adopted.The other regressions performed were not close to be the identity line (Figure 11).The error between the parallel lines in this case was estimated by the regression equation (equation 16).The populations of I ig and I ig; simplified are not significantly different after the removal of the variable N dm (dry months).However, the higher the I ig value, the greater the difference between I ig and I ig; simplified values obtained.We applied the linear regression (equation 15) for the ten lowest I ig values (Figure 11) and their respective I ig; simplified , to all I ig; simplified values in order to obtain I ig' for all cities considered in analysis.Thus, it was possible to obtain the estimated error in this case.10) obtained for each of the models did not show major differences.

CONCLUSIONS
By applying the initial model of degradation of wood in contact with the ground I ig , and performing the analysis of the variables "Temperature", "Precipitation" and "Dry Months" for populations of I ig < 2.50 and I ig > 2.50, we concluded that there were no statistical evidence of difference between these two populations for the variable "dry months".In order to simplify the initial model, and to determine I ig; simplified values, we compared the population of these values with the I ig population, but this did not achieving statistical significance, indicating that there is difference between these two populations of climatic indices.The best regression between I ig and I ig; simplified was that carried out with the lowest ten I ig values and their I ig; simplified , which is the closest to the identity line.Through this regression, it was possible to propose a final adjusted model of climate index for decaying wood in contact with the ground, the I LL-SP .This model was adjusted to data originated from the State of São Paulo, which did not require the variable N dm (dry months) to obtain values very close to the initial values obtained with the Leicester Model.So, two alternative models are possible to the initial Leicester model: the I ig; simplified and the I LL-SP .Therefore, for data analysis, an appropriate simplification of the Leicester model can be considered.

FIGURE 3 .
FIGURE 3. Histogram of absolute frequency of the average temperatures in the municipalities of the state of São Paulo.

FIGURE 4 .
FIGURE 4. Histogram of absolute frequency of the annual precipitations in the municipalities of the state of São Paulo.

FIGURE 6 .
FIGURE 6. Populations of I ig and I ig:simplified .

F
-test I ig and I ig; simplified H 0 : the variances of I ig and I ig; simplified are similar; H 1 : the variances of I ig and I ig; simplified are not similar.

FIGURE 11 .
FIGURE 11.Difference between the regressions considered .
(16) Subtracting the I ig' value from I ig, the error estimate (E) is obtained (equation 17).The value of average error (E m ) estimation was then determined (equation 18).mean error (E m ) was -0.0438074, in this case.We then obtained the final model adjusted to the data of São Paulo State (equation 16), which will be called the Climate Index of the State of Sao Paulo for decaying of wood in contact with the ground.models are proposed, the simplified model (equation 21), which was shown to have little variation with respect to the initial Leicester model, and the adjusted model to the State of São Paulo (equation 22).The comparison of the results (Table ).

TABLE 1 .
Parameters of decay of untreated external core of wood.

TABLE 2 .
Statistics of climate data.

TABLE 3 .
Statistics of the climate data.

TABLE 4 .
Representative Climate Index of four aggressiveness categories.

TABLE 5 .
Descriptive statistics for the variables of I ig population.: the variance of dry months for I ig<2.50 and I ig>2.50 are similar; H 1 : the variance of dry months for I ig<2.50 and I ig>2.50 are not similar.
F-Test for dry monthsH 0

TABLE 6 .
Descriptic statistics of I ig AND I ig;simplified .

TABLE 7 .
Climate index (representative) of four aggressiveness classes.

TABLE 8 .
Descriptive statistics of the parameters of the populations of I ig;simplified.

TABLE 9 .
Coefficient values of the regressions I ig x I ig;simplified.

TABLE 10 .
Comparation of I ig , I ig;simplified and I LL values.