FOR CALCULATING THE MODULUS OF ELASTICITY OF WOODEN BEAMS OF STRUCTURAL DIMENSIONS

This study aims to present an alternative calculation methodology based on the Least Squares Method for determining the modulus of elasticity in bending wooden beams of structural dimensions. The equations developed require knowledge of three or five points measured in displacements along the piece, allowing greater reliability on the response variable, using the statistical bending test at three points and non-destructively, resulting from imposition of measures from small displacements L/300 and L/200, the largest being stipulated by the Brazilian norm NBR 7190:1997. The woods tested were Angico, Cumaru, Garapa and Jatoba. Besides obtaining the modulus of elasticity through the alternative methodology proposed, these were also obtained employing the Brazilian norm NBR 7190:1997, adapted to the condition of non-destructive testing (small displacements) and for pieces of structural dimensions. The results of the modulus of elasticity of the four species of wood according to both calculation approaches used proved to be equivalent, implying the good approximation provided by the methodology of calculation adapted from the Brazilian norm.


INTRODUCTION
The use of wood has grown over the last few years, because it is a material of renewable source, low density and good mechanical performance, usually used in civil and rural buildings, performing the structural role as beam elements, columns and others (CHRISTOFORO, 2011).
The design of wooden structures, as well as of other materials, requires the knowledge of some variables, including the modulus of elasticity, obtained through experimental tests advocated by normative documents, which may be destructive or not.
Because the wood is an orthotropic and heterogeneous material, aiming to increase reliability, its characterization in bending is more appropriate if performed in parts with structural dimensions.In this context, only international normative documents can be cited (EN 789:1995, ASTM D4761:1996, ASTM D198:1997), since the Brazilian norm ABNT NBR 7190:1997 (Wood Structures Design), which deals with the wood characterization, contemplates only the condition of destructive testing on sample parts with small dimensions and free from defects (Pigozzo et al, 2000;FIORELLI, 2005;MIOTTO & DIAS, 2009).
Structural models contained in the normative documents mentioned above consist of the static bending tests at three and four-points, obtaining the modulus of elasticity from the knowledge of two measurements of force and successive displacements, defined for the stretch of elastic and linear behavior of the material, based on the value of the maximum force applied to the part.The mathematical models of calculations contained in these codes do not include criteria for optimality (idealized formulations in the search for optimal solutions -neighboring), with displacements in the trial obtained from one or two different points along the elements.
As mentioned before, the characterization of wooden pieces of structural dimensions can also be accomplished through non-destructive testing, aimed at determining the physical and mechanical properties of a structural element without changing its use capabilities (ROSS et al., 1998, WANG et al, 2008;LIANG & FU, 2007;DONG and HAI, 2011;SALES et al, 2011).The advantage of employing non-destructive testing constitutes waiving the extraction of sample parts, further enabling the study of structural integrity (OLIVEIRA & SALES, 2002;MINÁ et al, 2004;BURDZIK & NKWERA, 2002), commonly performed by means of tests with transverse and ultrasound vibration.
From the above, the mathematical models contained in normative documents for the calculation of the modulus of elasticity in bending do not include optimality criteria, and with respect to the usual non-destructive testing (transverse and ultrasound vibration), the need to acquire specialized equipment for determining the modulus of elasticity is emphasized.This paper proposes an alternative calculation methodology, based on the Least Squares Method and on the three-point bending test, non-destructively, for determining the modulus of elasticity for bending in pieces of lumber in structural dimensions, having evaluated the Angico , Cumaru , Garapa and Jatoba woods.

MATERIAL AND METHODS
For validity of the use of Euler Bernoulli beam theory used in the calculation of the modulus of elasticity by this method, the wooden beams must comply with the L/h≥21 relation (ROCCO LAHR, 1983), disregarding the effect of the shearing stress in the calculation of displacements, where L is the effective length of the piece (distance between supports -span) and h is the height of the cross section.
The experimental test used to determine the modulus of elasticity is considered nondestructive, because the highest values are in displacement in the experiments (midpoint) limited to the reasons L/200 and L/300 (L in cm), the largest of them being defined by the Brazilian norm ABNT NBR 7190:1997 as a measure of small displacements.Besides obtaining the modulus of elasticity with displacements restricted to these limits, they were also obtained with the use of the methodology prescribed in the norm ABNT NBR 7190:1997, adapted for nondestructive testing and for structural dimension parts, aiming to verify the differences between them.
Figure 1 illustrates the structural test schemes used to calculate the modulus of elasticity, where L is the span of the piece, F is the force applied at the midpoint of the span and b and e are the dimensions of the base and height of the rectangular cross section, FIGURE 1. Testing setup.
Figure 1a shows the test scheme used to calculate the elasticity modulus according to NBR 7190:1997 (simplified approach), using an only dial indicator positioned at the midpoint of the piece.Figures 1b and 1c respectively illustrate the structural designs used to calculate the modulus of elasticity through the alternative methodology, using three and five dial indicators equally spaced along the elements.
From the structural test design illustrated in Figure 1b, equidistant experimental displacements of the restraints (δ1and δ3) are measured when the displacement at mid-span (δ2) is equal either to L/300 or L/200.Similarly, the displacements (δ1, δ2, δ4 and δ5) of the dial indicators positioned in sixths of the span (Figure 1c) are obtained.
The alternative methodology used to calculate the effective modulus of elasticity is presented for the condition of three dial indicators (Figure 1b), being analogous to the condition of five (Figure 1c).
From the Euler Bernoulli beam theory, analytical displacements in the positions of dial indicators 1, 2 and 3 in Figure 1b are expressed by Equations 1 and 2, rewritten as a function of the longitudinal elastic modulus (δ 1 (E), δ 2 (E) e δ 3 (E)).
Equation 1 is equivalent to the proposal by the Brazilian norm NBR 7190:1997 for the calculation of the elastic modulus (E m ), also consisting of a (simplified) calculation methodology to be evaluated in this work.
The elastic modulus to be calculated with the information derived from the test model of Figure 1b starts out from the idea of least squares (Equation 3), aiming to determine the value of the modulus of elasticity so that the residue generated between the analytical (δ i (E)) and experimental (δ i ) values of displacements is the least possible.

 
( ) ( ) Substituting Equations 1 and 2 in Equation 3 and deriving and equating the latter to zero one can get to the effective modulus of elasticity (E m,3 ) for the structural design with three dial indicators, expressed by Equation 4, proving this the minimum and global point by the criteria of the second derivative ., , ( ) By using the same methodology for the calculation of Equation 4 we arrive at the expression for the calculation of the longitudinal modulus of elasticity (E m,5 ) in parts of lumber in structural dimensions using five dial indicators (Figure 1c), expressed by Equation 5.
Equations 1, 4 and 5 along with the displacement restrictions L/300 and L/200 were used to calculate the modulus of elasticity for Angico, Cumaru, Garapa and Jatoba woods.Woods had 35cm × 50cm × 130cm medium size, and 12 pieces of each species were evaluated.

RESULTS AND DISCUSSION
Tables 1, 2, 3 and 4 respectively show the results obtained for the modulus of elasticity (E m ; E m,3 ; E m,5 ) of the wood parts of the species Angico, Cumaru, Garapa and Jatoba for the displacement averages L/300 and L/200.Figure 2 shows the residual graphs of the modulus of elasticity obtained from the use of the Brazilian NBR 7190:1997 (E m ) for the wood species Jatoba and Angico, in order to verify that the samples are in accordance with the assumptions of hypothesis testing.At the employment of the hypothesis testing procedure we start from the premise that both samples are drawn from independent populations, described by a normal distribution, and that the standard deviations or variations of the populations are equal (MONTGOMERY, 2005).It is observed that the points distributed uniformly along the line for Angico wood meet the conditions of normality and homogeneity required for validation of this test, which does not occur with the other species (Cumaru, Garapa and Jatoba).Alternatively, the Johnson transformation for achieving standardization and homogeneity of data was applied to wood species Cumaru, Garapa and Jatoba.The result of the confidence interval (Probability Plot for Transformed Data) for Jatoba wood (Figure 3), as well as for Garapa and Cumaru woods, proves the normality of the transformed data by presenting P-value of 0.978, greater than 0, 05 (CHOU et al. 1998).After normalization of the modulus of elasticity of Cumaru, Garapa and Jatoba woods, in order to verify the statistical equivalence between the values of the effective modules (E m,3 a E m,5 ) with the ones coming from NBR 7190:1997 (E m ) for the four wood species, the hypothesis test of the means of two independent populations was used, with the results for the Angico timber shown in Tables 5.The P-values greater than 0.05 (5% significance) or the appropriateness of the zero in the confidence interval proves the statistical equivalence between the modulus of elasticity (MONTGOMERY, 2005), which are also equivalent for the other three wood species.

CONCLUSIONS
The present methodology enables obtaining the modulus of elasticity in bending wood parts of structural dimensions with higher reliability for being based on optimality concepts, allowing the use of three or five values in experimentally measured displacements.
The restriction of displacement in the bending test (L/300 and L/200) confers the present methodology nondestructive character, which is interesting for the possibility of being able to use the part after tested.
The results between the modulus of elasticity from the use of the Brazilian ABNT norm NBR 7190:1997 (E m ) equation, adapted for pieces of structural dimensions and methodology of nondestructive testing, with what derives from the approach proposed herein (E m,3 e E m,5 ) were both equivalent.However, the results obtained cannot be extrapolated to other woods of the same or different species, justifying the use of the present calculation approach proposed.
Thus, the simplified model, adapted from the Brazilian NBR 7190:1997 norm, was able to provide results close to those obtained with the use of the least squares-based methodology (alternative), presenting itself as an effective alternative calculation methodology.

FIGURE 2 .
FIGURE 2. Normality test of the E m modulus of elasticity for Angico (a) and Jatoba (b) timber.

FIGURE 3 .
FIGURE 3. Johnson transformation for data standardization of Jatoba wood.

TABLE 5 .
Hypothesis testing of the Angico timber.