1. Introduction
Wood is a renewable resource with great versatility in civil construction. From a structural point of view, it has an excellent strength-to-density ratio when compared to other materials used in civil construction, i.e., steel and concrete [
1,
2,
3,
4].
In Brazil, 60% of the land surface is made up of forests, which corresponds to 500 million hectares. Of this area, 98.03% is composed of natural forests, while the remaining 1.97% are planted forests [
5]. Wood species from native forests are commonly used in structural design, and the Brazilian standard ABNT NBR 7190 [
6] establishes the premises and calculation methods for the project of wooden structures and presents physical and mechanical properties of several species. Some species are no longer commercialized in the country, which motivates the use of new species for structural purposes, among other applications.
As stated in the Brazilian standard NBR 7190 [
6], several properties need to be known for the correct design and use of wood structures; e.g., tensile and compressive strength parallel (
ft0,k and
fc0,k) and normal to the fibers (
ft90,k and
fc90,k), shear strength parallel and normal to the fibers (
fv0,k and
fv90,k), embeddedness, apparent density (ρ), modulus of elasticity (E), and toughness, among others. However, the vast majority of these tests require the use of specific equipment that is only found in large research centers. This fact makes characterizations of this nature difficult, given the scarcity of laboratories of this magnitude in Brazil [
7]. Due to this scarcity, the standards establish numerical correlations that allow the determination of mechanical properties through mathematical models, e.g., Equations (1)–(3) of NBR 7190 [
6], where
fc0,k is the characteristic fiber parallel compressive strength,
ft0,k the characteristic fiber parallel tensile strength, and
fm,k the characteristic flexural tensile strength. Such a situation can be seen in the vast majority of structural design codes.
As can be seen, the characterization of a wood species can be carried out more quickly by not having to perform all the existing experimental tests. However, the Brazilian regulations present few predictive relationships, a factor that has motivated the development of research in this area. As an example, Lahr et al. [
7] used apparent density as an estimator of compressive strength, since density is an easily obtainable property. In their research, a coefficient of determination (R
2) of 74.87% was achieved in a logarithmic regression model. Lahr et al. [
8] analyzed the relationship between characteristic values of shear and compressive strength, both evaluated in the direction parallel to the fibers, concluding that the coefficient value 0.24 (
fv0,k = 0.24 × f
c0,k) is more accurate than the value of 0.12 established by the Brazilian standard [
6]. Other researches have also been undertaken with the same purpose using the same or other methodology [
9,
10,
11,
12,
13,
14].
Another important characteristic to understand the uncertainties of the materials and to enable the simulation of test situations for wood, or any material, is to know the statistical distribution that best represents its behavior. One way to perform this check is through likelihood (MLE), as performed by Barker [
15] in wood phylogenetic analyses. Thus, since wood strengths are important in many analyses, knowing their probabilistic distribution helps to find their characteristic value based on the percentile of probability of occurrence. Pang et al. [
16] used MLE to avoid misinterpretation in order to understand the meaning of the developed test results that are hidden by suffering an unintentional failure mechanism.
There is also the need to establish the reliability of the proposed equation. One methodology to ensure this probability of failure is that specified in Annex D of EN 1990 [
17] which uses the coefficients of variation of the variables, the predicted values, and experimental values to define the predictive model’s calibration coefficient (γ) that should be applied to the new expression to ensure its reliability.
Thus, this article has as its main objective the proposition of a new relationship between mechanical properties of wood from native Brazilian species. This study will allow future versions of the Brazilian standard NBR 7190 [
6] to have wider coverage of predictive numerical models. It also presents the distribution type test that can represent the strength properties of wood, in addition to the calibration of the equation by the methodology of EN 1990 [
17].
In the case of this article, the investigation was carried out by associating the resistance of wood to different loads involved in simple bending. This paper investigated prediction models through the use of various regression models based on ANOVA, and the accuracy of these models correlating characteristic values of flexural strength (fm,k) with shear (fv0,k) in the direction parallel to the fibers of hardwoods from native forests. Furthermore, the contribution of this new expression is due to the fact that the wood shear test does not require a displacement transducer, making it possible to perform it in a greater number of national laboratories that do not have such a transducer.
2. Materials and Methods
As previously mentioned, the objective of this work is to define a relationship between wood resistances, so that if the wood shear resistance is known, the wood flexural resistance can be obtained (
fm,k). This relationship is possible to determine since there are normal stresses and shear stresses involved in simple bending [
18].
Therefore, an experimental study was built that contemplates 17 hardwood species covering the five strength classes (C20, C30, C40, C50, and C60) of the Brazilian standard NBR 7190 [
6]. It is worth pointing out that the strength classes of the Brazilian standards are ordered according to the compressive strength parallel to the fibers (
fc0,k) of the wood: 20 MPa ≤
fc0,k < 30 MPa (C20); 30 MPa ≤
fc0,k < 40 MPa (C30); 40 MPa ≤
fc0,k < 50 MPa (C40); 50 MPa ≤
fc0,k < 60 MPa (C50); and 60 MPa <
fc0,k (C60). The wood batches were composed of 12 boards of each species, with dimensions 12 cm × 12 cm × 300 cm, free of defects and made up almost entirely of core wood (wood from the center of the tree trunk). The samples were taken randomly throughout the batches, according to the prescriptions of the Brazilian standard ABNT NBR 7190 [
6]. Furthermore, no more than one specimen was taken per test from the same piece of wood.
Table 1 presents the common and scientific names and origins of the species studied. It is worth noting that all the results obtained were for a humidity of 12%, as indicated by the Brazilian norms [
6].
2.1. Experimental Tests
According to Brazilian normative premises [
6], 12 specimens are required for the determination of each property analyzed. The properties obtained in this research consisted of apparent density, compressive strength parallel to the fibers (
fc0—
Figure 1a,b), flexural strength (
fm—
Figure 1c,d), and shear strength in the direction parallel to the fibers (
fv0—
Figure 1e,f). Of these properties, it is worth mentioning that the compressive strength parallel to the fibers had the purpose of classifying each wood species according to the strength classes of the Brazilian standard, a classification established based on the characteristic value of the compressive strength in the direction parallel to the fibers [
6]. All samples were cut following Brazilian normative premises [
6], in which a circular saw was used, always keeping the faces parallel and smooth.
As established by the Brazilian standard [
6], once the wood’s moisture content is determined, the values for mechanical properties should be corrected (Equation (4)) for 12% moisture content, which is considered the equilibrium value by this standard. It is noteworthy that, for the correct use of this equation, the moisture value needs to be less than 20% and can be used in all wood species. In this research, an oven (temperature ranging from 40 °C to 90 °C) was used to reach this humidity, being subsequently corrected.
In Equation (4), f12 represents the strength property corrected for 12% moisture content, fU% is the strength property obtained at the given humidity, and U% is the sample’s moisture content.
After correcting the strength properties for the 12% moisture content, it was possible to determine the characteristic strength values (
fk) for each species, as presented in Equation (5). From this equation, n is the number of samples (12) and f1 to fn are the strength values of each specimen arranged in increasing order. The characteristic value is considered to be the highest between the lowest value obtained experimentally (
f1), 70% of the average of the evaluated samples, or the value obtained considering “
n/2” specimens.
2.2. Statistical Analysis
All statistical analyses were conducted in Jupyter Notebook version 6.4.12. The Pearson correlation (−1 ≤ r ≤ 1) was evaluated between the properties considered in the present investigation (Equation (6)). From this test, the closer to −1 or 1, the higher the correlation between the properties. Additionally, analysis of variance (ANOVA, 5% significance) was used to verify the significance of the tested correlations. By the formulation of this test, a
p-value (probability) less than 0.05 indicates that the correlation is significant, and not significant otherwise.
Table 2 shows the regression models considered in the statistical analyses, where f
v0,k is the independent variable,
fm,k is the dependent variable,
αx consist of the constants fitted by the least squares method, and
ε is the random error. The ANOVA of the regression models was also calculated at the significance level of 5%. In this case, the selected null hypothesis was the non-representativeness of the tested models (H0: β = 0), whereas the representativity was an alternative hypothesis (H1: β ≠ 0). P-values higher than the level of significance implied accepting H0 (the model tested was not representative; variations of
fv0,k were unable to explain the variations in
fm,k). In addition, the values of the adjusted coefficient of determination (R
2) were obtained to evaluate the probability of changing the independent variable
fv0,k to explain the predictive variable
fm,k. Following this procedure, the model selection that presents the best fit to the experimental results was carried out.
Another statistical analysis performed was which probability distribution the resistance variable can be predicted from. For this, the maximum likelihood theory (MLE) was applied, which is, in a reduced way, a methodology to estimate which probability distribution can best represent the behavior of the data, managing to represent by its expression the data behavior [
19,
20].
2.3. Model Calibration
In order to improve the discussion and make the results more applicable at the level of an application in structural design, this section aims to present the theory of calibration of the prediction equation. This procedure is justified since the prediction equations will present a variability with the real value due to several model uncertainties. Therefore, it is necessary that the prediction and action equations be calibrated to present a minimum probability of failure (pf).
For this process, Annex D of the European standard Eurocode: Basis of structural design EN 1990 [
17] was used. Such calibration is called “Design assisted by testing” and is used to determine calibration coefficients in predictive models based on experimental values [
21].
For the calibration coefficient determination, the sample data of the tests performed according to the recommendation of EN1990 [
17] were used. In addition, it is necessary that some conditions are met; e.g., all variables in the model must follow a normal or log-normal distribution, as well as the resistance values found experimentally. Another consideration is that the variables involved in the resistance function must not be correlated, since this will not be considered.
For the predictive model calibration, it is necessary to build the resistance factors. In the case of EN 1990 [
17] these would be the design resistance values (
rd) and the characteristic resistance value (
rk), given by Equations (7) and (8).
where
Qrt is the logarithmic standard deviation of the prediction resistance,
Qδ is the logarithmic standard deviation of the error, and
Q is the logarithmic standard deviation of the product of rt (predictive model equation representing the resistance function) and
δ (scatter error).
αrt is the ratio of
Qrt and
Q and
αδ is the ratio of
Qδ and
Q. The coefficient
k∞ is worth 1.64 and
kd,∞ and
kd,n are worth 3.04 for
n > 30. In case of fewer samples, refer to Table D1 and D2 of the EN 1990 [
17].
The coefficient b of the line is found by least squares, as shown in Equation (9), which represents the experimental value from the database.
With all the data above, it is possible to obtain the predictive model’s calibration coefficient by means of Equation (10).