Tropical wood species: alternative model to determine the characteristic compressive strength perpendicular to grain

Brazil has a vast area of native forests with the potential to be sustainably exploited for application in civil construction. Density is a key factor when analysing the characteristics of different wood species and their future uses, and additionally it can be used as an estimator for other mechanical properties of wood. Furthermore, in a developing country like Brazil, carrying out various characterisation tests for different wood species is sometimes impossible because of the associated high costs. The Brazilian standard ABNT NBR 7190:1997, Design of Timber Structure, governs wood construction and timber structures in Brazil. However, this guideline lacks equations that link the majority of mechanical characteristics to density. Therefore, the main aim of this study was to propose regression models for estimating compressive strength perpendicular to grain, for 17 native species of Brazilian tropical wood, as a function of apparent density and the characteristic value of apparent density. From the results obtained, it is possible to conclude that all the regression models provided in this study were successful in estimating characteristic compressive strength perpendicular to grain (fc90,k), and that the model from Equation 3 in this work, a linear model with one independent variable, might be included in future revisions of ABNT NBR 7190:1997.


Introduction
Southern Forests is co-published by NISC (Pty) Ltd and Informa UK Limited (trading as Taylor & Francis Group) Wood is a natural, abundant, sustainable and versatile material that has long been used in different fields of construction and civil engineering (Almeida et al. , 2019Christoforo et al. 2020a;Mascarenhas et al. 2021;Nowak et al. 2021).
Brazil has ~12% of the world's forests, which represents roughly 493.5 million hectares (Lahr et al. 2021). Considering that and the variety of species that the Amazon rainforest contains, wood and timber constructions have relatively great potential application in Brazil (Wolenski et al. 2020a(Wolenski et al. , 2000b. According to Steege et al. (2016) the Amazon rainforest has over 11 600 tree species, as catalogued between 1707 and 2015. This substantial number has inspired research aimed at characterising species that might potentially replace those that are presently frequently utilised in civil construction (Peixoto et al. 2020).
Overall, density is the simplest wood property to estimate, and it is possible to establish the important relationship between a wood's density and its mechanical properties (Alteyrac et al. 2006;Baar et al. 2015;Almeida et al. 2016;Miyoshi et al. 2018). Krauss (2009) notes that over the last decades there have been different attempts to establish a mathematical relationship between wood density and mechanical properties. In this sense, numerous studies worldwide and in Brazil have tried to estimate the mechanical properties of new and existing wood structures, including the research of Steiger and Arnold (2009), Niklas and Spatz (2010), Steiger et al. (2010), Muñoz and Gete (2013), Christoforo et al. (2020aChristoforo et al. ( , 2020b, Sholadoye and Abubakar (2020), Wolenski et al. (2020a), Lahr et al. (2021) and Nowak et al. (2021).
The European standard EN 338:2003(CEN 2003 and the research developed by Almeida et al. (2014), Cavalheiro et al. (2016), Almeida et al. (2016), Christoforo et al. (2020a) and Soares et al. (2021) are relevant to the use of density as an estimator of wood mechanical properties for Brazilian tropical tree species. Although some previous studies used the density as an estimator for the mechanical properties of the wood, no works have yet established empirical mathematical relationships using the apparent density (ρ12) and the characteristic value of apparent density (ρ12,k) to estimate the characteristic compression strength perpendicular to grain (f c90,k ). Negrão and Faria (2009) explain that compression strength perpendicular to the grain can be understood as the crushing strength, and that it depends on the wood density.
In Brazil, the design of wood and timber structures is guided by the Brazilian standard NBR 7190 (ABNT 1997), 'Design of Timber Structure'. However, the guideline does not present equations that correlate most of the mechanical properties with density. However, EN 338 (CEN 2003) presents different mathematical relationships to estimate the characteristic values of the mechanical properties of wood.
Because considerable time and financial resources and laboratory infrastructure are required to determine the various mechanical properties of wood, the availability of reliable mathematical equations and relationships to do this would represent savings in time and money. Such mathematical relationships can be an incentive for using lesser-known tropical species in Brazilian construction, establishing a more-sustainable wood application. Furthermore, carrying out many tests to characterise the wood of a given tree species is somewhat inaccessible in a developing country like Brazil.
This study aimed to present regression models to estimate compressive strength perpendicular to grain, for 17 native tree species found in Brazilian forests, as a function of apparent density and its characteristic value. We also sought to use the mathematical model presented by EN 338 (CEN 2003) to estimate values of f c90,k for these Brazilian tropical hardwood species. Hence, we present and discuss the error found between the results obtained using the EN 338 model and eight newly developed models. Table 1 lists the 17 tree species native to Brazil studied for developing regression models to estimate the characteristic compressive strength perpendicular to grain (f c90,k ) as a function of apparent density (ρ 12 ) and the characteristic value of apparent density (ρ 12,k ). Notably, the choice of the number of species was meant to encompass a large range of densities to guarantee representativeness.

Material and methods
Wood pieces from homogeneous lots were properly stored, resulting in a moisture content close to 12%, as recommended by the Brazilian standard NBR 7190 (ABNT 1997). Premises and test methods of the Brazilian code were followed to obtain the values of ρ 12 and f c90,k . The Brazilian code also recommends that 12 specimens per species be manufactured and tested in perpendicular compression (Figure 1), and another 12 specimens per species should be used for determination of apparent density values, resulting in a total of 408 experimental determinations.
After the specimens were tested in an Amsler universal testing machine (250 kN load capacity), their moisture contents (MC) at the time of the tests were obtained using the Marrari M51 wood moisture meter (10.23 ≤ MC ≤ 12.74%). Once the samples' MC was determined, the value of f c90,k was corrected to 12% moisture content (f c90.12 ), using Equation 1 Based on the corrected value (f c90,12 ), Equation 2 of NBR 7190 (ABNT 1997) was used to determine the characteristic value (f c90,k ), where f 1 , f 2 and f n denote values in ascending order of the n tested specimens (n = 12 per wood species evaluated): f (2) Regression models (linear and nonlinear) with one free variable and two (β 0 , β 1 ) or three (β 0 , β 1 , β 2 ) parameters, expressed by Equations 3-10, were used to relate ρ 12 and ρ 12,k with f c90,k . It is important to highlight that ρ 12,k was also determined using Equation 2 adapted to density values, and is also the characteristic value of density according to EN 384.
Mean absolute percentage error (MAPE) (Equation 11) was used as a criterion (smallest error) for choosing the most accurate model (i.e., one out of eight models); the coefficient of variation (CV) (Equation 12) and the coefficient of determination (r 2 ) (Equation 13) were also obtained to measure the quality of the adjustment, as follows: where n is the number of samples considered; Y predicted i is the value estimated by the regression model; Y data i is the experimentally determined value; and Ȳ data is the mean value of the experimentally determined results.
Furthermore, analysis of variance (ANOVA), set at a significance level of 0.05, was used to verify each model's significance as well as the terms that make up the models, to help in choosing the best precision fit. By ANOVA formulation, a p value of ≤ 0.05 implies that the model or its terms are considered significant.
Equation 14 shows how to estimate the characteristic compressive strength perpendicular to grain (f c90,k ) based on the characteristic apparent density (ρ 12,k ) according to EN 338 (CEN 2003):

Results and discussion
Tables 2 and 3 summarise the results obtained for apparent density and compressive strength perpendicular to grain, respectively, where: X M denotes the mean value; X K is the characteristic value of the properties; and IC is the mean confidence interval, obtained at the 95% confidence level. The Brazilian standard NBR 7190 (ABNT 1997) establishes that in the estimation of strength values of compression perpendicular to grain, a CV of 28% is admitted. The results presented in Table 3 reveal a CV within 28% for all 17 wood species, with only the CV obtained for Hymenolobium petraeum (Angelim-pedra hardwood) being slightly higher at 28.33%.
Frequency distribution histograms (f r ) of the mean values of apparent density and compression strength perpendicular to grain are presented in Figure 2. The models obtained (and other statistics) for estimation of f c90,k as a function of ρ 12 and ρ 12,k , respectively, are presented in Tables 4 and 5, with the terms considered significant by ANOVA (p ≤ 0.05) highlighted in bold.
These results show that the adoption of the characteristic value of the apparent density (ρ 12,k ) instead of the mean value of the apparent density (ρ 12 ) had little effect on the accuracy of the models. However, there was a small tendency to increase the values of the coefficients of determination (r 2 ) in practically all cases, and this might be a possible explanation of the use of the characteristic value of the apparent density (rather than the mean value) by the European standard EN 338 (CEN 2003) in estimations of f c,90,k .
All eight models obtained were considered significant by ANOVA (p ≤ 0.05). Based on the value of the coefficient of determination and the mean absolute percentage error (MAPE), the model for greater adherence consisted of the quadratic, considering both the apparent density and its characteristic value as estimators of the compressive strength perpendicular to the grain. However, the quadratic term (ρ 12 2 , ρ 12,k 2 ) was not considered significant by the ANOVA, which indicates that the variations in f c90,k values promoted by density (ρ 12 , ρ 12,k ) occur linearly.
With the exception of Equation 3 (a one-parameter linear model), owing to the proximity found to values of the coefficient of determination (r 2 ) and coefficient of variation (CV), and also mean absolute percentage error of the other models estimated by both ρ ap and ρ ap,k , ANOVA was used to verify whether Equations 3-8 could effectively be considered equivalent. The p values in both cases were close to 1,  , employing the characteristic value of apparent density (estimator), is considered to be the most adequate in estimating f c90,k . Figure 3a illustrates the adjustment of a linear model to a parameter to the sets of results determined based on the experiments, and Figure 3b shows the error distribution histogram made using this model.
Approximately 76% of the data (i.e. for 13 of the 17 tree species) estimated by the model provided errors of < 25% (Figure 3b), which reinforces the good precision of the linear model equation. Figure 4 presents the experimental results, the European model, and the equation proposed by this study (i.e. model 3).
The European standard EN 338 (CEN 2003) and the linear model proposed by this study presented satisfactory results (Figures 3a,b and 4). Comparing model 3 with the equation proposed by the European standard, the similarity of the equations, both linear correlations, is observed. In Figure 4, the estimated values are predominantly superior to the experimental values, especially the values provided by EN 338 (CEN 2003). It is important to highlight that Brazilian tropical wood forests have a great variety of native species (hardwoods), whereas the species that might have been considered for the aforementioned determination in Europe are predominantly softwoods. Figure 5 shows the relative error between experimental values and those estimated using the equation provided by EN 338 (CEN 2003)  Thereby, the research results indicated it is possible to adopt model 3 to estimate the characteristic value of compressive strength perpendicular to grain (f c90,k ) as a function of apparent density (ρ 12 ) and its characteristic value (ρ 12,k ).
As discussed, Brazil has a huge variety of wood species, but a lack of knowledge about their properties makes their application as structural elements difficult. Thus, this work presents a relatively simple method to estimate one important mechanical property in structural projects based on wood density, which is a relatively easy property to be obtained. Accordingly, the research results indicate the possibility to adopt model 3, which may facilitate employing wood as structural elements. In this way, species that have been unsustainably exploited can be replaced, which would increase the sustainability of wood harvesting in Brazil's tropical forests.

Conclusions
Costs associated with the several tests for characterisation of different wood species become an impediment to their widespread use in a developing country such as Brazil. Thus, related literature has noted the search for mathematical models that correlate different mechanical properties of wood with its density, both nationally and internationally. The Brazilian standard NBR 7190 (ABNT 1997) still does not present mathematical equations that relate the different properties to density. In this sense, by using linear and nonlinear regression models, we propose eight equations to estimate characteristic compression perpendicular to the Species common name        The values found for each mathematical model were compared with estimates using the equation proposed by the European standard EN 338 (CEN 2003). From the results obtained in this research it can be concluded that:  Brazil has a vast number of native tree species with potential for use for timber; therefore, the existence of normative equations that correlate wood density with its mechanical properties is of great importance.  From the eight regression models proposed and analysed here, two presented the lowest MAPE values: Equation 9 using apparent density, and Equation 5 using characteristic density.  When considering the smallest mean absolute percentage error (MAPE) and quality of the fit of the curve r 2 , Equation 3 was observed to be the most satisfactory to determine compressive strength perpendicular to the grain for the 17 studied species, when considering either the apparent density or the characteristic density.  When comparing the values obtained through European standard EN 338 (CEN 2003) and those obtained through the eight regression models, we found that those estimated by Equation 3 were the ones with the lowest error in relation to EN 338 (CEN 2003).  Because the native Brazilian hardwood species are different from the European hardwoods, which have mostly originated from plantations, this generated differences in the mathematical models.  Finally, all regression models proposed in this research proved to be accurate to estimate f c90,k; ; especially, the model obtained from Equation 3 could be incorporated in future versions of the Brazilian standard NBR 7190 (ABNT 1997).