Abstract
Accurate estimation of average modulus of elasticity in compression parallel to the grain (Ec0) is of paramount importance for rational sizing of timber structures, given the use of this property in the estimation of stability of compressed parts (ultimate limit state, ULS) and in calculation of excessive strains (serviceability limit state, SLS). In Brazil, if values cannot be experimentally determined, ABNT NBR 7190 (1997) allows for estimation of Ec0 through relations to average modulus of elasticity both in tension parallel to the grain (Et0) (Ec0 = Et0) and in bending (EM) (Ec0 = EM/0.90). This research aimed to access the efficiency of these relations by testing 30 tropical wood species. The analysis of variance results showed that Ec0 and Et0 were statistically equal. However, Ec0 and EM/0.90 were not statistically equal, and the method of least squares resulted in a coefficient of 0.98, which was 8.89% higher than the one suggested by ABNT NBR 7190 (1997) and close to 1, thus, validating the results of ANOVA, which pointed on the equivalence between Ec0 and EM (Ec0 = EM). As an alternative to simplified equations of the standard, two-parameter regression models were used. The geometric model with R² = 91.67% proved to be the model of best fit, which demonstrated that Ec0 could be calculated as a function of EM.
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Analysis of Relations between the Moduli of Elasticity in Compression, Tension, and Static Bending of Hardwoods
João P. B. Almeida,a,* Vinícius B. M. Aquino,b Anderson R. V. Wolenski,c Cristiane I. Campos,d Julio C. Molina,d Eduardo Chahud,e Francisco A. R. Lahr,f and André L. Christoforo a
Accurate estimation of average modulus of elasticity in compression parallel to the grain (Ec0) is of paramount importance for rational sizing of timber structures, given the use of this property in the estimation of stability of compressed parts (ultimate limit state, ULS) and in calculation of excessive strains (serviceability limit state, SLS). In Brazil, if values cannot be experimentally determined, ABNT NBR 7190 (1997) allows for estimation of Ec0 through relations to average modulus of elasticity both in tension parallel to the grain (Et0) (Ec0 = Et0) and in bending (EM) (Ec0 = EM/0.90). This research aimed to access the efficiency of these relations by testing 30 tropical wood species. The analysis of variance results showed that Ec0 and Et0 were statistically equal. However, Ec0 and EM/0.90 were not statistically equal, and the method of least squares resulted in a coefficient of 0.98, which was 8.89% higher than the one suggested by ABNT NBR 7190 (1997) and close to 1, thus, validating the results of ANOVA, which pointed on the equivalence between Ec0 and EM (Ec0 = EM). As an alternative to simplified equations of the standard, two-parameter regression models were used. The geometric model with R² = 91.67% proved to be the model of best fit, which demonstrated that Ec0 could be calculated as a function of EM.
Keywords: Hardwood; Regression model; Mechanical properties; Timber structure
Contact information: a: Department of Civil Engineering, Federal University of São Carlos (UFSCar), São Carlos, Brazil; b: Institute of Engineering of Araguaia, Federal University of Southern and Southeastern Pará (UNIFESSPA), Santana do Araguaia, Brazil; c: Federal Institute of Santa Catarina (IFSC), São Carlos, Brazil; d: São Paulo State University (UNESP), Itapeva, Brazil; e: Federal University of Minas Gerais (UFMG), Belo Horizonte, Brazil; f: Department of Structural Engineering, University of São Carlos (EESC/USP), São Carlos, Brazil; *Corresponding author: boff.joaopaulo@gmail.com
INTRODUCTION
Considered the material of the future (Kuzman and Sandberg 2017; Żmijewki and Wojtowicz-Jankowska 2017), timber is becoming increasingly popular and widely applied in civil construction (Wieruszewski and Mazela 2017). This is not only because wood is a natural, biodegradable, renewable, recyclable and, hence, environmentally friendly raw material (Wang et al. 2014; Araujo et al. 2016; Lima, Jr. et al. 2018; Souza et al. 2018), but also due to characteristics that make it an efficient building material compared to traditionally used steel and concrete (Ramage et al. 2017).
One such characteristic of wood is its excellent mechanical strength-to-density ratio (Pries and Mai 2013; Ramage et al. 2017; Huber et al. 2018; Lima, Jr. et al. 2018) that favors the use of wood in construction applications where weight of the structure itself presents a considerable load (e.g., roofs, bridges, and tall buildings) as well as in buildings subjected to seismic loading, given that heavier structures are subjected to higher seismic load (Ramage et al. 2017).
Given the effectiveness of wood as a structural element, timber constructions have become the most common, practical, and economical housing solution for most countries in the northern hemisphere (Araujo et al. 2016), leading to widespread use of timber in countries such as Austria, Japan, Scotland, and New Zealand, where 40%, 45%, 83%, and 85% of houses are made of wood, respectively (Mahapatra et al. 2012; Hurmekoski et al. 2015; Araujo et al. 2018).
Nevertheless, in Brazil, despite having the largest biodiversity of species on the planet (Beech et al. 2017), with evident reforestation potential, and a growing demand for housing, the use of timber for dwelling construction is still low (Araujo et al. 2018). This motivates the development of research that disseminates, mainly to the consumer market, information regarding benefits of timber constructions and physical-mechanical properties of wood that are necessary for rational elaboration of structural design.
Among these properties, average value of modulus of elasticity in compression parallel to the grain (Ec0) is of paramount importance, given its use in checks of stability of compressed parts (buckling) in the ultimate limit state (ULS) and in calculation of excessive strains in compliance with the serviceability limit state (SLS).
In Brazil, Annex B of ABNT NBR 7190 (1997) “Design of wooden structures” provides experimental methods for the determination of physical-mechanical properties of wood. Given that there are 17 physical-mechanical properties that need to be estimated, a complete characterization of species requires an extensive number of tests. The execution of these tests is time-consuming, expensive, and implies expenditure with materials and labor.
Given the many catalogued tree species in the Amazonian region as a whole, and in the Brazilian Amazon specifically (12000 and 7696, respectively, according to Steege et al. (2016)), any procedure aimed at reducing the number of tests is greatly desirable.
To simplify the assessment of Ec0, ABNT NBR 7190 (1997) allows estimation of the Ec0 value through relation with the average value of modulus of elasticity in tension parallel to the grain (Et0) and static three-point bending (EM), as shown in Eqs. 1 and 2, respectively:
(1)
(2)
Several previous works have sought to determine correlations between wood properties, particularly, with bulk density (ρap – an easily determinable physical property), which proves the academic interest in simplifying the characterization of wood.
Igartúa et al. (2015) studied the Argentinian species Acacia melanoxyon and found a strong correlation (with coefficient of determination (R²) above 70%) between ρap and parallel and normal to the grain compressive strength, as well as between ρap and modulus of elasticity and conventional bending strength.
Silva et al. (2018) used regression models to study whether physical-mechanical properties of Goupia glabra Aubl. (popularly called Cupiúba in Brazil) can be estimated as a function of ρap. They obtained regression models with good precision (R² ≈ 70%) for 15 studied relations. The most significant relation (R² = 87.96%) was between ρap and hardness parallel to the grain.
Almeida et al. (2017) and Dias et al. (2019) studied wood shrinkage estimation as a function of ρap with regression models based on experimental results from 15 and 43 tropical wood species, respectively. In both works, analysis of variance (ANOVA) results demonstrated weak correlation between investigated parameters, showing that ρap is not a reliable estimator for dimensional stability of wood.
In recent years, research has assessed the accuracy of estimated properties obtained through relationships provided by ABNT NBR 7190 (1997) and determined coefficients that best fit the proposed relationships. Matos and Molina (2016) studied correlations between characteristic shear strength (fv0,k) and compressive strength parallel to the grain (fc0,k) of conifer (fv0,k = 0.15∙fc0,k) and dicot (fv0,k = 0.12∙fc0,k) woods. The authors evaluated Pinus elliotti and Eucalyptus saligna species and for conifer woods obtained a relation approximately 95% higher compared to that from ABNT NBR 7190 (1997).
Based on three- and four-point static bending tests, Lahr et al. (2017) determined a relation between longitudinal (E) and tangential (G) modulus of elasticity. From the results obtained of five different tropical wood species tested, the authors determined the relation E = 35∙G, with the coefficient being 75% higher than the one given by ABNT NBR 7190 (1997) (E = 20∙G).
Recently, Christoforo et al. (2019) studied relations between characteristic compression strength (fc0,k), characteristic tensile strength parallel to the grain (ft0,k), and characteristic shear strength (fv0,k). The coefficients (α) they obtained after testing five tropical wood species were 0.96 and 0.23 for relations fc0,k = α∙ft0,k and fv0,k = α∙fc0,k, respectively, which were 25% and 92% higher than the coefficients specified by ABNT NBR 7190 (1997).
These studies demonstrate that some relations between properties of wood prescribed by ABNT NBR 7190 (1997) need to be revised to obtain reliable estimates for structural design. Thus, the aim of this work was to investigate statistical equivalence between modulus of elasticity obtained in bending, compression, and tension parallel to the grain (Eq. 1 and Eq. 2) and, in case equivalence is not confirmed, to define correlations between these properties that would give more accurate Ec0 estimates.
EXPERIMENTAL
Materials
Thirty different wood species were used in this study (Table 1), and these were obtained, from local companies, in the same manner as timber used in Brazilian civil construction, in the form of boards sized approximately 6 cm × 11 cm × 200 cm. Therefore, it was not possible to identify origin and age for the trees.
The wood was properly stocked and tested in three different research labs in the country: the Laboratory of Wood and Timber Structure (LaMEM) of the University of São Paulo (USP); the laboratories of Federal University of Minas Gerais (UFMG), campus Belo Horizonte (State of Minas Gerais); and at the laboratories of São Paulo State University (UNESP), campus Itapeva (State of São Paulo).
Methods
To determine Ec0, Et0, and EM, the static three-point bending test (Fig. 1a), compression (Fig. 1b), and tensile test (Fig. 1c) parallel to the grain were performed, respectively.
Table 1. Brazilian Tropical Wood Species Used in the Study
For each species and mechanical property, 12 specimens were produced and tested, which gave a total of 1080 experimental results. All the tests were conducted on the universal testing machine AMSLER (250 kN loading capacity) (Shimadzu Corporation, Kyoto, Japan) following the procedure described in Annex B (Determination of wood properties for structural design) of ABNT NBR 7190 (1997).
 |
 |
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| (a) | (b) | (c) |
Fig. 1. Static bending test (a), compression test parallel to the grain (b), and tensile test parallel to the grain (c)
The wooden boards were ambient-dried. After drying, the boards presented moisture level around 12%. According to the ABNT NBR 7190 (1997), 12% is a reference moisture level for presentation of the experimental results. Values of stiffness properties (Ec0, Et0, and EM) obtained with moisture levels (U%) different from 12% were adjusted for 12% moisture level using Eq. 3, as prescribed by the ABNT NBR 7190 (1997), where E12% and EU% are values corresponding to moisture levels of 12% and U%, respectively.
(3)
The accuracy of relations proposed by ABNT NBR 7190 (1997) (Eqs. 1 and 2) was evaluated using ANOVA at the 5% significance level through BioEstat5.3® software (Mamirauá Institute, Belém, PA, Brazil). A null hypothesis (H0) was that the average of the groups (Ec0 and Et0, Ec0, and 0.90/EM) was equal, and the alternative hypothesis (H1) was non-equivalence. Hence, a p-value higher or equal than the selected significance level (p-value ≥ 0.05) implied accepting H0 (tested relation was accurate). Otherwise (p-value < 0.05), H1 should be accepted.
Upon discovering non-equivalence (p-value < 0.05), two-parameter (a e b) regression models (Eq. 4 to Eq. 7) were used to estimate Ec0 (dependent variable – y) as a function of Et0 and EM (independent variables – x):
(Linear) (4)
(Exponential) (5)
(Logarithmic) (6)
(Geometric) (7)
Regression models based on ANOVA at a 5% significance level were used in considering the grouping of species and respective average values of properties. For ANOVA of regression models, a null hypothesis (H0: β = 0) was that the tested models were not representative and an alternative hypothesis (H1: β ≠ 0) was that they were representative.
P-values higher than the selected significance level (p-value > 0.05) implied accepting H0 (tested regression model was not representative – variations of x did not explain variations of y). In the opposite case, this hypothesis would be rejected (p-value ≤ 0.05 – regression model was representative).
In addition to ANOVA, values of coefficient of determination (R²) were obtained, which allowed for evaluation of the quality of estimated fit and determination of the most accurate representative model (p-value ≤ 0.05), that is, the model that best described variations of dependent variable y as a function of independent variable x.
Along with regression models, the least squares method (Eq. 8 and Eq. 9 – used in studies by Christoforo et al. (2012), Icimoto et al. (2015), Ferro et al. (2015), Lahr et al. (2017), Almeida et al. (2018), and Christoforo et al. (2019)) using Newton’s method with quadratic approximation was applied for determination of the optimal coefficient (λ) for relations Ec0 = λ·Et0 and Ec0 = EM/λ:
(8)
(9)
RESULTS AND DISCUSSION
Table 2 shows the experimentally obtained average values (Xm) and coefficients of variation (Cv) of stiffness properties (Ec0, Et0, and EM) for each tested species.
Table 2. Stiffness Properties of 30 Studied Wood Species
Both coefficients of variation and average stiffness values were consistent with experimental results from Gonçalez and Gonçalves (2001), Grobério and Lahr (2002), Dias and Lahr (2004), Araújo (2007), Faria et al. (2012), Ferro et al. (2015), Jesus et al. (2015), Moreira et al. (2017), Lahr et al. (2017), Aquino et al. (2018), and Almeida et al. (2018) that determined some of the stiffness properties of the species studied here.
The values of Ec0 determined in this study and found in the literature were in agreement with values presented in Appendix E (Common average strength and stiffness values of some native and afforestation woods) of ABNT NBR 7190 (1997), which includes among its 50 hardwood species 18 wood species tested in this work (Vataireopsis araroba (Aguiar) Ducke, Hymenolobium cf. heterocarpum Ducke, Hymenolobium petraeum Ducke, Dinizia excelsa Ducke, Sebastiania commersoniana (Baill.) L.B. Sm. & Downs, Andira anthelmia (Vell.) Benth, Cassia ferruginea (Schrad.) Schrad. ex DC., Pouteria cf. pachyphylla T.D.Penn., Cedrela odotara L., Cedrela cf. fissilis Vell., Dypterix odotora (Aubl.) Willd., Goupia paraensis Huber, Luetzelburgia cf. guaissara Toledo, Peltophorum dubium (Spreng.) Taub., Hymenaea courbaril L., Ocotea neesiana (Mig.) Kosterm., and Manilkara cf. inundata (Ducke) Ducke). These comparisons supported the results shown in Table 2.
The ANOVA showed that group means of Ec0 and Et0 were statistically equal because the p-value was higher than the significance level (p-value ≥ 0.05). Hence, the relation Ec0 = Et0 was accurate and gave a good estimate of Ec0. For the relation between Ec0 and EM, the p-value was less than the significance level (p-value < 0.05), which indicated that group means of Ec0 and EM/0.90 were not statistically equivalent. Hence, the equation Ec0 = EM/0.90 did not estimate Ec0 value accurately.
The regression models (Eq. 4 to Eq. 7) and least squares method (Eq. 9) were used as an alternative to the equation Ec0 = EM/0.90, for formulation of equations that could accurately estimate Ec0 values as a function of EM. All models (linear, exponential, logarithmic, and geometric) were significant (p-value < 0.05), and the geometric model described by Eq. 10 showed the best fit (R² = 91.67%):
(10)
The least squares method gave the optimal coefficient for the relation Ec0 = EM/λ for the entire group of species equal to 0.98 (Eq. 11) that was 8.89% higher than the coefficient (0.90) provided by ABNT NBR 7190 (1997):
(11)
It should be mentioned that Ec0 = EM/0.90 ratio was established by the Brazilian standard ABNT NBR 7190 (1997) without an adequate statistical analysis that would prove the reliability in the comparison of the groups of values (Ec0 and EM). In the present work, ANOVA was applied in order to investigate the relationship between Ec0 and EM (Ec0 = EM), which showed equivalence between the two groups. This result shows that the Ec0 = EM ratio is more precise than the Ec0 = EM/0.90 equation, proposed by the Brazilian standard. This highlighting the need for revision of this item in future versions.
CONCLUSIONS
- The ANOVA at 5% significance level demonstrated an equivalence between group means of Ec0 and Et0, which indicated the accuracy of Ec0 estimation for hardwood species through the equation Ec0 = Et0 proposed by ABNT NBR 7190 (1997).
- The ANOVA at 5% significance level demonstrated that group means of Ec0 and EM/0.90 were not equal, which indicated that the coefficient of 0.90 did not give an accurate estimation of Ec0 through the equation Ec0 = EM/0.90.
- All regression models used for estimation of Ec0 as a function of EM were significant and with a good fit. The best fit was achieved by the geometric model, which showed that Ec0 could be estimated by Eq. 10.
- The optimal coefficient obtained by the least squares method (Eq. 11) for the relation between Ec0 and EM was higher than the one established by ABNT NBR 7190 (1997). The value of this coefficient was around 1, thus, validating the results of ANOVA that indicated equivalence between Ec0 and EM (Ec0 = EM).
- Given the significant number of species tested in this study, Ec0 = EM ratio appeared to be widely applicable model for estimation of Ec0.
ACKNOWLEDGEMENTS
The authors are grateful for Laboratory of Wood and Timber Structure (LaMEM) of the University of São Paulo (USP); the laboratories of Federal University of Minas Gerais (UFMG), campus Belo Horizonte (State of Minas Gerais); and the laboratories of São Paulo State University (UNESP), campus Itapeva (State of São Paulo), for providing facilities and inputs required for this study.
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Article submitted: December 18, 2019; Peer review completed: February 22, 2020; Revised version received and accepted: March 17, 2020; Published: March 23, 2020.
DOI: 10.15376/biores.15.2.3278-3288