Shear and longitudinal modulus of elasticity in wood : relations based on static bending tests

Improve quality of timber structures design is an aim that must be systematically sought by engineers in this area. An important topic that can contribute directly to be achieved in this subject is the more consistent knowledge related to structural properties of wood. Know values of longitudinal modulus of elasticity (E) and shear modulus (G) is essential for proper evaluation of plate structures performance, as example. It has been usual to adopt statistical equivalence for E and G values in plans longitudinal-radial and longitudinal-tangential, although experimental confirmation of this hypothesis is required. In this context, the aim of this work is to determine values of ELR, ELT, GLR and GLT, based on static bending tests, to five dicotyledonous species. Results showed statistical equivalence between the elastic properties in both plans, and the relation E = 35G was obtained for the five wood species here considered.


Introduction
Improve quality of timber structures design is a aim that must be systematically sought by professionals in this area.Among the important topics that can strongly contribute to be achieved this goal, a more consistent knowledge of structural properties of wood can be pointed out.
Some normative codes in this matter adopt arithmetic relations to describe wood properties in order to make simple and quick the evaluation of structural elements behavior.In the specific case of Brazilian Code NBR 7190 (Associação Brasileira de Normas Técnicas [ABNT], 1997), some relations between longitudinal modulus of elasticity (E) and shear modulus (G) are adopted, but without appropriate experimental basis.This can induce to doubts in structural design and someone can take calculation assumptions that lead to imprecise estimation of stresses, as asserted by Bodig and Jayne (1982), Calil Junior, Lahr, and Dias (2003), Karlsen (1967), Mateus (1961), and Ritter (1990).Know values of longitudinal modulus of elasticity (E) and shear modulus (G) is essential for proper evaluation of plate structures performance, as example, according to Christoforo, Panzera, Batista, Borges, and Lahr (2011), Herzog, Natterer, Schweitzer, Votz, and Winter (2000), among others.
Several studies have been conducted to optimize the theoretical basis aiming to determine shear modulus in wood, considering its features of orthotropy, being mentioned among them Gillis (1972), Holmberg, Persson, and Petersson (1999), Nairn (2007), Price (1929), andSchniewind (1959).These authors have contributed to better understanding of the problem, usually working with the well-known tests in clear specimens.
Researchers as Ballarin and Nogueira (2003) sought to obtain experimental values of G, although working mostly with small number of specimens, aspect that prevent generalization of the results obtained.
Moreover, it is noteworthy that the values of wood stiffness properties can be obtained from nondestructive testing techniques, according to Ballarin and Palma (2009).They point out that, although in some cases leading to high variability of results, nondestructive techniques are configured as an interesting alternative to characterize wood from planted forests, given the significant amount of defects present in them.Papers by Alves and Carrasco (2013), Bucur and Archer (1984), Gonçalves, Trinca, and Cerri (2011), Gonzales, Valle, and Costa (2001), Ross, Brashaw, and Pellerin (1998), Sandoz (1989), Stålne and Gustafsson (2002), Tallavo, Pandey, and Cascante (2013), Yang, Wang, Lin, and Tsai (2008) are other examples of the same propositions of the first mentioned authors.Mascia and Lahr (2006), evaluating aspects of wood as an orthotropic material, calculated E and G values in the two longitudinal planes.Results published by these researchers were object of statistical analysis.In a more superficial approach, it could not be ruled out a possible difference between E LT and E LR and between G LR and G LT for the tropical species Jatobá (Hymenaea stilbocarpa).Also from data contained in the cited article, it's possible to infer that relation E G -1 is close to 20, for Jatobá.Christoforo, Ribeiro Filho, Panzera, and Lahr (2013) presented an analytical methodology for determination longitudinal and shear moduli for structural dimension lumber (proper to wood coming from planted forests), using three-point static bending tests; adapted from Brazilian Code NBR 7190 (ABNT, 1997).Wood species used in these trials were Pinus elliottii and Corymbia citriodora.The related equations were developed according to virtual forces method and the shear shape coefficient (f s ) to rectangular cross section was adopted as 1.20 (6/5).Results of coefficients (α) between moduli (E = α·G) for the referred wood species were, respectively, 18.70 and 21.20, very close to the coefficient (20) set by the aforementioned Brazilian Code.
Simplifying, it has been usual to adopt statistical equivalence for values of G in the longitudinalradial (G LR ) and longitudinal-transversal (G LT ) directions, important parameters related to structural design requirements, as evidenced by Gillis (1972) and Kretschmann (2010), among others.Similar position is taken by the NBR 7190 (ABNT, 1997) that establishes a unique relationship between these properties, i.e., E = G 20 -1 .
Then, this work focuses on determining values of E LR , E LT , G LR and G LT , based on static bending tests, exclusively to some dicotyledonous species grown in Brazil, aiming to confirm its equivalence (E LR and E LT , G LR and G LT ) or to establish proper correlations.

Material and methods
To achieve the proposed objective, five hardwood species were considered, each one representing a strength class, according to the prescriptions of Brazilian standard document NBR 7190 (ABNT, 1997): - Inclusion of these wood species in strength classes stipulated by NBR 7190 (ABNT, 1997) is based on the characteristic values of compression strength parallel to grain.
For each species, results of 12 tests species were considered, for specimens with nominal dimensions: 5×5×115 cm, with growth rings parallel to two opposite faces.
Each specimen was tested four times in static bending: two with force applied on LR plane (longitudinal-radial) and two in the LT plane (longitudinal-transversal).In all situations, the specimens were initially tested according to the four-point bending model (Figure 1a 1b).M ad displacemen n of the struct Equation shifts ion 2) providin G.

Static bending
ving Equation g tests, it le us of elasticity respectively. The

Results and di
Table 1 an modulus of el for the five wo   -This situation suggests the need of adjusting coefficient E/ G for adequate design of timber structures.
It's tempestive to signal that the conclusions here commented are only pertinent to tropical wood species.