Abstract
In biomechanical analyses, computational models are essential tools for simulating the behavior of a tree subjected to a load. However, such models allow only approximation of the actual behavior of the tree if the elastic parameters of the wood in different tree parts (stem, branches, and roots) and at least orthotropic behavior are not considered. In addition, as the wood is green, the parameters of strength and stiffness must be adequate for this level of moisture. However, even for stem wood, knowledge of elastic properties is not available for most species used in urban tree planting, and this scarcity of information is even greater for wood branches. The objective of this research was to evaluate methodology, based on wave propagation, in characterizing the 12 elastic constants of wood from branches. Complementarily, compression tests were performed to characterize the strength. The obtained elastic parameters using ultrasound tests were comparable with the values expected based on theoretical aspects related to the behavior of the wood. The results of the compression test complemented the ultrasound characterization, but the application of this method for the complete characterization of the elastic parameters is not feasible for tree branches because of their small size.
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Methodology for the Characterization of Elastic Constants of Wood from Tree Branches
Raquel Gonçalves,a,* Gustavo Henrique Lopes Garcia,b Sergio Brazolin,c Cinthya Bertoldo,d and Monica Ruy b
In biomechanical analyses, computational models are essential tools for simulating the behavior of a tree subjected to a load. However, such models allow only approximation of the actual behavior of the tree if the elastic parameters of the wood in different tree parts (stem, branches, and roots) and at least orthotropic behavior are not considered. In addition, as the wood is green, the parameters of strength and stiffness must be adequate for this level of moisture. However, even for stem wood, knowledge of elastic properties is not available for most species used in urban tree planting, and this scarcity of information is even greater for wood branches. The objective of this research was to evaluate methodology, based on wave propagation, in characterizing the 12 elastic constants of wood from branches. Complementarily, compression tests were performed to characterize the strength. The obtained elastic parameters using ultrasound tests were comparable with the values expected based on theoretical aspects related to the behavior of the wood. The results of the compression test complemented the ultrasound characterization, but the application of this method for the complete characterization of the elastic parameters is not feasible for tree branches because of their small size.
Keywords: Biomechanics; Longitudinal modulus; Poisson ratio; Shear modulus; Strength; Ultrasound
Contact information: a: Professor – School of Agricultural Engineering (FEAGRI), University of Campinas (UNICAMP) Av. Cândido Rondon, 501 – Cidade Universitária, Campinas – SP, 13083-875- Brazil; b: PhD student– FEAGRI/UNICAMP; c: Researcher – Institute for Technological Research (IPT) Av. Prof. Almeida Prado 532 Cid. Universitária – Butantã. 05508-901 São Paulo/SP.- Brazil; d: Assistant Professor – FEAGRI/UNICAMP; * Corresponding author: raquel@feagri.unicamp.br
INTRODUCTION
Lack of knowledge about the mechanical properties of wood from species used in urban arborization and of green wood has been an important obstacle to the development of studies related to biomechanics (Cavalcanti et al. 2018). This lack of knowledge is related to the small or nonexistent commercial appeal of these species and of the green moisture condition because they are not important for the construction sector, which is the primary area of demand for mechanical properties. This lack of knowledge is even worse for wood branches (Casteren et al. 2013).
One aspect of great importance in biomechanical studies of trees is wood stiffness because this parameter is responsible for the response of wood to the strain and displacements of its limbs (trunk, branches, and roots) when subjected to actions such as self-weight, wind, or snow. Aspects related to stiffness are also important for the movement of animals, such as monkeys, in trees because branches with great flexibility hinder the movement of animals by requiring a greater energy expenditure (Casteren et al. 2013).
As in the case of stiffness, strength properties are important in biomechanical studies of trees because they are related to the rupture of branches, trunks, and roots. Casteren et al. (2013) note that this property is also greatly important for animals that use tree branches to build their nests and to move around.
Because of the current need for a better use of natural resources, research has been carried out to analyze the physical and mechanical properties of wood branches, including for structural utilization (Dadzie et al. 2016). This study was carried out with wood at equilibrium moisture content, consistent with most of structural applications. Nevertheless, information about the mechanical properties of branches under green conditions and from species used in urban arborization is scarce. In searching for literature on the stiffness of green branches, important contributions were found from studies of monkey behavior (Thorpe et al. 2007; Gilman et al. 2011). The flexibility of the branches has a great influence on the mobility of animals, but no literature data are available. Current studies related to biomechanics that aim to analyze a tree’s behavior as a structural element have proposed the use of computational models that allow simulation of this behavior (Lang and Kaliske 2013; Martinez and Dias 2016). However, the use of more complex models that are able to more closely approximate the actual condition of the tree requires knowledge of the complete elastic properties (compliance matrix), not the properties in only longitudinal direction as is generally found. If one considers wood to be an orthotropic material, this means knowing 12 elastic constants.
The 12 elastic constants of wood (three longitudinal modulus, three shear modulus, and six Poisson’s ratio) can be obtained using static tests but the methodology is expensive and laborious because it is necessary to use 6 specimens for one test – 3 specimens obtained in axes and 3 specimens obtained out of axes and around 36 strain-gages (Sinclair and Farshad 1987). So, researchers around the world were trying to obtain other techniques and methodologies to obtain these wood constants. The theoretical basis to obtain these constants using wave propagation was proposed by Christoffel in the 1800s and, driven by technological advances in transducers, authors have resumed studies with the goal of proposing methodologies based on this theory to obtain the complete characterization of wood (Preziosa et al. 1981; Preziosa 1982; Bucur and Archer 1984, Bucur and Perrin 1988; François 1995; Bucur and Rasolofosaon 1998; Gonçalves et al 2011a; Ozyhar et al. 2013; Gonçalves et al. 2014; Vázquez et al. 2015).
Considering the mentioned aspects, the objective of this paper was to present a methodology, associating ultrasound and compression test, and preliminary results for the complete elastic characterization of wood from tree branches of species used in urban arborization. The experimental design consisted of 80 specimens (37 for ultrasound tests and 43 for compression tests) collected from 16 pieces of branches obtained from 2 or 3 fork levels on six species of urban trees.
EXPERIMENTAL
Materials
For the seven trees sampled, pieces of branches were removed from the 2 or 3 first fork levels (Fig. 1). The trees sampled were obtained in urban areas of Campinas, São Paulo, Brazil. Campinas’ climate is tropical in altitude (type Cwa according to Köppen), with a decrease in winter rainfall and an average annual temperature of 20.7 ° C, with mild, dry winters and rainy summers with moderately high temperatures. The warmest month in February has an average temperature of 23.4 ° C and the coldest month in July is 17.2 ° C. Fall and spring are transitional seasons. The average rainfall is approximately 1350 mm annually, concentrated between October and March, with January having the most precipitation (226 mm).
Fig. 1. Schematic of the locations of the pieces removed from branches at different fork levels and of the ultrasound (polyhedral) and static compression (prismatic) test specimens
The adoption of 2 or 3 fork levels depended on the diameter of the branch because it was necessary that the branch size was sufficient for the removal of the specimens. Polyhedral and prismatic specimens were obtained from each branch section for ultrasound and static compression tests, respectively (Fig. 1), according to the sampling indicated in Table 1.
Table 1. Number of Specimens Used in the Ultrasound and Static Compression Tests for Each Species and Fork Level
The minimum dimension of the specimen for the ultrasound test (polyhedral) is limited by the diameter of the transducer, which needs to be circumscribed to the face and by the theoretical bases of the waves propagation infinite media; this depends on the relationship between the length of wave propagation by the wave length (Bucur 2006). For the compression tests (prismatic specimens) the dimension was established based on the Brazilian Standard (ABNT NBR 7190) that indicate length 3 times the dimension of the edges.
The polyhedral specimens were produced using firstly a lathe machine to make a cylinder, allowing have the axes (longitudinal, radial, and tangential) to be well targeted. A milling tool was used with the cylinder to cut the angles necessary to produce the 26 faces of the polyhedron.
Methods
Ultrasound tests
Ultrasound tests were performed according to methodology used by this research group in the characterization of timber under equilibrium conditions (Gonçalves et al. 2014; Vazquez et al. 2015). This methodology can be regarded as adequate for the characterization of timber because by using just one polyhedral specimen (Fig. 1) it is possible to obtain the complete stiffness matrix, whose inverse allows the calculation of the compliance matrix. The compliance matrix allows calculation of the 12 elastic constants of the wood: modulus of elasticity in the longitudinal (EL), radial (ER), and tangential (ET) directions; shear modulus in the radial-tangential (GRT), longitudinal-tangential (GLT) and radial-tangential (GRT) planes; and the 6 Poisson ratios (RL, TL, LR, TR, LT, and RT). For the wave propagation measurements, ultrasound equipment (Epoch 1000 series, Olympus, USA) and 1-MHz longitudinal and shear wave transducers were used.
The polyhedral specimen had nominal dimensions of 50 mm edges. These dimensions allow the transducer to completely bind to the face of the specimen, minimizing signal losses (Bucur 2006). Starch glucose was used as a coupling medium in all tests because it minimized signal losses, especially for shear waves (Gonçalves et al. 2011b).
For the test, the longitudinal transducers were positioned on the specimen faces parallel to the axis (Fig. 2a), allowing the propagation and polarization of the wave on the main axes: L (longitudinal), R (radial) or T (tangential). From these tests, the velocities VLL, VRR, and VTT were obtained. Similarly, the shear transducers were positioned on the same faces of the specimen, allowing propagation on one of the main axes, L, R or T, and perpendicular polarization. With these measurements, the velocities VLR, VLT, VRL, VRT, VTR, and VRT were calculated. The first index corresponds to the propagation direction and the second the polarization direction. Considering the theoretical aspects related to the symmetries of stresses and strain accepted in orthotropic materials, the velocities Vij should be equal to Vji. In practice, there are small differences because the growth rings are not perfectly positioned nor totally free of curvature in the transverse section of the specimens. Thus, for the calculations, the average of the velocities obtained in Vij and Vji is adopted. To obtain the velocities outside the symmetry axes, the transducers were positioned on the inclined faces to each of the planes (Fig. 2b).