Empirical Formulation of Shear Modulus Functions for Tubular Pinewood Specimens Under Torsion

Abstract

In this study, tubular pinewood (Pinus Sylvestris L.) specimens are tested and shear strain measurements are performed by applying torsion in z-direction. Strain gauge measurements are performed for the maximum shear stresses which develop on the tubular specimen, along the radial r (r_in, r_out), circumferential ϕ (ϕ_in, ϕ_out) and z directions, in a point-wise (pw) manner. The data is gathered and examined for the determination of the local variations of empirical shear modulus functions on transversely isotropic surfaces of the specimens. The coordinate dependent shear modulus functions of G_zϕ(r), G_zϕ(ϕ), G_zϕ(z) are derived for G_zϕ(r, ϕ, z) as the function of r, ϕ and z, respectively, by analyzing the gathered data. It is proposed to represent the shear modulus functions, G_zϕ(ϕ) and G_zϕ(z) with the parabolic polynomials, and, to represent the shear modulus function G_zϕ(r) with a linear equation.

Nomenclature

A_avg, B_avg, C_avg :

Average coefficients of parabolic equation

c.v.:

Spread of data (coefficient of variation)

C_ij :

Matrix of elastic constants

Deg:

Degree

G_zϕ(r), G_zϕ(ϕ):

Shear modulus functions generated in the r − ϕ planes along the r, ϕ and z-directions as

G_zϕ(z):

Modes of function

G_avg :

Average shear modulus (GPa)

pw:

Point-wise

MA:

Measurement axis

MAA:

Measurement axis angle

Mean:

Average

r_in, r_out :

Inner and outer radiuses of tubular wood (m)

S_Y :

Standard deviation

${\mathrm{T}}_{\mathrm{z}},\overrightarrow{\mathrm{T}}$ :

Applied z-directional torsion moment (N. m) and its vector notation

V_o :

Output voltage (Volt)

α_ref :

Grain angle of wood specimen on the surface of the cylindrical wood bar which is measured from the main axis-z (degree)

ϕ:

Rotation angle on (rϕ) plane of the specimen (rad)

ϕ_in, ϕ_out :

Inner and outer circumferential angles of tubular wood specimen (degree)

∑ (exp):

Total number of experiments

(τ_zϕ − γ_zϕ)_pw :

Applied shear stress and developing shear strain distribution along point-wise direction

τ_rz, τ_zϕ, τ_rϕ :

Shear stress components (MPa)

ε_r, ε_ϕ, ε_z :

Normal strain components

γ_rz, γ_zϕ, γ_rϕ :

Shear strain components (radian)

γ, γ_out, γ_in :

Average shear strain, shear strains at outer and inner surfaces of the specimen (radian)