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Influence of cross section dimensions on Timoshenko’s shear factor – Application to wooden beams in free-free flexural vibration
Ann. For. Sci.
Influence of cross section dimensions on Timoshenko's shear factor - Application to wooden beams in free-free flexural vibration
BRANCHERIAU 0
0 CIRAD - Forêt , TA10/16, Av. Agropolis, 34398 Montpellier Cedex 5 , France
- This study was designed to examine the influence of the cross section height-to-width ratio on Timoshenko's shear factor. This factor was introduced to account for the irregular shear stress and shear strain distribution over the cross section. A new theoretical formulation of the shear factor is thus proposed to assess rectangular cross sections of orthotropic material such as wood. Numerical simulations were performed to examine shear factor variations with respect to the height-to-width ratio. The influence of the cross section size on the first five flexural vibration frequencies is also discussed. Timoshenko's shear factor / flexural vibration / wood Résumé - Influence des dimensions de la section transverse sur le facteur de cisaillement de Timoshenko. Applications aux poutres en bois en vibration de flexion libre-libre. L'objectif de cette étude est d'examiner l'influence du ratio hauteur sur épaisseur de la section droite sur le facteur de cisaillement de Timoshenko. Ce facteur est utilisé afin de prendre en compte le fait que les contraintes et les déformations de cisaillement ne sont pas uniformément réparties dans la section droite. Une nouvelle formulation théorique du facteur de cisaillement est alors proposée dans le cas d'une section droite rectangulaire pour un matériau orthotrope comme le bois. Des simulations numériques sont réalisées de manière à examiner la variation du facteur de cisaillement en fonction du ratio hauteur sur épaisseur. L'influence de l'effet de dimension de la section droite sur les cinq premières fréquences de vibration de flexion est également discutée.
1. INTRODUCTION
Free-free flexural vibration tests can be used to accurately
determine the elastic constants of wooden beams such as
Young’s modulus and the shear modulus [4]. The shear effect
and rotary inertia can be taken into account to extend the range
of applicability of the Euler-Bernoulli theory of beams [7].
These effects are incorporated in Timoshenko’s beam equation.
This formula has been the focus of considerable attention in the
literature, but few studies have investigated the validity range
of Timoshenko’s beam equation when the height/width ratio of
the cross section varies between that of a thin vibrating beam
to that of a thick vibrating plate. Cowper [2] derived
Timoshenko’s beam equation by integration of equations based
on the three-dimensional elasticity theory. A new shear factor
formula was proposed which integrates Poisson’s ratio for an
isotropic material but the cross section size effect was not taken
into consideration [2]. In this paper, a theoretical formulation
for the shear factor is proposed for orthotropic materials such
as wood with a rectangular cross section. This new theoretical
formulation includes relations based on the three-dimensional
elasticity theory and takes the size effect of the cross section
into account. Numerical shear factor values are calculated
according to variations in the height/width ratio of the cross
section. The influence of the cross section size effect on the first
five vibration frequencies is also discussed.
2. THEORETICAL FORMULATION
Let us consider an orthotropic and homogeneous beam in free-free
bending vibration. The governing equation of motion, as formulated
by Timoshenko [7] for an isotropic material, is as follows:
∂4ν EX ∂4ν ρ 2IGz ∂4 ∂4ν
EXIGz-------- – ρ IGz1 + -------------- --------------- + -------------- ------ν-- + ρ S-------- = 0
∂x4 KGXY ∂x2∂t2 KGXY ∂t4 ∂t2
where Ex is the longitudinal modulus of elasticity, IGz the cross section
inertia, ρ the density, K the Timoshenko shear factor, GXY the shear
modulus, S the cross section area and v the particle motion on the axis
(OY). Equation (
1
) takes the effects of shear deflection and rotary
inertia into account. Coefficient K, which is a dimensionless quantity that
is dependent on the shape of the cross section, is introduced to account
for the irregular shear stress and shear strain distribution over the cross
(
1
)
L. Brancheriau
section [2]. Timoshenko [8] defined K as the ratio of the average shear
strain on a section to the shear strain at the centroid. The K value is
thus 2/3 for a beam with a rectangular cross section [8]. However,
several authors have proposed other K value estimates [2–4, 10] and the
consensus value seems to be 5/6 [11].
The K value for a rectangular cross section can be calculated
considering the elementary work dw associated with the shear stress σxy
acting on a beam element of length dx [5]:
1 2
dw = ----------- ∫∫ σxydS dx .
2Gxy S
Timoshenko [9], Lekhnitskii [6] and Laroze [5] developed
theoretical formulations on the basis of a three-dimensional state of stress.
The shear stress distributions σxy and σx (...truncated)