Timoshenko equation

The Timoshenko equation holds only for elastic films on elastic substrates. Polymer layers, however, are viscoelastic, in particular near their glass transition temperatures. The experimental raw data can be deconvoluted to obtain stress measurements, even if the viscoelastic effects are important. This is because we use the thin-film equation, where the viscoelastic modulus of the polymer is not needed for the stress computation. For... 

The radius of curvature (R) is determined by the Young s moduli of the two materials and the thickness ratio of the layers. It can be approximated using the Timoshenko equation [21] ... 

Axial Young s modulus obtained from Timoshenko equations. 

The analysis to find the fiber buckling load in each mode is based on the energy method described by Timoshenko and Gere [3-31], The buckling criterion is that the change in strain energy for the fiber, AUf, and for the associated matrix material, AUf, is equated to the work done by the fiber force, AW, during deformation to a buckled state, that is,... 

As in the simple 3- or 4-point bending of a beam the vibrating reed device assumes the validity of the differential Eq. (2.2) which is due to Euler. Timoshenko25 included both rotary inertia and shear deformation deriving a more exact differential equation which reduces to the Euler equation as a special case. Use of the Timoshenko beam theory for anisotropic materials has been made by Ritchie et al.26 who derive a pair of equations for torsion-flexure coupling (which will always occur unless the axis of the beam coincides with the symmetry axis of the anisotropic material). 

As mentioned above, when the transverse dimensions of the beam are of the same order of magnitude as the length, the simple beam theory must be corrected to introduce the effects of the shear stresses, deformations, and rotary inertia. The theory becomes inadequate for the high frequency modes and for highly anisotropic materials, where large errors can be produced by neglecting shear deformations. This problem was addressed by Timoshenko et al. (7) for the elastic case starting from the balance equations of the respective moments and transverse forces on a beam element. Here the main lines of Timoshenko et al. s approach are followed to solve the viscoelastic counterpart problem. 

A well known equation by Timoshenko (2A) is employed to determine the layer stresses from the measured radii of curvature of a bimaterial strip (Figure 1). The stress in layer 1 as a function of distance y from the center of the layer is ... 

It remains a quadratic function but the coefficients are uncertain. Equation (2.164) is applicable to all modal frequencies for Timoshenko beam models. Therefore, it is proposed to bridge the squared fundamental frequency and the ambient temperature by a quadratic function, and the coefficients can be estimated by Bayesian analysis. 

As described previously, FE models will be used for the development of the track and train structures. The tracks will be modeled using Timoshenko beams and the trains will be modeled as system with rigid cars and springs and dampers coimecting them to the bogies. The behavior of these structures can be modeled using the second order differential equation... 

For a ringplate loaded uniformly over its inner edge by a moment Af we have (from equation of Timoshenko s Strength of Materials ) The angular deflection ... 

The basics of charge transfer may also be presented in the form of two analogies. One involves using equations that describe the collision mechanics between particles and the wall, as presented by Timoshenko (1951) and developed by Soo (1967). This is quite similar to the basic heat-transfer analysis. The second approach is to use the penetration theory as given by Higbie (1935) and Danckwerts (1951) for heat, mass, and momentum transfer for the analysis of charge transfer. 

All the elasticity equations given by Eqs. 9.29 - 9.32 as well as the biharmonic stress function equation can be developed for cylindrical coordinates (see, Timoshenko and Goodier, (1970)). The biharmonic equation is written as,... 

An elastic stability analysis is presented in this paper for Timoshenko-type beams with variable cross sections taking into consideration the effects of shear deformations under the geometrically non-linear theory based on large displacements and rotations. The constitutive relationship for stresses and finite strains based on a consistent finite strain hyperelastic formulation is proposed. The generalized equilibrium equations for varying arbitrary cross-sectional beams are developed from the virtual work equation. The second variation of the Total Potentid is also derived which enables... 

In section Nonlinear Flexural Dynamic Analysis of Beams with Shear Deformation Effect of this chapter, the geometrically nonlinear dynamic flexural analysis of homogeneous prismatic beam members taking into account shear deformation and rotary inertia effects (Timoshenko beam theory) is presented. The differential equations of... 

In this application the free vibrations of Timoshenko beams with very flexible boundary conditions is examined, since in this case the natural frequencies and the corresponding modeshapes are highly sensitive to the effects of shear deformation and rotary inertia (Aristizabal-Ochoa 2004). Thus, the linear free vibration analysis of piimed-free beams of length L = 5.0 m of various cross sections is examined (p = 2.40 tn/m, E = 25.42 GPa, G = 11.05 GPa). In Table 3 the geometric and inertia constants as well as the shear correction factors of the examined cross sections are presented (re = 1/a). The differential equations for the linear free vibrations of this special case can be obtained by Eq. 14 for zero axial stress resultant and external force as... 

Timoshenko SP (1921) On the correction factor for shear of the differential equation for transverse vibrations of bars of uniform cross-section. Philos Mag 41 744-746 Timoshenko SP (1922) On the transverse vibrations of bars of uniform cross-section. Philos Mag 43 125-131... 

The governing equations taking into account the geometric nonlinearity of the overlap and the transverse shear stiffness of the adherends can be derived by using Eqs. 24.18-24.20 and the constitutive equations of the Timoshenko beam, and they are given by (Luo and Tong 2008) ... 

The more exact Timoshenko differential equations of flexural beam motion correct for these shear effects and moreover for the effects of... 

General remarks concerning the method Well defined cross-over points in the E-E/G-planes are obtained in evaluating the flexural spectra (Figures 3 and 4). This shows that the flexural frequency equations based on the Timoshenko theory can be applied even to extremely anisotropic and comparatively compact specimens. 

The second method is the Lame approach (Kashani and Young, 2008), which is based on displacement differential equations and is applicable to any cylindrical vessel with any diameter-to-wall-thickness ratio. The Lame method is often referred to as the solution for thick wall cylindrical pressure vessels. Equations for the hoop stress and radial stress in a thick-walled cylinder were developed by Lame in the early nineteenth century (Timoshenko and Goodier, 1969) ... 

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