Timoshenko beam

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). 

The Timoshenko beam model accounts for both shear deformation as well as the rotational inertia effect. To mimic the behavior of a building, the beam model is taken to be a uniform... 

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. 

The FRF, G(tw), can be obtained by various methods such as finite element method (FEM), Timoshenko beam theory (Cao and Altintas 2004), impulse response method (Ewins 2001), or widely used frequency response analysis (Ewins 2001). Hybrid methods such as the receptance coupling substructure analysis (RCSA) (Schmitz and Duncan 2005) have also been utilized in practice. For example, excitation force generated in milling process can be estimated by an analytical model of the milling process (Altintas 2000). 

Eringen s nonlocal elasticity theory and von Karman geometric nonlinearity us- — ing nonlocal Timoshenko beam... 

Free vibrations of MWCNTs using Timoshenko beam... 

Ke, L. L., Xiang, Y, Yang, J. Kitipomchai, S. (2009). Nonhnear Free Wbration of Embedded Double-Walled Carbon Nanotubes Based on Nonlocal Timoshenko Beam Theory. Comput. Mater. Sci., 47, 409-417. 

To consider the effect of a receptor layer, Timoshenko beam theory, which was originally developed to analyze a bimetal strip, can be used [38]. Based on the Timoshenko beam theory, an analytical model for the static deflection of a cantilever sensor coated with a solid layer was derived. In a simple cantilever coated with a solid receptor layer as shown in Fig. 4.3.10, the deflection of the cantilever is described as ... 

Furthermore, we suppose that the bending-torsion coupling and the axial vibration of the beam centerline are negligible and that the components of the displacement field u of the beam are based on the Timoshenko beam theory which, in turn, means that the axial displacement is proportional to z and to the rotation ir x, t) of the beam cross section about the positive y-axis and that the transverse displacement is equal to... 

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... 

Barbat, A.H., Oiler, S., Hanganu, A. Onate, E. 1997. Viscous damage model for Timoshenko beam structures. International Journal for Solids and Structures-, 34(30) 3953-3976. 

All integrations may be performed symbolically. To avoid the implications of the effect, which might appear in the context of the description of the Timoshenko beam, a reduced integration scheme is applied, see for example Hughes [101]. Alternatively, one may start off from four nodes per element for the transverse displacements and rotations using cubic Lagrange polynomials for the interpolation and then reduce the degrees of freedom by means of a static condensation, see Knothe and Wessels [113]. 

Samuels, J.C. and Eringen, A.C., Response of a Simply Supported Timoshenko Beam to a Purely Random Gaussian Process, J. Appl. Meek., Trans. ASME 25, (1958), pp. 496-500. 

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... 

In Table 4 the first three natural frequencies of the pinned-free Timoshenko beams of various cross sections and in Fig. 4 the corresponding modeshapes for the cross section with web... 

Aristizabal-Ochoa JD (2004) Timoshenko beam-column with generalized end conditions and nonclassical modes of vibration of shear beams. ASCE J Eng Mech 130(10) 1151-1159... 

Sapountzakis EJ, Dikaros IC (2011) Non-linear flexural-torsional dynamic analysis of beams of arbitrary cross section by BEM. Int J Non-Linear Mech 46 782-794 Sapountzakis EJ, Douiakopoulos JA (2009a) Nonlinear dynamic analysis of Timoshenko beams by BEM. Part I Theory and numerical implementation. Nonlinear Dyn 58 295-306... 

Sapountzakis EJ, Elourakopoulos JA (2009b) Nonlinear dynamic analysis of Timoshenko beams by BEM. Part n Applications and validation. Nonlinear Dyn 58 307-318... 

Ahmida KM, Arrada JRF (2001) Spectral element-based prediction of active power flow in Timoshenko beams. Int J Solids Struct 38 1669-1679... 

In the present section, the main beam models are recalled, focusing mainly on the Timoshenko model, which is able to describe the flexural behavior of the beam taking into account, besides the flexural stiffness and inertia due to deflections, also the shear deformability and the rotational inertia. Nevertheless, before introducing the Timoshenko beam model, the two models that can considered to be the ones on which it is based, the Euler-Bemoufli and the simple shear models, will be briefly recalled. 

In dynamics the D Alembert principle is used and considering that again. Moreover, in the Timoshenko beam model, the inertial forces associated to rotations 0 are also taken into account... 

Fig. 2 Pulsations of the first (blue), second (red), and third (green) mode of cantilever beam with K = 0.85 andE/G = 2.6 (—) Timoshenko beam model,...

The first 12 modal shapes are shown in Fig. 3, and the parameters of Eq. 8 are shown in Tables 1 and 2. In the tables also the phase velocity, defined as the ratio between the angular frequency, co, and the wavenumber, ki, is reported as well known, the phase velocity is unbounded for the Euler-Bernoulli beam, while in the case of Timoshenko beam the two different families of waves present when CO > COc, with wavenumbers kj and k2, tend to the following values (Hagedom and DasGupta 2007) ... 

Vibrations of Beams for Seismic Response Estimation, Fig. 3 First 12 modal shapes of the Euler-Bemoulli beam dashed red) and Timoshenko beam solid blue)... 

Bishop R, Price W (1977) Coupled bending and twisting of a Timoshenko beam. J Sound Vib 50 469 77 Cacciola P (2010) A stochastic approach for generating spectrum compatible fully nonstationary earthquakes. Comput Struct 88 889-901... 

When the Timoshenko beam is used to model the adherends, the adhesive shear and peel stresses can be expressed as ... 

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) ... 

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