Timoshenko beam theory

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 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). 

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... 

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... 

O Figures 24.9 and O 24.10 illustrate distributions of the shear and peel stresses for the composite SLJ predicted by the linear adhesive-beam model based on the Euler beam theory and the nonlinear adhesive-beam model based on the Timoshenko beam theory, respectively. 

In O Figs. 24.9 and O 24.10, the same edge moment factor at = 9 calculated using O Eq. 24.33 is used for both models. Figure 24.9 shows that there are slight differences in values of shear stress predicted by the linear adhesive-beam model based on the Euler beam theory and the nonlinear adhesive-beam model based on the Timoshenko beam theory. However, there exist significant differences in the adhesive peel stress for the two models as shown in Fig. 24.10, including peak values and distribution patterns. 

Nonlinear Overlap Based on the Timoshenko Beam Theory... 

Fully coupled nonlinear analytical solutions for asymmetric and unbalanced adhesive joints are very complicated. Nevertheless, when the updated loadings are given, linear analysis based on the Timoshenko beam theory can predict sufficiently accurate adhesive stresses except for the case of very large deflection. 

Analytical approaches for adhesively bonded structures are presented in this chapter. Stress analysis for adhesively bonded joints is conducted using the classical adhesive-beam model and the other adhesive-beam models. Closed-form solutions of symmetric joints are presented and analytical procedures of asymmetric and unbalanced joints are discussed. Load update for single lap joints is investigated in detail. Numerical results calculated using the classical and other formulations are illustrated and compared. It is shown that the nonlinear adhesive-beam model based on the Timoshenko beam theory provides enhanced results compared to the linear adhesive-beam model based on the Euler beam theory for adhesively bonded composite structures. Analytical solutions of energy release rates for cohesive failure and delamination are presented, and several failure criteria are reviewed and discussed. 

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. 

Timoshenko s beam model used by Wang et al. [60] for free vibrations of MWCNTs study it was shown that the frequencies are significantly over predicted by the Euler s beam theory when the aspect ratios are small and when considering high vibration modes. They indicated that the Timoshenko s beam model should be used for a better prediction of the frequencies especially when small aspect ratio and high vibration modes are considered. 

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

Prom the linear beam theory [Gere and Timoshenko (1997)], we have... 

Spread-plasticity models are classical finite elements where material nonlinearities are modeled at each integration point. Besides the classical two-node displacement-based beam elements, force-based two-node force-based elements have seen a widespread use both in research and commercial software (McKenna, 1997 MIDAS, 2006 Zimmermann, 1985-2007). The assumed force fields in a two-mode force-based element are exact within classical beam theories, such as the Euler-Bernoulli and Timoshenko theories (Marini Spacone, 2006 Spacone et al, 1996). This implies that only one element per structural member is used. The element implementation is not trivial and it implies element iterations, but these steps are transparent to the user. The section constitutive law is the source of material nonlinearities. 

Timoshenko, 1983, pp. 11-15. The author is kind to Galileo and highlights his sound conclusions on scale effect. The errors in his beam theory show up in the discussion of Edme Mariotte s (1620-1684), correct formulation made some fifty years later. 

Therein the warping function 0 y,z) accounts for the cross-sectional properties, while the lengthwise dependency is provided by the rate of twist cj> x x). Supplying Eq. (7.9) with Eq. (7.11), the components of the total displacement of the classic beam theory of Euler and Bernoulli with extension to shear flexibility and torsional warping usually associated with the names of Timoshenko, respectively Vlassov, are obtained ... 

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... 

Luo and Tong (2007, 2008) proposed fully coupled nonlinear formulation for isotropic SLJs based on the Euler beam theory and then extended to composite SLJs based on the Timoshenko... 

Analytical approaches for unbalanced adhesive joints based on the Euler beam theory have been developed by a number of researchers (Hart-Smith 1973 Delale et al. 1981 Bigwood and Crocombe 1989 Luo and Tong 2002 Shahin et al. 2008). Luo and Tong (2009b, c) modeled the adherends as Timoshenko beams. Since the analytical procedure for the Euler beam is similar to that for the Timoshenko beam, the formulations are presented here for the Timoshenko beam only. 

Comparing the results with the obtained results of MD simulation, it can be inferred from literature that for large aspect ratios (i.e., length to diameter ratio L/d >10) the simple Euler-Bemoulli beam is reliable to predict the buckling strains of CNTs while there fined Timoshenko s beam model or their nonlocal counterparts theory is needed for CNTs with intermediate aspect ratios (i.e., 8 thin shell theory... 

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... 

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