Euler Bernoulli beam

The framework developed in the previous sections is applied to the problem of a moving mass on an Euler-Bernoulli beam (Fryba 1996, An et al. 2001). In order to demonstrate the presented methodology, the weak statement of the problem is derived. 

No assumption regarding the flow field is required for the floating shear stress sensor. The displacement (3) of the floating element sensor as a function of shear stress (tw) can be derived from Euler-Bernoulli beam theory as... 

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) ... 

Moreover, owing to the shear forces, the slope of the neutral fiber of the beam will be greater than that predicted by the Euler-Bernoulli theory (see Fig. 17.7). For the elastic case, these shear forces are given by... 

The simpler Euler-Bernoulli theory which considers zero transverse shear deformation Yxz has also been tested. Using Hamilton s principle the equations of motion of the beam are derived. This model has been used in various investigations of our group (see, among others, Stavroulakis et al. 2005, 2007). Further applications of piezoelectric layers in control can be found in the review article (Irschik 2002). 

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. 

We now consider that each link is modelized with a deformable beam. Moreover, we suppose that the elastic displacement field of each beam verifies Euler-Bernoulli s hypothesis. 

Eq. 23 is the so-called Bernoulli-Euler beam equation. The solution to this fourth-order equation contains four constants and is written in the form,... 

This is discussed in standard tests on elasticity, and the anisotropic case in Lekhnitskii171. The simplest assumption is that of the Bernoulli-Euler theory in which plane sections remain plane and Poisson contraction is ignored. Young s modulus measured by the deflection of a simply supported beam under three point bending is given by... 

If the beam is bent by couples M then Bernoulli-Euler theory gives the relation... 

When considering the bending of a beam and attempting to extract a modulus value one must make several assumptions, the most important being that the modulus in tension is the same as in compression, and is independent of strain (at least for the range of strain involved). The simple Bernoulli-Euler theory is usually used to interpret the data. When performing resonance tests it is particularly useful to find a set of resonances and compare the measured frequency ratios with the theoretical ones given in the previous chapter. 

As has been shown in preceding paragraphs, the classical theory of Euler and Bernoulli describing transverse vibrations in elastic beams can be generalized for the viscoelastic case. According to Eq. (17.13b), and based on the same grounds as Eq. (16.6b), we can write... 

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 ... 

Park S K and Gao X-L (2006) Bernoulli-Euler beam model based on a modified couple stress theory, J Micromech Microeng 16 2355-2359. 

Flexural vibrations As a first approximation, the only elastic constant affecting the flexural vibrations of a quasi-homogeneous beam is the axial Young s modulus. The Bernoulli-Euler (BE) theory of flexural motions gives the relationship between the Young s modulus and the resonant flexural frequencies of the beam. Provided compressive and tensile moduli are equal and independent of the strain rate, the relationship for prismatic bars of rectangular cross-section and "free-free" end conditions is as follows [7] ... 

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