On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation

Abstract

The problem of free vibrations of the Timoshenko beam model is here addressed. A careful analysis of the governing equations allows identifying that the vibration spectrum consists of two parts, separated by a transition frequency, which, depending on the applied boundary conditions, might be itself part of the spectrum. For both parts of the spectrum, the values of natural frequencies are computed and the expressions of eigenmodes are provided. This allows to acknowledge that the nature of vibration modes changes when moving across the transition frequency. Among all possible combination of end constraints which can be applied to single-span beams, the case of a simply supported beam is considered. These theoretical results can be used as benchmarks for assessing the correctness of the numerical values provided by several numerical techniques, e.g. traditional Lagrangian-based finite element models or the newly developed isogeometric approach

Fiber rupture in sheared planar pantographic sheets: Numerical and experimental evidence

“…The advantage of the presented numerical model can be resumed as follows: (i) it is able to produce easily results in the regime of large deformations and displacements, (ii) it is based on the best fit of a limited number of constitutive parameters (between five and seven) to all of which a clear physical meaning can be attributed, (iii) all aforementioned constitutive parameters are easily related to specific geometrical and mechanical properties of considered fabrics so that their value, in a specific situation and for a specific specimen, can be easily estimated to belong to a well-determined range, (iv) it seems to be easily generalizable to the study of dynamic regimes, where inertial forces may play a relevant role, see some insight in [8,9].…”

An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams

“…and the corresponding equation which provides can be computed in terms of real‐valued quantities only; results will be presented separately for the three cases outlined above. The relevant details are still given in .…”

Contrasting three alternative versions of Timoshenko‐Ehrenfest theory for beam on Winkler elastic foundation – simply supported beam

“…Above numerical results are clearly referred to the case of soft foundation. Indeed, Since the numerical results for the beam on elastic foundation with differ slightly from those of the beam without elastic foundation belonging to Cazzani, Stochino and Turco …”

“…In order to investigate the behavior of the beam on soft foundation we study the effect of increasing values. Table list the natural frequencies for the case of and with rest of the data as in the study of Cazzani, Stochino and Turco . Fundamental first frequency for the system with equals 404.354 (see Table ), whereas for , consisting over the 257‐fold increase.…”

“…Let us focus the attention on a particular beam to gain some insight. In agreement with the paper by Cazzani, Stochino and Turco, we choose the following set of parameters: Straight uniform and homogeneous beam whose length is Square cross‐sectional area with side length Cross‐sectional area Area moment of inertia Material density Young modulus Poisson's ratio Shear modulus Shear correction factor Modulus of Winkler foundation will be varying; as the first case we consider ; in these circumstances we get the following results for the natural frequency: …”