A p‐version, first order shear deformation, finite element for geometrically non‐linear vibration of curved beams

Abstract

SUMMARYA p-version, hierarchical finite element for curved, moderately thick, elastic and isotropic beams is introduced. The convergence properties of the element are analysed and some results are compared with results published elsewhere or calculated using a commercial finite element package. It is verified that, with the proposed element, shear locking does not affect the computation of the natural frequencies and that low dimensional, accurate models are obtainable. Geometrically non-linear vibrations due to finite deformations, which occur for harmonic excitations with frequencies close to the first three natural frequencies of vibration, are investigated using Newmark's method. The influence of the thickness, longitudinal inertia and curvature radius on the dynamic behaviour of curved beams are studied.

On the predictability of elasto-plastic and geometrically non-linear oscillations of beams under harmonic excitation

“…The p-version FEM was here chosen, since it naturally (i.e., without any additional model reduction procedure) requires a reasonably small number of degrees-of-freedom for accuracy [17][18][19][20], even in the case of elasto-plastic oscillations [17]. In addition, this version of the FEM is not prone to shear locking [20] and provides for a simple computational implementation. The following paragraphs describe the main points of the formulation and references are given for further information.…”

A 2D Hopfield Neural Network approach to mechanical beam damage detection

“…More details on the above written virtual works can be found in [12]. The virtual work of the external forces is given in [15]. Without damping, the equation of motion in time domain are:…”

“…Superscript lr represents longitudinal-rotation coupling that occurs due to damage and is reflected in the mass and linear sti↵ness matrix. The mass and sti↵ness matrices are given in [12], the vector of generalized external forces in [15].…”

A parallel spectral element method for dynamic three-dimensional nonlinear elasticity problems

“…To the best of our knowledge, the only applications of p-FEMs to dynamic problems are those in [18,29]. In [18] linear elastodynamic problems were investigated and some formulations towards future finite-deformation implementation were briefly mentioned.…”

“…In [18] linear elastodynamic problems were investigated and some formulations towards future finite-deformation implementation were briefly mentioned. In [29] p-FEMs for curved elastic and isotropic beams taking into account geometric nonlinearities were used to investigate the vibrations occurring due to harmonic excitations. In regard to parallelism, to fully take advantage of present-day large-scale parallel computers which are usually equipped with 1000-10,000 processors, efficient parallelization of solid mechanics p/hp-FEM codes is required and remains an extremely challenging problem.…”