Isogeometric approach in dynamic analysis of spatial curved beams

Abstract

Accurate numerical modeling of curved beams is of significant importance in different engineering fields. Several challenges can be present during the curved beam formulation, primarily due to the issues regarding the beam geometry, discretization and beam theory assumptions. In this dissertation, the isogeometric approach is applied in the dynamic analysis of spatial curved beams. A novel beam element was formulated using the Bernoulli - Euler hypothesis and the fundamental relations of the differential geometry, as well as the Cauchy continuum beam model. The geometry of the beam, as well as the displacement, velocity and acceleration fields, were defined using the Non-Uniform Rational B-Spline (NURBS) basis functions, which present the basis concept of the isogeometric approach. Complex geometry of the curved beams can be modeled accurately using NURBS - based isogeometric approach. Formulation of the spatial beam is conducted for the linear case, while the geometrically nonlinear formulation is conducted only for the plane curved beam using an explicit integration procedure. Free and forced vibration analyses of the curved beams are studied. In the latter, the influence of the moving mass on the curved beams is analyzed. The presented approach had shown that, in comparison to the classical finite element method (FEM), a less number of degrees of freedom are required in order to obtain accurate results. Consequently, fewer computational resources are needed to reach the appropriate level of accuracy for the curved beams. This makes the presented approach competitive with the conventional FEM, especially in the analysis of the flexible spatial engineering structures with complex geometry.

Tacno numericko modelitanje krivolinijskih gredna nosac je od izuzetnog znacaja u mnogim inženjerskim oblastima. Geometrija krivolinijskog grednog nosaca, diskretizacija grede kao i potrebne pretpostavke grede, predstavljaju poteškoce prilikom formulacije krivolinijskog grednog elementa. U ovom radu je primenjen izogeometrijski pristup u dinamickoj analizi prostornih krivolinijskih grednih nosaca. Novi gredni element je definisan primenom Bernuli - Ojelerove pretpostavke, kao i osnovnih relacija diferencijalne geometrije i mehanike kontinuuma Košijeve grede. Geometrija grede, kao i polje pomeranja, brzine i ubrzanja su definisani primenom NURBS baznih funkcija, što predstavlja fundamentalnu osobinu isogeometrijskog pristupa. Složena geometrija krivolinijskog grednog nosaca može se tacno modelirati primenom izogeometrijskog pristupa zasnovanog na NURBS baznim funkcijama. Formulacija prostornog krivolinijskog grednog nosaca je izvedena u uslovina linearne teorije, dok je geometrijski nelinearna teorija primenjena samo na ravanskom krivolinijskom grednom nosacu, primenom eksplicitne metode integracije. Izvšena je analiza slobodnih i prinudnih vibracija. Analiza prinudnih vibracija je fokusirana na uticaj pokretnog opterecenja na krivolinijski gredni nosac. Primenom date formulacije dobijeni su rezultati zadovoljavajuce tacnosti sa manje stepeni slobode u poredjenju sa klasicnom metodom konacnih elemenata. U sladu sa tim, u cilju dobijanja rezultata zadovoljavajuce tacnosti krivolinijskog grednog nosaca nepohodna je primena manje resursa. Ovo cini prikazani pristup konkurentnijim klasicnoj metodi konacnih elemenata u analizi fleksibilnih inženjerskih konstrukcija sa složenom geometrijom.