On the analytical approach to the linear analysis of an arbitrarily curved spatial Bernoulli–Euler beam

Abstract

The equilibrium and kinematic equations of an arbitrarily curved spatial Bernoulli-Euler beam are derived with respect to a parametric coordinate and compared with those of the Timoshenko beam. It is shown that the beam analogy follows from the fact that the left-hand side in all the four sets of those equations are the covariant derivatives of unknown vector. Furthermore, an elegant primal form of the equilibrium equations is composed. No additional assumptions, besides those of the linear Bernoulli-Euler theory, are introduced, which makes the theory ideally suited for the analytical assessment of bigcurvature beams. The curvature change is derived with respect to both convective and material/spatial coordinates, and some aspects of its definition are discussed. Additionally, the stiffness matrix of an arbitrarily curved spatial beam is calculated with the flexibility approach utilizing the relative coordinate system. The numerical analysis of the carefully selected set of examples proved that the present analytical formulation can deliver valid benchmark results for testing of the purely numeric methods.

Dynamic Response Analysis of a Forced Fractional Viscoelastic Beam ∗

“…e books can provide a general overview about the Euler-Bernoulli beam theory, please see [1][2][3]. Some important studies related to beams modeled in the sense of classical beam theory are also summarized as follows, but not limited to [4][5][6][7][8][9][10][11][12][13][14][15]. e beam systems in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] have the integer order derivatives of the state function.…”

“…Some important studies related to beams modeled in the sense of classical beam theory are also summarized as follows, but not limited to [4][5][6][7][8][9][10][11][12][13][14][15]. e beam systems in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] have the integer order derivatives of the state function. In the beginning of 1930s, fractional derivative was introduced for describing the constitutive relation of some beam materials [16], and after 1980s, since fractional order equations have good memory and can be used to describe material properties more accurately with fewer parameters, they are considered to be good mathematical models for describing the dynamic mechanical behavior of materials [17].…”

Geometrically exact static isogeometric analysis of arbitrarily curved plane Bernoulli-Euler beam

“…where χ is the change of curvature with respect to the Frenet-Serret frame of reference [28,50]. From the condition of pure bending (N = 0), we can calculate the exact physical axial strain of beam axis as: since χ = K = LPF (2nπ/L) for this example.…”

Geometrically exact static isogeometric analysis of an arbitrarily curved spatial Bernoulli-Euler beam

“…The majority of them also do not consider truly geometric exact nonlinear analysis due to their lack of objectivity. Recently, Radenković and Borković contributed to the linear static analysis of arbitrarily curved BE beams [45,46], based on foundations given in [47]. The aim of this paper is to extend the developed formulation, which is applicable to strongly curved beams, to the nonlinear setting of geometrically exact beam theory.…”