The linearized three-dimensional beam theory of naturally curved and twisted beams: The strain vectors formulation

Abstract

This paper presents the equations of the linearized geometrically exact three-dimensional beam theory of naturally curved and twisted beams. A new finite-element formulation for the linearized theory is proposed in which the strain vectors are the only unknown functions. The linear form of the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal, is enforced to be satisfied at chosen points. An arbitrary curved and twisted axis of the beam is taken into account which demands proper consideration of the non-linearity of spatial rotations. The accuracy and the efficiency of the derived numerical algorithm are demonstrated by comparing present numerical results with various analytical and numerical results.

Assessment of 4-node EAS–ANS shell elements for large deformation analysis

“…Linearization of δ EAS (28) can be obtained by introducing the above interpolations into δ EAS , see (22). One gets…”

“…The remaining geometric, material and load characteristics are: width w = 1.1, length l = 12, and E = 29 × 10 6 , ν = 0.22. Analytical value of displacement under the force in the y direction is 0.00174274 (data A), see [28], and 0.3427 (data B), see [21]. Normalized value of this displacement (with respect to analytical solution) is for derived elements presented in Table 3.…”

“…Linear analysis is performed for width w = 1.1, radius R = 5, thickness h = 0.32, material constants E = 29 × 10 6 , ν = 0.22, and F x = F y = 0, F z = 1. Analytical solution for displacement under the force in z direction is 0.00626901, see [28]. Normalized value of this displacement (with respect to the analytical solution) is for derived elements presented in Table 4.…”

The wavelet-based theory of spatial naturally curved and twisted linear beams

“…In the present paper the wavelet-based discretization will be applied to the linearized finite-strain beam theory [24] which assumes small displacements, rotations and strains but is capable of considering an arbitrary initial geometry and material behaviour. In the numerical solution algorithm, we base our derivations on the vector of strain measures as the only unknown functions in a finite element.…”

On the deformation of almost cylindrical elastic beams

“…In the context of the classical elasticity, the mechanical behavior of the noncylindrical elastic bars has been studied in many papers (see, e.g., Dryden (2007), Zupan and Saje (2006), You et al (2002), and the references therein). These bodies are of interest both from the technical and mathematical point of view.…”