DOI: 10.14457/tu.the.2020.78
title: Geometrically nonlinear multi-patch isogeometric analysis of beams: Timoshenko and Euler-Bernoulli beam theories

description: Isogeometric analysis (IGA) is a recently developed numerical technique that aims at unifying geometric design and computational finite element analysis into one model. The technique is conducted by using the same basis functions for both computer-aided design and finite element analysis. In this study, two new efficient Timoshenko and Euler-Bernoulli beam formulations are presented in the context of IGA for geometrically nonlinear analysis of spatial beam structures.A novel Timoshenko beam formulation is proposed by means of the total Lagrangian description using the Green-Lagrange strain tensor and the second Piola-Kirchhoff stress tensor as the energy conjugate pair. This approach facilitates consideration of hyperelastic materials in analysis of highly flexible beam structures. Although in the context of the conventional finite element approach, many beam formulations have been developed by using the Green-Lagrange strain and second Piola-Kirchhoff stress tensors, there virtually exist no isogeometric Timoshenko beam formulations that are derived by using this conjugate pair. Three-dimensional beam configurations are reduced into one-dimensional structures using the beam axis and director vectors of the cross-sections. The displacements of the beam axis and the total cross-sectional rotation are considered as unknown kinematics. The cross-sectional rotation is represented by an orthogonal tensor, which is parameterized by a vector-like parameter. Updating the cross-sectional rotations is performed purely through natural exponentiation and superposition of relevant rotational quantities. This enables the proposed Timoshenko beam formulation to tackle beam structures undergoing large displacements and rotations without any restriction in magnitude.In this thesis, a novel Euler-Bernoulli beam formulation is also proposed. In the developed Euler-Bernoulli beam formulation, a geometric description that is similar to the one employed in the Timoshenko beam formulation is utilized. However, kinematics are described by the displacements of the beam axis and the axial cross-sectional rotation, not the total one, showing some advantages which will be discussed later in the thesis. The orthogonality between the cross-sections and the beam axis is satisfied by using the smallest rotation mapping for the description of finite cross-sectional rotations. The use of the smallest rotation mapping reduces the nonlinearity of the employed strain measurements with respect to the unknown kinematics and offers linearization that is much more efficient than those of existing isogeometric Euler-Bernoulli beam formulations. Furthermore, a penalty-free approach is introduced to deal with rigid connections in multi-patch beam structures in the context of geometrically nonlinear analysis. A novel nonlinear transformation between the total cross-sectional rotation and the unknown kinematics is derived, which facilitates the use of the total cross-sectional rotations at the ends of patches as discrete unknowns. This approach also allows straightforward enforcement of rotational boundary conditions.To show the accuracy and efficiency of the proposed beam formulations, some benchmark and well-established numerical examples with various types of beams, i.e., straight, curved, pre-twisted beams, and lattice-like beam structures, are analyzed. The obtained results are compared with those in the literature, obtained from both analytical and numerical methods.