Abstract
The formulation of a family of advanced one-dimensional finite elements for the geometrically nonlinear static analysis of beam-like structures is presented in this paper. The kinematic field is axiomatically assumed along the thickness direction via a Unified Formulation (UF). The approximation order of the displacement field along the thickness is a free parameter that leads to several higher-order beam elements accounting for shear deformation and local cross-sectional warping. The number of nodes per element is also a free parameter. The tangent stiffness matrix of the elements is obtained via the Principle of Virtual Displacements. A total Lagrangian approach is used and Newton-Raphson method is employed in order to solve the nonlinear governing equations. Locking phenomena are tackled by means of a Mixed Interpolation of Tensorial Components (MITC), which can also significantly enhance the convergence performance of the proposed elements. Numerical investigations for large displacements, large rotations, and small strains analysis of beam-like structures for different boundary conditions and slenderness ratios are carried out, showing that UF-based higher-order beam theories can lead to a more efficient prediction of the displacement and stress fields, when compared to two-dimensional finite element solutions.
Highlights
Many structural elements, such as aircraft wings, rotor blades, robot arms, or structures in civil construction, can be idealised as beams
Further works on large rotations and Mathematical Problems in Engineering large displacements analysis of shear-deformable beams by using an absolute nodal coordinate formulation accounting for a nonrigid cross-sectional kinematics were carried out by Dufva et al [9] and Omar and Sharana [10]
A family of refined one-dimensional finite elements derived through a Unified Formulation of the displacement field has been proposed for the geometrically nonlinear analysis of beam-like structures
Summary
Many structural elements, such as aircraft wings, rotor blades, robot arms, or structures in civil construction, can be idealised as beams. Further works on large rotations and Mathematical Problems in Engineering large displacements analysis of shear-deformable beams by using an absolute nodal coordinate formulation accounting for a nonrigid cross-sectional kinematics were carried out by Dufva et al [9] and Omar and Sharana [10]. As opposed to the Lagrange layer-wise approximation [20, 21], in both linear and nonlinear analyses, the use of Taylor polynomials allows an enrichment of the cross-sectional kinematics by increasing the order N and with no need for additional crosssectional nodes This feature makes Taylor-based refined models suitable for the analysis of multilayered structures in the framework of an equivalent single-layer approach that will be presented in a future work. A detailed description of the stress analysis under large displacements, not often encountered in the literature, has been provided, together with the discussion of some limitations of the proposed formulation with respect to finite elements with large strains capabilities
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