Extension of MITC to higher‐order beam models and shear locking analysis for compact, thin‐walled, and composite structures

Abstract

A class of mixed interpolated beam elements is introduced in this paper under the framework of the Carrera Unified Formulation to eliminate the detrimental effects due to shear locking. The Mixed Interpolation of Tensorial Components (MITC) method is adopted to generate locking-free displacement-based beam models using general 1D finite elements. An assumed distribution of the transverse shear strains is used for the derivation of the virtual work, and the full Gauss-Legendre quadrature is used for the numerical computation of all the components of the stiffness matrix. Linear, quadratic, and cubic beam elements are developed using the unified formulation and applied to linear static problems including compact, laminated, and thin-walled structures. A comprehensive study of how shear locking affects general beam elements when different classical integration schemes are used is presented, evidencing the outstanding capabilities of the MITC method to overcome this numerical issue. Refined beam theories based on the expansion of pure and generalized displacement variables are implemented making use of Lagrange and Legendre polynomials over the cross-sectional domain, allowing one to capture complex states of stress with a 3D-like accuracy. The numerical examples are compared to analytic, numerical solutions from the literature, and commercial software solutions, whenever it is possible. The efficiency and robustness of the proposed method demonstrated throughout all the assessments, illustrating that MITC elements are the natural choice to avoid shear locking and showing an unprecedent accuracy in the computation of transverse shear stresses for beam formulations.

Geometrically Nonlinear Analysis of Beam Structures via Hierarchical One-Dimensional Finite Elements

“…where denotes an implicit summation and varies from 1 to − 1. , , and are the strain components coming from the geometrical relations in (3) evaluated at the -th tying point and are the assumed interpolating functions. Their expressions as functions of the natural beam element coordinate ∈ [−1, 1] can be found in Carrera et al [36] and they are reported below. For linear elements, the interpolation is reduced to a point evaluation, since…”

Cross-sectional mapping for refined beam elements with applications to shell-like structures

“…Moreover, in this analysis case the MITC (Mixed Interpolation of Tensorial Components) approach has been used for the HLE models to compute the shear stresses. This formulation is based on the assumed interpolation of the shear components of the stress tensor and its implementation on CUF models as well as its numerical performances when applied to higher-order theories are presented in Carrera et al [49]. In view of the results obtained for the L-angle beam, the following remarks can be made:…”

“…A solid Nastran model has been included for comparison purposes. MITC4 beam models, developed in [49], have been also used to assess the solutions. The solutions of displacements and normal stresses show a good agreement with the ones obtained through the solid model as the polynomial order of the theory increases.…”

Farkli Uç Sinir Koşullarina Sahi̇p Ki̇ri̇şi̇n Carrera Bi̇rleşi̇k Formulasyon (Cuf) Çerçevesi̇nde Stati̇k Anali̇zi̇

“…Wang ve Wang [7] ve Gao ve Wang [8] tarafından önerilen geliştirilmiş kiriş teorisi, dördüncü mertebeden bir diferansiyel denklem ve ikinci mertebeden bir transandant denklemden oluşmasına karşın, bu kompleks kiriş teorisi ankastre uç için yer değiştirme kısıtlarını düzgün bir şekilde ele almakta yetersiz kalmaktadır. Literatürde, kayma düzeltme faktörü, varyasyonel asimptotik çözüm, genelleştirilmiş kiriş teorisi ve kesit çarpılması gibi yüksek mertebe etkileri dikkate alan diğer çalışmalar da mevcuttur [9][10][11][12][13][14].…”