Beams are very important and widely used structural members in engineering practice. Recently, appli-cations of beams have been rapidly extended from classical metallic structures to nano- and bio-structures, in which finite element method is a tool dominantly adopted for analysis and design. To cover such new applica-tions, the modeling capability and nonlinear performance become more important in finite element analysis of beams.
First, we develop continuum mechanics beam elements in which fully coupled 3-D behaviors among stretching, bending, shearing, twisting and warping are considered. The beam element is directly degenerated from an assemblage of 3-D solid elements. The element has cross-sectional discretization which provides en-hanced modeling capabilities for complicated 3-D geometries including curved and twisted geometries, varying cross-sections, and arbitrary cross-sectional shapes.
Second, we propose a new and efficient displacement model to ensure the continuity of warping in beams with discontinuously varying arbitrary cross-sections. The entire warping displacement field is construct-ed by a combination of the three basis warping function, one free warping function and two interface warping functions, with warping degrees of freedom (DOFs). A new method to simultaneously calculate the free warping function and the corresponding twisting center is also introduced. Based on this method, the interface warping functions and the twisting centers at the interface cross-sections are obtained by solving a set of coupled equa-tions at the interface of two different cross-sections.
Third, we present the nonlinear formulation and performance of continuum mechanics based beam el-ements, in which fully coupled 3D behaviors of stretching, bending, shearing, twisting, and warping are automat-ically considered. The beam elements are directly degenerated from assemblages of 3D solid elements under the assumptions of Timoshenko beam theory. Therefore, cross-sectional discretization is possible and the elements can model complicated 3D beam geometries including curved and twisted geometries, varying cross-sections, eccentricities, and arbitrary cross-sectional shapes. In particular, the proposed nonlinear formulation can accu-rately predict large twisting behaviors coupled with stretching, bending, shearing, and warping. Through various numerical examples, we demonstrate the geometric (and material) nonlinear performance of the continuum mechanics based beam elements.
Finally, we propose a new numerical method to improve nonlinear performance: the eigen recomposi-tion. We classify and investigate the miss leading phenomena in internal virtual work via eigenvalue analysis. In order to improve the internal virtual work, the obtained nonlinear stiffness matrix is recomposed by the assumed eigenvector and the corresponding estimated eigenvalue. The performance of the recomposed stiffness matrix is demonstrated through several beam element examples.
다양한 공학적 구조물의 거동을 수치적으로 해석하기 위해 빔 유한요소는 널리 이용되고 있다. 하지만 최근의 의료장비와 분자구조 등의 해석으로 빔의 응용분야가 넓혀짐에 따라, 여전히 더 뛰어난 모델링 성능을 가지는 빔 요소의 개발이 필요하다.
첫번째로, 신축, 전단, 휨, 비틀림, 와핑 변형이 완벽히 연계된 해석이 가능한 연속체 역학 기반 빔 유한요소를 개발한다. 빔요소는 3차원 솔리드 요소모델로 부터 직접적으로 감절점되어 유도된다. 또한 이산화된 단면함수를 내재하고 있기 때문에, 복잡한 3차원 형상에 대한 모델링 역량이 뛰어나다.
두번째로, 불연속적으로 변하는 임의의 단면에 대한 효과적인 와핑 변위모델을 제시한다.
세번째로, 연속체 역학 기반 빔요소의 비선형 성능에 대해 발표한다.
네번째로, 고유벡터 재구성법이라는 비선형 성능을 증대시킬 수 있는 새로운 수치해석 방법을 제시한다.
