Więckowski Zdzisław, Świątkiewicz Paulina
Department of Mechanics of Materials, Łódź University of Technology, 90-924 Łódź, Poland.
Materials (Basel). 2021 Jan 19;14(2):460. doi: 10.3390/ma14020460.
The stress-based finite element method is proposed to solve the static bending problem for the Euler-Bernoulli and Timoshenko models of an elastic beam. Two types of elements-with five and six degrees of freedom-are proposed. The elaborated elements reproduce the exact solution in the case of the piece-wise constant distributed loading. The proposed elements do not exhibit the shear locking phenomenon for the Timoshenko model. The influence of an elastic foundation of the Winkler type is also taken into consideration. The foundation response is approximated by the piece-wise constant and piece-wise linear functions in the cases of the five-degrees-of-freedom and six-degrees-of-freedom elements, respectively. An a posteriori estimation of the approximate solution error is found using the hypercircle method with the addition of the standard displacement-based finite element solution.
提出了基于应力的有限元方法来求解弹性梁的欧拉 - 伯努利模型和铁木辛柯模型的静态弯曲问题。提出了两种类型的单元——具有五个和六个自由度。所阐述的单元在分段常数分布载荷情况下能再现精确解。对于铁木辛柯模型，所提出的单元不会出现剪切锁定现象。还考虑了温克勒型弹性基础的影响。在分别针对五个自由度和六个自由度单元的情况下，基础响应分别由分段常数函数和分段线性函数近似。使用超圆法并结合基于标准位移的有限元解来对近似解误差进行后验估计。
