Abstract A geometrically nonlinear (3,2) unified zigzag beam element is developed with a reduced number of degree-of-freedom for the large deformation analysis. The main merit of the beam element model is the Kirchhoff and Cauchy shear stress solution for large deformation and large strain analysis is more accurate. The geometrically nonlinearity is considered in the calculation of the zigzag coefficients. Thus, the results of shear Cauchy stress are matching well with solid element analysis in case of the beam with aspect ratio greater than 20 under large deformation. The zigzag coefficients are derived explicitly. The Green strain and the second Piola Kirchhoff stress are used. The second Piola Kirchhoff shear stress is continuous at the interface between adjacent layers priori. The bottom surface second Piola Kirchhoff shear stress condition is used to determine the zigzag coefficient and the top surface second Piola Kirchhoff shear stress condition is used to reduce one degree-of-freedom. The nonlinear finite element equations are derived. In the numerical tests, several benchmark problems with large deformation are solved to verify the accuracy. It is observed that the proposed beam has accurate solution for beam with aspect ratio greater than 20. The second Piola Kirchhoff and Cauchy shear stress accuracy is also good. A convergence study is also presented.

中文翻译：

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用于大变形分析的几何非线性 (3,2) 缩减自由度统一锯齿形层压梁单元

摘要 为大变形分析开发了一种几何非线性（3,2）统一之字形梁单元，其自由度数减少。梁单元模型的主要优点是大变形和大应变分析的基尔霍夫和柯西剪应力解更准确。在计算锯齿形系数时考虑了几何非线性。因此，对于大变形下纵横比大于20的梁，剪切柯西应力结果与实体单元分析吻合较好。锯齿形系数是明确导出的。使用格林应变和第二 Piola Kirchhoff 应力。第二个 Piola Kirchhoff 剪切应力在相邻层之间的界面处是连续的。底面第二 Piola Kirchhoff 剪应力条件用于确定曲折系数，顶面第二 Piola Kirchhoff 剪应力条件用于减少一个自由度。推导出非线性有限元方程。在数值试验中，解决了几个大变形的基准问题，以验证精度。观察到提出的梁对于纵横比大于20的梁具有准确的解。第二Piola Kirchhoff和Cauchy剪应力精度也很好。还介绍了收敛性研究。解决了几个大变形的基准问题以验证准确性。观察到提出的梁对于纵横比大于20的梁具有准确的解。第二Piola Kirchhoff和Cauchy剪应力精度也很好。还介绍了收敛性研究。解决了几个大变形的基准问题以验证准确性。观察到提出的梁对于纵横比大于20的梁具有准确的解。第二Piola Kirchhoff和Cauchy剪应力精度也很好。还介绍了收敛性研究。