This paper presents a meshless model for quasi-static and dynamic analysis of a three-dimensional Timoshenko beam with geometric nonlinearity. A general mathematical formulation is constructed based on the corrective smoothed particle method (CSPM), which can correct the low precision and completeness deficiency of the standard smoothed particle hydrodynamics(SPH) method. The discrete governing equations as well as the boundary conditions in strong form for the three-dimensional beam are then derived by using the conservation conditions and the CSPM interpolation function. The developed model enables one to expediently discretize the geometric nonlinear beam with only a row of particles at the central axis and to automatically satisfy the free boundary condition without any additional treatment. Moreover, Lagrangian kernel function and stress points are adopted to eliminate tensile instability and instability induced by the rank deficiency within the particle methods. Finally, comparisons with several results obtained from the existing literature are provided to demonstrate the validity and potential of the present procedure.

本文提出了一种用于几何非线性的Timoshenko梁的准静态和动态分析的无网格模型。基于校正平滑粒子方法（CSPM），构建了通用的数学公式，可以纠正标准平滑粒子流体动力学（SPH）方法的低精度和完整性缺陷。然后，通过使用守恒条件和CSPM插值函数，得出三维梁的离散控制方程式以及强形式的边界条件。开发的模型使人们能够方便地离散几何非线性光束，其中在中心轴只有一排粒子，并且能够自动满足自由边界条件，而无需任何其他处理。而且，采用拉格朗日核函数和应力点来消除拉伸不稳定性和粒子方法中秩不足引起的不稳定性。最后，与从现有文献中获得的一些结果进行比较，以证明本方法的有效性和潜力。