A three-dimensional beam formulation for large deformation and an accurate implementation of the free boundary

Abstract 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.

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