دانلود رایگان مقاله فرمول بندی زنجیره معادل تحلیل غیرخطی دو بعدی ساختار خرپا

Abstract

In this work, an equivalent continuum multiscale formulation is presented for the geometrical nonlinear analysis of the structures with lattice truss materials. This formulation is established by combining the extended multiscale finite element method and the co-rotational approach. Firstly, the lattice truss unit cell is equivalent to a continuum coarse element by using a numerical constructed interpolation function in the local coordinate system. Then the tangent stiffness matrix of this coarse element is derived by employing the basic idea of the co-rotational approach in the global coordinate system. Thus, the global nonlinear equilibrium equations of the structure at the macroscopic level can be solved by using the general displacement control algorithm to capture the equilibrium path with multiple critical points. After performing all of the incremental steps and the iterative steps on the macroscopic scale, the microscopic information, such as the displacement, stress and strain, can be obtained easily by virtue of the afore-constructed numerical interpolation functions once again. In addition, several numerical examples are carried out to study the effects of the layout and size of unit cell, investigate the sensitivity of coarse-scale meshes and verify the validation and efficiency of the presented multiscale formulation.

6. Conclusions

In this work, an equivalent continuum multiscale method (ECMM) is developed by combining the extended multiscale finite element method (EMsFEM) and the co-rotational formulation for the geometrical nonlinear analysis of the structures with lattice truss unit cells. Firstly, the basic ideas of the EMsFEM is briefly reviewed. Then, the macroscopic equivalent quantities in the local system of the unit cell are determined. Next, the co-rotational formulation is adopted to obtain the tangent stiffness matrix in the global system of the equivalent continuum coarse element and the overall tangent stiffness matrix of the structure at the macroscopic level in the global system could be assembled naturally. Finally, the nonlinear equilibrium equations of the structure at the macroscopic level can be solved by employing the general displacement control method and the results of each microscopic node and truss element within the unit cell can also be calculated easily by using the downscaling computation. Several numerical examples were implemented to investigate the layouts and the size effects of unit cell. The results indicated that the effects of different unit cell layouts on the computation accuracy are not obvious for the case with fixed coarse-scale mesh. However, increasing the unit cell size will improve the accuracy significantly for the case with fixed fine-scale mesh. Furthermore, the validation and efficiency of the proposed multiscale formulation were also investigated by comparison with the results of the traditional FEM and ANSYS on the fine-scale mesh.