Study On Spatial Beam Based On Displacement Deformation Theory

Since Beam element is often used in the structural analysis, the exploration and improvement of its performance is very valuable. The existing beam theories include: classical Euler-Bernoulli beam theory and Timoshenko beam theory. The Timoshenko beam theory can directly consider the effect of shear deformation, but for the classic Euler-Bernoulli beam theory, it introduces an influence coefficient to its element stiffness matrix to take consideration of shear deformation.And yet for all that, both cannot output the value of shear displacement. Besides, since they both adopt the first-order shear deformation theory, we can also not get the solution of shear stress directly. We need to calculate it according to the formula of mechanics of materials. Apparently, the neglect of shear stress in the elastic-plastic analysis using the commercial software will have an impact on the accuracy of the results, especially for the short and deep beam.For these reasons, this paper proposes a new model of the beam. It breaks the vertical displacement into shear displacement and bending displacement.Using a simple higher-order shear deformation theory to consider the impact of shear deformation and based on the interpolation method of shear displacement, the beam model is divided into nine degrees of freedom model and twelve degrees of freedom model.Then, based on the displacement deformation theory, the paper gives the geometry and physical relationship of the new beam model and the incremental constitutive relationship in elastoplastic stage. A finite element model of space beam element is built under the combined effects of tension and compression, bending, torsion, shear. The element stiffness matrix for linear, nonlinear analysis is derived. And in accordance with the principle of minimum potential energy it derives the finite element basic formula in linear elastic and elastic-plastic stage and gets the expression of equivalent nodal force and space coordinate conversion formula. Finally, introducing the relevant non-linear algorithms used in the program. And using FORTRAN to program, through three examples to verify the correctness and advantages of the spatial beam model in aspect of the calculation of shear displacement, elastic-plastic solving and analysis of space structure.