| Almost all problems in engineering are nonlinear and in most cases they can not be simplified to linear problems and thus we have to use nonlinear theories to find reasonable solutions. Geometric nonlinearity is caused by the large displacement and the large rotating of the structure and is pretty common in engineering. Because of the complexity of the structures in most practical engineering projects, the mathematical models in those projects are difficulty in solving precisely; therefore, the finite element numerical analysis suitable for all kinds of structures with complex shapes and boundary conditions is the plausible method to solve such problems.Solving practical engineering problems by traditional geometric nonlinear FEM has such problems such as complicated formula, large amount of calculation, low efficiency and low precision. Positional FEM is different from traditional nonlinear finite element methods because it uses the position rather than the displacement as the main variable and uses the position to directly calculate the stress and the strain. Thus it is more advantageous to solve problems with large displacement and large rotating; it can solve all geometric nonlinear problems such as those with large displacement small strain or those with large displacement large strain:it can be widely applied and has simple principles, smaller amount of calculation and high precision. But the research on positional FEM started pretty late and is not complete now, this paper creates a new kind of three-node beam element with constant corss-section and variable cross-section beam element, enriches the element library, studies the static and dynamic problems of positional FEM and use positional FEM to make dynamic analysis of complex structures. The main research contents and research results are as following: The static positional FEM for three-node beam element with equal cross-section is researched. To deal with the problems such as low’precision in calculation and the need to be divided into many elements, this paper creates a new kind of three-node beam element; the degree of freedom of the node is consisted with the coordinates of the node and the rotation angle of the cross-section. By the means of dimensionless parameterξ, the curvature of any point at the beam cross-section in any shape is deduced, the strain of longitudinal line of any point is calculated and the stain energy and the strain energy potential of the beam element are gained. Based on the lowest potential energy theory, the static positional FEM equation is derived and the program for calculating positional FEM is written. From examples such as purely curve Euler beam and cubic diamond-shaped rigid frame, the computational performance and the precision of this element is examined. The calculation results show that compared with traditional nonlinear FEM, the positional FEM uses fewer elements and gains high calculation precision, which shows its high calculation efficiency.The dynamic positional FEM for three-node beam element is researched. The lumped mass matrix of three-node beam element is derived, and the positional FEM equation for the dynamic problems of variable cross-section beam element is derived. The equation is solved by Newmark method; the velocity and the acceleration are derived from the position rather than the displacement of the node. The results from the calculation show that it is applicable to use simplified lumped mass matrix to calculate structure.The positional FEM of variable cross-section beam element is researched. According to the stress characteristics, the main shape and size characteristics of common projects and structures, the model of variable cross-section beam element is postulated; the longitudinal construction strain of any researching point on the cross-section is calculated; the strain energy potential, the strain energy and the area energy potential of the element is derived. Based on the theory of the lowest potential energy, the balance equation for static problems of geometric nonlinear positional FEM is deduced; this equation can be solved by Newton-Raphson iteration method. The lumped mass matrix and the consistent mass matrix of variable cross-section beam element are derived, and the positional FEM equation for the dynamic problems of variable cross-section beam element is derived. The equation is solved by Newmark method. For classic examples, the flexible variable cross-section beam with a solid bearing end is calculated. The results from the calculation show that the rotational inertia of the cross-section has little influence on the thin bar; the results from the usage of different mass matrix have few differences, so it is applicable to use simplified lumped mass matrix to calculate structure. Through calculating variable cross-section beams of different section shapes and getting the following results:if the coefficient is changed, the postulated model for variable cross-section can be applied to different shapes; compared with traditional segment elements of equal sections, this element needs smaller number of elements to achieve the required precision, and thus can lower the calculations and is applicable in practical engineering.The dynamic characteristics under modifying-amplitude of the crawler crane boom system are simulated by positional FEM. Based on the stress characteristics of space truss structure, the simplified FEM model for boom system is established by simulating the main jib of the boom with variable cross-section beam elements and three-node beam elements; the dynamic response time history of the boom system under modifying-amplitude is derived; these provide crucial theories for the design and analysis of hoisting machinery. It not only verifies the theoretical reasonability of the simplified model, but also shows that the application of positional FEM in complicated structural dynamics analysis is feasible. |
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