This paper initially discusses the dynamics of discrete structural systems of geometricaly nonlinear behaviour costituted by linear elastic materials. Two formulations are derived, namely global and incremental. They are both suitable to general FE modelling, as the matrix equations of motion are written in explicit form. Matrices and vectors involved are characterized in terms of constraint equations defined within the continuum discretization. In principle, such formulations are applicable to any structural theory, as the theories of beams, plates and shells. As an example, the Bernoulli-Euler beam element is studied herewith. Both global and incremental formulations capture the effect geometrical nonlinearities have upon inertial and elastic forces alike. The ANDROS FEM program, developed by the authors, which is based upon the global formulation, has been successfully used in several nonlinear analyses. From this general background, the paper proceeds to consider the effect statical loading may have upon the free undamped vibration frequencies of a structure. It is shown that the tangent stiffness matrix of the incremental formulation should be used in the resultant eingenvalue problem. In some cases, axial forces are seen to have a strong influence on the internal resonance tuning. It is shown, in a sample structure thus tuned and subjected to dynamical loading, that a nonlinear regime may appear in the response.
A General FEM Formulation of Nonlinear Dynamics Applied to Accessing the Statical Loading Effect Upon the Dynamic Response of Planar Frames
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Brasil, R. M. L. R. F., and Mazzilli, C. E. N. (November 1, 1993). "A General FEM Formulation of Nonlinear Dynamics Applied to Accessing the Statical Loading Effect Upon the Dynamic Response of Planar Frames." ASME. Appl. Mech. Rev. November 1993; 46(11S): S110–S117. https://doi.org/10.1115/1.3122625
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