The parametric response of beam-columns of variable cross-section resting on an elastic foundation
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This investigation studied the problem of the parametric instability of a beam-column with variable cross-section resting on an elastic foundation. The problem was analyzed both theoretically and experimentally. The equation of motion for the system was developed following the classical equilibrium approach. The terms arising from consideration of the distributed longitudinal inertia were incorporated into the equation. The equation of motion was reduced to a system of ordinary differential equations of the second order by means of a slightly modified Galerkin method developed by the author. Periodic solutions of these equations, which represent the boundaries of the regions of parametric instability, were sought in the form of a Fourier series with periods T and 2T. This approach reduced the problem to determination of the eigenvalues of a set of matrices. The QR transformation was used to compute the eigenvalues of these matrices. This transformation was preferred over other methods of extracting eigenvalues mainly because of its versatility, numerical stability and less computer time. A part of this investigation was devoted to the study of the steady-state amplitudes of parametric vibrations within the principal region of instability. The problem of determining the amplitudes was also reduced to an eigenvalue problem. An experimental investigation was conducted to verify the validity of the theoretical results. The experiment was designed so as to permit an independent variation of the constant and dynamic components of the load and the excitation frequency parameters. A comparison of the experimental and the theoretical results showed close agreement both for the boundaries of the principal and second region of instability and the steady-state parametric response within the principal region of instability. The experimental and theoretical findings revealed that the slope factor of a column with linearly variable cross-sectional depth has a pronounced effect upon the boundaries of the principal region of instability. The upper boundary, as compared to that of a uniform column, moved closer to the lower boundary as the dynamic load was increased. This in effect narrowed the width of the principal instability region. The addition of an elastic foundation also narrowed the width of the regions of instability to some extent, and altered their location as well. The amplitudes of the steady-state parametric response in the regions of instability were observed to be relatively small in presence of an elastic foundation. The modified Galerkin method used in this study greatly simplified the computations and can effectively be used to compute the natural frequencies and the critical loads of beam-columns with variable cross-sectional dimension.
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