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In this paper, the exact frequency and mode shape expressions are derived for universal bellows type of expansion joint in lateral and rocking modes of vibration. The effect of equivalent support stiffness and mass on the natural frequencies and mode shapes are studied in detail and the results for a range of non-dimensional parameters are presented in graphical forms which should be useful for piping and bellow designers.

This paper studies buckling instability of columns with variable bending stiffness subjected to an axially compressive load. An analytic approach has been presented to determine critical buckling loads of a nonuniform column with or without continuous elastic restraint along its length. We transform this problem into a Fredholm equation, and then to a system of linear equations. The desired buckling loads can be easily obtained by solving the least positive eigenvalue of the resulting system. The validity and efficiency of the method is confirmed by comparing our numerical results with those available. The influences of variable cross-section or elastic restraint stiffness on the buckling loads of a simply-supported column are analyzed. A suboptimal design of a tapered cylindrical bar with fixed weight is given. The present results are of benefit to the optimal design of beam/column structures.

Approximate numerical solutions are obtained for the vibration response of a functionally graded (FG) micro-scale beam entrapped within an axially-directed magnetic field using the differential transformation method (DTM). Idealized as a one-dimensional (1D) continuum with a noticeable microstructural effect and a thickness-directed material gradient, the microbeam’s behavior is studied under a range of nonclassical boundary conditions. The immanent microstructural effect of the micro-scale beam is accounted for through the modified couple stress theory (MCST), while the microscopic inhomogeneity is smoothened with the classical rule of mixture. The study demonstrates the robustness and flexibility of the DTM in providing benchmark results pertaining to the free vibration behavior of the FG microbeams under the following boundary conditions: (a) Clamped-tip mass; (b) clamped-elastic support (transverse spring); (c) pinned-elastic support (transverse spring); (d) clamped-tip mass-elastic support (transverse spring); (e) clamped-elastically supported (rotational and transverse springs); and (f) fully elastically restrained (transverse and rotational springs on both boundaries). The analyses revealed the possibility of using functional gradation to adjust the shrinking of the resonant frequency to zero (rigid-body motion) as the mass ratio tends to infinity. The magnetic field is noted to have a negligibly minimal influence when the gradient index is lower, but a notably dominant effect when it is higher.

This research presents a numerical approach to address the moving load problem of functionally graded (FG) beams with rotational elastic edge constraints, in which the regularized Dirac-delta function is used to describe a time-dependent moving load source. The governing partial differential equations of the system, derived in accordance with the classical Euler–Bernoulli beam theory, are approximated by the discrete singular convolution (DSC) method. The resulting set of algebraic equations can then be solved by the Newmark-β integration scheme. Such a singular Dirac-delta formulation is also employed as the kernel function of the DSC method. In this work, the material properties of FG beams are assumed to be changed in the thickness direction. A convergence study is performed to validate the accuracy and reliability of the numerical results. In addition, the effects of moving load velocity and material distribution on the dynamic behavior of elastically restrained FG beams are also studied to serve as new benchmark solutions. By comparing with the available results in the existing literature, the present results show good agreement. More importantly, the major finding of this work indicates that the DSC regularized Dirac-delta approach is a good candidate for moving load problems, since the equally spaced grid system adopted in the DSC scheme can achieve a preferable representation of moving load sources.