M.A. De Rosa

Free vibrations of Timoshenko beams on two-parameter elastic foundation

The free vibration frequencies of Timoshenko beams on two-parameter elastic foundation are examined. Two variants of the equation of motion are deduced, in which the second foundation parameter is a function of the total rotation of the beam or a function of the rotation due to bending only, respectively. Both axial flexibilities and rotational flexibilities of the constraints are taken into account, and some numerical examples allow us to draw certain conclusions about the different behaviour of the two proposed foundation models.

The influence of an intermediate support on the stability behaviour of cantilever beams subjected to follower forces

Abstract A non-uniform undamped beam, which is simply supported at one end, resting on an elastically flexible support at an intermediate abscissa q, is examined. The beam is subjected to a generic set of follower forces along the axis, so that Beck, Leipholz and Hauger problems can be easily recovered as particular cases. In all these systems a location for the intermediate support exists, which corresponds to passage from flutter to divergence.

Free vibrations of tapered beams with flexible ends

The dynamic behaviour of beams with linearly varying cross-section is examined, in the presence of rotationally and axially flexible ends. The equation of motion is solved in terms of Bessel functions, and the boundary conditions lead to the frequency equation which is a function of four flexibility coefficients. For some particular cases of perfect constraints some known results of the literature can be recovered. Numerical results end the paper.

On the dynamic behaviour of slender beams with elastic ends carrying a concentrated mass

The dynamic behaviour of a slender beam carrying a concentrated mass at an arbitrary abscissa is examined. The beam is supposed to be elastically restrained against the rotation and the translation at both the ends, so that it is possible to study all the common boundary conditions. First, the exact solution is calculated, by solving the differential equations of motion and by imposing the corresponding boundary conditions. The resulting frequency equation is numerically solved. Subsequently, various approximate results are given, using the optimized Rayleigh-Schmidt approach with trigonometric and static shape functions, so that some comparison becomes possible. Finally, an application of the Morrow method allows us to obtain a lower bound to the true results.

Exact and approximate dynamic analysis of circular arches using DQM

Abstract A modified version of the differential quadrature method is applied to two versions of the sixth-order differential equation of motion governing free in-plane inextensional vibrations of circular arches (see Henrych, 1981). All the boundary conditions can be imposed exactly, without introducing δ points (see e.g. Bert and Malik, 1996). Consequently, the results are calculated with high precision, and a comparison between exact and approximate frequencies becomes possible. The convergence rate of the discretization method is shown to be very fast, even for the higher eigenvalues, so that a small number of Lagrangian coordinates permits a good approximation to the true results. It is shown that the approximate formulation leads to noticeable errors for the first frequencies of deep arches, whereas shallow arches and higher-order frequencies can be safely calculated with the simplified approach. The paper ends with some tables in which the first ten free vibrations frequencies for clamped arches, two-hinged arches and cantilever arches are compared with some known results from the literature.

Stability and dynamic analysis of two-parameter foundation beams

Abstract A dynamic and stability analysis of a foundation beam resting on a two-parameter elastic soil is performed. The beams cross-section can vary according to an arbitrary law, all the boundary conditions can be examined, and the presence of intermediate constraints and concentrated masses can be easily dealt with. Both the soil parameters can vary along the span with an arbitrary law. The structure is reduced to a n degrees of freedom system, the kinetic energy and the total potential energy are readily written, and the resulting eigenvalue problems can be solved to give the free vibration frequencies or the critical loads. Numerical examples and comparisons show the effectiveness of the proposed approach.

STABILITY AND DYNAMICS OF BRIDGE GIRDERS UNDER MULTIPLE AXIAL LOADS

Abstract The static and dynamic behaviour of a three-span continuous beam subjected to multiple independent axial forces is examined. The two central supports can be considered elastically flexible, in order to simulate the real piers of a girder bridge, while the cross section and inertia of the beam can vary according to a generic (even discontinuous) law. The actual value of the stiffness of the central supports cannot be exactly detected, hence the evolution of stability boundaries and fundamental characteristic surfaces is examined for various stiffness values. The forced dynamics of the system are also studied, where the forcing term is given by a synchronous motion of the four supports, or by a simple motion of a single support. This last forcing term can easily represent a seismic excitation if the span of the bridge is large. Finally, a numerical example is given, in which a recently built three-span continuous bridge is examined.

Plate bending analysis by the cell method: Numerical comparisons with finite element methods

Abstract A recently developed discretization method is applied to some classical problems in plate bending analysis, in order to check the accuracy of the solutions and the rate of convergence for various discretization levels. Numerical comparisons with available finite element solutions are reported, and it is shown that the proposed method can be considered competitive with the more recent finite element techniques.

Free Vibrations of a Cantilevered SWCNT with Distributed Mass in the Presence of Nonlocal Effect

The Hamilton principle is applied to deduce the free vibration frequencies of a cantilever single-walled carbon nanotube (SWCNT) in the presence of an added mass, which can be distributed along an arbitrary part of the span. The nonlocal elasticity theory by Eringen has been employed, in order to take into account the nanoscale effects. An exact formulation leads to the equations of motion, which can be solved to give the frequencies and the corresponding vibration modes. Moreover, two approximate semianalytical methods are also illustrated, which can provide quick parametric relationships. From a more practical point of view, the problem of detecting the mass of the attached particle has been solved by calculating the relative frequency shift due to the presence of the added mass: from it, the mass value can be easily deduced. The paper ends with some numerical examples, in which the nonlocal effects are thoroughly investigated.

Eigenvalues' behaviour of continuous beams on elastic supports in the presence of damping and follower forces

The frequencies of free vibration of a continuous beam on elastic supports, in the presence of multiple conservative and follower axial loads are examined, both for damped and undamped systems. The structure is discretized according to the so-called cell procedure (V. Franciosi, Le Situazioni Semilineari in Scienze delle Costruzioni. Liguori editore, Naples, Italy (1987) [1]), in which the structure is reduced to a set of rigid bars, elastic springs and concentrated dashpots. The resulting finite-degree-of-freedom system allows us to examine in some detail the parameter influence on the frequency behaviour. Extensive numerical calculations are also performed, for a three span continuous beam, in which statical and dynamical critical loads are plotted vs the stiffness of the elastic supports.