Raymond H. Plaut

Parametric instability of laminated composite plates with transverse shear deformation

Abstract The instability of composite laminated plates under uniaxial, harmonically-varying, in-plane loads is investigated. Both symmetric cross-ply laminates and antisymmetric angle-ply laminates are analyzed. The first-order shear deformation plate theory is used to model composite laminates. The resulting linear equations of motion are transformed into small, uncoupled sets of equations, and instability regions in the plane of load amplitude versus load frequency are determined using the finite element method. The effects of damping, ratio of edge length to thickness of the plate, orthotropy, boundary conditions, number of layers and lamination angles on instability regions are examined.

Sensitivity Analysis and Optimal Design of Nonlinear Beams and Plates

ABSTRACT A uniform formulation of sensitivity analysis for beams and plates is presented in terms of generalized stresses and strains. Both physical and geometric nonlinearities can be treated within this formulation. Next, optimal design problems for stress and deflection constraints are formulated and the relevant optimality conditions are derived using the concept of a linear adjoint structure. Finally, several numerical solutions of optimal design problems of beams are presented.

The influence of an internal resonance on non-linear structural vibrations under combination resonance conditions

The response of structural elements to a simple harmonic, transverse excitation is considered. The effects of both initial curvature and midsurface stretching are included; thus, the governing equations contain both quadratic and cubic terms. A perturbation technique, the method of multiple scales, is used to determine the response. Attention is focused on the subharmonic resonance Ω≅2ω2, where Ω is the forcing frequency and ω2 is a natural frequency. If the system possesses an internal resonance of the form ω2≅2ω1, energy may be transferred from the second mode to the first mode. The structural response is investigated in the presence, and in the absence, of this internal resonance. A comparison of the results reveals that the amplitude of the response can be significantly reduced by the presence of such an internal resonance. This suggests a means of passive vibration control. Also, the internal resonance causes a saturation phenomenon and a role reversal between the directly and indirectly excited modes.

Design of laminated plates for maximum buckling load

The buckling load of laminated plates having midplane symmetry is maximized for a given total thickness. The thicknesses of the layers are taken as the design variables. Buck ling analysis is carried out using the finite element method. The optimality equations are solved by a homotopy method which permits tracing optima as a function of total thick ness. It is shown that for any design with a given stacking sequence of ply orientations, there exists a design associated with any other stacking sequence which possesses the same bending stiffness matrix and same total thickness. Hence, from the optimum design for a given stacking sequence, one can directly determine the optimum design for any rearrangement of the ply orientations, and the optimum buckling load is independent of the stacking sequence.

Detection of Cracks in Rotating Timoshenko Shafts Using Axial Impulses

A rotating Timoshenko shaft with a single transverse crack is considered. The crack opens and closes during motion. The shaft has simply supported ends, and the six coupled, piecewise-linear equations of motion are integrated numerically after application of Galerkins method with two-term approximations for each of the six displacements. Time histories and frequency spectra are compared for shafts with no crack and with a crack

The effects of initial thrust and elastic foundation on the vibration frequencies of a shallow arch

A shallow elastic arch subjected to a static load is considered. Plots of load magnitude versus the squares of the vibration frequencies (i.e., characteristic curves) have been obtained previously. Here, the effects of initial thrust and elastic foundation on the characteristic curves are investigated. For simplicity, results are derived for an arch with pinned ends and a sinusoidal initial shape, and the static load is assumed to have a sinusoidal distribution.

Post-bauckling and Vibration of Heavy Beam on Horizontal or Inclined Rigid Foundation

A slender, straight beam resting on a flat, rigid foundation does not buckle when subjected to a compressive load, since the load cannot overcome the effect of the beam’s weight. However, it buckles if its ends are moved toward each other. Post-buckling of such a beam is examined, both theoretically and experimentally, for horizontal and inclined foundations. The beam is modeled as an elastica, and equilibrium states with large deflections are computed, including cases in which self-contact occurs. Frequencies and mode shapes for small vibrations about equilibrium are also determined. Agreement between the theoretical and experimental results is very good.

Free Vibration Analysis of an Inflated Toroidal Shell

Free vibration analysis of a free inflated torus of circular cross-section is presented. The shell theory of Sanders, including the effect of pressure, is used in formulating the governing equations. These partial differential equations are reduced to ordinary differential equations with variable coefficients using complete waves in the form of trigonometric functions in the longitudinal direction. The assumed mode shapes are divided into symmetric and antisymmetric groups, each given by a Fourier series in the meridional coordinate. The solutions (natural frequencies and mode shapes) are obtained using Galerkins method and verified with published results. The natural frequencies are also obtained for a circular cylinder with shear diaphragm boundary condition as a special case of the toroidal shell. Finally, the effects of aspect ratio, pressure, and thickness on the natural frequencies of the inflated torus are studied.

Two-dimensional analysis of stacked geosynthetic tubes on deformable foundations

Stacked geosynthetic tubes resting on a deformable foundation such as soil are analyzed. The tubes contain a slurry which applies hydrostatic pressure. The material of the tubes is assumed to act like an inextensible membrane and to have negligible weight, and the foundation is assumed to exert a normal upward pressure which is proportional to the downward deflection. Friction is neglected between tubes and at the foundation interface. Two configurations are considered: (a) one tube on top of another and (b) one tube straddling two tubes underneath it. For the latter formation, the case of external fluid acting on one side is analyzed, to simulate an application as a dike, and rigid blocks are utilized to prevent sliding of the tubes. Equilibrium shapes of the tubes are obtained numerically from a closed-form integral formulation, and the tension in each tube is computed.

Non-linear structural vibrations involving a time delay in damping

Time delays occur in many physical systems. In particular, when automatic control is used with structural or mechanical systems, there typically exists a delay between measurement of the system state and corrective action. In this paper, non-linear vibrations are considered and a time delay is present in the damping. The system has one degree of freedom with weak quadratic and cubic non-linearities in the restoring force. Cases of external (forcing) excitation and of parametric excitation are treated. Six resonance conditions are analyzed by using the method of multiple scales, including primary, subharmonic, and superharmonic resonances. The effect of the time delay is investigated and the results are presented as plots of the steady state response amplitude versus the excitation frequency and amplitude.