Boundary Conditions Of Beam Research Articles

A mixed variational principle and its application to the nonlinear bending problem of orthotropic tubes—I. Development of general theory and reduction to cylindrical shells

A procedure is presented for obtaining mixed, nonlinear variational principles for elastic shells based on the intrinsic formulation of the shell equations. The applicability of the procedure is demonstrated by developing specific principles for shells of weak curvatures and for circular cylindrical shells in regular and extended forms. Other cases are also discussed. The principles are developed within the scope of small-strain, large-rotation theory for shells under the Kirchhoff-Love hypothesis and require the availability of curvature functions for the given classes of shells. No other restrictions need be placed, except for those related to the geometries of the shells under investigation. Specifically, subject to the limitation of small extensional strains, the displacements and rotations may be large and no particular mode of shell behavior is postulated. The variational functional basically contain the strain energy of bending and the complementary energy of the membrane force resultants. These functionals are formulated in terms of curvature and stress functions and their Euler-Lagrange equations are those of normal equilibrium, Gauss compatibility and associated boundary conditions. All may be nonlinear. Using the extended principle as a starting point, approximate principles and equations are developed in Part II for the nonlinear, nonuniform bending of orthotropic circular cylindrical tubes of finite length (extended Brazier effort). The semi-membrane approximation, with membrane-type shear deformation retained, is used in the analysis, plus some added restrictions of the Rayleigh-Ritz type on the curvature and stress fields. The results can be used for problems involving tubes subjected to various beam and shell type boundary conditions. The specific example of a clamped tube subjected to pure beam bending is calculated, using solutions of the equations for weak nonlinearity and a Rayleigh method for strong non-linearity. Application of some of the results to the nonlinear “local buckling” analysis of a finite-length tube subjected to bending compare favorably with published results. Besides the interest in the specific problem, this demonstrates the applicability of the mixed principle for obtaining direct, approximate nonlinear solutions to useful ongoing problems, as a complement to more exact, but cumbersome, finite element or series solutions.

Differential quadrature analysis of deflection, buckling, and free vibration of beams and rectangular plates

A new approach to apply the differential quadrature method to the deflection, buckling, and free vibration analysis of beams and plates with various boundary conditions is presented. A different method for application of the beam boundary conditions, proposed earlier by Wang and Bert, is extended to clamped-fixed, simply supported-fixed, and fixed-fixed beams and excellent results are obtained. It is found that the differential quadrature method gives less accurate results for the buckling load when the plate aspect ratio ( a b ) exceeds 2.45 for the cases considered and possible reasons for this are discussed.

Derivation of linear beam equations using nonlinear continuum mechanics

Familiar linear elastic and viscoelastic beam equations (Euler-Bernoulli, Rayleigh, Kelvin-Voigt, Timoshenko, and Shear Diffusion) and boundary conditions are derived from a nonlinear theory of large motions rather than the usual variational techniques. Also included is a fairly detailed derivation of the nonlinear theory and a careful discussion of the hypotheses.

Laser-interferometric detectors of gravitational waves

A laser-interferometric detector of gravitational waves is studied and a complete solution (to first order in the metric perturbation) of the coupled Einstein-Maxwell equations with appropriate boundary conditions for the light beams is determined. The phase shift, the light deflection and the rotation of the polarization axis induced by gravitational waves are computed. The results are compared with previous literature, and are shown to hold also for detectors which are large in comparison with the gravitational wavelength.

Variational analysis of thin beam excitation using asymmetric piezoelectric actuators.

This paper presents a dynamic model as well as an experimental study of the response of a beam to excitation by a single piezoelectric ceramic glued to the structure. Such an actuator, when excited, produces both flexural and extensional motion in the structure. Based on an analytical approach using a variational method, the model takes into account the dynamic coupling between the structure and the actuator, includes the free stress conditions at the piezoelectric actuator boundaries and allows one to predict with accuracy the beam displacement and thus the extensional strain on the surface of the structure for any beam boundary condition. Although essentially devoted to the study of asymmetric piezoelectric actuators, the accurate model presented here allows one to discuss the simplifying hypothesis usually made in past studies on symmetric actuators.

Random response of symmetric cross-ply composite beams with arbitrary boundary conditions

A generalized modal approach is presented to solve the equations of motion of a laminated composite beam obtained with a third-order shear deformation theory. The biorthonormal eigenfunctions of the differential equations expressed in the state form are used to decouple the equations. To obtain these eigenfunctions for beams with any arbitrary beam boundary conditions, a method is presented. The solution obtained by this approach is used to calculate the beam response for spatially and temporally correlated random loads. Several sets of numerical results are presented to demonstrate the importance of shear deformations in the dynamic analysis of composite beams.

Diagnosis of changes in elastic boundary conditions in beams by adaptive vibrational testing

This paper develops a vibrational diagnosis which identifies the boundary stiffnesses, mass density and flexural stiffness of a uniform beam. Dynamic excitation of the beam at one point and displacement measurements at a single position are sufficient. The need for measurements or excitations at different points along the beam is eliminated by on-line choice of the excitations, thus dynamically adapting the excitations to the assayed beam. Two different types of adaptive measurements are employed. The diagnosis is initiated by measuring three eigenfrequencies. It is shown that fairly large uncertainty (∼0.5%) in these measurements is acceptable. The adaptive diagnosis is tested by simulation.

Buckling and postbuckling behavior of delaminated sandwich beams

An investigation was performed to study the buckling and postbuckling behavior of sandwich beams containing lengthwise and depthwise through-the-width delaminations. An analytical beam model was developed to predict the buckling load of the beam and to describe its postbuckling response for arbitrarily situated delaminations and various combinations of boundary conditions. Griffith's energy release rate model was employed to predict the stability of delamination propagation under external loading and to determine the direction of delamination growth. Parametric studies over a wide range of beam geometries, damage sizes and locations, composite facings and beam boundary conditions were carried out to study their effects on the overall behavior of the sandwich structure, as well as its damage tolerance. The results demonstrated that a sandwich construction is very ‘sensitive’ to the presence of delaminations situated at the core-faceplate interface. Premature buckling failure occurs at external loads which are significantly lower than the buckling load for a ‘perfect’ sandwich beam; in ‘imperfect’ beams with composite faceplates, the layup sequence affects significantly the load-carrying capacity of the beam; varying either the boundary conditions in a sandwich beam or the lengthwise location of a delamination has a small effect on the postbuckling behavior of the beam. Delaminations located within composite faceplates have less pronounced influence, and as the defect is moved outwards the limit load may reach the buckling load corresponding to that of the ‘perfect’ beam. The proposed model is capable of analyzing the postbuckling behavior of both sandwich and composite laminated beams for arbitrary locations of the delamination, and various combinations of boundary conditions.

An investigation of elastic impact problems on beams and plates

When a system is struck by an impacting object, the system will be set into vibration and, in the vicinity of the contact point, local deformation will occur. An analog approach is used to simulate four kinds of boundary conditions for beams and simply supported rectangular plates, and two kinds of boundary conditions for circular plates, considered as lumped systems. By using the lumped parameter method, the impact forces and the deflections of the beams and plates can be easily calculated. This method is much simpler than using conventional complex impact formulas. Furthermore, the maximum impact velocity—the cause of the elastic deformation in the contact region—is obtained through a trial and error process. Finally, results of a simple experiment are presented to verify whether or not the calculated maximum impact velocity which caused the elastic deformation in the contact region is acceptable.

Application of the Ritz method to the analysis of non-linear free vibrations of beams

This paper presents an analytical solution for geometrically non-linear free vibrations of beams with elastically supported ends in the horizontal direction. The equation of motion is obtained by employing Hamilton's principle and assuming that horizontal inertia forces can be neglected. The Ritz method, with a continuum solution and an iterative procedure, are used for determining the frequencies and non-linear modes of vibrations. The orthogonality conditions for these modes are also discussed. Numerical results for various beam boundary conditions are presented and compared with available results.

Analogy for Beam‐Foundation Elastic Systems

An analogy between two systems, each consisting of an elastic beam on elastic foundation, is given. Through this analogy, forces and displacements of one of two beams correspond to the displacements and forces, respectively, of the conjugate one, and field equilibrium and mechanical boundary conditions of one beam correspond to field compatibility equations and kinematical boundary conditions of the other beam. As the beam and foundation stiffnesses of the real system correspond to the foundation and beam compliances of the conjugate one, the problems of statically indeterminate beams not lying on foundation, having uniform or nonuniform cross section can be solved by using simple equilibrium equations on the conjugate beam. The analogy can be applied also when the beam on elastic foundation experiences inelastic deformations both distributed and concentrated. This enables us to solve several practical problems, e.g., temperature change effects, influence lines, secondary moments on statically indeterminate prestressed structures, etc. Applications to cylindrical tanks subjected to rise of temperature and influence lines of nonprismatic shear beams are reported.

On Asymptotic Approximations to Beam Model Shapes

Recently the author had occasion to investigate inter alia the asymptotic character of the mode shapes of uniform beams. These results do not seem to have been presented before in the literature and so are given here as they appear to be of general interest. Clamped-free and free-free beams are considered, although other beam boundary conditions may be treated in a similar manner.

Linear electrostatic instability of the electron beam ion source

Linear plasma fluid theory is used to study the stability of a cold electron beam in Brillouin equilibrium which passes through a stationary cold ion background, with particular interest in stability for parameters relevant to EBIS devices. Dispersion is studied both analytically and numerically. For l = 0, the usual infinite medium two stream instability condition is shown to correspond to a requirement that beam perveance exceed a minimum value, P > 33 μperv; hence, this mode is stable for EBIS ( P ≈ 1 μperv). The Brillouin equilibrium rotation is shown to cause an electron-ion rotating stream instability, which is convectively unstable. The l = 1 mode is also found to be unstable. Higher modes numbers, l > 1, are unstable, but have reduced growth. Instability is only weakly affected by finite beam radius and boundary conditions.

Vibratory Identification of Beam Boundary Conditions

In previous work, by means of stochastic and deterministic models, the authors presented a system identification theory for performing nondestructive testing of elastic systems. The procedure requires identification of the structure’s boundary conditions. Herein, the mathematical models are improved, and vibration data are presented for the determination of the boundary parameters. These experimentally derived results are shown to validate the models.

The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions

The sixth-order differential equation of motion is derived in terms of the transverse displacement, w, for a three-layer sandwich beam with a viscoelastic core. Mathematical expressions in terms of w are found for a variety of beam boundary conditions. The solution of the differential equation by the method of Di Taranto is shown to yield a special class of complex, forced modes of vibration which are completely uncoupled. These complex modes can only exist when the beam is externally excited by specific “damped normal loadings” which are also complex and which are proportional to the local transverse inertia loading on the beam. Use of these modes in the analysis of the forced vibration problem leads to a simple series form of solution. The orthogonality of these complex modes is briefly discussed and proved.