Flexural Stress Waves in an Infinite Elastic Plate Due to a Suddenly Applied Concentrated Transverse Load

1960 ◽  
Vol 27 (4) ◽  
pp. 681-689 ◽  
Author(s):  
Julius Miklowitz
Keyword(s):  
Shear Force ◽  
Plate Theory ◽  
Three Dimensional ◽  
Infinite Plate ◽  
Present Theory ◽  
Transverse Load ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Flexural Stress ◽  
The Moment

The problem solved is that of an infinite plate subjected to a suddenly applied concentrated transverse shear load. The solution is derived from a plate theory that incorporates, in addition to bending, the effect of shear force and rotatory inertia on the deflection. These added effects give the present theory true wave character along with greater accuracy in the waves predicted. Numerical evaluation of the solution brings out the effects of dispersion and distortion on the moment and shear-force response of the plate. A criterion is developed for judging the accuracy of this response. It is based on a comparison, employing the stationary phase method, of the present approximate and exact (three-dimensional) theories.

Flexural Wave Solutions of Coupled Equations Representing the More Exact Theory of Bending

1953 ◽  
Vol 20 (4) ◽  
pp. 511-514
Author(s):  
Julius Miklowitz
Keyword(s):  
Boundary Conditions ◽  
Present Method ◽  
Shear Force ◽  
Beam Element ◽  
Transverse Load ◽  
Flexural Wave ◽  
Infinite Beam ◽  
Coupled Equations ◽  
Wave Solutions ◽  
The Moment

Abstract Presented here is a new method for deriving flexural wave solutions for the Timoshenko bending theory. The method is based on a breakdown of the total deflection into its bending and shear components. Instead of treating the full Timoshenko equation (1) an equivalent set of coupled equations, representing the rotational and translatory motions of the beam element, is solved. The advantages of this method stem from (a) the simplicity of the associated expressions for the moment and shear force, which are the elementary bending theory relations, and (b) the well-defined nature of the related boundary conditions. The latter is particularly important since it is difficult to define the proper boundary conditions associated with the full Timoshenko equation. This is evidenced in the works of Uflyand (2) and Dengler and Goland (3), both of which are concerned with wave solutions for the infinite beam under the action of a concentrated transverse load. The quoted work (3) points out the erroneous boundary conditions used in the Uflyand work (2). The present method is applied to the same case treated in the works (2, 3). Agreement is shown with the Dengler and Goland solution. The Uflyand solution is shown to have meaning when interpreted properly. The derivation of transforms for other beam cases, both finite and infinite, by the present method has also been included in this work.

The Effect of Transverse Shear Deformation on the Bending of Elastic Plates

1945 ◽  
Vol 12 (2) ◽  
pp. A69-A77 ◽  
Author(s):  
Eric Reissner
Keyword(s):  
General Solution ◽  
Rectangular Plate ◽  
Transverse Shear ◽  
Plate Theory ◽  
Infinite Plate ◽  
Present Theory ◽  
System Of Equations ◽  
Elastic Plates ◽  
Classical Plate Theory

Abstract A system of equations is developed for the theory of bending of thin elastic plates which takes into account the transverse shear deformability of the plate. This system of equations is of such nature that three boundary conditions can and must be prescribed along the edge of the plate. The general solution of the system of equations is obtained in terms of two plane harmonic functions and one function which is the general solution of the equation Δψ − (10/h2)ψ = 0. The general results of the paper are applied (a) to the problem of torsion of a rectangular plate, (b) to the problems of plain bending and pure twisting of an infinite plate with a circular hole. In these two problems important differences are noted between the results of the present theory and the results obtained by means of the classical plate theory. It is indicated that the present theory may be applied to other problems where the deviations from the results of classical plate theory are of interest. Among these other problems is the determination of the reactions along the edges of a simply supported rectangular plate, where the classical theory leads to concentrated reactions at the corners of the plate. These concentrated reactions will not occur in the solution of the foregoing problem by means of the theory given in the present paper.

Fully-Coupled Modeling of Composite Piezoelectric Plates

Aerospace ◽  
2005 ◽  
Author(s):  
Wenbin Yu ◽  
Lin Liao
Keyword(s):  
Plate Theory ◽  
Three Dimensional ◽  
Coupled Model ◽  
Composite Plates ◽  
Present Theory ◽  
Geometrically Nonlinear ◽  
Coupled Modeling ◽  
Classical Plate Theory ◽  
Variational Asymptotic ◽  
Fully Coupled

A fully coupled model considering both sensing and actuation for composite plates with embedded piezoelectrics is constructed using the variational asymptotic method. Without invoking any ad hoc kinematic assumptions, we take advantage of the geometric small parameter inherent in the structure to mathematically split the original three-dimensional, geometrically nonlinear, piezoelectricity problem into: a coupled, linear, one-dimensional piezoelectric through-the-thickness analysis and a coupled, geometrically nonlinear, two-dimensional piezoelectric plate analysis. Two asymptotically correct models of multi-layer plates are developed for two different types of electrode arrangements. The constructed models are of the form of classical plate theory having a layerwise displacement and electric potential distribution. The present theory is implemented using the finite element method into the computer program VAPAS (variational-asymptotic plate and shell analysis). Simple examples are used to demonstrate the application of the present theory.

Elasticity Theory of Plates and a Refined Theory

1979 ◽  
Vol 46 (3) ◽  
pp. 644-650 ◽  
Author(s):  
Shun Cheng
Keyword(s):  
Shear Deformation ◽  
Plate Theory ◽  
Three Dimensional ◽  
Fundamental Equation ◽  
Present Theory ◽  
Refined Theory ◽  
Refined Plate Theory ◽  
Elasticity Equations ◽  
Theory Of Plates ◽  
Fundamental Equations

A method for the solution of three-dimensional elasticity equations is presented and is applied to the problem of thick plates. Through this method three governing differential equations, the well-known biharmonic equation, a shear equation and a third governing equation, are deduced directly and systematically from Navier’s equations. It is then shown that the solution of the second fundamental equation (the shear equation) is in fact related to the shear deformation in the bending of plates, hence it may be appropriately called the shear solution and the equation the shear equation. Moreover, it is found that the solution of the third fundamental equation does not yield transverse shearing forces. Because of these results, a refined plate theory which takes into account the transverse shear deformation can now be explicitly established without employing assumptions. With the present theory three boundary conditions at each edge of the plate and all the fundamental equations of elasticity can be satisfied. As an illustrative example, the present theory is applied to the problem of torsion resulting in exactly the same solution as the Saint Venant’s solution of torsion, although the two approaches are appreciably different. The second example also illustrates that accurate solutions, as compared with exact solutions, can be obtained by means of the refined plate theory.

Asymptotic Construction of Reissner-Like Models for Composite Plates With Accurate Strain Recovery

2001 ◽  
Author(s):  
Wenbin Yu ◽  
Dewey H. Hodges ◽  
Vitali V. Volovoi
Keyword(s):  
Plate Theory ◽  
Three Dimensional ◽  
Composite Plates ◽  
Present Theory ◽  
Laminated Plates ◽  
Constitutive Law ◽  
Two Dimensional ◽  
Normal Line ◽  
Plate Analysis ◽  
Line Analysis

Abstract The focus of this paper is to develop an asymptotically correct theory for composite laminated plates when each lamina exhibits monoclinic material symmetry. The development starts with formulation of the three-dimensional, anisotropic elasticity problem in which the deformation of the reference surface is expressed in terms of intrinsic two-dimensional variables. The Variational Asymptotic Method is then used to rigorously split this three-dimensional problem into a linear one-dimensional normal-line analysis and a nonlinear two-dimensional “plate” analysis accounting for transverse shear deformation. The normal-line analysis provides a constitutive law between the generalized, two-dimensional strains and stress resultants as well as recovering relations to approximately express the three-dimensional displacement, strain and stress fields in terms of plate variables calculated in the “plate” analysis. It is known that more than one theory that is correct to a given asymptotic order may exist. This nonuniqueness is used to cast a strain energy functional that is asymptotically correct through the second order into a simple “Reissner-like” plate theory. Although it is true that it is not possible to construct an asymptotically correct Reissner-like composite plate theory in general, an optimization procedure is used to drive the present theory as close to being asymptotically correct as possible while maintaining the beauty of Reissner-like formulation. Numerical results are presented to compare with the exact solution as well as a previous similar yet very different theory. The present theory has excellent agreement with the previous theory and exact results.

scholarly journals The near-resonant regimes of a moving load in a three-dimensional problem for a coated elastic half-space

2016 ◽  
Vol 22 (1) ◽  
pp. 89-100 ◽  
Author(s):  
Bariş Erbaş ◽  
Julius Kaplunov ◽  
Danila A Prikazchikov ◽  
Onur Şahin
Keyword(s):  
Half Space ◽  
Moving Load ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
Point Load ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Dispersive Effect ◽  
Mach Cones ◽  
Integral Solutions

This paper deals with the three-dimensional analysis of the near-resonant regimes of a point load, moving steadily along the surface of a coated elastic half-space. The approach developed relies on a specialized hyperbolic–elliptic formulation for the wave field, established earlier by the authors. Straightforward integral solutions of the two-dimensional perturbed wave equation describing wave propagation along the surface are derived along with their far-field asymptotic expansions obtained using the uniform stationary phase method. Both sub-Rayleigh and super-Rayleigh cases are studied. It is shown that the singularities arising at the contour of the Mach cones typical of the super-Rayleigh case, are smoothed due to the dispersive effect of the coating.

A Method to Evaluate Three-Dimensional Time-Harmonic Elastodynamic Green’s Functions in Transversely Isotropic Media

1992 ◽  
Vol 59 (2S) ◽  
pp. S96-S101 ◽  
Author(s):  
H. Zhu
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Isotropic Case ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Transversely Isotropic Media ◽  
Time Harmonic ◽  
Isotropic Media

The three-dimensional time-harmonic elastodynamic Green’s functions in infinite transversely isotropic media have been derived explicitly. The Green’s functions consist of the corresponding static Green’s functions and double integral representations over a finite domain with the integrands being continuous. The Green’s functions will reduce to those for the isotropic case when the isotropic elastic constants are substituted. The singular parts of the Green’s functions have been shown to be the same as those of the static ones. The far-field approximations have been obtained by using the stationary phase method. In addition, a simpler method to construct wave front curves has been presented.

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