On the static and dynamic equilibrium of concentrated loads in linear acoustics and linear elasticity

The displacement field in an unbounded linear elastic fluid subjected to a time-dependent point force is obtained by using integral transform techniques. Differentiation of the displacement field yields the pressure field. It is shown that the pressure on the surface of a spherical ball B _r of radius r centered at the point where the force is applied is statically equivalent in the limit as r0 to only one-third of the force. The remaining two-thirds are carried by the inertia terms. It is also shown, by an independent reasoning, that a point force cannot be carried in static equilibrium by a linear elastic fluid.The displacement field corresponding to an unbounded isotropic linear-elastic solid subjected to a time-dependent point force (the Stokes solution) is also obtained by using integral transform techniques. As is well-known, the tractions of the Stokes solution on the surface of a spherical ball B _r are statically equivalent in the limit as r0 to the force itself; consequently, the inertia terms do not contribute to the dynamic equilibrium of B _r. The contrast between the response of a fluid and that of an isotropic solid under the action of a point force is discussed.     

Kirchhoff and thomson-tait transformations in the classical theory of plates

The transformation of the torque into the transverse force is considered; this transformation is traditional in the educational literature [1] and was proposed by Kirchhoff [2] and Thomson and Tait [3] to match the order of the differential equation of the classical theory of plates with the number of boundary conditions. It is shown that this transformation is not universal and its mathematical and physical justification depends on the conditions of the plate fixation and loading. It is shown that this justification is absent for the most widely used problems of bending of a rectangular plate freely supported and fixed on the contour.     

Refined theory of orthotropic plates subject to tangential surface loads

The existence of two correction coefficients traditionally introduced to account for the effect of the distribution of tangential stresses over the thickness of a plate is discussed. The virtual-work principle is used to generalize the expressions for the coefficients to the case of arbitrary loading. These expressions and hypotheses for displacements help to derive equations for orthotropic rectangular plates subject to tangential surface loads. These equations account for the effect of the distribution of tangential stresses over the thickness of the plate. Numerical examples are given. The results obtained are compared with those produced by other theories __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 107–119, April 2008.     

Vibrations of multilayer plates under the effect ofimpulse loads. Three-dimensional theory

A new analytical–numerical approach to investigation of the response of multilayerplates to impulse loading is described in this paper. The plates behaviour is described by theequations of the three-dimensional elasticity theory. According to the approach being proposed,the sought for functions included in the system of equations and the boundary and initialconditions are presented as Fourier series expansions in the tangential directions. The derivativesof these functions in the transverse direction are replaced by their finite-difference presentations.As a result of such transforms, the problem of vibration of a multilayer plate is reduced tointegration of a system of ordinary differential equations with constant coefficients. Integration isperformed by expansion into the Taylors series. The possibilities of the approach proposed andthe validity of results obtained is illustrated by several examples of calculating vibration processesand the processes of propagation of elastic waves. A comparison of the results obtained on thebasis of other approaches has been performed.     

A boundary integral equation formulation for the Reissner's plates resting on two-parameter foundation

In this paper a boundary integral equation formulation for the Reissner's plates resting on a two-parameter foundation is established. With the aid of the Hormander Operator method, the equations of the corresponding fundamental solutions are converted into a sixth order partial differential equation with a scale function as an unknown. In order to reduce the equation further, two auxiliary functions are introduced. They satisfy a second and a fourth order equation respectively. The expressions of the auxiliary functions can be derived easily. The fundamental solutions of the Reissnei's plates on the two-parameter foundation arc expressed by a linear combination of the auxiliary functions and their derivatives. The boundary integral equations are formulated by the use of the weighted residual procedure. The fundamental solutions obtained are taken as the kernel functions of the boundary integral equations. A few examples are studied. The numerical results show high accuacy and efficiency of the present formulation.This work was supported by the National Natural Science Foundation of China.     

Application of Lie transformation group methods to classical linear theories of rods and plates

In the present paper, a class of partial differential equations governing various rod and plate theories of Bernoulli–Euler and Poisson–Kirchhoff type is studied by Lie transformation group methods. A system of equations determining the generators of the admitted point Lie groups (symmetries) is derived and the general statement of the associated group-classification problem is given. A simple relation is deduced allowing to recognize easily the variational symmetries among the “ordinary” symmetries of a self-adjoint equation of the class examined. Explicit formulae for the conserved currents of the corresponding (via Bessel-Hagen’s extension of Noether’s theorem) conservation laws are suggested. Solutions of group-classification problems are given for subclasses of equations of the foregoing type governing stability and vibration of rods, fluid conveying pipes and plates resting on variable elastic foundations. The obtained group-classification results are used to derive conservation laws and group-invariant solutions readily applicable in rod dynamics and plate statics and dynamics. New generalized symmetries and conservation laws for the theories of Timoshenko beams, Reissner–Mindlin plates and three-dimensional elastostatics are presented.     

A refined bound on the error of Reissner's plate theory

Summary The accuracy of two versions of Reissner's plate theory as compared with three-dimensional elasticity is evaluated. The relative mean square error is shown to consist of two parts: one related to transverse normal rigidity and proportional to the plate thickness squared and the other related to transverse shear rigidity and proportional to the thickness cubed. This refines a previous estimate by Nordgren where both of those parts were proportional to the thickness squared.

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Verbesserte Fehlerschranke für die Reissnersche Plattentheorie

Übersicht Für zwei Varianten der Reissnerschen Plattentheorie werden im Vergleich mit dreidimensionaler Elastizitätstheorie globale Fehlerschranken erhalten. Es wird gezeigt, daß die L ₂-Norm des relativen Fehlers aus zwei Teilen besteht: Einem von der Dehnungssteifigkeit in Dickenrichtung bedingten und dem Quadrat der Plattendicke proportionalen und aus dem zweiten Teil, der von der Schubsteifigkeit der Querschnitte bedingt und der 3. Potenz der Plattendicke proportional ist. Dieses Ergebnis verbessert eine von Nordgren erhaltene Abschätzung in der beide Teile dem Quadrat der Dicke proportional sind.

     

Theory and refined theory of elasticity for transversely isotropic plates and a new theory for tnick plates

A theory of elasticity for the bending of transversely isotropic plates has been developed from the basic equations of elasticity in terms of displacements for transversely isotropic bodies, which takes into account the loads distributed over the surfaces of the plates. Based on this theory, a refined theory of plates which can satisfy three boundary conditions along each edge of the plates and a new theory of thick plates are established. The solution of the refined theory for simply supported polygonal plates has been obtained; and its numerical result is very close to the exact solution of the three-dimensional theory of elasticity. A systematic comparison with the former theories of thick plates shows that the present theory of thick plates is closest to the result of the theory of elasticity.     

Impulsive stress analysis of plates at the concentrated impact loading point by the three-dimensional dynamic theory of elasticity

In this paper, a method of stress analysis is proposed to analyze impulsive stresses at an impact loading point, since use of the classical plate theory cannot lead to a convergent result. In particular, under these impact conditions (regardless of mass of impactor, velocity of impactor, stiffness of plate, etc.), we used the three-dimensional dynamic theory of elasticity and potential theory of displacement to analyze the impulsive strains at an impact loading point. Also, by using Hertzian contact theory to account for local deformation, the impact loading could be analyzed, and this loading was applied to the impulsive stress analysis by allowing the impact loading to be approximated to an analyzable function. In the numerical analysis, we used a fast Fourier transform (FFT) algorithm and applied the numerical inverse Laplace transformation.     

Geometrically-exact,intrinsic theory for dynamics of moving composite plates

A geometrically-exact and fully intrinsic theory is presented for dynamics of composite plates undergoing large deformation. To say that the formulation is intrinsic means that it is without displacement and rotation variables. Although the equations are geometrically-exact, the highest degree nonlinearities are quadratic; there are no singularities associated with finite rotation. Methods for posing problems in this framework along with advantages of the formulation are discussed.     

A dynamic method for measuring the critical loads of elastic flat plates

A dynamic method is described for determining the linear buckling loads of elastic, perfectly flat, rectangular plates. The proposed method does not require the application of in-plane loads; it requires only vibrational excitation of the plate. The buckling load is determined from the measured normal modes of vibration. The method is applicable to isotropic as well as anisotropic plates with any type of edge support. The accuracy of the dynamic method was evaluated by tests in which buckling loads of aluminum and graphite fiber-reinforced-epoxy composite plates were determined both by the dynamic method and by imposing static in-plane loads on the plates. The results of the dynamic and static tests agree closely. A. Segall (on leave from RAFAEL, Israel)     

The suspended elastic cable under the action of concentrated vertical loads

This investigation treats the static response of a single elastic cable which is suspended between two points that are not necessarily at the same level. The cable is loaded by its self-weight and any number of concentrated vertical loads which may be arbitrarily placed along its length. The analysis presented uses a Lagrangian approach. For the strained cable profile, the tension and displacements are given as functions of a single Lagrangian co-ordinate. A specific application of the general analysis is made and compared with a simple experiment.     

Influence of concentrated lateral loads on the elastic stability of cylinders in bending

The character of the instability and the degradation of the moment-carrying capacity are found by Mylar model experiments for cylinders in bending when subjected to concentrated lateral loads. Lateral loads can seriously degrade the moment capability of cylinders. Critical combinations of moment and lateral load cause two distinct modes of failure—collapse and snapping. Collapse modes exhibit buckles which cover the compression half of the cylinder and are critical for large values of moment and small values of lateral load. Snapping modes of failure involve a single dimple and exist for smaller values of moment and larger values of lateral load.     

An asymptotic theory for the vibration of anisotropic plates

Summary A two-dimensional theory for the motion of anisotropic plates is derived in a systematic and consistent manner without making any of the usual a priori assumptions regarding the dependence of field variables on the thickness coordinate. The method of asymptotic integration of the three-dimensional elasticity equations is used. It is shown that only six elastic constants are present in the first approximation, which also includes the effect of rotatory inertia.

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Übersicht Es wird eine zweidimensionale Theorie anisotroper Platten angegeben, wobei auf die üblichen Annahmen bezüglich der Abhängigkeit der Variablen von der Plattendicke verzichtet wird. Die dreidimensionalen Elastizitätsgleichungen werden durch asymptotische Integration gelöst. Es wird gezeigt, daß nur 6 elastische Konstanten in den Gleichungen der ersten Näherung auftreten, durch die auch der Effekt der Drehträgheit erfaßt werden kann.