Hydrodynamic analogy for direct shear force and bending moment evaluation in a plate

A brand new interpretation of the plate bending equations is given using hydrodynamic analogy. It permits one to determine directly the shear forces and bending moments of a plate without the need of finding deflections. In engineering design of a plate it is more important to know shear forces and bending moments than the deflections. The existing numerical methods of solution of plate problems consist in determining deflections; then shear forces and bending moments are obtained by differentiating the deflection three and two times which produces great loss of accuracy. The hydrodynamic analogy method has the advantage over other numerical methods because the shear forces and bending moments are obtained directly, without the need of finding deflections and because they are obtained with better accuracy. The hydrodynamic analogy can be applied to a plate of arbitrary shape, with arbitrary boundary conditions under an arbitrary loading.     

关于简支厚板与薄板的关系

厚板的数学理论是建立在与薄板不同的力学假定的基础上的。本文分析了厚板与薄板之间静力学方面的关系。对于任意的简支多边形板,得到了厚板解通过薄板解的显式表达式,从而证明了:Reissner模型的厚板解与薄板解具有相同的剪力,但弯矩、转角、挠度有差别;而washizu模型的厚板解则与薄板解不仅剪力相同,连弯矩与转角亦相同,只是挠度有差别。     

FEM Analysis of the Three-Dimensional Buckling Problem for a Clamped Thick Rectangular Plate Made of a Viscoelastic Composite

The buckling instability of a thick rectangular plate made of a viscoelastic composite material is studied. The investigation is carried out within the framework of the three-dimensional linearized theory of stability. The plate edges are clamped and the plate is compressed through the clamps. Moreover, it is assumed that the plate has an initial infinitesimal imperfection, and, as a buckling criterion, the state is taken where this imperfection starts to increase indefinitely at fixed finite values of external compressive forces. From this criterion, the critical time is determined. The corresponding boundary-value problems are solved by employing the three-dimensional FEM and the Laplace transform. The material of the plate is assumed orthotropic, viscoelastic, and homogeneous. Numerical results related to the critical time are presented.     

A method for analysis of linear dynamic systems driven by stationary non-Gaussian noise with applications to turbulence-induced random vibration

A method is developed for approximating the properties of the state of a linear dynamic system driven by a broad class of non-Gaussian noise, namely, by polynomials of filtered Gaussian processes. The method involves four steps. First, the mean and correlation functions of the state of the system are calculated from those of the input noise. Second, higher order moments of the state are calculated based on Itô’s formula for continuous semimartingales. It is shown that equations governing these moments are closed, so that moment of any order of the state can be calculated exactly. Third, a conceptually simple technique, which resembles the Galerkin method for solving differential equations, is proposed for constructing approximations for the marginal distribution of the state from its moments. Fourth, translation models are calibrated to representations of the marginal distributions of the state as well as its second moment properties. The resulting models can then be utilized to estimate properties of the state, such as the mean rate at which the state exits a safe set. The implementation of the proposed method is demonstrated by numerous examples, including the turbulence-induced random vibration of a flexible plate.     

Optimum control of the forced motions of systems with continuously distributed parameters : PMM vol. 38, n 6, 1974, pp. 1056–1062

We propose one of the possible versions of the optimum control of the forced motions of elastic systems of the type of rods, plates, and shells. We apply the procedure developed to elementary problems on the transition of a freely-supported rod or plate from an initial state φ, ψ to the rest state in the least possible time T in the presence of a constraint on the forcing load. We use the elementary results of theory of the l-problem of moments of Krein [1–3].     

Delamination buckling of a rectangular orthotropic composite plate containing a band crack

The delamination buckling problem for a rectangular plate made of an orthotropic composite material is studied. The plate contains a band crack whose faces have an initial infinitesimal imperfection. The subsequent development of this imperfection due to an external compressive load acting along the crack is studied through the use of the three-dimensional geometrically nonlinear field equations of elasticity theory for anisotropic bodies. A criterion of initial imperfection is used in determining the critical forces. The corresponding boundary-value problems are solved by employing the boundary-form perturbation technique and the FEM. Numerical results for the critical force are presented.     

Computation of worst geometric imperfection profiles of composite cylindrical shell panels by minimizing the non-linear buckling load

In this article the “most unfavorable” shape of initial geometric imperfection profile for laminated cylindrical shell panel is obtained analytically by minimizing the limit point load. The partial differential equations governing the shell stability problem are reduced to a set of non-linear algebraic equations using Galerkin's technique. The non-linear equilibrium path is traced by employing Newton–Raphson method in conjunction with the Riks approach. A double Fourier series is used to represent the initial geometric imperfection profile for the cylindrical shell panel. The optimum values of these Fourier coefficients are determined by minimizing the limit point load using genetic algorithm. The results are determined for simply supported composite cylindrical shell panel. Numerical results show that more number of terms is needed in Fourier series representation to obtain the “worst” geometric imperfection profile which gives lower limit load compared to single term representation of imperfection. We have incorporated constraints on the shape of imperfection to avoid unrealistic limit point loads (due to imperfection shape) as we have assumed that the imperfection is due to machining/manufactuting.     

Free vibration analysis of cracked postbuckled plate

In this paper, a methodology is introduced to address the free vibration analysis of cracked plate subjected to a uniaxial inplane compressive load for the first time. The crack, assumed to be open and at the edge is modeled by a massless linear rotational spring. The governing differential equations are derived using the Mindlin theory, taking into account the effect of initial imperfection. The response is assumed to be consisting of static and dynamic parts. For the static part, differential equations are discretized using the differential quadrature element method and resulting nonlinear algebraic equations are solved by an arc-length strategy. Assuming small amplitude vibrations of the plate about its buckled state and exploiting the static solution in the linearized vibration equations, the dynamic equations are converted into a non-standard eigenvalue problem. Finally, natural frequencies and modal shapes of the cracked buckled plate are obtained by solving this eigenvalue problem. To ensure the validity of the suggested approach an experimental setup and a numerical finite element model have been made to analyze the vibration of a cracked square plate with simply supported boundary conditions. Also, several case-studies of cracked buckled plate problem have been solved utilizing the proposed method, and effects of selected parameters have been studied. The results show that the applied load and geometric imperfection as well as the position, size and depth of the crack have different impact on natural frequencies of the plate.     

The torsion of a viscoelastic plate in an unsteady flow

The stability of the equilibrium position of a viscoelastic plate, subjected to torsional strain and effect of the free airflow is investigated. The unsteadiness of the flow is taken into account by introducing integral terms into the moments of the aerodynamic forces acting on the plate. In a neighbourhood of the equilibrium position, a general solution of a Volterra-type integro-differential equation with partial derivatives is constructed in the form of a Fourier series, as a function of the longitudinal coordinate of the plate with coefficients that are the power series in the small parameters introduced. The stability of the plate equilibrium in the unstrained state is analysed in the case when there are small perturbations (possibly, discontinuous) of the flow velocity. The stability under persistent perturbations of the equilibrium of the strained plate with respect to non-linear perturbing forces and perturbations of its shape at the instant of time preceding the specified initial instant is also investigated.     

Numerical analysis of a locking‐free mixed finite element method for a bending moment formulation of Reissner‐Mindlin plate model

This article deals with the approximation of the bending of a clamped plate, modeled by Reissner‐Mindlin equations. It is known that standard finite element methods applied to this model lead to wrong results when the thickness t is small. Here, we propose a mixed formulation based on the Hellinger‐Reissner principle which is written in terms of the bending moments, the shear stress, the rotations and the transverse displacement. To prove that the resulting variational formulation is well posed, we use the Babu?ka‐Brezzi theory with appropriate t ‐dependent norms. The problem is discretized by standard mixed finite elements without the need of any reduction operator. Error estimates are proved. These estimates have an optimal dependence on the mesh size h and a mild dependence on the plate thickness t. This allows us to conclude that the method is locking‐free. The proposed method yields direct approximation of the bending moments and the shear stress. A local postprocessing leading to H¹ ‐type approximations of transverse displacement and rotations is introduced. Moreover, we propose a hybridization procedure, which leads to solving a significantly smaller positive definite system. Finally, we report numerical experiments which allow us to assess the performance of the method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Moments of Change

There is a strong intuition that for a change to occur, there must be a moment at which the change is taking place. It will be demonstrated that there are no such moments of change, since no state the changing thing could be in at any moment would suffice to make that moment a moment of change. A moment in which the changing thing is simply in the state changed from or the state changed to cannot be the moment of change, since these states are respectively before and after the change; moreover, to select one of these moments over the other as the moment of change would be arbitrary. A moment in which the changing thing is neither in the state changed from nor in the state changed to cannot be the moment of change, since there are changes for which it is impossible for something to be in neither state. Finally, the moment of change cannot be a moment in which the changing thing is in both the state changed from and the state changed to, as suggested by Graham Priest and others. Even if, like proponents of this view, we are willing to accept the contradictions that the account entails, it is demonstrated that on such a model, every change would require an infinite number of other changes, every change would take an infinite amount of time, and some changes would occur without occurring at any time. Further, the model is grossly counterintuitive, with the exact nature of the counterintuitive element depending on what model of time and space one endorses. Finally, it is demonstrated that this model is incompatible with the Leibniz Continuity Condition.     

Rectangular anisotropic (orthotropic) plates on a tensionless elastic foundation

The behavior of anisotropic (orthotropic) elastic plates of rectangular shape on a tensionless Winkler foundation is analyzed. The tensionless character of the foundation is taken into account by using an auxiliary function. The displacement function of the plate is approximated by using the eigenfunctions of the completely free beam. The difference between the free-end boundary conditions of the plate and the beam is compensated for by considering a differential operator in addition to the governing equation of the plate. Using Galerkin's method, the problem is reduced to the solution of a system of algebraic equations. The governing equations of the plate are derived under action of external uniformly distributed load, concentrated load, and moments. However, the influence of the mechanical properties on the configurations of the contact region and on the distribution of the displacements is investigated for concentrated load and moments for various values of the mechanical properties characterizing the anisotropy of the plate material. Considered problems are solved within the framework of Kirchhoff-Love hypothesis.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 3, pp. 378–386, May–June, 1995.     

关于MZ1元,MZ2元和MB1元的误差估计

本文给出MZ1元、MZ2元和MB1元的收敛阶,并且证明MZ1元与MB1元等价。     

On the internal path length of d‐dimensional quad trees

It is proved that the internal path length of a d‐dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a fixed point of a random affine operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limiting distribution can be evaluated from the recursion and lead to first order asymptotics for the moments of the internal path lengths. The analysis is based on the contraction method. In the final part of the paper we state similar results for general split tree models if the expectation of the path length has a similar expansion as in the case of quad trees. This applies in particular to the m‐ary search trees. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 5: 25–41, 1999     

The construction of exact solutions in the floating-plate problem

It is shown that it is possible to construct, by an inverse method, exact solutions of the problem of the flexural-gravitational oscillations of a floating elastic plate. The results obtained are used to check the accuracy of numerical solutions of the problem. It is shown that the numerical algorithm given in Ref. [Khabakhpasheva TI. The plane problem of an elastic floating plate. In Continuum Dynamics. Inst. Gidrodinamiki SO Ross Akad Nauk 2000;16:166–9.], predicts, with high accuracy, the values of the amplitudes of the oscillations of the plate and the distributions of the bending moments and hydrodynamic pressure for a wide frequency range.     

Behaviour of queueing approximations based on sample moments

We investigate moment–based queueing approximations in the presence of sampling error. Let L be the steady–state mean number in the system for a GI/M/1 queue. We focus on the estimation of L under the assumption that only sample moments of the interarrival–time distribution are known. A simulation experiment is carried out for several interarrival–time distributions. For each case, sample moments from the interarrival–time distribution are matched to an approximating phase–type distribution and the corresponding estimate L is obtained. We show that the sampling error in the moments induces bias as well as variability in L. Based on our simulation experiment, we suggest matching only two moments when the sample coefficient of variation is low or when sample size is low; otherwise, matching three moments is preferable.     

Finite Codimension Subfields of a Field Complete with Respect to a Real Valuation

Let K be a field complete with respect to a real valuation v and not algebraically closed. We will show that every finite codimension subfield of K is closed in the v-adic topology if and only if the degree of imperfection of K is finite. It follows that there are incomplete finite codimension subfields of K when the degree of imperfection of K is infinite. These examples exhibit other interesting pathologies. We are able to give a necessary (and in the case of a discrete real valuation also sufficient) condition for a given finite codimension subfield to be complete. Finally, we give some applications to fields of Laurent series.

Communicated by A. Prestel.     

Approximate Solution to Inverse Scattering Problem for Potentials With Small Support

Let q(x) be a real-valued function with compact support D⊂ℝ³. Given the scattering amplitude A(α′, α, k) for all α′, α∈S² and a fixed frequency k>0, the moments of q(x) up to the second order are found using a computationally simple and relatively stable two-step procedure. First, one finds the zeroth moment (total intensity) and the first moment (centre of inertia) of the potential q. Second, one refines the above moments and finds the tensor of the second central moments of q. Asymptotic error estimates are given for these moments as d = diam(D)→0. Physically, this means that (k²+sup∣q(x))d²<1 and sup∣q(x)∣d<k. The found moments give an approximate position and the shape of the support of q. In particular, an ellipsoid D̃ and a real constant q̃ are found, such that the potential q̃ (x) = q̃, x∈D̃, and q̃ (x) = 0, x∉ D̃, produces the scattering data which fit best the observed scattering data and has the same zeroth, first, and second moments as the desired potential. A similar algorithm for finding the shape of D given only the modulus of the scattering amplitude A(α′,α) is also developed.     

On one contact problem of plane elasticity theory with partially unknown boundary

Applications of elastic plates weakened with full-strength holes are of great interest in several mechanical constructions (building practice, in mechanical engineering, shipbuilding, aircraft construction, etc). It's proven that in case of infinite domains the minimum of tangential normal stresses (tangential normal moments) maximal values will be obtained on such contours, where these values maintain constant(the full strength holes). The solvability of these problems allow to control stress optimal distribution at the hole boundary via appropriate hole shape selection. The paper addresses a problem of plane elasticity theory for a doubly connected domain S on the plane z = x + iy, which external boundary is an isosceles trapezoid boundary; the internal boundary is required full-strength hole including the origin of coordinates. In the provided work the unknown full-strength contour and stressed state of the body were determined. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)