Higher order vibrations of thin periodic plates

A problem of higher order vibrations of thin periodic plates related to a plate periodicity is discussed. The applied model, based on the Kirchhoff plate theory assumptions and additional hypothesis of tolerance averaging [Wo?niak Cz, Wierzbicki E. Averaging techniques in thermomechanics of composite solids. Tolerance averaging versus homogenization. Poland, Cz?stochowa: Cz?stochowa University of Technology; 2000], makes it possible to take into account the effect of the periodicity cell size on the overall plate behaviour. Assuming more functions describing oscillations of the periodicity cell, we can investigate within the model not only fundamental vibrations but also higher order vibrations related to a periodic plate structure. These functions have to be properly chosen approximations of solutions to eigenvalue problem for natural vibrations of separated cell with periodic boundary conditions, [J?drysiak J. On vibrations of thin plates with one-dimensional periodic structure. Int J Eng Sci 2000;38:2023-2043]. Using the known two-dimensional plate models—the orthotropic plate model and finite element method as well as the discrete model, certain justifications of the proposed model are made.     

薄板的高阶振动

对薄板的高阶振动进行了讨论。应用模型基于Kirchhoff平板理论的假定和公差平均值的假定[Wo■ niak Cz,Wierzbicki E.Averaging techniques in thermomechanics of composite solids.Tolerance averaging versus homogenization.Poland,Czestochowa:Czestochowa University of Technology;2000],这使得模型可以考虑到周期单元对于整块板性能的影响。假设对周期单元振动的影响参数过多,可以对模型进行比基本振型更高阶的振动分析。这些函数是通过从边界条件中分离周期单元,对其自振时的特征值方程分析选取的近似值[JedrysiakJ.On vibrations of thin plates with one-dimensional periodic structure.Int J Eng Sci 2000;38:2023-2043]。通过应用平面板模型-正交各向异性模型、有限元模型和离散模型,对提议模型进行了论证。     

A non-asymptotic model for the stability analysis of thin biperiodic cylindrical shells

The object of considerations is a thin linear-elastic cylindrical shell with arbitrary cross-section having a periodic structure in both directions tangent to the shell midsurface. The aim of this paper is to propose a new averaged non-asymptotic model of dynamical stability problems for such shells. As a tool of modeling we shall apply the tolerance averaging technique, given for periodic structures in Woźniak and Wierzbicki [Averaging techniques in thermomechanics of composite solids, Wydawnictwo Politechniki Częstochowskiej, 2000]. In contrast with models obtained by means of asymptotic homogenization methods, the proposed one makes it possible to describe the effect of the period lengths on the global shell stability, called a length-scale effect.     

Tolerance modelling of vibrations of thin functionally graded plates

A problem of a modelling of vibrations of thin plates with a functionally graded macrostructure and a tolerance-periodic microstructure in planes parallel to the plate midplane is analysed. The proposed model, based on the Kirchhoff plate theory assumptions and additional hypothesis of the tolerance averaging technique [Wo?niak et al., editors. Thermomechanics of heterogeneous solids and structures. Tolerance averaging approach. ?ód?, University Press, Technical University of ?ód?; 2008], describes the effect of the microstructure size on dynamic behaviour of the plate. In this paper, as an example there are analysed free vibration frequencies of a functionally graded plate band. These frequencies are obtained in the framework of two proposed models—the tolerance model and the asymptotic model, using the known Ritz method.     

Static and dynamic analysis of thin plates in bending— A novel finite element model

The static and dynamic behavior of thin flat plates in bending have been studied by means of a recently developed¹ doubly curved triangular shell element. The element's formulation is based on a modified mixed variational principle, wherein the primal variable σ (vector of shell stress resultants) and (boundary displacement vector) are required to satisfy a priori: (1) the complete shallow shell equilibrium equations, and (2) interelement C¹ displacement continuity. Several well-selected plate structures have been analyzed and the numerical results obtained indicate that the new element scheme competes most favorably with recently developed as well as with well-established elements included in commercial general-purpose finite element codes.     

Nonlinear dynamic stability of composite plates with piezoelectric layers subjected to periodic in-plane load

Nonlinear dynamic stability characteristics of composite plates subjected to periodic in-plane load are investigated via the finite element method with dynamic response analysis. Piezoelectric actuator layers are embedded at the top and bottom of the laminated composite plate. The theoretical formulation is based on the modified first order shear deformation theory (MFSDT) incorporating the von Kármán type nonlinear strains. The structural system is considered to be undamped. The nonlinear governing equations obtained are solved using the Newmark's direct integration method coupled with the direct iteration method. The boundaries of dynamic instability regions are obtained using Bolotin's approach. Effects of in-plane forcing frequency and applied voltage on the characteristic features of dynamic stability behaviour are investigated using both linear and nonlinear dynamic response analyses.     

Elastic stability of bi-axially loaded rectangular plates with a single circular hole

The finite-element method has been employed to determine the elastic buckling stresses of a bi-axially loaded perforated rectangular plate with dimensions a and b in the x- and y-directions, respectively. The considered perforation is a single circular hole whose center is located along the longitudinal axis of the plate. The considered plate has simply supported edges in the out-of-plane direction and is subjected to bi-axial uniformly distributed end loads (compressive load σ_x in the x-direction, and compressive or tensile load σ_y in the y-direction). Parameters considered in the study are the plate's aspect ratio a/b, the stress ratio ξ between the applied stresses in the y- and x-directions (ξ=σ_y/σ_x), the circular hole size d and location e_x.The study shows that, in most of the considered cases, the bigger the hole size d, the lesser the plate stability and the lesser the buckling stresses. It also shows that the plate aspect ratios, a/b, between 0.6 and 1.2 should be avoided for plates with large holes and negative ξ, due to the large reduction in the buckling stresses. The hole location should also be selected to be away from the loaded edge of the plate as much as possible (better to have e_x/b>1.0) to increase the buckling stresses and improve stability. The study demonstrates also that the increase in tension in the y-direction in bi-axially loaded plate with large hole (d/b>0.4) reduces its stability. This is in contrary to the expected increase in the stability due to the increase in tension which can be seen clearly in the cases of solid plates and plates with small holes.     

The stability of eccentrically compressed thin plates with a longitudinal free edge and with stress variation in the longitudinal direction

This paper presents results of the investigation of the stability of eccentrically compressed thin plates with one longitudinal free edge with the participation of loadings generating the variation of stresses in the direction of the length of a plate. A deflection function (5) is proposed to enable modelling of boundary conditions on the second longitudinal edge from a simply supported, through elastically restrained, to a built-in edge. Formulas are derived for work done by external force under loading generating the variation of stress according to a linear function and according to parabola 2°. Tables and plots of buckling coefficient (k) for variously supported and variously loaded plates with a free edge, which are not found in the literature, are determined. The effect of a longitudinal stress variation on the plate buckling modes is analyzed.     

Static and dynamic interactive buckling of isotropic thin-walled closed columns with variable thickness

The present paper deals with static and dynamic analysis of interactive buckling of thin-walled closed columns with variable thickness subjected to in-plane constant and/or pulse loading. This investigation is concerned with thin-walled structures with corners bevelled at the angle of 45° under axial compression. The plate model is adopted for the structures. The material, all plates are made of, is subject to Hooke's law. The structures are assumed to be simply supported at the ends. The differential equations of motion have been obtained from Hamilton's principle. In this paper the static solution has been obtained by Koiter's asymptotic method in the second-order approximation. The study is based on the numerical method of the transition matrix using Godunov's orthogonalization. The interaction of an overall mode with two local modes having the same wavelength has been considered (i.e. three-mode approach). The nonlinear equations of dynamic stability are solved with the Runge–Kutta method. The calculations are carried out for settled imperfections.     

Periodicity in the response of nonlinear plate, under moving mass

The dynamics of nonlinear thin plates under influence of relatively heavy moving masses is considered. By expansion of the solution as a series of mode functions, the governing equations of motion are reduced to an ordinary differential equation for time development of vibration amplitude, which is Duffing's oscillator with time varying coefficients. Through the application of Banach's fixed-point theorem, the periodic solutions are predicted. The method presented in this paper is general so that the response of plate to moving force systems can also be considered.     

An equivalent plasticity theory for jointed rock masses

A non-representative volume element (NRVE) approach to equivalent rock mass properties shows that the form of the elastic–plastic constitutive equations is the same for homogeneous material elements and multiple-material elements, subsequently homogenized. Thus the average stress and strain increments in an arbitrary jointed rock mass volume are related by {dσ}=([C*_ep]){dε} where σ is effective stress. The equivalent elastic-plastic properties matrix [C*_ep] is the sum of an equivalent elastic moduli matrix [C^*] and a plastic ‘correction’ matrix [C*_p, as usual. However, there are no equivalent plastic potentials Y^* or yield functions, failure criteria F^* or strengths. The equivalent elastic-plastic properties are constructed from the elastic moduli and strengths of the rock mass joints, the intact rock between and strain influence functions that relate local to overall average strains. Numerical examples that simulate laboratory-like tests on jointed rock cubes illustrate the approach.     

Dynamic buckling of thin isotropic plates subjected to in-plane impact

The dynamic stability behaviour of imperfect simply supported plates subjected to in-plane pulse loading is investigated. For the calculation of dynamic buckling loads a stress failure criterion is applied. The large-deflection plate equations are solved by a Galerkin method by using Navier's double Fourier series. In this paper the dynamic load factor (DLF) is redefined and plots that are useful for the design of plate structures are presented. Parametric studies are performed in which the influences of the pulse duration, shock function, imperfection, geometric dimensions and limit stress of the material are discussed. Comparison between the dynamic buckling loads, which are obtained by the commonly used criterion of Budiansky and Hutchinson [Proceedings of the 11th International Congress of Applied Mechanics (1964) 636], and the dynamic elastic limit loads, which are computed by the stress failure criterion, shows that the latter criterion is more useful for the design of lightweight structures.     

Investigation into thin-walled structures at the technical University of ódź

The behaviour of thin-walled structures has been a very important area of research activity in the Department of Strength of Materials and Structures (previously the Institute of Applied Mechanics) of the Technical University of ódź in Poland for over 30 years.The investigations concerned many kinds of thin-walled structures: plates of different shapes including trapezoidal and circular, girder and column structures composed of flat plates, cylindrical and conical shells, and composite multi-layer structures subjected to a variety of load types.The analyses included buckling and post-buckling behaviour, assessment of ultimate loads and the behaviour of the structure in post post failure range. This necessitated usage of the theory of elasticity and elasto plasticity in the constitutive relations describing phenomena of the materials behaviour. All the works done at our Department resulted in a number of papers and books, achieved academic degrees and titles (PhD, DSc, Professor). They were also presented in numerous conferences, symposia and seminars.It is our academic duty and a pleasure to mention that for about 20 years we have been accompanied in our thin-walled research activity by our sister unit from the University of Strathclyde-the Department of Mechanical Engineering. Some of our achievements would not be possible without this valuable collaboration, which was part of a more general scheme of academic link initiated between our universities almost 30 years ago.     

Buckling of imperfect functionally graded plates under in-plane compressive loading

Buckling behavior of rectangular functionally graded plates with geometrical imperfections is studied in this paper. The equilibrium, stability, and compatibility equations of an imperfect functionally graded plate are derived using the classical plate theory. It is assumed that the nonhomogeneous mechanical properties of the plate, graded through thickness, are described by a power function of the thickness variable. The plate is assumed to be under in-plane compressive loading. Simultaneous solving of the stability and compatibility equations in conjunction with the equilibrium equations leads to the buckling relation of the plate. The critical buckling load of a sample plate is obtained and compared for different geometrical ratios. The results are reduced and compared with the results of perfect functionally graded and imperfect isotropic plates.     

Stability analysis of stiffened plates by finite strips

The stability analysis of stiffened plates by means of the finite strip method is presented. The studies are based on the thin shallow theory, giving nonlinear strain displacement relations, but linear curvature displacement relations. The nonlinear equilibrium equations are obtained by the principle of incremental virtual work, using finite strip discretization. The higher order strip with one internal nodal line is applied. It is shown that considerable improvements can be obtained using this kind of strip. It is especially true for the postbuckling analysis. Numerical examples of the strength of stiffened plates in compression are carried out, covering a range of plate and stiffener slenderness.     

Solution to the problem of axisymmetric and asymmetric dynamic instability of three-layered annular plates

This paper presents the solution to the dynamic stability problem of three-layered, annular plate loaded by compressive stress increasing in time. The solution enables the evaluation of the critical, dynamic loads corresponding to the various modes of plate buckling. The symmetrical cross-section structure of plate is described by the classical theory of sandwich plate with the broken line hypothesis, the nonlinear Kármán’s plate equations and linear physical relations. The solution is based on Bubnov–Galerkin method and finite difference method. The values of critical dynamic loads have been calculated by means of the stability criterion presented by (Volmir (1972) [1]). The comparison of values of critical dynamic and static loads is presented, using the dynamic ratio. The obtained results have been compared with those obtained for plate model built by means of finite element method. The calculations were carried out using the ABAQS system. The dynamic response of plate models examined by two methods is consistent.     

Measurement-related uncertainties in overcoring data at the Äspö HRL, Sweden. Part 2: Biaxial tests of CSIRO HI overcore samples

This paper is the second dealing with measurement-related uncertainties of overcoring data undertaken at the Äspö Hard Rock Laboratory and focuses on the biaxial test data from CSIRO HI overcore samples. The first paper dealt with measurement-related uncertainties in connection with the overcoring phase [1].The uncertainties identified in connection to biaxial tests of CSIRO HI overcore samples include too large applied pressures and poor sampling frequency. At the Äspö HRL, the results yield that most overcore samples fractured during biaxial testing, meaning that a significant part, 56%, of available strain gauge combinations were removed from calculations of the elastic parameters. Remaining strain gauge combinations indicate average values of 62±5 GPa for Young's modulus and 0.25±0.01 for Poisson's ratio, which are considerably lower than previously published values [2–5], and are in good agreement with results from biaxial tests on Borre Probe overcore samples [6–10]. The stress calculations were obtained from re-analyzed elastic parameters and strains, and show primarily a reduction in stress magnitudes. Overall, the stress field obtained with different stress measurement methods and its variation with depth is now quite well resolved. The overcoring data suggest that the principal stresses are inclined with a vertical component dipping about 65° from the horizontal over the investigated rock volume (140–420 m depth). This is interpreted as a result of influence from the sub-vertical NE-2 Fracture Zone that divides the stress data into two stress domains [11], although it may also be an artefact because the σ₂- and σ₃-magnitudes are of the same order of magnitude.     

Mathematical modeling and investigation of nonlinear vibration of perfectly conductive plates in an inclined magnetic field

Nonlinear differential equations and boundary conditions describing the vibrations and stability of perfectly conductive, viscoelastic plates in an inclined homogeneous external magnetic field have been obtained. On the basis of obtained equations two various problems of wave propagation in the plate are considered:The problems of stability and propagation of modulated waves. The problem is reduced to the nonlinear equation of Shrodinger type. In the particular case when viscosity of the material is not taken into account, the condition for magnetic field is obtained when the modulated waves are unstable.Problems of nonlinear resonance interaction of magnetoelastic waves. For simplicity three-wave resonance is studied. In this case a system of nonlinear ordinary differential equations for amplitude and phase of interacting waves are obtained. In a particular case, when the viscosity of material is neglected, the obtained system was solved analytically. In the general case it is solved numerically.     

Buckling of simply supported rectangular plates under combined bending and compression using eigensensitivity analysis

In this paper, an approximate quadratic closed-form expression is presented for the critical buckling analysis of a plate subjected to combined bending and compression. The formula is developed by expanding the eigenvalue, the critical buckling load, for a plate under combined bending and compression in a Mauclaurin's series about a plate subjected only to compression. The general expression can be used for all combinations of simply supported and clamped rectangular plates boundary conditions. An explicit formula in terms of the plate aspect ratio R and plate load parameter α is evaluated for simply supported plates. Compared with the Rayleigh–Ritz method, this approximate expression provides an excellent comparison when the load parameter α1.52 for plate aspect ratio between 0.2R2.8.