Random vibrations of orthotropic plates clamped or simply supported all round

Acta Mechanica - An approximate method is presented for determining the probabilistic response of rectangular orthotropic plates clamped all round. For solving the stochastic boundary value...     

Flexure of composite plates

A new higher order shear deformation theory of laminated composite plates is developed. The basic displacement variables in this theory are two partial normal displacements and two in-plane displacement parameters. The governing equations are presented in the form of four simultaneous partial differential equations. The shear deformation theories of Bhimareddy and Stevens, and of Reddy are special cases of this formulation. In their models, transverse shear strains will become zero at points in the plate where displacements are constrained to be zero such as those on fixed edges. This limitation has been overcome in the present formulation.     

On a new symplectic geometry method for exact bending solutions of orthotropic rectangular plates with two opposite sides clamped

A novel symplectic geometry method is presented for exact bending solutions of orthotropic rectangular thin plates with two opposite sides clamped. In the proposed mathematical method, it starts with the basic governing equations for the bending of orthotropic plates, and there are no predetermined functions, which overcome the deficiency of conventional semi-inverse methods; therefore, the straightforward implementation procedure serves as a completely rational model in plate bending analysis. The proposed method can be extended to more problems of plates such as buckling and vibration. Tabulated accurate results are included in this paper, which are expected to be valuable for future comparison.     

Indentation of an orthotropic layer by a semi-infinite punch

In this paper, a plane elasticity problem treated involving punch indentation of an orthotropic strip resting on a rigid frictionless foundation. A semi-infinite punch and an infinitely long layer were considered, so that the problem was amenable to the Wiener-Hopf technique. Applying Fourier transforms to the governing differential equations and the imposed boundary conditions, we succeeded in formulating a Wiener-Hopf relationship which then yielded the asymptotic estimation of the stress and displacement field in the immediate vicinity of the edge of stamp.     

A bi-orthogonality relation for clamped sector plates

Summary Constructing a bi-orthogonality relation it is indicated how the solutions for the bending of a sector plate clamped along the radial edges and with physically meaningful boundary functions prescribed on the curved edge can be directly reduced to an infinite system of linear algebraic equations. Some numerical results are presented for the bending of a uniformly loaded clamped sector plate.     

Pulse pressure loading of clamped mild steel plates

Some results of a large series of pulse pressure loading tests on 1 m×1 m, clamped mild steel plates with different in-plane edge conditions are presented. A novel experimental device was used to produce uniform, repeatable pulse pressure loads of an approximate triangular form. The behaviour of the restrained plates was dominated by membrane effects leading to substantial spring-back. An assumed-modes, elastic–plastic analytical approach was used to predict the maximum and residual deflections of the test plates under dynamic loading with reasonably good success. Multi-element models of the test plates were developed in which the bi-axial behaviour of the plate was represented by one-dimensional finite strips. The in-plane restraint of the plate was approximated using non-linear translation springs and a slippage allowance. Satisfactory results were achieved owing to the careful determination and definition of the boundary conditions, which can significantly influence the results particularly if non-linear membrane forces and slippage occur. It has been shown that a simple methodology such as the one proposed in this paper can provide adequate predictions of the dynamic, large-deflection behaviour of pulse-loaded plates.     

Nonlinear adhesive behavior effects in a cracked orthotropic sheet stiffened by a semi-infinite orthotropic sheet

The stress-intensity factors are determined for a cracked orthotropic sheet adhesively bonded to an orthotropic stringer where the adhesive layer is modeled with a nonlinear stress-strain curve. Since the stringer is modeled as a semi-infinite sheet, the solution is most appropriate for a crack tip located near a stringer edge. By the use of Green's functions and the complex variable theory of orthotropic elasticity developed by Lekhnitskii, a set of integral equations is obtained. The integral equations are replaced by an equivalent set of algebraic equations, which are solved to obtain the shear stress distribution in the adhesive layer. With these adhesive stresses, the crack-tip stress-intensity factors are found.When the adhesive was modeled with a nonlinear stress-strain curve, the peak shear stresses in the adhesive were considerably reduced in comparison to the solution for the linear elastic adhesive. This resulted in increases in the stress-intensity factors for the nonlinear adhesive solution compared to the linear adhesive solution. The nonlinear adhesive did not have a significant effect on the stress-intensity factor unless the near crack tip was beneath the stringer. The present investigation assumes that the adhesive bond remains intact. Onset of adhesive failure is predicted to occur at decreasing levels of applied stress as the crack propagates beneath the stringer.     

Axisymmetric postbuckling of orthotropic thick annular plates

Summary This study deals with the geometrically nonlinear postbuckling response of cylindrically orthotropic thick annular plates subjected to inplane radial compressive load at the outer edge. Clamped and simply supported plates with a free hole and a plugged hole have been considered. Governing differential equations are expressed in terms of transverse displacementw, shear rotation and stress function . Polynomial expansions are employed for these three field variables and the orthogonal point collocation method has been used to obtain the discretised equations. Buckling loads have been determined by solving a linear eigen-value problem using the method of inverse iterations. Postbuckling loads for different central deflections have been obtained by solving the nonlinear differential equations. The effect of thickness ratio, orthotropic parameter, annular ratio and the edge conditions has been investigated. The effect of shear deformation on the buckling loads increases with the orthotropic parameter and the thickness-to-radius ratio.Notations a, b, h Outer and inner radii and thickness of the plate - E ,E ,v ,v Young's moduli, Poisson's ratios - Orthotropic parameter:E /E _r - G _rz /E _r - u ^*,u ₀ ^* Radial displacement, radial displacement at mid plane - r, , z Cylindrical polar coordinates - , ^* Strains and stresses - w ^*,w ,w Transverse displacement, shear rotation and stress function - w, , Dimensionless deflection, shear rotation and stress function - N _r ,N ,Q _r Inplane stress resultants and transverse shear - M _r ,M Moment resultants - k ² Shear correction factor - , (r–b)/(a–b), b/(a–b) - a/h - _r ^b , ^m Radial bending stress, circumferential membrane stress - N, _i Number and radii of collocation points - Eigenvalue for buckling With 6 Figures     

A three-dimensional analysis of laminated orthotropic plates

A mathematical apparatus is developed for the analysis of the three-dimensional stress–strain state with small deflections of laminated orthotropic plates of any thickness and non-symmetrical layer structure through the thickness. This theory is based on the representation of the displacement vector in terms of products of the given functions in the direction of the axes x₁, x₂ and the unknown functions over the axis x₃. Both the real and complex roots of the characteristic equation describing the representation of the sought functions through the thickness of the plate are taken into account. Such an approach allows us to expand the scope of physical and mechanical relationships among the material characteristics.     

Large amplitude vibration of orthotropic elliptical plates

Summary An analytical investigation of the large amplitude flexural vibration behaviour of clamped, rectilinearly orthotropic elliptical plates is carried out. Dynamic von Kármán-type equations of orthotropic plates are expressed in terms of the three displacement components. A solution is obtained on the basis of a single-mode approach. Period ratios and stresses during vibration are graphically presented and discussed for three types of high modulus fibre-reinforced plates. In the case of isotropic circular plates, the present solution is in excellent agreement with those available in literature. Also, present results closely agree with those of Prabhakara and Chia for static cases.

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Zusammenfassung Es wird eine analytische Abschätzung des Biegeschwingverhaltens bei großen Amplituden von eingespannten, geradlinig orthotropen, elliptischen Platten durchgeführt. Dynamische Gleichungen vom Kármán-Typ der orthotropen Platte werden durch die drei Verschiebungskomponenten dargestellt. Eine Lösung wird unter Benutzung einer einzigen Eigenfunktion erhalten. Periodenverhältnisse und Spannungen während der Schwingung werden graphisch dargestellt und für drei Typen von faserverstärkten Platten mit hohem Modul besprochen. Im Fall isotroper, kreisrunder Platten steht die hier geschilderte Lösung in ausgezeichneter Übereinstimmung mit den bekannten Lösungen der Literatur. Es passen die vorliegenden Ergebnisse auch sehr gut zu denen von Prabhakara und Chia für den statischen Fall.

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Notations 2a, 2b axes of the elliptical plate - elastic orthotropic material constants referred tox andy directions - h plate thickness - q ₀ intensity of uniformly distributed transverse load - T non-linear period - T ₀ linear period - u,v components of in-plane displacements - w transverse deflection - w ₀ central deflection - x,y,z rectangular cartesian co-ordinates - membrane stresses in the plate - extreme-fibre bending stresses in the plate - mass density - non-dimensional time, With 5 Figures     

Non-axisymmetric instability of polar orthotropic annular plates

The non-axisymmetric instability of polar orthotropic annular plates under inplane uniform radial pressure is studied by use of the shooting method. The characteristic equations and eigenvalues under a variety of edge conditions are given. Under two appropriate hypotheses, we prove that all eigenvalues are bifurcation points. Hence, it is possible that non-axisymmetric buckled and post-buckled states branch from axisymmetric unbuckled states of an annular plate. Asymptotic formulae for buckled states are obtained and curves for the deflection and stress are shown.     

Stresses in an orthotropic semi-infinite strip due to edge loads

Summary The stresses in an orthotropic elastic semi-infinite strip subject to plane strain are investigated. Symmetrical distributions of surface tractions are prescribed on the sides of the strip, while along the end the boundary conditions are arbitrary. By using an integral transform method the problem is reduced to a singular integral equation. The dependence of the stress singularity and the stress-intensity factors on the orthotropic properties of the strip is investigated. Stress distributions over the strip end are evaluated numerically.With 5 Figures     

Tearing of blast loaded plates with clamped boundary conditions

Extensive experimental results for the prediction of the onset of thinning (necking) and hence subsequent tearing at the boundary of clamped circular plates subjected to uniformly loaded air blasts are presented. The experiments include: plates of diameter ranging from 60mm to 120mm; different clamped edge conditions; (sharp edge; and radii of 1.5mm and 3.2mm) and an examination of the microstructural behaviour. Observations of thinning at the boundary show similar trends for all plate diameters with sharp edge conditions, and where the boundary edge condition is relaxed the deflections are larger before thinning and tearing occurs. Three phases within Mode I (large inelastic deformation) are identified as; Mode I (no visible necking at the boundary); Mode Ia (necking around part of the boundary); and Mode Ib (necking around the entire boundary). Predictions using the code ABAQUS provide, in particular for Mode I, favourable correlations with the experiments.     

Boundary discontinuous Fourier analysis of thick cross-ply clamped plates

A new analytical solution to the problem of deformation of a finite-dimensional general cross-ply thick rectangular plate, modeled using a higher order shear deformation theory (HSDT), is presented. A solution methodology, based on a boundary-discontinuous generalized double Fourier series approach, is used to solve a system of five highly coupled linear partial differential equations, generated by the HSDT-based laminated plate analysis, with the C3-type clamped boundary condition prescribed at all four edges. The numerical accuracy of the solution is ascertained by studying the convergence characteristics of deflections and moments of a cross-ply plate. The primary focus of the present study is to investigate the effect of end clamping on the response of a thick laminated plate, while keeping the in-plane end constraints unaltered. Important numerical results presented include sensitivity of the predicted response quantities of interest to lamination, material property, thickness effects and end clamping as well as their interactions.     

Steady-state transverse vibrations of compound rectangular orthotropic plates

A method is proposed for examining the steady-state vibrations of a compound orthotropic plate under a transverse load. The small-parameter method is used. A numerical study is presented for a plate consisting of three bands. The deflection at the center of the plate is presented as a function of the loading frequency.Translated from Problemi Prochnosti, No. 7, pp. 67–70, July, 1994.