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In this paper, the Ritz method is developed for the analysis of thin rectangular orthotropic plates undergoing large deflection. The trial functions approximating the plate lateral and in-plane displacements are represented by simple polynomials. The nonlinear algebraic equations resulting from the application of the concept of minimum potential energy of the orthotropic plate are cast in a matrix form. The developed matrix form equations are then implemented in a Mathematica code that allows for the automation of the solution for an arbitrary number of the trial polynomials. The developed code is tested through several numerical examples involving rectangular plates with different aspect ratios and boundary conditions. The results of all examples demonstrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Many types of engineering structures can be effectively modelled as orthotropic plates with opposite free edges such as bridge decks. The other two edges, however, are usually treated as simply supported or fully clamped in current design practice, although the practical boundary conditions are intermediate between these two limiting cases. Frequent applications of orthotropic plates in structures have generated the need for a better understanding of the dynamic behaviour of orthotropic plates with non-classical boundary conditions. In the present study, the transverse vibration of rectangular orthotropic plates with two opposite edges rotationally restrained with the remaining others free was studied by applying the method of finite integral transforms. A new alternative formulation was developed for vibration analysis, which provides much easier solutions. Exact series solutions were derived, and the excellent accuracy and efficiency of the method are demonstrated through considerable numerical studies and comparisons with existing results. Some new results have been presented. In addition, the effect of different degrees of rotational restraints on the mode shapes was also demonstrated. The present analytical method is straightforward and systematic, and the derived characteristic equation for eigenvalues can be easily adapted for broad applications. [ABSTRACT FROM AUTHOR]

The problem of flexural characteristics of rectangular orthotropic thin plates with opposite rotationally restrained and free edges subjected to arbitrary transverse loads is investigated in present study. By introducing a rotational fixity factor to simulate the elastic rotation restraint at plate edges, the finite integral transform method is employed to obtain exact analytical solution of the problem. Numerical solutions for rectangular plates with various aspect ratios and edge flexibility are investigated and compared to the results reported in previous research. The accuracy and convergence of the present formulation are assessed by comparing with previously reported results. The excellent agreement on the results between the proposed method and previous studies indicates that the proposed method is accurate and effective. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of the origin of in-plane stresses in continuous, two-dimensional high-speed webs. In the case of thin, slender webs, a typical modelling approach is the application of a stationary in-plane model, without considering the effects of the in-plane velocity field. However, for high-speed webs this approach is insufficient, because it neglects the coupling between the total material velocity and the deformation experienced by the material. By using a mixed Lagrange–Euler approach in model derivation, the solid continuum problem can be transformed into a solid continuum flow problem. Mass conservation in the flow problem and the behaviour of free edges in the two-dimensional case are both seen to influence the velocity field. We concentrate on solutions of a steady-state type, and study briefly the coupled nature of material viscoelasticity and transport velocity in one dimension. Analytical solutions of the one-dimensional equation are presented with both elastic and viscoelastic material models. The two-dimensional elastic problem is solved numerically using a nonlinear finite element procedure. An important new fundamental feature of the model is the coupling of the driving velocity field to the deformation of the material, while accounting for small deformations of the free edges. The results indicate that inertial effects produce an additional contribution to elastic contraction in unsupported, free webs. [ABSTRACT FROM AUTHOR]

For the first time, the double half-Sinusoidal series is adopted as integral kernel to construct integral transform pair for new precise vibration analysis of orthotropic rectangular thin plates, supported with two adjacent edges rotationally-restrained and the other edges are free. Impose the defined single/double transforms on both the vibration governing partial differential equation (PDE) and some of the boundaries investigated, which gives the Fourier coefficient for the deflection expressed by some specific unknown constants. Utilizing the inverse formular to satify the remaining boundaries leads to a set of linear algebraic simultaneous equations, which yields the solutions of the natural frequency parameter and the associated mode shapes. Moreover, plates under classical boundary conditions also can be investigated via changing the rotating fixed coefficients introduced. Finally, more than 500 comprehensive analytical solutions were well validated by numerical results presented by finite element method (FEM), which can be treated as new reference results for further studies. Since its straightforward and rational solution procedure, the presented approach is promising to be further extended to solve more complicated boundary value problems (BVPs) of plates. [ABSTRACT FROM AUTHOR]