New analytic buckling solutions of rectangular thin plates with two free adjacent edges by the symplectic superposition method

On the symplectic superposition method for new analytic free vibration solutions of side-cracked rectangular thin plates

The bending problem of free orthotropic rectangular thin plate (RTP) on two‐parameter elastic foundation under a concentrated load is studied by the symplectic superposition method. Firstly, the original bending problem is decomposed into three subproblems by analyzing load effects and boundary conditions, each of which is the bending problem of the plate with two opposite edges slidingly supported. In order to solve the three sub‐problems based on the separation of variables method in Hamiltonian system, the Hamiltonian system for the orthotropic RTP with two opposite edges slidingly supported is studied, and then the eigenvalues and eigenfunctions of the Hamiltonian operator are obtained by combining the separation of variables and symbolic computation. Secondly, according to the symplectic orthogonality and the completeness of the eigenfunctions, the general solution of the Hamiltonian system with the two opposite edges slidingly supported is obtained. Furthermore, the solutions in the form of series of the three subproblems are obtained respectively. Finally, the symplectic superposition solution of the original bending problem is obtained by superposing the solutions of the three subproblems and the correctness of the symplectic superposition solution is verified by two numerical examples.

Remarks of some problems for rectangular thin plates with free edges on elastic foundations

Bending of rectangular orthotropic plates with rotationally restrained and free edges: Generalized integral transform solutions

Orthotropic thin plates are essential in aerospace applications due to their high strength-to-weight ratio, which is critical for efficient structural performance. The mechanical properties of these structures are influenced by numerous factors, with free vibration characteristics being particularly significant. However, analytical solutions for free vibration are often limited to plates with simple boundary conditions (BCs). Few exact analytical solutions exist for plates with complex non-Lévy-type or rotationally restrained BCs. This study aims to analyze the free vibration behavior of orthotropic thin plates featuring three rotationally restrained edges and one free edge, employing the finite integral transform (FIT) method. A key advantage of this method is its simplicity and versatility, as it does not require the predefinition of a deflection function. By setting different values of the rotating fixed coefficient, the study offers analytical free vibration solutions for plates with various combinations of rotationally restrained, simply supported, clamped, and free BCs without additional derivations. The solutions derived using the FIT method are validated by comparison with numerical results obtained from ABAQUS software and available analytical solutions, confirming the method’s accuracy and reliability. Parameter analyses are conducted to examine the effects of aspect ratio, BCs, material properties, and rotating fixed coefficient on non-dimensional frequency parameters and corresponding vibration modes. The results reveal that these parameters significantly influence the frequency characteristics of orthotropic thin plates. The results can be used as a benchmark for validating other analytical and numerical methods.

Free vibration analysis of beams and rectangular plates with free edges by the discrete singular convolution

Using the superposition and small deflection theories, thermal change perpendicular to a surface is considered. A CRTP (concrete rectangular thin plate) with two adjacent CEDs (clamped edges) and one SSED (simply supported edge) and one FED (free edge) under thermal change is seen as a RTP’s superposition with three SSEDs and one FED in the case of temperature change and a RTP with three SSEDs and one FED under the bending moment on two adjacent edges. Adopting the Levy method, by supposing the deflection function with undetermined parameter on the FED, the theoretical solution of a CRTP (concrete rectangular thin plate) with three SSEDs and one FED is obtained; the theoretical solution of a CRTP with three SSEDs and one FED is obtained under thermal change; taking the theoretical solution of a CRTP with four SSEDs under the bending moment on two adjacent edges, the theoretical solution of a CRTP with three SSEDs and one FED under the bending moment on two adjacent edges is obtained; the theoretical solution for the bending moment and deflection of a CRTP with two adjacent CEDs and one SSED and one FED under temperature change is obtained.

Theoretic solution of rectangular thin plate on foundation with four edges free by symplectic geometry method

Analytical solutions for elastoplastic buckling of plates play a crucial role in providing benchmark results and facilitating fast structural analyses for preliminary designs. However, the analytical solutions for elastoplastic buckling of plates remain incomplete due to the inherent mathematical difficulties associated with higher-order partial differential equations and material nonlinearity. Consequently, the existing analytical solutions for rectangular plates are only applicable to those with Lévy‐type boundary conditions. To address the limitation, this study extends a novel symplectic superposition method to obtain new analytical elastoplastic buckling solutions of non-Lévy‐type rectangular plates, where both the incremental theory (IT) and deformation theory (DT) are adopted. Comprehensive benchmark elastoplastic buckling loads are presented and validated by the modified differential quadrature method. The plastic buckling paradox is explicitly observed, which highlights a significant disparity between the IT and DT in predicting buckling loads for relatively thick plates. Furthermore, our analysis reveals that the DT provides an optimal load ratio for buckling resistance, while no such finding is observed with the IT. The stability criterion curves are plotted using the elastic theory and plastic theories (IT and DT) to further reveal the importance of incorporating the effect of plasticity as well as to provide a useful guideline for the relevant analyses and designs.

Review of solution methodologies for structural analysis of composites

The “free-corner effect” in thermally loaded laminates

This paper aims to obtain the analytical free vibration solution of an orthotropic rectangular thin plate using the finite integral transformation. Due to the mathematical difficulty of complex boundary value problems, it is very hard to solve the title problem with common analytical methods. By imposing the integral transformation, the high-order partial differential equation with specified boundary conditions is converted into linear algebraic equation and the exact solutions with high precision and fast convergence are obtained elegantly. The main advantage of the proposed method is that it is simple and general, and it does not need to pre-determine the deflection function, which makes the solution procedure more reasonable. The method has a wide range of applications and can deal with other elastic plate problems, such as buckling and bending. The present results are validated by comparison with the existing analytical solutions that showed satisfactory agreement. The new analytical solution obtained can be used as a benchmark for validating other numerical and approximate methods.

Free vibrations of rectangular plates with interior point supports

The flexibility function approach to vibration analysis of rectangular plates with arbitrary multiple point supports on the edges

This paper presents an innovative series expansion method for the accurate solution of biaxial symmetry problems of rectangular thin plates. This method surpasses traditional techniques such as the Navier and Lévy series solutions, the Timoshenko and Woinowsky-Krieger superposition method, and the symplectic geometry method, offering improved efficiency, easier implementation, and faster convergence. It effectively handles corner singularities arising from different boundary conditions and demonstrates superior accuracy. Key contributions include the application of particular closed-form solutions, the supplementation of corner conditions at intersections of free and clamped edges, the classification of corner singularities, and the selection of appropriate series expansions based on the inherent nature of the boundary conditions. The robustness of this method in dealing with complex boundary conditions and different loading scenarios is emphasized, providing deep insight and exceptional accuracy in structural analysis. Validation against peer results confirms its superiority and identifies potential problems in existing “exact” solutions, highlighting its practical value and potential for widespread adoption.