Efficient large deflection analysis of rectangular orthotropic plates by direct energy minimisation

This paper focuses on investigating the mechanical buckling behavior of composite and orthotropic classical rectangular plates using the Galerkin theory. By utilizing the classical plate theory of elasticity, the governing equations for the functionally graded plate material subjected to a uniaxially distributed load along the x and y axes are derived based on the principle of work energy. Consequently, the displacement function is determined as the solution to a differential equation that satisfies specific boundary conditions. The equilibrium expression derived from the work‐energy equation is then minimized and analytically solved to determine the stresses and critical buckling loads of an orthotropic composite rectangular plate. The critical buckling loads of all edges, including clamped and simply supported orthotropic and isotropic thin rectangular plates with aspect ratios ranging from 0.1 to 3, are calculated and compared with existing literature to assess the stiffness of isotropic and orthotropic plates. The findings reveal that the critical buckling load of a clamped plate decreases as the aspect ratio increases from 0.1 to 1 but increases for aspect ratios between 1 and 3. The average percentage variation of critical buckling load values obtained from previous studies using orthotropic and isotropic (classical lamination plate theory [CLPT]) materials is found to be 1.93% and 3.65%, respectively. Additionally, it is observed that the critical buckling load of clamped edges in isotropic plates is lower than that of orthotropic plates at lower aspect ratios (0.4–1.0) but higher at higher aspect ratios (above 1.0). The results indicate that plates with all‐round clamped boundaries exhibit higher buckling loads due to their effective fixation at all boundaries, which provides greater resistance to deformation and buckling compared to plates with all edges simply supported, where there is more flexibility for deformation under loads. This highlights the influence of boundary conditions on the critical buckling load of the plate. Furthermore, a significant percentage difference is noted when comparing approximate solutions, while a low percentage difference of 0.43% with the author’s derived displacement function indicates close agreement, demonstrating the high level of accuracy and reliability of the proposed model in predicting buckling load of plates. Hence, it can be recommended for stability analysis, especially where the material properties need to be controlled in a specific direction to optimize performance.

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

Analysis of stress concentration in isotropic and orthotropic plates with hole under uniformly distributed loading conditions

Simply supported orthotropic elastic rectangular plates in biaxial compres sion are considered, and it is proved that such plates will buckle with m or n (or both) equal to 1, where m and n are the number of half waves in the two prin cipal directions. While this theorem may be known to many as an empirical computational fact, it does not appear to have been explicitly stated or proved heretofore in the literature on orthotropic plates

The article proposes a methodology for calculating CLT panels as composite beams. This approach allows for the analysis of panels with any number of layers. It is shown that, unlike existing approaches, the behavior of the panel is considered in two directions, rather than as a beam system. Initially, the stiffness characteristics of the panel are determined in each direction, taking shear into account. Then, the slab is analyzed as an orthotropic plate with different stiffness characteristics in mutually perpendicular directions. This calculation is performed either as an orthotropic solid plate or as a cross-laminated beam system. To determine the stiffness characteristics in each direction while considering the shear of the cross-layer boards, the theory of composite beams is used. Taking into account the boundary conditions, the system of differential equations is reduced to a system of algebraic equations by expanding the unknowns and external load in Fourier series.

The article deals with a laminated orthotropic plate. Between the layers there is rigid contact as well as contact with frictionless slippage. On the end faces at the level of the neutral surface of the slipping layers the plate is loaded by tangential loads. The model of loss of stability is based on the hypotheses on displacements of the plate that take transverse shear strain into account. Normal displacements of the plate do not depend on the transverse coordinate. The hypotheses derived by iteration make it possible to satisfy the conditions of interlaminar contact, both rigid and with frictionless slippage, with respect to displacements and stresses, and also the conditions on the outer surfaces of the plate. With the aid of Lagrange's variational principle we obtained differential equations under conditions of the precritical and the postcritical state, and also the corresponding boundary conditions. The equations of the postcritical state are written in terms of displacements. On the basis of the presented equations the problem of the stability of a homogeneous orthotropic rectangular plate with pin support along the periphery is solved. It is shown that the results obtained with the suggested model coincide with the known results. The article also examines a square three-layered orthotropic plate under conditions of frictionless slippage contact between the layers and pin support along the periphery. The central layer is loaded by tangential loads on the end faces at the level of its neutral surface. It is shown that when slippage is taken into account, the critical forces are substantially reduced.

Effect of fibre orientation on the vibration behaviour of orthotropic rectangular plates

This paper deals with determination of stress, strain and displacement state of a circular annulus which is made of material with Cartesian orthotropic rheological behaviour. There is a continuous and constant load on the inner and/or outer edge. Stress and strain state in the circular annulus is in the elastic domain. Stress state is determined on the basis of an Airy stress function, where all boundary conditions have to be fulfilled. The Ritz's method was applied to determine unknown parameters of the Airy function. References F. Kosel. The stress and deflection state in the circular annulus with Cartesian orthotropy. Proceedings 14th Canadian Congress of Applied Mechanics , volume 2, pages 493--494. CANCAM, 1993. Pistonesi, C. and Laura, P. A. A., Forced vibrations of a clamped, circular plate of rectangular orthotropy, J. Sound Vibr. , 228 , 1999, 712--716. Tahan, N., PavloviE, M. N. and Kotsovos, M. D., Orthotropic rectangular plates under in-plane loading part 1: closed-form solutions for stresses, Compos. Struct. , 33 , 1995, 35--48. Wu, Z. J. and Wardenier, J., Further investigation on the exact elasticity solution for anisotropic thick rectangular plates, Int. J. Solids Struct. , 35 , 1998, 747--758. Adewale, A. O., Application of the singularity function method to semi-infinite orthotropic rectangular plates on an elastic foundation, Int. J. Mech. Sci. , 43 , 2001, 2261--2279. Bhaskar, K. and Kaushik, B., Simple and exact series solutions for flexure of orthotropic rectangular plates with any combination of clamped and simply supported edges, Compos. Struct. , 63 , 2004, 63--68. S. G. Lekhnitskii. Theory of elasticity of an anisotropic body . Mir Publishers, Moscow, 1981.

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.

Many engineering structures are built by assembly of plates, and in recent years there has been a widespread use of composite plates, and in particular of orthotropic plates, to improve dynamic structural performance and reduce structural weight. Depending on the manufacturing process used, orthotropic plates might be affected by variability of the mechanical properties, which in turn will affect their performance. The four elastic constants of orthotropic plates can be obtained via direct static or dynamic tests, or via inverse methods which combine dynamic experiments with mathematical models. However these approaches may not be appropriate for establishing manufacturing variability across a number of manufactured plates, since they might require expensive testing equipment and they may be time consuming. In this paper an inverse procedure based on the Chladni patterns is investigated to analyse thin orthotropic rectangular plates. A detailed mathematical derivation of the equations needed for estimating the elastic constants based on the knowledge of five natural frequencies is presented. The repeatability and reproducibility of the results obtained with the proposed procedure is first assessed, and the evaluation of the manufacturing variability of an ensemble of “nominally identical” plywood plates with the proposed approach is then investigated.

The SAFE/RAS and STRAW codes were used to analyze the static response of the TREAT MK III loop subjected to thermal and mechanical loadings arising from an accident situation for the purpose of determining the deflections and stresses. The analysis was subdivided into four tasks: (1) an analysis of the complete loop under uniform temperature with constant material properties to validate the model, (2) an elastic analysis simulating a heatup condition of 70°F to 300°F in the test leg, again for model verification, (3) an analysis of a flow blockage accident (Cases A, Band C), where all the energy is deposited in the test leg, resulting in a temperature increase from 925° to 1730° F, with a small internal pressure throughout the loop and {4) an analysis of a fiow blockage accident (Case D), where only one-third of the energy is assumed to be deposited in the test leg, resulting in temperature rise from 925° to 1200°, and a pressure of 1250 lb/in{sup 2}. The purpose of the last two tasks was to determine if loop failure can occur with the thermal differential across the pump and test legs. Also of interest is whether undesirable amount of loop lateral deflection will be caused by the thermal differential. A two and three dimensional analysis of the TREAT MK III loop was performed. The analysis accounted for material nonlinearities, both as a function of temperature and stress, and geometric nonlinearities arising from large deflections. Straight beam elements with annular cross sections were used to model the loop. The analyses show that the maximum strains are less than 42% of their failure strains for all cases considered. The failure strain is 8.3% for the material at 925°F and 33.3% at 1730°F. Large lateral deflections are observed when the loop is not constrained except at its clamped support--as much as 11.8 inches. However, by accounting for the constraint of the concrete biological shield, the maximum lateral deflection was reduced to 0.2 inches at the points of concern.

Non-linear principal resonance of an orthotropic and magnetoelastic rectangular plate

This research article considers the flexural rigidity influence along both x and y axes on the dynamic response of orthotropic rectangular plate resting on constant elastic foundation with elastic end conditions. The orthotropic rectangular plate model is a coupled fourth order partial differential equation having variables and singular coefficients. The solutions to this model are arrived at by reconstructing the fourth order partial differential. This partial differential equation model is converted to a set of coupled second order ordinary differential equations by using a special technique adopted by Shadnam et al., [11]. This set of second order ordinary differential equations is then reduced using modified asymptotic method of Struble. The closed form solution is evaluated, resonance conditions are obtained and the results are showed in plotted curves to depict the influence of flexural rigidities along both x and y axes on the dynamic response of orthotropic rectangular plate resting on constant elastic foundation with elastic end conditions for both cases of moving distributed mass and moving distributed force.

The study presents a continuous analysis method for determining the buckling load of delaminated beams and plates. Investigations include beam-plates with end rotational restraints using the Stokes' transformation technique and orthotropic plates with simple supports. The analysis allows structures to have a multiple number of delaminations of various shapes and sizes, and at any locations. The method treats the discontinuous structure as a body without delamination but with a force system added at a number of discrete points in the delaminated region so as to make the net interracial tractions to vanish at these points. Hence, the procedure solves problems of delaminated structures directly as a continuous system. Fourier series in conjunction with the Stokes' transformation is used which provides a unified solution for problems with different end conditions. Numerical results accounting for the effects of delamination length, location and shape; boundary conditions and stiffness of adhesive layer, to the local buckling load are presented for illustrative purposes. The method of analysis presented in the study is simple in concept, yet so effective and accurate in solving such discontinuous structures which would otherwise be extremely difficult if not impossible to analyze

The solution to the problem of the stability of a rectangular orthotropic plate is described by the numerical-analytical method of boundary elements. As is known, the basis of this method is the analytical construction of the fundamental system of solutions and Green’s functions for the differential equation (or their system) for the problem under consideration. To account for certain boundary conditions, or contact conditions between the individual elements of the system, a small system of linear algebraic equations is compiled, which is then solved numerically. It is shown that four combinations of the roots of the characteristic equation corresponding to the differential equation of the problem are possible, which leads to the need to determine sixty-four analytical expressions of fundamental functions. The matrix of fundamental functions, which is the basis of the transcendental stability equation, is very sparse, which significantly improves the stability of numerical operations and ensures high accuracy of the results. An analysis of the numerical results obtained by the author’s method shows very good convergence with the results of finite element analysis. For both variants of the boundary conditions, the discrepancy for the corresponding critical loads is almost the same, and increases slightly with increasing critical load. Moreover, this discrepancy does not exceed one percent. It is noted that under both variants of the boundary conditions, the critical loads calculated by the boundary element method are less than in the finite element calculations. The obtained transcendental stability equation allows to determine critical forces both by the static method and by the dynamic one. From this equation it is possible to obtain a spectrum of critical forces for a fixed number of half-waves in the direction of one of the coordinate axes. The proposed approach allows us to obtain a solution to the stability problem of an orthotropic plate under any homogeneous and inhomogeneous boundary conditions.