Buckle Pattern of Biaxially Compressed Simply Supported Orthotropic Rectangular Plates

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.

On buckling loads for edge-loaded orthotropic plates including transverse shear

An analysis of a crack with bridging fibers in an orthotropic elastic infinite plate is carried out. Based on Lekhnitskii's anisotropic elastic theory, the distributions of the stress and the displacement of the orthotropic elastic infinite plate with a crack are analyzed under a uniform remote tension stress. As well as utilizing the model of fiber pull-out, the relationship between the traction force at a bridging fiber and the crack opening displacement is established. The stress intensity factor KI and the strain energy release rate GI of the crack are obtained by linear elastic fracture mechanics, and the influence factor BK of the bridging for the stress intensity is defined. Finally, in order to discuss the effectiveness of the present method, an example problem, the finite element method solution of which was given, is numerically solved.

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.

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

Instrumented impact testing of sintered metal test specimens: R.A. Shimansky et al (Bucknell University, Lewisburg, Pennsylvania, USA), Int. J Powder Metallurgy, Vol 28, No 4, 1992, 401–408

Sintering behaviour of copper coated iron powder with 3% copper: H.S. Yang and D.J. Lee (Chonnam National University, Kwang-Ju, Korea), J Korean Inst of Metals, Vol 30, No 1, 1992, 98–103 (in Korean)

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

The behavior of thin, rectangular, orthotropic elastic plates, with immovable edges and undergoing large deflections, is investigated by the numerical technique of differential quadrature. Approximate results are obtained, using the Newton-Raphson method and, alternatively, a finite-difference-based method to solve the nonlinear systems of equations. Bending stresses, membrane stresses, and deflections are calculated for plates with fully clamped and simply supported flexural edge conditions under uniform pressure loading. Results are compared with existing analytical, numerical, and experimental ones. The present method gives good accuracy and is computationally efficient.

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

This paper is the second part of free vibration and stability analysis of thick isotropic and orthotropic plates. The support condition CCCC and CCFC support was considered by applying the alternative II theory based on polynomial shape function. The work was based on the application of the formulation earlier derived using Ritz method. The frequency values of the first mode and critical loads obtained were compared with those obtained using first order shear deformation theory. For span to depth ratio of 10, the fundamental linear frequency for orthotropic CCFC plate corresponding to modulus of elasticity ratios (E1/E2) of 10, 25 and 40 were 0.00312Hz, 0.00373 Hz and 0.00398Hz.

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 double finite cosine integral transform method is exploited to obtain explicit solutions for the natural frequencies and mode shapes of the orthotropic rectangular plate on foundation with four edges free. In the analysis procedure, the classical Kirchhohh orthotropic rectangular plate is considered and the Winkler elastic foundation is utilized to represent the elastic foundation. Because only are the basic dynamic elasticity equations of the orthotropic thin plate on elastic foundation adopted, it is not need prior to select the deformation function arbitrarily. Therefore, the solution developed by present paper is reasonable and theoretical. In order to illuminate the correction of formulations, the numerical results are also presented to comparing with that of the other references.