Analytical Buckling Loads for Rectangular Orthotropic and Symmetrically Laminated Plates

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.

New exact solutions for free vibrations of thin orthotropic rectangular plates

Buckling resistance is often a controlling criterion in the design of structures composed of plate elements. Design concepts that lead to increased buckling loads (or strains) of these plate elements can directly lower the structural cost and/or weight by a number of means. This study quantifies the improvements that can be achieved in compression buckling loads of rectangular composite plates by using a simple stiffness-tailoring concept. The approach is to position the unidirectional lamina through the thickness and over the planform of the plate so that the buckling load is increased with no loss in in-plane stiffness or increase in weight. Finite element analyses have been used to determine the effects of tailoring on the buckling loads of plates with various boundary conditions, aspect ratios, thicknesses, and membrane stiffnesses. Increases in buckling loads (or strains) of 200% or more compared to the uniform plate-buckling loads are shown possible with this tailoring concept. HE design of aircraft primary structural elements is often strongly influenced by a plate-buckling criterion. For ex- ample, the wing cover surface is normally restricted from buckling at loads up to maximum operating or even ultimate levels due to aerodynamic smoothness or global stiffness re- quirements. The fuselage skin is often restricted from buckling up to a relatively low load level to control out-of-plane defor- mations at normal service conditions and to limit these defor- mations and the loss of effective in-plane stiffnesses at higher postbuckling load levels. Similar requirements exist for struc- tural elements in many other types of applications. Design concepts that lead to increased plate-buckling loads (or strains) can directly lower the structural cost and/or weight by 1) allowing supporting substructure to be more widely spaced, thus reducing substructure, fastener, and attachment part count, 2) allowing the use of simpler substructure offering little rotational restraint to the surface structural elements, or 3) reducing the total structural weight by permitting design to a higher strain level governed by the material strength or damage tolerance rather than to lower values governed by buckling. This study quantifies the improvements that can be achieved in compression buckling loads of rectangular composite plates by using a simple stiffness-tailoring concept. It provides guide- lines that designers can use to achieve the improvement. Here stiffness tailoring is defined as the precise placement of lamina with various orientations through the thickness and across the planform of the laminated plate. The local membrane and bending stiffnesses of the tailored plate become nonuniform over the plate. When the loading is applied to the plate in the form of uniform imposed end displacements, stiffness tailor- ing brings about a redistribution of the in-plane loads that can directly benefit buckling response. At the same time, the dis- tribution and magnitudes of the bending and twisting stiff- nesses can also be modified to further improve the buckling load. The properly tailored design will be a function of the type of loading, the plate geometry, the boundary conditions, the relative material properties, and the basic configuration of the tailoring concept. Bulson1 shows that the type of tailoring

In this study, the vibration correlation technique was introduced to determine the buckling load of rectangular thin plates. It is theoretically shown that the natural frequency approaches zero when the applied compressive load approaches the buckling load of the plate. To avoid the effects of premature out-of-plane deformation, it is proposed in this study that the buckling load is to be identified using the natural frequencies of plates under tensile loading. A set of aluminum plates was tested for natural frequencies using an impact test method. Specimens with two types of boundary conditions, i.e., CCCC and CCCF, were included in the experiment. The square of the measured natural frequency was plotted against the applied load and extrapolated to determine the predicted buckling load. The buckling loads from vibration data compare closely with numerical solutions. The average percentage differences between the measured buckling loads and the numerical solutions are 1.24 % and -1.14 % for specimens with CCCC and CCCF boundary conditions, respectively. In conclusion, the buckling load of rectangular thin plates can be experimentally identified with acceptable accuracy using vibration data. This approach is very useful especially for structures with unknown or imperfect boundary conditions where analytical or numerical solutions to the problem are not available.

Thin rectangular plates are widely used in civil, mechanical, and aeronautical engineering, where accurate prediction of buckling behaviour is critical for structural reliability. While most classical analyses focus on small-deflection conditions, such approaches neglect the nonlinear effects induced by large out-of-plane displacements. This study aims to apply the new general mathematical model for biaxial buckling of a thin isotropic plate with large deflection to formulate new specific equations for six plate boundary conditions. The nonlinear buckling behaviour is investigated for plates with boundary conditions CCCC (clamped - clamped - clamped - clamped), CSSS (clamped - simply supported - simply supported - simply supported), CSCS (clamped - simply supported - clamped - simply supported), CCSS (clamped - clamped - simply supported - simply supported), CCCS (clamped - clamped - clamped - simply supported) and SSSS (simply supported all-round)) subjected to biaxial compressive loads. The new specific equations of this work allow for the evaluation of buckling load coefficients across varying aspect ratios, biaxial buckling ratios (n), and deflection-to-thickness ratios (w/t). The results obtained reveal that the biaxial buckling load coefficient (and load) decreases with increasing ‘n’ but increases with an increase in w/t. This highlights the combined influence of loading distribution and geometric nonlinearity. Deducing from the new biaxial equations for the uniaxial loading case for the purpose of comparison in square plates, the large-deflection buckling coefficients compared showed negligible deviation (~0%) from prior works, validating the proposed equations’ accuracy. Comparative analysis on values of buckling and postbuckling loads of CCCS plates against existing work under biaxial loading shows percentage differences between 0% and 5.56%, with the present results consistently upper bound but within acceptable engineering tolerances. The further comparison of biaxial buckling and post-buckling load coefficients of Sthe SSS plate with existing works has also shown the adequacy of this new model. The findings demonstrate that the proposed linear and nonlinear buckling equations not only align with established studies but also offer enhanced predictive capability for large deflection analysis of thin isotropic rectangular plates.

In general, the method used to obtain the tensile strength from a Brazilian test depends on the elastic theory for a pair of concentrated forces. However, the contact areas between the specimen and loading platens are observed in experiments. Several attempts have been tried for the boundary conditions in modeling of an isotropic or orthotropic circular disk in the Brazilian test. The forces in only the normal direction to the loading platen or specimen have been dealt with in most of these models. Therefore, a dent in the contact area between specimen and loading platens is observed in numerical results from most models. However, no dents have ever been observed in experiments using loading platens. In this study, not only normal loads but also shear loads on the loading platens are adopted as boundary conditions. The depth of the resulting dent from the present model is smaller than those seen in previous studies. In addition, huge tensile stresses are observed near both edges of the contact area in the present model. Furthermore, cracks are also observed near both edges in experiments. In brittle materials, cracks are caused by tensile stress. Therefore, the accuracy of calculations is expected to be improved using the present model for Brazilian tests.

This study assesses the one-dimensional (1D) consolidation of unsaturated soil subjected to time-dependent loading based on the 1D consolidation theories of unsaturated soil. The differential quadrature method (DQM) is used to produce a general solution that considers various boundary conditions, various initial pore-water and pore-air distributions, and complex time-dependent loading. A special case in which the analytical solution is available in the literature is used for verification and accuracy analysis. It is found that the DQM solution can deliver more accurate results compared with the finite-difference method with a small number of sampling points. The general solution can avoid cumbersome computations in solving eigenequations encountered with the analytical solution. In addition, the proposed solution is more suitable for practical engineering because of its generality in complex initial, boundary, and loading conditions. Finally, the characteristics of the 1D consolidation of unsaturated soils under various initial pore pressure distributions, boundary conditions, and complex time-dependent loading conditions are investigated, and it is found that the initial and boundary conditions have a significant influence on the consolidation of unsaturated soils.

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

A new global spatial discretization method is developed to accurately calculate natural frequencies and dynamic responses of two-dimensional continuous systems such as membranes and Kirchhoff plates. The transverse displacement of a two-dimensional continuous system is separated into a two-dimensional internal term and a two-dimensional boundary-induced term; the latter is interpolated from one-dimensional boundary functions that are further divided into one-dimensional internal terms and one-dimensional boundary-induced terms. The two- and one-dimensional internal terms are chosen to satisfy predetermined boundary conditions, and the two- and one-dimensional boundary-induced terms use additional degrees of freedom at boundaries to ensure satisfaction of all boundary conditions. A general formulation of the method that can achieve uniform convergence is established for a two-dimensional continuous system with an arbitrary domain shape and arbitrary boundary conditions, and it is elaborated in detail for a general rectangular Kirchhoff plate. An example of a rectangular Kirchhoff plate that has three simply-supported boundaries and one free boundary with an attached Euler-Bernoulli beam is investigated using the developed method and results are compared with those from other global and local spatial discretization methods. Natural frequencies and dynamic responses that include the displacement, the velocity, rotational angles, a bending moment, and a transverse shearing force are calculated using both the developed method and the assumed modes method, and compared with results from the finite element method and the finite difference method, respectively. Advantages of the new method over local spatial discretization methods are much fewer degrees of freedom and much less computational effort, and those over the assumed modes method are better numerical property, a faster calculation speed, and much higher accuracy in calculation of the bending moment and the transverse shearing force that are related to high-order spatial derivatives of the displacement of the plate with an edge beam.

Summary The need to improve completion reliability has prompted a second, deeper look at the analysis of tubing buckling. The conventional-buckling analysis is assumed to hold "far from the packer", but what happens "near" the packer? The original development by Lubinski and Woods used the method of virtual work to determine a constant pitch helix. All attempts to connect a constant pitch helix to boundary conditions, such as a packer, have failed. The conventional wisdom says that the virtual work solution applies "far from the packer". How far is this, and what happens between the packer and this point? Even worse is the case of tapered strings. The generally accepted answer to the tapered string problem is a "stacked" set of Lubinski solutions, with each solution far from the interconnection of the different sized tubulars (1962). Perhaps there are nonconstant pitch solutions? Because the buckling-differential Eq. is nonlinear, it is not surprising that no other analytic solutions have been discovered until recently. Once these solutions are known, the initial value problem can be posed and solved. Two possible boundary conditions are considered in this study: a cantilever boundary condition to model a packer, and a "pinned" boundary condition to model a centralizer. For the first time, analytic solutions were found for both problems, and interestingly, both solutions converged rapidly to Lubinski's constant pitch solution at a specified distance from the boundary. For this reason, many of the original Lubinski calculations remain essentially correct, (such as length change and contact forces). Alternatively, the bending moment near the boundary exceeds the bending moment calculated from the Lubinski solution by approximately 20%. For this reason, conventional pipe-buckling stress analysis must be revised. Further applications of these solutions allow different sized tubulars to be interconnected, solving the tapered-string problem for the first time. An additional, useful result is the solution to the helix "reversal" problem. The principal applications for these solutions are the stress analysis and design of tubing and casing strings. The boundary solutions are simple formulas, suitable for spreadsheet or hand calculations.

The most fundamental question about helically buckled pipe is "What is the pitch?" The original development by Lubinski and Woods used the method of virtual work to determine a specific constant pitch. All attempts to connect a constant pitch helix to boundary conditions, such as a packer, have failed. The conventional wisdom says that the virtual work solution applies "far from the packer". How "far" is this, and what happens between the packer and this point? Even worse is the case of tapered strings. The generally accepted answer to the tapered string problem is a "stacked" set of Lubinski solutions, with each solution "far" from the interconnection of the different sized tubulars. Perhaps there are non-constant pitch solutions? Since the buckling differential equation is non-linear, it is not surprising that no other analytic solutions have been discovered until recently. Once these solutions are known, the initial value problem can be posed and solved. Two possible boundary conditions are considered in this study: a cantilever boundary condition to model a packer, and a "pinned" boundary condition to model a centralizer. For the first time, analytic solutions were found for both problems, and interestingly, both solutions converged rapidly to Lubinski's constant pitch solution at a specified distance from the boundary. For this reason, many of the original Lubinski calculations remain essentially correct, such as length change and contact forces. On the other hand, the bending moment near the boundary exceeds the bending moment calculated from the Lubinski solution by as much as 20%. For this reason alone, conventional pipe buckling stress analysis must be revised. Further applications of these solutions allow different sized tubulars to be interconnected, solving the tapered string problem for the first time. An additional useful result is the solution to the helix "reversal" problem. The principal applications for these solutions are the stress analysis and design of tubing and casing strings. The boundary solutions are simple formulas, suitable for spreadsheet or hand calculations.

Conformal theory of the two-dimensional O( N) model with ordinary, extraordinary, and special boundary conditions

For the first time, an accurate analytical solution, based on coupled three-dimensional (3D) piezoelasticity equations, is presented for free vibration analysis of the angle-ply elastic and piezoelectric flat laminated panels under arbitrary boundary conditions. The present analytical solution is applicable to composite, sandwich and hybrid panels having arbitrary angle-ply lay-up, material properties, and boundary conditions. The modified Hamiltons principle approach has been applied to derive the weak form of governing equations where stresses, displacements, electric potential, and electric displacement field variables are considered as primary variables. Thereafter, multi-term multi-field extended Kantorovich approach (MMEKM) is employed to transform the governing equation into two sets of algebraic-ordinary differential equations (ODEs), one along in-plane (x) and other along the thickness (z) direction, respectively. These ODEs are solved in closed-form manner, which ensures the same order of accuracy for all the variables (stresses, displacements, and electric variables) by satisfying the boundary and continuity equations in exact manners. A robust algorithm is developed for extracting the natural frequencies and mode shapes. The numerical results are reported for various configurations such as elastic panels, sandwich panels and piezoelectric panels under different sets of boundary conditions. The effect of ply-angle and thickness to span ratio (s) on the dynamic behavior of the panels are also investigated. The presented 3D analytical solution will be helpful in the assessment of various 1D theories and numerical methods.

Many disciplines in science and engineering, heat transfer plays a vital role and the problems associated with heat transfer are of great importance. Generally, two-dimensional steady state conduction is governed and transformed into a second order partial differential equation (PDE). Satisfying the differential equation along with four boundary conditions is essential for a solution to be valid. There are analytical solutions available, but only for simple boundary conditions and these are not suitable for complex boundary conditions. A technique for solving partial differential equations is Method of Lines (MOL), in which one dimension is discretized. In this study an analytical approach to a two-dimensional slab with steady state heat conduction under different types of boundary conditions is considered. The complete description of heat flow through slab using MOL is presented and the obtained PDEs are solved. Temperature distribution profiles in the slab by using MOL were plotted and compared with the profiles of analytical solutions by using MATLAB. The steady state analysis of temperature distribution in a slab with specified Dirichlet boundary conditions and Neumann boundary conditions was developed by using Method of Lines. From the profiles it is observed that as the number of lines increases, the error between analytical and MOL (semi analytical) solutions decreases, and the profiles of analytical and MOL converged which indicates the preference of higher number of lines in order to obtain accurate values.

The use of solar energy has limitations due to its periodic availability: solar plants do not operate at night and are ineffective in dull weather. The solution of this problem involves the introduction of energy storage and duplication systems into the conversion loop. Among the energy storage systems, solid–liquid phase transition modules have significant energy, ecologic, and cost advantages. Physical processes in modules of this type are described by a system of non-stationary nonlinear partial differential equations with specific boundary conditions at the phase interface. The verification of a method for solving the Stefan problem for a heat-storage material is presented in this paper. The use of the mushy layer method made it possible to simplify the classical mathematical model of the Stefan problem by reducing it to a nonstationary heat conduction problem with an implicit heat source that takes into account the latent heat of transition. The phase transition is considered to occur in an intermediate zone determined by the solidus and liquidus temperatures rather than in in infinite region. To develop a Python code, use was made of an implicit computational scheme in which the solidus and liquidus temperatures remain constant and are determined in the course of numerical experiments. The physical model chosen for computer simulation and algorithm verification is the process of ice layer formation on a water surface at a constant ambient temperature. The numerical results obtained allow one to determine the temperature fields in the solid and the liquid phase and the position of the phase interface and calculate its advance speed. The algorithm developed was verified by analyzing the classical analytical solution of the Stefan problem for the one-dimensional case at a constant advance speed of the phase interface. The value of the verification coefficient was determined from a numerical solution of a nonlinear equation with the use of special built-in Python functions. Substituting the data for the physical model under consideration into the analytical solution and comparing them with the numerical simulation data obtained with the use of the mushy layer method shows that the results are in close agreement, thus demonstrating the correctness of the computer algorithm developed. These studies will allow one to adapt the Python code developed on the basis of the mushy layer method to the calculation of heat storage systems with a solid-liquid phase transition with account for the features of their geometry, the temperature level, and actual boundary conditions.