Analytical Solutions For Bending Research Articles

Analytical and isogeometric solutions of flexoelectric microbeams based on a layerwise beam theory

The theory of layerwise beam, which considers both microstructure and flexoelectric effects, is used in the curvature-based flexoelectricity theory. The displacement field function of the layerwise beam model is given. The governing equations and boundary conditions are obtained using a variational formulation based on Hamilton's principle. Analytical solutions for static bending and free vibration of a simply supported layerwise beam are derived. Isogeometric analysis (IGA) with higher-order continuity considering the flexoelectric effect is presented to numerically solve the static bending and free vibration problems. The results obtained from the analytical and IGA approaches are compared through several examples in which the microstructure and flexoelectric effects on the mechanical responses of layerwise nanobeams are analyzed.

The analytical bending solutions of orthotropic rectangular plates with four clamped edges by the symplectic superposition method

The analytical bending solutions of orthotropic rectangular plates with four clamped edges by the symplectic superposition method

Analytical solutions for bending, buckling and vibration of nanobeams on Winkler–Pasternak foundation

The nonlocal elasticity theory and the Euler–Bernoulli (EB) beam theory are used to present closed-form analytical expressions for static bending, axial buckling, and free vibration of nanosized beams resting on an elastic foundation. The differential constitutive relations of Eringen are utilized to represent the small-scale effects of the nanobeam’s mechanical response. The governing equation of motion is derived by employing Hamilton’s principle. Utilizing the Laplace transform approach, analytical expressions of the bending displacements, the critical buckling force, and the vibration frequency of nanobeams with simply supported (S-S), clamped, cantilevered, and propped cantilevered boundary conditions are produced. In order to confirm the correctness of the offered closed-form equations, their outputs are compared to those of the available numerical method solutions. The effects of the Winkler parameter, the Pasternak parameter and the nonlocal parameter on bending, buckling, and vibration characteristics of nanobeams have been explained. Presented analytical expressions and graphical representations demonstrate how increasing Winkler and Pasternak parameters reduce bending displacements while raising the critical buckling load and the natural frequency of nonlocal nanobeams. Benchmark numerical results are also presented to investigate and discuss the effects of all parameters on bending deflections, buckling loads, and natural frequencies of nanobeams.

Analytical Solution for Bending and Free Vibrations of an Orthotropic Nanoplate based on the New Modified Couple Stress Theory and the Third-order Plate Theory

In the present work, the equations of motion of a thin orthotropic nanoplate were obtained based on the new modified couple stress theory and the third-order shear deformation plate theory. The nanoplate was considered as a size-dependent orthotropic plate. The governing equations were derived using the dynamic version of Hamilton’s principle and natural boundary conditions were formulated. An analytical solution in the form of a double Fourier series was obtained for a simply supported rectangular nanoplate. The eigenvalue problem was set and solved. It was analytically shown that the displacements of the median surface points in the plane of the plate do not depend on the material length scale parameters in the same directions; these in-plane directional displacements depend on the material length scale parameter in the out-of-plane direction only. On the other hand, the out-of-plane directional displacement depends on the length scale parameter in the plane directions only. The cross-section rotation angles depend on all length scale parameters. It was shown that the size-dependent parameters only have a noticeable effect on the deformed state of the plate if their order is not less than the order (plate height)-1.

Theories and Analysis of Functionally Graded Beams

This is a review paper containing the governing equations and analytical solutions of the classical and shear deformation theories of functionally graded straight beams. The classical, first-order, and third-order shear deformation theories account for through-thickness variation of two-constituent functionally graded material, modified couple stress (i.e., strain gradient), and the von Kármán nonlinearity. Analytical solutions for bending of the linear theories, some of which are not readily available in the literature, are included to show the influence of the material variation, boundary conditions, and loads.

Theories and analyses of functionally graded circular plates

This paper presents the governing equations and analytical solutions of the classical and shear deformation theories of functionally graded axisymmetric circular plates. The classical, first-order, and third-order shear deformation theories are presented, accounting for through-thickness variation of two-constituent functionally graded material, modified couple stress effect, and the von Kármán nonlinearity. Analytical solutions for bending of the linear theories, some of which are not readily available in the literature, are included to show the influence of the material variation, boundary conditions, and loads.

Closed-form expressions for bending and buckling of functionally graded nanobeams by the Laplace transform

This paper presents analytical solutions for bending and buckling of nonlocal functionally graded (FG) Euler–Bernoulli (EB) nanobeams. Material gradation along the thickness direction could be defined by a power function (P-FG), a sigmoidal function (S-FG), and an exponential function (E-FG). Laplace transform is applied to the differential form of the equation of motion of the nonlocal elasticity theory. Closed-form expressions for bending deflection and critical buckling load of FG nanobeams are derived. Effects of material gradations as well as the nonlocal parameter are examined. It is found that bending displacements and critical buckling loads could be controlled by an appropriate choice of material distribution parameter for P-FG nanobeams. The presented results also demonstrate the influences of factors such as the choice of material gradation, power-law index, and nonlocal parameter on bending and buckling behavior.

Elasticity Solution for Bending and Frequency Behavior of Sandwich Cylindrical Shell with FG-CNTRC Face-Sheets and Polymer Core Under Initial Stresses

This paper explores the high-accuracy analysis of bending and frequency response of the sandwich cylindrical shell with functionally graded (FG) carbon nanotubes reinforced composite (FG-CNTRC) face-sheets and polymeric core under the effect of initial axial stress and various mechanical loading based upon the three-dimensional theory of elasticity for various sets of boundary conditions. The sandwich structure is composed of multilayers with uniformly dispersed carbon nanotubes (CNT) in each fictitious layer of face-sheets, but its weight fraction changes layer-by-layer along the thickness direction. With the aid of compatibility conditions, the sandwich structure with three layers is modeled. Analytical bending and frequency solutions are obtained for simply supported shells. Also, the state-space based differential quadrature method (SS-DQM) is employed to determine the bending and frequency response of the sandwich cylindrical shell by considering various boundary conditions. The bending response of the sandwich cylindrical shell is obtained under the impact of various mechanical loadings. The influences of several parameters, such as initial stress and various mechanical loadings are investigated on the bending and frequency of the structures. The results of the presented study can be served as benchmarks to assess the validity of the conventional two-dimensional theory.

New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method

New analytic bending, buckling, and free vibration solutions of rectangular nanoplates with combinations of clamped and simply supported edges are obtained by an up-to-date symplectic superposition method. The problems are reformulated in the Hamiltonian system and symplectic space, where the mathematical solution framework involves the construction of symplectic eigenvalue problems and symplectic eigen expansion. The analytic symplectic solutions are derived for several elaborated fundamental subproblems, the superposition of which yields the final analytic solutions. Besides Lévy-type solutions, non-Lévy-type solutions are also obtained, which cannot be achieved by conventional analytic methods. Comprehensive numerical results can provide benchmarks for other solution methods.

Analytical bending solutions of thin plates by two‐dimensional generalized integral transform method

AbstractIn this paper, analytical bending solution of a rectangular thin plate with different boundary conditions was obtained by using the two‐dimensional generalized finite integral transform method. During the solution procedure, the vibrating beam functions satisfying different boundary conditions of the plate are taken as integral kernels to form the integral transform pairs. This two‐dimensional integral transform is used to the partial differential governing equations of the plate, to transform them to a system of linear algebraic equations from which the accurate analytical solution is obtained easily. The superiority of the present method is that it does not need to seek the deformation function in advance, and avoids the complex superposition processes. Thus, the proposed method is reasonable and feasible. The good agreement of present results and the analytical solution from the literature shows the validity of the method.

A Levy solution for bending, buckling, and vibration of Mindlin micro plates with a modified couple stress theory

Based on the modified couple stress and first-order shear deformation plate theories, a Levy-type solution is presented for bending, buckling, and vibration analyses of rectangular isotropic micro plates with simple supports at opposite edges and different boundary conditions at the other two ones. The governing equations are derived using the Hamilton's principle, and then solved by a single Fourier series expansion and the state-space method, which its implementation has not been straightforward. The results are verified with the existing ones for a fully simply supported micro plate in the literature. Finally, the effect of geometric parameters and length scale parameter on bending, buckling, and vibration behaviors of micro plates is studied. Since, there are no analytical solutions for bending, buckling loads, and natural frequencies of Mindlin micro plates with different boundary conditions in the literature and the Navier method is the only available analytical solution for the Mindlin and higher order shear deformable micro plates within the modified couple stress theory, the results presented here can be used as a benchmark in future studies. In addition, it is shown that the difference between results of the Kirchhoff and the Mindlin plate models depends not only on the plate thickness but also on the length scale parameter to thickness ratio (\(l/h)\) as well as the boundary supports. This result emphasizes the significance of analytical solutions for shear deformation models of micro plates with different boundary supports.

On the symplectic superposition method for new analytic bending, buckling, and free vibration solutions of rectangular nanoplates with all edges free

This paper extends an up-to-date symplectic superposition method for analytic bending, buckling, and free vibration solutions of rectangular nanoplates based on the nonlocal theory. By transferring the problems to the Hamiltonian system expression, the symplectic approach becomes applicable, which involves generation of symplectic eigenvalue problems and symplectic eigen expansion. An original problem is then converted into superposition of several elaborated subproblems that are solved by the symplectic approach. The analytic bending deflection, natural frequency, buckling load, and associated mode shape solutions are obtained by the equivalence between the original problem and the superposition. As a representative class of non-Levy-type nanoplates that were not solved by conventional analytic methods, the nanoplates with all edges free are focused on the rapid convergence of the present solutions confirming comprehensive results tabulated with the accuracy of five significant figures, which can serve as benchmarks for future comparison. Quantitative parameter analyses are performed with the analytic solutions to investigate the effects of the nonlocal parameter and plate dimension on the mechanical behaviors of free nanoplates. The present method has merits on tackling complex boundary-value problems of higher-order partial differential equations represented by those for nanoplates and may be further developed for more analytic solutions in future studies.

Exact analytical solutions for bending of bi-directional functionally graded nanobeams by the nonlocal beam theory using the Laplace transform

This paper presents a comprehensive analytical method to solve the fourth-order differential equation relevant to bending analysis of bi-directional functionally graded (BDFG) nanobeams obtained by the Eringen's nonlocal elasticity theory. Unlike usual numerical solutions (e.g., by following the Differential Quadrature method), analytical solutions following the Laplace transform are adopted. The material properties of the nanobeam vary along both the thickness and the axial directions by following the power law and the exponential law, respectively. Explicit equations are presented for bending displacements of BDFG nanobeams under various distributions of applied loads in combination with a few common boundary conditions. The effects of material inhomogeneity parameters along both the axial and the thickness directions, as well as that of the nonlocal parameter, are investigated. Comparisons with a few available results in the literature demonstrate the accuracy and unravel the versatility of the presented analytical method.

Bending and free vibrational analysis of bi-directional functionally graded beams with circular cross-section

The bending and free vibrational behaviors of functionally graded (FG) cylindrical beams with radially and axially varying material inhomogeneities are investigated. Based on a high-order cylindrical beam model, where the shear deformation and rotary inertia are both considered, the two coupled governing differential motion equations for the deflection and rotation are established. The analytical bending solutions for various boundary conditions are derived. In the vibrational analysis of FG cylindrical beams, the two governing equations are firstly changed to a single equation by means of an auxiliary function, and then the vibration mode is expanded into shifted Chebyshev polynomials. Numerical examples are given to investigate the effects of the material gradient indices on the deflections, the stress distributions, and the eigenfrequencies of the cylindrical beams, respectively. By comparing the obtained numerical results with those obtained by the three-dimensional (3D) elasticity theory and the Timoshenko beam theory, the effectiveness of the present approach is verified.

Analytical bending solutions of thin plates with two adjacent edges free and the others clamped or simply supported using finite integral transform method

The finite integral transform method is developed to explore the bending analysis of thin plates with the combination of simply supported, clamped, and free boundary conditions. Previous solutions mostly focused on simply supported and clamped boundary conditions, but the existence of free boundary conditions makes the solving process more complex, because it is difficult to find the exact solution which satisfies both deflection and internal force by conventional inverse/semi-inverse method or approximate method. Using this method, the plate high-order partial differential equation is simplified to a linear algebraic equation by the integral transformation. Then, through some mathematical manipulation, the analytical solution is elegantly achieved in a straightforward procedure. Compared with other methods, the present method is much simpler and general and does not need to pre-determine the deflection function, which makes it very attractive for calculating the mechanical responses of the plates. Comprehensive analytical results obtained in this paper illuminate the validity of the proposed method by comparison with the existing literature and finite-element method using (ABAQUS) software.

Analytical Bending Solutions of Orthotropic Rectangular Thin Plates with Two Adjacent Edges Free and the Others Clamped or Simply Supported Using Finite Integral Transform Method

For the first time, the finite integral transform method is introduced to explore the accurate bending analysis of orthotropic rectangular thin plates with two adjacent edges free and the others clamped or simply supported. Previous solutions mostly focused on plates with simply supported and clamped edges, but the existence of free corner makes the solution procedure much complex to solve by conventional inverse/semi-inverse methods. Compared with the conventional methods, the employed method eliminates the need to preselect the deflection function, which makes it more reasonable and theoretical for calculating the mechanical responses of the plates. Moreover, the approach used can also analyze static problems of moderately thick plates and thick plates with the same boundary conditions investigated in this article. Finally, comprehensive analytical results obtained in this paper illuminate the validity of the proposed approach by comparing with the previous literature and finite element method by using (ABAQUS) software.

Finite integral transform method for analytical solutions of static problems of cylindrical shell panels

In this paper, a double finite integral transform method is developed for analytical bending solutions of non-Lévy-type cylindrical shell panels without a free edge that were not obtained by classical semi-inverse methods. Three double finite integral transforms are imposed on the governing high-order partial differential equations, which, with some boundary conditions, yields the relationship between the transformed quantities and specific unknowns. Incorporating the inversions into the remaining boundary conditions leads to systems of linear algebraic equations, which determine the final analytical solutions. Comprehensive benchmark results for representative cylindrical shell panels with combinations of clamped and simply supported edges are presented, which are well validated by satisfactory agreement with other solution methods. Due to its rigorous and straightforward solution procedure, the developed method provides a solid easy-to-implement approach for exploring new analytical solutions.

Exact solution for bending analysis of two-directional functionally graded Timoshenko beams

This paper studies the bending behavior of two-dimensional functionally graded (TDFG) beam based on the Timoshenko beam theory, where the material properties of the beam vary both in the length and thickness directions. By introducing a new auxiliary function, we simplify the coupled governing equations for the deflection and rotation to a single governing equation. Moreover, all physical quantities of interest can be expressed in terms of the auxiliary function. Then, the exact analytical solutions for bending of TDFG Timoshenko beams are derived for various boundary conditions. The influence of gradient indexes on the deflection and stress distribution of TDFG Timoshenko beams is discussed subjected to different transverse loadings, including uniformly distributed loading, linearly distributed loading and concentrated external loading. The introduced approach is of benefit to exact bending analysis of TDFG beams by employing other beam theories.

Analytical Solutions for Bending of Nanoscaled Bars Based on Eringen’s Nonlocal Differential Law

The one-dimensional nanoscaled bar is commonly seen in current nanodevices, and it plays a significant role in nanoelectromechanical system (NEMS) and related nanotechnology. Motivated by this, the paper is concerned with the bending and stability of a nanoscaled bar especially a flexible bar subjected to vertical concentrated, vertical linearly distributed, and horizontal concentrated forces simultaneously. The theoretical model is developed by employing Eringen’s nonlocal differential law. Hence, the nonlocal differential constitutions in terms of stress and bending moment at a nanoscale are applied to the classical equilibrium formulation in order to construct the nonlocal constitutive model and then determine the analytical solutions for bending of such a nanoscale bar. Subsequently, the effect of a nonlocal scale on the midpoint deflections and internal forces is shown numerically, and the internal and external characteristic length scales are also examined. It is revealed that the capacity of resisting compression decreases with increasing the nonlocal effect since the internal characteristic scale cannot be neglected by comparing with the external characteristic scale. There exists a threshold value for bending stiffness. Both the midpoint deflection and bending moment vary monotonously when the material bending stiffness exceeds the threshold value, while the mechanical quantities fluctuate when the bending stiffness is less than its threshold value, namely, for a flexible nanoscaled bar. Accordingly, two different kinds of nonlocal predictions and the related disputes are resolved in the context of Eringen’s nonlocal differential law. The work is expected to be useful for the design, application, and optimization of nanoscaled bars sustaining bending with linear vertical and horizontal forces.

An analytical solution for bending of transversely isotropic thick rectangular plates with variable thickness

In this study, the bending solution of simply supported transversely isotropic thick rectangular plates with thickness variations is provided using displacement potential functions. To achieve this purpose, governing partial differential equations in terms of displacements are obtained as the quadratic and fourth order. Then, the governing equations are solved using the separation of variables method satisfying exact boundary conditions. The advantage of the purposed method is that there is no limitation on the thickness of the plate or the way the plate thickness is being varied. No simplifying assumption in the analysis process leads to the applicability and reliability of the present method to plates with any arbitrarily chosen thickness. In order to confirm the accuracy of the proposed solution, the obtained results are compared with existing published analytical works for thin variable thickness and thick constant thickness plate. Also, due to the lack of analytical research on thick plates with variable thickness, the obtained results are verified using the finite element method which shows excellent agreement. The results show that the maximum displacement of the plates with variable thickness is moved from the center toward the thinner plate edge. In addition, results exhibit the profound effects of both thickness and aspect ratio on stress distribution along the thickness of the plate. Results also show that varying thickness has not a profound impact on bending and twisting moments in transversely isotropic plates. Five different materials consist of four transversely isotropic and one isotropic, as a special case, are considered in this paper, which it is shown that the material properties have a more considerable impact on higher thickness plate.