Bending Solutions Research Articles

On simplified deformation gradient theory of modified gradient elastic Kirchhoff–Love plate

A concise modified gradient elastic Kirchhoff–Love plate model with two length-scale parameters is proposed based on simplified deformation gradient theory. A sixth-order basic differential equation and boundary conditions applicable to arbitrary shapes are obtained by applying the principle of minimum potential energy and combining with the generalized strain energy related to classical strain, strain gradient and rotation gradient. By introducing couple stress, the classical bending stiffness corresponding to the fourth-order terms and force-related boundary conditions in the typical gradient elastic Kirchhoff–Love plate model are modified, and the bending deformation of thin plates can be described more flexibly. Under certain conditions, the modified gradient elastic Kirchhoff–Love plate model can be reduced to typical gradient elastic thin plate model, couple stress thin plate model and classical Kirchhoff–Love plate model. The bending boundary value problems of gradient elastic thin plates are further studied, and the specific modified classical boundary conditions and non-classical higher-order boundary condition acceptable to the sixth-order basic equation in Cartesian coordinates are derived. The analytical and numerical bending solutions to gradient Navier-type and Levy-type thin plates with various boundary conditions, including simply supported, clamped and free boundaries are presented, and the size-dependent bending stiffness that is jointly determined by geometric dimensions and length-scale parameters is defined to describe the size effect of thin plates comprehensively.

Arbitrary temperature gradient in thermal-postbuckling behavior of polymer nanocomposites reinforced by boron nitride nanotubes

Polymer nanocomposites reinforced by boron nitride nanotubes (BNNTs) have great potential for use in a wide variety of thermal environments. A closed-form solution for nonlinear bending and thermal-postbuckling behavior of BNNT-reinforced nanocomposite beams under arbitrary temperature gradient in the thickness and length directions is presented for the first time. The uniform and functionally graded (FG) distributions of BNNTs through the thickness of macro-or micro-sized beams are considered. For constant temperature or constant heat flux conditions at two ends of the beam, thermal analysis is carried out to obtain temperature field ΔTx,z using the two-dimensional heat conduction equation and the assumption of arbitrary variation in the z direction. The governing equations of modified couple stress Timoshenko beam theory with von Kármán geometric nonlinearity are derived for BNNT-RC beams subjected to thermomechanical loads. Applying change of variable technique, the governing equations are exactly solved to develop closed-form expression for nonlinear transverse deflection. Finally, parametric study is performed to investigate the effects of length scale parameter, aspect ratio (L/h), distribution of BNNTs, and temperature gradient on the nonlinear bending and postbuckling equilibrium paths. It is shown that the thermal-buckling behavior of clamped and cantilever beams is completely different. Under thermomechanical loading, an initial snap-through buckling behavior is observed in the cantilever beam, while there exist both stable and unstable configurations in the static equilibrium path of clamped beam. Moreover, it is found that the assumption of parabolic variation of temperature through thickness is not compatible with real situations, since the temperature field is obtained as harmonic functions with very short wavelengths in the length direction.

A wavelet method for large-deflection bending of irregular plates

A very rare sixth-order multiresolution method based on wavelet theory for solving irregular domain problems is proposed and applied to the large-deflection bending of irregular plates. High accuracy approximation of multiple integrals of functions defined in the irregular domain is realized by suggesting a new method of boundary extension of functions, which is associated with a boundary node removal technique to avoid singular matrices in the integration process. Each derivative of the unknown function is defined as a new function in the solution of the governing differential equations of the problem. The original equations containing derivatives of different orders can then be transformed into a system of algebraic equations with only discrete nodal values of the highest-order derivative, based on integral relations between these functions and excellent accuracy features on the wavelet approximation of multiple integrals. Various boundary conditions for irregular domains can be automatically and almost exactly included in the integration operation. The proposed method is applied to the solution of large-deflection bending of various irregular plates. The displacement and stress results demonstrate that the accuracy of the proposed method is still as high as the rare sixth order even when the problems are strongly nonlinear and defined in irregular regions, showing a much better accuracy and efficiency than those of classical methods such as the finite element method.

A Legendre-Ritz solution for bending, buckling and free vibration behaviours of porous beams resting on the elastic foundation

By using high-order deformation theory, this paper presents a novel approach for analysing the mechanical behaviours of functionally graded porous beams resting on a Winkler-Pasternak elastic foundation. The governing equations are derived from the Lagrange principle. Legendre-Ritz polynomial functions, which possess the simplicity of algebraic polynomials and the efficiency of orthogonal polynomials, are developed to solve the problem. Three boundary conditions accompanied by two porosity types, including symmetric and asymmetric distributions of beams, are considered. The effects of porosity, boundary condition, foundation parameter, span-to-height ratio, and distribution type on the mechanical behaviours of porous beams are investigated. Some new results are presented for the first time as a benchmark for future studies.

Classical and homogenized expressions for the bending solutions of FGM plates based on the four variable plate theories

Exact correspondence relations between the bending solutions of a simply supported rectangular functionally graded material plate based on the four variable refined higher-order shear deformation theory and those of the corresponding reference homogenous Kirchhoff plate based on the classical plate theory are derived analytically for the material properties varying continuously in the thickness direction. The deflection, stress components, the resultant forces and bending moments of a thick functionally graded material plate are expressed analytically in terms of the deflection of the reference homogenous Kirchhoff plate with the same geometry, loadings and boundary constraints. Consequently, the bending solution of a functionally graded material plate based on the higher-order shear deformation theory is simplified as calculations of three scaling factors which can be easily determined analytically for the specified material gradient profile, the shear stress shape function and the aspect ratio of the functionally graded material plate, because the solution of the reference homogenous Kirchhoff plate can be easily found even in the text book. As examples, particular solutions for a functionally graded material plate subjected to both uniformly and sinusoidally distributed loads are presented, which illustrate the validity of this new approach. Accuracy of the present solutions are demonstrated by comparing them with those obtained by different palate theories with different shear stress shape functions available in the literature. The analytical solutions can be used as benchmarks to check numerical solutions of static bending of functionally graded material plates based on different higher-order shear deformation theories.

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

Highly Accurate Wavelet Solution for Bending and Free Vibration of Circular Plates Over Extra-Wide Ranges of Deflections

Abstract The wavelet multiresolution interpolation Galerkin method in which both the unknown functions and nonlinear terms are approximated by their respective projections onto the same wavelet space is utilized to implement the spatial discretization of the highly coupled and nonlinear Von Karman equation for thin circular plates with various types of boundary conditions and external loads. Newton’s method and the assumption of a single harmonic response are then used for solving the static bending and free vibration problems, respectively. Highly accurate wavelet solutions for an extremely wide range of deflections are finally obtained by the proposed method. These results for moderately large deflections are in good agreement with existing solutions. Meanwhile, the other results for larger deflections are rarely achieved by using other methods. Comparative studies also demonstrate that the present wavelet method has higher accuracy and lower computational cost than many existing methods for solving geometrically nonlinear problems of thin circular plates. Moreover, the solutions for large deflection problems with concentrated load support the satisfactory capacity for handling singularity of the proposed wavelet method. In addition, a trivial initial guess, such as zero, can always lead to a convergent solution in very few iterations, even when the deflection is as large as over 46 times thickness of plate, showing an excellent convergence and stability of the present wavelet method in solving highly nonlinear problems.

Numerical Solution for the Mechanical Bending of Two Elastically Coupled Plates Problem

This paper uses the finite difference method (FDM) to provide a high-accuracy numerical approach to solve for the governing equations of two elastically coupled plates under symmetrical bending. The solution strategy, including governing equations and boundary conditions, is thoroughly discussed, and presented. Finally, this solution was used to test the effect of the key parameters of the coupled plates structure on the mechanical response of the system and relate that to the reliability performance of electronic packages under mechanical bending.

2‐D Analytical solutions for the multisegmented panel subjected to arbitrary boundary condition

AbstractA closed‐form bending solution of the multisegmented panels is presented for different support conditions. The panels can have multiple no. of segments along the x‐axis with varying material properties. The governing equations are formulated using the extended Kantorovich method (EKM) in the mixed‐form, and a set of ordinary differential equations have been obtained. The continuity of displacement and stresses are satisfied at the interface of each segment. The panels can have simply‐support, clamp, and free types of support conditions. The panel interface at each segment along the x‐axis is assumed to be perfectly bonded. In this work, the stress and deformation variations for four types of panel configuration have been studied. Two segmented panels having aluminum and Graphite–Epoxy with equal and unequal segments are considered. A four‐segmented Al/SiC panel having gradual material variation is also analyzed. The effect of ply‐angles on the longitudinal variables is also studied for the equal segmented Al/GrEp panel. The deflection and stresses are compared with the finite element solution and found in good agreement with the EKM results. The current development will lead to developing the solution for stepped panels, dissimilar plates, and more complex cases used in aerospace applications.

Semi-Analytical Solution for Thermo-Piezoelectric Bending of FG Porous Plates Reinforced with Graphene Platelets

This research is devoted to investigating the thermo-piezoelectric bending of functionally graded (FG) porous piezoelectric plates reinforced with graphene platelets (GPLs). A refined four-variable shear deformation plate theory is utilized considering the transverse shear strain to describe the displacement components. The porous nanocomposite plate is composed of polymer piezoelectric material containing internal pores and reinforced with FG GPLs. In accordance with modified distribution laws, the porosity and GPLs volume fraction are presumed to vary continuously through the plate thickness. Four GPLs and porosity distribution types are presented. By applying the Halpin–Tsai model, the elastic properties of the nanocomposite plate are calculated. The governing equations are derived based on the present theory and the principle of virtual work. The deduced partial differential equations are converted to ordinary equations by employing Levy-type solution. These equations are numerically solved based on the differential quadrature method (DQM). In order to determine the minimum grid points sufficient to gain a converging solution, a convergence study is introduced. Moreover, the accuracy of the present formulations are examined by comparing the obtained results with those published in the literature. Additional parametric analyses are introduced to investigate the influences of the GPLs weight fraction, distribution types, side-to-thickness ratio, external electric voltage and temperature on the thermal bending of FG GPLs porous nanocomposite piezoelectric plates.

A homotopy-based wavelet method for extreme large bending analysis of heterogeneous anisotropic plate with variable thickness on orthotropic foundation

A novel homotopy-based decoupled wavelet numerical strategy is developed to investigate the large-deflection bending of variable-thickness anisotropic thin plate constituted by heterogeneous material resting on orthotropic foundation. The anisotropic plates are described undergoing sinusoidally initial imperfections with inhomogeneous thickness and exponentially varying elastic modules, while the subgrade reaction involves the nonlinear support effects by Winkler-type foundation along with Pasternak-type orthotropy. Geometric nonlinearity by uneven thickness and large deformation, material nonlinearity by heterogeneity and contact nonlinearity of orthotropic foundation are simultaneously considered with the highly coupled and strongly nonlinear variable-coefficient Von Kármán equations derived. The accuracy and convergence of the present formulation are validated in excellent agreement with published results, while the new results of extreme bending solutions are provided, which are few given by other methods and further verify the effectiveness and superiority of proposed wavelet procedures. Parametric investigations of influences of material orthotropy and inhomogeneity, unevenness of thickness, amplitude of initial imperfection along with orthotropy of nonlinear Winkler–Pasternak foundation have been studied, which are of great significance on the extremely large bending properties.

Effect of thickness stretching and electro-mechanical loading on sandwich FGM plate with piezoelectric face sheet with different boundary conditions: A semi-analytical approach

Functionally graded material (FGM) belongs to the class of advanced composite materials with material properties that vary according to the volume fraction. FGM is the future material for smart structures, being used in place of conventional composite materials. The article aims at performing the static analysis of different types of sandwich FGM plates with a piezoelectric face sheet taking into consideration the porosity and thickness stretching effect. The problem is formulated using a variational approach and the bending solution is obtained for the first time using Galerkin Vlasov’s method. Three different types of porous sandwich plates are considered for the analysis when subjected to sinusoidally distributed mechanical and electrical loading. For the first time, a new uneven non-symmetric porosity model is put forward to capture the realistic bending behavior of the plate. The effect of boundary conditions, thickness stretching, and applied voltage on stress and deflection distribution along and across the porous plate’s surface is analyzed. It is observed that effect of the applied potential difference when assumed to be a function of position, the change in stresses and deflection are approximately 30%. This infers that voltage distribution plays a significant role in the distribution of stresses in the plate and cannot be neglected which is otherwise assumed as a constant by various researchers. To check the exactness and validity of the current formulation, various examples are discussed. The estimated results can be used as a validation study for the upcoming research on porous sandwich FGM plates with piezoelectric facesheets. Communicated by Francesco Tornabene

Bent Crystal Design and Characterization for High-Energy Physics Experiments

Bent crystal are widely used as optics for X-rays, but via the phenomenon of planar channeling they may act as waveguide for relativistic charged particles beam as well, outperforming some of the traditional technologies currently employed. A physical description of the phenomenon and the resulting potential for applications in a particle accelerator is reported. The elastic properties of the anisotropic crystal lattice medium are discussed, introducing different types of curvature which can enable a wide array of bending schemes optimized for each different case features. The technological development of machining strategy and bending solutions useful for the fabrication of crystals suitable in high energy particle manipulations are described. As well as the high precision characterization processes developed in order to satisfy the strict requirements for installation in an accelerator. Finally, the characterization of channeling phenomenon in bent crystal is described, pointing out several experimental setups suitable to comply each specific case constrains.

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.

Gradient-based optimisation of rectangular honeycomb core sandwich panels

When subjected to bending loads, sandwich panels are highly efficient structural components with the potential to achieve substantial weight reduction. A successful design methodology for sandwich panels should aim at maximising this potential for weight reduction while considering the various possible failure mode in a simple yet accurate manner. This paper investigates the application of steel sandwich panels as two-way deck systems. Near-optimal designs for all-steel Rectangular Honeycomb Core Sandwich Panels (RHCSPs) under general out-of-plane loading are achieved using a gradient-based optimisation method. The method relies on continuously optimising the design limit state constraints while the response constraints are considered a priori in the analysis stage using simplified analytical assessment. Plate bending solutions and sandwich bending solutions are used as alternatives to estimate the internal stresses on each layer of the sandwich panel under out-of-plane loads, where comparisons are made between these two analysis methods in terms of computational efficiency and accuracy. The internal stresses are then used to formulate design limit state equations for each relevant failure mode, including material yielding, plate buckling and deformation control. The Method of Moving Asymptotes is used for the optimisation of RHCSPs, considering the limit states as the constraints of the optimisation problem and weight as the objective function to be minimised. The proposed methodology for simplified assessment is verified against detailed nonlinear finite element models for optimal design solutions. The implications of the results of the proposed optimisation strategy on the development of a systematic design methodology for RHCSPs are also highlighted, making specific reference to critical failure modes.

Wavelet solution for hygrothermomechanical bending of initially defected plate undergoing large deformation on nonlinear elastic foundation

An efficient wavelet approach has been developed for geometrically nonlinear bending analysis of largely deformed plate with arbitrary initial curvature on nonlinear Winkler–Pasternak foundation loaded with sinusoidally hygrothermal stresses. Translational and rotational constraints have been analyzed and equivalently transformed into the Cauchy-type boundary conditions of Airy stress function, which are efficiently approached by interpolating Coiflet-type wavelet. A newly hygrothermomechanical bending model of plate subjected to the adiabatically impermeable edges or with boundaries of sinusoidal temperature and humidity has been firstly formulated. The highly coupled and nonlinear governing equations have been decomposed into linear differential terms by homotopy transformation and successfully solved by the Coiflet-type wavelet Galerkin method. The wavelet strategy has been validated superior in dealing with extremely nonlinear bending and high-precision reconstitution of initial deflection in hygrothermal environment, while the obtained solutions are in excellent agreement with published results. Effect of initial deflection, hygrothermal load and elastic foundation has been investigated, which implies the hygrothermal effect on plate in linear bending is prominent but gradually negligible in large deformation, while the nonlinear effects of elastic foundation become significant by the increase of deflection but not obvious at smally deformed state.

Nonlinear Bending and Vibration Analysis of Imperfect Functionally Graded Microplate with Porosities Resting on Elastic Foundation Via the Modified Couple Stress Theory

This paper represents the nonlinear bending and free vibration analysis of a simply supported imperfect functionally graded (FG) microplate resting on an elastic foundation based on the modified couple stress theory and the Kirchhoff plate theory (KPT) together with the von-Kármán’s geometrical nonlinearity. The FG microplates with even and uneven distributions of porosities are considered. Analytical solutions for the nonlinear bending and free vibration are obtained. Comparing the obtained results with the published one in the literature shows the accuracy of the current solutions. Numerical examples are further presented to investigate the effects of the material length scale parameter to thickness ratio, the length to thickness ratio, the power-law index and the elastic foundation on the nonlinear bending and free vibration responses of the FG microplate.

Static Electro-Mechanical Response of Axisymmetric One-Dimensional Piezoelectric Quasicrystal Circular Actuator.

The piezoelectric effect of piezoelectric quasicrystalline materials is coexcited by phonon and phason fields. Piezoelectric quasicrystalline materials have excellent properties of both piezoelectric materials and quasicrystalline materials, which are expected to be used as actuators in the fields of aerospace, automotive, and intelligent manufacturing. Based on the three-dimensional elastic theory of piezoelectric quasicrystals, the state space equation for axisymmetric piezoelectric quasicrystal circular plate actuators is derived by using the state space method. Afterwards, the finite Hankel transformation is performed on the state equation, and a system of ordinary differential equations and corresponding boundary conditions are obtained. Finally, the exact solution of axisymmetric bending of one-dimensional hexagonal piezoelectric quasicrystal circular actuators under generalized elastic simply supported boundary conditions is obtained by using the propagator matrix method. Numerical results are given to compare the degradation results in this paper with those in the literature, and present the influences of the thickness-to-span ratio and stacking sequence on the phonon, phason, and electric fields when the surface of the laminated circular actuators is subjected to mechanical load. The exact solution obtained does not introduce any deformation assumption; therefore, the exact solution can provide references for numerical calculations of the mechanical behavior of piezoelectric quasicrystals.

2D closed form solution for bending of edge bonded dissimilar beams: An application of EKM

Dissimilar material like composite laminated materials combined with metals like aluminium/steel have been extensively used to reduce the weight in automotive industries and aerospace industries. Moreover, prosthetic limbs are also designed to have varying material along the length for better suitability. At these interfaces, 3D state of stress will exist. Therefore, accurate estimation of deflection and bending stresses is very important and will lead to better design of structures. The objective of this paper is to develop a closed-form bending solution for edge-bonded beams in two dimensions (2D). Beams can be segmented along x-axis and can have any materials. Using Extended Kantorovich method, governing equations are formulated in mixed form and two sets of ordinary differential equations are obtained. Continuity of displacement and stresses are satisfied at interface of each segment. Two segmented beam having aluminium and Gr/Ep equal and unequal segment are considered. A Four segmented Al/SiC beam having gradual material variation is also analysed. The deflection and stresses are compared with the finite element solution and found in good agreement. The formulation is thoroughly verified by comparing the results with 2D FE. The present development will lead to developed solution for dissimilar plates and more complex cases.

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.