Clamped Edges Research Articles

Accurate Nonlinear Bending Analytical Solutions Setting New Benchmarks for Rectangular Thin Plates with Four Clamped Edges

The von Kármán plate equations have been challenging to solve accurately, especially for two-dimensional problems. This paper presents more accurate nonlinear analytical solutions for rectangular thin plates with four clamped edges, providing a new benchmark beyond Lévy’s solutions. The main features of the present method are that the deflection is expanded by the double series of the classical beam eigenfunctions, the Airy stress function satisfying the geometric deformation compatibility equation corresponds to the nonlinear coupling relationships between the plate deflection and the in-plane force and/or displacement boundary conditions. These solutions can verify existing nonlinear numerical and approximate analytical solutions, offering a robust basis for engineering design. The central deflection error is less than 0.01%, and the central stress error is below 0.6%, exceeding the accuracy of the existing literature.

Three-dimensional elasticity solutions for doubly-curved composite shells by extended differential quadrature method

This paper presents the three-dimensional (3D) stress analysis of doubly-curved composite shells with general boundary conditions using the strong sampling surfaces (SaS) formulation. The SaS method is based on the choice of SaS parallel to the middle surface and located at Chebyshev polynomial nodes to introduce the displacements of these surfaces as unknown functions that leads to a non-conventional shell formulation, in which strain–displacement and stress–strain relationships are represented in terms of SaS displacements. This is due to the use of Lagrange polynomials in approximating displacements, strains and stresses in the thickness direction. The outer surfaces are not included into a set of SaS that makes it possible to uniformly minimize the error used by Lagrange interpolation. The strong SaS formulation, based on direct integration of elasticity equilibrium equations in the thickness direction by the extended differential quadrature (EDQ) method, can be applied efficiently for high-precision calculations of doubly-curved composite shells with clamped and free edges. This is because in the SaS/EDQ formulation, displacements, strains and stresses of SaS are interpolated in a rectangular domain specified in a curvilinear coordinate system using a Chebyshev-Gauss-Lobatto grid and Lagrange polynomials are also used as basis functions. The proposed approach deals with equilibrium equations in terms of SaS stresses, avoiding the integration of second order differential equations in terms of SaS displacements, that greatly simplifies the implementation of the EDQ method.

Designing Moiré Patterns by Shearing.

We analyze the elastic properties, structural effects, and low-energy physics of a sheared nanoribbon placed on top of graphene, which creates a gradually changing moiré pattern. By means of a classical elastic model we derive the strains in the ribbon and we obtain its electronic energy spectrum with a scaled tight-binding model. The size of the sheared region is determined by the balance between elastic and van der Waals energy, and different regimes are identified. Near the clamped edge, moderate strains and small twist angles lead to one-dimensional channels. Near the sheared edge, a long region behaves like magic angle twisted bilayer graphene (TBG), showing a sharp peak in the density of states, mostly isolated from the rest of the spectrum. We also calculate the band topology along the ribbon and we find that it is stable for large intervals of strains and twist angles. Together with the experimental observations, these results show that the sheared nanoribbon geometry is ideal for exploring superconductivity and correlated phases in TBG in the very sought-after regime of ultralow twist angle disorder.

A study on the influence of material gradient index on bending and stress responses of FGM rectangular plates using the Finite Element Method

Functionally graded materials (FGMs) are advanced materials designed to achieve specific property gradients. The unique characteristic of these materials—variations in spatial dimensions—allows for integrating the advantages of different materials within a single component, where a combination of properties, such as mechanical strength, thermal resistance, and others, is needed. This paper utilizes finite element analysis to examine the deflection and stress responses of FGM rectangular plates with different material gradient profiles. Various boundary conditions, including clamped, simply supported, and free edges in different configurations, are considered. The plates are subjected to uniformly distributed, sinusoidally distributed, and concentrated loads. The study investigates the effects of boundary and loading conditions, along with the impact of the material gradient, on the deflections and stress responses of FGM rectangular plates. The results indicate variations in deflection and stress values for different material gradients, under varying boundary and loading conditions.

Experimental and numerical buckling analysis of CNT reinforced polymer composite plates

A combined experimental and numerical buckling analysis is performed for Carbon nanotube-reinforced composite (CNTRC) laminates under uniaxial in-plane compressive load. A finite element simulation program is developed in MATLAB environment and the recorded buckling responses are verified with the experimental results. Closely aligned numerical and experimental results exhibits that inclusion of 0.3% CNT increases buckling strengths of CNTRC composite laminates, regardless of their geometrical parameters. Clamped edge laminates exhibit higher buckling mode shapes, influenced by aspect ratio and edge restraint. Microscopic analysis of CNTRC's fractured surface under compression shows enhanced buckling strength attributed to CNTs’ bridging and pullout effects in the matrix, suppressing crack initiation and propagation. Field Emission Scanning Electronic Microscope (FESEM) is utilized for this investigation.

The effect of different edge conditions on the motion of a submerged elastic disc

The present study examines the effects of various edge conditions of a submerged flexible disc on wave propagation for infinite-depth water within the framework of linear water waves theory. Three types of edge conditions (namely Free edge, simply supported edge, and clamped edge) are taken into consideration for the analysis. The governing boundary value problem (BVP) has been solved by reducing it to a two-dimensional hypersingular integral equation. The notion of modal analysis has been adopted to examine the structural response of the disc on the propagation of waves. Later, the two-dimensional hypersingular integral has been transformed into a second kind one-dimensional Fredholm integral equation by applying Fourier series expansion. Finally, the Nystrom technique based on Gauss–Legendre quadrature nodes is used to obtain an approximate solution of the one-dimensional integral equation. The computed solution is used to evaluate the numerical estimates of the physical quantities such as added mass, damping coefficient, hydrodynamic force, and surface elevation for different edge conditions mentioned earlier.

On maintaining bistability of prestressed laminates after clamping

Bistable composite laminates exhibit a high degree of shape change and stiffness variation between their stable configurations, making them suitable for applications in morphing structures and energy harvesting. However, integration of these laminates into larger systems often imposes different boundary conditions, which can eliminate one of their stable states. Moreover, clamping one or more edges of a rectangular bistable laminate causes a drastic change in its strain energy landscape, indicating a strong interplay between the laminate geometry, boundary conditions, and prestress. In this work, we investigate the effect of clamping on the bistability of rectangular prestressed laminates. An analytical approach is proposed to examine the deflection decay imposed by the boundary condition along the laminate’s length. Different prestress values, laminate dimensions, and material properties are analyzed to establish their effect on the curvature change due to the localized clamp effect. A length criterion is determined to guarantee bistability after clamping the bistable laminate, suggesting the need to utilize complementary techniques to retain the bistable behavior for orthotropic prestressed laminates. Different strategies to counter the clamped edge effect and thereby retain the bistability of these types of laminates are then examined. The proposed analytical model is expanded to consider multi-section composite laminates, showing the role of the symmetric regions in bistability retention. Finally, the results from the model are validated against experiments.

Investigation of the effect of exposed area on blast mitigation response of structures

In this study, the effect of exposed area on the failure response of large plates subjected to large scale explosions are investigated using coupled Eulerian-Lagrangian (CEL) approach. A new failure due to multiple necking between plate center and the clamped edge is reported. The fragment velocity in the plate configuration with square exposed area is higher than the circular exposed area at higher impulses. Further, the square exposed configuration exhibited higher mid-deflection than the circular exposed configuration and this difference increases with the increasing impulse.

Effects of non-uniformity in thickness and volume fraction of nanofillers on the flutter characteristics of nanocomposite cantilever trapezoidal plates

The aim of this work is to investigate the flutter characteristics of nanocomposite cantilever trapezoidal plates with non-uniform thickness enriched with either carbon nanotubes (CNTs), graphene nanoplatelets (GNPs), or graphene oxide powders (GOPs) which are distributed functionally graded (FG) in the axial direction. It is assumed that the thickness of the plate and the volume fraction of the nanofillers vary in one direction from the wider clamped edge of the plate to the outer narrower free one. The modeling of the plate is done using the first-order shear deformation theory (FSDT) and the aerodynamic pressure generated by the aerodynamic pressure is modeled using the linear approximation of the piston theory. The material properties of the plate are calculated using the mixing rule (ROM) and the Halpin–Tsai model. The governing equations and boundary conditions at the clamped and free edges of the plate are derived via Hamilton’s principle. An approximate solution is applied using the differential quadrature method (DQM) to calculate the natural frequencies and the damping ratios of the plate. Numerical examples show that it is possible to find an optimal thickness variation profile that provides the greatest aeroelastic stability. It is concluded that by considering the same value for the mass fractions of the nanofillers, the highest aeroelastic stability can be attained by utilizing the GNPs as the reinforcers. It is found that to attain further improvement in aeroelastic stability, most nanofillers should be distributed near the clamped edge and away from the outer free edge.

Собственные колебания упругой прямоугольной CFCF-пластинки при одноосном растяжении-сжатии ее плоскости

The influence of a uniaxial distributed load in the plane of a thin rectangular plate on the value of the first natural frequency of its oscillations has been studied. We considered a plate, two opposite faces of which are clamped, and the other two are free (CFCF-plate, C-clamped edge, F-free edge). The load is applied to the clamped edges. The main differential equation for the coordinate component of the deflection function and all the boundary conditions of the problem are fulfilled exactly using two hyperbolic-trigonometric series in two coordinates and an additional function depending on one variable x. The problem was reduced to an infinite system of linear algebraic equations with respect to one sequence of uncertain coefficients, containing as parameters the magnitude of the load and the frequency of oscillations. For a number of load values, the natural frequencies of oscillations were found by enumerating frequency values in combination with the method of successive approximations when solving a reduced system of linear algebraic equations. To ensure acceptable accuracy of calculations, the number of terms in the series (the size of the reduced system), the number of iterations and the number of significant digits in the mantissa when calculating non-trivial coefficients of the system were changed. A square plate was considered as an example. Based on the calculation results, a graph was constructed of the dependence of the first natural frequency of oscillations on the magnitude of tension-compression forces, which is a curve close to a parabola. Under Eulerian compressive load, the oscillations stop. The shapes of natural vibrations changed slightly and were similar to the shape of the curved surface of a plate under the action of a uniform transverse load. The purpose of this study is to create an effective algorithm for calculating the natural frequencies of oscillations of a CFCF-plate when the uniaxial tensile-compression load of its clamped faces changes.

NGHIÊN CỨU PHÂN BỐ ỨNG SUẤT TẠI BIÊN NGÀM CỦA VỎ TRỤ FGM CHỊU TẢI TRỌNG TĨNH SỬ DỤNG LÝ THUYẾT BIẾN DẠNG CẮT BẬC CAO QUASI-3D

This article presents the results of investigating FGM closed cylindrical shells using quasi-3D higher-order shear deformation theory. The law of volume distribution of materials according to the power function. In this article, the results of the analysis of the stress state at the clamped boundary area of the FGM cylindrical shell subjected to radial local loads distributed according to different laws. The mathematical model and the calculation program in Maple are compared and verified with the published results. The article has shown that at the clamped boundary edge, the values of stress components increase sharply, and at the same time, the strong influence of the load parameters (the variation law, the size of the load area), the power-law index on the values and distribution of stress components in the boundary area is demonstrated.

An Analytical Solution for the Bending of Anisotropic Rectangular Thin Plates with Elastic Rotation Supports

The mechanical analysis of thin-plate structures is a major challenge in the field of structural engineering, especially when they have nonclassical boundary conditions, such as those encountered in cement concrete road slabs connected by transfer bars. Conventional analytical solutions are usually limited to classical boundary conditions—clamped support, simple support, and free edges—and cannot adequately describe many engineering scenarios. In this study, an analytical solution to the bending problem of an anisotropic thin plate subjected to a pair of edges with free opposing elastic rotational constraints is found using a two-dimensional augmented Fourier series solution method. In the derivation process, the thin-plate problem can be transformed into a problem of solving a system of linear algebraic equations by applying Stoke’s transform method, which greatly reduces the mathematical difficulty of solving the problem. Complex boundary conditions can be optimally handled without the need for large computational resources. The paper addresses the exact analytical solutions for bending problems with multiple combinations of boundary conditions, such as contralateral free–contralateral simple support (SFSF), contralateral free–contralateral solid support–simple support (CFSF), and contralateral free–contralateral clamped support (CFCF). These solutions are realized by employing the Stoke transformation and adjusting the spring parameters in the analyzed solutions. The results of this method are also compared with the finite element method and analytical solutions from the literature, and good agreement is obtained, demonstrating the effectiveness of the method. The significance of the study findings lies in the simplification of complex nonclassical boundary condition problems using a simple and reliable analytical method applicable to a wide range of engineering thin-plate structures.

On the optimal planform of a cantilever unimorph piezoelectric vibrating energy harvester

This work relates to piezoelectric vibrating energy harvesters, that are constructed from a unimorph cantilever with a massive edge block. The dynamic response of the cantilever is considered when it is excited into vibrations at its natural frequency, where its deformation amplitude is maximal. The optimal response of such a harvester is achieved when the amplitude of the axial strain in the piezoelectric layer, is uniform. Practical technological considerations dictate the thickness of the unimorph, but its planform geometry (i.e. the vareation of the width along the cantilever) is a design choice. The optimal planform of such a unimorph cantilever has been the focus of many previous studies, which included extensive simulations and experimental investigations. In these previous studies it was concluded that the optimal planform is a trapeze, where the cantilever tapers from its clamped edge towards the edge block. However, to date, no model with explicit predictive capabilities was proposed. In the present study we derive an analytic expression of the planform, that ensures a uniform axial strain over the top surface of a cantilever unimorph with an edge block. Our analysis provides a rational explanation why a trapeze planform is optimal, and provides an explicit functional form of the optimal geometrical parameters of this planform. The predictive capabilities of our model are validated by comparison to finite element simulations.

Snap-through dynamics of a buckled flexible filament in a channel flow

The flow-induced snap-through dynamics of a buckled flexible filament in a channel flow was explored using the penalty immersed boundary method. Two edge condition distributions were considered for comparison. One edge condition was a simply supported leading edge and a clamped trailing edge (SC); the other condition was two clamped edges. The effects of channel height and bending rigidity on the energy harvesting performance were systematically examined. The presence of the channel wall compresses the activation space of the vortex, leading to the formation of a high shear flow near the wall, which, in turn, strongly influences the wake pattern. In the snap-through oscillation mode, a wake pattern of 2S + 2P is observed in both narrow and broad channels, whereas the mechanism of vortex shedding varies between the two cases. Both cases demonstrate greater critical rigidity and greater elastic energy compared with conditions under external flow, suggesting an enhancement of the energy harvesting performance. The greater energy harvesting ability of the SC case is derived from both the larger deflection and the higher strain energy in this system. The wall effect is inversely proportional to the channel height, becoming nearly negligible when the nondimensional channel height exceeds 2. These findings provide valuable insights into the dynamics of flexible filaments in the channel flow and their potential for energy harvesting applications.

Vibrational analysis of orthotropic rectangular plate having combination of circular thickness and parabolic temperature

The objective of this study is to investigate the natural vibration of a rectangular plate made of orthotropic material with circular thickness (two dimensions) and temperature variation on the plate is parabolic (two dimensions) in nature. The solution to the problem is obtained by utilizing the Rayleigh-Ritz technique and the first four frequency modes are obtained under clamped edge conditions. The study aims to provide numerical data that demonstrate how circular variation in tapering parameters of plate can effectively control and optimized vibrational frequencies of the plate. Orthotropic rectangular plate, thermal gradient, circular tapering, aspect ratio.

Boundary Discontinuous Fourier analysis of clamped isotropic and cross-ply laminated plates via Unified Formulation

This paper presents an analytical solution for the static analysis of plates with clamped boundary conditions prescribed at the edges. The displacement field is expressed via the Carrera Unified Formulation (CUF) where an Equivalent-Single-Layer (ESL) approach is adopted. The governing equations are obtained by employing the principle of virtual displacements (PVD) statement. The main novelty is the use of the boundary-discontinuous Fourier-based approach to provide accurate numerical solutions. From thick to thin isotropic, cross-ply laminated and sandwich plates with different side-to-thickness ratios and stacking sequences are studied. Furthermore, the out-of-plane stresses are calculated via both the constitutive relation and the stress recovery technique. The accuracy of the proposed solution is verified by comparing the numerical results obtained with those from the literature and 3D FEM solutions. The present approach seems capable of handling not just fully clamped conditions but also mixed external conditions, which may include clamped and simply-supported edges. The solution approach provided in this article is unique, hence the proposed results might be useful as a benchmark for validating new plate theories and finite elements.

The free vibration analysis of a cantilever trapezoidal plate with non-uniform thickness reinforced with non-uniformly distributed nanofillers

The presented work is devoted to studying the dynamics of cantilever trapezoidal plates with non-uniform thickness and enriched with either carbon nanotubes (CNTs) or graphene nanoplatelets (GNPs). It is assumed that the volume fractions of nanofillers (CNTs or GNPs) and thickness of the plate vary in one direction from the clamped edge of the plate to the outer free edge. The mathematical modeling of the plate is performed according to the first-order shear deformation theory (FSDT), and the density and the effective values of elastic constants of the plate are calculated utilizing the rule of mixture (ROM), the Halpin-Tsai model, and the Eshelby-Mori-Tanaka scheme. Hamilton’s principle is employed to derive the governing equations and boundary conditions at a clamped edge and three free edges of the plate. An approximate solution is provided via the differential quadrature method (DQM) to estimate the natural frequencies of the plate and achieve the corresponding mode shapes. It is concluded that to attain higher natural frequencies in the low vibrational modes, it is more effective to distribute most of the nanofillers as close as to the clamped edge and reduce their volume fraction from the clamped edge to the outer free edge.

Snap-through dynamics of a buckled flexible filament with different edge conditions

The flow-induced snap-through dynamics of a buckled flexible filament under different edge conditions were explored using the penalty immersed boundary method. Three filament edge conditions were simulated: a simply supported leading edge and a clamped trailing edge (SC), a clamped leading edge and a simply supported trailing edge, and both edges clamped. The effects of the bending rigidity and density ratio on the energy harvesting performance were systematically examined. Two different modes were observed: an equilibrium mode and a snap-through oscillation mode. The parameter range under which the modes were observed changed depending on the edge conditions. Mode transitions, induced by an increase in transverse fluid force, occurred when the bending rigidity was low. A clamped leading edge enhanced filament stability, whereas a simply supported leading edge reduced stability. Among the three configurations, the SC case showed the highest critical bending rigidity and oscillation frequency, resulting in superior energy harvesting performance. The greater energy harvesting ability of the SC case derives from the larger deflection and the higher strain energy in this system. The strain energy in the filament with SC edges tended to concentrate in two regions of the filament: the rear part and the section near the clamped end. The SC case, coupled with low density and high rigidity, offers favorable conditions for energy harvesting purposes.

FREE VIBRATIONS OF THIN ELASTIC ORTHOTROPIC CYLINDRICAL PANEL WITH RIGID – CLAMPED EDGE GENERATOR

Using the system of equations corresponding to the classical theory of orthotropic cylindrical shells, the free vibrations of thin elastic orthotropic cylindrical panel with rigid – clamped edge generator are investigated. In order to calculate the natural frequencies and to identify the respective natural modes, the generalized Kantorovich-Vlasov method of reduction to ordinary differential equations is used. Dispersion equations for finding the natural frequencies of possible types of vibrations are derived. An asymptotic relation between the dispersion equations of the problem at hand and the analogous problem for a cantilever rectangular plate is established. An algorithm for separating possible boundary vibrations is presented. As an example, the values of dimensionless characteristics of natural frequencies are derived for an orthotropic cylindrical panel.

Closed-Form Analytical Solutions for the Deflection of Elastic Beams in a Peridynamic Framework

Peridynamics is a continuum theory that operates with non-local deformation measures as well as long-range internal force/moment interactions. The resulting equations are of the integral type, in contrast to the classical theory, which deals with differential equations. The aim of this paper is to analyze peridynamic governing equations for elastic beams. To this end, the strain energy density is formulated as a function of the non-local curvature. By applying the Lagrange principle, the peridynamic equations of motion are derived. Examples of non-local boundary conditions, including simple support, clamped edge, roller clamped edge, and free edge, are presented by introducing the interaction domain. Novel closed-form analytical solutions to integral equations are presented for beams with various boundary conditions, including clamped—simply supported, clamped–clamped, simply supported–roller-clamped, and clamped–roller-clamped beams. Furthermore, different types of loadings, including uniformly distributed load, concentrated force, and concentrated moment, are considered. The results are validated by comparing the derived solutions against solutions to the classical Bernoulli–Euler beam theory. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes, which shows the capability of the derived equations of motion and proposed boundary conditions.