Deflection Of Plate Research Articles

Machine learning assisted nonlinear deflection analysis of agglomerated carbon nanotube core smart sandwich plate with three-phase magneto-electro-elastic skin

This research focuses on evaluating the combined effects of the interphase of the three-phase magneto-electro-elastic (TPMEE) composites and carbon nanotubes (CNTs) agglomeration on the nonlinear deflection of the multifunctional sandwich plate, using an artificial neural network (ANN) assisted finite-element (FE) approach. The data points collected from the in-house developed FE computational tool are used to train the ANN model. To this end, a backpropagation-based Levenberg–Marquardt algorithm is used. The core of the plate is made of agglomerated CNTs, and the facesheets are made of TPMEE composites. Two different agglomeration states, partial and complete, are considered for evaluation. Also, three variations of CNTs arrangement are assumed. On the other hand, the interphase effects are incorporated through its volume fractions and compositions. The plate kinematics is based on the higher-order shear deformation theory, and the nonlinearity is assumed to follow von Karman's strain-displacement relation. The equations of motion are derived using the total potential energy principle. Finally, the direct iterative method is used to arrive at the solutions. The numerical examples are also provided to understand the influence of coupling fields associated with the parameters, such as agglomeration, interphase volume fraction, interphase compositions and CNTs arrangement.

Static/dynamic analyses of sandwich micro-plate based on modified strain gradient theory

In this work, bending and dynamic analyses of a three-layered functionally graded micro annular plate whit piezomagnetic layers are investigated based on FSD circular plate theory. The micro plate is subjected to magnetic field. It is assumed that the micro annular plate is resting on an orthotropic elastic medium. The stress relations which cooperate both mechanical and magnetic interactions are calculated based on MSGT and the governing equations are extracted through Hamilton’s principles. In order to solve the governing equations of micro annular plate, the numerical Ritz method is applied. After convergence of Ritz solving domain the obtained results are compared and adapted with the output results of the previous studies. Effects of magnetic field, thickness of metal-to-ceramic ratio, length-to-width ratio of micro annular plate, small scale parameter and thickness of piezomagnetic layers ratio are studied on the frequencies and deflection of micro annular FGM plate. The results show that applying the magnetic field lead to increase of micro structure stiffness and enhances the natural frequencies, while the maximum deflection of micro structure decreases with increasing micro annular plate stiffness. Also, it is concluded that that due to the high ceramic stiffness, the natural frequencies and the deflection of the micro annular plate reduces and increases with increasing of the thickness of the metal to the ceramic ratio, respectively.

Bending of Bi-directional inhomogeneous nanoplates using microstructure-dependent higher-order shear deformation theory

In contrast of experimental analysis in the mechanical behavior of the complicated structures, the progress of theoretical research has many difficulties, especially with regard to the modelling of structures. The main objective of this research is to analyze analytically the static problem of a new model of functionally graded materials, namely the “bi-coated FGM” plate. The materials are graded continuously through 2-directional by using a power law function. Two types of coated FG plates are investigated, Hardcore and Softcore FG plates. Based on the generalized field of displacement, a Quasi-3D higher-order shear deformation plate theory is proposed in this work by reducing the number of variables from six to five variables. The equilibrium equations are performed based on the principle of virtual work and solved using the Galerkin method to cover different boundary conditions. The nonlocal strain gradient theory is employed to capture the microstructure-dependent effect. The model is validated compared to recent relevant research interests. Numerical results are performed to present the impact of hardcore and softcore distributions, gradation indices, and boundary conditions on static bending deflection and stresses of bi-coated FGM plate.

Attenuation in hydroelastic response of floating-elastic-plate by porous membrane in two-layer fluid with bottom undulation

The prime intention of this article is to study the effects of bottom undulation and a porous membrane barrier on the hydroelastic response of an elastic plate floating in a two-layer fluid with changing bottom topography assuming the small amplitude waves and potential flow theory. Finite and semi-infinite plates coupled with porous membrane barriers are considered in this study. As a mathematical technique, the eigenfunction expansion method along with an approximation method mild-slope are adopted for the uniform bottom and variable bottom topographies, respectively. In the variable bottom topography, a system of differential equations is solved. By using matching and jump conditions, the solution is expressed as a linear algebraic system, from which the unknown constants of interest are computed. The effects of fluid density ratio, water depth ratio, bottom topography, and porous membrane on the bending moment, shear force, and deflection of the elastic plate are studied. It is found that when the density ratio becomes closer to one, the bending moments and shear forces acting on the elastic plates tend to diminish. The variations in the bending moments, shear forces, and plate deflection with respect to changing density ratios are found to be in opposite trends caused by surface and interfacial waves, respectively. With higher tension and a less porous membrane, the bending moment, shear forces, and pate deflection are reduced. Concave-up bottoms reflect more surface wave energy, whereas concave-down bottoms reflect more interfacial wave energy. The observations made in this article may help in analyzing the response of very-large floating structures in the presence of an undulating seabed.

Hermitian numerical manifold method for large deflection of irregular Föppl-von Kármán plates

We proposed the Hermitian manifold numerical method (HNMM) fitted for finite strain analysis of thin plates with irregular domains. The large deflection of elastic thin plates is generally characterized by Föppl-von Kármán (FvK) equations, coupled with nonlinear fourth-order partial differential equations. The corresponding primal variational formulation requires the solution function to be globally C1 continuous but piecewise C2 continuous (namely H2 regular). Hermitian numerical manifold method (HNMM) can easily construct an approximation to solutions that satisfy the H2 regular requirements with structured meshes. Furthermore, those classical plate elements in finite element history can be embedded in the framework of HNMM and be more flexible and adaptable to solve plates with irregular domains. Thus, HNMM is formulated based on the Föppl-von Kármán (FvK) model using the dual cover system and a triplet attributes group on the physical patches. The numerical results demonstrate the accuracy of HNMM in large deflection of Föppl-von Kármán plate with complex domain shape.

A local meshfree radial point interpolation method for Berger equation arising in modelling of thin plates

In this study, two-dimensional (2D) Berger equation arising in deflection of thin plates is solved numerically. A local meshfree collocation technique based on radial point interpolation is developed for the Berger equation. As radial basis function thin plate splines are used to avoid selecting optimal shape parameter. The method is used to solve both linear and non-linear Berger equation on regular and irregular geometries. Obtained results during numerical simulations, are compared with different methods exist in literatrue such as method of fundamental solutions, local Kansa’s method and Pascal polynomial based meshless method. The comparisons affirm the accuracy and reliability of the suggested method.

GDQ computation for thermal vibration of thick FGM plates by using third-order shear deformation theory

The third-order shear deformation theory (TSDT) effects on the thick functionally graded material (FGM) plates under thermal vibration are investigated by using the generalized differential quadrature (GDQ) method. The nonlinear coefficient term of displacement field of TSDT is used to derive the equations of motion for thermal vibration of thick FGM plates. The simpler form stiffness of FGM plates under linear temperature rise is considered in the thermo-elastic stress–strain relations. The dynamic equilibrium differential equations of FGM plates in terms of partial derivatives of displacements and shear rotations subjected to partial derivatives of thermal loads, mechanical loads and inertia terms are derived. Two parametric: environment temperature and FGM power law index effects on the thermal stress and center deflection of FGM thick plates are investigated.

Nonlinear bending of FG metal/graphene sandwich microplates with metal foam core resting on nonlinear elastic foundations via a new plate theory

Nonlinear bending of functionally graded metal/graphene (FGMG) sandwich rectangular plate with metal foam core resting on nonlinear elastic foundations is elucidated in this article. A new shear deformation plate theory is presented to formulate the displacement field. The nonlinear partial differential equations considering the small size effect are established via the principle of virtual work. The nonlinearity is considered by using Von Karman’s strain-displacement relations. While, the size effect is captured by employing the modified couple stress theory. The upper and lower layers are made of aluminum as a matrix that reinforced with graphene platelets (GPLs). The GPLs are functionally graded through the thickness of the face layers according to a new cosine rule. Moreover, the metal foam core is also made of aluminum containing porosities that uniformly distributed or functionally graded through the core thickness. The governing equations are solved based on the Galerkin and Newton’s methods. The obtained results are examined by introducing some comparison examples. In addition, several parametric examples are discussed including the effects of the porosity type, GPLs distribution type, core-to-face thickness, elastic foundation stiffness, side-to-thickness ratio, plate aspect ratio and material length scale parameter on the nonlinear deflection and stresses of FGMG sandwich plate with metal foam core.

FEM-ANN approach to predict nonlinear pyro-coupled deflection of sandwich plates with agglomerated porous nanocomposite core and piezo-magneto-elastic facings in thermal environment

The present work deals with evaluating the nonlinear deflections of the smart sandwich plate with agglomerated Carbon Nanotubes (CNTs) porous core and piezo-magneto-electric (PME) facings, using a novel finite element method (FEM) – artificial neural network (ANN) approach. For the first time, an ANN-based computational tool that integrates the effects of agglomeration of CNTs, porosity and pyro-coupling of the PME materials is presented. Firstly, an in-house finite element (FE) computational tool is proposed and developed using the principle of virtual work in association with higher-order shear deformation theory (HSDT) and von-Karman’s nonlinearity. The data points owing to the nonlinear deflections are collected using the proposed FE formulation, which trains the ANN model using Levenberg–Marquardt algorithm. The externally applied thermal loads are assumed to vary uniformly and linearly across the thickness of the plate. The primary focus of this work is to assess the variation in the degree of pyro-coupling associated with agglomeration and porosity. Two states of agglomeration, such as partial and complete; three forms of porosity, such as uniformly distributed, and two variants of functionally graded porosity, are considered for investigation. Numerical examples are solved to understand the interrelated effects of these material properties. A significant variation in the deflection of the plate, which refers to its actuation capability, is witnessed when the parameters of agglomeration and porosity change.

Time-dependent modelling of thin poroelastic films drying on deformable plates

AbstractUnderstanding the generation of mechanical stress in drying, particle-laden films is important for a wide range of industrial processes. One way to study these stresses is through the cantilever experiment, whereby a thin film is deposited onto the surface of a thin plate that is clamped at one end to a wall. The stresses that are generated in the film during drying are transmitted to the plate and drive bending. Mathematical modelling enables the film stress to be inferred from measurements of the plate deflection. The aim of this paper is to present simplified models of the cantilever experiment that have been derived from the time-dependent equations of continuum mechanics using asymptotic methods. The film is described using nonlinear poroelasticity and the plate using nonlinear elasticity. In contrast to Stoney-like formulae, the simplified models account for films with non-uniform thickness and stress. The film model reduces to a single differential equation that can be solved independently of the plate equations. The plate model reduces to an extended form of the Föppl-von Kármán (FvK) equations that accounts for gradients in the longitudinal traction acting on the plate surface. Consistent boundary conditions for the FvK equations are derived by resolving the Saint-Venant boundary layers at the free edges of the plate. The asymptotically reduced models are in excellent agreement with finite element solutions of the full governing equations. As the Péclet number increases, the time evolution of the plate deflection changes from $t$ to $t^{1/2}$ , in agreement with experiments.

Experimental investigations of wave interaction with semi-submerged horizontal rectangular cylinder of elastic bottom

Laboratory experiments were conducted to investigate the interaction of gravity waves with a semi-submerged rectangular cylinder of elastic bottom. Laboratory investigations were conducted for a wide range of wave parameters and a structure of different drafts. The analysis of results focuses on wave transmission at the structure, wave-induced pressures excreted on the plate, plate deflections, etc. The results shows that the nonlinear pressure and plate deflection components arising from the interaction of incoming waves with the structure are often significant. The results show that the nonlinear components often exceed their linear counterparts even for waves of low steepness. This is an original and surprising outcome of significant practical importance as the large nonlinear pressure and plate deflection components may spark a resonance. The problem is serious because plate natural vibration frequencies are often lower than the frequency of fundamental waves or their nonlinear components and the vibration of a plate may be amplified by incoming waves or their nonlinear products. The occurrence of a resonance amplified by incoming waves may lead to the failure of a structure. This outcome is of significant practical importance because it may occur for a wide range of wave parameters.

Wave resonances and hydroelastic wave motion

Three different types of resonances, i.e. blocking, trapping and Bragg resonance, associated with flexural gravity wave motion in the presence of current and permeable bottom are discussed. Also, using the different modes of wave propagation, the interconnection among all these resonances is analysed. An uneven pattern in the plate deflection is seen inside the blocking resonance zone. The uniform convergence of the infinite series associated with the trapped mode is proved. We have found that the trapped mode does not exist if the magnitude of the opposing current is higher or whenever we move the cylinder away from the platecovered surface. It is also observed that two propagating modes within the blocking frequencies create the environment for the occurrence of trapped waves. At the same time, all three modes have a remarkable contribution to Bragg resonance. The Bragg reflection increases in the blocking zone and decreases outside of it. We have seen that the amplitude of the Bragg reflection increases with the increase in current speed and porosity parameter. Also, the discontinuities in Bragg reflection coefficients are seen at blocking frequencies. The model is also carefully validated numerically for trapped and Bragg resonances against the results available in the existing literature.

Експериментальні дослідження величини прогину пластини для корекції лійкоподібної деформації грудної клітки під впливом згинаючого навантаження

One of the leading methods of reconstruction of the funnel-shaped deformity of the chest is nonresection thoracoplasty, or Nuss surgery. For its implementation, implanted systems consisting of one or two corrective plates are used. The technique is minimally invasive, does not require large incisions and resection of ribs. But there are complications related to the insufficient strength of bone tissue, which according to the literature is defined in a very wide range. Purpose - to determine the dependence of the amount of deflection of single and double plates for the correction of sternum deformity on the amount of bending load; to develop a method of determining the corrective load that the plate exerts on bone structures during correction. Materials and methods. Experimental studies of plates for correction of sternum deformities were performed. We used 6 plates. In the first part of the experiment, each plate was tested separately. In the second part, the plates were connected in pairs, and each pair was tested for bending. In total, 3 pairs of plates were tested. The bending force was gradually increased from 0 N in increments of 50 N. At each step, the deflection of the plate was recorded. Results. The results of experimental studies of single plates showed that the deviation of deflection values did not differ statistically significantly (p>0.05) from the calculated values over the entire range of bending loads. The studies of the twin plates for the correction of sternum deformities also did not reveal statistically significant (p>0.05) differences in the deflection values of the samples obtained in the experiment from the calculated values. The results of the study made it possible to develop a methodology for calculating the force with which the plate acts on the sternum when correcting its deformation. Conclusions. A methodology has been developed for determining the force with which the plate acts on the sternum when correcting a deformation of a certain magnitude. It is confirmed that the calculated data on the deflection of the plate under the influence of bending load and the data obtained in the experiment do not differ statistically significantly. The data obtained can be useful for selecting the size of the corrective plate, the method of its attachment to the ribs, as well as for modeling the options for applying plates to correct various sternum deformities. The research was carried out in accordance with the principles of the Helsinki Declaration. The study protocol was approved by the Local Ethics Committee of the institution mentioned in the paper. The informed consent of the patient was obtained for conducting the studies. No conflict of interests was declared by the authors.

TNT equivalency method in confined space based on steel plate deformation

TNT equivalency has been a widely accepted concept in engineering. However, as a fuel-rich explosive, the afterburning effect of detonation products would be remarkable when TNT is detonated in a confined space. The additional energy release from the burning of detonation products makes the traditional TNT equivalency to be more complicated. In this paper, in order to reveal the influence of afterburning effect of detonation products under the condition of confined blast, a series of TNT detonation tests in a confined chamber filled with four different atmospheres, air, nitrogen, water mist, and combined nitrogen and water mist were performed. The temperature, pressure-time histories and deflection of blast loaded steel plate were recorded. It is found that the pressure, temperature and the dynamic response of the blast loaded plate in nitrogen and water mist atmosphere were significantly lower than those in the air conditions, due to different mechanisms of energy release and absorption. In addition, the relation between deflection of target plate and the released energy from explosive was established. Furthermore, the method of determining TNT equivalency in confined explosion considering the afterburning effect was proposed and was employed to determine the TNT equivalency of a cased charge. By comparing the present TNT equivalency method with the method based on quasi-static pressure, the potential application of the method has been illustrated.

Interaction of a Flexural-Gravity Wave with a Vertical Rigid Plate Built in a Floating Elastic Plate

The linear two-dimensional problem of interaction between an hydroelastic wave propagating along an elastic floating ice plate with built-in vertical rigid plate is studied. The fluid under the ice is inviscid and incompressible. The fluid depth is finite. The deflection of the ice plate is described by the linear theory of thin elastic plates. The flow under the ice is potential. The total velocity potential is decomposed into the potential of the incident wave, even potential caused by the vertical motion of the rigid plate, and an odd potential caused by the rotation of the rigid plate. The vertical mode method is used. The third potential is obtained by solving a mixed boundary-value problem numerically using Chebyshev polynomials. The solution is validated by analysis of its convergence. The first and second potentials, and the corresponding deflections and strains of the ice plate, are obtained analytically. The motions of the rigid plate, as well as deflection and strains in the floating plate, are numerically analyzed. It is shown that the rotation of the rigid plate due to the incident wave is the main factor of increasing strains in the ice plate.

Single and double-layered kirigami corrugated sandwich panels against impact loads

The influences of geometric configurations on the impact resistance performances for single and double-layered kirigami corrugated (KC) sandwich panels are investigated in this study. By folding a portion of the cell wall of a conventional corrugated unit, a kirigami corrugate unit is formed with significantly improved resistance, which is then used to form single and double-layered sandwich panels. The numerical model is first constructed and validated against the pendulum impact test results of the KC panel. To evaluate the effect of the geometric configurations on their impact resistance performance, four factors are considered, including the number and location of the fold-ins of the single-layered KC panel, the stacking and layer orientation of the double-layered KC panels. In total, 38 configurations of the panels are numerically studied, and their impact performances including energy absorption, peak and residual back plate deflection, and impact force–displacement curves are evaluated and compared. Up to 84% reduction in the residual back plate deflection is observed for single-layered KC panel compared to the conventional corrugated panel. Superior impact resistance is demonstrated for single-layered KC panels with more fold-ins, especially near the impacting area. It is also found that the stacking configuration and layer orientation are not directly linked to the impact resistance performance of double-layered KC panels. However, enhanced impact resistance could be achieved by arranging the stacking and orientation of the two core layers to provide better supports to the top cores. This work provides general design guidelines for the proposed single and double-layered KC panels.

Piezoelectric Patches for Deflection Control of Functionally Graded Carbon Nanotube-Reinforced Composite Plates

In the present work, a smart structure is being investigated, where a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plate is equipped with piezoelectric actuators to provide vibration control. Due to their high mechanical properties coupled with lightweight, FG-CNTRCs are mainly used in the aerospace industry and in advanced engineering applications. The CNTs have a linear and non-linear distribution along the thickness of the plate and are distributed according to five configurations, namely: UD, FG-X, FG-O, FG-A and FG-V. The first order shear deformation (FOSD) theory is considered in the formulation of a 9-node quadratic finite element with 5 degrees-of-freedom per node, and an additional degree of freedom is provided for the piezoelectric layer. The model developed in this study assesses the free vibration behavior and controls the nanocomposite plate deflection through the electromechanical coupling factor piezoelectric. In addition, it investigates: (i) the effect of the plate configuration, (ii) the CNT volume fraction, (iii) the CNT destruction patterns, (iv) the linear and nonlinear distribution of CNTs, (v) the number of CNTRC ply, (vi) the boundary conditions and (vii) the dimensions with different locations of actuators. The results obtained show the first natural frequencies for all configurations, which are considered to be in good agreement with those available in the literature and illustrate that the effective stiffness of the nanocomposite plates can be improved further when the reinforcement is dispersed according to the FG-X pattern. In addition, for the case of the deflection control analysis, results indicate that the distributed piezoelectric layers (actuators) attenuate the deflection of the CNTRC to the desired tolerance. It is noted that patches with partial coverage compared to the case of total coverage of piezoelectric layers require more electrical power to reach the same level of attenuation. The developed numerical model is intended to be used in a variety of potential advanced engineering applications.

Closed‐form buckling analysis of unsymmetrically laminated plates

AbstractThe local stability of thin‐walled fibre‐reinforced plastic composite beams can be reduced to individual laminates using discrete plate theory. These individual plates receive a supporting effect from their surrounding structure, which is modelled with rotational restraints. In the present investigation, this buckling problem is described by a closed‐form solution. The energy‐based method works with the Rayleigh quotient and the principle of the stationary value of the elastic potential energy. For the analysis of unsymmetrically laminated plates, the classical laminated plate theory (CLPT) considers both the plate deflection and the in‐plane displacements. The first‐order shear deformation theory (FSDT) and third‐order shear deformation theory (TSDT) additionally describe the cross‐sectional rotations and thus take transverse shear deformations into account. In addition to the direct consideration of the bending‐extension couplings, these have also been investigated using the reduced bending stiffness (RBS) method. The investigation shows the influence of bending‐extension coupling on the stability of compressively loaded unsymmetrically laminated plates. Moreover, it is found that the transverse shear stiffness reduces the critical load at relatively high plate thicknesses. The closed‐form analytical solution and the RBS method show good agreement with finite element analyses. The presented closed‐form analytical methods provide explicit solutions for the critical compressive load of unsymmetric laminates under different boundary conditions. Due to the explicit solution, this method is significantly more computationally efficient than numerical, semi‐analytical or exact methods. The present methods are characterised by a simple applicability as well as a very high computational efficiency and are very suitable for preliminary design as well as optimisation of laminated structures.

Quantification of various reduced order modelling computational methods to study deflection of size-dependent plates

Quantification of various reduced order modelling computational methods to study deflection of size-dependent plates

Multiwall Rectangular Plates under Transverse Pressure—A Non-Linear Experimental and Numerical Study

Large deflection of rectangular plates under transverse pressure is described by Föppl-von Kármán equations, which have only approximated solutions. One of these methods is the separation into a small deflection plate and a thin membrane described by a simple third order polynomial expression. The present study presents an analysis to obtain analytical expressions for its coefficients by using the plate's elastic properties and dimensions. To validate the non-linear relationship between the pressure and the lateral displacement of the multiwall plate, a vacuum chamber loading test is used to measure the plate's response, with a large number of plates and length-width combinations. In addition, to further validate the analytical expressions, several finite element analyses (FEA) were performed. It has been found that the polynomial expression fairly describes the measured and calculated deflections. This method allows the prediction of plate deflections under pressure as soon as the elastic properties and the dimensions are known.