Simply-supported Plate Research Articles

Complex Trapped Modes of the Helmholtz Equation in Cross, S, and U Bends

The Helmholtz equation with zero boundary conditions is important in the vibration of membranes, simply-supported plates, and electromagnetic and quantum waveguides. Trapped modes, with a lowered frequency and limited local effect, may exist for infinite strips with local bends or enlargements. Infinite strips with a cross bulge, with an S bend and with a U bend are studied using an efficient eigenfunction expansion and domain decomposition method. As the bulge become larger, more trapped modes with more complexity appear. Some geometries do not support trapped modes at all. These results are new.

A Nonlinear Mathematical Model of Thermoelastic Thin Plates with Voids and Its Applications

Following the linear theory of thermoelastic materials with voids, a generalized Hamilton variational principle is extended to the thermoelastic plates with voids under the case of large deflection, and a 3D nonlinear mathematical model is presented. In this process, the balance equation of the entropy is converted to an equivalent form without the first-order time-derivative by integral, and the concept of the moments for the volume fraction of voids and temperature field is introduced. As application, the nonlinear dynamic and aerodynamic characteristics of simply-supported rectangular thermoelastic plates with voids for four different materials are studied and compared by using a Galerkin approach. The effects of the initial deflections and material parameters are considered in detail. In addition to providing a generalized Hamilton variational principle and a 3D nonlinear mathematical model, it is also provided a valuable numerical method to solve the dynamic problem directly in the paper. The theory and the method can be applied to solving various problems of the thermoelastic plates with voids easily.

Modal approach for nonlinear vibrations of damped impacted plates: Application to sound synthesis of gongs and cymbals

This paper presents a modal, time-domain scheme for the nonlinear vibrations of perfect and imperfect plates. The scheme can take into account a large number of degrees-of-freedom and is energy-conserving. The targeted application is the sound synthesis of cymbals and gong-like musical instruments, which are known for displaying a strongly nonlinear vibrating behaviour. This behaviour is typical of a wave turbulence regime, in which the wide-band spectrum of excited modes is observable in the form of an energy cascade. The modal method is selected for its versatility in handling complex damping laws that can be implemented easily by selecting appropriate damping values in each one of the modal equations. In the first part of the paper, the modal method is explained in its generality, and it will be seen that the method is valid for plates with arbitrary geometry and boundary conditions as long as the eigenmodes are known. Secondly, a time-integration, energy-conserving scheme for perfect and imperfect plates is presented, and implementation comments are given in order to treat efficiently the high-dimensionality of the resulting dynamical system. The scheme is run with appropriate parameters in order to produce sound samples. A simple impact law is considered for the excitation, whereas the flexibility of the method is highlighted by showing simulations for free-edge circular plates and simply-supported rectangular plates, together with various damping laws.

Nonlinear Dynamics of a Functionally Graded Thin Simply-Supported Plate Under a Hypersonic Flow

An analysis on the nonlinear dynamics of an FGM plate in hypersonic flow subjected to an external excitation and uniform temperature change is presented in this article. Material properties of the constituents are assumed to be temperature-dependent and the effective properties of FGM plate are graded in thickness direction according to a simple power law function in terms of the volume fractions. Based on von Karman type nonlinear strain–displacement relationship, the nonlinear governing equations of motion are derived by using Hamilton's principle. Galerkin's method is then utilized to discretize the governing partial equations to a two-degree-of-freedom nonlinear system, including the quadratic and cubic nonlinear terms. A third-order piston theory is applied for the hypersonic aerodynamic load. The Mach number and in-plane loading are used as bifurcation parameters. Numerical results including the bifurcations, waveform, phase plots, and Poincare maps are presented, for an FGM plate to show the influences of material gradient index, Mach number, and in-plane load on the nonlinear dynamics.

Post-buckling behavior of functionally graded circular plates under asymmetric transverse and in-plane loadings

Based on the first-order nonlinear von Karman theory, the post-buckling analysis of FG circular plates under asymmetric transverse and in-plane loadings are presented. The nonlinear governing equations are reformulated using a stress function and a boundary layer function, and solved by employing a multi-parameter perturbation technique and Fourier series method. The material properties are assumed to vary through the plate thickness according to a power-law distribution of the volume fractions of the constituents. The results are verified with the known results in the literature. The effects of loading, material properties, and boundary conditions on different response quantities are studied. It is observed that the buckling load in clamped FG plates under asymmetric in-plane loading is lower than that of the same plate under axisymmetric in-plane loading. It is concluded that under downward transverse loading, the snap buckling occurs in simply-supported FG plates with an initial considerable upward deflection caused by an in-plane load. In addition, the behavior of boundary layer in FG plates under in-plane loading is studied and compared with that in transverse loading.

Optimisation of the structural modes of automotive-type panels using line stiffeners and point masses to achieve weak acoustic radiation

In this paper, an optimisation method is presented that turns the structural modes of an automotive-type panel into weak acoustic radiators. The advantage of the proposed method is that the optimum design does not depend upon a specific excitation mechanism. Hence, the optimised panel can be used in the early stages of vehicle product design when specific knowledge about the excitation forces is not available. In the proposed method the boundary conditions of the panel are specified such that the panel can be considered in isolation from the rest of the structure. The optimisation of the structural modes of the panel is then achieved by placing a number of constraint masses and stiffeners of optimum size and weight at optimum locations on the plate. Firstly, a theoretical model of a simply-supported plate is developed that takes into account the constraints at arbitrary positions and orientations. Then a genetic algorithm is used to minimise the sound power radiated by different modes on the plate by finding the optimum design for the constraints. Secondly, the use of added masses and stiffeners is then applied to a floor panel of a simplified model of a vehicle body structure in order to reduce the radiated sound. Predicted and experimental results of the radiated sound power from a reference flat panel and from various optimised panel designs are presented to illustrate the effectiveness of the optimised designs in reducing radiated sound from the structure. A combination of masses and stiffeners is shown to be an effective way of constructing weakly radiating structural modes on the panel.

Proper orthogonal decomposition method for analysis of nonlinear panel flutter with thermal effects in supersonic flow

The proper orthogonal decomposition (POD) method for analysis of nonlinear panel flutter subjected to supersonic flow is presented. Optimal POD modes are extracted from a chaotic Galerkin mode responses. The aeroelastic equations of motion are constructed using von Karman plate theory, first-order piston theory and quasi-steady thermal stress theory. A simply-supported plate with thermal loads from a uniformly distributed temperature is considered. Many types of panel behaviors, including stable flat, dynamically stable buckled, limit cycle oscillation, nonharmonic periodic motion, quasi-periodic motion and chaotic motion are observed. Our primary focus is on chaos and the route to chaos. It is found that a sudden transition from the buckled state to chaos occurs. Time history, phase portrait, Poincaré map, bifurcation diagram and Lyapunov exponent are employed to study chaos. The POD chaotic results obtained are compared with the traditional Galerkin solutions. It is shown that the POD method can obtain accurate chaotic solutions, using fewer modes and less computational effort than the Galerkin mode approach; additionally, the POD method converges faster in the analysis of chaotic transients. Effects of length-to-width ratios and thermal loads are presented. It is found that a smaller width for fixed length will produce more stable flutter response, while the thermal loads degrade the flutter boundary and result in a more complex evolution of dynamic motions. The numerical simulations show that the robustness of the POD modes depends on the dynamic pressure but not on temperature.

Shear Capacity of Steel Plates with a Square Opening

Steel plate elements are frequently encountered in structural applications, where thin plates are susceptible for local buckling at low loads, while thick plates may reach their strength limit. An opening of any size and shape on such structural plates may significantly affect the stiffness, strength, and the failure characteristic of the entire plate. The objective of this study is to quantify the shear capacity of plates containing a centrally placed square opening. The finite element method based numerical investigation considered the behavior of simply-supported rectangular steel plates, with and without a square opening and subjected to in-plane shear loading, experiencing buckling, post-buckling, and yielding until failure. The model employed representative material model, initial geometric imperfections and large deflection analysis. The investigation established the shear strength reduction factors due to opening of different parametric dimensions, and compared such results with the reduction factors based on the current cold-formed steel design code provisions. This paper establishes an enhanced shear strength reduction factor design equation, incorporating the impact of a centrally located square opening, for the shear capacity of un-stiffened steel plates.

Layer-wise and equivalent single layer models for smart multilayered plates

Layer-wise and equivalent single layer plate models for magneto-electro-elastic multiphysics laminates are presented in a unified framework. They are based on variable kinematics and quasi-static behavior of the electromagnetic fields. The electromagnetic state of each single layer is preliminary determined by solving the corresponding governing equations coupled with the proper interface continuity and external boundary conditions. By so doing, the electromagnetic state is condensed into the plate kinematics and the layer governing equations are inferred by the principle of virtual displacements. This approach identifies effective mechanical layers, which are kinematically equivalent to the original smart layers. These effective layers are characterized by stiffness, inertia and load properties which take the multifield coupling effects into account as their definitions involve the electromagnetic coupling material properties. The layers governing equations are finally assembled enforcing the mechanical interface conditions. This allows to obtain the smart plate resolving system, which involves primary mechanical variables only. Results for thick simply-supported multilayered plates are obtained by an exact closed-form Navier-type solution and compared with benchmark 3D solutions to investigate the features and accuracy of the proposed modeling approach.

A higher-order hyperbolic shear deformation plate model for analysis of functionally graded materials

This paper presents a new higher-order hyperbolic shear deformation theory for analysis of functionally graded plates. In this theory, the transverse shear stresses account for a hyperbolic distribution and satisfy the free-traction boundary conditions on the top and bottom surfaces of the plate. By making a further assumption, the present theory contains only four unknowns and its governing equations is therefore reduced. Equations of motion are derived from Hamilton’s principle and Navier-type analytical solutions for simply-supported plates are compared with the existing solutions to verify the validity of the developed theory. The material properties are continuously varied through the plate thickness by the power-law and exponential form. Numerical results are obtained to investigate the effects of the power-law index and side-to-thickness ratio on the deflections, stresses, critical buckling load and natural frequencies.

Optimal design of glass plates loaded transversally

This paper focuses on glass used in buildings and presents a criterion for fail-safe optimal design of glass plates.First, the paper shows that the glass plate is fail-safe only if the load-bearing system is composed of two glass layers bonded to one another with an elastomeric interlayer (Laminated Glass), and if the live loads act upon a sacrificial glass ply (tri-layer system).Then, activity was directed at analyzing the simply-supported fail-safe glass plate loaded out-of-plane, and carrying out research targeted at reducing the incidence of weight and cost (optimal design). The results, obtained using an analytical exact model, show that the limit states are always dictated by the maximum deflection and not by the load-carrying capacity. Thus, optimal design requires finding the thickness of the glass layers and the stiffness of the interlayer that provide the plate with exactly the minimum allowable stiffness, while the stress verifications are fulfilled automatically.Finally, for every span and load that is found in building applications of glass, the paper provides the thicknesses and the materials of the glass layers and interlayers that adjust capacity to match demand. These results may also replace structural analysis and assessment of Laminated Glass plates.

Multipulse Chaotic Dynamics for Nonautonomous Nonlinear Systems and Application to a FGM Plate

The extended Melnikov method is improved to investigate the nonautonomous nonlinear dynamical system in Cartesian coordinate. The multipulse chaotic dynamics of a simply supported functionally graded materials (FGM) rectangular plate subjected to transversal and in-plane excitations is investigated in this paper for the first time. The formulas of the FGM rectangular plate are two-degree-of-freedom nonautonomous nonlinear system with coupling of nonlinear terms including several square and cubic terms. The extended Melnikov method is improved to enable us to analyze directly the nonautonomous nonlinear dynamical system of the simply-supported FGM rectangular plate. The results obtained here indicate that multipulse chaotic motions can occur in the simply-supported FGM rectangular plate. Numerical simulation is also employed to find the multipulse chaotic motions of the simply-supported FGM rectangular plate.

Trapped modes due to narrow cracks in thin simply-supported elastic plates

We consider an elastic plate of infinite length and constant width supported simply along its two parallel edges and having a finite length crack along its centreline. In particular, we look for and find trapped modes (localised oscillations) in the presence of the crack. An explicit wide-spacing approximation based on the Wiener–Hopf technique applied to incident wave scattering by semi-infinite cracks is complemented by an exact formulation of the problem in the form of integro-differential equations. An application of a Galerkin method for the numerical calculation of results from the latter method leads to a novel explicit ‘small-spacing’ approximation. In combination with the wide-spacing results this is shown to provide accurate results for all lengths of crack.

VIBRATION ANALYSES OF CRACKED PLATES BY THE RITZ METHOD WITH MOVING LEAST-SQUARES INTERPOLATION FUNCTIONS

The solutions for the vibrations of cracked thin plates are obtained by the Ritz method with admissible functions. Based on the classical plate theory, the basis functions comprising polynomials and crack functions are adopted to generate the admissible functions by the moving least-squares approach for a set of nodes randomly distributed in the domain. The crack functions account for the singular behaviors of stress resultants at crack tip(s), which are discontinuous in displacement and slope across the crack. The present solutions are validated through convergence tests of frequencies and by comparison with the published results for simply-supported cracked rectangular plates. The solutions are further employed to determine the natural frequencies of cantilevered skewed rhombic and isosceles triangular plates and completely free circular plates, each with a crack of varying length, location and orientation. The numerical results are tabulated and some corresponding mode shapes are also presented, by means of nodal patterns. Most of the results shown here are new to the literature.

A nonlinear plate theory for the monolayer graphene

In the present paper, the kinematic equation of a monolayer graphene is proposed based on a plate theory, and the nonlinear elasticity stress-strain relations are obtained from experiments. The equation includes cubic and quintic nonlinearities. The bending produced when subjected to a concentrated force at the center of the plate and the static buckling arising from edge in-plane axial uniform loads are investigated using Ritz methods for a simply-supported rectangular plate. Results suggest that the plate theory with nonlinear constitutive equation may characterize the mechanical property of a monolayer graphene appropriately, and the quintic nonlinearities have a significant effect on the bending deformations of the graphene.

Estimating ensemble average power delivered by a piezoelectric patch actuator to a non-deterministic subsystem

Abstract Engineering systems such as aircraft, ships and automotive are considered built-up structures. Dynamically they are taught of as being fabricated from many components that are classified as ‘deterministic subsystems’ (DS) and ‘non-deterministic subsystems’ (Non-DS). Structures' response of the DS is deterministic in nature and analysed using deterministic modelling methods such as finite element (FE) method. The response of Non-DS is statistical in nature and estimated using statistical modelling technique such as statistical energy analysis (SEA). SEA method uses power balance equation, in which any external input to the subsystem must be represented in terms of power. Often, input force is taken as point force and ensemble average power delivered by point force is already well-established. However, the external input can also be applied in the form of moments exerted by a piezoelectric (PZT) patch actuator. In order to be able to apply SEA method for input moments, a mathematical representation for moment generated by PZT patch in the form of average power is needed, which is attempted in this paper. A simply-supported plate with attached PZT patch is taken as a benchmark model. Analytical solution to estimate average power is derived using mobility approach. Ensemble average of power given by the PZT patch actuator to the benchmark model when subjected to structural uncertainties is also simulated using Lagrangian method and FEA software. The analytical estimation is compared with the Lagrangian model and FE method for validation. The effects of size and location of the PZT actuators on the power delivered to the plate are later investigated.

Large deflection of magneto-electro-elastic laminated plates

A model for the large deflection analysis of magneto-electro-elastic laminated plates is derived. The first order shear deformation theory and the von Karman stress function approach are employed. A set of resolving partial differential equations involving kinematical variables and the stress function is obtained as a consequence of the preliminary condensation of the electro-magnetic state to the plate kinematics. A closed form solution for simply-supported plates is presented. Numerical results are carried out for plates consisting of piezoelectric BaTiO3 and piezomagnetic CoFe2O4 layers. These results show the influence of large deflections on the plate response and could be useful in the analysis and design of layered magneto-electro-elastic composite plates.

Enhanced damage index method using torsion modes of structures

A growing need has developed in the United States to obtain more specific knowledge on the structural integrity of infrastructure due to aging service lives, heavier and more frequent loading conditions, and durability issues. This need has spurred extensive research in the area of structural health monitoring over the past few decades. Several structural health monitoring techniques have been developed that are capable of locating damage in structures using modal strain energy of mode shapes. Typically in the past, bending strain energy has been used in these methods since it is a dominant vibrational mode in many structures and is easily measured. Additionally, there may be cases, such as pipes, shafts, or certain bridges, where structures exhibit significant torsional behavior as well. In this research, torsional strain energy is used to locate damage. The damage index method is used on two numerical models; a cantilevered steel pipe and a simply-supported steel plate girder bridge. Torsion damage indices are compared to bending damage indices to assess their effectiveness at locating damage. The torsion strain energy method is capable of accurately locating damage and providing additional valuable information to both of the structures\' behaviors.

Degenerate mode shapes for rectangular membranes and simply supported rectangular plates

The accuracy of the nodal patterns of combined degenerate mode shapes for a square membrane by previous researchers is investigated, and the nodal patterns of superimposed degenerate modes for square and non-square membranes and simply-supported plates are presented in a full range of amplitude ratios.

Analysis of classical and first-order shear deformable cracked orthotropic plates

The Kirchhoff and Mindlin plate theories are applied in this study to calculate the stresses and the energy release rates in delaminated orthotropic composite plates. A novel double-plate system is developed with the imposition of the kinematic continuity constraints in the interface plane. The governing equations of the system were derived in both cases. As a demonstrative example a simply-supported plate subjected to a point force was analyzed using Lévy plate formulation and the problem was solved by a state-space model. The distribution of the stress resultants and the interlaminar stresses in the uncracked part were also determined. Moreover, the distributions of the mode-II and mode-III energy release rates along the crack front were calculated by the J-integral. The 3D finite element model of the plate was created providing reference data for the analytical model. The results show that the displacement and stress fields obtained from the Kirchhoff and Mindlin theories are quite similar, but in the case of the energy release rates, transverse shear effect is necessary to consider to obtain reasonably good agreement between the analytical and numerical results.