Terms Of Displacement Components Research Articles

Vibration and damping analysis of sandwich electrorheological fluid deep arches with bi-directional FGM containers

In this project, for the first time, the dynamic features of a deep sandwich arch with an electrorheological fluid (ERF) core are evaluated. The retentive face sheets of this structure are composed of novel bi-directionally functionally graded materials (FGMs) according to a dual-index power law. The curved beam's displacement field is predicted by applying the sandwich theory, which takes into account shear deformations and rotating inertias for all layers. When calculating the equivalent features of FGM face layers, the rule of mixtures is utilized as a homogenization technique. The ERF core's constitutive law is found to be in the pre-yield zone by considering the magnitudes of the electric field applied. Employing Hamilton's principle, the dynamic equations in terms of displacement components are derived. The discrete version of these equations is achieved by the use of the Chebyshev collocation tool. Parametric investigations are carried out after analyzing the efficacy and precision of the used approach and formulation via the presentation of verification cases. In the parametric section, the contribution of each geometric and material characteristic, as well as different boundary conditions, on the system frequencies and related mode shapes and loss factors, will be explored.

Response Suppression of FGM Plate Using Piezoelectric Layers Under Parametric Uncertainty Conditions with Markovian Jump Approach

It should be noted that in addition to the geometry, constituent material also affects the strength and rigidity of the cylindrical shell, some factors that determine the transient response are its geometry and the constituent material. The capability of piezoelectric materials to adept their properties in reaction to environmental factors including electricity and loading is one of the major reasons for using in this work. Therefore, in this study, the transient response of a symmetric annular sandwich plate incorporating functionally graded core and piezoelectric layers under external harmonic force and electrical voltage is investigated. The properties of the core material vary along its thickness according to a power law model. The displacement field is represented by the third-order shear deformation theory. With the aid of Hamilton’s principle, the structural equations are obtained in terms of displacement components, then solved using the differential quadrature method. In addition, the time response is evaluated with respect to effective parameters including the internal radius, power law index, core thickness, and external voltage. According to the simulation results, the oscillation amplitude decreases as the internal radius of the plate increases over the desired time interval. Also, a higher index parameter is associated with a wider time response range. Moreover, the stability analysis of a piezoelectric system with [Formula: see text] performance is considered based on the theory of Markovian jump systems. To this end, a Markovian jump state-space model of the piezoelectric system obtained using system identification under the effect of external disturbance is presented. The [Formula: see text] stability index is selected based on a candidate Lyapunov function that leads to a set of linear matrix inequalities for each region. The uncontrolled and controlled transient responses of the coupled system under external disturbance are calculated and compared, indicating the satisfactory controller performance in the presence of external disturbance and jump in the sensor and system dynamics.

Free vibration analysis of rotating functionally graded conical shells reinforced by anisogrid lattice structure

This paper investigates the free vibration behavior of a rotating functionally graded conical shell, reinforced by an anisogrid lattice structure. The material properties of the shell are assumed to be graded in the thickness direction. The governing equations have been derived based on classical shell theory and considering the effects of centrifugal and Coriolis accelerations as well as initial hoop tension due to shell rotation. The smeared method is also employed to superimpose the stiffness contribution of the stiffeners with those of the shell to obtain the whole structure's equivalent stiffness parameters. The resulting equations, which are the coupled set of three variable coefficient partial differential equations in terms of displacement components, are solved by the Galerkin method for different boundary conditions. The obtained frequencies are compared with the available literature and the finite element software results. Finally, new results are discussed to show the effect of various parameters such as shell geometrical and material properties, stiffeners, rotating speed, and boundary conditions on natural frequencies.

Closed-form solutions for thermomechanical buckling of orthotropic composite plates

A closed-form solution is derived for the buckling of orthotropic composite plates under the effect of thermal and mechanical loads. The plates are subjected to constant temperature increment and length variation while the width expansion is constrained. The problem is formulated in terms of displacement components, studied using classical plate theory in combination with classical lamination theory. An analytical formula that relates critical temperatures to applied plate displacements is obtained. The buckling of heated, fully restrained plates is also derived as a particular case. Examples of plates made of different materials and lay-ups are presented in graphical form, and are verified by finite element analysis. The obtained formula can be used during initial design, for sensitivity analysis and also for obtaining desired buckling shapes.

Nonlinear Instability Analysis of Functionally Graded Sandwich Truncated Conical Shells Reinforced by Stiffeners Resting on Elastic Foundations

This paper investigates the nonlinear instability of eccentrically stiffened functionally graded (ES-FG) sandwich truncated conical shells subjected to the axial compressive load. The core of the FG sandwich truncated conical shells, assumed to be thin, is made of pure metal or ceramic materials and the two skin layers are made of a FG material. The shell reinforced by orthogonal stiffeners (stringers) is also made of FG materials. The change of spacing between the stringers in the meridional direction is considered. The governing equations are derived using the Donnell shell theory with von Karman geometrical nonlinearity along with the smeared technique for stiffeners. The resulting coupled set of three nonlinear partial differential equations with variable coefficients in terms of displacement components are solved by the Galerkin’s method. The closed-form expressions for determining the critical buckling load and for analyzing the postbucking load–deflection curves are obtained. The accuracy of present formulation is verified by comparing the results obtained with available ones in the literature. The effects of various parameters such stiffeners, foundations, material properties, geometric dimensions on the stability of the shells are studied in detail.

Nonlinear thermomechanical postbuckling analysis of ES-FGM truncated conical shells resting on elastic foundations

ABSTRACTThis paper examines an analytical investigation into nonlinear thermomechanical buckling and postbuckling of ES-FGM truncated conical shells. Shells are reinforced by orthogonal stiffeners. The change of spacing between stringers are taken into account. The governing equations, which are the coupled set of three variable coefficients nonlinear partial differential equations in terms of displacement components, are solved by Galerkin method. The explicit expressions for determining the critical buckling load and analyzing the postbuckling load–deflection curves are obtained. The influences of various parameters such as temperature, stiffeners, foundations, material properties, geometric dimensions on the stability of shells are considered and highlighted.

Comparison study on hysteretic energy dissipation and displacement components between cast‐in‐place and precast piers with high‐strength bars

Precast piers incorporated with high‐strength bars (≥500 MPa) have great potential to accelerate bridge construction and enhance seismic performance. Four large‐scale pier column specimens using high‐strength bars by cast‐in‐place (CIP) and precast construction were experimentally studied. The distinctions of damage pattern between CIP and precast piers were characterized by plastic hinge and toed hinge, which expounded the smaller damage in precast piers. In contrast with CIP piers, the energy dissipation of precast pier was reduced by approximately 50–60% after pier yielding. An appropriate reinforcement ratio was essential for CIP piers to maintain enough displacement ductility and prevent premature fracture of energy dissipation bars. In terms of displacement components, the plastic term of precast pier was much smaller than that of CIP pier, and the gap opening‐induced displacement predominates after pier yielding. The toed hinge was elaborated to disclose the displacement mechanism in precast piers. The experimental findings in this paper can provide additional input for seismic design of precast piers with high‐strength bars.

Thermal buckling analysis of functionally graded cylindrical shells

Thermal buckling behavior of cylindrical shell made of functionally graded material (FGM) is studied. The material constituents are composed of ceramic and metal. The material properties across the shell thickness are assumed to be graded according to a simple power law distribution in terms of the volume fraction rule of mixtures. Based on the Donnell shell theory, a system of dimensionless partial differential equations of buckling in terms of displacement components is derived. The method of separation of variables is used to transform the governing equations to ordinary differential equations (ODEs). A shooting method is used to search for the numerical solutions of the differential equations under two types of boundary conditions. Effects of the power law index, the dimensionless geometrical parameters, and the temperature ratio on the critical buckling temperature are discussed in detail.

Postbuckling nonlinear analysis of FGM truncated conical shells reinforced by orthogonal stiffeners resting on elastic foundations

This paper presents an analytical investigation on buckling and postbuckling of eccentrically stiffened functionally graded thin truncated conical shells subjected to axial compressive load and resting on elastic foundations. Shells are reinforced by rings and stringers attached to their inside. The shell material properties are graded in the thickness direction according to a volume fraction power-law distribution. The change in spacing between stringers in the meridional direction is taken into account. The theoretical formulations based on the Donnell shell theory with von Karman geometrical nonlinearity and the smeared stiffeners technique are derived. The resulting equations, which are a coupled set of three variable coefficients nonlinear partial differential equations in terms of displacement components, are solved by the Galerkin method. The closed-form expressions for determining the critical buckling load and for analyzing the postbuckling load–deflection curves are obtained. The influences of various parameters such as stiffeners, foundations, material properties and geometric dimensions on the stability of shells are considered in detail.

Nonlocal transient electrothermomechanical vibration and bending analysis of a functionally graded piezoelectric single-layered nanosheet rest on visco-Pasternak foundation

ABSTRACTThis study develops the transient thermoelectromechanical vibration and bending analysis of a functionally graded piezoelectric nanosheet rest on visco-Pasternak’s foundation. Nonlocal elasticity theory as well as classical plate theory is used to implement basic equations of the nanosheet. The plate is resting on visco-Pasternak’s foundation and subject to mechanical, thermal, and electrical loadings. Hamilton’s principle is used for derivation equations of motion in terms of displacement components. As a first case study and for validation of the responses of the system, nonlocal vibration analysis of nanosheet is studied. The effects of nonlocal parameter and nonhomogeneous index of nanosheet are studied on the fundamental frequencies of the system. As the main objective of this study, the electrothermal bending results of the nanosheet are studied. The effects of some important parameters such as nonlocal parameter, nonhomogeneous index, thickness, distribution of electric potential, and damping are calculated on the maximum deflection of the sheet under various thermal and electrical loadings.

Analytical investigation on mechanical buckling of FGM truncated conical shells reinforced by orthogonal stiffeners based on FSDT

This work presents an analytical investigation for analyzing the mechanical buckling of truncated conical shells made of functionally graded materials, subjected to axial compressive load and external uniform pressure. Shells are reinforced by closely spaced stringers and rings. The change of spacing between stringers in the meridional direction also is taken into account. Using the adjacent equilibrium criterion, the first order shear deformation theory (FSDT) and Lekhnitskii smeared stiffener technique, the linearization stability equations have been established. The resulting equations which they are the system of five variable coefficient partial differential equations in terms of displacement components are investigated by Galerkin method. The closed-form expression for determining the critical buckling load is obtained. The effects of material properties, dimensional parameters, stiffeners and semi-vertex angle on buckling behaviors of shell are considered. Shown that for thick conical shells, the use of FSDT for determining their critical buckling load is necessary and more suitable.

Buckling analysis of functionally graded material (FGM) sandwich truncated conical shells reinforced by FGM stiffeners filled inside by elastic foundations

An analytical solution for buckling of an eccentrically stiffened sandwich truncated conical shell is investigated. The shell consists of two functionally graded material (FGM) coating layers and a core layer which are metal or ceramic subjected to an axial compressive load and an external uniform pressure. Shells are reinforced by stringers and rings, in which the material properties of shells and stiffeners are graded in the thickness direction following a general sigmoid law distribution. Two models of coated shell-stiffener arrangements are investigated. The change of the spacing between stringers in the meridional direction is taken into account. A couple set of three-variable-coefficient partial differential equations in terms of displacement components are solved by the Galerkin method. A closed-form expression for determining the buckling load is obtained. The numerical examples are presented and compared with previous works.

Theory of equivalent staggered‐grid schemes: application to rotated and standard grids in anisotropic media

ABSTRACTThe previous finite‐difference numerical schemes designed for direct application to second‐order elastic wave equations in terms of displacement components are strongly dependent on Poisson's ratio. This fact makes theses schemes useless for modelling in offshore regions or even in onshore regions where there is a high Poisson's ratio material. As is well known, the use of staggered‐grid formulations solves this drawback. The most common staggered‐grid algorithms apply central‐difference operators to the first‐order velocity–stress wave equations. They have been one of the most successfully applied numerical algorithms for seismic modelling, although these schemes require more computational memory than those mentioned based on second‐order wave equations. The goal of the present paper is to develop a general theory that enables one to formulate equivalent staggered‐grid schemes for direct application to hyperbolic second‐order wave equations. All the theory necessary to formulate these schemes is presented in detail, including issues regarding source application, providing a general method to construct staggered‐grid formulations to a wide range of cases. Afterwards, the equivalent staggered‐grid theory is applied to anisotropic elastic wave equations in terms of only velocity components (or similar displacements) for two important cases: general anisotropic media and vertical transverse isotropy media using, respectively, the rotated and the standard staggered‐grid configurations. For sake of simplicity, we present the schemes in terms of velocities in the second‐ and fourth‐order spatial approximations, with second‐order approximation in time for 2D media. However, the theory developed is general and can be applied to any set of second‐order equations (in terms of only displacement, velocity, or even stress components), using any staggered‐grid configuration with any spatial approximation order in 2D or 3D cases. Some of these equivalent staggered‐grid schemes require less computer memory than the corresponding standard staggered‐grid formulation, although the programming is more evolved. As will be shown in theory and practice, with numerical examples, the equivalent staggered‐grid schemes produce results equivalent to corresponding standard staggered‐grid schemes with computational advantages. Finally, it is important to emphasize that the equivalent staggered‐grid theory is general and can be applied to other modelling contexts, e.g., in electrodynamical and poroelastic wave propagation problems in a systematic and simple way.

Thermodynamic behaviour of microstretch viscoelastic solids with internal heat source

The dynamical interactions caused by a line heat source moving inside a homogeneous isotropic thermo-microstretch viscoelastic half space, whose surface is subjected to a thermal load, are investigated. The formulation is in the context of generalized thermoelasticity theories proposed by Lord and Shulman (J. Mech. Phys. Solid, 15, 299 (1967)) and Green and Lindsay (Thermoelasticity, J. Elasticity, 2, 1 (1972)). The surface is assumed to be traction free. The solutions in terms of displacement components, mechanical stresses, temperature, couple stress, and microstress distribution are procured by employing the normal mode analysis. The numerical estimates of the considered variables are obtained for an aluminium–epoxy material. The results obtained are demonstrated graphically to show the effect of moving heat source and viscosity on the displacement, stresses, and temperature distribution.

Free vibration of simply supported rectangular plates on Pasternak foundation: An exact and three-dimensional solution

This paper deals with exact solution for free vibration analysis of simply supported rectangular plates on elastic foundation. The solution is on the basis of three dimensional elasticity theory. The foundation is described by the Pasternak (two-parameter) model. First, the Navier equations of motion are replaced by three decoupled equations in terms of displacement components. Then, these equations are solved in a semi-inverse method. The obtained displacement field satisfies all the boundary conditions of the problem in a point wise manner. The solution is in the form of a double Fourier sine series. Then free-vibration characteristics of rectangular plates resting on elastic foundations for different thickness/span ratios and foundation parameters are studied. The numerical results are compared with the available results in the literature. Important parameters on the accuracy of plate theories and freevibration characteristics of rectangular plates resting on elastic foundations are discussed.

On the stability of functionally graded truncated conical shells reinforced by functionally graded stiffeners and surrounded by an elastic medium

This paper presents an analytical approach to investigate the mechanical buckling load of eccentrically stiffened functionally graded truncated conical shells surrounded by elastic medium and subjected to axial compressive load and external uniform pressure. Shells are reinforced by stringers and rings in which material properties of shell and stiffeners are graded in the thickness direction according to a volume fraction power-law distribution. The elastic medium is assumed as two-parameter elastic foundation model proposed by Pasternak. The change of spacing between stringers in the meridional direction is taken into account. The equilibrium and linearized stability equations for stiffened shells are derived based on the classical shell theory and smeared stiffeners technique. The resulting equations which they are the couple set of three variable coefficient partial differential equations in terms of displacement components are investigated by Galerkin method and the closed-form expression for determining the buckling load is obtained. Four cases of stiffener arrangement are analyzed. Carrying out some computations, effects of foundation, stiffener and input factors on stability of shell have been studied. The effectiveness of FGM stiffeners in enhancing the stability of cylindrical shells comparing with homogenuos stiffener is shown.

Instability of eccentrically stiffened functionally graded truncated conical shells under mechanical loads

This paper is concerned with the mechanical buckling load of an eccentrically stiffened truncated conical shells made of functionally graded materials and subjected to axial compressive load and external uniform pressure by analytical method. Shells are reinforced by stringers and rings. The change of spacing between stringers in the meridional direction is taken into account. Material properties of shell are graded in the thickness direction according to a volume fraction power-law distribution. The equilibrium and linearized stability equations for stiffened shells are derived based on the classical shell theory and smeared stiffeners technique. The resulting equations which they are the couple set of three variable coefficient partial differential equations in terms of displacement components are investigated by Galerkin method and the closed-form expression for determining the buckling load is obtained. The effects of stiffeners, material and dimensional parameters are analyzed in detail.

Three-dimensional vibration analysis of a finite cylinder using the generalised differential quadrature method

Using the Navier’s equations of the motion in cylindrical coordinate, vibration analysis of a finite cylinder is accomplished. These equations are transformed into the dimensionless forms in terms of non-dimensional displacements and time. The different boundary conditions are considered for this problem which the former analytical and semi-analytical methods could not treat them. Two ends boundary condition namely fixed-fixed and fixed-free are examined. The dimensionless governing equations of the motions in conjunction with the boundary condition are discretised making use of the generalised differential quadrature (GDQ) method. The ensuing system of algebraic equations in terms of displacement components leads to an eigenvalue problem to find the natural frequencies of the problem. The comparison of the results with some earlier works validates the methodology and formulation of the problem.

Geometrically induced stress singularities in a piezoelectric body of revolution

An eigenfunction expansion approach is combined with a power series solution technique to establish the asymptotic solutions for geometrically induced electroelastic singularities in a piezoelectric body of revolution, with its direction of polarization not parallel to the axis of revolution. The asymptotic solutions are obtained by directly solving the three-dimensional equilibrium and Maxwell’s equations in terms of displacement components and electric potential. When the direction of polarization is not along the axis of revolution, the assumption of axisymmetric deformation that is often made in the published literature is not valid, and the direction of polarization and the circular coordinate variable can substantially affect the singularities. The numerical results related to singularity orders are shown in graphical form for bodies of revolution that comprise a single material (PZT-4 or PZT-5H) or bonded piezo/piezo (PZT-4/PZT-5H) or piezo/isotropic elastic (PZT-4/Al or PZT-5H/Al) materials. This is the first study to present results for the direction of polarization not along the axis of revolution.

A novel analytical approach for the buckling analysis of moderately thick functionally graded rectangular plates with two simply-supported opposite edges

In this article, a novel analytical method for decoupling the coupled stability equations of functionally graded (FG) rectangular plates is introduced. Based on the Mindlin plate theory, the governing stability equations that are coupled in terms of displacement components are derived. Introducing four new functions, the coupled stability equations are converted into two independent equations. The obtained equations have been solved for buckling analysis of rectangular plates with simply-supported two edges and arbitrary boundary conditions along the other edges (Levy boundary conditions). The critical buckling loads are presented for different loading conditions, various thickness to side and aspect ratios, some powers of FG materials, and various boundary conditions. The presented results for buckling of moderately thick FG plates with two simply-supported edges are reported for the first time.