Terms Of Displacement Function Research Articles

Three-dimensional active-control hygrothermal dynamic behavior of PFMLs structure subjected to laser heat treatment with dual-ellipse distribution form

Based on three-dimensional transient thermal conduction theory, the refined plate theory and Hamilton variational principle, the constitutive equations, geometrical equations, and governing equations for fiber metal laminates with piezoelectric layer (PFMLs) under hygrothermal environment is presented. Subsequently, the Newmark method and Galerkin method are used to solve the governing equations which in terms of displacement functions. The results show that electric potential function, fiber orientation, external load as well as thickness ratio have great influence on the deflection as well as normal stress. The selection of (45°/0°) fiber orientation is more resistance to deformation and normal stress.

Nonlocal nonlinear higher-order finite element model for static bending of curved nanobeams including graphene platelets reinforcement

The aim of this work is concerned with the nonlocal nonlinear bending characteristics of non-classical isotropic and composite curved beams based on graphene platelets reinforcement, viz., nano size/micro beams. The size-dependant effect prominent in micro/nanobeams is incorporated in the formulation based on the nonlocal theory using size-dependent continuum approach. The proposed structural model includes both moderately large displacement and rotations, respectively. The system governing equations are established in terms of displacement functions using the principle of virtual work statements. A three-noded shear deformable curved beam finite element using a higher-order sinusoidal function based beam theory is employed; the solutions are obtained by adopting direct iterative procedure. The efficacy of the model is tested against the analytical studies available for the nano-size beams. A detailed analysis is made to highlight the influence of various design variables like thickness ratio, curvature of beam, nonlocal parameter, and the support conditions on the nonlinear bending behavior of curved nano/micro-size beams.

Hygrothermal dynamic behavior of fiber metal laminated structure under dual-ellipse distribution heat source

Refined plate theory could account for parabolic variation of transverse shear stress through the thickness, and satisfy the zero traction boundary conditions on the top and bottom surfaces of the plate without introducing shear correction factor, of which all displacement variables can be depicted in functions of shear and bending components. Based on three-dimensional transient thermal conduction theory and the refined plate theory, an analytical solution of fiber metal laminated (FML) plate under hygrothermal environment is presented. In present work, distinct from other literatures, the separate variable method and Newton-Cotes method is applied to solve the three-dimensional transient temperature field (subjected to moving dual-ellipse distribution laser heat source) firstly. Secondly, the constitutive equations and geometrical equations are derived. Subsequently, the governing equations in hygrothermal environment are obtained. Finally, the Newmark method and Galerkin method are used to solve the governing equations of which in terms of displacement functions. The results show that fiber orientation affects the bending stiffness, the selection of (45°/ 0°) fiber orientation is more resistance to deformation and normal stress, and bending part is more sensitive to shear factor between layers.

An analysis and experimental validation of natural frequencies of elastic ellipsoids with the Rayleigh–Ritz method

The Rayleigh–Ritz method is widely employed to analyze free vibrations of elastic solids and structures. For a simple formulation and efficient evaluation of vibrations of elastic ellipsoids, which also include spheres, the Cartesian coordinate system is utilized. It is hoped that the formulation with lengthy expressions can be solved with fewer terms of displacement functions in Chebyshev polynomials for simple evaluations of stiffness and mass matrices of the elastic ellipsoids with the procedure. The vibrations of elastic ellipsoids are calculated with geometric parameters for the validation of the procedure and formulation with known results and the analysis from this study.

Axisymmetric Free Vibration of Layered Cylindrical Shell Filled with Fluid

The aim of the study is to analyse the axisymmetric free vibration of layered cylindrical shells filled with a quiescent fluid. The fluid is assumed to be incompressible and inviscid. The equations of axisymmetric vibrations of layered cylindrical shell filled with fluid, on the longitudinal and transverse displacement components are obtained using Love’s first approximation theory. The solutions of displacement functions are assumed in a separable form to obtain a system of coupled differential equations in terms of displacement functions. The displacement functions are approximated by Bickley-type splines. A generalized eigenvalue problem is obtained and solved numerically for a frequency parameter and an associated eigenvector of spline coefficients. Two layered shells with three different types of materials under clamped-clamped boundary conditions are considered. Parametric studies are made on the variation of the frequency parameter with respect to length-to-radius ratio and length-to-thickness ratio.

Free Vibration of Angle-ply Laminated Conical Shell Frusta with Linear and Exponential Thickness Variations

Free vibration of laminated conical shell frusta of variable thickness is studied using spline approximation. This problem includes first order shear deformation and considers shells as antisymmetric angle-ply orientation. The governing differential equations of the shells are resolved in terms of displacement functions and rotational functions. These functions are approximated using splines and the method of collocation is adopted for simultaneous algebraic equations. These equations become generalized eigenvalue problems and are solved numerically to avail eigenfrequencies and the corresponding eigenvectors. The variation of frequencies is analysed with respect to the cone angle, aspect ratio, material properties, number of layers, and thickness variation.

Free vibration of layered truncated conical shells filled with quiescent fluid using spline method

Free vibration of layered truncated conical shells filled with quiescent fluid using spline method is studied. Love’s first approximation theory is used to formulate the equations of motion of truncated conical shells. Velocity potential and Bernoulli’s equations have been applied for the expression of the pressure of the fluid. The fluid is assumed to be incompressible, inviscid and quiescent. The solutions of displacement functions are assumed in a separable form to obtain a system of coupled differential equations in terms of displacement functions. The displacement functions are approximated by Bickley-type splines to obtain the generalized eigenvalue problem by combining with boundary conditions. A generalized eigenvalue problem is obtained and solved numerically for frequency parameter and an associated eigenvector of spline coefficients. Two layered shells are considered. Parametric studies are made to investigate the effect of fluid on the frequencies with respect to the relative layer thickness, semi cone angle and length ratio.

Free vibration of symmetric angle ply truncated conical shells under different boundary conditions using spline method

Free vibration of symmetric angle-ply laminated truncated conical shell is analyzed to determine the effects of frequency parameter and angular frequencies under different boundary condition, ply angles, different material properties and other parameters. The governing equations of motion for truncated conical shell are obtained in terms of displacement functions. The displacement functions are approximated by cubic and quintic splines resulting into a generalized eigenvalue problem. The parametric studies have been made and discussed.

Vibration analysis of cross-ply laminated truncated conical shells using a spline method

Free vibration of symmetric and antisymmetric cross-ply composite laminated truncated conical shells using the spline function technique is studied. The equilibrium equations for a truncated conical shells are formulated including first-order shear deformation theory. The equations of motion are derived in terms of displacement functions and rotational functions using stress–strain and strain–displacement relationships. The coupled differential equations are solved using Bickley-type splines to obtain the generalized eigenvalue problem by combining suitable boundary conditions. The convergence and comparative results are presented. Both symmetric and anti-symmetric cross-ply shells are considered using various types of material properties. Parametric studies are made to investigate the effect of transverse shear deformation on the frequency parameter with respect to the thickness ratio, length ratio, cone angle, and circumferential mode number using different numbers of layers under various types of boundary conditions.

Asymmetric free vibrations of laminated annular cross-ply circular plates including the effects of shear deformation and rotary inertia: spline method

Asymmetric free vibrations of annular cross-ply circular plates are studied using spline function approximation. The governing equations are formulated including the effects of shear deformation and rotary inertia. Assumptions are made to study the cross-ply layered plates. A system of coupled differential equations are obtained in terms of displacement functions and rotational functions. These functions are approximated using Bickley- type spline functions of suitable order. Then the system is converted into the eigenvalue problem by applying the point collocation technique and suitable boundary conditions. Parametric studies have been made to investigate the effect of transverse shear deformation and rotary inertia on frequency parameter with respect to the circumferential node number, radii ratio and thickness to radius ratio for both symmetric and anti-symmetric cross-ply plates using various types of material properties.

Stress singularities in bimaterial bodies of revolution

An eigenfunction expansion solution is first developed for determining the stress singularities of bimaterial bodies of revolution by directly solving the equilibrium equations of three-dimensional elasticity in terms of displacement functions. The characteristic equations are explicitly given for determining the stress singularities in the vicinity of the interface corner of two intersecting bodies of revolution having a sharp corner with free boundary conditions along the corner. The characteristic equations are found to be equivalent to a combination of the characteristic equations for plane elasticity problems and St. Venant torsion problems. The strength of stress singularities varying with the interface angles is also investigated. The first known asymptotic solutions for the displacement and stress fields are also explicitly shown for some interface angles. The present results will be useful not only for understanding the singularity behaviors of stresses in the vicinity of a revolution interface corner, but also for developing accurate numerical solutions with fast convergence for stress or vibration analysis of a body of revolution having an interface corner.

Analysis of Stress Singularities at Bi-Material Corners in Reddy's Theory of Plate Bending

AbstractThe order of stress singularity at a sharp corner of a plate needs to be known before a numerical approach can be taken to determine accurately the stress distribution of a plate with irregular geometry (such as a V-notch) under loading. This work analyzes the order of the stress singularity at a bi-material corner of a thick plate under bending, based on Reddy's third-order shear deformation plate theory. An eigenfunction expansion technique is used to derive the asymptotic displacement field in the vicinity of the sharp corner by solving the equilibrium equations in terms of displacement functions. This paper explicitly shows the first known characteristic equations for determining the order of the stress singularity at the interface corner of a bonded dissimilar isotropic plate. Moreover, the numerical results are given in graphic form for the order of stress singularity at the interface corner in bonded dissimilar isotropic plates and at the vertex of a bi-material wedge with free radial edges. The results presented herein fill some of the gaps in the literature

Analytical solution for cylindrical thin shells with normally intersecting nozzles due to external moments on the ends of shells

The stress analysis based on the theory of a thin shell is carried out for cylindrical shells with normally intersecting nozzles subjected to external moment loads on the ends of shells with a large diameter ratio(ρ 0 «0. 8). Instead of the Donnell shallow shell equation, the modified Morley equation, which is applicable toρ 0(R/T)1/2 »1, is used for the analysis of the shell with cutout. The solution in terms of displacement function for the nozzle with a nonplanar end is based on the Goldenveizer equation. The boundary forces and displacements at the intersection are all transformed from Gaussian coordinates (α, β) on the shell, or Gaussian coordinates (ζ, θ) on the nozzle into three-di-mensional cylindrical coordinates(ρ,θ, z). Their expressions on the intersecting curve are periodic functions ofθ and expanded in Fourier series. Every harmonic of Fourier coefficients of boundary forces and displacements are obtained by numerical quadrature. The results obtained are in agreement with those from the three-dimensional finite element method and experiments.

A general nonlinear theory of sandwich panels

The present work can be considered as an extension to sandwich panels of the moderately large deflection theory of plates formulated by the author in [1, 2] and as a refinement of the contributions to the theory of sandwich panels by Eringen in [3] and by Yu in [4, 5]. Variational integrals of the stress equations of motion and boundary conditions (23) consistent with the simplified nonlinear strain-displacement relations (1) are obtained from the Hamilton principle. Each layer of the sandwich panel is of different thickness and of a different anisotropic material having one plane of elastic symmetry. Transverse shear and transverse normal strain as well as the inertia and thermal effects are included in the analysis. Four special cases of practical importance are considered. Finally the equilibrium equations in terms of displacement functions are obtained and compared with the equilibrium equations obtained by Eringen in [3].