J. N. Reddy

An introduction to the finite element method

1 Introduction 2 Mathematical Preliminaries, Integral Formulations, and Variational Methods 3 Second-order Differential Equations in One Dimension: Finite Element Models 4 Second-order Differential Equations in One Dimension: Applications 5 Beams and Frames 6 Eigenvalue and Time-Dependent Problems 7 Computer Implementation 8 Single-Variable Problems in Two Dimensions 9 Interpolation Functions, Numerical Integration, and Modeling Considerations 10 Flows of Viscous Incompressible Fluids 11 Plane Elasticity 12 Bending of Elastic Plates 13 Computer Implementation of Two-Dimensional Problems 14 Prelude to Advanced Topics

A Simple Higher-Order Theory for Laminated Composite Plates

A higher-order shear deformation theory of laminated composite plates is developed. The theory contains the same dependent unknowns as in the first-order shear deformation theory of Whitney and Pagano (1970), but accounts for parabolic distribution of the transverse shear strains through the thickness of the plate. Exact closed-form solutions of symmetric cross-ply laminates are obtained and the results are compared with three-dimensional elasticity solutions and first-order shear deformation theory solutions. The present theory predicts the deflections and stresses more accurately when compared to the first-order theory.

ANALYSIS OF FUNCTIONALLY GRADED PLATES

Theoretical formulation, Naviers solutions of rectangular plates, and finite element models based on the third-order shear deformation plate theory are presented for the analysis of through-thickness functionally graded plates. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The formulation accounts for the thermomechanical coupling, time dependency, and the von Karman-type geometric non-linearity. Numerical results of the linear third-order theory and non-linear first-order theory are presented to show the effect of the material distribution on the deflections and stresses. Copyright

Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates

Abstract The response of functionally graded ceramic metal plates is investigated using a plate finite element that accounts for the transverse shear strains, rotary inertia and moderately large rotations in the von Karman sense. The static and dynamic response of the functionally graded material (fgm) plates are investigated by varying the volume fraction of the ceramic and metallic constituents using a simple power law distribution. Numerical results for the deflection and stresses are presented. The effect of the imposed temperature field on the response of the fgm plate is discussed. It is found that in general, the response of the plates with material properties between those of the ceramic and metal is not intermediate to the responses of the ceramic and metal plates.

THERMOMECHANICAL ANALYSIS OF FUNCTIONALLY GRADED CYLINDERS AND PLATES

The dynamic thermoelastic response of functionally graded cylinders and plates is studied. Thermomechanical coupling is included in the formulation, and a finite element model of the formulation is developed. The heat conduction and the thermoelastic equations are solved for a functionally graded axisymmetric cylinder subjected to thermal loading. In addition, a thermoelastic boundary value problem using the first-order shear deformation plate theory (FSDT) that accounts for the transverse shear strains and the rotations, coupled with a three-dimensional heat conduction equation, is formulated for a functionally graded plate. Both problems are studied by varying the volume fraction of a ceramic and a metal using a power law distribution.

A refined nonlinear theory of plates with transverse shear deformation

Abstract A higher-order shear deformation theory of plates accounting for the von Karman strains is presented. The theory contains the same dependent unknowns as in the Hencky-Mindlin type first-order shear deformation theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate. Exact solutions of simply supported plates are obtained using the linear theory and the results are compared with the exact solutions of 3-D elasticity theory, the first order shear deformation theory, and the classical plate theory. The present theory predicts the deflections, stresses, and frequencies more accurately when compared to the first-order theory and the classical plate theory.

Vibration of functionally graded cylindrical shells

Abstract Functionally gradient materials (FGMs) have attracted much attention as advanced structural materials because of their heat-resistance properties. In this paper, a study on the vibration of cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of configurations of the constituent materials on the frequencies. The properties are graded in the thickness direction according to a volume fraction power-law distribution. The results show that the frequency characteristics are similar to that observed for homogeneous isotropic cylindrical shells and the frequencies are affected by the constituent volume fractions and the configurations of the constituent materials. The analysis is carried out with strains–displacement relations from Love’s shell theory and the eigenvalue governing equation is obtained using Rayleigh–Ritz method. The present analysis is validated by comparing results with those in the literature.

Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory

Abstract A higher-order shear deformation theory is used to determine the natural frequencies and buckling loads of elastic plates. The theory accounts for parabolic distribution of the transverse shear strains through the thickness of the plate and rotary inertia. Exact solutions of simply supported plates are obtained and the results are compared with the exact solutions of three-dimensional elasticity theory, the first-order shear deformation theory, and the classical plate theory. The present theory predicts the frequencies and buckling loads more accurately when compared to the first-order and classical plate theories.

Nonlocal continuum theories of beams for the analysis of carbon nanotubes

The equations of motion of the Euler–Bernoulli and Timoshenko beam theories are reformulated using the nonlocal differential constitutive relations of Eringen [International Journal of Engineering Science 10, 1–16 (1972)]. The equations of motion are then used to evaluate the static bending, vibration, and buckling responses of beams with various boundary conditions. Numerical results are presented using the nonlocal theories to bring out the effect of the nonlocal behavior on deflections, buckling loads, and natural frequencies of carbon nanotubes.

A new beam finite element for the analysis of functionally graded materials

A new beam element is developed to study the thermoelastic behavior of functionally graded beam structures. The element is based on the first-order shear deformation theory and it accounts for varying elastic and thermal properties along its thickness. The exact solution of static part of the governing differential equations is used to construct interpolating polynomials for the element formulation. Consequently, the stiffness matrix has super-convergent property and the element is free of shear locking. Both exponential and power-law variations of material property distribution are used to examine different stress variations. Static, free vibration and wave propagation problems are considered to highlight the behavioral difference of functionally graded material beam with pure metal or pure ceramic beams.