Separated Solution Procedure for Bending of Circular Plates With Circular Holes

1991 ◽  
Vol 44 (11S) ◽  
pp. S27-S35 ◽  
Author(s):  
M. D. Bird ◽  
C. R. Steele
Keyword(s):  
Boundary Conditions ◽  
Processing Time ◽  
Solution Procedure ◽  
Trigonometric Fourier Series ◽  
Two Dimensional ◽  
Circular Plates ◽  
Arbitrary Boundary ◽  
Circular Holes ◽  
Circular Domains ◽  
Harmonic Equation

A separated solution procedure is presented for the two-dimensional bi-harmonic equation on circular domains with circular holes and arbitrary boundary conditions. The solutions use the traditional trigonometric Fourier series on the boundaries with a power series decay into the domain. The interaction of the boundaries is expressed simply and exactly resulting in quick processing time. The only simplification made is the use of a finite number of terms in the boundary conditions.

A Solution Procedure for Laplace’s Equation on Multiply Connected Circular Domains

1992 ◽  
Vol 59 (2) ◽  
pp. 398-404 ◽  
Author(s):  
M. D. Bird ◽  
C. R. Steele
Keyword(s):  
Boundary Conditions ◽  
Transformation Method ◽  
Solution Procedure ◽  
Laplace’S Equation ◽  
Shape Functions ◽  
Arbitrary Boundary ◽  
Laplace's Equation ◽  
Multiply Connected ◽  
Circular Holes ◽  
Circular Domains

A solution procedure is presented for the two-dimensional Laplace’s equation on circular domains with circular holes and arbitrary boundary conditions. The shape functions use the traditional trigonometric Fourier series on the boundaries with a power series decay into the domain thereby satisfying the governing equation exactly. The interaction of the boundaries is expressed simply and exactly resulting in quick processing time. The only simplification made is the use of a finite number of terms in the boundary conditions. The results are compared with a Green’s function method due to Naghdi (1991) and a Mo¨bius transformation method due to Honein et al. (1991).

Free vibration analysis of three-layer thin cylindrical shell with variable thickness two-dimensional FGM middle layer under arbitrary boundary conditions

2021 ◽  
pp. 109963622110204
Author(s):  
Xue-Yang Miao ◽  
Chao-Feng Li ◽  
Yu-Lin Jiang ◽  
Zi-Xuan Zhang
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Volume Fraction ◽  
Ritz Method ◽  
Admissible Function ◽  
Free Vibration Analysis ◽  
Middle Layer ◽  
Two Dimensional ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions

In this paper, a unified method is developed to analyze free vibrations of the three-layer functionally graded cylindrical shell with non-uniform thickness. The middle layer is composed of two-dimensional functionally gradient materials (2D-FGMs), whose thickness is set as a function of smooth continuity. Four sets of artificial springs are assigned at the ends of the shells to satisfy the arbitrary boundary conditions. The Sanders’ shell theory is used to obtain the strain and curvature-displacement relations. Furthermore, the Chebyshev polynomials are selected as the admissible function to improve computational efficiency, and the equation of motion is derived by the Rayleigh–Ritz method. The effects of spring stiffness, volume fraction indexes, configuration on of shell, and the change in thickness of the middle layer on the modal characteristics of the new structural shell are also analyzed.

A New Global Spatial Discretization Method for Calculating Dynamic Responses of Two-Dimensional Continuous Systems With Application to a Rectangular Kirchhoff Plate

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
K. Wu ◽  
W. D. Zhu
Keyword(s):  
Boundary Conditions ◽  
Continuous System ◽  
Dynamic Responses ◽  
Kirchhoff Plate ◽  
Continuous Systems ◽  
Discretization Method ◽  
Two Dimensional ◽  
Spatial Discretization ◽  
Arbitrary Boundary ◽  
Discretization Methods

A new global spatial discretization method (NGSDM) is developed to accurately calculate natural frequencies and dynamic responses of two-dimensional (2D) continuous systems such as membranes and Kirchhoff plates. The transverse displacement of a 2D continuous system is separated into a 2D internal term and a 2D boundary-induced term; the latter is interpolated from one-dimensional (1D) boundary functions that are further divided into 1D internal terms and 1D boundary-induced terms. The 2D and 1D internal terms are chosen to satisfy prescribed boundary conditions, and the 2D and 1D boundary-induced terms use additional degrees-of-freedom (DOFs) at boundaries to ensure satisfaction of all the boundary conditions. A general formulation of the method that can achieve uniform convergence is established for a 2D continuous system with an arbitrary domain shape and arbitrary boundary conditions, and it is elaborated in detail for a general rectangular Kirchhoff plate. An example of a rectangular Kirchhoff plate that has three simply supported boundaries and one free boundary with an attached Euler–Bernoulli beam is investigated using the developed method and results are compared with those from other global and local spatial discretization methods. Advantages of the new method over local spatial discretization methods are much fewer DOFs and much less computational effort, and those over the assumed modes method (AMM) are better numerical property, a faster calculation speed, and much higher accuracy in calculation of bending moments and transverse shearing forces that are related to high-order spatial derivatives of the displacement of the plate with an edge beam.

Exact Perturbation for the Vibration of Almost Annular or Circular Plates

1995 ◽  
Author(s):  
R. G. Parker ◽  
C. D. Mote
Keyword(s):  
Circular Domain ◽  
Circular Plates ◽  
Irregular Domain ◽  
Third Order ◽  
Element Analysis ◽  
Shape Deviation ◽  
Arbitrary Boundary ◽  
Boundary Shape ◽  
Circular Domains ◽  
And Control

Abstract Using perturbation analysis, the eigensolutions for plate vibration problems on nearly annular or circular domains are determined. The irregular domain eigensolutions are calculated as perturbations of the corresponding annular or circular domain eigensolutions. These perturbations are determined exactly. The simplicity of these exact solutions allows the perturbation to be carried through third order for distinct unperturbed eigenvalues and through second order for degenerate unperturbed eigenvalues. Furthermore, this simplicity allows the resulting orthonormalized eigenfunctions to be readily incorporated into response, system identification, and control analyses. The clamped, nearly circular plate is studied in detail, and the exact eigensolution perturbations are derived for an arbitrary boundary shape deviation. Rules governing the splitting of degenerate unperturbed eigenvalues at both first and second orders of perturbation are presented. These rules, which apply for arbitrary shape deviation, generalize those obtained in previous works where specific, discrete asymmetries and first order splitting are examined. The eigensolution perturbations and splitting rules reduce to simple, algebraic formulae in the Fourier coefficients of the boundary shape asymmetry. Elliptical plate eigensolutions are calculated and compared to finite element analysis and, for the fundamental eigenvalue, to the exact solution given by Shibaoka (1956).

Polynomial Based Elasticity Solutions of Beams and their Numerical Validation: A Review

2011 ◽  
Vol 311-313 ◽  
pp. 2315-2321
Author(s):  
Sebin Jose ◽  
Sunil Bhat
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Boundary Conditions ◽  
Two Dimensional ◽  
Element Analysis ◽  
Stress Problem ◽  
Elasticity Solutions ◽  
Harmonic Equation ◽  
Complex Cases ◽  
Good Agreement

Solution of two-dimensional stress problem is reduced to integration of bi-harmonic equation[1].A polynomial is chosen as Airy’s stress function.Constants of the polynomial[2] are found by fulfilling the boundary conditions. Stress solutions are obtained from.The paper presents polynomial based stress solutions of beams for complex cases involving offset loads and other combinations with offset loads.The results are compared with those obtained from finite element analysis[3] and conventional methods.The results are in good agreement with each other.

Electric–elastic analysis of multilayered two-dimensional decagonal quasicrystal circular plates with simply supported or clamped boundary conditions

2021 ◽  
pp. 108128652098161
Author(s):  
Yunzhi Huang ◽  
Min Zhao ◽  
Miaolin Feng
Keyword(s):  
Boundary Conditions ◽  
Electric Fields ◽  
Radial Coordinate ◽  
Elastic Analysis ◽  
Two Dimensional ◽  
Circular Plates ◽  
State Equations ◽  
Element Analysis ◽  
Decagonal Quasicrystal ◽  
Simply Supported

A three-dimensional (3D) electric–elastic analysis of multilayered two-dimensional decagonal quasicrystal (QC) circular plates with simply supported or clamped boundary conditions is presented through a state vector approach. Both perfect and imperfect bonds between the layers are considered by adjusting the parameter sets in the model. Governing equations for the plates subjected to electric or elastic load on the bottom surfaces are derived using the state equations and the propagator matrix method. We explicitly obtain the analytical solution by writing the physical variables as Bessel series expansions and polynomial functions with respect to the radial coordinate. The solution is validated by comparing the numerical results with the 3D finite element analysis. The basic physical quantities of the plates in the phonon, phason, and electric fields are computed in the numerical examples. Result shows that the QC layers as coatings decrease the deflection in the phonon and phason fields of plates. The phonon–phason coupling elastic modulus and piezoelectric constant produce positive and negative effects on the magnitudes of stresses. Besides, compliance coefficients of the weak interface in the phonon field contribute more to the variations than those in the phason field.

Elastic deformations of infinite strips

1950 ◽  
Vol 46 (1) ◽  
pp. 164-181 ◽  
Author(s):  
H. G. Hopkins
Keyword(s):  
Boundary Conditions ◽  
Mathematical Problem ◽  
The Other ◽  
Transverse Displacement ◽  
Two Dimensional ◽  
Airy Stress Function ◽  
Elastic Deformations ◽  
Isotropic Plates ◽  
Edge Force ◽  
Harmonic Equation

ABSTRACTIn this paper, Fourier integrals are used to solve some elastic problems of generalized plane stress and small transverse displacements in infinitely long, rectangular, isotropic plates stressed only at their edges. The Airy stress function and the transverse displacement satisfy the two-dimensional bi-harmonic equation, and the basic mathematical problem is to solve this equation subject to different sets of boundary conditions. Little attention has been given hitherto to problems in which some of the boundary conditions depend directly upon displacements. Here the general problem is solved when one long edge is fixed, and stresses or displacements are arbitrarily prescribed at the other, with no stresses and displacements at infinity. The problem of a concentrated edge force is discussed in detail and numerical values of the stresses at the fixed edge are given.

scholarly journals Behavior of Uniform Anisotropic Beams of Rectangular Section under Transverse Impact of a Mass

1997 ◽  
Vol 4 (2) ◽  
pp. 125-141 ◽  
Author(s):  
Lu Chun ◽  
K. Y. Lam
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Laminated Composite ◽  
Solution Procedure ◽  
Beam Deflection ◽  
Transverse Impact ◽  
Lagrange Principle ◽  
Arbitrary Boundary ◽  
Force Contact ◽  
Laminated Composite Beam

A numerical method is presented to investigate the dynamic response of uniform orthotropic beams subjected to an impact of a mass. Higher order shear deformation and rotary inertia are included in the analysis of the beams. The impactor and laminated composite beam are treated as a system. The nonlinear differential governing equations of motion are then derived based on the Lagrange principle and modified nonlinear contact law, and solved numerically. The solution procedure is applicable to arbitrary boundary conditions. Numerical results are compared with those available in the literature to demonstrate the validity of the method, and very good agreement is achieved. The effects of boundary conditions on the contact force, contact duration, stress distributions, and beam deflection are discussed.

Sign in / Sign up

Export Citation Format

Share Document