THE ELASTIC SOLUTION OF CONCENTRATED FORCE ACTING IN ORTHOGONAL ANISOTROPIC HALF-PLANE AND CONSTANT ELEMENT FUNDAMENTAI FORMULAE OF BOUNDARY ELEMENT METHOD

In this paper, the elastic solutions of concentrated force acting in orthogonal anisotropic half-plane are derived by imaginal method and the formulae of coefficient matrix for constant element are put forward. To solve half-plane problems numerically by BEM, this paper provides the necessary formulae. Because the expressions of fundamental solutions are very simple, the. object functions could be obtained for every integral of constant element and higher order element of indirect BEM. Thus, the procedure of integration could be avoided in calculation program

Revisiting the elastic solution for an inner-pressured functionally graded thick-walled tube within a uniform magnetic field

In this paper, the mechanical responses of a thick-walled functionally graded hollow cylinder subject to a uniform magnetic field and inner-pressurized loads are studied. Rather than directly assume the material constants as some specific function forms displayed in pre-studies, we firstly give the volume fractions of different constituents of the functionally graded material(FGM) cylinder and then determine the expressions of the material constants. With the use of the Voigt method, the corresponding analytical solutions of displacements in the radial direction, the strain and stress components, and the perturbation magnetic field vector are derived. In the numerical part, the effects of the volume fraction on the displacement, strain and stress components, and the magnetic perturbation field vector are investigated. Moreover, by some appropriate choices of the material constants, we find that the obtained results in this paper can reduce to some special cases given in the previous studies.

Elasticity solution of clamped-simply supported beams with variable thickness

This paper studies the stress and displacement distributions of continuously varying thickness beams with one end clamped and the other end simply supported under static loads. By introducing the unit pulse functions and Dirac functions, the clamped edge can be made equivalent to the simply supported one by adding the unknown horizontal reactions. According to the governing equations of the plane stress problem, the general expressions of displacements, which satisfy the governing differefitial equations and the boundary conditions attwo ends of the beam, can be deduced. The unknown coefficients in the general expressions are then determined by using Fourier sinusoidal series expansion along the upper and lower boundaries of the beams and using the condition of zero displacements at the clamped edge. The solution obtained has excellent convergence properties. Comparing the numerical results to those obtained from the commercial software ANSYS, excellent accuracy of the present method is demonstrated.

AN ELASTICITY SOLUTION FOR AXISYMMETRIC PROBLEM OF FINITE CIRCULAR CYLINDER

In this paper the complete double-series in the closed region expressing the double-variable functions and their partial derivatives are derived by the H-transforniution and Stockes transformation. Using the double-series, a series solution for the axisyinmetric boundary value problem of the elastic circular cylinder with finite length is presented.In a numerical example, the cylinder subjected to the axisymmetric traellens with various loaded regions is investigated and the distributions of the displacement sand stresses are obtained.It is possible to solve the axisymmetric boundary value problems in the eylinderical coordinates for other scientific fields by use of the method presented in this paper.

Three-dimensional elastostatic solutions for transversely isotropic functionally graded material plates containing elastic inclusion

Based on the generalized England-Spencer plate theory, the equilibrium of a transversely isotropic functionally graded plate containing an elastic inclusion is studied. The general solutions of the governing equations are expressed by four analytic functions α（ζ）, β（ζ）, φ（ζ）, and ψ（ζ） when no transverse forces are acting on the surfaces of the plate. Axisymmetric problems of a functionally graded circular plate and an infinite func- tionally graded plate containing a circular hole subject to loads applied on the cylindrical boundaries of the plate are firstly investigated. On this basis, the three-dimensional （3D） elasticity solutions are then obtained for a functionally graded infinite plate containing an elastic circular inclusion. When the material is degenerated into the homogeneous one, the present elasticity solutions are exactly the same as the ones obtained based on the plane stress elasticity, thus validating the present analysis in a certain sense.

Elasticity solutions for functionally graded plates in cylindrical bending

The plate theory of functionally graded materials suggested by Mian and Spencer is extended to analyze the cylindrical bending problem of a functionally graded rectangular plate subject to uniform load. The expansion formula for displacements is adopted. While keeping the assumption that the material parameters can vary along the thickness direction in an arbitrary fashion, this paper considers orthotropic materials rather than isotropic materials. In addition, the traction-free condition on the top surface is replaced with the condition of uniform load applied on the top surface. The plate theory for the particular case Of cylindrical bending is presented by considering an infinite extent in the y-direction. Effects of boundary conditions and material inhomogeneity on the static response of functionally graded plates are investigated through a numerical example.

THERMOELASTIC PROBLEMS IN THE HALF SPACE—AN APPLICATION OF THE GENERAL SOLUTION IN ELASTICITY

In this paper, some thermoelastic problems in the half space are studied by using the general solutions of the elastic equations. The method presented here is extremely effective for the axisymmetric problems of the half space as well as the half plane problems.

An Elasticity Solution of a Nonhomogeneous Half-Plane Problem

We employ fundamental equations of non-homogeneous elasticity and Fourierintegral transformations to obtain the general solutions of the stress function.On thebasis of these points of view and when the forces on the boundary are arbityary for nonhomogeneous half-plane problems with the Young’s modulus E(x)-E_0θxp[βx].accurate solutions are obtained At last with the degeneracy it is obtained that thefamous Boussnesq solution and this method is successful.

ON THE METHOD OF RECIPROCAL THEOREM TO FIND SOLUTIONS OF THE PLANE PROBLEMS OF ELASTICITY

In this paper the method of reciprocal theorem is extended to find solutions of plane problems of elasticity of the rectangular plates with various edge conditions.First we give the basic solution of the plane problem of the rectangular plate with four edges built-in as the basic system and then find displacement expressions of the actual system by using the reciprocal theorem between the basic system and actual system with various edge conditions.When only displacement edge conditions exist, obtaining displacement expressions by means of the method of reciprocal theorem is actual. But in other conditions, when static force edge conditions or mixed ones exist, the obtained displacements are admissible. In order to find actual displacement, the minimum potential energy theorem must be applied.Calculations show that the method of reciprocal theorem is a simple, convenient and general one for the solution of plane problems of elasticity of the rectangular plates with various edge conditions. Evidently, it is a new method.

ANALYSIS SOLUTION OF BULGING CONVEXITY OF THE HYDRAULIC ELASTIC BULGING ROLL

The hydraulic elastic bulging roll system is a new roll system being developed in recent years.It is the first time in home that dead load test and study on this system has been made.Based on the experiment,the analysis solution of the bulging convexity of the hydraulic elastic bulging roll was obtained by selecting appropriate displacement function and adopting inverse solution of the elasticity theory.In evaluating,a variable displacement function and its solving method was advanced.The analysis results are in accord with the experiment's.Some conclusions have laid the foundation of establishing close-loop controlled mathematical model of the hydraulic elastic bulging roll system.

Boundary element analysis for elastic and elastoplastic problems of 2D orthotropic media with stress concentration

Both the orthotropy and the stress concentration are common issues in modem structural engineering. This paper introduces the boundary element method （BEM） into the elastic and elastoplastic analyses for 2D orthotropic media with stress concentration. The discretized boundary element formulations are established, and the stress formulae as well as the fundamental solutions are derived in matrix notations. The numerical procedures are proposed to analyze both elastic and elastoplastic problems of 2D orthotropic me- dia with stress concentration. To obtain more precise stress values with fewer elements, the quadratic isoparametric element formulation is adopted in the boundary discretization and numerical procedures. Numerical examples show that there are significant stress concentrations and different elastoplastic behaviors in some orthotropic media, and some of the computational results are compared with other solutions. Good agreements are also observed, which demonstrates the efficiency and reliability of the present BEM in the stress concentration analysis for orthotropic media.

Analytical algorithm for longitudinal deformation profile of a deep tunnel

To investigate the longitudinal deformation profile(LDP)of a deep tunnel in non-hydrostatic condition,an analytical model is proposed in our study.In this model,the problem is considered as a superposition of two partial models,and the displacement field of the second partial model is the same as that of the concerned problem.Therefore,the problem can be solved by a model with simple boundary conditions.We obtain the solutions for the stress and displacement fields of an infinite body caused by arbitrary surface tractions on the boundary of the coming tunnel(zone inside the tunnel before excavation)by integrating the extended Kelvin solution over the boundary.The obtained stress solution is used to solve the specific surface tractions,which can satisfy the boundary conditions of the second partial model,on the boundary of the coming tunnel in an infinite body.Then,the specific surface tractions are substituted into the obtained displacement solution to solve the displacement on the wall and face of the tunnel.Therefore,the LDP can also be calculated.The proposed solution is verified by both numerical simulation and the LDP functions recommended by other researchers.The major advantage of our analytical model is that it can consider the effects of the axial and horizontal lateral pressure coefficients.It is revealed that the horizontal lateral pressure coefficient majorly affects the LDP behind the tunnel face,while the axial lateral pressure coefficient dominates the LDP ahead of the tunnel face.Furthermore,the deformation characteristics of the LDPs ahead of the face and the unexcavated core are investigated.The axial displacements of the excavation face can be used to predict the crown displacements ahead of the face.

On asymmetric bending of functionally graded solid circular plates

Based on the three-dimensional elasticity equations, this paper studies the elastic bending response of a transversely isotropic functionally graded solid circular plate subject to transverse biharmonic forces applied on its top surface. The material prop- erties can continuously and arbitrarily vary along the thickness direction. By virtue of the generalized England＇s method, the problem can be solved by determining the expres- sions of four analytic functions. Expanding the transverse loarl in Fourier series along the circumferential direction eases the theoretical construction of the four analytic functions for a series of important biharmonic loads. Certain boundary conditions are then used to determine the unknown constants that are involved in the four constructed analytic functions. Numerical examples are presented to validate the proposed method. Then, we scrutinize the asymmetric bending behavior of a transversely isotropic functionally graded solid circular plate with different geometric and material parameters.