Several Elliptical Punches on an Elastic Half Space

1986 ◽  
Vol 53 (2) ◽  
pp. 390-394 ◽  
Author(s):  
V. I. Fabrikant
Keyword(s):  
Half Space ◽  
General Theorem ◽  
Elastic Half Space ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Generalized Displacements

A general theorem is established which relates the resulting forces, acting on a set of arbitrary punches, with their generalized displacements through a system of linear algebraic equations. The theorem is applied to the case of arbitrarily located elliptical punches. Several specific examples are considered.

scholarly journals Two Contact Problems for Annular Rigid Stamp on an Elastic Half Space

2018 ◽  
Vol 17 (6) ◽  
pp. 458-464
Author(s):  
S. V. Bosakov
Keyword(s):  
Functional Equation ◽  
Half Space ◽  
Bessel Functions ◽  
Infinite System ◽  
Annular Plate ◽  
Contact Problems ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

The paper presents solutions of two contact problems for the annular plate die on an elastic half-space under the action of axisymmetrically applied force and moment. Such problems usually arise in the calculation of rigid foundations with the sole of the annular shape in chimneys, cooling towers, water towers and other high-rise buildings on the wind load and the load from its own weight. Both problems are formulated in the form of triple integral equations, which are reduced to one integral equation by the method of substitution. In the case of the axisymmetric problem, the kernel of the integral equation depends on the product of three Bessel functions. Using the formula to represent two Bessel functions in the form of a double row on the works of hypergeometric functions Bessel function, the problem reduces to a functional equation that connects the movement of the stamp with the unknown coefficients of the distribution of contact stresses. The resulting functional equation is reduced to an infinite system of linear algebraic equations, which is solved by truncation. Under the action of a moment on the annular plate  die, the distribution of contact stresses is searched as a series by the products of the Legendre attached functions with a weight corresponding to the features in the contact stresses at the die edges. Using the spectral G. Ya. Popov ratio for the ring plate, the problem is again reduced to an infinite system of linear algebraic equations, which is also solved by the truncation method. Two examples of calculations for an annular plate die on an elastic half-space on the action of axisymmetrically applied force and moment are given. A comparison of the results of calculations on the proposed approach with the results for the round stamp and for the annular  stamp with the solutions of other authors is made.

The Scattering of SH-Wave Caused by Subsurface Circular Cavities in a Layered Half-Space

2011 ◽  
Vol 488-489 ◽  
pp. 226-229
Author(s):  
Dong Ni Chen ◽  
Hui Qi ◽  
Yong Shi
Keyword(s):  
Half Space ◽  
Elastic Layer ◽  
Complex Function ◽  
Expansion Method ◽  
Elastic Half Space ◽  
Wave Functions ◽  
Large Radius ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Linear Algebraic Equations

The scattering of SH-wave caused by the subsurface circular cavities in an elastic half-space covered with an elastic layer was discussed, which was based on the complex function method ,wave functions expansion method and big circular arc postulation method in which the circular boundary of large radius was used to approximate straight boundary of surface elastic layer. By the theory of Helmholtz, the general solution of the Biot’s wave function was achieved. Utilizing the complex series expansion technology and the boundary conditions, we could transform the present problem into the problem in which we needed to solve the infinite linear algebraic equations with unknown coefficients in wave functions. Finally, the dynamic stress concentration factors around the circular cavities were discussed in numerical examples.

The Effect on Scattering of SH-Wave in a Layered Half-Space with the Subsurface Circular Cavities

2011 ◽  
Vol 194-196 ◽  
pp. 1908-1911
Author(s):  
Dong Ni Chen ◽  
Hui Qi ◽  
Yong Shi
Keyword(s):  
Half Space ◽  
Elastic Layer ◽  
Complex Function ◽  
Expansion Method ◽  
Elastic Half Space ◽  
Wave Functions ◽  
Large Radius ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Linear Algebraic Equations

The scattering of SH-wave caused by the subsurface circular cavities in an elastic half-space covered with an elastic layer was discussed, which was based on the complex function method and wave functions expansion method. The solution of scattering of SH-wave was given by using circular boundary of large radius to approximate straight boundary of surface elastic layer. According to boundary conditions, we needed to solve the infinite linear algebraic equations with unknown coefficients in wave functions. Finally, the dynamic stress concentration factors around circular cavities were discussed in numerical examples.

Approximate solutions to the half-space integral transport equation near a plane boundary

1980 ◽  
Vol 58 (9) ◽  
pp. 1291-1310 ◽  
Author(s):  
Michael S. Milgram
Keyword(s):  
Transport Equation ◽  
Half Space ◽  
Matrix Multiplication ◽  
Solution Space ◽  
Plane Geometry ◽  
Plane Boundary ◽  
Infinite Plane ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Integral Transport

A set of functions spanning the solution space of the integral transport equation near a boundary in semi-infinite plane geometry is obtained and used to reduce the problem to that of a system of linear algebraic equations. Expressions for the boundary angular flux are obtained by matrix multiplication, and the theory is extended to adjacent half-space problems by matching the angular flux at the boundary. Thus a unified theory is obtained for well-behaved arbitrary sources in semi-infinite plane geometry. Numerical results are given for both Milne's problem and the problem of constant production in adjacent half-spaces, and albedo problems in semi-infinite geometry. The solutions for the flux density are best near the boundary, and for the angular flux are best for angles near the plane of the boundary; it is conjectured that the theory will prove most useful when extended to arrays of finite slabs.

The Dynamic Response of Elastic Half Space Including a Semi-Lining Hill

2013 ◽  
Vol 753-755 ◽  
pp. 1712-1718
Author(s):  
Jing Fu Nan
Keyword(s):  
Dynamic Response ◽  
Half Space ◽  
Line Source ◽  
Complex Function ◽  
Arbitrary Point ◽  
Elastic Half Space ◽  
Displacement Function ◽  
Algebraic Equations ◽  
Out Of Plane ◽  
Horizontal Interface

The dynamic response of elastic half space including semi-cylindrical lining hill while bearing out-of-plane harmonic line source loading on horizontal interface is investigated using the method of complex function and Greens function. the displacement function of incidence wave is given while a out-of-plane harmonic line source is loaded on arbitrary point of horizontal interface;and the solution domain is divided into two domains, an elastic half space with the semi-circular canyon and a cylindrical lining; the scattering wave of semi-cylindrical canyon and the standing wave of cylindrical lining are constructed. Finally, it conjoins the two domains, and a series of infinite algebraic equations can be obtained to settle this problem. In the end, the numerical expressions of the ground motion in the horizontal surface are discussed.

scholarly journals To Solution of Contact Problem for Rectangular Plate on Elastic Half-Space

2020 ◽  
Vol 19 (3) ◽  
pp. 224-229
Author(s):  
S. V. Bosakov
Keyword(s):  
Contact Problem ◽  
Rectangular Plate ◽  
Half Space ◽  
Chebyshev Polynomials ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Rectangular Region ◽  
Double Row ◽  
Linear Algebraic Equations ◽  
Boussinesq Solution

Until the present time there is no exact solution to the contact problem for a rectangular plate on an elastic base with distribution properties. Practical analogues of this design are slab foundations widely used in construction. A lot of scientists have solved this problem in various ways. The methods of finite differences, B. N. Zhemochkin and power series do not distinguish a specific feature in contact stresses at the edges of the plate. The author of the paper has obtained an expansion of the Boussinesq solution for determining displacements of the elastic half-space surface in the form of a double series according to the Chebyshev polynomials of the first kind in a rectangular region. For the first time, such a representation for the symmetric part of the Boussinesq solution was obtained by V. I. Seimov and it has been applied to study symmetric vibrations of a rectangular stamp, taking into account inertial properties of the half-space. Using this expansion, the author gives a solution to the problem for a rectangular plate lying on an elastic half-space under the action of an arbitrarily applied concentrated force. In this case, the required displacements are specified in the form of a double row in the Chebyshev polynomials of the first kind. Contact stresses are also specified in the form of a double row according to the Chebyshev polynomials of the first kind with weight. In the integral equation of the contact problem integration over a rectangular region is performed while taking into account the orthogonality of the Chebyshev polynomials. In the resulting expression the coefficients are equal for the same products of the Chebyshev polynomials. The result is an infinite system of linear algebraic equations, which is solved by the amplification method. Thus the sought coefficients are found in the expansion for contact stresses.

Problems in calculating the flexure of beams on elastic foundation

1987 ◽  
Vol 14 (4) ◽  
pp. 581-584 ◽  
Author(s):  
Dajun Ding
Keyword(s):  
Half Space ◽  
Elastic Foundation ◽  
Elastic Characteristic ◽  
Elastic Half Space ◽  
Algebraic Equations ◽  
Improved Method ◽  
Concentrated Loads ◽  
Dimensionless Coefficient ◽  
Loading Case ◽  
Beams On Elastic Foundation

This paper deals with some problems concerning the flexure of beams on an elastic half space. Based on the link method of Zemochkin, the author gives an improved method by which the number of simultaneous algebraic equations can be reduced by 2. Following the author's proposal, the universal tables of reaction coefficients for any loading case are compiled. The author has obtained dimensionless coefficients of reaction, shear, and moment between the infinite and semi-infinite beams subjected to concentrated loads and those under moments, so the coefficient tables of long beams under concentrated loads can be used for calculating those under moments. Key words: elastic foundation, settlement, half-plane and half-space, link method, pressure, segment, typical equation, matrix, dimensionless coefficient, long (infinite and semi-infinite) beam, finite beam (beam with finite length), elastic characteristic value.

scholarly journals Dynamic Analysis of a Shallow Buried Tunnel Influenced by a Neighboring Semi-cylindrical Hill and Semi-cylindrical Canyon

2019 ◽  
Author(s):  
Xiaotang Lv ◽  
Cuiling Ma ◽  
Meixiang Fang
Keyword(s):  
Stress Concentration ◽  
Dynamic Analysis ◽  
Half Space ◽  
Dynamic Stress ◽  
Mixed Boundary ◽  
Scattered Wave ◽  
Dynamic Stress Concentration ◽  
Algebraic Equations ◽  
Sh Waves ◽  
Linear Algebraic Equations

This paper provides a dynamic analysis of the response of a subsurface cylindrical tunnel to SH waves influenced by a neighboring semi-cylindrical hill and semi-cylindrical canyon in half-space using complex functions. For convenience in finding a solution, the half-space is divided into two parts and the scattered wave functions are constructed in both parts. Then the mixed boundary conditions are satisfied by moving coordinates. Finally, the problem is reduced to solving a set of infinite linear algebraic equations, for which the unknown coefficients are obtained by truncation of the infinite set of equations. The effects of the incident angles and frequencies of SH waves, as well as of the radius of the tunnel, hill, and canyon on the dynamic stress concentration of the tunnel are studied. The results show that the hill and canyon have a significant effect on the dynamic stress concentration of the tunnel.

Load Transfer From a Finite Cylindrical Fiber Into an Elastic Half-Space

1994 ◽  
Vol 61 (4) ◽  
pp. 971-975 ◽  
Author(s):  
Ven-Gen Lee ◽  
Toshio Mura
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Load Transfer ◽  
Transfer Problem ◽  
Numerical Procedure ◽  
Elastic Half Space ◽  
Equivalent Condition ◽  
Algebraic Equations ◽  
Equivalent Inclusion Method ◽  
Cylindrical Fiber

Based on the equivalent inclusion method, the load transfer problem of a finite cylindrical fiber embedded in an elastic half-space of different elastic properties is presented. The equivalent condition of inhomogeneity and inclusion problems simulates the fiber to an inclusion with chosen eigenstrains, and the problem is formulated to a set of integral equations with the unknown strength of eigenstrains. A numerical procedure is developed using a discretizing scheme by which the set of integral equations is reduced to a system of algebraic equations.

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