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 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.

A concave indenter on a piezoelectromagneto-elastic substrate or a layer elastically supported

2012 ◽  
Vol 47 (6) ◽  
pp. 362-378 ◽  
Author(s):  
Bogdan Rogowski
Keyword(s):  
Half Space ◽  
Elastic Foundation ◽  
Contact Region ◽  
Approximate Solutions ◽  
Elastic Half Space ◽  
Solution Technique ◽  
Elementary Functions ◽  
Full Field ◽  
Full Contact ◽  
Two Parameter

Indentation of piezoelectromagneto-elastic half-space or a layer on a two-parameter elastic foundation by a cylindrical indenter with a slightly concave base is considered. Full-field magnetoelectro-elastic solutions in elementary functions are obtained for the case of full contact and half-space. If the axial load is small, the contact area will be an annulus the outer circumference of which coincides with the edge of the punch. The inner circumference will shrink with increasing load and there will be a critical load above which the stratum makes contact with the entire punch base. The contact problem for high loads can therefore be treated by classical methods. The more interested case in which the load is less the critical value and the contact region is annulus remains. By use the methods of triple integral equations and series solution technique the solution for an indentured substrate over an annular contact region is also given. For parabolic and conical concave punches the exact or approximate solutions are obtained for full contact or annular contact region, respectively. For the layer on two-parameter elastic foundation and concave punch approximate solution is established.

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.

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.

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.

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.

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.

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