Hankel transformation method for solving the Westergaard problem for point, line and distributed loads on elastic half-space

Ike, Charles Chinwuba

doi:10.1590/1679-78255313

Cited by 3 publications

(10 citation statements)

References 18 publications

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“…To a limited extent, the studies in [16][17][18] can be an option of overcoming the above difficulties. Transformations and additional functions associated with basic dependences were considered.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…Transformations and additional functions associated with basic dependences were considered. In the problem considered in [16], the Hankel's transform was applied to the basic differential Cauchy-Navier equilibrium equation to reduce the problem to an ordinary differential equation. In the case of the argument functions, transitions are used as well, however, the ones between partial differential equations.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

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Invariant differential generalizations in problems of the elasticity theory as applied to polar coordinates

Chigirinsky¹,

Naumenko²

2020

EEJET

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The method of argument functions has become famous for solving problems of continuum mechanics. The solution of problems of the elasticity theory in polar coordinates was the further development of this method. The same approaches are applied to solving problems of the theory of plasticity, the theory of elasticity, and the theory of dynamic processes. If regularities of the solution are determined correctly, then they should be continued in other fields including the problems of the theory of elasticity in polar coordinates. The proposed approach features finding not the solution itself but the conditions for its existence. These conditions may include differential or integral relations which make it possible to close the solution in a general form. This becomes possible when additional functions are introduced into consideration or the argument functions of coordinates of the deformation zone. Basic dependences that satisfy the boundary or edge conditions as well as the functions that simplify the solution of the problem in general should be the carriers of the proposed argument functions. For various reasons, two basic dependences were used in the solution: trigonometric and exponential. Their arguments are two unknown argument functions. In the process of transformations, a mathematical connection was established between them in a form of the Cauchy-Riemann relations which had a stable tendency to be repeated in problems of the continuum mechanics. From these positions, the flat problem was solved in the most detailed way, tested, and compared with the studies of other authors. By reducing the solution to a particular result, a way to classical solutions was found which confirms its reliability. The result obtained is useful and important since it becomes possible to solve an extensive class of axisymmetric applied problems using the method of argument functions of a complex variable

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Section: Literature Review and Problem Statementmentioning

confidence: 99%

Section: Literature Review and Problem Statementmentioning

confidence: 99%

Invariant differential generalizations in problems of the elasticity theory as applied to polar coordinates

Chigirinsky¹,

Naumenko²

2020

EEJET

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show abstract

“…Ike [37] also used Fourier integral transform method to obtain closed-form solutions to 2D elastic half-plane problems using Love stress function approach. Ike [38] used the Hankel transformation method to solve the closed-form Westergaard half-space problem for concentrated and line load. He distributed real loads acting on the surface of the Westergaard half-space.…”

Section: Introductionmentioning

confidence: 99%

“…He distributed real loads acting on the surface of the Westergaard half-space. Ike [38] found that the Hankel transform method is ideal for solving the problem because of the axisymmetric nature of the vertical point load at the origin of Westergaard halfspace problem and the simplification offered by the Hankel transformation for axisymmetric problems. The point load solution was then used as the integral kernel in [38] to develop line solutions and distributed real loads.…”

Section: Introductionmentioning

confidence: 99%

Solving Westergaard Half-Space Problems Using Potential Theory

Ike

2023

ETJ

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 The governing differential equation is derived for an elastic half-space problem with horizontal inextensibility.  The potential theory is applied to solve the Westergaard problem for a point load on the boundary.  The approach adopts first principles to derive the governing equations, highlighting limitations and scope.  Results are validated by comparison with literature sources.The Westergaard half-space problem has been solved using the potential theory in this work. It is a classical theme in elasticity theory that seeks to find the displacements and stresses in the half-space caused by known boundary loads. It has important applications in analyzing stresses and displacement fields in soil due to applied points and distributed loads on the boundary caused by structures placed on the soil. It is governed by stress-strain, strain-displacement, and equilibrium equations. Horizontal inextensibility is assumed in developing the problem, simplifying the displacement formulation to a three-dimensional (3D) Laplace equation.The potential theory is applied to find the vertical displacement. Stressdisplacement equations obtained from the simultaneous use of the kinematic and stress-strain equations are used to obtain the stress fields.The specific problem of point load at the origin was considered and solved. The equilibrium of internal vertical stresses and the external vertical load is used to find the integration constant. Hence, vertical displacements were found. The stress fields were found from the stressdisplacement equations. The expressions for the vertical displacements and stresses were found to be exact within the framework of the theory used, as they satisfied all the governing equations of the problem. However, the solutions become unbounded at the origin due to the singularity of the vertical displacement and stresses. The obtained solutions are identical to previously obtained solutions.

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“…The advantages and merits offered by the displacement and stress-based methods have led researchers to develop solutions to the governing equations of the displacement and stressbased methods. Such solutions, which satisfy the governing equations of the displacement and stress-based methods are called respectively displacement functions and stress functions [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. Some displacement functions of the theory of elasticity are: Green and Zerna potential (harmonic) displacement function, Boussinesq displacement functions.…”

Section: Introductionmentioning

confidence: 99%

Fourier Integral Transformation Method for Solving Two Dimensional Elasticity Problems in Plane Strain Using Love Stress Functions

Ike¹

2021

MMEP

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The Fourier integral method was used in this work to determine the stress fields in a two dimensional (2D) elastic soil mass of semi-infinite extent subject to line and strip loads of uniform intensity acting on the boundary. The two dimensional plane strain problem was formulated using stress-based method. The Fourier integral was used to transform the biharmonic stress compatibility equation to a fourth order linear ordinary differential equation (ODE) in terms of the stress function. The ODE was solved subject to the boundedness condition to obtain the bounded stress function. Cartesian stress components were obtained using the Love stress functions. Application of the stress boundary conditions for the case of line load of uniform intensity and the cases of uniformly distributed load on a strip of finite width gave the respective unknown constants of the Love stress functions; and hence the complete determination of the Cartesian stress components for the two cases considered. Inversion of the Fourier integral expressions obtained for the normal and shear stresses in the Fourier parameter gave respective expressions for the normal and shear stress fields for line and finite strip loads of finite width in the physical domain variables. The results obtained agreed with the results from previous studies which used displacement based methods.

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Hankel transformation method for solving the Westergaard problem for point, line and distributed loads on elastic half-space

Cited by 3 publications

References 18 publications

Invariant differential generalizations in problems of the elasticity theory as applied to polar coordinates

Invariant differential generalizations in problems of the elasticity theory as applied to polar coordinates

Solving Westergaard Half-Space Problems Using Potential Theory

Fourier Integral Transformation Method for Solving Two Dimensional Elasticity Problems in Plane Strain Using Love Stress Functions

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