Hankel Tranform Method for Solving Axisymmetric Elasticity Problems of Circular Foundation on Semi-infinite Soils

Ike, Charles Chinwuba

doi:10.21817/ijet/2018/v10i2/181002111

Cited by 3 publications

(8 citation statements)

References 6 publications

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“…Evaluating this double integration, we obtain the following expression, which is identical to the expressions obtained previously by Egorov and Screbrjanyi [30], [22] and Harr [31]:…”

Section: Solution For Vertical Fields Due To Uniformly Distributed Load Over Circular Foundation Areassupporting

confidence: 67%

Trefftz Displacement Potential Function Method for Solving Elastic Half-Space Problems

Ike¹,

Mama²,

Onah³

et al. 2021

cea

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The elastic half-space problem has been solved previously using Boussinesq, Papkovich, Love, and Green and Zerna, potential function methods. In this work, the Trefftz displacement potential function method is used to obtain the stress and displacement fields in an elastic half-space subjected to boundary loads. Point load and various distributed loads are considered. The problem is presented using displacement formulation as Navier-Lamé equations. It is proved that the Trefftz functions are solutions of the Navier-Lamé displacement equations. Strain fields are derived in terms of the Trefftz function using the strain-displacement relations. The stress fields are similarly derived. The Trefftz function for the case of a point load acting at the origin of the elastic half-space is derived using the double exponential Fourier transformation technique. Stress equilibrium boundary conditions are used to fully determine the Trefftz function. Stress and displacement fields for the point load are then determined. The solutions to stress and displacement fields for point load are then used as Green functions to obtain stress and displacement fields for uniformly distributed load over a finite line, circular area and rectangular area. It is found that the solutions obtained for the stress and displacement fields in the elastic half-space due to point and distributed loads are identical with previously obtained expressions, thus validating this work.

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“…Evaluating this double integration, we obtain the following expression, which is identical to the expressions obtained previously by Egorov and Screbrjanyi [30], [22] and Harr [31]:…”

Section: Solution For Vertical Fields Due To Uniformly Distributed Load Over Circular Foundation Areassupporting

confidence: 67%

Trefftz Displacement Potential Function Method for Solving Elastic Half-Space Problems

Ike¹,

Mama²,

Onah³

et al. 2021

cea

Self Cite

View full text Add to dashboard Cite

show abstract

“…The determination of the normal and shear stress distributions due to point, line and distributed loads applied to the surface of linear elastic media are problems of the classical mathematical theory of elasticity (Ike, 2006(Ike, , 2018a(Ike, , 2018b. Such problems are extensively encountered and applied in the elastic stress and settlement analysis of solids and structural footings (Rocscience, 2018;Westergaard, 1938).…”

Section: Introductionmentioning

confidence: 99%

“…Stress methods, introduced by Michell and Beltrami involve the reformulation of the system of fifteen governing equations such that only the six Cauchy stress components (in a three dimensional formulation) become the primary variables (Ike, 2006(Ike, , 2018a(Ike, , 2018bRocscience, 2018;Westergaard, 1938;Ojedokun and Olutoge, 2012;Westergaard, 1964;Barbar, 2010;Sokolnikoff, 1956;Timoshenko and Goodier, 1970;Green and Zerna, 1954). The advantage is the simplification inherent in the reduction of the number of governing equations from fifteen to six.…”

Section: Introductionmentioning

confidence: 99%

“…Westergaard sought to solve the theory of elasticity problem of finding stress distributions at any point under a foundation that would cater for alternating horizontal layers of soft and stiff materials (Ike, 2006(Ike, , 2018a(Ike, , 2018bRocscience, 2018;Anyaegbunam, et al, 2011). For such materials with alternating soft and stiff layers, the Boussinesq stress distribution theory cannot correctly give the stresses, since the fundamental assumptions for constructing the Boussinesq theory have been violated by the material properties of the half-space.…”

Section: Introductionmentioning

confidence: 99%

“…Westergaard presented solutions for the stress fields for such problems by making the following fundamental assumptions (Ike, 2006(Ike, , 2018a(Ike, , 2018bRocscience, 2018):…”

Section: Introductionmentioning

confidence: 99%

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

Ike

2019

Lat. Am. j. solids struct.

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The Hankel transformation method was used in this work to determine the normal and shear stress distributions due to point, line and distributed loads applied to the surface of an elastic media. The elastic media considered in this study was assumed to be inextensible in the horizontal directions, and the only non-vanishing displacement component is the vertical component. Such materials were first considered by Westergaard as models of elastic half-space with alternating layers of soft and stiff materials with the stiff materials of negligible thickness and so closely spaced that the composite material characteristics is idealised as isotropic and homogeneous. The study adopted a displacement formulation. The Hankel transformation was applied to the governing Cauchy-Navier differential equation of equilibrium to reduce the problem to a second order ordinary differential equation (ODE) in terms of the deflection function in the Hankel transform space. Solution of the ODE subject to the boundedness condition yielded bounded deflection function in the Hankel transform space. Equilibrium of the internal vertical forces and the external applied load, was used to obtain the constant of integration, for the deflection function in the Hankel transform space. Inversion yielded the deflection in the physical domain variable. The stress displacement equations were then used to determine the Cauchy stresses. The vertical stress distributions due to line and distributed load over a rectangular area were also determined by using the point load solution for vertical stresses as Green functions, and then performing integration along the line and over the rectangular area of the load. The results obtained for vertical stresses due to point, line and distributed loads were determined in terms of dimensionless influence coefficients which were presented. The results obtained for the deflection, normal and shear stresses due to point, line and distributed load agreed with the solutions originally presented by Westergaard who used a stress function method.

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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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Hankel Tranform Method for Solving Axisymmetric Elasticity Problems of Circular Foundation on Semi-infinite Soils

Cited by 3 publications

References 6 publications

Trefftz Displacement Potential Function Method for Solving Elastic Half-Space Problems

Trefftz Displacement Potential Function Method for Solving Elastic Half-Space Problems

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

Solving Westergaard Half-Space Problems Using Potential Theory

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