A Remark on the Computation of the Gravitational Potential of Masses with Linearly Varying Density

D’Urso, Maria Grazia

doi:10.1007/1345_2015_138

Cited by 16 publications

(5 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Due to the large number of integration formulas to derive, we shall explicitly address only the linear case, represented by the first two terms of the sum in (11). This case is of particular interest in contact mechanics [26] where linear distributions of pressures are used as firstorder approximation to interpolate contact pressures by means of triangular tessellation of the loaded domain.…”

Section: Definition Of the Problemmentioning

confidence: 99%

“…However, without lack of generality, Q mz can be assumed to fulfill all kinds of symmetries since it contracts with ρ ⊗m in formula (11). For instance, observing that the second order tensor ρ ⊗ ρ has the following matrix representation…”

Section: Definition Of the Problemmentioning

confidence: 99%

“…To this end we shall make use of a generalized version of the Gauss theorem, recently exploited with success in the geodesy literature [9][10][11][12][13], i.e., a version which correctly takes into account the singularities of the integrand function.…”

Section: Definition Of the Problemmentioning

confidence: 99%

“…The rationale of the approach outlined in the paper is based on a generalized version of Gauss theorem, recently applied in the geodesy literature by D'Urso [9][10][11][12][13][14]. Accordingly, the surface integrals expressing the mechanical fields are transformed into the sum of an integral extended to the boundary of the loaded region and of an additional term accounting for the singularity of the field to be integrated.…”

mentioning

confidence: 99%

See 3 more Smart Citations

A General Approach to the Solution of Boussinesq’s Problem for Polynomial Pressures Acting over Polygonal Domains

Marmo

Rosati

2015

J Elast

View full text Add to dashboard Cite

We outline a general approach for extending the classical Boussinesq's solution to the case of pressures distributed according to a polynomial law of arbitrary order over a polygonal domain. To this end we exploit a generalized version of the Gauss theorem and recent results of potential theory which consistently take into account the singularities affecting the expressions of the fields of interest, an issue which seems to have been overlooked in the existing literature. For linearly varying pressures we derive analytical expressions of displacements, strains and stresses at an arbitrary point of the half-space as a function of the loading function and of the position vectors which define the boundary of the loaded region. We briefly discuss how bilinear and more general pressure distributions can be accommodated in our formulation since the paper is mainly motivated by the interest in developing efficient computational tools for solving 3D problems in foundation engineering and contact mechanics. Finally, comparisons with existing solutions and numerical examples are discussed.

show abstract

Section: Definition Of the Problemmentioning

confidence: 99%

Section: Definition Of the Problemmentioning

confidence: 99%

Section: Definition Of the Problemmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A General Approach to the Solution of Boussinesq’s Problem for Polynomial Pressures Acting over Polygonal Domains

Marmo

Rosati

2015

J Elast

View full text Add to dashboard Cite

show abstract

“…Analogously to up-to-date application of potential theory in geodesy [11][12][13][14], this has been possible by suitable variants of Gauss theorem, some of which are proved for convenience. The same rationale has been exploited to solve the companion problem of evaluating the displacements within the half-space induced by surface loads [15].…”

mentioning

confidence: 99%

Vertical stress distribution in isotropic half-spaces due to surface vertical loadings acting over polygonal domains

D’Urso

Marmo

2013

Z. Angew. Math. Mech.

View full text Add to dashboard Cite

By integrating the classical Boussinesq expression we derive analytically the vertical stress distribution induced by pressures distributed with arbitrary laws, up to the third order, over polygonal domains. Thus, one can evaluate in closed form either the vertical stress produced by shell elements, modelling raft foundations by finite elements, acting over a Winkler soil or those induced by a linear pressure distribution simulating axial force and biaxial bending moments over a pad foundation. To this end we include charts and tables, both for rectangular and circular domains, which allow the designer to evaluate the vertical stresses induced by linear load distributions by hand calculations. The effectiveness of the proposed approach is witnessed by the comparison between the analytical results obtained with the proposed formulas and the numerical ones of a FEM discretization of the soil associated with the loading distribution induced by a foundation modeled by plate elements resting on a Winkler soil.The classical approach to finding stress and displacements in an elastic, homogeneous and isotropic half-space due to surface tractions has been first developed by Boussinesq [4] who provided the solution for a point load by making use of the potential theory [3].The case of uniform pressure applied within a circular domain was addressed by Lamb [19] while Love [20] considered the case of isotropic uniform pressure applied within a rectangular region.However, even in many natural soils deposited through a geologic process of sedimentation over a period of time stiffness becomes greater with depth. Analytical continuum models reflecting this property were first studied by Gibson [16] by assuming incompressibility and a linear increase of soil stiffness with depth. In later papers Gibson further refined his model and presented on account of his results in a Rankine lecture [17].Subsequently Vrettos [30] solved the Boussinesq problem for an half-space exhibiting an exponential behavior of stiffness yet with a bounded value at infinite depth. Half-spaces of this kind were later studied by Selvadurai in the framework of contact static problems and a survey account of his result were presented in [24].Since the pioneering work by Gibson an abundant literature has been published on the evaluation of stresses and displacements in inhomogeneous and/or anisotropic half-spaces, see e.g. [31] for a fairly complete account.Nevertheless, Boussinesq solution is still widely used mainly for its elegance and conciseness, for instance to evaluate the order of magnitude of the depth of influence of a foundation load [7] and the related settlement. Actually, it is usually assumed that the variation of vertical stress with depth below a loaded region on the surface can be predicted with sufficient accuracy using linear, homogeneous, isotropic elastic theory.Further applications of this theory range, to quote only a few, from space geodesy to contact mechanics, rock mechanics and geomechanics.For instance, in the former case position...

show abstract

Complete solutions of three-dimensional problems in transversely isotropic media

Marmo

Sessa

Vaiana

et al. 2018

Continuum Mech. Thermodyn.

View full text Add to dashboard Cite

A Remark on the Computation of the Gravitational Potential of Masses with Linearly Varying Density

Cited by 16 publications

References 24 publications

A General Approach to the Solution of Boussinesq’s Problem for Polynomial Pressures Acting over Polygonal Domains

A General Approach to the Solution of Boussinesq’s Problem for Polynomial Pressures Acting over Polygonal Domains

Vertical stress distribution in isotropic half-spaces due to surface vertical loadings acting over polygonal domains

Complete solutions of three-dimensional problems in transversely isotropic media

Contact Info

Product

Resources

About