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

Abstract

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...

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

“…A similar example is provided in [16] where a third order distribution of vertical pressures has been considered for the computation of the vertical normal stress.…”

“…It is based on the use of a generalized version of the Gauss theorem [49] which has been recently applied in geodesy [12,14] to evaluate gravity effects of polyhedral bodies, and subsequently extended to several problems in geophysics [15] and geomechanics [16,45]. The necessity of adopting a generalized form of Gauss theorem stems from the singularity of the fields to be integrated in (22) which, in turn, is associated with the singularity of the functions in (1), (4) and (7) when z = 0 and ρ = 0.…”

“…In particular, we evaluate the full set of displacement and stress components of the wall foundation already analyzed in [16], where only the vertical normal stresses had been computed. The material parameters of the halfspace are assigned by means of its shear modulus G = 833.3 MPa and Poisson ratio ν = 0.2.…”

“…In particular we discuss at length the singularities which can affect the expressions of the mechanical fields and the way in which they enter the final formulas, an issue which seems to have been overlooked in the literature. Actually, differently from previous works, see, e.g., [15,16], we here provide a complete solution which allows for the evaluation of the whole set of displacement, strain and stress components.…”

The Gravity Anomaly of a 2D Polygonal Body Having Density Contrast Given by Polynomial Functions

“…Aim of this paper is to derive an analytical expression of the gravity anomaly for polygonal bodies whose density contrast is expressed as a polynomial function of arbitrary degree in both the horizontal and vertical directions. The result is obtained by reducing the original domain integral to a boundary integral by virtue of the generalized Gauss theorem first presented in D'Urso (2012, 2013a), and subsequently applied to several problems ranging from geodesy, (D'Urso, 2014a(D'Urso, ,b, 2015bD'Urso and Trotta, 2015c), to geomechanics, (Sessa and D'Urso, 2013;D'Urso and Marmo, 2015a;Marmo and Rosati, 2015), to geophysics (D'Urso and Marmo, 2013b) and to heat transfer (Rosati and Marmo, 2014). The generalized Gauss theorem referred to above does allow one not only to derive an expression of the gravity anomaly which is expressed in terms of a boundary integral but also to prove that the singularity of the gravity anomaly, arising when the observation point does belong to the integration domain, is eliminable.…”