Abstract
New solutions of potential functions for the bilinear vertical traction boundary condition are derived and presented. The discretization and interpolation of higher-order tractions and the superposition of the bilinear solutions provide a method of forming approximate and continuous solutions for the equilibrium state of a homogeneous and isotropic elastic half-space subjected to arbitrary normal surface tractions. Past experimental measurements of contact pressure distributions in granular media are reviewed in conjunction with the application of the proposed solution method to analysis of elastic settlement in shallow foundations. A numerical example is presented for an empirical ‘saddle-shaped’ traction distribution at the contact interface between a rigid square footing and a supporting soil medium. Non-dimensional soil resistance is computed as the reciprocal of normalized surface displacements under this empirical traction boundary condition, and the resulting internal stresses are compared to classical solutions to uniform traction boundary conditions.
Highlights
The classical solution to the problem of a semi-infinite homogeneous, elastic body subjected to vertical loads on its boundary surface was first developed by Boussinesq [1] and discussed in more depth by Love [2]
We develop a complete set of closed-form solutions for the bilinear hyperbolic–paraboloidal potential functions and their required derivatives
Where D = ∂u/∂x + ∂v/∂y + ∂w/∂z is the strain dilation; λ and μ are the Lame’s constants; and is the Laplacian differential operator with respect to the spatial coordinates. This problem reduces to a problem of potential theory when either surface displacements or tractions are given
Summary
The classical solution to the problem of a semi-infinite homogeneous, elastic body subjected to vertical loads on its boundary surface was first developed by Boussinesq [1] and discussed in more depth by Love [2]. Dydo & Busby [11] further discussed linear and bilinear variations in vertical traction fields over a rectangular contact domain and provided one of the most comprehensive sets of closed-form solutions of the potential functions to date. Contact traction fields relevant to modern engineering applications may not be of such a low order As it stands, a new set of closed-form solutions must be developed for higher-order boundary conditions, for which in most cases the calculations are intractable. A surface traction function is empirically formulated by curve fitting pointwise contact-pressure data from past foundation experiments in the literature This is prescribed as a boundary condition for the half-space problem. A unique resistance distribution results directly from the corresponding non-uniform vertical boundary traction, which appears to evolve with increased applied load [21,22]
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