The paper presents a new refined-modified solution of the fundamental two-dimensional problem of the elasticity theory on the perpendicular application to the boundary of the half-plane of a concentrated-linear constant load. In contrast to the similar classical Fleman problem, which is a special case of a simple radial stress state, all three stress components, two normal and one tangent, as well as an additional geometric parameter characterizing the width of the site of the external local force’s actual distribution have been taken into consideration. In addition, on the basis of the classical interpretation of plane deformation, the known contradictions are eliminated that are associated with the uncertainty of the angular displacement at the boundary of the half-space and with the constancy of the second kinematic component in the pursuit of the infinity coordinates of an arbitrary point of the base material. In the course of the research, it is proved that there are cylindrical surfaces where equal tensile stresses act which trajectories have the shape of circles. In a simplified Fleman solution of such curves- isobars are Boussinesq circles with constant the principal compressive stresses. The derived analytical dependences are presented in a rectangular frame of reference, which allows to quantify the following with a high accuracy: 1) stresses in the depth of the base in horizontal and vertical sections; 2) contact pressure and draft of the soil elastic surface under the sole of a rigid long foundation when the base, within the generally accepted assumptions, is assumed to be linearly deformable, homogeneous, isotropic, solid, experiencing a one-time load. The results of the developed generalized physical and mathematical model can serve as a conceptual basis used in solving special fundamental and applied problems of mechanics directly related to the refined calculation of the bearing capacity of various parts and structures, widely used in modern engineering and construction such as bearings, cylindrical rollers, gears, foundations strip foundations, pavements in their steel compaction rolls, etc.

The inapplicability of the reference formula for determining the convergence of two statically compressed parallel cylinders made of a homogeneous, isotropic and physically linear material has been proved due to a well-known logarithmic feature in the plane classical problem of mechanics of elastic solids. In the special case of the elastic interaction of a cylinder with a halfplane, when one of the radii has an infinite length, it has been found that the convergence also becomes equal to infinity. This paradoxical result contradicts not only the physical and mechanical meaning of the process under study, but also confirms the inadequacy of Flamant model of a simple radial stress state in determining displacements. The authors have proposed an algorithm for eliminating the contradictions based on the solution of Fredholm integral equation of the first kind. In the future, it can be considered as a new fundamental and applied problem of the theory of elasticity, which is of a great importance for a refined assessment of the contact strength and stiffness of the cylindrical parts of load-bearing structures taking into account the general and local deformations (cylindrical rollers, gears, road surfaces, when they are compacted wit h steel rollers, etc.) on the basis of a flat Flamant calculation scheme considering three stress components and the width of the cylinder contact area previously developed and mathematically approximated by the authors.

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