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In many engineering constructions, the foundations should be built adjacent to each other. Therefore, the effect of interfering of close foundations should be considered in the design stage. In this research work, the effect of interference of closely separated foundations resting on a slope on the elastic settlement is investigated by considering a semi-analytical solution. The distribution of stress due to the footing pressure in the slope is computed by a proposed Airy stress function, and then by employing the finite difference scheme, the displacement of the footings is calculated. The results obtained show that by increasing the distance between the foundations, the interference influence on the ratio of settlement will be diminished. However, this behavior is highly linked to the slope characteristics. For a slope with a height of 10 times of footing width, beyond an S/B ratio larger than 10, the effect of interference is not tangible, and the footings behave like an isolated foundation. By decreasing the slope height, this behavior will occur at a lower S/B.

Using function notation is very common in science and technology. The vast use of Function notation prompts philosophical analysis to investigate it and brings upon many theories that try to explain this concept. Most of these investigations consider Function notation usage in natural sciences, especially in biology, and they either neglect or disregard this concept in the Technological domain. The main consideration of this paper is Use-plan based function theory (or more commonly known as “ICE Theory”) which is built primarily for the Technical domain. In the beginning, the paper presents a short historical review of the development of function theories. Then, the usual problems of some of these theories in the technology realm are mentioned; and as a result, the motivations for developing a theory of function which is appropriate for the technology domain are described. Origin and foundation of The ICE theory is a theory about Using and Designing of technical artifacts, which is described briefly and suitable for needs of our argument in this article. The main method in this paper and for assessing function theories is, comparing these theories with the intuitions regarding artifacts in particular and technology in general. Then for evaluating function theories, four standards (or four intuitions) are introduced. Moreover, three general technical function theories are introduced. It is shown that each of these theories fails to meet at least one of these criteria. Finally, after presenting a formulation of the ICE theory, it is shown that this theory is consistent with the intuitions presented and has a comparative advantage over the other three theories.

Summary
In this study, horse-shoe tunnels have been analyzed with respect to the ratio of stiffness, relative thickness, and the ratio of in situ stress under the presence of earth pressure, and water pressure inside tunnels. Surrounding rock mass and concrete have been assumed as elastic materials, and complex potential functions along with conformal mapping have been used in the investigation. it was observed that an enhance in concrete stiffness resulted in increasing circumferential stress at key points. A critical value for lining thickness was obtained that for values smaller than which the circumferential stress exceedingly increased.
Tunnels are the main infrastructures widely used for transportation, water passage, and other purposes such as underground mining. Among various tunnels configurations, horse-shoe tunnels are more popular for their merit of suitable stress distribution in surrounding rock, which makes them capable of maintenance in a wide variety of rocks, from soft to hard ones.
Here a sensitivity analysis was conducted on the most important parameters for lining design of hydraulic horse-shoe tunnels. These parameters consist lining thickness, rock and concrete stiffness, initial in situ stress ratio and inside water pressure. To this aim, Muskhilishvili complex potential functions combined with conformal mapping were used. Concrete lining and the surrounding rock mass were assumed as linearly elastic materials, and the problem was tackled based on a plane strain scheme.
Methodology and Approaches
The impact of variation of input parameters were investigated on circumferential stress produced through lining and rock mass regions. Muskhilishvili complex potential functions along with conformal mapping were used in order to implement this investigation.
Results and It is demonstrated that increasing the ratio of stiffness will enhance circumferential stress at key points in the concrete, while reduce that at the roof in the rock mass. It is observed that for high ratio of stiffness when relative thickness is less than 0.04, the circumferential stress boosts, which should be evaded in design. An enhancement in the ratio of in situ stresses also raised circumferential stress at the roof, and reduced that at the wall. The investigations implied that the amount of the variations of circumferential stress corresponding to a ratio of in situ stress, where the ratio of stiffness is less than 1, were exceedingly lower than that when the ratio of stiffness is more than 1.

Foundation loading in mines typically occurs on pillars in underground mines which may not have necessarily a regular shape. The previous studied have mostly focused on the stress distribution within the pillar, rather than on the stresses induced in the underlying layers. In other words, a mining pillar has been mainly regarded as a column rather than a footing. The determination of foundation-induced stresses in rock and soil is of great importance when designing mine pillars. Hence, different methods have been presented to analyze the stresses induced in the ground due to the foundation loading. The influence chart is a graphical analysis method based on the elasticity theory which is capable of analyzing foundations with arbitrary shape. However, since this method suffers from difficulties associated with drawing and calculation, it has not been considered as an efficient method of foundation analysis. In this paper, a computer program is introduced and evaluated which determines the mining pillar-induced pressure using the influence chart method combined with image processing techniques. The software presents a graphical user interface which facilitates creating and editing of the foundation geometry. The developed program is capable of estimating the over-pressure value at given coordinates below the foundation. The modeling procedure in the developed software is much faster compared to other sophisticated numerical codes, while no discrepancy in results is observed. Hence, it would be considered as an efficient and handy numerical tool in preliminary studies.

Using Muskhelishvili and Kolosov complex potential functions, an elastic solution is presented in this study in order to investigate stress components around circular tunnels reinforced by concrete lining with constant thickness. It was assumed that rock mass and concrete behave as isotropic linearly elastic materials. The rock mass undergoes an in situ stress field. It was also supposed that rock and concrete interface is in no-slip condition so that they have common displacement. Due to complexity of the problem for concrete reinforced layer, conformal mapping functions were utilized in order to find a solution. Supposing plane strain condition, the problem was solved, and a closed-form solution was obtained. The solution was compared to Kirsch solution, in which the lining thickness was reduced to zero, and also ABAQUS finite element software results, which showed a good agreement, except for ABAQUS software predictions around crown of tunnel lining periphery where some discrepancies were found; also it was demonstrated that this solution predicts stress components at inner lining periphery much more accurately than ABAQUS software. Finally, a sensitivity analysis based on rigidity and thickness of liner was conducted and some propositions were made on design of concrete liner. The advantage of this solution lays in the fact that it has quicker and more accurate calculation process compared to numerical methods.

A new analytical solution for stress state around gas storage cavities، under constant internal pressure، is presented in this study. Rock mass was assumed to behave as isotropic، linear elastic materials. Due to complexity of the problem for noncircular cross-section configuration of cavities، Muskhelishvili complex potential functions combined with conformal mapping was used. Finally، the results predicted by analytical solution were compared with those obtained by phase 2 finite element software for both weighted and weightless models which showed a good agreement except for some points of the cavity wall where large curvature and high stress concentration exist.

Convergence-confinement is one of the most popular methods which is used for analysis of stress and strain fields around a circular opening excavated in rock masses. It is assumed that the circular tunnel is excavated in a continuous، homogeneous، isotropic، initially elastic rock masses under hydrostatic loading. Selection of a suitable failure criterion is extremely important in the analysis since it affects the plastic zone and obtained displacement and stress field around the opening. Brown et al (1983) proposed a stepwise procedure based on Hoek-Brown criterion to obtain stress and displacement around a circular opening. A similar stepwise procedure is implemented in this study for Mohr-Coulomb criterion. Then the effect of Mohr-Coulomb and Hoek-Brown criteria on ground response curve is investigated for various models of rock mass behaviors. Finally a sensitive analysis is implemented for Mohr-Coulomb and Hoek-Brown criteria. By a comparison of Hoek-Brown and Mohr-Coulomb criteria، it can be concluded that the relative displacement in Mohr-Coulomb criterion is at least 5 times more sensitive than that in Hoek-Brown criterion to variation of strength parameters. It is obtained that varying rock mass strength parameters develops more relative deformation for medium-quality rock masses. Therefore، it should be carried out to preserve the inherent strength of rock masses by controlled blasting or mechanical excavation، especially for medium quality of rock masses (often with elastic-strain softening behavior).

Nowadays, convergence-confinement method is one of the most important methods in determining support systems for both underground circular and non-circular spaces. Selecting an appropriate constitutive model including elastic-perfectly plastic, elastic-brittle-plastic and elastic-strain softening models has a significant role in the application of this method. In some cases, the difference of displacements resulting from choosing different constitutive models is more than 100%. In this research, we attempted to investigate the effect of different constitutive models on the ground reaction curve and plastic zone made around a circular opening. It has been supposed that a rock mass is initially carries an elastic behavior and undergoes an isotropic pressure,. The failure criterion used here is non-linear Hoek-Brown yield criterion. The problem is solved in a stepwise procedure proposed by Brown et al (1983), in which rock mass strength parameters change in a stepwise method from the peak to the residual values. Finally, a sensitive analysis is performed on the Hoek-Brown strength parameters.