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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.

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Because of the continuous changes of mechanical properties of functionally graded materials and therefore reducing the effects of stress concentration, many researchers are interested in studying the behavior and use of these materials in various industries. For the correct design of perforated inhomogeneous plate is needed to know the accurate information about the deformation and stress distribution in different points of the plate especially around the hole. In this paper, is tried to present the analytical solution to calculate the 2D stress distribution around the circular hole in long FG plate, by using the complex potential functions method. The plate subjected to constant uniaxial or biaxial stress. One of the most important goal of this research is to study the effect of compression load applied to the hole boundary on stress distribution around the hole. The variation of material properties, especially Young's modulus is in a radial direction and concentric to the hole. The special exponential function is used to describe the variation of mechanical properties. The finite element method has been used to check the accuracy of analytical results for homogeneous and heterogeneous plates, also for all loading cases. In the presence of applied load at the boundary of circular hole, amount of radial stress in addition to hoop stress is considerable. Therefore the Von Mises stress is used to study the stress around the hole. The results showed that inhomogeneous plate with increased modulus of elasticity has greater load bearing capacity with respect to homogeneous plate.

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The beam theory is used in the analysis and design of a wide range of structures, from buildings to bridges to the load-bearing bones of the human body. Beams resting on elastic foundation have wide application in many branches of engineering problems namely geotechnics, road, railroad and marine engineering and bio-mechanics. The foundation is very often a rather complex medium; e.g., a rubberlike fuel binder, snow, or granular soil. The key issue in the analysis is modelling the contact between the structural elements and the elastic bed. Since of interest here is the response of the foundation at the contact area and not the stresses or displacements inside the foundation material, In most cases the contact is presented by replacing the elastic foundation with simple models, usually spring elements. The most frequently used foundation model in the analysis of beam on elastic foundation problems is the Winkler foundation model. In the Winkler model, the elastic bed is modeled as uniformly distributed, mutually independent, and linear elastic vertical springs which produce distributed reactions in the direction of the deflection of the beam. However since the model does not take into account either continuity or cohesion of the bed, it may be considered as a rather crude representation of the elastic foundation. In order to find a physically close and mathematically simple foundation model, Pasternak proposed a so-called two-parameter foundation model with shear interactions. The first foundation parameter is the same as the Winkler foundation model and the second one is the stiffness of the shearing layer in the Pasternak foundation model.
Dynamic analysis is an important part of structural investigation and the results of free vibration analysis are useful in this context. Vibration problems of beams on elastic foundation occupy an important place in many fields of structural and foundation engineering.With the increase of thickness, existence of simplifying hypotheses in beam theories such as the ignorance of rotational inertial and transverse shear deformation in classic theory, application of determination coefficient in first-order shear theory and expression of one or few unknown functions based on other functions in higher-order shear theories is accompanied by reduction in accuracy of these theories. This represents the necessity of precise and analytical solutions for beam problems with the least number of simplifying hypotheses and for different thicknesses.
In the present study, the analytical solution for the problem of free vibration of homogeneous prismatic simply supported beam with rectangular solid sections and desired thickness resting on Pasternak elastic foundation is provided for completely isotropic behaviors under two-dimensional theory of elasticity and functions of displacement potentials. Characteristic equations of natural vibration are defined by solving one partial differential equations of fourth order through separation of variables and application of boundary conditions. The major characteristics of present study are lack of limitation of thickness and its validity for beams of low, medium and large thickness. To verify, the results of present study were compared with those of other studies. The results show that increases of foundation parameters is associated with an increased natural frequency, The intensity by increasing the ratio of thickness to length and in values larger than 0.2 and in the higher modes of vibration is reduced considerably.