Deformable structures such as beams and arches are used widely in modern engineering applications. These structures have found vast applications in different field such as mechanical engineering, robotics, and civil engineering. Besides the mentioned applications in macro scale, these structures are also used in micro and nano scales. For instance they are used in biosensors, atomic force microscope, and micro/nano electromechanical systems. Due to the importance of arches and nano-arches because of their wide application in different engineering fields and also the importance of the stability concepts in designing different configurations of structures, the present study is concerned with the stability analysis of arches and nano-arches. It should be noted that some recently published articles shows that the reinforcing nanotubes used in composite materials are curved in their own planes. So modeling them using curved beam models would yield in more realistic results than the results obtained from straight beam models. The governing equations are derived using the variational principles and the principle of stationary value of the total potential energy. In the nano-arch problem the effect of small scale is included in the governing equation using nonlocal elasticity theory. Analytical solution of governing differential equations of structures is only possible for specified loading, geometric, and boundary conditions and sometimes these analytical solutions is so complicated that it is difficult to use them in daily engineering applications. So it is needed to use general approaches to solve easily the problems with arbitrary loading and boundary conditions. Thus the obtained equations are tried to be solved numerically in order to be able to study different states of the considered problem. The equations are descretized using finite difference method and a displacement control method called the GDC method is used to solve the system of nonlinear algebraic equations. In this method two parameters named the general stiffness parameter (GSP) and the incremental load parameter are used to detect the change in loading direction. Based on the solution used, a program is developed in MATLAB R2011a and the stability of arches and nano-arches has been studied. The solution is verified through comparing the results with the available analytical solution for the specified situations. Finally the buckling and post buckling behavior of arches and nano-arches is studied and the effect of different parameters such as the Winkler and Pasternak modulus, the initial shape, boundary conditions, load distribution, and initial imperfections on buckling load and critical initial height is investigated. Keywords Arch, Nano-arch, Shallow, Buckling, Post buckling, Finite difference Method, Nonlocal elasticity theory
