Dynamic Vibration in Deep Beams and Solution of Governing Nonlinear Differential Equations by new Approach MrAM and AGM methods

In this paper, we are going to investigate what reactions vibration causes on deep beams. For example, examples of such behaviors can be found in the vibration of beams during an earthquake. Deep beams are widely used as transfer girders in high-rise buildings, can beams in bridges, pile caps in foundations, and other important structural members. Due to their small shear-span-to-effective-depth ratios (a/d ≤ ~۱/۸), deep beams carry heavy shear forces and develop more complex deformation patterns than slender beams. Since the ۱۹۸۰'s, researchers have carried out a large number of experiments and have proposed different types of models in order to understand the shear behavior and provide reliable predictions of the shear capacity of deep beams. However, the majority of these studies have been focused on members without web openings, while in practice openings are frequently present in deep beams to accommodate doors, windows, various conduits.The openings can significantly reduce the shear capacity of deep beams, particularly if they interrupt the direct flow of compressive stresses from the loading points to the supports (strut or arch action).[۲۱] Due to the low shear span of deep beams, high shear forces must generate to develop their flexural strength, which are sufficient to demolish the shear resistance mechanism before achieving their flexural capacity. Therefore, the shear capacity of deep beams represents a major issue in their design. Test investigations revealed that a significant amount of load is directly transmitted to the supports by a compression strut. In contrast slender beams behavior, nonlinear strain distribution dominates the response of deep beams and the plane section does not remain plane, hence, plane sectional theory is not valid in the analysis. The vibration behavior of all components of a building during vibration according to its physical properties is based on non-linear differential equations of the ode, pde type. It is very difficult to analytically solve such non-linear differential equations, which are the behaviors of each component of the structure.In this paper, we analytically solve the vibration dynamic behavior that includes nonlinear partial differential equations for deep beams. [۲۲] In this article, we present three innovative methods called Akbari Ganji Method(AGM) , Mohammadreza Akbari (MrAM).These virgin methods can be successfully applied in various engineering fields such as petroleum industry (solid,fluid,mass and heat transfer, dynamic,quantum physic,chemical reactors,electronics,applied sciences, economics, and etc.). Besides, the methodology behind these techniques are completely understandable, easy to use, and users with common knowledge of mathematics will be capable of solving the most complicated equations at low calculation cost.Our goal is to present these practical methods in this article to solve nonlinear differential equations, familiar to scientific researchers.

کلیدواژه ها:

deep beams ، vibration ، nonlinear differential equations ، MrAM ، AGM