The Shallow Foundation Behavior with Elastic Bed Affected by Random Vertical Loading

Abstract eng:
In many cases, the loading on the foundation is random vertical, such as foundation of mechanical machines and loading on the foundation while earthquake and vertical acceleration. The most important index of random loading is that it’s not periodic, so we can’t understand their exact analytical amount based on the previous data of system input and output. We can just guess through probability of their amount in the future by use of random vibration theory. This article aims at studying the shallow foundation behavior with elastic bed affected by random vertical loading in order to determine the most probable resulted amount of deflection and the stress. Regarding to the dynamic balance of an element, differential equation of motion of plate with damping, inertia and spring force, was obtained. Then regarding to boundary Conditions and choosing appropriate functions, this equation was solved. Having movement equation, Self–correlation function of deflection and stress were obtained based on input intersecting spectral density, supposing permanent loading process. Then we obtained the square average of maximum deflection and stress, supposing whitening of input density and then we show the results in graphs. Finally, the different parameters such as reaction coefficient of elastic bed, foundation dimensional ratio and the damping coefficient were studied. We consider here a Shallow foundation on the elastic bed affected by a random loading that is a function of time with uniform distribution. It aims at finding out the most probable output response of system, like deflection and stress in foundation and determining the effect of different parameters on that. Based on white spectral density and the supposing permanent process, and considering continuum system for a concrete foundation with an elastic bed affected by the random vertical loading, we obtained the output autocorrelation function (deflection) and also stress from bending in foundation, as well as the most probable amounts of vertical deflection and bending stress. To survey the parameters effecting, we studied damping coefficient, bed soil hardening, and the ratio of width to length and thickness, and in every cases based on the obtained functions they were solved by MATLAB numerically and we offered the effect of parameters amounts of most probable vertical deflection and bending stress in foundation in the third part of paper. Using probability and random vibration theory, we can find the maximum of deflection and stress and by extending this idea we can put other structures under random loading, survey the spectral density and its special conditions in order to guess their responses based on probability and random vibrations of structure against earthquake or any random loading kind.

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