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Subgrade reaction coefficient is one of the important parameters in the design of structures that for the first time was presented by Winkler (1867). In the Winkler method, in order to evaluate and calculate the vertical displacement of the beam, it is assumed that the deformation at each point of the beam fits with the contact pressure of the reactive bed at that point and for mathematical defining of this ratio, the term ”Subgrade reaction coefficient” was used. After Winkler so many researchers such as Baiot, Terazeghi, Vesic and Bowles have studied on this parameter. Researches show that this parameter is not a constant characteristic of soil and depends on many factors. Factors such as size, shape of foundation, soil type, bending stiffness of soil and foundation, level of under ground water and water content are effective on Subgrade reaction coefficient. One  of this factor that less attention has been in prior research is the depth of foundation. In this study by using  of numerical method, the effect of the depth of foundation on the amount  of  Subgrade reaction coefficient was studied. For this purpose the results of 4 samples of plate load test at four depths 10, 12 ,14 and 20 meters and an numerical models that consider the depth of foundation placement have been used. Plate load tests have been conducted on sandy soils of Qom city. Numerical model was done by PLAXIS 2D V8.2 software and hardening soil behavior model was used for sandy soil. The results of numerical model in this study show significant effect of the depth of foundation on the value of Subgrade reaction coefficient and that by increasing the depth, this coficient is increased and changes in foundations with smaller dimention is more and with increasing the dimention of foundation, the percentage of increasing of Subgrade reaction coefficient decreases. The increase in the depth of the foundation from a given value to the next has a small effect on the amount of response coefficient, and almost immediately afterwards, the coefficient of reaction of the bed has reached a constant. In the following, results of numerical model is compared with two formula of Terzaghi and Bowles. For the comparing with Terzaghi formula, the results of plate load test is used and for different width of foundation this amount was calculated and compared with results of numerical model. Tezaghi formula for sandy soils first presents smaller amount and then larger amount than the numerical model. Based on the results of four studied soils,  with the fewer depth of test implimentation, results of Tarzaghi formula and numerical model have adaptation in fewer width of foundation. In depth of 10 to 20 meters of current research, almost in foundation width of 5 meter, the results of numerical model is adapted to Tarzaghi formula. The results of comparing the numerical model with Bowles formula shows, to a definite depth for foundation, there is good adaptation between results of  this formula and numerical model and with increasing the depth, the differences between Bowles formula and numerical model increases. Finally by using of all states data and nonlinear regression method and SPSS V20 software, for studied sand soils a proposed formula was presented. 

Keywords: Subgrade reaction coefficient, Sandy soil, Depth of foundation, Numerical model

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