The Calculus Of Variations Applied To Stability Of Slopes

The analysis of the stability of slope is extremely importanrt to prevent the earth slide which is common and grave natural hazard. Methods used at the present time for analysing the stability of slope consist of determing the factors of safety associated with a series of sliding lines previously defined by the engineer and then obtaining the minimum of these, which is taken as rhe safety factor of the slope. However, as this trial procedure does not cover all the possible curves it is obvious that other lines could lead to a smaller factor of safety. But since the number of curves which are actually used in the trial is very high, it can be hoped that the difference between the absolute minimum and the computed one would be small enough not to give serious consequences. The analysis of stability of a slope by limiting equilibrium criteria reduces to the search for the critical slip line, giving the minimum safety factor, and to the calculation of this value. The common procedure of searching for this line, which has been previously described, is arduous and in addition becomes terribly cumbersome, owing to the high number of calculations actually involved. However, the problem can be solved by successive approximations. This means that it is necessary to know the resultes of the first trial to start the second one and so on until a clear idear of the shape of the sliding line is obtained. This involves a long time for the engineer to analyse the slope. For these and other reasons it would be very useful to have a method available to provide this sliding line without the shortcomings just described. The method presented here meets these requirements, allowing a solution of the problem using the calculus of variations.Firstly, this work introduces some method in common use of analysing the stability of slope, construing them elementarily at the same time. Secondly, some basic theorem and deducibility of the theory of the calculus of variations are explained simply, and the two things based on the problems of variations of the changeless boundary and the movable one are discussed. Thirdly, the expression of the factors of the stability of slope (F) is educed, applying the calculus of variations to the method of analysing the stability of slope (Janbu) and assuming the theory of Janbu and the critical slip line calculated by the equation of Eular and the conditions of the boundary and the transversality. In addition, the means dealed with the thing of existing water outside the slope are argued. Finally, the calculus of variations applied to the stability of slope is validated by a case.The result obtained with the calculus is discussed.