The Generalized Variational Principle And Approximate Calculation For Beams And Plates On Half Space Elastic Foundatoins

The elastic foundation beam is a familiar component in civil engineering and different fundamental equations of which are corresponded to different foundation models. In this paper the infinite elastic half space body is taken as the foundation model, then the equations are established. It includes two parts that one is a differential equation which is composed of the external loads q, the deflection w of beams and the reaction p of foundation,and the other is a integral equation composed of the w and the p. To get the exact solution of the equations is difficult, so it is necessary to look for the approximate one. The variatioanal method is an effective means in approximate calculation,but the variatioanal functional with the foundation model should be established above all.Based on the functional of generalized potential energy of beams, the integral connection between deflection and reaction is introduced in the functional by using of Lagrange multiplier and a new generalized functional is established. The new functional is composed of two independent variables that are generalized displacement and the reaction of foundation. Its stationary value is equivalent to the above integral-differential equations and their corresponding boundary conditions. The approximate means with the Ritz method for solving the beam problems are also proposed. By the method, the results under the function of uniform load are compared with the ones by the Zemochkin method. Meanwhile, the advantage and the disadvantage of approximate solution are analyzed further. In chapter five, emulating the variatioanal procession of beams, the generalized variatioanal principle of plates is also established.