Solving The Problem Of Beams On Elastic Foundation By Using The Principle Of Minimum Complementary Energy

Foundation beams are common foundation structures in construction engineering,and are widely used in various civil and industrial buildings,such as road construction,strip foundations,beam foundations,and bridge structures in railway track design,and most of the engineering structures Calculations can be simplified into calculations for elastic foundation beams.Although the analysis and calculation of elastic foundation beams is a classic research field,with the advancement of the times,new foundation models and calculation methods have emerged endlessly.Scholars’ research on this subject is still constantly improving and developing.After consulting a large number of documents,it is found that most researchers are based on the Winkler foundation model,and the methods used are mostly based on the principle of minimum potential energy.For the foundation beam system,if the foundation reaction force is basically unknown,then the external force received by the beam and the foundation reaction force satisfy the balance condition of the force,and the boundary differential condition of the foundation beam meets the force and displacement boundary conditions.The process of solving elastic foundation in this paper is based on the Boussinesq’s solution and set the deformation of the surface of the foundation to be equal to that of the beam,and the force and displacement at the boundary at infinity approach zero when the self-weight of the foundation is ignored,indicating that the force and displacement are satisfied everywhere in the foundation Boundary conditions and equilibrium equations.It can be seen that the foundation beam system can be solved using the principle of minimum residual energy.Therefore,in this paper,a half-space elastic foundation model is selected,and the elastic foundation beam is systematically studied based on the Boussinesq’s solution and the principle of minimum residual energy.Firstly,the settlement deformation formula of the foundation model is derived by means of distributed loading,and the deformation residual energy generated by the foundation under external force is obtained;the residual energy functional of the foundation beam system is obtained according to the elastic superposition principle and the energy principle,based on the minimum The principle of residual energy establishes a linear equation system with the basic unknown quantity of the foundation reaction force,combined with the stress boundary conditions to solve the foundation reaction force;combined with the specific working conditions,the mathematical software is used to write the calculation program to solve the free elastic foundation beam at both ends and the two ends have Constrained foundation beams,and compared with other solving methods;solving Winkler elastic foundation beams and half-space elastic foundation beams under the same conditions.The reaction curves and bending moment distribution curves of concrete foundations are analyzed,and the respective characteristics of the two foundation models are compared and analyzed.The main research result of this paper is to deduce the residual energy functional of elastic foundation and elastic foundation beam in half space.Based on the principle of minimum residual energy,the ground contact reaction force is solved directly.A method for solving the ground reaction force and settlement deformation of free elastic foundation beams at both ends and simply supported and fixed elastic beams at both ends under concentrated load and uniform load is given.Comparing the calculation results of the Winkler foundation beam,it reflects the limitations of the Winkler foundation model.The method used in this article can be used as a theoretical basis for studying the principle of interaction between building structure and foundation.