Application Of Ordinary Differential Equation Method In Calculation Of Statically Indeterminate Structure

With the development of computer technology and mechanics theory, the methodsfor solving statically indeterminate structures have been a big progress, lots of analyticalmethods and numerical calculation methods have emerged. Especially the wideapplication of the finite element method, the finite element analysis software hasbecome increasingly mature. But with the continuous application of new material andnew technology, calculation of the engineering structure is more complex, and theaccuracy requirements are also getting higher and higher. Therefore, the method notonly can meet the requirement of fast computerized calculation, but also can guaranteethe results’ accuracy is favored.Based on the hypothesis of member structure in structural mechanics and thedifferential equation of the deflection member, a new method to solve staticallyindeterminate structure--ordinary differential equations method was put forward. Dividethe structure into member elements, and establish ordinary differential equations. Theinternal forces and displacements would be obtained by Maple with boundaryconditions, deformation compatibility conditions and force equilibrium conditions.Three examples were calculated to introduce the solving approaches with ordinarydifferential equation method for plane statically indeterminate beam and rigid frameunder static load, temperature load and support translation load. Compared with thetraditional calculation method, the new method’s results were verified accurate. Theordinary differential equations considering the effects of shear deformation wereintroduced and an example was calculated by Maple. Based on ordinary differentialequation method, modeled on the analytic geometry method, a more simple method--algebraic equations method was proposed which was only suitable for plane structureunder simple uniform load. The algebraic equations would be solved with presenting theconstraints conditions, and analytical expressions of internal forces and displacementsof the structure would be obtained. Based on the solution of plane statically indeterminate structure theory, the methodwas extended to solve the space frame. Torsional differential equation was added to getthe analytic solution of internal forces and displacements, and an example was given byusing the software of Maple.In bridge engineering application, with a three-span continuous beam and singletower cable-stayed structure as example, introduced the treatment method of elasticsupport and cable in ordinary differential equations.The bridge superstructure and the substructure could be analyzed together by usingthe ordinary differential equation method. Based on the elastic foundation beam method,established the deflection differential equations of the piles and girders, the horizontaldisplacement of the pile could be obtained. In this paper, by calculating a single–spanjointless integral bridge using the "m" method, introduced the application of ordinarydifferential equation method in the pile foundation calculation.The ordinary differential equation method not only provided a new idea for solvingthe problems of structural mechanics, but also made an important step for theapplication of ordinary differential equation method in bridge and structure engineering,and provided a new direction for engineering calculation software.