传统弯桥微分方程求解中一个理论问题

在假定平曲线梁的平面内及平面外弯曲符合直梁的物理关系,平曲梁的扭矩与扭率的关系与直梁相同的条件下,根据微分几何理论,推导平曲线梁桥微单元轴向应变、弯曲曲率增量、扭转曲率与梁段单元位移、扭转关系的几何微分方程,构建曲梁的微分物理方程。理论分析结果表明:微分方程中曲梁的竖向弯曲曲率增量仅与曲梁的竖向变形有关,与梁体扭率无关;曲梁的扭率与曲梁的竖向变形无关。该方程与相关文献中推导和引用的曲梁微分物理方程(基本假定相同)结论不同。其原因是,推导过程中没有采用传统的几何方法,而是采用微分几何建立各变量之间的联系。

A Theoretical Problem in Obtaining Solution for Curved Bridge by Traditional Differential Equation

Under the consumption that,in and outside the plane of the plain curved beams,the curve conforms to the physical relationship of the straight beam and the relationship between the torque and coefficient of torsion in plain curved beam and straight beam is the same,according to the differential geometry theory,the geometry differential equation is inferred for the relationships of the plain curved beam micro unit axial strain,curving curvature increase,torsion curvature increase and beam unit displacement and torsion.The differential physical equation of the curved beam is also constructed.Theoretical analysis results indicate that vertical curvature increase of the curved beam is only related to its vertical distortion and has nothing to do with its coefficient of torsion.The coefficient of torsion of the curved beam has nothing to do with its vertical distortion.The equation is different from other conclusions inferring and quoting curved beam differential physical equation(with same fundamental assumptions) in the correlative literatures.The reason is that the differential geometrical method is used to construct the relation of the different variant in this inference other than the traditional method in other papers.