不共面四边形为边界的双直纹曲面找形与主曲率分析

Form Finding and Principal Curvature Analysis of Double Ruled Surface with Non-Coplanar Quadrangle As Boundary

不共面四边形为边界的双直纹曲面建筑结构具有造型优美、传力路径清晰等优点,已经在建筑结构中被广泛应用.吕梁新城体育场屋面主体结构为钢结构,屋面双向密肋钢梁之间形成了800多个平面投影边长4352mm的正方形单元,每个单元屋面为不共面四边形边界的双直纹曲面预制混凝土壳体.结合吕梁新城体育场屋面单元混凝土壳体的设计、制作和安装,研究了不共面四边形边界双直纹曲面的找形和曲面曲率.以参数-1≤u≤1、-1<v≤1为局部坐标,两组对边等比例分点连线为母线,建立了双直纹曲面方程.根据整体坐标和局部参数之间的关系,利用微分几何理论,求出了曲面第一、第二基本形式二次型系数,推导了主曲率及其方向的计算公式.通过吕梁体育场预制混凝土屋面壳体找形计算,不仅验证了找形方法和曲率计算公式的正确性,也验证了找形方法和主曲率与坐标选取的无关性.可为建筑空间设计、曲面结构设计和施工提供理论依据.

Curved surface structures with non-coplanar quadrangle boundary have been widely used in building structures because of their beautiful shape and clear force transmission path.The main structure of the roof of the New City Stadium in Luliang is steel structure,and more than 800 square units with a plane projection side length of 4352mm are formed between the bidirectional ribbed steel beams on the roof,each unit is a precast concrete shell with non-coplanar quadrilateral boundary.Combining with the design,fabrication and installation of the concrete shell of the roof unit of the new City Stadium in Luliang,the paper studies the form finding and surface curvature of the curved surface with the boundary of non-coplanar quadrangle.The parameters-1≤u≤1 and-1≤u≤1 are taken as local coordinates,line connected the equal proportion points of opposite sides is taken as generatrix,the curved surface equations of two families of straight generatrices are established.According to the relation between global coordinates and local parameters,the quadratic coeficients of the first and second basic forms of hyperboloids are obtained by differential geometry theory,and the formulas of principal curvature and its direction are derived.Through the form finding calculation of the precast concrete roof slab of Luliang Stadium,not only the correctness of the form finding method and the calculation formula of curvature,but also the independence of the form finding method and the principal curvature and the selection of coordinates are verifed.The paper can provide theoretical basis for architectural space design,curved surface roof structure design and construction.