Finite difference analysis of curved lattice space structures
Peter Norman Davenport · University of Canterbury Research Repository (University of Canterbury) · 1977
An analytic technique, the finite difference calculus, is presented as a method that can be used, subject to certain restrictions, to produce mathematical models for the response of curved lattice space structures when subject to load. The restrictions are that the structure is linear elastic and has a regular lattice layout on a shallow curved surface. The method produces partial difference equations, which together with their boundary conditions may be solved analytically to give a formulae for the unknown joint deformations. Having this solution, the structural analyst need only evaluate the formulae for the values of the parameters that describe a particular structure. Solutions are given for three types of structures which all lie on a shallow second order surface, i.e. cylinders and elliptic or hyperbolic paraboloids. Single layer structures with pin jointed rods or rigid jointed beams and a double layered structure with pin jointed members are analysed. In all cases the structures are supported by gables on all four sides of a rectangular boundary. For these structures the solution for the displacements takes the form of a double trigonometric series with a finite number of terms. Numerical results obtained with a digital computer are compared with those from other methods, principally the direct stiffness method. The finite difference calculus method proved to be accurate and more economic to use. The effect of varying structural and geometrical parameters is discussed and of particular interest is the interaction between the rises of the curved surface in the two directions. A critical geometry for the hyperbolic paraboloid which has a rise and sag of equal magnitude is found. At this critical shape, the response of the structure shows an undesirable feature that load is carried by bending action rather than by membrane action.