A comparison of analytical procedures for grid structures

In this paper three methods which take into account the torsional effect in analysis of rectangular two-way grid structures arc introduced and compared. The first method in the "Plate-Analogy Method" which treats the grid as an unisotropic plate. The solution of the basic equation of the equivalent unisotropic plate, by analogy, is the solution for the grid. In this study, the finite difference technique was used to naive the plate equation. Special treatment for the case of the fixed edge grid was introduced* The second method is the " Slope-Deflection Method" which adopts three unknowns at each intersection joint of the grid: the deflection, and two orthrogonal elopes at that joint. The moments torques, and shear at the joint can be expressed in terms of the slopes and deflections of the surrounding joints. Prom equilibrium relationships, three equations can be written at each joint. Solution of the resulting sets of simultaneous equations yields as results the unknown slopes and deflections. Then the moments, torquers, and shears can be calculated. The third method considered is the " Deflection Method" proposed by the writer. The basic idea is to treat the grid as composed of discrete rigid bars connected by elastic two-way joints. The moments, torques and shears can be expressed by the deflections of the surrounding joints. From equilibrium relationships one equation can be written at each joint. Solution of resulting simultaneous equations gives the unknown deflections. Then the moments, torquers, and cheers can be calculated An a 8 X 8 square grid, uniformly loaded, with edges either simply supported, fixed,or supported at four corners was studied. The Slope-Deflection Method yields the exact answer, Whereas the Plate-Analogy Method and Deflection Method are approximate methods. The study shows that both the Plate-Analogy Method and Deflection Method give results which are acceptable for most engineering applications. The Deflection Method is much easier to apply than the other two methods. The effect of torsional stiffness of the member should not be ignored in some grid analysis. This study shows that if a grid is made of pipes, the resulting error in deflection and stresses will be 100 %; whereas, for the case of open rolled sections the error will be less than 10 % for the case studied. By assigning a special member section with the property the " Slope-Deflection Method" can be adapted as the " Grid-Analogy Method" to solve plate problems. Molecular equations for various boundary conditions for each method are developed fully in this work. These equations are developed in a very general manner, so that they could be applied to a grid with member sizes and spacings not necessarily the same in two orthogonal directions