Zastosowanie Różnic Skończonych w Przypadkach Dwukierunkowych Stanów Naprężeń w Budowlach - W. Wierzbicki (Engineering Transactions, 1953)
This is a discussion of problems of structural mechanics concerning processes essentially continuous but assumed uncontinuous for the sake of simplicity or for mathematical reasons. In structures characterized by two-dimensional states of stress this procedure is applied principally to reinforced concrete plates, girders of finite height and dams. Chapter I concerns application of linear difference equations to the theory of plates, in which the theory of linear difference equations is commonly used, especially in reinforced concrete plates subjected to various kinds of load and supported in various manners. Sec. 1 of Chapter I contains a discussion of the various ways, in which finite differences may be applied to plate computation. The assumption that reinforced concrete plates should be treated as isotropic plates is justified. Sec. 2 brings an example of the application of linear difference equations to an isotropic cantilever plate. In Chapter II the problem of isostatic lines in girders and dams is discussed. Sec. 1 contains the derivation of differential equations of isostatic lines taking into consideration the expressions for stress directions and stress diagrams. Lines of equal stresses are represented for dams of various shapes, computed on the basis of stresses in separate points. In sec. 2 the equations of isostatic line of the type (71) are integrated. This type of equation is reduced to a nonlinear differential equation and then to a nonlinear difference equation. The equation is solved by means of Runge's numerical method. The isostatic lines are determined in several cases. A graphical method of solution of Eqs. (71) is discussed and both methods are compared.