Vízügyi Közlemények, 1953 (35. évfolyam)
2. szám - XI. Kisebb közlemények
(39). From a structural point of view the circular storage basin, water tower tank consists two of components : 1. Circular (or annular) plate, 2. Cylindrical wall. The analysis of circular plates by the moment distribution method was treated in a previous paper of the author (Hydraulic Proceedings, 1952/Nr. I.). In this paper the deflection theory of the shell forming surface of revolution, of uniform wall thickness, circle-symmetrically loaded, further the cylindrical wall and its different loading cases are treated in detail ; formulae are derived by which the cylindrical wall can also be treated as a member of the frame consisting of plates. Finally the application of the procedure is shown on some examples. At the end of the paper handy diagrams are presented, supposing reinforced concrete structure (/г = 1/6), for the determination of the ring force and moment diagrams of the stiffness factor of the cylindrical wall. 1. Deflection theory of the circlc-symmetrically loaded shell forming surface of revolution On the shell element of uniform wall thickness (h — const.) the forces illustrated in fig. 1. are acting. The equilibrium of the components of forces parallel to the directions of the tangent of the meridional cut and those normal to the surface, as well as the equilibrium of moments acting in the meridional section are expressed by equations (1), (2) and (3). The variation of the angle of the tangent to the meridional cut at point P may be expressed by equation (4) (fig. 2.). For the deduction of the general equations the specific changes, of length of the middle surface in directions x an (i <P and of that at distance; (5), (8), (9) and (14) are needed. The variation of the principal radii of curvature is shown by equations (16) and (17). Internal forces of the shell element (normalforce 1 JV, moment M) can be determined by equations (21), (22) and (24) respectively. With knowledge of boundary conditions the eight, parameters necessary for the determination of the force pattern N x, N<p, M x, M<p, p, iB, V and w can be calculated by the above equation. 2. Cylindrical tank under hydrostatic load The general equations established in the previous chapter assume simpler forms in consequence of the special shape of the cylindrical shell. In case of membrane stress conditions the force pattern of the cylindrical wall is expressed by equations (25) — (30). The differential equation of the side wall of the tank (fig. 5.) is written down under (45) after simplifications; its complete solution is presented under (46). The force pattern of the side wall of the tank can be determined by (48). Further simplifications of these are given by formulae (61). The numerical values of functions f occurring here are determined with the aid of boundary conditions depending on the design of the tank. The integral constants necessary for calculation are given to various cases of loading (fig. 6-13) : (62)-(70). Values of f N(X, »,) and of necessary for the I» determination of the diagrams of the ring forces and moments can be easily found with aid of tables 1—VIII and of diagrams 1 — 8. 3. Cylindrical tank, loaded by earth pressure Tanks sunk in earth are loaded by earth pressure. Because of earth cover at the tank;" the shape of the load is generally trapeziodal, but this can be resolved into two triangular loads (fig. 14.) by the formulae (71) already presented. 4. Cylindrical tank loaded by edge moments In case of moments acting on the edges there acts no external load on the cylindrical wall. Therefore from the inhomogeneous differential equation (45) remains only the homogeneous part (72). Its solution with the first, second, and third derivatives are written down under (75). Stresses are given by formulae (77). With aid of boundary conditions depending on conditions of support the integral constants и г ... 4 (fig. 15 — 17, loading 9 — 11.) are determined. The numerical values of functions oi/jv (a,IJ) and ar e determined by tables IX —XI and diagrams 9—11. 5. Carry-over and stiffness factors The initial fixed-end moment necessary for moment distribution is determined for loadings 1, 2, 3, 7 and 8. The carry-over factor is determined from loading 10 (82). Attention has to be paid in calculation to the carry-over factor being a number with positive or negative sign !
