9 Various Applications of the Membrane Theory

The quantity N_ϕr₂ sin² ϕ in Eq. (8-12), when multiplied by 2π, represents the total applied force acting on a structure at a given parallel circle of angle ϕ. Hence, for complicated geometries, the value of N_ϕ in Eq. (8-12) at any given location can be obtained by taking a free-body diagram of the structure. The value of N_θ at the same location can then be determined from Eq. (8-11). This method is widely used (Jawad and Farr 1989) in designing pressure vessels, flat-bottom tanks, elevated water towers, Fig. 9-1, and other similar structures. Example 9-1 illustrates the application of this method to the design of a water tower. The American Petroleum Institute (API 620 1991) Standard has various equations and procedures for designing components by the free-body method. This method is also useful in obtaining an approximate design at the junction of two shells of different geometries. A more accurate analysis utilizing bending moments may then be performed to establish the discontinuity stresses of the selected members at a junction if a more exact analysis is needed.