2 Various Applications of the Membrane Theory
The quantity Nφr2 sin2φ in Eq. (1.12), 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 Nφ in Eq. (1.12) at any given location can be obtained by taking a free-body diagram of the structure. The value of Nθ at the same location can then be determined from Eq. (1.11). This method is widely used in designing pressure vessels, flat bottom tanks, elevated water towers (Figure 2.1), and other similar structures. Example 2.1 illustrates the application of this method to the design of a water-holding tank. The ASME nuclear code paragraphs NC-3932 and ND-3932 have 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.