The membrane shell theory is used extensively in designing such structures as flat-bottom tanks, pressure vessel components (Fig. 8-1) and dome roofs. The membrane theory assumes that equilibrium in the shell is achieved by having the in-plane membrane forces resist all applied loads without any bending moments. The theory gives accurate results as long as the applied loads are distributed over a large area of the shell such as pressure and wind loads. The membrane forces by themselves cannot resist local concentrated loads. Bending moments are needed to resist such loads as discussed in Chapter 10. The basic assumptions made in deriving the membrane theory (Gibson 1965) are
1. The shell is homogeneous and isotropic.
2. The thickness of the shell is small compared to its radius of curvature.
3. The bending strains are negligible and only strains in the middle surface are considered.
4. The deflection of the shell due to applied loads is small.