Shell geometry The shell element

z C q P r a b C P z N r f r 1 r 2 c d r r 2 df df f r r 2 Figure 7.3 a Surface of revolution swept out by rotation of a curve C about the z axis. b Principal radii of curvature at the point P. c Geometric relations at P. length of the element along the meridian is ds = ρ 2 dφ, and from Figure 7.3b andc, the following geometric relations can be identified, r = ρ 1 sin φ 7.1 and dr = ρ 2 dφ cos φ 7.2

7.3 Equilibrium equations

As shown in Figure 7.4, we consider a shell element of sides r dθ and, ρ 2 dφ. The pressure acting on this element exerts an outward force along the surface normal of pr dθρ 2 dφ Due to the curvature of the shell, the forces on the element T θ ρ 2 dφ in the hoop or circumferential direction exert an inward force in the horizontal direction of T θ ρ 2 dφ dθ The component of this force along the normal is T θ ρ 2 dφ dθ sin φ 110 Mechanics of Sheet Metal Forming d q r T f + d T f r + dr d q T f r 2 d q d f T q r 2 df d q T f r d q T q × r 2 × d f p f Figure 7.4 Forces acting on a shell element. and the component tangential to the surface in the direction of the meridian is T θ ρ 2 dφ dθ cos φ Due to the curvature of the shell the forces along the meridian T φ r dθ exert a force in the direction normal to the surface of T φ r dθ dφ The equilibrium equation in the direction normal to the surface is pr dθρ 2 dφ = T θ ρ 2 dφ dθ sin φ + T φ r dθ dφ Combining with Equation 7.1, this reduces to p = T θ ρ 1 + T φ ρ 2 7.3 The equilibrium equation in the direction of the meridian is T φ + dT φ r + drdθ − T φ r dθ − T θ ρ 2 dφ dθ cos φ = 0 Combining with Equation 7.2, this reduces to dT φ dr − T θ − T φ r = 0 7.4

7.4 Approximate models of forming axisymmetric shells

Analytical models of some sheet forming processes are developed here using a number of simplifying assumptions. These are summarized as below. • The shell is symmetric about the central axis and all variables such as thickness, stress and tension are constant around a circumference. • The thickness is small and all shear and bending effects are neglected. • Contact pressure between the tooling and the sheet is small and friction is negligible. • The shell is bounded by planes normal to the axis and boundary loads are uniform around the circumference and act tangentially to the surface of the shell. There are no shear forces or bending moments acting on the boundaries. The total force acting at a Simplified analysis of circular shells 111