d z
s
q
s
f
r
T
f
T
q
d q
Figure 7.1 An element of an axisymmetric shell.
T
T
q
T
f
Figure 7.2 Yield diagram for principal tensions where the locus remains of constant size and the
effective tension T is constant.
first quadrant, strain-hardening is needed in the sheet to avoid local necking and tearing. As discussed in Section 3.3.1, thinning will accompany stretching processes and while
the stresses increase due to strain-hardening, the sheet will thin rapidly and, to a first approximation, the product of stress and thickness will be constant.
7.2.1 Shell geometry
An axisymmetric shell, or surface of revolution, is illustrated in Figure 7.3a. A point on the surface, P, can be described in terms of the cylindrical coordinates r, θ , z as shown.
The curve generating the shell, C, is illustrated in Figure 7.3b and the outward normal to the curve and the surface at P is N
P . This makes an angle φ with the axis. The ordinary
curvature of the curve at P is ρ
2
, and this is also one of the principal radii of curvature of the surface. The other principal radius of curvature of the surface is ρ
1
, as shown. The arc Simplified analysis of circular shells 109
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
boundary will therefore be a central force along the axis, as shown in Figure 7.5. The magnitude of the axial force is,
Z = T
φ
2π r sin φ
7.5 where the subscript zero refers to the boundary conditions.
• At any instant during forming, all elements of the shell are assumed to be deforming plastically.
• The sheet obeys a plane stress Tresca yield condition and, as mentioned, strain-hardens in such a way that the product of flow stress and thickness remains constant, i.e.
σ
f
t = T = constant.
f Z
T
f
r
Figure 7.5
Boundary force conditions for a shell.
7.5 Applications of the simple theory
7.5.1 Hole expansion
We consider a circular blank stretched over a domed punch as shown in Figure 7.6a. At any instant, there is a circular hole at the centre of radius r
i
, and a meridional tension is applied at the outer radius r
. At the edge of the hole, the meridional tension must be zero and a state of uniaxial tension in the circumferential direction would exist. We expect
that the meridional tension would become more tensile towards the outer edge and in the yield locus, the tensions would fall in the first quadrant of the diagram as illustrated in
Figure 7.6b.
T
f
T
q
r
i
r
i
r r
T
f
T
a b
Figure 7.6 a Hole expansion process with the sheet stretched over an axisymmetric punch.
b Region on the tension yield locus for this process.
112 Mechanics of Sheet Metal Forming
