6.1 Shell equations

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The midsurface of the shell is represented by the position vector

x(θ1, θ2) = [x1(θ1, θ2), x2(θ1, θ2), x3(θ1, θ2)]T , (6.1)

which depends on the two parameters θ1 and θ2. If either of these is kept fixed,

the position vector traces out parameter curves θi = c on the shell midsurface;

see Fig. 6.2. The two tangent vectors

a1 = ∂x

∂θ1

, a2 = ∂x

∂θ2

(6.2)

and the associated normal vector

the coupling between these two stress states due to the curvature of the element;

see Fig. 6.1. The topic is so complex that not all aspects of FE shell

486 6 Shells

Fig. 6.2. Representation of the shell midsurface by a function x(θ1, θ2)

a3 = a1 × a2

|a1 × a2| (6.3)

form the basis vectors of a system of curvilinear coordinates. The symmetric

tensor

aik = ai • ak

_

a11 a12

a21 a22

_

=

_

E F

F G

_

(6.4)

is called the first fundamental form of the surface, and the symmetric tensor

bαβ = ∂aα

∂θβ

• a3 = aα,β • a3 (6.5)

is the second fundamental form of the surface (curvature tensor).

The basic property of a shell, as can be seen from the following equations,

which are based on Koiters shell model,

− (￣n↑

αβ − bβ

λ ￣mλα)|α + bβ

α ￣mλα|λ = pβ β = 1, 2

−bαβ (￣nαβ − bβ

λ ￣mλα) − ￣m↑

αβ|αβ = p3 , (6.6)

is that membrane and bending effects are coupled. Without the curvature

terms bαβ and bβ

α = bβ ρ aρα, the system would be decoupled. The displacements

of the midsurface would be the solutions of a system of second-order

differential equations, and the deflection w—in this model—would be the solution

of the biharmonic equation ( ￣mαβ|αβ = p or KΔΔw = p). Other

shell models adopt the Reissner–Mindlin theory for the lateral deflection, but

basically many aspects of 2-D elasticity theory can be carried over to shell

theory—in particular, what was said about about concentrated forces, point

supports, and infinite strain energy.

6.1 Shell equations 487

The strain energy product

a(u,ˆu) =

_

S

_

￣nαβ(u) γαβ(ˆu) + ￣mαβ(u) ραβ(ˆu)

_

ds (6.7)

contains strains

γαβ = γβ α =

1

2

(uα|β + uβ|α) − bαβ u3 (6.8)

and curvature terms

ραβ = ρβ α = −

_

u3|αβ − bλ

α bλβ u3 + bλ

α uλ|β + bλβ

uλ|α + bλβ

|α uλ

_

(6.9)

that are multiplied by the conjugate resultant stresses

￣nαβ = tCαβ λδ γλδ ￣mαβ = t3

12 Cαβ λδρλδ , (6.10)

where t is the shell thickness. The elasticity tensor

Cαβ λδ = Cλδ αβ = μ

_

aαλ aβ δ + aαδ aβ λ +

2 ν

1 − ν

aαβ aλδ

_

(6.11)

depends like the strain and curvature terms on the metric tensor aik = ai • ak

of the shell midsurface.

What is different in shell theory is that in general the geometry of the shell

midsurface must be approximated as well.

Fig. 6.3. The membrane stress state of a hyperboloid

is governed by a system of hyperbolic differential equations

Membrane stresses

In a membrane stress state the load is carried solely by normal forces and

plane shear forces:

488 6 Shells

nxx =

_ t/2

−t/2

σxx dz nyy =

_ t/2

−t/2

σyy dz nxy =

_ t/2

−t/2

σxy dz . (6.12)

Whereas in structural mechanics the differential equations are normally of

elliptic type, the type of the equations that govern membrane stress states

depend on the curvature of the shell. At each point of the shell midsurface,

two orthogonal directions exist with respect to which the curvature κ = 1/R

attains its maximum (κ1) and minimum (κ2) value, respectively. The Gaussian

curvature

K = det bβ

α =

1

κ1 κ2

(6.13)

determines the type of the differential equations that relates the displacements

to the load.

In cooling towers the Gaussian curvature is negative, K <0 (see Fig. 6.3)

and therefore the differential equations are of hyperbolic type. In cylindrical

shells the Gaussian curvature is zero (K = 0), so that the equations are of

parabolic type and only in a sphere where K >0 are the equations of elliptic

type; see Table 6.1. The problem is that St. Venant’s principle holds only

Table 6.1. Gauss curvature and type of differential equations

Gauss curvature type of equations example

positive elliptic sphere

zero parabolic cylindrical shell

negative hyperbolic cooling tower

for systems of elliptic equations, i.e., in a cooling tower, local disturbances

at the lower edge of the shell propagate along a generator straight up to the

rim of the shell. (In reality, cooling towers are not pure hyperbolic shells.) In

bending-dominated problems the situation is different, because the equations

are of elliptic type [189] and St. Venant’s principle applies.

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