6.3 Volume elements and degenerate shell elements

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If shells are approximated by volume elements, the number of degrees of freedom

easily becomes very large, and the large differences in the membrane and

bending stiffnesses make the element sensitive to rounding errors.

A better strategy is to design special degenerate shell elements (see Fig.

6.5) by modifying volume elements. Because these shell elements inherit their

properties from 3-D elements, they are of Reissner–Mindlin type, and are also

called Mindlin shell elements.

Fig. 6.5. Degenerate shell element, reduction

of a volume element with 20 nodes to a shell

element with 8 nodes

The reduction is essentially done by mapping all terms to the shell midsurface

while maintaining contact with points outside by means of a vector

v3 _ n:

x(ξ, η, ζ) =

_

i

xi ϕi(ξ, η) +

_

i

ϕi(ξ, η) ζ

2 v3i . (6.24)

6.4 Circular arches 491

The first sum is an expansion in terms of the intrinsic shell coordinates ξ, η of

the nodes, and the second sum is the part that extends beyond the midsurface.

In the same sense the displacement field of the shell is developed by starting

at the midsurface (ζ = 0)

u(ξ, η) =

_

i

ui ϕi(ξ, η) +

_

i

ϕi(ξ, η) ζ ti

2

[v1i αi − v2i βi] , (6.25)

and letting the second part translate the rotations αi and βi (axes v1i and

v2i in the tangential plane) into displacements at levels ζ ti/2 above the midsurface.

Next one can derive a stiffness matrix for a shell element by letting σ33 = 0:

Ke =

_ +1

−1

_ +1

−1

_ +1

−1

BT EBdet J dξ dη dζ . (6.26)

Here too one must be careful, because as t → 0 shear-locking might set in,

and if the element is curved, then so might membrane locking. But there is a

whole catalog of countermeasures with which to improve the situation [26].

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