4.3 Shape functions

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An element Ωe with n nodes xi is usually equipped with n local shape functions

ψe

i , i = 1, 2, . . . , n with the property ψe

i (xj) = δij. By continuing these

shape functions across interelement boundaries, global shape functions ψi are

generated.

With these global shape functions, “monochrome” displacement fields can

be generated which represent unit displacements fields of the nodes:

ϕ1 =

_

ψ1

0

_

ϕ2 =

_

0

ψ1

_

← horizontal displacement

← vertical displacement (4.26)

These are called monochrome because to portray a horizontal displacement

of a node, no vertical component is needed, and vice versa, even though such

monochrome displacement fields will also cause stresses in the other direction.

The FE displacement field is an expansion in terms of these 2n unit displacement

fields ϕi:

uh(x, y) =

_2n

i=1

ui ϕi(x, y)

=

node 1  __ _

u1

_

ψ1

0

_

_ _ _

ϕ1

→

+u2

_

0

ψ1

_

_ _ _

ϕ2

↑

+

node 2  __ _

u3

_

ψ2

0

_

_ _ _

ϕ3

→

+u4

_

0

ψ2

_

_ _ _

ϕ4

↑

+. . . (4.27)

The unit load case pi which can be associated with a unit displacement field

ϕi is simply the set of all forces necessary to force the plate into the shape

ϕi; see Fig. 4.9.

338 4 Plane problems

Furthermore the superposition of all these load cases pi is the FE load

case

ph =

_2n

i=1

ui pi , (4.28)

which is tuned in such a way—by adjusting the nodal displacements ui—

that it is work-equivalent to the original load case p with respect to all unit

displacement fields:

δWe(p,ϕi) = δWe(ph,ϕi) for all ϕi . (4.29)

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