1.13 Equivalent nodal forces

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No concept better expresses the nature of the FE method than the notion of

an equivalent nodal force, because an FE program does not think in terms of

forces but in terms of work. This is the medium whereby an FE program establishes

contact with the outside world, and forces that contribute the same

work when acting through the same displacement are the same for an FE program.

They all belong to the same equivalence class, and the representatives

of these equivalence classes are the equivalent nodal forces:

fi =

_

Ω

pϕi dΩ (kNm) = (kN/m2)(m)(m2) . (1.145)

How much a load contributes to an equivalent nodal force depends on how

much of the movement of the node is felt at the location of the load. The

influence of a node extends precisely as far as the nodal unit displacements;

see Fig. 1.29. Hence nodal unit displacements are influence functions. They

are influence functions for equivalent nodal forces (Fig. 1.30).

Now virtual work is a fuzzy measure, because given any load p there is obviously

a second, (third, fourth, ...), load ˜p not identical to p, that contributes

the same amount of work as p:

1.13 Equivalent nodal forces 43

44.32 25.04 15.167.30 2.32 0.31

30.31 28.29 17.47 9.92 4.80 1.63

47.45 23.55 14. 48 7.22 2.34 0.32

Fig. 1.31. All three

load cases are equivalent:

a) The original

load case p; b) the

FE-load case ph; c)

the equivalent nodal

forces represent the

equivalence class to

which the two load

cases belong

_ l

0

p ϕi dx = fi =

_ l

0

˜p ϕi dx . (1.146)

Hence a single nodal force fi represents a whole class of loads, namely all the

loads that contribute the same work acting through ϕi. Because they are all

equivalent with respect to ϕi, we call fi an equivalence class of loads (see Fig.

1.31) and we come to understand that the accuracy of the FE results cannot

exceed the resolution of the FE mesh.

44 1 What are finite elements?

Reverse engineering

In soil mechanics the soil is not stress free and so we must associate with the

stress state Ss (stress tensor) of the soil a set of equivalent nodal forces. But

this is easy because the equivalent nodal forces are simply

fi = a(us,ϕi) =

_

Ω

Ss •Ei dΩ (1.147)

where Ei is the strain tensor of the nodal unit displacement ϕi and us (which

actually is not required) is the deformation of the soil.

This example also demonstrates that there is an “external” or an “internal”

approach to calculating equivalent nodal forces. According to Green’s

first identity which is here formulated for an elastic solid

G(u,ϕi) =

_

Ω

p •ϕi dΩ +

_

Γ

t •ϕi ds

_ _ _

fi =δWe

−a(u,ϕi)

_ _ _

δWi

= 0 (1.148)

both approaches yield the same result. If the volume forces p and surface

tractions t are known then fi = p(ϕi) but because of (1.148) we have as well

fi = a(u,ϕi). This is what we do in reverse engineering.

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