1.16 Practical consequences

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Now there is no need to put up stop signs, because if a node is kept fixed,

it is not a point support, and a nodal force is not a point load. This is a

consequence of the inherent fuzziness of the FE method. For an FE program,

a nodal force fi is always an equivalent nodal force. It represents a load which

upon acting through a nodal unit displacement ϕi contributes work fi × 1.

The FE program neither knows nor cares whether the load is a line load, a

surface load, or a point load, and therefore nodal forces lose much of their

seemingly dangerous nature.

The same is true of point supports. For a node to be a point support it

must not only be fixed, but the support reaction must also resemble the action

of a truly concentrated force. But if the stress field near such a fixed node

is studied, it soon becomes apparent that the support reaction more closely

resembles a diffuse cloud of volume forces, surface loads, and line forces (jumps

in the stresses at interelement boundaries) than a distinct point force fi.

Near and far

It makes no sense to refine the mesh beyond a certain limit in the neighborhood

of a point support or a point load, as this can actually force the node out of

the region of interest. And it also makes no sense to design the structure

for the stresses that appear in the FE output, because these numbers are

“random” numbers whose magnitude indicates the presence of a hot spot in

the structure but which, in and of themselves, provide no lower or upper

bound for the stresses.

56 1 What are finite elements?

Fig. 1.37. We cannot make

a 3-D model of a hinged plate

But at some distance from the point load P, it no longer matters whether

the applied load is a point load or a volume force p, as may be illustrated by

the following (somewhat simplified) equations of 2-D elasticity:

ui(x) = ln rP ·P rP = |yP

− x| (1.182)

ui(x) =

_

Ω

ln r p(y) dΩy _ ln rP

_

Ω

p dΩ = ln rP · P . (1.183)

In words, at some distance from the source, the effect of a point load is essentially

identical to a one-point quadrature of the influence integral of the

volume forces.

Capacity

The integrals

_ π

0

sinxdx = 2

_ π

0

_ π

0

sin(xy) dx dy = 2.90068 (1.184)

do not change if the integrand sin x is changed at one point x0 or if sin(xy)

is changed along a whole curve. Similar results hold true in the theory of

elasticity because

• in plates (2-D elasticity) points have zero capacity,

• in elastic solids (3-D elasticity) curves (line supports!) have zero capacity.

This means that point supports or line supports (in 3-D problems) are simply

ignored by a structure. No force is necessary to displace a single point in a

plate, which implies that the displacement cannot be described at a point

support. And in makes no sense to specify the displacement field of an elastic

Virtually the same can be said about Reissner–Mindlin plates. A point load

effects an infinite deflection, w = ∞, and no force is necessary to displace a

single point in vertical direction.

A Kirchhoff plate would not tolerate this. But if a single moment M = 0

(or almost zero) is applied at a point, the point will start to rotate infinitely

rapidly. Hence if a slab is coupled with a beam via a torsional spring, the

neighborhood of the spring should not be refined too much (Fig. 1.38) lest the

rotational stiffness of the slab be lost.

Fig. 1.37.

solid along a curve. That is, an elastic solid ignores line supports; see

1.17 Why finite element results are wrong 57

Fig. 1.38. The FE model

cannot be refined too much,

Summary

What is possible:

• Kirchhoff plate: point loads, line moments m kN m/m, surface loads

• Reissner–Mindlin plate: line loads, surface loads

• plates (2-D elasticity): line loads, surface/volume forces

• elastic solids (3-D elasticity): surface loads, volume forces

What is not allowed (theoretically):

• Kirchhoff plate: single moments

• Reissner–Mindlin plate: point loads, single moments

• plates (2-D elasticity): point loads

• elastic solids (3-D elasticity): point loads, line loads

Hence it is also clear which supports are admissible and which are not (theoretically).

In actual practice, an FE structure can be placed on point supports, and it

is legitimate to work with point forces and single moments. Only the stresses

close to such points are not reliable, and depend on the mesh size: the finer

the mesh, the more singular the stresses.

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