1.32 St. Venant’s principle

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According to St. Venant’s principle, the difference between the stresses due

to statically equivalent loads becomes insignificant at distances greater than

the largest dimension of the area over which the loads are spread.

St. Venant’s principle is valid for elliptic differential equations, i.e., for

most of the equations of structural mechanics. Typically, for static or harmonic

loads, the solutions decay very rapidly outside of the loaded region, as can be

seen for example, from the influence function for the bending moment mxx of

a hinged slab:

mxx(x) =

_

Γ

_

g2 · vν + mν · (g2)∂w

∂ν

_

dsy +

_

Ω

g2 · pdΩy

+

_

c

g2(yc) · Fc . (1.488)

Note that the subscript ν indicates that the functions depend on the normal

vector ν = [ν1, ν2]T at the integration point y. This vector must be distinguished

from the normal vector n at the observation point x.

The contributions to this influence function come from the support reaction

vν, the slope ∂w/∂ν, the surface load p, and the corner forces Fc; the

influence decays as ln r or r−2:

g2(y, x) = O(ln r) mν(g2(y, x)) = O(r

−2) . (1.489)

moment:

In a typical FE solution, many more sources contribute to the bending

1.32 St. Venant’s principle 173

Fig. 1.120. a) Influence functions for the support reaction A; b) only the average

value matters; c) antisymmetric loads are orthogonal to the kernel of the influence

functions

mh

xx(x) =

_

Γ

[ g2 · vh

ν + mν(g2) · ∂wh

∂ν

]

_ _ _

influence of boundary values

dsy

+

_

e

_

Ωe

g2 · ph dΩy +

_

i

_

Γi

[ g2 · vΔ

h

− ∂g2

∂ν

· mΔh

] dsy

_ _ _

influence of sources in the domain

+

_n

k=1

g2(yk) · Fh

k

_ _ _

influence of nodal forces

+

_

c

g2(yc) · Fh

c

_ _ _

influence of corner forces

(1.490)

namely the element load ph, the jumps in the Kirchhoff shear, vΔ

h , the discontinuities

mΔh

in the bending moments, the nodal forces Fh

k (due to the

corner discontinuities of the twisting moment mh

xy) and the corner forces Fh

c .

All these forces together constitute the load case ph.

This influence function looks very complicated, but in the end it is the

same polynomial that is obtained when the shape functions are differentiated

directly:

mh

xx = 3.14 + 2.72 x + 9.81 y =

_

Γ

[ g2 vh

ν

− ∂

∂ν

g2mhν

− . . . (1.491)

The strange thing is that n data cells (n = number of degrees of freedom of

the plate element) obviously suffice to store all the influence that the distant

sources have on a single element.

If the two expressions (1.488) and (1.490) are subtracted, a representation

of the FE error is obtained:

174 1 What are finite elements?

Fig. 1.121. The fastest connection between two points A and B is a cycloid. Because

the initial acceleration at the points A1 or A2 is less than at A, it takes the same

time to travel to B from A1 or A2 as from A [231]

mxx(x) − mh

xx(x) =

_

Γ

[ g2(vν − vh

ν) + mν(g2) ∂

∂ν

(w − wh) ] dsy − . . .

(1.492)

The error cannot be calculated because the support reactions vν and the slope

on the edge, ∂w/∂ν, are not known, but the formula provides a glimpse into

how the error propagates, which depends on the nature of the kernel functions:

g2 = O(ln r) = deflection surface due to Mx = 1 at x

∂

∂ν

g0 = O(r

−1) = slope at the edge

mν(g2) = O(r

−2) = bending moment at the edge

vν(g0) = O(r

−3) = Kirchhoff shear

These kernel functions decay very rapidly. The later increase of the logarithm

comes too late to be of any significance (ln 100 = 4.6).

St. Venant’s principle depends on this rapid decay of the kernels and the

averaging effect of integration, as can be illustrated by a simple example.

The support reaction A of the cantilever beam in Fig. 1.120 is the scalar

product of the influence function ηA = 1 and the distributed load p:

A =

_ l

0

ηA(x) p(x) dx . (1.493)

Because ηA is constant the support reaction A is simply the average value pa

of the distributed load p times the length l:

A =

_ l

0

p(x) dx = pa × l . (1.494)

That is, the kernel ηA = 1 eliminates all “harmonics” of p which are antisymmetric

with respect to the center x = l/2 of the beam (see Fig. 1.120 c)

because they are orthogonal to the kernel ηA.

1.33 Singularities 175

1.122. Debounded

punch the stresses in the soil

strains are infinite

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