1.15 Green’s functions

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Point solutions or Green’s functions represent the response of a structure to

point loads P = 1. The best known point solution is perhaps the triangular

shape G0(y, x) of a guitar string (Fig. 1.35 a) when the string is plucked:

52 1 What are finite elements?

Fig. 1.35. a) Green’s

function; b) the curve

dP = pdy

G0(y, x) = deflection at x, force P = 1 at y . (1.169)

The importance of the point solutions is that any distributed load p can be

approximated by a series of equally spaced (Δy) point loads ΔPi = p(yi) Δy,

and the envelope of all these triangles G0(yi, x)ΔPi (as the subdivision Δy

tends to zero) is the integral

w(x) =

_ l

0

G0(y, x) p(y) dy . (1.170)

This holds for any structure, and is the reason that Green’s functions play

such a central albeit hidden role in structural mechanics. A Green’s function

is the visible embodiment of the differential equation. The structural analysis

of a string or a taut rope could begin as well with the triangular shape and

only in the second step would we search for the differential equation that is

solved by this unit response.

Weak and strong influence functions

The right-hand side of (1.170) is an expression of external virtual work

_ l

0

0 e 0 w solves the load case p (1.171)

and because of Green’s first identity

G(w,G0) = δWe(w,G0) − δWi(w,G0) = δWe(w,G0) − a(w,G0) = 0

(1.172)

the external virtual work can be expressed as well by internal virtual work,

the strain energy product between G0 and w,

δWe(w,G0) =

_ l

0

G0(y, x) p(y) dy = a(G0, w) = δWi(w,G0) (1.173)

and so there is a “strong” and a “weak” influence function for the deflection

w(x) of the guitar string

w(x) is the envelope

of all the Green’s

functions of the different

point loads

G (y, x) p(y) dy = δW (G ,w)

1.15 Green’s functions 53

w(x) =

_ l

0

G0(y, x) p(y) dy

_ _ _

strong

= a(w,G0[x])

_ _ _

weak

=

_ l

0

Hw

_

G

_

0(y, x) dy . (1.174)

The proof of Tottenham’s equation (1.210) p. 64 is based on this switch. For

more on the subject see Sect. 7.7, p. 535.

Singularities of Green’s functions

Green’s functions being point solutions are of course subject to the conditions

set forth by physics: the singularity of the stresses must be consistent with

the dimension n of the continuum. To study the consequences this entails, let

a force Pj = 1 act at a point y = (y1, y2, y3) in 3-D space. A force in the yj

direction will cause the displacements

Uij(y, x) =

1

8πG(1 − ν) r

[(3 − 4 ν) δij + r,i r,j ] i = 1, 2, 3 (1.175)

at a remote point x = (x1, x2, x3) where the first index i indicates the direction

of the displacement. The term δij is the Kronecker delta, and the r,i are

the directional derivatives of the distance r = |y − x| with respect to the

coordinates yi of the source point

δij =

_

1 i = j

0 i        = j

r,i := ∂r

∂yi

= yi − xi

r

. (1.176)

Of course there is only one system of coordinates and therefore only one origin

but to distinguish the source point y (the load) from the field point x (the

observer) the coordinates are labeled differently.

As can be seen from (1.175), the displacements Uij behave as 1/r, because

in 3-D the stresses and therefore the strains must tend to infinity as 1/r2 and

the antiderivative of 1/r2 (the strains) is 1/r (the displacement).

The 3 × 3 functions Uij form a symmetric matrix U = [Uij ], the columns

of which are the three Green’s functions (displacement fields), corresponding

to the three unit forces acting at y

U = [G(1)

0 ,G(2)

0 ,G(3)

0 ] G(i)

0 =

⎡

⎣

U1i

U2i

U3i

⎤

⎦ . (1.177)

Line loads l(y) can be simulated by a succession of point forces, and volume

forces p(y) can be simulated by a 3-D grid of point forces, so that the corresponding

displacements can be considered the scalar product of the Green’s

functions and these loads:

ui =

_

Γ

G(i)

0 (y, x) • l(y) dsy ≡

_

Γ

1

r

l(y) dsy (1.178)

ui =

_

Ω

G(i)

0 (y, x) • p(y) dΩy ≡

_

Ω

1

r

p(y) dΩy (1.179)

54 1 What are finite elements?

1.36.

’

Dirac deltas

do exist, don’t they?’ (La-

Fert´e pedestrian bridge in

Stuttgart, Germany)

where in the second part of each equation we have retained only the characteristic

singularity 1/r. The factor 1/r is the reason that a block of concrete

cannot be prestressed with a (mathematical) wire.

To explain this behavior, let us assume that on the y1-axis (this is the

normal x-axis) and within the interval [0, 1] there acts a constant line load

l = [0, 0, l3]T pointing in the y3 direction (the z-axis), and the observer x is

located at the origin of the coordinate system, at the front of the line load.

The observer will not be able to remain in place, because he will experience

a shift of infinite magnitude in vertical direction

u3(0) =

_ 1

0

U33(y, 0) l3 dy1 = l3(3 − 4ν)

8πG(1 − ν)

_ 1

0

1

y1

dy1 = ∞. (1.180)

(Note that 1/r = 1/y1, ds = dy1 and r,3= (y3 − x3)/r = 0, because the load

and the observer are on the same level, x3 = y3 = 0).

This holds at any point x that happens to lie in the load path [0, 1], while

outside the load path the displacements are bounded because r > 0.

Hence a line element dr = O(1) cannot counterbalance a singularity that

goes as 1/r. Only an area or surface element dΩ = r dr dϕ = O(r) can cope

with such a singularity. This is why a rope that exerts only the lightest pressure

cuts through the thickest concrete, while a surface load (theoretically at least)

cannot crush a block of concrete.

Fig.

1.16 Practical consequences 55

In plane elasticity the situation is not so dramatic, because in 2-D the

stresses are O(1/r) and the displacements are O(ln r), so that a line element

dr sustains the attack of the prestressing forces l within a rope

ui(x) = l ·

_ 1

0

ln r dr = O(1) . (1.181)

The more the load spreads, point → line → surface, the weaker the singularity,

r−1 → ln r → O(1), as Fig. 1.35 so aptly illustrates: the peak under the

point load vanishes immediately if the load is evenly spread.

The importance of Green’s functions for structural mechanics probably

cannot be overestimated. Engineers often claim that point loads are imaginary

quantities which do not exist in reality, see however Fig. 1.36. This is true of

a mechanical model, but it must also be recognized that line loads or surface

loads are simply iterated point loads, and that therefore any displacement field

is just a superposition of infinitely many Green’s functions, each representing

the influence of an infinitesimal portion dP of the total load p.

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