1.28 The influence of a single element

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The influence of a single element Ωe on, say, the displacement uhx

(x) at a point

x in a plate is the contribution

_

Ωe

Gh0

(y, x) • p(y) dΩy (1.387)

of the element to the sum total

Fig. 1.88. Influence

of an element: The

shape u of the single

ent from the shape

Figure a). The new

shape u would fill

the void in Figure b)

it originally had in

1.28 The influence of a single element 127

uhx

(x) =

_

e

_

Ωe

Gh0

(y, x) • p(y) dΩy . (1.388)

If we employ a weak form of the influence function (“Mohr’s integral”), see

Sect. 7.7 p. 535,

uhx

= a(Gh0

,uh) =

_

Ω

[σxx · εxx + 2σxy · εxy + σyy · εyy] dΩy (1.389)

then the contribution of a single element (see Fig. 1.86)

_

Ωe

[σxx · εxx + 2σxy · εxy + σyy · εyy] dΩy (1.390)

is the strain energy product between the stress field of the Green’s function

Gh0

and the strains of the FE solution uh in this element or vice versa, because

the strain energy product is symmetric, a(Gh0

,uh) = a(uh,Gh0

).

The important point to note is that influence depends on two quantities.

The strains (or stresses) from the load case ph are weighted with the stresses

(or strains) from Ghi

. Only if both quantities are large will the contribution

be significant. And typically influence depends on the distance r = |y − x|

between the two points, G0(y, x) = G0(y − x), (and the angular orientation

between the two points x/|x| and y/|y| on the unit sphere) so that influence

functions act like convolutions.

But the influence of a single element could also be understood in the

following sense: how would the results change if the element were removed

from the structure? Or stated otherwise: how important is a certain element

for a structure?

This question can be answered with the same formulas as before, it is only

that the displacement field uh of the element must be replaced by the field

uc

h which is the shape of the element if it were drained of all its stiffness.

If a frame element [x1, x2] is removed from a structure then the change in

any output functional J(w) at any point x is, see Sect. 3.8,

J(ew) := J(wc) − J(w) =

_ x2

x1

McMi

EI

dx (1.391)

where

• Mi is the bending moment of the influence function Gi for J(w) .

• Mc is the bending moment in the spline wc that reconnects the released

nodes.

The spline wc is that curve that bridges the gap after the frame has found its

new equilibrium position. It attaches seamlessly to the two released nodes.

Equation (1.391) holds true for other structures as well; see Fig. 1.88. The

change in any output functional J(u) due to the loss of an element Ωe is

128 1 What are finite elements?

Fig. 1.89. The influ-

Gauss point depends

c of

the Gauss point

J(eu) := J(uc) − J(u) = a(uc,Gi)Ωe (1.392)

where uc is the shape of the “phantom element” that bridges seamlessly the

void left by the original element Ωe.

Because of

|J(eu)| = |a(uc,Gi)Ωe

| ≤ |a(uc,uc)Ωe

| · |a(Gi,Gi)Ωe

| (1.393)

we conclude that the strain energy of the displacement field uc and the strain

energy of the influence function Gi (which also is a displacement field) provide

an upper bound for the change.

To repeat: the displacement field uc is that displacement field which reconnects

the edges of the void after the structure has found its new equilibrium

position. It is the shape the element would assume if it were slowly drained of

all its stiffness but would cling to the structure. Alternatively one could imagine

a single element with the original stiffness which by prestressing forces is

bent into the shape uc so that it gives the impression as if it would bridge the

gap.

The energy a(uc,uc) is just the strain energy in this element. If the prestressing

forces on the edge of the element would be applied in opposite direction

on the edge of the void the structure would assume the shape it had

before the element was removed.

Hence the importance of a single element Ωe for a single value J(u) depends

on the strain energy of the fields uc and Gi inside that element. The

more the phantom element must stretch to fill the void and the more intense

the strain energy of the Green’s function Gi is in the element the more

ence of the element

on the stresses at the

(1) on the shape u

leaves if it is removed

and (2) the magnitude

of the stresses

in the element caused

by the dislocations at

the void the element

important is Ωe for J(u), see Fig. 1.89.

1.28 The influence of a single element 129

settlement of a pier

Every structural engineer knows that the elements that get stretched or

bent the most are the most important for a structure. But with (1.392) we

can quantify this feeling and give it a precise mathematical expression. And

as it turns out it is not exactly the shape u of the element we see on the

screen but the shape uc of the element when it is drained of all its stiffness

which is decisive. The larger the gap the drained element must bridge the

more important the element is.

The logic can also be applied to a planned excavation if we want to know

how much the cavity will affect the foundation of a nearby pier; see Fig. 1.90.

Do the following:

1. Apply a vertical point load P = 1 at the foot of the foundation and

calculate the strain energy a(G0,G0) of the region ΩX which is to be

excavated with a one-point quadrature that is

a(G0,G0) _ [σxx(xc) εxx(xc) + . . . + σyy(xc) εyy(xc)] × ΩX

(1.394)

where xc is the center of the cavity and ΩX is the area of the cross section.

2. Excavate ΩX, determine the edge displacement (= uc) of the cavity under

load and apply these displacements to a plate which has the same

extension as ΩX.

3. Calculate as before the strain energy in this plate. The product of these

two energies provides a rough upper bound for the additional settlement

of the pier due to the excavation.

Of course this is purely theoretical and impractical because in step 2 we

very nearly have the answer. But these steps may provide a clue as to how we

excavation of a tunnel

Fig. 1.90. Planned

and influence on the

130 1 What are finite elements?

Fig. 1.91. Retrofitting a Vierendeel girder a) bending moments from the traffic

load b) bending moments from the point load P = 1 (Dirac delta δ0) c) proposed

modifications

argue when we try to make predictions. Foremost it is the distance between the

foundation and the cavity which interests us—the energy a(G0,G0) depends

on this distance—and the shape uc of the cavity when the load is applied to

the surface of the halfspace.

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