1.29 Retrofitting structures

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Green’s functions are also an ideal tool to find the zones in a structure where

retrofit measures will be the most effective. The Vierendeel girder in Fig. 1.91

may serve as an introductory example. Which parts of the girder should be

1.29 Retrofitting structures 131

Fig. 1.92. Continuous beam, fixed on the left side a) momentM(x) from distributed

load b) moment M2(x) of the influence function for M(0.5 l) at the center of the

first span c) moment M3(x) of influence function G3 for the shear force V (0.5 l) at

the same point; ⊕ and _ indicate where ΔEI must be positive or negative if the

bending moment M(0.5 l) is to be increased by a change EI +ΔEI in the stiffness

retrofitted, EI → EI + ΔEI, to reduce the deflection (at the bottom of the

girder) due to a uniform load on the upper part of the girder?

The bending moment distribution caused by the load is plotted in Fig.

1.91 a and the bending moment of the Dirac delta is plotted in Fig. 1.91 b.

According to the equation

J(ew) = wc(x) − w(x) = −ΔEI

EI

_ x2

x1

McM0

EI

dy M0 = −EI

d2

dy2 G

__

0

(1.395)

132 1 What are finite elements?

Fig. 1.93. Retrofitting a plate to reduce the vertical displacement and the horizontal

stress σxx at the bottom of the plate. Depicted are the principal stresses a) of the

original load case (gravity load), b) of the Dirac delta δ0 and c) of the Dirac delta

δ1. The plate is fixed on both sides. In the last figure the stresses near the source

point have been masked out because they would outshine all other stresses

the parts where both contributions are large should be retrofitted and these

are obviously the joints of the frame; see Fig. 1.91 c. (As is customary we

approximate Mc by the bending moment distribution M of the unmodified

frame).

This result is typical for frame analysis. In a clamped one-span beam which

carries a uniform load p the bending moment at mid-span is p l2/24 while the

bending moment at the ends is double that value, p l2/12. So that retrofitting

measures at the ends of frame elements will be more effective in general.

But note that the change has a also direction which depends on the sign

of Mc ×Mi

J(wc) − J(w) = −ΔEI

EI

_ x2

x1

Mc(y)Mi(y, x)

EI

dy (1.396)

1.29 Retrofitting structures 133

Fig. 1.94. Cantilever plate, fixed on the left side. To reduce the deflection the

thickness t should be increased where indicated a) main stresses from edge load

0

a(G0, u)Ωe

Where the sign is negative an increase ΔEI will effect a positive change,

J(wc) − J(w) > 0 and vice versa.

In the case of the Vierendeel girder things are simple because the correlation

between M and M0 is very strong. In the case of the continuous beam in

Fig. 1.92, which carries a uniform load in the first two spans, things are a bit

more complicated. Plotted are the bending moment distribution M from the

traffic load, Fig. 1.92 a, the moment M2 of the influence function G2 for the

bending moment at the center of the first span, Fig. 1.92 b, and the moment

M3 of the influence function G3 for the shear force V (0.5 l) at the same point,

Fig. 1.92 c. Assume the goal is to increase the bending moment M(0.5 l) at

the center of the first span

b) main stresses from point load (δ ) c) contour lines of element strain energy product

134 1 What are finite elements?

Mc(0.5 l) −M(0.5 l) = −ΔEI

EI

_ x2

x1

Mc(y)M2(y, 0.5 l)

EI

dy (1.397)

then EI must be increased, ΔEI > 0, where the product of Mc × M2 is

negative and EI must be decreased, ΔEI < 0, where Mc × M2 is positive.

The symbols ⊕ and _ respectively indicate these regions.

Though it is almost too trivial to mention: a change ΔEI in the last

span, the cantilever beam, would effect nothing in the first span because the

indicator function M2 = 0 is zero in that part of the beam.

Note also that these predictions are based on the simplifying assumptions

Gci

∼ Gi and therewith Mc

i

∼ Mi. So if the change ΔEI becomes too large

the indicator functions Mi may be too far off from the true Mc

i .

As a third example we consider the plate in Fig. 1.93. A local (Ωe) change

in the thickness t → t + Δt effects a change

J(eu) = −Δt

_

Ωe

σG

ij

· εc

ij dΩy (1.398)

in any quantity J(u) of the plate in Fig. 1.93. Hence the regions where the

strain energy product of the field u ∼ uc and the Green’s function is the

largest are the most important. According to Fig. 1.93 these are the regions

near the fixed edges and the bottom of the plate. Similar considerations hold

true for the cantilever plate in Fig. 1.94 where the aim is the reduction of the

deflection at the end of the plate.

To evaluate the strain energy product, (we let uc = u),

Δt

_

Ωe

σG

ij

· εij dΩy = Δt · fe •uG (1.399)

in a single element Ωe we could either use Gaussian quadrature or we could

calculate the vector of element nodal forces fe = Ke ue and multiply this

vector with the vector uG of the nodal displacements of the Green’s function.

In some cases the solution u itself is the Green’s function for the output

functional. Consider for example a cantilever plate to which an edge load

p = {0, 1}T is applied at the upper edge Γu so that

u ∈ V : a(u, v) =

_

Γu

p •uds =

_

Γu

uy ds v ∈ V . (1.400)

If the output functional is the average value of the edge displacement

J(u) =

_

Γu

uy ds (1.401)

then J(u) = a(u,u) and so to reduce the deflection

J(eu) =

_

Γu

uc

y ds −

_

Γu

uy ds = −

_

e

Δt · a(u,u)Ωe (1.402)

1.29 Retrofitting structures 135

1.95.

exact solution at the

eglob

h = uI − uh = 0

the thickness, t → t+Δt, should be increased in all those elements Ωe where

a(u,u) is large.

We can generalize this: the displacement field u of a structure is the

Green’s function for the weighted average—the load p is the weight—of the

displacement field taken over the region Ωp where the load p is applied

J(u) =

_

Ωp

u• p dΩ =

_

Ωp

(ux · px + uy · py) dΩ = a(u,u) . (1.403)

Because in this case the indicator function G is identical with u, the correlation

between G = u and u is of course optimal. But also in the first two

examples—the Vierendeel girder and the cantilever plate—the Green’s function

and the original displacement field are of similar type and so in these two

cases a(u,u) would be nearly as good an indicator as a(G,u): simply where

the stresses are large the thickness t should be increased.

Fig. Local error

and global error

in two shafts, u is the

axial displacement a)

the FE solution does

not interpolate the

nodes while in b) the

drift is zero and so

136 1 What are finite elements?

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