1.12 Scalar product and weak solution

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In classical structural mechanics the deflection curve w of a beam is determined

by solving the differential equation EI wIV = p and adjusting the

solution to the boundary conditions. According to the principle of virtual displacements

(Green’s first identity), the classical solution is also a solution of

a variational problem: find a function w such that

_ l

0

M δM

EI

dx =

_ l

0

p δwdx for all δw ∈ V . (1.140)

The variational form and the differential equation are equivalent formulations.

The differential equation EI wIV = p is the Euler equation of the variational

principle. The variational solution is called a weak solution, because for the

variational statement

_ l

0

MhMi

EI

dx =

_ l

0

pϕi dx , i = 1, 2, . . . n , (1.141)

to make sense the solution must only have square-integrable second derivatives,

Mi = −EI ϕ__

i , while the Euler equation requires the solution w to have

fourth-order derivatives.

This is the official (?) version. But we think that the person who first spoke

of a weak solution had more in mind than counting derivatives.

In mathematics there is the concept of weak convergence, and this concept

is closely related to the scalar product (or principle of virtual displacements),

and ultimately to the way the shopkeeper checks the arm of a balance and

modern structural engineers argue.

To determine the mass of a brick we throw it in the air. Sensing the force

f, the acceleration a and knowing that f = ma we guess the mass m of the

brick. Basically we draw our conclusion indirectly4.

And this is how an FE program proceeds. To judge the load on a structure

an FE program “shakes” the structure. It applies virtual displacements and it

measures the virtual work done by the load. This is what the scalar product

is for.

With the scalar product duality enters the stage, and therewith the distinction

between displacements and forces. An A is tested by holding it against a

B, where A(= p) might be a distributed load and B(= δw) a virtual displacement,

and the work done by p acting through the displacement δw provides

a measure to judge p.

If we drive a truck over a bridge and then shake the bridge by applying

a series of virtual deflections δw, the truck performs virtual work. If in this

scalar product

4 According to a quote in [74] p. 172 Germain [93] expressed similar ideas:

’

When we

wish to see if a suitcase is heavy, we lift it. To estimate the tension in a (stationary)

transmission belt, we try to draw it aside from its equilibrium position. The

essential underlying mathematical idea is that of “duality”’.

1.12 Scalar product and weak solution 41

1.29.

nodal force (= work) of truck

B is the wheel load × the

deflection under the wheel.

The influence of truck A is

zero

_

Ω

p δwdΩ =: p (δw) p = truck (1.142)

the load p is kept fixed and the virtual displacement δw is varied, the scalar

product becomes a functional p (δw)5. This is an expression into which a

function δw is substituted and which returns a number. Any truck and any

load case p constitutes a functional in this sense.

If p is the original truck and ph the FE truck, then the FE method consists

in replacing the functional p () on Vh by a functional ph () in such a way

that the real truck p() and the pseudo-truck ph(), the two functionals, are

equivalent with respect to all virtual displacements ϕi ∈ Vh:

p (ϕi) = ph (ϕi), i= 1, 2, . . . , n , (1.143)

and the FE truck ph eventually will converge to the real truck p (if the mesh

size h tends to zero) if in the limit the functional ph agrees with the functional

p with respect to all virtual displacements:

lim

h→0

ph (δw) = p (δw) for all δw of the structure . (1.144)

This is what weak convergence means, and in this sense the FE solution is a

weak solution.

The distance between p and ph, the original truck and the FE truck, is

not judged directly, i.e., by comparing the pressure per square inch on the

bridge |ph(x) − p(x)|, but by studying the effects which the two trucks ph

and p trigger with regard to the same virtual displacements. Our judgement

is based on the belief that if the effects are the same then the agents behind

these effects must be the same.

This conclusion is—if the reader will allow this remark—typical of our

time where substance has been replaced by function. We no longer care what

something is, but are only interested in how it interacts with other objects.

5 Usually we denote the functional by the same letter as the load.

Fig. The equivalent

42 1 What are finite elements?

Fig. 1.30. Reduction of the load into the nodes. The equivalent nodal forces are

equal to the work which the two forces P contribute acting through the unit displacements

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