1.26 The Dirac energy

Back

Energy plays a fundamental role in mechanics. We speak of the strain energy,

U = uT Ku, that is stored in a truss and we know that it must be equal to

the work done by the exterior load, uT f,

U = uT Ku = uT f . (1.311)

with

Fig. 1.80. Principal

stresses in a triangle

volume forces p =

1.26 The Dirac energy 111

Here we want to show that also each single displacement u(x), each single

stress σ(x) is an energy quantum which represents a specific amount of

energy—the Dirac energy11

The pulley in Fig. 1.82 is momentarily at rest and also the scale of the

market woman in Fig. 1.18, p. 24, has come to a stop. But how is the balance

maintained when the forces are not the same, when G = H or Pl         = Pr? The

answer is: because each side knows that it cannot win. If the weight Pl on the

left side of the scale moves down by δu units Pr moves up by hr/hl δu units

and so the total effort Pl · δu − Pr · (hr/hl δu) = 0 is zero. In the classical

sense equilibrium means actio = reactio or ut tenso sic vis (Hooke). The force

that pulls at a rod or a spring is equal to the force that holds the rod or

proportional to the elongation of the spring (Hooke). But in a more precise

sense equilibrium is defined by zero virtual work, which means that the forces

must be orthogonal to the rigid-body motions r, these are the functions r such

that a(r, u) = 0 for all u,

G(u, r) =

_ l

0

−EAu

__

r dx + [N r]l

0

− a(u, r) =

_ l

0

−EAu

__

r dx + [N r]l

0 = 0.

(1.312)

So we may conclude that work = force × displacement is the common denominator

in mechanics.

Sure, we say we calculate displacements or stresses, but what we actually

calculate is work

u(x) ·1 = . . . σxx ·1 = . . . (1.313)

because behind each quantity stands an influence function which is based

on energy principles and so the result is of the dimension work = force ·

displacement.

To repeat: for to calculate the shear force V (x) in a beam we apply a

dislocation δ3 = 1at x. The beam tries its best to lessen the strain by assuming

the shape G3(y, x). According to Betti’s theorem the work done by the two

shear forces Vl(x−) and Vr(x+) on both sides of x, which equals V (x) · 1, plus

the work done by the applied load p on acting through G3 is zero

δWe = −V (x) · 1 +

_ l

0

G3(y, x) p(y) dy = 0 (1.314)

and so V (x) must be equal to the work done by the load on acting through

the Green’s function G3. We call this work or energy the Dirac energy

V (x) ·1 = Dirac energy =

_ l

0

G3(y, x) p(y) dy . (1.315)

11 We understand that there is also a Dirac energy in quantum mechanics but because

structural mechanics operates on a very different length scale we think there

is very little danger of getting things mixed up.

112 1 What are finite elements?

Fig. 1.82. Pulley

So to each displacement u(x), each stress σ(x), belongs a certain mechanism

which, if released, induces a certain displacement in the structure and the

work done by the load on acting through this displacement is u(x), is σ(x), is

the Dirac energy. The Dirac energy is specific for each point x and each value

u(x), σ(x).

This energy balance δWe = 0—which is a natural extension of Newton’s

law

actio = reactio ⇒ δ u · actio = reactio · δ u (1.316)

is a basic law of structural mechanics and a simple application of the idea

behind a pulley. A pulley is characterized by its transmission ratio. The hand

H that pulls the rope downward by one unit length moves the weight G

upward by η units

δWe = H · 1 − G · η = 0. (1.317)

It is only that in structures the ratio, η = G3(y, x), is not constant but depends

on y, that is where the load is applied.

In a well designed structure the maximum values of the influence functions

for the support reactions are less than one, |GR(y, x)| ≤ 1, because otherwise

the structure amplifies the load. Archimedes knew this: ’Give me a place to

stand and I will move the Earth’ .

Statics is not static

Statics is static, isn’t it? Nothing is supposed to move. Otherwise it would

be called dynamics’. No—statics is not static, it is kinematics’.

The tourist gazing in wonder at the Eiffel tower does not realize this. The

mighty tower does not move. How then should the tourist understand that

‘

‘

1.27 How to predict changes 113

the forces in each member are well tuned, that they reflect the kinematics of

the tower from the foundation up to the very top. If the tourist drives up to

the uppermost platform then each frame element of the tower will support

her or his weight G with a fraction f which is equal to the movement of the

platform in vertical direction if a corresponding hinge is introduced in the

frame element and spread by one unit in vertical direction and obviously all

members have decided to bear the load jointly and in fair shares because in

each cross section the sum of all factors fi equals 1.0

G = (f1 + f2 + . . . fn) · G. (1.318)

That is a structure consists of infinitely many mechanisms—all bolted and

fixed so that the structure can carry the load. But if we release one mechanism,

apply a unit rotation or dislocation, then this will induce a movement in the

structure and the work done by the load on acting through this movement is

equal to M(x) · 1 or V (x) · 1, is equal to the Dirac energy. In FE analysis we

hinder the movements of the structure and so the mechanism gets the wrong

signal of how large the Dirac energy really is, and consequentlyMh(x) = M(x)

and Vh(x)        = V (x).

So the kinematics of a mesh determines the accuracy of an FE solution

• mesh = kinematics = accuracy of influence functions = quality of results .

Remark 1.12. Actually, once a structure has found the equilibrium position

we could remove the bolts and nuts from all the internal mechanisms without

having to fear that the structure would collapse because for any movement

that is compatible with the kinematics of the structure δWe would be zero,

so the structure would be “safe”.

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