1.3 Potential energy

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To see these principles applied, we analyze a very simple structure, a taut

rope (see Fig. 1.5).

Imagine that the rope is pulled taut by a horizontal force H and that it

carries a distributed load p. The distribution of the vertical force V within

the rope and the deflection w of the rope are to be calculated. The deflection

w is the solution of the boundary value problem

− Hw

__(x) = p(x) 0< x < l w(0) = w(l) = 0. (1.7)

The vertical (or transverse) force T is proportional to the slope w_

T = Hw

_

, (1.8)

and the vector sum of H and T is the tension S in the rope

S =

_

H2 + T2 . (1.9)

The potential energy of the rope is the expression

Π(w) =

1

2

_ l

0

H(w

_)2 dx −

_ l

0

pwdx =

1

2

_ l

0

T2

H

−

_ l

0

pwdx . (1.10)

For completeness we also note Green’s first identity for the operator −Hw__:

G(w, ˆ w) =

_ l

0

−Hw

__ ˆ wdx + [T ˆ w]l

0

−

_ l

0

T ˆ T

H

dx = 0 (1.11)

because it encapsulates the structural mechanics of the rope.

To approximate the deflection w(x) of the rope, the rope is subdivided

into four linear elements: see Fig. 1.5. The first and the last node are fixed

so that only the three internal nodes can be moved. Between the nodes the

deflection is linear, that is the rope is only allowed to assume shapes that

can be expressed in terms of the three unit displacements ϕi(x) of the three

internal nodes (see Fig. 1.5)

wh(x) = w1 · ϕ1(x) + w2 · ϕ2(x) + w3 · ϕ3(x) . (1.12)

The nodal deflections, w1, w2, w3, play the role of weights. They signal how

much of each unit deflection is contained in wh.

All these different shapes—let the numbers w1, w2, w3 vary from −∞ to

+∞—constitute the so-called trial space Vh.

The space Vh itself is a subset of a greater space, the deformation space

V of the rope. The space V contains all deflection curves w(x) that the rope

can possibly assume under different loadings during its lifetime. It is obvious

that the piecewise linear functions wh in the subset Vh represent only a very

small fraction of V .

1.3 Potential energy 7

The next question then is: what values should be chosen for the three

nodal deflections w1, w2, w3 of the FE solution? What is the optimal choice?

According to the principle of minimum potential energy, the true deflection

w results in the lowest potential energy on V

Π(w) =

1

2

_ l

0

H(w

_)2dx −

_ l

0

pwdx . (1.13)

But if the exact solution w wins the competition on the big space V , it

seems a good strategy to choose the nodal deflections wi in such a way that

the FE solution

wh(x) =

_3

i=1

wi ϕi(x) (1.14)

wins the competition on the small subset Vh ⊂ V . Then Π(wh) is as close as

possible to Π(w) on Vh.

Because each function wh in Vh is uniquely determined by the nodal deflections

wi at the three interior nodes, i.e. the vector w = [w1, w2, w3]T, the

potential energy on Vh is a function of these three numbers only

Π(wh) = Π(w) =

1

2 wTKw − fTw

=

1

2

[w1, w2, w3]

4H

l

⎡

⎣

2 −1 0

−1 2−1

0 −1 2

⎤

⎦

⎡

⎣

w1

w2

w3

⎤

⎦ − [f1, f2, f3]

⎡

⎣

w1

w2

w3

⎤

⎦

=

4H

l

[w2

1

− w1 w2 + w2

2

− w2 w3 + w2

3] − f1 w1 − f2 w2 − f3 w3 ,

(1.15)

where the matrix K and the vector f have the elements

k ij =

_ l

0

Hϕ

_

i ϕ

_

j dx f i =

_ l

0

p ϕi dx = p le = p

l

4 . (1.16)

Finding the minimum value of Π on Vh is therefore equivalent to finding the

vector w—the “address” of wh ∈ Vh—for which the function Π(w) becomes

a minimum. A necessary condition is, that the first derivatives of the function

Π(w) vanish at this point w:

∂Π

∂wi

=

_3

j=1

k ij wj − f i = 0, i= 1, 2, 3 , (1.17)

which leads to the system of equations

Kw = f (1.18)

8 1 What are finite elements?

Fig. 1.6. The error e is orthogonal to

the plane

or

4H

l

⎡

⎣

2 −1 0

−1 2−1

0 −1 2

⎤

⎦

⎡

⎣

w1

w2

w3

⎤

⎦ = p l

4

⎡

⎣

111

⎤

⎦ , (1.19)

which has the solution w1 = w3 = 1.5 p l2/(16H) , w2 = 2.0 p l2/(16H). Hence

the deflection

wh(x) = p l2

16H

[1.5 · ϕ1(x) + 2.0 · ϕ2(x) + 1.5 · ϕ3(x)] (1.20)

is the best approximation on Vh.

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