1.7 Set theory

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In their lowest level, many systems are at their most stable position. Many

processes in physics are governed by a minimum principle. The same holds

in beam analysis: the deflection curve w of a continuous beam minimizes the

potential energy of the beam

Π(w) =

1

2

_ l

0

M2

EI

dx −

_ l

0

pwdx → minimum (1.56)

1.7 Set theory 17

Fig. 1.12. The potential energy Π(wh) of the FE solution always lies to the right

of the exact potential energy Π(w)

on V , which is the set of all functions w that satisfy the support conditions,

i.e., that have zeros, w = 0, at all supports. All such functions w compete for

the minimum value of Π(w).

The winner is the deflection curve w of the continuous beam. According

to Green’s first identity, G(w,w) = 0 (see Sect. 7.2, p. 508)

_ l

0

M2

EI

dx =

_ l

0

pwdx (2Wi = 2We) , (1.57)

hence the minimum of the potential energy is

Π(w) =

1

2

_ l

0

M2

EI

dx −

_ l

0

pwdx = −1

2

_ l

0

pwdx . (1.58)

Obviously the potential energy is negative in the equilibrium position, because

the integral (p,w) itself is positive. It is the work done by the distributed load

p inducing its own deflections, and such work (eigenwork) is always positive.

If no load p is applied, but instead displacements δ are prescribed at one

or more supports then the potential energy is

Π(w) =

1

2

_ l

0

M2

EI

dx > 0 , (1.59)

(support displacements δ never enter into the potential energy—they only

appear in the definition of the space V ), i.e., the minimum value of Π must

be greater than zero, because the integral of M2 is positive. Hence the two

types of load cases differ by the sign of the potential energy:

• load casesp Π<0

• load casesδ Π>0 .

Now if a continuous beam is placed on additional supports as in Fig. 1.13, the

set V “shrinks” because the candidates—the deflection curves w that compete

for the minimum value of Π(w)—must have zeros, w = 0, at additional points.

In contrast if supports are removed, then V increases, because the numbers

of constraints w = 0 shrinks. Therefore the “size” of V is proportional to

18 1 What are finite elements?

ports, the smaller the space

V

and |Π(u)|

increase

potential energy must decrease (V shrinks) or increase (V grows).

increases, because then also those displacement fields that are discontinuous at

the faces of the crack can compete for the minimum value of Π(u) whereupon

the minimum value of Π(u) decreases, which actually means that |Π(u)|

increases [115].

The opposite tendency is observed in FE analysis where one seeks the

minimum value of Π(u) only on a subset Vh of V . On the subset the minimum

value cannot be less than the minimum on the whole space V .

A second observation can be added to this: in a load case p, the strain

energy of the FE solution is always less than the strain energy of the exact

solution, see (1.36),

_ l

0

M2

h

EI

dx ≤

_ l

0

M2

EI

dx (load case p) , (1.60)

while in a load case δ the situation is just the opposite, because the strain

energy of the FE solution exceeds the strain energy of the exact solution

_ l

0

M2

EI

dx ≤

_ l

0

M2

h

EI

dx (load case δ) . (1.61)

Both effects suggest that an FE solution tends to overestimate the stiffness of

a structure.

Fig.

the number of constraints and consequently the absolute value |Π(w)| of the

both the space V

1.13.

Or imagine that a crack develops in a plate; see Fig. 1.14. Then the space V

The more sup-

Fig. 1.14. With each crack,

1.7 Set theory 19

The potential energy of the exact solution is always less than the potential

energy of the FE solution:

Π(w) < Π(wh) becauseVh ⊂ V (1.62)

or if we identify Π with numbers on the x-axis, the point Π(wh) will always

lie to the right of the point Π(w); see Fig. 1.12.

This implies that in a load case p the potential energy of the FE solution

will not be as low as the potential energy of the true solution and the structure

will not deflect as much—the displacements will be smaller.

The fact that Π(wh) lies to the right of Π(w) means in a load case δ

that more strain energy is “stored” in the FE solution than the true solution.

Obviously because more energy must be supplied, to displace the support of

a stiffer structure. To sum it up we have:

• in a load casep Π(wh) is closer to zero than Π(w)

• in a load caseδ Π(wh) lies farther from zero than Π(w)

But these observations do not imply that FE displacements are smaller than

the exact displacements! This certainly will be true for some nodes, but in

general it cannot be guaranteed to be true for all nodes.

There is only one example where this conclusion—at least for one node—

holds, namely if a single force P acts at a point xP of a Kirchhoff plate. In

the equilibrium position the potential energy is just the (negative) work done

by the force P

− 1

2P w(xP) = Π(w) < Π(wh) = −1

2P wh(xP ) (1.63)

and this inequality can only be true if the FE deflection at xP is less than the

exact value, wh < w.

A similar result can be observed in a beam which is loaded at midspan,

x = l/2, with a single force P, so that

Π(w) = −1

2P w( l

2

) . (1.64)

What happens next is exactly what is predicted by set theory. The more

supports that are added (see Fig. 1.15), the smaller the deflection w(l/2) at

the center of the beam. Then V decreases, as does the absolute value |Π(w)|

of the potential energy and thus the deflection w(l/2).

The same effect can be observed if the beam is placed on one or two

additional elastic supports. Springs are different, in that they do not change

the size of V , because springs have no hard supports such as w(0) = 0.

Braces and diaphragms also enable the absolute value of the potential

energy of a structure to decrease. The more plates, beams, columns and slabs

a structure contains per cubic meter, the closer the absolute value of the

potential energy of the structure in a load case p will be to zero, while in a

20 1 What are finite elements?

Fig. 1.15. The greater the number of supports, the smaller the value of |Π|, the

smaller the deflection w, and the smaller the size of the space V

load case δ the opposite will be true. If in addition such a complex structure

is modeled with just the bare minimum of elements, the structure will be very

stiff.

Minimum or maximum ?

In some sense the principle of minimum potential energy could also be called a

maximum principle—at least for load cases p. Calling it a minimum principle is

attractive, because many processes in nature follow a principle of least action,

but in reality the load p on a beam tends to push the beam downwards as far

as possible, transforming positional energy into potential energy:

Π(w) = −1

2

_ l

0

pwdx , at w = equilibrium point (1.65)

in mathematical terms, it pushes the point |Π(w)| as far away from zero as

possible.

The movement stops at the equilibrium point. This is the point at which

the external work We equals the internal energy Wi,

We =

1

2

_ l

0

pwdx =

1

2

_ l

0

M2

EI

dx = Wi at the equilibrium point w .

(1.66)

The more the load presses the beam down (We increases), the more resistance

the load feels because the beam bends; the bending moments increase, thereby

increasing the internal energy Wi; see Fig. 1.16. The equilibrium point is the

point at which the two trends balance.

Only in load cases δ does the minimum keep its original meaning. Then

the structure tries to avoid any excess strain energy, and follows with as little

1.7 Set theory 21

Fig. 1.16. Because the internal energy Wi increases quadratically with u, while the

external work We increases only linearly, Wi always catches up with We, and there

will always be an equilibrium point where Wi = We

resistance as possible, i.e., Π(w) is as close to zero as possible the movements

imposed by the displaced supports.

But regardless of whether the minimum is positive or negative the potential

energy is in any case a concave-up parabola meaning that energy must be

added to the structure to move the structure out of its equilibrium position.

In both types of load cases, p and δ, the equilibrium is stable.

Structural mechanics in a nutshell

Figure 1.16 is a nice illustration of the so-called ellipticity of a spring. Because

the stiffness k of the spring is greater than zero, the strain energy is positive

definite

a(u, u) = k u2 > 0 u      = 0, (1.67)

and if P is bounded, the external work P · u is a continuous, linear function

of u. This guarantees that there is always a solution u = P/k, because the

parabola 1/2k u2 will ultimately rise faster than the straight line 1/2P u. The

parabola will catch up with the straight line; that is, there is always a balance

between internal energy and external work:

Wi =

1

2 k u2 =

1

2 P u = We at u = P/k . (1.68)

Note that at first the external work 1/2P u grows faster (and must grow

faster!) than the parabola 1/2k u2 of the internal energy. If that were not

the case, we would see no movement! Hence in some sense—let P = k = 1—

structural mechanics is rooted in the fact that u > u2 in the interval (0, 1),

and that beyond the end point the opposite is true. The transition point u = 1

is the equilibrium point.

22 1 What are finite elements?

Fig. 1.17. The support

beam with it

Skew projection

The attentive reader will recall our remark that in a load case δ the strain

energy of the FE solution exceeds the energy of the exact solution. When the

potential energy is minimized in a load case δ

Π(w) =

1

2

_ l

0

M2

EI

dx → minimum (1.69)

the solution is sought in the solution space S. This space contains all deflections

w which at the proper point, say the end of the beam, exhibit the correct

deflection w(l) = δ; see Fig. 1.17.

In contrast to the space S the test or trial space V consists of all those

functions w which have zero displacement, w(l) = 0, at the end of the beam,

and which of course also satisfy the other support conditions w(0) = w_(0) = 0

of the beam.

Normally, that is in load cases p, the two spaces coincide, S = V , because

normally support conditions are of homogeneous type, w(0) = 0 or w_(0) = 0

etc.. Only in load cases δ the two spaces are different. Then S is simply a shift

of V in a certain direction.

The setup of this space S can be illustrated as follows: one deflection wδ

with the property w(l) = δ is chosen and a curve from V is repeatedly added

to this curve until the whole space S is generated from wδ plus V . This may

be denoted by

S = wδ ⊕ V . (1.70)

In one regard, the space S is different from V . The sum of two curves w1 and

w2 from S is a curve with double the deflection at the end of the beam, 2 δ;

that is, the sum w1+w2 does not lie in S. Hence, if a test function ˆ w is added

to w to see whether the value of Π(w) can be reduced, Π(w + ˆw) < Π(w),

then ˆ w must be from V if the property w(l) = δ is to be retained.

Hence the minimum problem can be formulated as follows: given a fixed

but otherwise arbitrary function wδ, find the minimum of

settles and takes the

1.8 Principle of virtual displacements 23

Π(wδ + w) =

1

2

_ l

0

(Mδ +M)2

EI

dx

=

1

2

_ l

0

M2

δ

EI

dx +

_ l

0

MδM

EI

dx +

1

2

_ l

0

M2

EI

dx

=

1

2

_ l

0

M2

δ

EI

dx

_ _ _

I1

+

_ l

0

pδ wdx +

1

2

_ l

0

M2

EI

dx

_ _ _

˜Π

(w)

(1.71)

by varying the additive term w ∈ V , where pδ = EI wIV

δ is the load which belongs

to wδ. (Depending on the character of wδ, the work integral (pδ, w) must

eventually be supplemented with contributions of single forces or moments.)

Now the FE solution wh of the subproblem: find the extreme point wex of

˜Π

(w) on V

˜Π

(w) =

1

2

_ l

0

M2

EI

dx −

_ l

0

(−pδ)wdx → minimum (1.72)

satisfies the inequality ˜Π (wex) < ˜Π (wh). Hence, given that

0 < Π(wδ + wex) = I1 + ˜Π (wex) < I1 + ˜Π (wh) = Π(wδ + wh) (1.73)

the strain energy of the FE solution of a load case δ exceeds the strain energy

of the exact solution, precisely because the “FE shadow” (of the homogeneous

part) has a shorter length than the original function. The respective contributions

at wex ∈ V and wh ∈ Vh are negative:

˜Π

(wex) < ˜Π (wh) < 0 , (1.74)

because they are solutions of a load case p.

In 2-D and 3-D problems, things are a little more complicated. To satisfy

the geometric boundary condition w = ￣w      = 0 on a part ΓD of the boundary,

a deflection surface wδ must be constructed (mainly out of the nodal unit

deflections of the nodes xi which happen to lie on ΓD) which interpolates ￣ w

on ΓD. Now if ￣w is too complex, wh

δ will only be an approximation, and the

trial space will have a slightly different focus, because Sh = wh

δ

⊕V is different

from S = wδ ⊕ V .

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