1.11 Distance inside = distance outside

Back

The central equation of the FE method—in the case of a beam for example,

is

a(e, e) =

_ l

0

(M −Mh)2

EI

dx → minimum. (1.128)

The FE solution is scaled in such a way that the mean squared error attains

the smallest possible value; see Fig. 1.28. This is achieved by projecting the

exact solution onto the subspace Vh in such a way that the error in the bending

moments is orthogonal to all ϕi in Vh:

a(e, ϕi) =

_ l

0

(M −Mh)Mi

EI

dx = 0, i= 1, 2, . . . n . (1.129)

But how can the error be controlled if the exact bending moment distribution

M(x) is not known? How does the program measure M −Mh? The answer is

simple: the virtual internal energy is equal to the virtual external work. Hence

if the error in the bending moments is orthogonal to the curvature produced

by each ϕi

δWi =

_ l

0

(M −Mh)

_ _ _

unknown

Mi

EI

dx =

_ l

0

(p − ph)

_ _ _

known

ϕi dx = δWe = 0, (1.130)

the residual forces p − ph must be orthogonal to each ϕi as well. The terms

on the right-hand side are

_ l

0

(p − ph) ϕi dx =

_ l

0

pϕi dx −

_ l

0

ph ϕi dx

= fi −

_n

j=1

k ij uj = fi − fh

i = 0, (1.131)

i.e., from the equivalent nodal force fi of the load case p, the equivalent nodal

forces fh

i of the load case ph are subtracted

_ l

0

ph ϕi dx =

_ l

0

(

_n

j=1

wj pj) ϕi dx =

_n

j=1

_ l

0

pj ϕi dx wj

=

_n

j=1

k ij wj = fh

i (1.132)

and because fi = fh

i it follows that δWe(e, ϕi) = δWi(e, ϕi) = 0.

In a plate, the same equations are

38 1 What are finite elements?

Fig. 1.28. Original

load and substitute

load

δWi =

_

Ω

(σ − σh)

_ _ _

unknown

• εi dΩ =

_

Ω

(p − ph)

_ _ _

computable

•ϕi dΩ

+

_

Γ

(t − th)

_ _ _

computable

•ϕi ds +

_

k

_

Γk

_t_Δ_

computable

• ϕi ds = δWe (1.133)

where the integrals

_

k

_

Γk

tΔ •ϕi ds (1.134)

are the virtual work done by line loads tΔ, which represent the jumps in the

stresses on the interelement boundaries Γk.

The orthogonality in the stresses is equivalent to the fact that the unit

load cases pi—the action behind the displacement fields ϕi—contribute no

work on acting through the error e(x) = u(x) − uh(x)

δWi =

_

Ω

(σ − σh) • εi dΩ =

_

Ω

(p − ph) •ϕi dΩ + . . . = δWe = 0.(1.135)

This is the right occasion to recall how we argue in the force method. The

bending moment distribution M = M0+X1M1+X2M2+. . . of a continuous

1.11 Distance inside = distance outside 39

beam is orthogonal to all redundants Xi

_ l

0

MMi

EI

dx = 0 ⇒ w(xi) = 0 or Δw

_(xi) = 0, etc. (1.136)

which means that the previously eliminated constraints at the supports, i.e.,

conditions such as w(xi) = 0 or Δw_(xi) = 0 (relative rotation), are satisfied

by the exact solution.

Now it might be assumed that the orthogonality (1.133) means just this:

that the error in the displacements is zero at the nodes

_

Ω

(σ − σh) • εi dΩ = 0

?⇒

u(xi) − uh(xi) = 0 . (1.137)

But this is not true. The FE solution does not interpolate the exact solution

at the nodes.

Only in one-dimensional problems, as in beam problems, is the condition

δWi =

_ l

0

(M −Mh) Mi

EI

dx = 0, (1.138)

equivalent to the fact that the error in the deflection is zero at the nodes. Here

Mi is the bending moment corresponding to the unit deflection ϕi, and the

conclusion is correct, because (due to δWe = δWi) the orthogonality (1.138)

is equivalent to

0 = δWi = δWe = (w(xi) − wh(xi)) · P = 0· P . (1.139)

The nodal force P is the force that causes the unit nodal deflection at xi. (If

the distance from the node to the neighboring nodes is not the same, then an

additional moment M can appear, but this moment does not contribute any

work, because the rotation of the node is zero, ϕ_

i(xi) = 0.)

In plates, slabs and shells, it is not possible to associate a single point force

with a unit nodal displacement ϕi. Instead such a displacement is generated

by a diffuse cloud of surface loads and line forces in the neighborhood of

the nodes, and therefore the sharp point condition u(xi) = uh(xi) is not

guaranteed.

It is not the intention of an FE program to interpolate the exact displacement

field u at the nodes, but rather to minimize the error in the stresses.

From an engineering point of view, this certainly makes more sense than to

interpolate the true displacement field u at the nodes. Only in 1-D problems

do we get both: interpolation at the nodes + minimal distance in terms of

energy.

Remark 1.3. By 1-D problems are meant here and in the following the classical

differential equations −Hw__ (rope), −EAu__ (bar), and EI wIV (beam) of

structural mechanics. With regard to extended equations such as −EAu__+cu

or EI wIV + cw, see the remark at the end of Sect. 3.1 on p. 292.

40 1 What are finite elements?

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