1.8 Principle of virtual displacements

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The preceding text began with the principle of minimum potential energy

and derived the FE method and the structural system Ku = f from this

principle. But there are other variational formulations which lead to the same

equations.

24 1 What are finite elements?

checks the equilibrium of the

arm of a scale with the principle

of virtual displacements

The most prominent of these is probably the principle of virtual displacements.

It is more general, because there are situations where a potential energy

expression does not exist, while a variational statement can still be formulated.

Furthermore, the concept of an equivalent nodal force, or more generally, of

the equivalence of different load cases, is based on the principle of virtual

displacements, and therefore this principle is central to the FE method.

From a formal perspective the principle of virtual displacements is trivial.

If 3 × 4 = 12, then δu × 3 × 4 = 12 × δu for arbitrary δu. This basically

is the principle of virtual displacements. Things become more exciting if a)

integration by parts (with functions) is performed and b) the equation δu ×

3 × u = 12 × δu is read as a variational problem to find the “strong solution”

u = 4.

Consider first a spring with stiffness k = 3 kN/m to which a force f = 12

kN is applied. The elongation u of the spring satisfies the equation 3 u = 12.

But if 3 u = 12 then δu × 3 u = 12 × δu as well, whatever the value of δ u.

Or if the vector u of the nodal displacements of a truss solves the system

Ku = f, then δuT Ku = δuT f, for any vector δu.

Or if the deflection curve w of a beam solves the differential equation

EI wIV = p, then (δw,EI wIV ) = (δw, p) for arbitrary virtual deflections δw.

3 u = 12 ⇒ δu × 3 u = 12 × δu for all δu

Ku = f ⇒ δuT Ku = δuT f for all δu

EI wIV (x) = p(x) ⇒

_ l

0

δwEI wIV dx =

_ l

0

δw p dx for all δw .

(1.75)

Fig. 1.18. A shopkeeper

placements, these are displacements which are compatible with the support

If the virtual displacements δw of the beam are admissible virtual dis1.8

Principle of virtual displacements 25

2

_ l

0

δwEI wIV dx =

_ l

0

M δM

EI

dx (1.76)

and the third equation in (1.75) becomes

EI wIV (x) = p(x) ⇒

_ l

0

M δM

EI

dx

_ _ _

δWi

=

_ l

0

δw p dx

_ _ _

δWe

. (1.77)

The equations on the left-hand side in (1.75) are the Euler equations. The

ments, i.e., equilibrium in the weak sense:

If a structure is in a state of equilibrium, then for any virtual displacement

δu, the virtual internal work δWi is the same as the virtual

external work δWe.

In classical structural mechanics conclusions are drawn from left to right,

while in modern structural mechanics the opposite is true:

Euler equation ⇒ δWi = δWe classical structural mechanics

Euler equation ⇐ δWi = δWe modern structural mechanics

Today the search for the equilibrium position is cast in the form of a variational

problem. The elongation u of the spring, the vector u of the nodal

displacements of the truss, the deflection w of the beam are solutions of the

following variational problems: find a number u, a vector u, a function w ∈ V

such that

δu × 3u↑

= 12 × δu for all δu ,

δuT Ku

↑

= δuT f for all δu ,

_ l

0

↓

M δM

EI

dx =

_ l

0

δw p dx for all δw ∈ V .

The next question then is: under what conditions is a variational solution also

a classical solution?

3 u = 12 ⇐ δu × 3 u = 12 × δu , (1.78)

Ku = f ⇐ δuT Ku = δuT f , (1.79)

EI wIV (x) = p(x) ⇐

_ l

0

M δM

EI

dx =

_ l

0

δw p dx . (1.80)

2 This is Green’s first identity G(w, δw) = 0 with M(0) = M(l) = 0; see Sect. 7.2,

p. 508.

equations on the right-hand side formulate the principle of virtual displace-

(no springs), then

conditions, and if for simplicity all the supports are assumed to be rigid

26 1 What are finite elements?

Fig. 1.19. A mechanic checks the eccentricity of a cylinder by rolling the cylinder

across a flat surface

Fig. 1.20. The approximate shape is equivalent

to a cylinder for all rotations which are

multiples of 45

How many times must the spring be moved or a virtual displacement δu be

applied to the truss or a virtual deflection δw to the beam before it is correct

to say that the variational solution is also a solution in the classical sense?

The spring has one degree of freedom, so a test with one virtual displacement

δu        = 0 suffices. If the truss has n degrees of freedom, then δWi = δWe

must be verified for at least (and at most) n linear independent virtual displacements

δu to draw the conclusion that equilibrium holds in the classical

sense, Ku = f. But a beam has infinitely many degrees of freedom, so

δWi = δWe must be verified for infinitely many virtual deflections δw before

we can claim that the variational solution w(x) satisfies the differential

equation EI wIV (x) = p(x) at any point 0 < x < l.

Elementary examples

When a shopkeeper checks the equilibrium of the arm of a scale

Pl hl = Pr hr , (1.81)

she lightly tips the arm with her finger (Fig. 1.18), and if this slight disturbance

does not start a hefty rotation of the arm then she gathers that the work done

by the two loads Pl and Pr must be the same, and she concludes that (1.81)

must hold:

Pl hl = Pr hr ⇐= Pl hl tan ϕ = Pr hr tan ϕ for all ϕ . (1.82)

The same principle is applied by a mechanic who checks the eccentricity of a

cylinder by rolling the cylinder back and forth on a flat surface. He knows that

0

1.8 Principle of virtual displacements 27

Fig. 1.21. The trial

space Vh ⊂ V and its

“dual”Ph ⊂ P

perfect cylinder ⇒ vertical deviation vanishes for all rotations ϕ

and he concludes that if he senses no vertical movement with his fingers, the

cylinder must be perfect:

perfect cylinder ⇐ vertical deviation vanishes for all rotations ϕ .

Like the mechanic, the shopkeeper uses the principle of virtual displacements,

which originally had the direction

Pl hl = Pr hr =⇒ δWe = δWi (1.83)

in the opposite direction. If the two loads Pl and Pr satisfy the variational

statement δWe = δWi then they must also satisfy the equilibrium conditions

in the classical sense

Pl hl = Pr hr ⇐= δWe = δWi . (1.84)

This is also the approach of modern structural analysis. The FE method begins

with the principle of virtual displacements, and it constructs an equivalent load

case ph which is work-equivalent to the original load case p with respect to a

finite number of virtual displacements.

Likewise the mechanic would start with an iron bar with a quadratic cross

section double the radius R of the cylinder, (2R×2R). This shape is equivalent

to the cylinder (= maintains the vertical position of its center) with regard to

all rotations which are a multiple of 900, that is ϕi = i×(360/4)0, i = 1, 2, . . ..

By refining the shape, 4 → 8 → 16 → . . . (sides), the mechanic enlarges the

“test and trial space Vh” and consequently the shape more and more begins

to resemble a true cylinder; see Fig. 1.20.

for any rotation angle ϕ of a perfect cylinder the axis maintains its (vertical)

position, (see Fig. 1.19),

28 1 What are finite elements?

Fig. 1.22. The virtual work done by the distributed load p and the equivalent nodal

forces is the same for any unit displacement ϕi: δWe(ph, ϕi) = δWe(p, ϕi)

Remark 1.2. The virtual internal work δWi of the arm of the balance is zero,

because the arm is a rigid body, but this does not change the logic. A rigid

body is in a state of equilibrium if and only if the virtual external work is

zero:

Pl hl = Pr hr ⇐⇒ δWe = δWi = 0. (1.85)

Equivalence

The set of shapes w that a taut rope can assume over its lifetime constitute

the deformation space V . The shapes wh that the shape functions ϕi can

generate constitute a small subspace Vh within V .

Now with any shape w of the rope (Fig. 1.21), we can associate a load case

p. If w ∈ C2(0, l), the load is simply the right-hand side corresponding to w,

i.e., p = −Hw__. If w is not that smooth, point forces can appear where w_ is

discontinuous. The set of all these load cases constitutes the “dual space” P.

By this mapping

deflection curve w ⇒ load case p (1.86)

the subspace Vh is also mapped onto a subspace Ph of P, and true to the nature

of the shape functions ϕi the load cases in the subspace Ph ⊂ P consist of

nodal forces only.

Because the original load case p (Fig. 1.5), is not among these load cases

in Ph, the FE method chooses a substitute load case ph in Ph, that can be

solved on Vh.

1.9 Taut rope 29

• That load case ph in Ph is chosen which is work-equivalent to the original

load case with regard to the virtual displacements ϕi ∈ Vh:

δWe(ph, ϕi) = δWe(p, ϕi) for all ϕi ∈ Vh . (1.87)

In other words, the substitute loads, the nodal forces, must—upon acting

through any virtual displacement ϕi ∈ Vh—contribute the same virtual work

as the original load p; see Fig. 1.22.

The simplest application of this idea can be seen in Fig. 1.23. Over every

cycle of the see-saw, father (load case p) and son (load case ph) (or is it vice

versa?) contribute the same virtual work. With regard to all possible cycles

of the see-saw, father and son represent two equivalent load cases. They are

indistinguishable on Vh.

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