1.9 Taut rope

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In the following we will show how this idea (1.87) is applied to a taut rope.

The FE solution is a linear combination of the three unit deflections

wh(x) = w1 · ϕ1(x) + w2 · ϕ2(x) + w3 · ϕ3(x) . (1.88)

Each unit deflection ϕi represents a particular load case p i ∈ Ph, i.e., a

particular arrangement of nodal forces. Take for example the unit deflection

ϕ1 of the first node (Fig. 1.24). The rope assumes this shape if at the first

node a force f1 = −P = H/le pointing upward is applied, at the next node a

force double that size pointing downward f2 = 2P, and at the third node a

force f3 = −P again pointing upward:

f1 = −H

le

↑ f2 = 2H

le

↓ f3 = −H

le

↑ (1.89)

where H is the horizontal force that pulls the rope taut. In the same sense,

two load cases p2 and p3 can be associated with the other two unit deflections

ϕ2 and ϕ3. Hence, given any shape

wh = w1 · ϕ1(x) + w2 · ϕ2(x) + w3 · ϕ3(x) (1.90)

there is a load case ph

Fig. 1.23. The work

the see-saw

done by father and

son is the same over

every cycle turn of

30 1 What are finite elements?

Fig. 1.24. The three unit load

cases p1, p2, p3 and the three unit

deflections ϕ1, ϕ2, ϕ3

ph = w1 · p1 + w2 · p2 + w3 · p3

= w1 · (↑ ↓ ↑)1,2,3 + w2 · (↑ ↓ ↑)2,3,4 + w3 · (↑ ↓ ↑)3,4,5 ,

(. . .)1,2,3 = (. . .) at the nodes 1,2,3 (1.91)

that produces the shape wh.

By an appropriate choice of the weights wi (the nodal deflections!), the

FE load case ph can be scaled in such a way that it is work-equivalent to the

distributed load p in the sense of the principle of virtual displacements. This

is the basic idea.

Because the substitute load case ph consists of only three nodal forces, the

balance between those forces and the original load p cannot be maintained

with regard to all possible virtual deflections of the rope. Rather, the number

of tests must also be restricted to three, and therefore the three unit deflections

of the three nodes are chosen as test functions (virtual deflections).

The virtual work done by the distributed load p acting through unit deflection

ϕi is the integral

δWe(p, ϕi) =

_ l

0

pϕi dx , (1.92)

and the virtual work of the three nodal forces fi at the three nodes x1, x2, x3

acting through the same unit deflection is the sum

δWe(ph, ϕi) = f1 · ϕi(x1) + f2 · ϕi(x2) + f3 · ϕi(x3) . (1.93)

The virtual work must be the same for any virtual deflection ϕi, hence

1.9 Taut rope 31

δWe(p, ϕi) = δWe(ph, ϕi), i= 1, 2, 3 . (1.94)

Next comes an important idea:

• The FE solution wh is itself an equilibrium solution, and therefore it too

satisfies the principle of virtual displacements.

Hence, for any ϕi, the virtual internal work of the FE solution wh

δWi(wh, ϕ1) =

_ l

0

ThT1

H

dx Th = Hw

_

h, T1 = Hϕ

_

1 , (1.95)

is equal to the virtual external work done by the nodal forces:

δWi(wh, ϕ1) = δWe(ph, ϕ1) . (1.96)

This means that the internal virtual energy of the FE solution can be added

to the string of equations (1.94)

δWe(p, ϕ1) = δWe(ph, ϕ1) = δWi(wh, ϕ1)

_ _ _

principle of virtual displacements

(1.97)

or if the term in the middle is dropped

δWe(p, ϕ1) = . . . = δWi(wh, ϕ1) , (1.98)

and the whole equation turned around

δWi(wh, ϕ1) = δWe(p, ϕ1) (1.99)

and the original notation used, then the result is

_ l

0

ThT1

H

dx = f1 . (1.100)

If these steps are repeated with ϕ2 and ϕ3, a system of three equations

Kw = f , (1.101)

is obtained, where K is just the stiffness matrix of the rope

K =

4H

l

⎡

⎣

2 −1 0

−1 2−1

0 −1 2

⎤

⎦ . (1.102)

The element k ij of K is the strain energy product between the two unit

deflections ϕi and ϕj

k ij =

_ l

0

Hϕ

_

i ϕ

_

j dx , (1.103)

32 1 What are finite elements?

Fig. 1.25. Taut rope:

with regard to a sineh

equivalent to the load

case p

while the component fi of the vector f is the work done by the load p acting

through ϕi

fi =

_ l

0

p ϕi dx . (1.104)

Because the distributed load p is constant, all the equivalent nodal forces are

the same

f1 = f2 = f3 = p l

4 . (1.105)

Hence the nodal deflections are

w1 = 1.5 p l2

16H

, w2 = 2.0 p l2

16H

, w3 = 1.5 p l2

16H

, (1.106)

and the solution is

wh = p l2

16H

[1.5 · ϕ1(x) + 2.0 · ϕ2(x) + 1.5 · ϕ3(x)] , (1.107)

which coincides with (1.20).

What we did in the end is that we replaced the original load case p with a

load case ph. A reviewing engineer who checks the FE solution with the unit

deflections would not find any difference between the two load cases p and

ph. On each test with one of the unit deflections, he would recognize that the

response of the rope, measured in units of virtual work, is the same.

Only by refining the tools and testing the FE solution with a sine wave

δw = sin πx

l

(1.108)

_ l

0

p sin πx

l

dx        =

_3

i=1

fi sin πxi

l

xi = location of fi . (1.109)

wave, the FE load

case p is not workwill

he realize that the two load cases cannot be the same, because the virtual

work is not the same, (see Fig. 1.25),

1.10 Least squares 33

Force method

To complete the picture, note that the force method of structural mechanics

is based on the principle of minimum complementary energy

Πc(w) = −1

2

_ l

0

M2

EI

dx + V · δ → minimum. (1.110)

(The term δ denotes a possible displacement of a support.) This principle

is essentially the opposite of the principle of minimum potential energy. The

minimum value of Πc is sought among all functions w = w0+X1 w1+. . . Xn wn

that are particular solutions of the equation EI wIV = p. The solution consists

of a curve w0, which is a particular solution of the equation EI wIV

0 = p, and

the deflections wi caused by the redundants Xi, EI wIV

i = 0. Better known

is the bending moment distribution M of this solution

M = M0 + X1M1 + X2M2 + . . . + XnMn . (1.111)

The condition ∂Πc/∂ Xi = 0 leads to the system of equations

F x = −δ0 (1.112)

where F is the flexibility matrix with elements fij = (Mi,Mj ), the vector

x = [X1,X2, . . . , Xn]T contains the redundants, and the vector δ0 embodies

the interaction between M0 and the Mi, because in the principle of minimum

complementary energy the “equivalent nodal forces” are just the scalar

product3 of the bending moment M0 (moment distribution of the statically

determinate structure) with the curvature Mi/EI of the redundants

δi0 =

_ l

0

M0Mi

EI

dx . (1.113)

Theoretically one could write an FE program that starts with a particular

state w0 enriched with n redundants wi. In frame analysis, where n is equal

to the degree of static indeterminacy this method would always yield the exact

solution, while in plate or shell analysis infinitely many Xi would be needed.

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