1.42 Infinite energy

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If a bar is stretched by two opposing forces which increase in magnitude with

decreasing distance Δx between the two forces (Fig. 1.151)

P =

1

Δx

P = − 1

Δx

(1.587)

then as the distance Δx tends to zero the horizontal displacement u of the bar

becomes discontinuous; see Fig. 1.151. At the collision point of the two forces

a gap of size u(x+) − u(x−) = 1/EA opens up. It is no surprise that in the

final stage of this experiment the strain energy becomes infinite (EA = 1):

1

2

_ l

0

N2

EA

dx =

1

2

_ l

0

1

Δx2 dx =

1

2

1

Δx2Δx =

1

2

1

Δx

= ∞ Δx _→ 0 .

(1.588)

M =

1

Δx

M = − 1

Δx

(1.589)

then in the limit Δx _→ 0 a plastic hinge will form at the collision point and

the strain energy will become infinite:

1

2

_ l

0

M2

EI

dx =

1

2

_ l

0

1

Δx2 dx =

1

2

1

Δx

= ∞ Δx _→ 0 . (1.590)

What these examples are saying is that infinite forces are necessary to tear a

bar apart or to form a plastic hinge in a beam, and by virtue of the energy

balance, Wi = We = ∞×gap, the strain energy Wi must also be infinite.

In mathematical terms, a fracture or a plastic hinge is a discontinuity in

a displacement, and the message is that discontinuous displacements mean

Fig. 1.151. Effect of

tal displacements

are plotted vertically

Similar things happen in a beam, see Fig. 1.152. If two opposing moments

are applied to a beam, and if these moments increase in magnitude as the

distance Δx between the two moments shrinks,

infinite energy, see Fig. 1.153.

1.43 Conforming and nonconforming shape functions 209

Fig. 1.152. The closer the moments,

the larger they become

Fig. 1.153. A linear

interpolation requires

plastic hinges

Remark 1.18. Note that when the work done by these forces or moments 1/Δx

via functions u or w is calculated and we take the limit, we are actually

differentiating u or w_, because

lim

Δx→0

u(x + 0.5Δx) − u(x − 0.5Δx)

Δx

= u

_(x) (1.591)

lim

Δx→0

w_(x + 0.5Δx) − w_(x − 0.5Δx)

Δx

= w

__(x) . (1.592)

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