Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 666

FIGURE 11.13
Element of a bar with state variables.

N 0

Bar in Compression ( < 0 )

| N 0 |

ε = ·

EI

.

EA

10 0 0 0

. 3

EI (ε −

sin ε)

2
EI (

1 − cos ε)

− sin ε
ε

− −

01 0

ε

3

ε

2

.

2
EI (

1 − cos ε) EI sin ε

0 0 cos ε 0

ε

2

ε

U x = = (11.47)

··· ··· ··· ··· ··· ··· ···

.

00 0 1 0 0

.

00 0 0 1 0

.

EI sin ε

00 −

ε

sin ε 0 cos ε

ε

N 0

Bar in Tension ( > 0 )

N 0
EI

ε = ·

.

EA

10 0 0 0

.

− sinh ε
ε

3
EI (ε −

sinh ε)

2
EI (

1 − cosh ε)

01 0

ε

3

ε

2

.

EI (

− ε)

−

2

1 cosh

0 0 cosh ε 0

ε

2

U x = = (11.48)

··· ··· ··· ··· ··· ··· ···

.

00 0 1 0 0

.

00 0 0 1 0

.

EI
ε

sinh ε

00 − sinh ε 0 cosh ε

ε