Energy potential of the system

(u1, u2) =U W=EA

2

((1)2 + (2)2

)+ Pu2, (1)

where

(e) =L(e)

L(e)0

1, e = 1, 2,

L(1)0 =a,

L(2)0 =b,

L(1) =

(a + u1)

2+ (b + u2)

2,

L(2) =

u21 + (b + u2)

2. (2)

The denitions (2) are valid for large strains (assuming linear elastic constitutive behavior). In case of small

displacements can be used the following approximations for the strains, which are obtained from linearization of (2)

(1) =au1 + bu2a2 + b2

,

(2) =b

u2. (3)

Equilibrium equations

These are obtained from Principle of Minimum Potential Energy. The residual form of the equilibrium equations is

r

u1

u2

= 0, (4)which can be recast in the following form (more convenient from a computational point of view)

F = R, (5)

where

F =

U

u1U

u2

, (6)

R =

{0P}. (7)

Note (5) is a nonlinear system of equations.

1

References

[1] R.L. Bisplingho, J.W. Mar, T.H. Pian. Statics of Deformable Solids. Dover Publications, 1990.

2