The Finite Element Method for the Analysis of Non-Linear and ...

Truss and Cable Elements - Example
Then equation (10) can be rewritten as:
[ t
]
τ 0
R
R
0 0
T t ρ
=
0 ρ RU t 0 S RUT ⇒ R
[ t
]
τ 0
t [
ρ
t0
=
0 0 0 ρ U t 0 S S 0
UT ⇒
0 0
[ t τ 0
0 0
]
=
]
R T t ρ
=
0 ρ RU t 0 S UT R T ⇒
0 ρ
t ρ U−1 [ t τ 0
0 0
]
(U −1 ) T
Ultimately, carrying out the calculations yields:
t
0 0 S 11 = ρ 0
t ρ ( L t P
0 L + ∆L )2 t A
Since mass=const
0 ρ 0 L 0 A = t ρ 0 L + ∆L t A ⇒ t 0 0 S 11 = L t P
0 L + ∆L 0 A
Note how the components of t 0S do not depend on rotation, hence the tensor retains only
the S 11 one component. Then, the nonlinear part of the stiffness matrix is derived as:
∫
t
0 K t T
NL = 0 B NL 0 S t 0 B NLd 0 V
0 V
[
where t 0 B NL = ( 0 J −1 −1 0 1 0
) N ,ξ =
0 −1 0 1
]
Institute of Structural Engineering Method of Finite Elements II 24