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

Truss and Cable Elements - Example
In order to derive the nonlinear part of the stiffness matrix we first need to evaluate the
Piola-Kirchhoff stress. We know that the Cauchy stress at time t is equal to t t P
τ =
t A ,
directed along the axis of the element at time t. Using the rotational matrix we can
rotate the stress tensor from the element axis system to the original reference system
( t x 1 , t x 2 ). Denoting the rotated tensor as t ¯τ we have:
t ¯τ = R
[ t τ 0
0 0
]
[
R T cosθ −sinθ
, R =
sinθ cosθ
Then, from Lecture 4 we know that the Piola-Kirchhoff stress is given as:
t
0 [
0 S = ρ
t
0
t t X t ¯τ 0 t X T τ 0
or R
ρ
0 0
]
R T =
]
t ρ t
0 0
ρ
X t 0 S t 0 XT (10)
where the deformation gradient, t 0X, can be obtained as the product of a rotational and
a stretch component as t 0X = RU. For this example the stretch matrix is obviously
(elongation along x):
⎡
⎤
⎡
0 L + ∆L
U = ⎣ 0 0 ⎦
L
⇒ U −1 = ⎣
0 1
0 L
0 L + ∆L 0
0 1
⎤
⎦
Institute of Structural Engineering Method of Finite Elements II 23