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

The Beam Element in Large Displacements
Example:
For the calculation of the incremental displacements we will need to
calculate the rotational components V s t (since V k t = 0). Using the
following formula:
V k s = θ k × t V k s + 1 2 θ k × (θ k × t V k s)
we get:
θ k × t V k s =
∣
e 1 e 2 e 3
0 0 θ
−sin( t θ k ) cos( t θ k ) 0
⎡
∣ = ⎣
−θ k cos( t θ k )
−θ k sin( t θ k )
0
⎤
⎦
and θ k × (θ k × t V k s) =
∣
e 1 e 2 e 3
0 0 θ
−θ k cos( t θ k ) −θ k sin( t θ k ) 0
⎡
∣ = ⎣
θ 2 k sin(t θ k )
−θ 2 k cos(t θ k )
0
⎤
⎦
Institute of Structural Engineering Method of Finite Elements II 45