98
5 Finite Element Formulations
rium equation as
1 ¯
K
(i)
uu q
(i)
= Δλ
(i) F
(i)
ue .
(5.111)
Further, we define a tangent vector of the equilibrium path at 0th and ith iteration
as
t
(0)
=
q
(0)
Δλ
(0)
, t
(i)
=
Δλ
(i)
q I
(i)
Δλ
(i)
(i 1) .
(5.112)
The searching orientation vector n
(i) , normal to the tangent vector t
(i) , can be defined
as
n
(i)
=
q
(i+1)
−Δλ
(i+1)
.
(5.113)
The initial increment of the loading factor Δλ
(0) is prescribed, and the next incremental loading factor can be obtained by using the constraints of n
(i)
· t
(i)
= 0 as
Δλ
(i)
=
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
((q
(0)
)
T
q II
(1)
((q (0) ) T q I
(1) + Δλ (0) i = 1
((q I
(i−1)
)
T
q II
(i)
((q I
(i−1) ) T q I
(i) + 1
i 2
(5.114)
with
q I
(i) = (
1 K
(i)
uu )
−1 F
(i)
ue , sq II
(i)
= (
1 K
(i)
uu )
−1
1 K
(i−1)
uu q
(i−1) +
1 F
(i−1)
ui
−
1 F
(i)
ui
.
(5.115)
The arc length of the first loading case can be calculated by
ΔS 0 = =t
(0)
· t
(0)
=
(Δλ (0) ) 2 + ((q (0) ) T · q (0) .
(5.116)
The arc length can be either fixed during all the loading cases, or updated according
to the desired and the actual number of iterations by using the updating equation
given as
ΔS i = ΔS i−1
I des
I i−1
,
(5.117)
where ΔS i−1 is the current arc length, and ΔS i represents the updated one. Furthermore, I des and I i−1 are the desired and the current number of iteration, respectively. Therefore, the first incremental loading factor for the next loading step can be
obtained as
Δλ
(0)
i =
±ΔS i
1 + ((q I
(0) )
T
i · ((q I
(0) ) i
.
(5.118)
Here, the sign of ΔS i can be determined by the stiffness matrix.