5.1 Basics of the Layered Approach
109
u
( j)
2 − u
( j−1)
2
2 +
u
( j)
3 − u
( j−1)
3
2 +
u
( j)
4 − u
( j−1)
4
2
u
( j)
2
2 +
u
( j)
3
2 +
u
( j)
4
2
≤ t end ,
(5.21)
where t end is a small number. Before applying the iteration scheme of Eq. (5.20), it
is appropriate to have a closer look at the tangent stiffness matrix K T . In general, the
tangent stiffness matrix can be expressed as [2]
K T =
∂ r((u)
∂∂u
= K +
∂ K
∂∂u
u ,
(5.22)
which can be written in our case of three unknowns as:
K T = K
3×3
+
∂ K
∂∂u 2
u
∂ K
∂∂u 3
u
∂ K
∂∂u 4
u
3×3 matrix
,
(5.23)
or in components:
K T =
⎡
⎢
⎢
⎣
K 11 K 12 K 13
K 21 K 22 K 23
K 31 K 32 K 33
⎤
⎥
⎥
⎦ +
⎛
⎜
⎜
⎝
⎡
⎢
⎢
⎣
∂ K 11
∂∂u 2
∂ K 12
∂∂u 2
∂ K 13
∂∂u 2
∂ K 21
∂∂u 2
∂ K 22
∂∂u 2
∂ K 23
∂∂u 2
∂ K 31
∂∂u 2
∂ K 32
∂∂u 2
∂ K 33
∂∂u 2
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
u 2
u 3
u 4
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
∂ K 11
∂∂u 3
∂ K 12
∂∂u 3
∂ K 13
∂∂u 3
∂ K 21
∂∂u 3
∂ K 22
∂∂u 3
∂ K 23
∂∂u 3
∂ K 31
∂∂u 3
∂ K 32
∂∂u 3
∂ K 33
∂∂u 3
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
u 2
u 3
u 4
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
∂ K 11
∂∂u 4
∂ K 12
∂∂u 4
∂ K 13
∂∂u 4
∂ K 21
∂∂u 4
∂ K 22
∂∂u 4
∂ K 23
∂∂u 4
∂ K 31
∂∂u 4
∂ K 32
∂∂u 4
∂ K 33
∂∂u 4
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
u 2
u 3
u 4
⎤
⎥
⎥
⎦
⎞
⎟
⎟
⎠ .
(5.24)
The partial derivatives in Eq. (5.24) can be identified under consideration of Eq. (5.17)
as partial derivatives of the bending stiffness (E I Y ) i with respect to one of the
unknown displacement increments. Since the second moment of area can be considered in our specific case as a constant, the partial derivatives take the form:
∂ K kl
∂∂u
∼
∂ E
∂∂u
.
(5.25)
Looking at Fig. 5.6, one can conclude that these partial derivatives are in the case of
an ideal plastic—or even linear hardening—material equal to zero and the tangent
stiffness matrix reduces for this special case to the stiffness matrix: K T = K .
Thus, the iteration scheme of Eq. (5.20) reduces in this special case to:
u
( j+1)
= u
( j)
−
K
( j)
−1 (K ((u))u − F)
( j)
(5.26)
=
K
( j)
−1 F
( j)
,
(5.27)