104
E. Chircan et al.
N 3 = 1 − 10ζ
3
+ 15ζ
4
− 6ζ
5
; N 4 = 10ζ
3
− 15ζ
4
+ 6ζ
5
;
N 5 = l
ζ − 6ζ
3
+ 8ζ
4
− 3ζ
5
; N 6 = l
−4ζ
3
+ 7ζ
4
− 3ζ
5
;
N 7 =
l
2
2
ζ
2
− 3ζ
3
+ 3ζ
4
− ζ
5
; N 8 =
l
2
2
ζ
3
− 2ζ
4
+ ζ
5
.
(11)
The shape function matrix is:
[N] =
⎡
⎣
N (u)
N (v)
N (w)
⎤
⎦
=
⎡
⎣
N 1 0 0 0 0 0 0 0 0 N 2 0 0 0 0 0 0 0 0
0 N 3 0 0 0 N 5 0 0 N 7 0 N 4 0 0 0 N 6 0 0 N 8
0 0 N 3 0 −N 5 0 0 −N 7 0 0 0 N 4 0 −N 6 0 0 −N 8 0
⎤
⎦ (12)
The rotations of the beam ends can be obtained as:
β = −
d
dx
([N w ]{δ e }) = −
N
w
{δ e }; γ =
d
dx
([N v ]{δ e }) =
N
v
{δ e };
(13)
Let’s also note:
N
∗
=
⎡
⎢
⎣
N
∗
(α)
N
∗
(β)
N
∗
(γ )
⎤
⎥
⎦
=
⎡
⎣
0 0 0 N 1 0 0 0 0 0 0 N 2 0 0 0 0 0
0 0 −N
3 0 N
5 0 N
7 0 0 0 −N
4 0 N
6 0 N
8 0
0 N
3
0 0 0 N
5 0 N
7 0 N
4
0 0 0 N
6 0 N
8
⎤
⎦
(14)
N
∗∗
=
N
∗∗
(z)
N
∗∗
(y)
=
0 0 −N
3 0 N
5 0 −N
7 0 0 0 −N
4 0 N
6 0 −N
8 0
0 N
3
0 0 0 N
5
0 N
7 0 N
0 0 0 N
6
0 N
(15)
So: