mkb04macsztpsn4(1).pdf

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270039681 UNPDF
Macierz sztywności PSN4
i 03
:=
..
ξ
:=
(
− 11 1
1
−
) T
ξξ
:=
η 1
:=
(
− 1
− 11
) T
ηη
:=
Funkcje kształtu
Ni x y
(
, i
,
)
:=
1
4
⋅
(
1 ξ i x
+
⋅
)
⋅
(
1 η i y
+
⋅
)
NI x y
(
, i
,
) i x y
:=
(
, i
,
) identity ()
⋅
1
4
⋅
(
1 ξ i x
+
⋅
)
⋅
(
1 η i y
+
⋅
)
0
NI x y
(
, i
,
) simplify
→
1
4
(
)
(
)
0
⋅
1 ξ i x
+
⋅
⋅
1 η i y
+
⋅
Nx x y
(
, i
,
)
:=
d
d
Ni x y
(
, i
,
)
Nx x y
(
, i
,
)
→
1
4
⋅ 1 η i y
ξ i
⋅
(
+
⋅
)
x
Ny x y
(
, i
,
)
:=
d
d
Ni x y
(
, i
,
)
Ny x y
(
, i
,
) simplify
→
1
4
⋅
(
1 ξ i x
+
⋅
)
⋅
η i
y
Bloki macierzy odkształceń
Nx x y
(
, i
,
)
0
Bxy
, i
,
)
:=
0
Ny x y
(
, i
,
)
Ny x y
(
, i
,
)
Nx x y
(
, i
,
)
1
4
⋅ 1 η i y
ξ i
⋅
(
+
⋅
)
0
Bxy
, i
,
) simplify
→
0
1
4
⋅
(
1 ξ i x
+
⋅
)
⋅
η i
1
4
⋅
(
1 ξ i x
+
⋅
)
⋅
η i
1
4
⋅ 1 η i y
ξ i
⋅
(
+
⋅
)
(
(
270039681.003.png
Macierz konstytutywna
E1
:=
EE
:=
ν 0.3
:=
νν
:=
t 1
:=
t
:=
t
1
ν
0
wspłlczynnik skalujący
D
:=
E
1 ν 2
⋅
ν
1
0
de
:=
⋅
41 ν 2
Et
( )
−
0
0
1 ν
−
2
⋅
−
1
ν
0
ν
1
0
ν
1
0
C ()
:=
ν
1
0
C ()
1 ν
−
2
→
0
0
1
2
1
2
0
0
−
⋅
ν
Bloki macierzy sztywności
1
1
E
t
k
ij
11
k
ij
12
k
t
∫ ∫
− −
B
T
D
B
dx
dy
K
ij
i
j
4
2
)
k
ij
21
k
ij
22
1
1
bcb x y
(
, i
, j ,ν
,
)
:=
4Bxy
⋅
(
(
, i
,
)
) T
⋅ Bxy
C ()
⋅
(
, j
,
)
1
4
⋅ 1 η i y
ξ i
⋅
(
+
⋅
)
⋅ 1 η j y
ξ j
⋅
(
+
⋅
)
+
4
⋅
1
4
+
1
4
⋅ x
ξ i
⋅
⋅
η i
⋅
1
2
−
1
2
⋅
ν
⋅
1
4
+
(
)
bcb x y
, i
, j ,ν
,
→
1
4
1
4
η i
1
2
1
2
1
4
1
4
(
)
(
)
+
⋅ x
ξ i
⋅
⋅ ⋅ξ j
⋅ 1 η j y
⋅
+
⋅
+
ξ i 1 η i y
⋅
+
⋅
⋅
−
⋅
ν
⋅
+
bcb x y
(
, i
, j ,ν
,
) 00
,
→
1
4
⋅ 1 η i y
ξ i
⋅
(
+
⋅
)
⋅ 1 η j y
ξ j
⋅
(
+
⋅
)
+
4
⋅
1
4
+
1
4
⋅ x
ξ i
⋅
⋅
η i
⋅
1
2
−
1
2
⋅
ν
⋅
1
4
+
270039681.004.png 270039681.005.png 270039681.006.png 270039681.001.png
1
1
E
t
k
ij
11
k
ij
12
k
t
∫ ∫
− −
B
T
D
B
dx
dy
K
ij
i
j
4
2
)
k
ij
21
k
ij
22
1
1
⌡
1
⌡
1
(
) 00
k11 i j
()
:=
bcb x y
, i
, j ,ν
,
,
d
x
d
y
−
1
−
1
⌡
1
⌡
1
(
) 01
k12 i j
()
:=
bcb x y
, i
, j ,ν
,
,
d
x
d
y
−
1
−
1
⌡
1
⌡
1
(
) 10
k21 i j
()
:=
bcb x y
, i
, j ,ν
,
,
d
x
d
y
−
1
−
1
⌡
1
⌡
1
(
) 11
k22 i j
()
:=
bcb x y
, i
, j ,ν
,
,
d
x
d
y
−
1
−
1
Ostatecznie:
k11 i j
() ξ i ξ j
→
⋅
−
1
2
⋅ η j
η i
⋅ ν
⋅
+
1
2
⋅ η j
η i
⋅ i
⋅ ξ j
⋅
+
1
2
⋅ η j
η i
⋅
−
1
6
⋅ η j
η i
⋅ ⋅ξ i
⋅ ξ j
⋅
k12 i j
() ξ i ⋅η j
→
⋅
+
1
2
⋅ ξ j
η i
⋅
−
1
2
⋅ ξ j
η i
⋅ ν
⋅
k21 i j
() η i ξ j
→
⋅ ν
⋅
+
1
2
⋅ η j
ξ i
⋅
−
1
2
⋅ ⋅η j
ξ i
⋅
k22 i j
() 1
2
→
⋅ η j
η i
⋅ i
⋅ ξ j
⋅ i η j
+
⋅
+
1
2
⋅ ξ j
ξ i
⋅
−
1
6
⋅ η j
η i
⋅ ⋅ξ i
⋅ ξ j
⋅
−
1
2
⋅ ξ j
⋅ ν
⋅
Kij
()
:=
⋅
41 ν 2
Et
⋅
k11 i j
()
k12 i j
()
( )
k21 i j
()
k22 i j
()
⋅
−
ξ i ξ j
⋅
−
1
2
⋅ η j
η i
⋅ ν
⋅
+
1
2
⋅ η j
η i
⋅ i
⋅ ξ j
⋅
+
1
2
⋅ η j
⋅
−
1
6
⋅ η j
η i
⋅ ⋅ξ i
⋅ ξ j
⋅
E
⋅
⋅
(
t
44ν
⋅
Kij
()
→
1
2
1
2
E
t
44ν 2
η i ξ j
⋅ ν
⋅
+
⋅ η j
ξ i
⋅
−
⋅ ⋅η j
⋅
⋅
⋅
(
)
−
⋅
UWAGA: porwnaj powyższe rozwiązanie z rozwiązaniem prezentowanym w
książce M.Witkowskiego "Met. Komp. w Bud." str. 54.
ξ i
η i
−
ξ i
270039681.002.png
 

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