Elementos finitos
!
" %
#
$
&
'
(
& ' & '
)
(
& ' & ' ) )
*
(
+
&$&
,
&$&
[ ke ] = [ Be ] [ D ][ Be ] dVe
Ve
T
-
[ D] = E
dVe = A( x)dx
.
[ ke ] =
L
E [ Be ] [ Be ] A( x)dx
T
(
[ Be ] =
Ne = .
dN e dx
/
# 2 #
0+ $ &$'1 # $
+
&$'
.
ui = 1 xi
Ni
i = 0,1, 2,3,........., n
!
$
# #
N1u2 = 0 N2 ( x1 ,1)
+
&$'
#
u1
u2 N1 ( x2 , 0) u1 = 1
-
N1
#
m= ( y2 − y1 ) ( x2 − x1 )
( y − yi ) = m( x − xi )
y = Ni y1 = u1 y2 = u 2
.
m= (u2 − u1 ) ( x2 − x1 )
( N i − ui ) = m( x − xi ) N i = m( x − xi ) + ui
m= (0 − 1) −1 = ( x2 − x1 ) ( x2 − x1 ) x2 − x1 = L −1 L
m=
N1 =
−1 ( x − x1 ) + 1 L ( x1 − x) +1 L
N1 =
( .
( x2 ,1)
N2
N2u2 = 1 u2 u1 = 0
!
( x1 , 0)
m=
(1 − 0) 1 = ( x2 − x1 ) ( x2 − x1 ) x2 − x1 = L 1 L
m=
N2 =
1 ( x − x2 ) + 1 L ( x − x2 ) +1 L
N2 =
(
N1 = ( x1 − x) +1 L ( x − x2 ) +1 L
N2 =
-
&
x1 = 0 x2 = L
N1 = 1 −
x L
N2 =
x L
3
x L x L
Ne = 1 −
( .
Ne
Be =
%
&$&
!
Be
dN e dx
/
Ne = 1 − x L x L
d 1− Be = dx
x L
d
xL dx
Be =
−1 L
1 L
3
−1 L
[ Be ]
T
= 1 L
−1 L
[ Be ] [ Be ] =
1 L
T
T
−1 L
1 L
[ Be ] [ Be ] =
4
1 1 −1 L2 −1 1
[ ke ] =
L
E [ Be ] [ Be ] A( x)dx
T
/
[ ke ] = EA [ Be ] [ Be ] dx
L
T
[ ke ] = EA
L
1 1 −1 dx L2 −1 1
3
#
[ Be ] [ Be ] =
T
1 1 −1 L2 −1 1
[ ke ] =
EA 1 −1 L2 −1 1
dx
L
[ ke ] =
EA1 −1 L L2 −1 1
[ ke ] =
EA 1 −1 L −1 1
5
# ! 6 29 x10 psi $ 0/
7
6
62 p lg 2
#
1
&
'
( (
x1 = 0 x2 = 5 ft = 60 p lg N1
& '
u1 = 1 u2 = 0
8
( x1 , u1 ) = (0,1) ( x2 , u2 ) = (60, 0)
N1
m=
(u2 − u1 ) ( x2 − x1 )
( N i − ui ) = m( x − xi ) N i = m( x − xi ) + ui
m=
(0 − 1) −1 = ( x2 − x1 ) (60 − 0) −1 60
m=
N1 =
−1 ( x − 0) + 1 60−x +1 60 x 60
N1 =
N1 = 1 −
(
N2
' &
u1 = 0 u2 = 1
8
( x1 , u1 ) = (0, 0) ( x2 , u2 ) = (60,1)
N2
m=
(1 − 0) 1 = ( x2 − x1 ) (60 − 0) x2 − x1 = 60 1 60
m=
N2 =
1 ( x − 60) + 1 60 ( x − 60) +1 60
N2 =
x 60 x N1 = 1 − 60 N2 =
N2 =
x 60
Ne = 1 −
x 60
x 60
(
Be
Ne
d 1− Be =
x 60 dx
d
x 60 dx
Be =
−1 60
1 60
−160
[ Be ]
T
= 1 60
−1 60
[ Be ] [ Be ] =
1 60
T
T
−1 60
1 60
[ Be ] [ Be ] =
1 −1 1 −1 1 1 = 2 (60) −1 1 3, 600 −1 1
4
[ ke ] =
L
E [ Be ] [ Be ] A( x)dx
T
/
[ ke ] = EA [ Be ] [ Be ] dx
L
T
[ ke ] = EA
1 −1 1 dx 3, 600 −1 1 60
3
#
[ Be ] [ Be ] =
T
1 −1 1 3, 600 −1 1
[ ke ] =
EA 1 −1 L2 −1 1
dx =
L
EA 1 −13, 600 −1 1
dx
60
[ ke ] =
EA 1 −1 EA 1 −1 60 L= 2 3, 600 −1 1 L −1 1
[ ke ] =
EA = (29 x106 )(62) = 1798 x106 lbs
EA 1 −1 60 −1 1
[ ke ] =
1798 x106 1 −1 −1 1 60
.
[ ke ] = (29,966, 666.67)
!" # $ $
1 −1
−1 1
.
[ ke ] =
5 # 2
kij = EA( x) [ Bi ]
L
T
B j dx
+
&$)
(
L2
& # ! ! $
'
L1
'
)
9
+
&$*
+
&$6
+&$:
. %
N1
Bi
Bj
N kn
N
2 1
Nk
N1
'
&$
N kn
. ! 9
N1
& &$*
x1 = 0, x2 = L1 , x3 = L2 (0,1)( L1 , 0)( L2 , 0)
N1
1 N1
m=
(u2 − u1 ) ( x2 − x1 )
( N kn − un ) = m( x − xn ) N kn = m( x − xn ) + un
m= (0 − 1) −1 = ( x2 − x1 ) ( L1 − 0) −1 L1
m=
1 N1 =
−1 ( x − 0) + 1 L1 −x +1 L1
1 N1 =
N1
N12
m= (u3 − u2 ) ( x3 − x2 )( N kn − un ) = m( x − xn ) N kn = m( x − xn ) + un
m= (0 − 0) =0 ( x2 − x1 ) m=0 N12 = 0( x − x2 ) + 0 N12 = 0
N2
/
'
(0, 0)( L1 ,1)( L2 , 0)
N2
1 & N2
m=
(u2 − u1 ) ( x2 − x1 )
( N kn − un ) = m( x − xn ) N kn = m( x − xn ) + un
m= (1 − 0) 1 = ( x2 − x1 ) ( L1 − 0) 1 L1
m=
1 N2 =
1 ( x − L1 ) + 1 L1 ( x − L1 ) +1 L1 x L1
1 N2 =
1 N2 =
N2
'...
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