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Steel Designers' Manual - 6th Edition (2003)
and
ÈD
nD
[ D]=
Í
nD1
Í
ÎÍ
0
D1
0
3
Et
Dx = Dy = D1=
12 1 -
1
Dxy = ( - ) D1
2 1
n
n
Finally the strain energy can be written as
1 T
Ub= Ú [ c] [ D][ c]
dA
2
Denoting the generalized nodal displacement by the vector {d e— },
d e { }= [ C]{ A}
Finite element method 323
(9.38)
(9.39)
where {A – } is a vector of polynomial constants. From Equations (9.36) and (9.39),
-
{ c}= [ B]{ A}= [ B][ C] { d }
(9.40)
1
e
Using Equations (9.38) and (9.40), the following equation for Ub in terms of nodal
point displacement parameters {d e— } is obtained:
e UbC B D B A C
(9.41)
A
Differentiation of U with respect to the nodal displacements yields the stiffness
e matrix [ K ]:
T 1
-1, T È T ˘ -1
e
= { d } [ ] ÍÚ
[ ] [ ][ ] d d
2
Î
˚
e
-1, T È T ˘ -1
[ K ]= [ C] ÍÚ
[ B] [ D][ B] dA˙[
C]
ÎA
˚
Equation (9.42) can now be written as
e
, T
[ K ]= [ C] [ Q][ C]
where
T
[ ]= [ ] [ ][ ]
Ú
Q B D B dA
A
A
1 1
0
0
D xy
2 ( )
-1 -1
˘
˙
˙
˚˙
˙[ ] { }
(9.42)
(9.43)
(9.44)
As mentioned before, the accuracy is dependent on choosing a large number of
elements. Many refined elements giving greater accuracy are described in standard
books on finite element methods, which also provide details of assembling the