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FIGURE 4.7
Beam elements for computing the second, third, and fourth columns of stiffness coefficients.
where
C
i
=
w
i
,i
are the unknown constants of the assumed series that are chosen
so as to obtain a good approximation. Such trial functions are sometimes referred to as
basis
functions
. Often the variable representation by a trial series will be supplemented with a
superscript tilde, e.g., ˜
=
1
,
2
,
...
, to indicate that it is being approximated. For this beam element,
we choose to retain only the first four terms. Write the polynomial in the form
w
=
w
1
w
xx
2
x
3
]
2
w
T
N
u
N
u
=
w
=
[1
w
(4.41)
w
3
w
4
where
xx
2
x
3
]
N
u
=
[1
and
4
]
T
=
w
w
w
w
w
[
1
2
3
It is convenient to rewrite the assumed series in terms of physically meaningful parame-
ters (unknown displacements) at the ends of the beam element, rather than in terms of the
unknown constants
w
i
as in Eq. (4.41). Thus, the vector of unknowns
w
can be transformed
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