Civil Engineering Reference
In-Depth Information
The interpolation functions can now be obtained by substituting the coeffi-
cients given by Equation 6.14 into Equation 6.5 and collecting coefficients of the
nodal variables. However, the following approach is more direct and alge-
braically simpler. Substitute Equation 6.14 into Equation 6.7 and equate to Equa-
tion 6.6 to obtain
1
0
0
0
0
1
0
0
v
1
1
v
2
2
3
L
2
2
L
3
L
2
1
L
xx
2
x
3
]
v
(
x
)
=
[1
−
−
−
2
L
3
1
L
2
2
L
3
1
L
2
−
v
1
1
v
2
2
=
[
N
1
N
2
N
3
N
4
]
(6.15)
The interpolation functions are
1
0
0
0
0
1
0
0
3
L
2
2
L
3
L
2
1
L
xx
2
x
3
]
[
N
1
N
2
N
3
N
4
]
=
[ 1
−
−
−
(6.16)
2
L
3
1
L
2
2
L
3
1
L
2
−
and note that the results of Equation 6.16 are identical to those shown in
Equation 4.26.
The reader may wonder why we repeat the development of the beam element
interpolation functions. The purpose is twofold: (1) to establish a general proce-
dure for use with polynomial representations of the field variable and (2) to re-
visit the beam element formulation in terms of compatibility and completeness
requirements. The general procedure begins with expressing the field variable as
a polynomial of order one fewer than the number of degrees of freedom exhib-
ited by the element. Using the examples of the truss and beam elements, it has
been shown that a two-node element may have 2 degrees of freedom, as in the
truss element where only displacement continuity is required, or 4 degrees of
freedom, as in the beam element where slope continuity is required. Next the
nodal (boundary) conditions are applied and the coefficients of the polynomial
are computed accordingly. Finally, the polynomial coefficients are substituted
into the field variable representation in terms of nodal variables to obtain the
explicit form of the interpolation functions.
Examination of the completeness condition for the beam element requires a
more-detailed thought process. The polynomial representation of the displace-
ment field is such that only the third derivative is
guaranteed
to have a constant
