Civil Engineering Reference
In-Depth Information
6.2.2 Completeness
In the limit as element size shrinks to zero in mesh refinement, the field variable
and its partial derivatives up to, and including, the highest-order derivative
appearing in the integral formulation must be capable of assuming constant
values.
Again, because of the discretization, the completeness requirement is
directly applicable to the interpolation functions.
The completeness requirement ensures that a displacement field within a
structural element can take on a constant value, representing rigid body motion,
for example. Similarly, constant slope of a beam element represents rigid body ro-
tation, while a state of constant temperature in a thermal element corresponds to
no heat flux through the element. In addition to the rigid body motion considera-
tion, the completeness requirement also ensures the possibility of constant values
of (at least) first derivatives. This feature assures that a finite element is capable of
constant strain, constant heat flow, or constant fluid velocity, for example.
The foregoing discussion of convergence and requirements for interpolation
functions is by no means rigorous nor greatly detailed. References [1-5] lead the
interested reader to an in-depth study of the theoretical details. The purpose here
is to present the requirements and demonstrate application of those requirements
to development of appropriate interpolation functions to a number of commonly
used elements of various shape and complexity.
6.3 POLYNOMIAL FORMS:
ONE-DIMENSIONAL ELEMENTS
As illustrated by the methods and examples of Chapter 5, formulation of finite
element characteristics requires differentiation and integration of the interpola-
tion functions in various forms. Owing to the simplicity with which polynomial
functions can be differentiated and integrated, polynomials are the most com-
monly used interpolation functions. Recalling the truss element development of
Chapter 2, the displacement field is expressed via the first-degree polynomial
u
(
x
)
a
1
x
(6.2)
In terms of nodal displacement, Equation 6.2 is determined to be equivalent to
=
a
0
+
1
u
1
+
x
L
x
L
u
2
u
(
x
)
=
−
(6.3)
The coefficients
a
0
and
a
1
are obtained by applying the nodal conditions
u
(
x
u
2
. Then, collecting coefficients of the nodal
displacements, the interpolation functions are obtained as
=
0)
=
u
1
and
u
(
x
=
L
)
=
x
L
x
L
N
1
=
1
−
N
2
=
(6.4)
Equation 6.3 shows that, if
u
1
=
u
2
, the element displacement field corre-
sponds to rigid body motion and no straining of the element occurs. The first
