The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 367

Then a complex variation of strain within the structure can be approximated. For
this constant strain representation, the displacement function must contain those
terms that can eventually result in constant strain states. If the structure is actu-
ally in a constant strain state, the functions must be able to represent this constant
strain.

To satisfy the condition of compatibility, the trial functions should be chosen such that
(1) they are continuous within the element, and (2) at the element interfaces at least the first
r derivatives are continuous, where r 1 is the highest derivative appearing in the func-

tional of the principle of virtual work, i.e., the highest derivative appearing in the D u matrix

+

in the principle of virtual work expression V δ( = V δ

T ED u u dV

u T u D T ED u u dV . D u u )

For linear elastic elements where r = 0 ( D u contains first order derivatives for linear

elastic solids [see Chapter 1], hence r 0), the compatibility condition

requires that the trial function be continuous both inside the element and on the interele-
ment boundaries. For bending elements of the sort needed for beams and plates where
r

+ 1 = 1 and r =

D u contains second order derivatives), compatibility means that the slope of the
trial function must be continuous inside the element and on its boundaries. This require-
ment of r continuity will ensure that no contribution is made from the element inter-
face to the total functional of the variational principle [Zienkiewicz, 1977]. This condi-
tion is satisfied by a complete polynomial of degree r

= 1 (

+ 1. This polynomial is defined in

Section 6.5.5.
Trial functions are said to exhibit C r continuity [Bathe, 1996] if their derivatives of order
r are continuous. The completeness and compatibility requirement for an element can be
stated in terms of the continuity conditions. If the trial function has C r + 1 continuity inside the
element, the element is complete. If the trial function has C r continuity at the interelement
boundaries, the element is compatible. The elements which satisfy both of these continuity
conditions are called C r elements . The requirement for C r

continuity at the boundaries

was explained above. The C r + 1

continuity requirement means that a derivative of order

r 1 in the element is continuous, so that it can approach a constant value as the element

size approaches zero. A complete polynomial which has the C r + 1 continuity satisfies the
complete conditions of an element. The constant and linear terms of the polynomial ensure
that rigid body motion is permitted, while all constant and linear single terms below the
order r

+

1 ensure that the solution and its derivatives in each element can approach a
constant value as the elements are refined further [Bathe, 1996].
Elements that do not satisfy the compatibility requirement are called incompatible or
nonconforming [Bazeley, et al., 1965]. If the incompatibility disappears with increasing mesh
refinement, the elements can still be acceptable as they may lead to convergence to the
correct solution.

+

EXAMPLE 6.3 Completeness and Compatibility
Investigate the completeness and compatibility of the displacement functions used in
Section 6.4.
First consider the completeness. From Eqs. (6.11) and (6.12) the displacement functions
show that a rigid body displacement in the x direction can be achieved if u 1 = 0, and that

in the y direction can occur if 0.

A rigid body rotation can be achieved if (Fig 6.10) u x 4

u 5 =

(ξ = 0 , η = 1 ) =− u y 2 (ξ = 1 , η = 0 ) ,

=−

i.e., u 3 is not equal to zero, the

movement of the element is a translation plus rotation. If they are zero, the movement is a
rotation.

u 4 u 6 , where u 4 and u 6 are non-zero. If either u 1 or