The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 374

4 matrix N u 1 can be inverted readily, giving

The 4 ×

1000

− 1100

N − 1
u 1

=

− 1001

1 − 11 − 1

Then, from Eq. (6.53) u 1 is given by

u 1 = N − 1
u 1

v x − N − 1

u 1 N u 2

u 2 (6.54)

Substitute Eq. (6.54) into Eq. (6.52), yielding

N u 1 N − 1 u 2 +

u 1 v x − N − 1 u 1 N u 2

u x = N u 2 u 2

N u 2

N u 1 N − 1 N u 1 N − 1

u 1 N u 2

=

u 1 v x

+ − + u 2

= N 1 v x + N 2 u 2 (6.55)

where

N 1 = [ ( 1 − ξ)( 1 − η) ξ( 1 − η) ξη ( 1 − ξ)η ]

and

N 2 = [ ( 1 − ξ)ξ η( 1 − η) ]

u 2 do not represent nodal displacements, they are called nodeless
variables. Sometimes they are referred to as “extra shapes.”
From Eq. (6.55), using the principle of virtual work, the element stiffness equations can
be formed as

Since the terms N 2

k 11 v x p

r

k 12

= (6.56)

k 21 k 22 u 2

where r contains the load parameters associated with N 2 . The variables u 2 can be condensed

out before the element stiffness matrix is assembled into the global stiffness matrix. In the
condensation process, the second equation of Eq. (6.59) is used to express

u 2 in terms of

k 21 , k 22 , v x , and r . Then this u 2 is substituted into the first equation of Eq. (6.56), resulting

p . In Eq. (6.56), k 11 is the same as k i in Eq. (6.31). That is,
the inclusion of extra shapes only expands the stiffness matrix and the original stiffness
matrix remains intact. As this higher order approximation proceeds, the element matri-
ces computed at the previous step of the approximation are used and, hence, need not be
re-established. The introduction of

in a stiffness matrix kv x =

u 2 serves the purpose of introducing more DOF, and
thereby, the accuracy of an element can be improved. Using the extra shapes can permit, for
example, a parabolic deformation along the element edge, and a more realistic deformation
shape may be achieved. But because

do not represent the nodal displacements,
it may have different values for the adjacent elements. This raises the question of com-
patibility. A gap or overlap may develop at the interelement boundaries, but incompatible
elements are still acceptable if the incompatibilities disappear and a constant strain state is
approached as the mesh is refined. Some modifications on the extra shapes can also help
reduce the incompatibility problem.

u i ( i = 5 , 6 )