The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 395

In such cases, the element stiffness matrix and loading vector are the summation of the
matrices of the triangular or tetrahedral elements. After the stiffness matrix and the loading
vector are formed, DOF associated with the nodes not on the element boundary can be
condensed out using the technique that is employed to process Eq. (6.56).

Coordinate Transformation
Although most of the derivations thus far in this chapter refer to the global xy coordinate
system, some elements, e.g., the triangular element, can be more conveniently formulated in
a local coordinate system. Then the stiffness matrices and loading vectors are transformed
from the local system to the global system. The transformation for a triangular element
will be discussed here. In Chapter 5, XY Z and xyz indicate the global and local coordinate
systems, respectively. In this chapter, however, xyz are used for the global system, whereas

x u zj ] T

will be used to describe the nodal variables in the global and local coordinates at node j .
The relationship between the local

y z represent the local system. Accordingly, v j = [ u xj u yj u zj ] T and v j = [ u xj u yj

y and global xy coordinate system is shown in
Fig. 6.36a and that between the nodal variables in the two systems is shown in Fig. 6.36b.
Note that in Fig. 6.36a one node is located at the origin of the local coordinate system and
the

x

x axis is aligned along one side of the element. This makes the shape functions of the
element much simpler to form, i.e., the expressions for 1 , 2 , 3 , of Eq. (6.75)

are simpler. As a consequence, it takes less effort to create the stiffness matrix using the
procedure in Section 6.5.8. From Fig. 6.36b, the transformations between the displacement
variables (or forces if u is replaced by p )atthe j th node of element i is

α i , β i , and γ i ,i =

cos u xj

u yj

u xj α sin α

= =

− α α

u yj sin cos

(6.107)

v i j T i jj v i j

=

FIGURE 6.36
Coordinate transformation between the local and global systems.