Structural Analysis Methods II: Structural Systems - Mechanics of Structures: Variational and Computational Methods - page 256

x

a

X

FIGURE 5.10
Right-handed global

y,Y

( X, Y, Z ) and local ( x, y, z ) coordi-

z

nate systems. The angle α is positive as shown (vector

Z

corresponding to α is positive along positive Y axis).

Global to Local Forces and Displacements:

i T aa 0

0 bb

i p a
p b

i

p a

v i T i v i = and = (5.22)

p b

p i T i p i =

with, in more general notation, the transformation matrix T i

T aa i

T ab

T i = (5.23)

T ba T bb

cos

α − sin α

T ba = T ab = 0 and T aa = T bb = (5.24)

sin α cos α

where is the angle between the X (global coordinate) and the x (local) coordinate.

Observe that for T i jj ,j

α

= a or b ,defined by Eq. (5.24)

T i jj T i jj = T iT T i

=

I , I (5.25)

where superscript T designates the transpose and I is the unit diagonal matrix. The coor-
dinate transformation for orthogonal coordinate axes always possesses this property. Since
( T i ) − 1 T i = I , it follows that

T i ) − 1 T iT ( = (5.26)

Hence, the transformation from local to global coordinates can be expressed as
Local to Global Forces and Displacements:
p a
p b

i T aa 0

0 bb

i i

p a

v i T iT v i

= and = (5.27)

p b

p i T iT p i =

Stiffness matrices are readily transformed from one coordinate system to another. This is
accomplished by noting that

k i k i T i v i

p i T iT p i T iT v i T iT = = =

Since p i

=

k i v i , it follows that

k i T i (5.28)
A triple matrix product of this form is called a congruent transformation . The product k i
will be a symmetric matrix if the k i

k i T iT

=

is symmetric. As indicated in the previous two chapters,

stiffness matrices are symmetric.