Structural Analysis Methods II: Structural Systems - Mechanics of Structures: Variational and Computational Methods - page 272

resulting alignment of the forces along the bar axis, the element force and displacement
vectors in the local xz coordinates will be

N xa
N xb

i N a

N b

i i i

u xa u a

p i v i = = = = (5.79)

u xb u b

where coordinate x is along the bar axis. However, these same forces and displacements in
the global XZ coordinate systems will appear as

i i

F Xa
F Za
F Xb
F Zb

u Xa
u Za
u Xb
u Zb

p i v i

= = (5.80)

The expanded vectors are necessary, since there will be more components in the global
system than in the local system, where the forces and displacements are aligned with the
x axis. In Section 5.2.1 element end forces in both the local and global coordinates are
designated by lower case letters and capital letters are used for the system (nodal) forces.
Unfortunately, tradition dictates that capital letters should be used for the element forces
of truss members, e.g., Eqs. (5.79) or (5.80). The same situation occurs with beam elements,
where traditionally the bending moment and shear force are represented by upper case
letters.
The uniaxial characteristics of the forces and displacements referred to the local coordi-
nate system simplify the coordinate transformations. For example, the displacements (or
forces) would transform as

u i xa = u i Xa cos u i Za sin α − α

or, more generally,

i

u Xa
u Za
u Xb
u Zb

i cos

u xa α − sin α | 0 0

=

(5.81)

u xb 0 0 | cos α − sin α

v i T i v i =

with

T aa 0
0 bb

i

T i

=

(5.82)

and

]
Similarly, T iT can be employed to transform variables from local to global coordinates.
Thus, for example,

T aa = T bb = [cos α − sin α

i

F Xa
F Za
......
F Xb
F Zb

cos α 0

N xa
N xb

i

− 0

...... ......
0

sin α

= (5.83)

cos α

0 − sin α

p i T iT p i

=