Civil Engineering Reference
In-Depth Information
Z
Y
z
y
x
X
Figure 4.1.
Coordinate systems.
the following sections are transmission and rotations.
Transmission
is
moving effects from one point to another.
Rotations
are used to re-orient
axes about a specific point. These techniques will be used in determining
the stiffness of a member (local) and developing the global (joint) stiff-
ness solution.
4.2
ROtAtiOn MAtRiX
Rotation can take place about any of the three global axes. The following
three examples derive the rotation transformations. The alpha,
a
, rotation
is a rotation about the global
Z
axis from the global system to the local
system. Rotation about the
Y
axis is a beta,
b
, rotation and rotation about
the
X
axis is a gamma,
g
, rotation.
Example 4.1
Rotation matrix,
a
Derive the alpha,
a
, rotation matrix
.
The following variables are represented in Figure 4.2 and are used to
develop
a
. The location of
z
remains unchanged since rotation is occurring
about that axis.
a
= rotation about global
Z
axis from the global to the local system
(
x
0
,
y
0
) = global coordinate location
(
x
l
,
y
l
) = local coordinate location
