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FIGURE 5.9
Forces and displacements at the two ends of a line element.
5.2.2
Coordinate Transformations
In order to combine element stiffness matrices to perform an analysis of an entire structure,
it is helpful to be able to have all forces and displacements for all elements referred to a
common reference frame, the global coordinates. To accomplish this, those nodal forces and
displacements expressed in the local coordinates must first be transformed to the global
coordinates.
A typical coordinate transformation is illustrated in Fig. 5.10.
Global to Local Coordinates:
Local to Global Coordinates:
x
z
cos
X
Z
X
Z
cos
x
z
α
sin
α
α
sin
α
=
=
(5.21)
sin
α
cos
α
sin
α
cos
α
Forces and displacements are transformed in the same manner. Moments and rotations
may have a different form of transformation.
The transformation for forces and displacements appears as
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