Civil Engineering Reference
In-Depth Information
cHAPtER 4
m
Atrix
s
tructurAL
s
tiffness
Building structures are made up of columns, beams, girders, joists, slabs,
walls, shells, and many other components that act together to resist the
loads placed on them. These members can be at various orientations, but
they must be represented in a common mathematical form. The structural
stiffness method is used to represent members, loads, support constraints,
and other components of a building structure in a consistent manner. This
chapter will focus on linearly elastic members subjected to axial, bending,
shear, and torsional forces.
4.1
MAtRiX tRAnSfORMAtiOnS AnD
cOORDinAtE SYStEMS
When using the structural stiffness method, the
global
Cartesian right-
hand coordinate system will be used to organize the system. This sys-
tem was developed in 1637 by René Descartes (Descartes 1637). We will
denote these three orthogonal axes as X, Y, and Z. You could represent
these as your thumb, fore finger, and middle finger on your right hand.
Individual members may not be at the same orientation as the global sys-
tem. All members have their own
local
system represented by
x
,
y
, and
z
.
This coordinate system is also a Cartesian right-hand system with the
x
axis running along the member length. Both of these systems are shown
in Figure 4.1.
This chapter provides techniques to reference causes and effects by
coordinate systems and how to manipulate these coordinate systems.
These manipulations allow changes from one system to another and are
called
transformations
. The two basic transformations that are done in
