Civil Engineering Reference
In-Depth Information
cHAPtER 2
s
oLutions
of
s
imuLtAneous
L
ineAr
A
LgebrAic
e
quAtions
u
sing
m
Atrix
A
LgebrA
Matrix algebra is commonly utilized in structural analysis as a method of
solving simultaneous equations. Each motion, translation or rotation, at
each discrete location in a structure is normally the desired variable. This
chapter explores matrix terminology, matrix algebra, and various methods
of linear algebra to determine solutions to simultaneous equations.
2.1
SiMuLtAnEOuS EQuAtiOnS
The solutions of simultaneous equations in structural analysis normally
involve hundreds and even thousands of unknown variables. These solu-
tions are generally
linear algebraic equations
. A typical linear algebraic
equation with
n
unknowns is as follows, where
a
is the coefficient,
x
is the
unknown, and
C
is the constant. In some cases,
x
,
y
, and
z
are used in lieu
of
x
1
,
x
2
, etc.
…
ax ax ax
++++ =
ax C
nn
11 22 33
Equation sets can be separated into two categories,
homogeneous
and
non-homogeneous
. Homogeneous equation sets are those in which all the
C
s are zero and all other equation sets are known as non-homogeneous.
A unique solution to a non-homogeneous set exists only if the equations
are independent or non-singular (determinant is non-zero), and a non-trivial
solution set exists to a homogeneous set only if the equations are not inde-
pendent (determinant is zero). The determinant is further discussed in
