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These are so-called rigid body displacements of the bar. That is, the beam can undergo rigid
body motion without introducing elastic forces.
As a result of these dependent variables, the stiffness matrix is singular. This is evident
from
p
a
k
aa
v
a
k
ab
=
(4.18)
p
b
k
ba
k
bb
v
b
p
i
k
i
v
i
=
which is a system of linear equations. Because
v
a
and
v
b
are dependent, the solution to Eq.
(4.18) is not unique. From Cramer's
1
rule, a set of simultaneous linear equations
k
i
v
i
p
i
(
=
)
has a unique solution if, and only if, the determinant of the coefficient matrix
k
i
is not zero,
i.e., det
k
i
0. Thus, if
k
i
v
i
p
i
=
=
does not have a unique solution, the determinant of
k
i
must be zero. It follows that matrix
k
i
is singular, and its rows (columns) are linearly
dependent.
The conclusion of the previous paragraph can also be reached by scruitiny of Eq. (4.12).
Note from the stiffness matrix of Eq. (4.12) that the sum of rows 1 and 3 is [0 0 0 0]. Thus,
the determinant of
k
i
is zero and
k
i
is singular. Also, rows 1 and 3 are linearly dependent.
×
The singular property of the 4
4 stiffness matrix is also evident by noting that the sum of
columns 1 and 3 is zero. Again the determinant of
k
i
will be zero. Furthermore, the sum of
columns 2 and 4 is equal to
times column 1. This too leads to a singular matrix.
It is possible to ascertain analytically the number of rigid body displacements contained
in an elemental stiffness matrix. This is accomplished by transforming the stiffness matrix
into diagonal form. The number of rigid body displacements is then equal to the number of
zero terms in the diagonal. The transformation involves establishing the eigenvalues and
eigenvectors of the stiffness matrix, with the number of zero eigenvalues being equal to the
number of rigid body motions. It is also possible to scrutinize the strain energy, since the
lack of deformation of a rigid body motion should correspond to zero strain energy. Thus,
if the strain energy contributed by a certain eigenvector of the stiffness matrix is zero, then
the associated eigenvalue is zero which corresponds to rigid body motion.
Elimination of the rigid body displacements will produce a nonsingular stiffness matrix
and permit the flexibility matrix to be formed by inversion. This elimination of the rigid
body displacements, e.g., through the consideration of supports or constraints, reduces the
number of degrees of freedom of the element. Thus, a flexibility matrix
f
relates a reduced
set of forces
p
R
at
a
and
b
to a reduced set of displacements
v
R
at
a
and
b
through
v
R
=
fp
R
(4.19)
The flexibility matrix is defined only for restrained systems. Otherwise rigid body motion
would occur and the magnitude of the displacements could be unlimited. This follows from
the definition of a flexibility coefficient
f
ij
being the displacement at
i
due to a unit load at
j
. Because of the need for a restraining condition, the flexibility matrix for a beam element
is not unique. Rather, it depends on which DOF are chosen to be unrestrained.
To illustrate these principles, consider a beam element simply supported at both ends.
The displacement boundary conditions are
w
a
=
0
,
w
b
=
0. With these restraints taken into
1
Gabriel Cramer (1704-1752) was an early 18th century Swiss mathematician and professor who was probably
not the originator of the popular rule named after him. Cramer's rule is a simple scheme for solving a set of linear
equations. Maclaurin, whose name is attached to a series (Maclaurin's series) which he did not discover, appears
to have originated Cramer's rule. Cramer is also known for Cramer's paradox which he also did not originate.
Cramer's paradox deals with the intersection of cubic curves.
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