Civil Engineering Reference
In-Depth Information
the matrix notation is used extensively. A general matrix is designated by
brackets [ ] and a column matrix (vector) by braces { }.)
Equation 2.6 shows that the element stiffness matrix for the linear spring
element is a 2
×
2 matrix. This corresponds to the fact that the element exhibits
two nodal displacements (or degrees of freedom) and that the two displacements
are not independent (that is, the body is continuous and elastic). Furthermore, the
matrix is symmetric. A symmetric matrix has off-diagonal terms such that
k
ij
=
k
ji
.
Symmetry of the stiffness matrix is indicative of the fact that the body is lin-
early elastic and each displacement is related to the other by the same physical
phenomenon. For example, if a force
F
(positive, tensile) is applied at node 2
with node 1 held fixed, the
relative
displacement of the two nodes is the same as
if the force is applied
symmetrically
(negative, tensile) at node 1 with node 2
fixed. (Counterexamples to symmetry are seen in heat transfer and fluid flow
analyses in Chapters 7 and 8.) As will be seen as more complicated structural
elements are developed, this is a general result: An element exhibiting
N
degrees
of freedom has a corresponding
N
×
N
, symmetric stiffness matrix.
Next consider solution of the system of equations represented by Equa-
tion 2.4. In general, the nodal forces are prescribed and the objective is to solve
for the unknown nodal displacements. Formally, the solution is represented by
u
1
u
2
[
k
e
]
−
1
f
1
f
2
=
(2.7)
where
[
k
e
]
−
1
is the inverse of the element stiffness matrix. However, this inverse
matrix does not exist, since the determinant of the element stiffness matrix is
identically zero. Therefore, the element stiffness matrix is
singular,
and this also
proves to be a general result in most cases. The physical significance of the
singular nature of the element stiffness matrix is found by reexamination of
Figure 2.1a, which shows that no displacement constraint whatever has been im-
posed on motion of the spring element; that is, the spring is not connected to any
physical object that would prevent or limit motion of either node. With no con-
straint, it is not possible to solve for the nodal displacements individually.
Instead, only the
difference
in nodal displacements can be determined, as this
difference represents the elongation or contraction of the spring element owing
to elastic effects. As discussed in more detail in the general formulation of inter-
polation functions (Chapter 6) and structural dynamics (Chapter 10), a properly
formulated finite element must allow for constant value of the field variable. In
the example at hand, this means rigid body motion. We can see the rigid body
motion capability in terms of a single spring (element) and in the context of sev-
eral connected elements. For a single, unconstrained element, if arbitrary forces
are applied at each node, the spring not only deforms axially but also undergoes
acceleration according to Newton's second law. Hence, there exists not only
deformation but overall motion. If, in a connected system of spring elements, the
overall system response is such that nodes 1 and 2 of a particular element dis-
place the same amount, there is no elastic deformation of the spring and therefore
