Information Technology Reference
In-Depth Information
TABLE 5.3
Notation for Classical Matrix Techniques of Structural Analysis
Indices: Superscript for element index, subscript for node index, e.g., v i k is the displacement v of the
i th element at the k th node.
Local coordinate system: Whenever it is essential to make the distinction, variables referred to a local
coordinate system will be designated by a tilde
, e.g., k i
is the local stiffness matrix of the i th
element.
Prescribed (applied) variables will be indicated with a line over the letter, e.g., V k is the prescribed
displacement at node k .
Symbol
Definition
Remarks
V
Vector of nodal displacements of the
V and P are considered only in the
complete system
global coordinate system
P
Vector of nodal forces of the complete system
V
Vector of applied displacements of the
complete system
Vector of applied forces of the complete system
P
v i
Vector of the end displacements of the
i th element, e.g.,
v 3 is for element 3 in
the local coordinates
p i
Column matrix of end forces of the i th element
v i , p i
Element vectors containing the effects
of applied loading
Stiffness matrix of the i th element e.g., k 3
k i
is the
stiffness matrix of element 3 in local
coordinates
K
Stiffness matrix of the complete system
f i
Flexibility matrix of the i th element
F
Flexibility matrix of the complete system
a
Kinematic transformation matrix
v
=
aV
b
Static transformation matrix
p
=
bP
T
Transformation matrix, T i
corresponds to the i th
e.g.,
v i
=
T i v i
element
= T a 0
T bb
i
T i kk
Transformation submatrix, element i , node k
T i
0
. For nodes a , b of
element i
5.2.1
Basic Definitions of Elements, Nodes, Forces, Displacements, and Coordinate Systems
To achieve computational tractability, network structures are usually modeled by a finite
number of elements connected at nodes. A solution procedure is established to compute
forces and displacements at the nodes. Since only these discrete nodal variables will appear
in the governing equations, the structure is said to be discretized spatially. As indicated in
Fig. 5.6, the element may be one, two or three-dimensional, as required by the structure.
These structural models are also referred to as finite element models, and the associated
structural analysis methodology forms the basis of the finite element method. In the finite
element method the element characteristics may be obtained by numerical approximation.
System State Variables at the Nodes
The elements (Fig. 5.6) of a system may possess a finite number of common nodal points.
It is convenient to describe the location of the nodes and elements in a single global coor-
dinate system
.Define at each node forces and displacements as shown in Fig. 5.7.
These system or global forces and displacements serve as the unknowns in the respective
(
X, Y, Z
)
Search WWH ::




Custom Search