Civil Engineering Reference
In-Depth Information
between element displacements and global displacements as well as the contribu-
tions of individual elements to overall structural stiffness. In the
direct stiffness
method, the stiffness matrix of each element is transformed from the element
coordinate system to the global coordinate system. The individual terms of each
transformed element stiffness matrix are then added directly to the global stiffness
matrix as determined by element connectivity (as noted, the connectivity relations
ensure compatibility of displacements at joints and nodes where elements are
connected). For example and simply by intuition at this point, elements 3 and 7 in
Figure 3.1b should contribute stiffness only in the global
X
direction; elements 2
and 6 should contribute stiffness in both
X
and
Y
global directions; element 4
should contribute stiffness only in the global
Y
direction. The element transfor-
mation and stiffness matrix assembly procedures to be developed in this chapter
indeed verify the intuitive arguments just made.
The direct stiffness assembly procedure, subsequently described, results in
exactly the same system of equations as would be obtained by a formal equilib-
rium approach. By a
formal equilibrium approach,
we mean that the equilibrium
equations for each joint (node) in the structure are explicitly expressed, including
deformation effects. This should
not
be confused with the method of joints [2],
which results in computation of forces only and does not take displacement into
account. Certainly, if the force in each member is known, the physical properties
of the member can be used to compute displacement. However, enforcing com-
patibility of displacements at connections (global nodes) is algebraically tedious.
Hence, we have another argument for the stiffness method: Displacement com-
patibility is assured via the formulation procedure. Granted that we have to
“backtrack” to obtain the information of true interest (strain, stress), but the back-
tracking is algebraic and straightforward, as will be illustrated.
3.2 NODAL EQUILIBRIUM EQUATIONS
To illustrate the required conversion of element properties to a global coordinate
system, we consider the one-dimensional bar element as a structural member of a
two-dimensional truss. Via this relatively simple example, the
assembly
procedure
of essentially any finite element problem formulation is illustrated. We choose
the element type (in this case we have only one selection, the bar element); spec-
ify the geometry of the problem (element connectivity); formulate the algebraic
equations governing the problem (in this case, static equilibrium); specify the
boundary conditions (known displacements and applied external forces); solve
the system of equations for the global displacements; and back-substitute dis-
placement values to obtain
secondary
variables, including strain, stress, and reac-
tion forces at constrained locations (boundary conditions). The reader is advised
to note that we use the term
secondary
variable only in the mathematical sense;
strain and stress are secondary only in the sense that the values are computed after
the general solution for displacements. The strain and stress values are of
primary
importance
in design.
