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CEE 371 Direct Stiffness Procedure, page 1 of 2
A PROCEDURE FOR THE DIRECT STIFFNESS METHOD
The Direct Stiffness Method is by far the most widespread
approach for computerizedstructural analysis. The following direct
stiffness procedure, representative of computerimplementation in
finite element programs, emphasizes three aspects:
A formalized use ofmatrices to organize calculations A
systematic, element-by-element treatment of structures
The assembly of the global arrays by addition, which is based on
node-by-nodeequilibrium
1. Create an idealization or model of the structure in terms of
geometry, boundaryconditions, constitutive behavior, and kinematic
behavior.
2. Establish a global (overall, structural) coordinate system.
Discretize the structural modelinto elements interconnected at
nodes. Number all nodes and elements (essentiallyproviding each
with indices, names, or labels).
[For convenience in illustrative and hand-calculated examples,
we may wish to numberthe free nodes prior to the restrained
nodes.]
3. Determine the total number of nodal DOF,N, and the degree of
kinematic indeterminacy,n. Then the number of supported or
restrained degrees of freedom iss =Nn. Numberboth the free and
restrained DOF. Set up a null (zero, empty)NxNglobal
stiffnessmatrix [K] and a nullNx 1 global load vector {P} with the
proper number of entries andpartitions to distinguish free and
supported DOF. These are represented in the globalequilibrium
(stiffness) equations as:
{ } [ ] { } partitioned to give
1 1
ff fsf f
sf sss s
K KPP K
K KP
N N N N
= =
in which the subscriptfindicates free, subscripts indicates
supported, [Kff] is n x n,[Ksf] iss x n, etc.[Again for
convenience, one procedure for numbering DOF is to start from node
1 andgo through the nodes in ascending order, numbering the free
DOF sequentially 1 throughn. Then cycle through the nodes again,
numbering the supported DOF sequentially fromn + 1 toN.]
4. Begin to assemble the global external load vector by adding
the values of the nodal(concentrated) loads to the appropriate
locations in {P}.
5. Assemble the global stiffness matrix [K] by performing the
following steps for eachelement:
(a) Evaluate the element stiffness [k'] in local (element)
coordinates and, if there areloads applied to the element between
nodes, evaluate the element reversed fixed-end
forces {FF '} in local (element) coordinates.(b) Transform to
global coordinates to obtain [k] and {FF}:
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CEE 371 Direct Stiffness Procedure, page 2 of 2
[ ] [ ] [ ] [ ] { } [ ] { }' 'T TF Fk k F F = =
(c) Assemble contributions to [K] and {P} by adding the elements
of [k] and {FF} toappropriate locations in the respective global
arrays.
6. Noting that {s} is generally zero, solve the equilibrium
equations for the unknown nodal
displacements, symbolically (although inversion is never used
instead of more efficientsolution methods that take into account
the sparsity and symmetry of [Kff]), and then use{f} to solve for
the reactions at the supports .[It is possible to generalize this
to cases where the displacements at some supports isnon-zero.]
7. Determine the forces within elements by performing the
following for each element:
(a) From the global nodal displacement vector {}, select those
nodal displacements thatoccur at the nodes of the element
(b) Transform to obtain { } [ ] { } = .(c) Find internal element
end forces from the element equilibrium (stiffness)
equations:
{ } [ ] { } { }'FF k F =
