Building Design and Construction Handbook - Merritt - Ventech!

5.114 SECTION FIVE
5.13.3 Displacement (Stiffness) Method
With the stiffness or flexibility matrix of each finite element of a structure known,
the stiffness or flexibility matrix for the whole structure can be determined, and
with that matrix, forces and displacements throughout the structure can be computed
(Art. 5.13.2). To illustrate the procedure, the steps in the displacement, or stiffness,
method are described in the following. The steps in the flexibility method are similar.
For the stiffness method:
Step 1. Divide the structure into interconnected elements and assign a number,
for identification purposes, to every node (intersection and terminal of elements).
It may also be useful to assign an identifying number to each element.
Step 2. Assume a right-handed cartesian coordinate system, with axes x, y, z.
Assume also at each node of a structure to be analyzed a system of base unit
vectors, e 1 in the direction of the x axis, e 2 in the direction of the y axis, and e 3 in
the direction of the z axis. Forces and moments acting at a node are resolved into
components in the directions of the base vectors. Then, the forces and moments at
the node may be represented by the vector P ie i, where P i is the magnitude of the
force or moment acting in the direction of e i. This vector, in turn, may be conveniently
represented by a column matrix P. Similarly, the displacements—translations
and rotation—of the node may be represented by the vector � ie i, where � i is the
magnitude of the displacement acting in the direction of e i. This vector, in turn,
may be represented by a column matrix �.
For compactness, and because, in structural analysis, similar operations are performed
on all nodal forces, all the loads, including moments, acting on all the
nodes may be combined into a single column matrix P. Similarly, all the nodal
displacements may be represented by a single column matrix �.
When loads act along a beam, they should be replaced by equivalent forces at
the nodes—simple-beam reactions and fixed-end moments, both with signs reversed
from those induced by the loads. The final element forces are then determined by
adding these moments and reactions to those obtained from the solution with only
the nodal forces.
Step 3. Develop a stiffness matrix k i for each element i of the structure (see Art.
5.13.2). By definition of stiffness matrix, nodal displacements and forces for the i
the element are related by
S � k � i � 1,2,...,n (5.160)
i i i
where S i � matrix of forces, including moments and torques acting at the nodes
of the ith element
� i � matrix of displacements of the nodes of the ith element
Step 4. For compactness, combine this relationship between nodal displacements
and forces for each element into a single matrix equation applicable to all the
elements:
S � k� (5.161)
where S � matrix of all forces acting at the nodes of all elements
� � matrix of all nodal displacements for all elements