Advanced Methods of Structural Analysis

382 11 Matrix Stiffness Method

presented as vectorPE. Let the structure hasnunknown internal forces; these
unknowns according toS-ediagram may be presented as vectorES. Both of vectors
are connected by static matrixAby formula

PEDASE: (11.1)

This equation is called the static matrix equation. The number of columnsnof
the matrixAequals to the number of the unknown internal forces; the number of
rowsmof the static matrixAequals to the number of the possible displacements.
Ifm>n, then a structure is geometrically changeable; ifmDn, then a structure is
statically determinate; ifm<n, then a structure is statically indeterminate. In fact,
the matrix equation (11.1) describes the structure, its supports, type of joints, and
order of the elements connections.
The static matrixA.mn/may be constructed on the basis of a set of equilibrium
conditions for specific parts of a structure in ordered form. Equilibrium equations
for frames must be constructed for each joint which have the angular displacement
and for part of the frame which containsthe joints with linear displacements. In
case of possible angular displacement, the equation

P
MD 0 should be used; if a
displacement is a linear one, then uses equation

P

XD 0.

In case of trusses the equilibrium of each joint in form

X

XD 0 and/or

X

YD 0

should be considered.
Each possible load (each component of the vectorP) must be presented in terms
of all unknowns (all components of the vectorSE). Left part of each equation of equi-
librium should contains only a possible load, while the right part unknown internal
forces. The type of possible load corresponds to the type of possible displacement
according to diagramZ-P. If a possible displacement is the angle of rotation then
corresponding load is a moment. If possible displacement is linear one, then a cor-
responding load is force.
Figure11.17a presents the continuous beam. Unknown angular displacementsZ
of intermediate supportsAandBand corresponding momentsP 1 andP 2 are
labeled on theZ-Pdiagramby1and2(Fig.11.17b). Positive unknown internal
momentsMi(iD 1 –5) at the ends of each portions are labeled on theS-ediagram
by 1–5 (Fig.11.17c).
Equilibrium equations for intermediate supports can be rewritten as follows

P 1 D 0 M 1 C 1 M 2 C 1 M 3 C 0 M 4 C 0 M 5 ;

P 2 D 0 M 1 C 0 M 2 C 0 M 3 C 1 M 4 C 1 M 5 :