390 11 Matrix Stiffness Method
11.5 Set of Formulas and Procedure for Analysis..........................
The behavior of any structure can be described by following three groups of
equations:
Equilibrium equationsThis matrix equation establish relationships between exter-
nal possible joint loadsPEand unknown internal forcesSE
PEDAES; (11.8)whereAis a static matrix.
Geometrical equationsThis matrix equation establish relationships between
deformation of elementsEeand possible global displacementsZEof the joints
EeDBZEDATZE; (11.9)whereBis a matrix of deformation.
Physical equationsThis matrix equation establish relationships between required
internal forcesSEand deformation of elementsEe
SEDkQEe; (11.10)wherekQis a stiffness matrix of a structure in local coordinates (internal stiffness
matrix).
These three groups of equations describe completely any structure (geometry,
distribution of stiffness of separate members, types of connections of the members),
types of supports, and external exposure.
11.5.1 Stiffness Matrix in Global Coordinates......................
Rearrangement of (11.8)–(11.10) allows to obtain the equation for vector of un-
known internal forcesS. For this purpose, let us apply the following procedure.
The vectoremay be eliminated from (11.9)and(11.10). For this (11.9) should
be substituted into (11.10). This procedure allows us to express the vector unknown
internal forces in term of vector of unknown displacementsZ
SEDkQEeDkAQ TZE: (11.11)Now vectorScan be eliminated from (12.8) and (12.11). For this (12.11) should
be substituted into (11.8); this procedure allows us to express the vector of possible
joint external loads in term of unknown displacements
PEDASEDAkAQ TZE: (11.12)