370 11 Matrix Stiffness Method
11.1.1 Finite Elements...............................................
Each structure may be subdivided into separate elements of simple geometrical con-
figuration called the finite elements. This step has no theoretical justification. For
each finite element, the stress–strain analysis is preliminarily investigated in detail;
the result of such analysis is presented in existing handbooks. For presentation of the
given structure as a set of the finite elements, the features of the structure as well as
required accuracy of analysis should be taken into account. Engineering experience
is an important factor for choosing the type and number of the finite elements.
In case of a truss, the separate members of the truss may be adopted as finite
elements. Therefore, the discrete model of the truss in terms of finite elements co-
incides with design diagram of the truss.
In case of the frame with uniform members, separate members of the frame also
may be adopted as the finite elements (Fig.11.1a). If the frame contains the member
with variable cross-section, then this member may be divided into several portions
with constant stiffness along each element (Fig.11.1b).
Fig. 11.1 Frames and their
presentation by the set of
finite elements
a P1
2b P1
32The uniform beams in TablesA.3–A.8, subjected to displacements of supports
may be considered as the simplest finiteelements. The finite elements can be one,
two, or three-dimensional. This chapter deals with planar bar structures only, so the
finite elements are straight thin bars with three types of constraints at the ends. They
are hinged-hinged (truss member), fixed-pinned, and fixed-fixed (frame member).
General idea of MSM.At the end points of each finiteelement, the some displace-
ments and interaction forces arise. For structure in whole these forces are internal,
while for each finite element these forces should be considered as the external loads.
For all finite elements, we can write three groups of equations. They are the (1) equi-
librium equations, (2) physical equations, and (3) geometrical ones. Equilibrium
equations take into account external forcesfor each finite element. Physical equa-
tions relate forces and displacements atend points of each element. Geometrical
equations describe continuity conditionsbetween ends of the elements. Solving of
these equations allows determining displacements and forces at end points of each
element.
11.1.2 Global and Local Coordinate Systems.......................
The local coordinate system is referred to as the specified element, while global sys-
tem is related to the whole structure. To understand these concepts, let us consider
a truss, subjected to forceP(Fig.11.2).