Advanced Methods of Structural Analysis

302 8 The Displacement Method

In this case, the extended fibers at the center support are locatedabovethe neu-
tral line and the corresponding moment at the support isql^2 =8. If the external load
is absent and the center support is placed below the beam level on, then the
extended fibers at the support will be locatedbelowthe neutral line and the corre-
sponding moment at the support will be



3 EI=l^2


. If beam is subjected to load
qand displacementsimultaneously, then we can determine parameterfor the
required distribution of internal forces. For example, whatshould be in order that
the bending moment at the center support be zero, or whatshould be in order
that the maximum bending moment at the span and bending moment at the center
support be equal, etc.

8.6 Analysis of Symmetrical Structures...................................

Symmetrical structures are used often in structural engineering.Symmetrical struc-
turesmean their geometrical symmetry, symmetry of supports, and stiffness sym-
metry of the members. For analysis of symmetrical statically indeterminate frames
subjected toany loads, the Combined Method (the combination of the force and
displacement methods) canbe effectively applied.

8.6.1 Symmetrical and Antisymmetrical Loading.................

We start consideration of the combined method from concept of resolving a total
load on symmetrical and antisymmetrical components. Symmetrical frame sub-
jected to horizontal loadPis shown in Fig.8.14a. This load may be presented as a
sum of symmetrical and antisymmetrical components (Fig.8.14b, c). In general,any
loadmay be presented as the sum of symmetrical and antisymmetrical components.

P

Axis of

symmetry

+

P/2 P/2 P/2 P/2

=

AS AS

Symmetrical

load

Antisymmetrical

load

ab c

Fig. 8.14 Presentation of the loadPas a sum of the symmetrical and antisymmetrical components

Symmetrical frame subjected to symmetrical and antisymmetrical components
allows simplifying the entire design diagram. This simplification is based on the
change of the entire design diagram by its equivalenthalf-frame(the physical basis
of such changing will be discussed below). This procedure leads to the following