Advanced Methods of Structural Analysis

294 8 The Displacement Method

showninFig.8.9d, then such a structure can be easily analyzed by the displacement
method in canonical form. Indeed, the number of unknowns by the force method
is six, while by the displacement method itis just one (the angular displacement
of the rigid joint). Detailed tables for parabolic uniform and nonuniform arches are
presented in TablesA.19andA.20.
Figure8.9e shows a statically indeterminate beam. The number of unknowns by
the force method is four. The number of unknowns by the displacement method is
one. It is obvious that the displacement method is more preferable in this case.
Figure8.9f shows a statically indeterminate frame with an absolutely rigid cross-
bar. By the displacement method this structure has only one unknown no matter
how many vertical elements it has; analysis of frames of this type is considered in
Sect.8.4.

8.4 Sidesway Frames with Absolutely Rigid Crossbars..................

So far it has been assumed that the each rigid joint of the frame can rotate. Such joint
corresponds to one unknown of the displacement method. This section describes
analysis of a special type of frame: sidesway frames with absolutely rigid crossbars
(flexural stiffnessEID1).
Let us consider the frame shown in Fig.8.10a. The connections of the crossbar
and the vertical members are rigid. A feature of this structure is that the crossbar is
an absolutely rigid body; therefore, even if the joints are rigid, there are no angles of
rotation of the rigid joints. Thus, the frame has only one primary unknown, i.e., the
linear displacementZ 1 of the crossbar. The primary system is shown in Fig.8.10b.
Introduced constraint 1 prevents horizontal displacement of the crossbar. Flexural
stiffness per unit length isiDEI=h.
The canonical equation of the displacement method isr 11 Z 1 CR1PD 0.
The bending moment diagram caused by unit linear displacementZ 1 is shown
in Fig.8.10c.
Unit reactionr 11 is calculated using the equilibrium equation for the crossbar.
The bending moment for the clamped-clamped beam due to lateral unit displace-
ment isMD6i=h; therefore the shear force for vertical elements isQD2M= hD
12i= h^2. This force is transmitted to the crossbar (Fig.8.10d) and after that the unit
reaction can be calculated as follows:

r 11!

X

XD 3

12i

h^2

Cr 11 D 0 !r 11 D

36i

h^2

:

Applied loadPdoes not produce bending moments in the primary system. Nev-
ertheless, the free term is not zero and should be calculated using the free-body
diagram for the crossbar (Fig.8.10e).
Equilibrium condition

P

XD 0 leads to the following result:R1PDP.