Analysis of Statically Indeterminate Structures - Structural and Stress Analysis - page 529

sway. Let us suppose that the final endmoments in the members of the frame are M AB ,
M BA , M BC , etc. Since we are assuming a linearly elastic system we may calculate the
end moments produced by the applied loads assuming that the frame does not sway,
then calculate the end moments due solely to sway and superimpose the two cases.
Thus

M NS M AB M NS M BA M AB =

AB +

M BA =

BA +

and so on, in which M NS

AB is the end moment at A in the member AB due to the applied
loads, assuming that sway is prevented, while M AB is the end moment at A in the
member AB produced by sway only, and so on for M BA , M BC , etc.

We shall now use the principle of virtual work (Section 15.2) to establish a relationship
between the final end moments in the member and the applied loads. Thus we impose
a small virtual displacement on the frame comprising a rotation, θ v , of the members
AB and DC as shown in Fig. 16.46(b). This displacement should not be confused with
the sway of the frame which may, or may not, have the same form depending on the
loads that are applied. In Fig. 16.46(b) the members are rotating as rigid links so that
the internal moments in the members do no work. Therefore the total virtual work
comprises external virtual work only (the end moments M AB , M BA , etc. are externally
appliedmoments as far as each framemember is concerned) so that, from the principle
of virtual work

M AB θ v + M BA θ v + M CD θ v + M DC θ v + Ph θ v = 0

Hence

M AB + M BA + M CD + M DC + Ph =

0 (16.44)

Note that, in this case, the member BC does not rotate so that the end moments M BC
and M CB do no virtual work. Now substituting for M AB , M BA , etc. in Eq. (16.44) we
have

M NS M AB + M NS M BA + M NS M CD + M NS M DC +

AB + BA + CD + DC +

Ph = 0 (16.45)

in which the no-sway end moments, M NS

AB , etc., are found in an identical manner to

those in the frame of Ex. 16.21.

Let us now impose an arbitrary sway on the frame; this can be of any convenient mag-
nitude. The arbitrary sway and moments, M AS

AB , M AS BA , etc., are calculated using the

moment distribution method in the usual way except that the FEMs will be caused
solely by the displacement of one end of a member relative to the other. Since the
system is linear the member end moments will be directly proportional to the sway so
that the end moments corresponding to the actual sway will be directly proportional to
the end moments produced by the arbitrary sway. Thus, M AB =

kM AS

AB , M BA =

kM AS

BA ,