Page 144 - Handbook of Civil Engineering Calculations, Second Edition

STRUCTURAL STEEL DESIGN 1.127

and M p2 , and the composite mechanism therefore lacks significance. But if the basic
mechanisms produce rotations of opposite sign at any section whatsoever, M p3 may ex-
ceed both M p1 and M p2 .
In summary, a composite mechanism is significant only if the two basic mechanisms
of which it is composed produce rotations of opposite sign at any section. This theorem,
which establishes a necessary but not sufficient condition, simplifies the analysis of a
complex frame by enabling the engineer to discard the nonsignificant composite mecha-
nisms at the outset.

ANALYSIS OF AN UNSYMMETRIC
RECTANGULAR PORTAL FRAME

The frame in Fig. 30a sustains the ultimate loads shown. Compute the plastic moment and
ultimate-load reactions.

Calculation Procedure:
1. Determine the solution method to use
Apply the mechanism method. In Fig. 30b, indicate the basic mechanisms.
2. Identify the significant composite mechanisms
Apply the theorem of the previous calculation procedure. Using this theorem, identify the
significant composite mechanisms. For mechanisms 1 and 2, the rotations at B are of op-
posite sign; their composite therefore warrants investigation.
For mechanisms 1 and 3, there are no rotations of opposite sign; their composite there-
fore fails the test. For mechanisms 2 and 3, the rotations at B are of opposite sign; their
composite therefore warrants investigation.
3. Evaluate the external work associated with each mechanism

Mechanism W E
1 80 1 80(10 ) 800
2 2 2 20(15 ) 300
3 300
4 1100
5 600

4. List the sections at which plastic hinges form; record the angular
displacement associated with each mechanism
Use a list such as the following:

Section
Mechanism B C D F
1 – +2 –
2 + . . –1.2
3 –1.5 . . . . . +2.5
4 . . +2 –2.25
5 –0.5 . . –1.25 +2.5