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Steel Designers' Manual - 6th Edition (2003)
w/unit length
Design of common structural forms 25
Parallel-flange universal sections, subject to meeting certain physical constraints
regarding breadth-to-thickness ratios of both flanges and webs, lend themselves to
rapid investigation by the plastic methods of structural analysis. The basis of the
plastic method is the need to determine the load applied to the frame which will
induce a number of ‘plastic hinges’ within the frame, thereby causing failure of the
frame as a mechanism.
This requirement is best illustrated by the following simple example.
Considering the pinned-base frame shown in Fig. 1.21(a), subject to a uniform
vertical load, w, per unit length: the reactions at the foundations are shown in
Fig. 1.21(b). The frame has one degree of indeterminacy. In order that the frame
fails as a mechanism, at least two plastic hinges must form (i.e. the degree of
indeterminacy + 1) as shown in Fig. 1.21(c). (It should be noted that although four
hinges are shown in Fig. 1.21(c), due to ‘theoretical’ symmetry only one pair either
side of the apex will in fact form, due to the obvious imperfections in both loading
and erection conditions.)
In many structures, other than the most simple, it is not clear where the plastic
hinges will form. There are several methods available to the design engineer which
greatly assist the location of these hinge positions, not least the abundance of proprietary
software packages specifically relating to this form of analysis. Prior to the
use of these packages, however, it is imperative that the engineer fully understands
the fundamentals of plastic analysis by taking time to calculate, by hand, several
design examples. The example which follows uses a graphical construction as a
means of illustrating the applications of the method to a simple portal frame.
Further information on plastic analysis is given in Chapter 11.
The frame shown in Fig. 1.22(a) has one degree of indeterminacy. It is made
statically determinate by assuming a roller at the right-hand base as shown in
Fig. 1.22(b), and the free-bending moment diagram drawn as shown in Fig. 1.22(c).
The reactant line for the horizontal force ‘removed’ to achieve a statically determinate
structure must now be drawn as follows:
V V
(a) (b) (c)
Fig. 1.21 Structural behaviour of pinned-base portal
plastic hinges