F. K. Kong MA, MSc, PhD, CEng, FICE, FIStructE, R. H. Evans CBE, DSc, D ès Sc, DTech, PhD, CEng, FICE, FIMechE, FIStructE (auth.)-Reinforced and Prestressed Concrete-Springer US (1987)

Hil/erborg's strip method 319
can be dealt with by the work method. However, in many instances,
an alternative technique in yield-line theory, namely, the so-called
equilibrium method or nodal-force method, would give a solution
more quickly. Readers interested in this alternative approach should
consult the works of Jones and Wood, Morley and others [7-10].
8.6 Hillerborg's strip method
In comment (a) following Example 8.4-1 on yield-line analysis, the upperbound
theorem was mentioned. In plastic theory [5, 6, 11] there is another
theorem called the lower-bound theorem [5, 6, 11], which states that, for
a structure under a system of external loads, if a stress distribution
throughout the structure can be found such that (a) all the conditions of
equilibrium are satisfied and (b) the yield condition is nowhere violated,
then the structure is safe under that system of external loads. It helps to
consider the application of this theorem to a simple case: say a frame
structure-in which only bending moments need be considered. The lowerbound
theorem then states that, if a distribution of bending moments can
be found such that the structure is in equilibrium under the external
loading, and such that nowhere is the yield moment of resistance of any
structural member exceeded, then the structure will not collapse under that
loading, however 'unlikely' that distribution of moments may appear [12].
To illustrate the application of the lower-bound theorem to design
[12], consider the span AB of a continuous reinforced concrete beam
in Fig. 8.6-1(a), in which the load for the ultimate limit is 20 kN/m.
According to the theorem, any of the bending moment diagra111s in
Figs 8.6-1(b), (c) and (d) may be used as a basis for design, provided we
are concerned only with strength capacity and provided the beam is
sufficiently under-reinforced to exhibit plastic behaviour at collapse.
Adopting Fig. 8.6-1(b) as the design bending moment diagram will
A 20kN/m 9
~Y.Y.W?
r 5m 1
(a)
11m. • 3m • ""!
~kNm ~~
,---IE>/
_ EfZ·5kNm
40~v _L
(c)
[V ~~kNm
(d)
Fig. 8.6-l