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)

Principles of column interaction diagrams 249
Concrete
section
(a)
Fig. 7.1-1
Strain
distribution
(b)
Concrete stress
distribution
(c)
where the word 'concrete' emphasizes that theN and M values refer to a
plain concrete section.
Equations (7.1-1) and (7.1-2) can be expressed in dimensionless form:
N(conc)
Uconc = fcubh
X
= 0.405/i
[de]
= 0.45 h
M(conc) [de][ de]
fJconc = fcubh2 = 0.225 h 1 - h
= 0.203~[ 1 - 0.9~]
(7.1-3)
(7.1-4)
Thus for xl h = 0.1, aconc = 0.0405 and Pconc = 0.0185; for xl h = 0.2, aconc
= 0.0810 and fJconc = 0.0333 and so on. Therefore by successively assigning
different values to xlh, the complete acunc - Pcunc curve may be
constructed (Fig. 7.1-2). Each point on this curve represents a state of
incipient failure; points inside the curve represent safe combinations of N
and M, while those outside represent unacceptable combinations. (It
should be noted here that eqns 7.1-3 and 7.1-4 are based on BS 8110's
simplified stress block as described in Section 4.4. The stress block is
intended primarily for design so that the stress intensity of 0.45/cu includes
an allowance for the partial safety factor. Therefore at incipient failure, the
N and M values will be higher than those given by these equations.
However, as long as this point is understood, there is no objection to using
BS 8110's stress block, since our emphasis is on principles.)
Now consider the effect of an area As 2 of reinforcement, as in Fig.
7.1-3(a). The contribution of As 2 to the axial force is
N(Asz) = Aszf,z (7.1-5)
where /,2 is the compressive stress in the reinforcement which corresponds