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)

Stresses in service: elastic theory 339
Z1} Mimax - Mimin
Z2 ~ famax - famin
(9.2-8)
that is, the minimum Z's are independent of Mct, f 1 and fz.
(g) The critical section of the beam is where the imposed-load moment
range Mr(= Mimax-Mimin) is a maximum. In particular, Mimax at the
critical section is not necessarily larger than that at another section,
and Mimin at the critical section is not necessarily smaller than that at
another section.
(h) If, at the critical section, the actual Z 1 (or Z2) is exactly equal to the
minimum value of eqn (9.2-8), then the prestressf1 (or fz) must have
that unique value which makes eqns (9.2-4) and (9.2-5) (or eqns
9.2-6 and 9.2-7) identities. Referring to Fig. 9.2-2, for such a case
the points F and J (or H and E) fall on a1 and b1 (or a2 and b2)
respectively.
(i) In practice it is rare for the Z's actually provided to be exactly the
minima required. Therefore, f 1 and fz may vary within limits. The
minimum / 1 required is that at which point F coincides with point a 1;
the minimum f 2 required is that at which H coincides with a2.
Similarly, the maximum permissible /J is that which makes J coincide
with b1, and the maximum permissible f 2 makes E coincide with b2.
In other words, the minimum required prestresses are those that
make eqns (9.2-4) and (9.2-7) identities:
· d f I' Mimax + Md
mm. req l = Jamin + zl (9.2-9)
. d f I' Mimin + Md (9 O)
mm. req 2 = Jamin - z 2 .2-1
Similarly, the maximum permissible prestresses are those that make
eqns (9.2-5) and (9.2-6) identities:
f
I' Mimin + Md
max. perm. 1 = Jamax + z 1 (9.2-11)
f
I' Mimax + Md
max. perm. 2 = Jamax - 22 (9.2-12)
(j) Referring to Fig. 9.2-2, the prestress at the centroid of the section is
/cp = ~c (9.2-13)
where A is the cross-sectional area. Minimum /cp is compatible with
minimumf1 andfz; hence, from eqn (9.2-13), the required minimum
prestressing force, Pemin' is that which gives the minimum f 1 and f 2 •
From eqns (9.2-2) and (9.2-3),
p = UtZt + !zZ2)A
e zl + Zz
_ (ft - fz)ZtZ2
es - UtZl + fzZz)A
(9.2-14)
(9.2-15)