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)

354 Prestressed concrete simple beams
Comments
Examples 9.4-1 to 9.4-5 all deal with prestressed beams with a single
tendon. The loss of prestress in members with multi-level tendons is dealt
with in Problems 9.5 and 9.6.
9.5 The ultimate limit state: flexure (BS 8110)
Having designed the member for the service conditions and checked the
stresses at transfer, it is still necessary to check that the ultimate limit state
requirements are satisfied; for this latter purpose, the partial safety factors
should be those for the ultimate limit state (see Tables 1.5-1 and 1.5-2).
First consider the flexural strength. Using BS 8110's rectangular stress
block (see Fig. 4.4-5) the resistance moment of a rectangular beam is
immediately seen to be
Mu = [pbAps(d- 0.45x) (9.5-l(a))
where [pb = the tensile stress in the tendons at beam failure;
Aps = the area of the prestressing tendons in the tension zone.
(Prestressing tendons in the compression zone should be
ignored in using this equation.);
d = the effective depth to the centroid of Aps; and
x = the neutral axis depth.
This then is BS 8110's equation for the ultimate flexural strength; it applies
to rectangular beams and to flanged beams in which the neutral axis lies
within the flange. Values of [pb and x for such beams may be taken from
Table 9.5-1 for bonded tendons.
Table9.5-l Conditions at the ultimate limit state for pre-tensioned beams or
bonded post-tensioned beams (BS 8110: Clause 4.3. 7.3)
Ratio of depth of neutral axis
Design stress in tendons as a to that of the centroid of the
proportion of the design tendons in the tension zone,
strength, fpb/0.87fpu xld
fpuAps
fcubd
fpe/fpu = fpe/fpu
0.6 0.5 0.4 0.6 0.5 0.4
0.05 1.0 1.0 1.0 0.11 0.11 0.11
0.10 1.0 1.0 1.0 0.22 0.22 0.22
0.15 0.99 0.97 0.95 0.32 0.32 0.31
0.20 0.92 0.90 0.88 0.40 0.39 0.38
0.25 0.88 0.86 0.84 0.48 0.47 0.46
0.30 0.85 0.83 0.80 0.55 0.54 0.52
0.35 0.83 0.80 0.76 0.63 0.60 0.58
0.40 0.81 0.77 0.72 0.70 0.67 0.62
0.45 0.79 0.74 0.68 0.77 0.72 0.66
0.50 0.77 0.71 0.64 0.83 0.77 0.69