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)

372 Prestressed concrete simple beams
Noting that ( P - o P) I P is the prestress loss ratio a we have
r (prestess, long term) = E) a + - 2-cp (hogging) (9.8-4)
1 Pe ( 1 + a )
where P = the prestressing force immediately after transfer;
e 5 = the tendon eccentricity at the section considered;
cp = the creep coefficient for the time interval;
Ee = the modulus of elasticity of the concrete at transfer (Table
2.5-6);
I = the second moment of area of the uncracked section;
a = the prestress loss ratio, i.e. a = (P- oP)I p = Us- Ofs)lfs (see
Examples 9.4-1 to 9.4-4 for 0/ 5 ).
Of course, the curvatures due to the applied load are simply
~ (load, short term) = f: 1 (sagging) (9.8-5)
r (load, creep) = cp Eel (sagging) (9.8-6)
1 M
Adding together,
r (load, long term) = (1 + cp) Eel (sagging)
1 M
The right-hand side of eqn (9.8-7) is sometimes written as
M
(long-term Ee)I
(9.8-7)
where the long-term or effective modulus Ee is Ee/(1 + cp ), as in eqn
(5.5-3).
Comments
(a) See Problem 9.3 for the legitimacy of eqn (9.8-6) (which is
occasionally questioned by the brighter students!).
(b) When calculating the long-term deflections of ordinary reinforced
concrete beams, it helps to use the concept of an effective modulus,
as in eqn (5.5-3). However, for prestressed concrete beams, it is
necessary to consider the effects of the loss of prestress (see eqns
(9.8-2) and (9.8-3)) and the use of an effective modulus can lead to
confusion. Indeed, the important eqn (9.8-4) cannot be conveniently
expressed in terms of an effective modulus of elasticity.
Example 9.8-2
Calculate the long-term deflections of the prestressed concrete beam in
Example 9.8-1 if:
concrete creep coefficient cp = 2.0
concrete shrinkage fes = 450 X 10- 6
tendon relaxation fr= 10% of fs = 129 N/mm 2