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)

350 Prestressed concrete simple beams
df, = /r
s
1 + EsAps[1 + e;]
EcA k 2 (9.4-1)
For practical prestressed concrete sections, the quantity EsAps(1 + e;!k 2 )!
EcA is small compared with unity; a numerical example, on p. 110 of
Reference 2, gives 0.05 as a typical value. Therefore, in design the
relaxation loss is usually taken simply as
dfs (relaxation) = /r (9.4-2)
Example 9.4-2
For the pretensioned concrete section shown in Fig. 9.4-1, determine the
loss of prestress due to the elastic deformation of the concrete, if the initial
prestress in the tendon is /s immediately before transfer.
SOLUTION
Let dfs = loss of prestress in the tendon due to elastic deformation;
fc = concrete compressive stress at tendon level after transfer.
The equilibrium condition is, from statics,
F = (fs - dfs)Aps[1 + e;]
Jc A e
where k = ~(II A) is the radius of gyration. The compatibility condition is
/c = dfs
Ec Es
Eliminating /c from the equilibrium and compatibility conditions,
fsae(! [ 1 + p. e;J
dfs = [ 2]
1 + aee 1 + ~~
(9.4-3)
where ae = Esl Ec and (! = A psi A. As explained in the paragraph
following eqn (9.4-1), the denominator on the right -hand side of eqn
(9.4-3) is nearly equal to unity. Therefore, in design, the elasticity loss is
usually taken as
dfs (elasticity) = fsaee[ 1 + ~~] (9.4-4)
Example 9.4-3
For the prestressed concrete section in Fig. 9.4-1, determine the loss of
prestress due to a concrete shrinkage Ecs·
SOLUTION
Let dfs = loss of prestress in the tendon due to concrete shrinkage;
dfc = the corresponding reduction in the concrete compressive stress
at tendon level.