Creep and shrinkage of concrete and relaxation of steel - Concrete Structures: Stresses and Deformations - page 5

time in ε cs0 depends upon the quality of concrete and the

ambient air humidity. The function

fi nity. The value of

t s ) adopted by MC-90 depends
upon the size and shape of the element considered (see Appendix A).
The free shrinkage,

β s ( t −

ε cs ( t 2 , t 1 ) occurring between any two instants t 1 and t 2
can be determined as the di erence between the two values obtained by

Equation (1.3), substituting t 2 and t 1 for t .

ff

1.4 Relaxation of prestressed steel

The e ect of creep on prestressing steel is commonly evaluated by a relaxa-

tion test in which a tendon is stretched and maintained at a constant length
and temperature and the loss in tension is measured over a long period. The
relaxation under constant strain as in a constant-length test is referred to as
intrinsic relaxation,

ff

∆ σ pr . An equation widely used in the US and Canada for
the intrinsic relaxation at any time τ of stress-relieved wires or strands is: 4

∆ σ pr
σ p0 =−

log( τ − t 0 ) σ p0

f py −

0.55 (1.4)

10

where f py is the 'yield' stress, de ned as the stress at a strain of 0.01. The ratio

f py to the characteristic tensile stress f ptk varies between 0.8 and 0.90, with the
lower value for prestressing bars and the higher value for low-relaxation
strands ( (

fi

t 0 ) is the period in hours for which the tendon is stretched).
The amount of intrinsic relaxation depends on the quality of steel. The
MC-90 5 refers to three classes of relaxation and represents the relaxation as a
fraction of the initial stress

τ −

rst class include cold-drawn
wires and strands, the second class includes quenched and tempered wires
and cold-drawn wires and strands which are treated (stabilized) to achieve
low relaxation. The third class, of intermediate relaxation, is for bars.
For a given steel and duration of relaxation test, the intrinsic relaxation
increases quickly as the initial stress in steel approaches its strength. In the
absence of reliable relaxation tests, MC-90 suggests the intrinsic relaxation
values shown in Fig. 1.3 for duration of 1000 hours and assumes that the
relaxation after 50 years and more is three times these values.
The Eurocode 2-91 6 (EC2-91) allows use of relaxation values di

σ p0 . Steels of the fi

ering
slightly from MC-90. The values of EC2-91 are given between brackets in the
graphs of Fig. 1.3.
The following equation may be employed to give the ratio of the ultimate
intrinsic relaxation to the initial stress:

ff

∆ σ pr ∞
σ p0

=− η ( λ − 0.4) 2 (1.5)

where