A typical stress-strain graph for concrete is shown in Figure 6.1. In service condition, where the compressive stress rarely exceeds 50 per cent of concrete strength, it is a common practice to express the immediate strain, occurring within seconds of stress application, as follows:
(6.1)
where σc(t0) is the concrete stress introduced at time t0; and Ec(t0) is the modulus of elasticity of concrete at the
age t0. The value of Ec, the secant modulus, defined in Figure 6.1, depends upon the magnitude of the stress, but
A stress increment σc(t0) introduced at time t0. and sustained, without change in magnitude, until time t (Figure
6.2a) produces at time t a total strain (instantaneous plus creep) given by (see Figure 6.2b):
(6.2)
where is creep coefficient, a function of ages of concrete t and t0. representing the ratio of creep to
instantaneous strain. The value of increases with the decrease in t0 and the increase in length of the period (t−t0)
during which the stress is sustained.
The dashed line in Figure 6.2(a) represents a stress increment whose magnitude is gradually increased from zero at t0. to a final value σc at time t. When this case is represented by a number of small stress increments introduced
at variable time τ, such that t0<τ<t, the total strain at time t can be expressed as the sum of the effect of individual
increments. Using equation (6.2) for each:
(6.3)
Figure 6.2 Definitions of creep and aging coefficients: (a) stress versus time; (b) strain versus time.
where n is the total number of increments; and Δσci is the ith stress increment introduced at τi. In practice this
equation is approximated (see Figure 6.2b):
(6.4)
or
where χ, referred to as aging coefficient, is a function of (t,t0). Variations of Ec and with time also affect the
value of χ1. However, in many practical situations χ is not substantially different from 0.8; thus the value is frequently adopted in practice.
When (t−t0)≥ one year, the aging coefficient can be more accurately estimated by
(6.6)
referred to as age-adjusted elasticity modulus for concrete, is defined by
(6.7)
Equations (6.1), (6.2), (6.4) and (6.5) are linear stress-strain relationships applicable when the stress increment σc
is compressive or tensile. The stress σc is considered here positive when tensile. The same equations can be
combined to give the strain due to a stress increment whose magnitude varies between a non-zero value σc(t0) at
t0 and a final value σc(t) at t. Here the total strain at time t can be determined by superposition of the strain due to
sustained stress σc(t0) (equation 6.2) and the strain due to a gradually introduced stress varying from zero at time
t0 to [σc(t)−σc(t0)] at time t (equation 6.4).
Drying of concrete in air causes shrinkage, while concrete in contact with water swells. When free to occur, the volumetric change, the shrinkage or the swelling, develops gradually with time without stress. The symbol εcs(t,
t0) will be used for the free (unrestrained) strain due to shrinkage or swelling in the period t0 to t; εcs is considered
positive when it represents elongation; thus for shrinkage εcs is a negative value.
In tanks and silos, shrinkage or swelling of concrete is always restrained; thus stresses develop. The restraint can be caused by the presence of reinforcing steel, by the supports at the wall edges or by the differential values of εcs
for the wall, the base and the cover. The stress due to shrinkage εcs(t, t0) develops gradually in the period t0. to t;
thus shrinkage is always accompanied by creep. This substantially alleviates shrinkage stresses; analysis of shrinkage stresses will always account for creep.
The strain that develops due to free shrinkage between ts and later instant t may be expressed by:
εcs(t, t0)=εcoβs(t−ts), (6.8)
where εco is the total shrinkage that occurs after concrete hardening up to time infinity. The value εco (between
−300×10−6 and −800×10−6) depends upon the quality of concrete, the ambient humidity and the thickness of the element. βs is a function of the length of time (t−ts), with ts being the time at which curing stops. The free
shrinkage, εcs(t,t0), occurring between any two instants t0 and t is given by the difference between εcs(t,ts) and
εcs(t0,ts) determined by equation (6.8).
The concrete parameters required for the time-dependent analyses are: Ec(t0), χ(t,t0) and εcs(t,t0). Codes2 and
technical committee reports3 give guidance on the values to be used in practice. 6.3 Relaxation of prestressed steel
length and temperature over a period of time; ∆σpr is a negative value, representing loss of tension. The intrinsic
relaxation magnitude depends upon the quality of steel and the value of the initial stress. For initial stress of 70 per cent of tensile strength, ∆σpr that takes place in 1000h varies between 2.5 and 8.0 per cent of the initial stress.
Three times these values are expected after 50 years or more.
The magnitude of the intrinsic relaxation depends heavily on the value of the initial stress. A tendon in a prestressed concrete member does not have a constant length as is the case in a relaxation test to determine the intrinsic relaxation. Owing to creep and shrinkage, the distance between anchorages in the prestressed member is shortened, thus causing faster reduction of tension compared to the reduction in a constant-length test. This has a similar effect on the magnitude of relaxation as if the initial stress were smaller. Thus, the relaxation value to be used in prediction of the loss of prestress in concrete tanks and silos should be smaller than the intrinsic
relaxation obtained from a constant-length test. For this purpose, we define the reduced relaxation value to be used in analysis of concrete structures:
(6.9)
where Δσpr is the intrinsic relaxation; this is the value of relaxation that would occur in a constant-length
relaxation test; χr is a dimensionless coefficient, commonly smaller than 1.0. The value of χr can be expressed as
a function of the initial stress and other parameters depending upon creep and shrinkage4. For many practical cases χr is not substantially different from 0.8; the value is often adopted.