3-6 TIME-DEPENDENT VOLUME CHANGES
3. Relative shrinkage strain and expected crack width
Using the shrinkage strain values calculated in the prior steps for the slab and the wall after the slab was cast, the net differential shrinkage strain between the slab and the wall is,
With this value, if the observed cracks in the slab are occurring at a spacing of 6 ft, the expected crack widths would be,
This is an approximate value for the crack width because it assumes a uniform spac- ing between the cracks in the slab and does not account for the effect of reinforcement re- straining shrinkage strains in the concrete. If reinforcement is present the shrinkage strains
would be from 75 to 90 percent of the calculated values. ■
Creep of Unrestrained Concrete
When concrete is loaded in compression, an instantaneous elastic strain develops, as shown in Fig. 3-22b. If this load remains on the member, creep strains develop with time. These occur because the adsorbed water layers tend to become thinner between gel particles
0.031 in.
Crack width 6 ft * 12 in./ft * 430 * 10-6
Net differential strain = (630 - 200) * 10-6 = 430 * 10-6 strain Net wall strain = (560 - 360) * 10-6 = 200 * 10-6 strain
= 355 * 10-6 360 * 10-6 (esh)t = 53
35 + 53 * 590 * 10-6
= 564 * 10-6 560 * 10-6strain (esh)t = t
35 + t(esh)u = 783
35 + 783 * 590 * 10-6 (esh)u
t = 2 * 365 + 2 * 30 - 7 = 783 days
= 0.90 * 0.84 * 780 * 10-6 = 590 * 10-6 strain (esh)u = grh * gns * 780 * 10-6
gns = 1.2-0.12V/S = 1.2-0.893 0.84
transmitting compressive stress. This change in thickness occurs rapidly at first, slowing down with time. With time, bonds form between the gel particles in their new position. If the load is eventually removed, a portion of the strain is recovered elastically and another portion by creep, but a residual strain remains (see Fig. 3-22b), due to the bonding of the gel particles in the deformed position.
Creep strains, which continue to increase over a period of two to five years, are on the order of one to three times the instantaneous elastic strains. Increased concrete compression strains due to creep will lead to an increase in deflections with time, may leadto a redistribution of stresses within cross sections, and cause a decrease in prestress- ing forces.
The ratio of creep strain after a very long time to elastic strain, is called thecreep coefficient, The magnitude of the creep coefficient is affected by the ratio of the sustained stress to the strength of the concrete, the age of the concrete when loaded, the humidity of the environment, the dimensions of the element, and the composition of the concrete. Creep is greatest in concretes with a high cement–paste content. Concretes con- taining a large aggregate fraction creep less, because only the paste creeps and because creep is restrained by the aggregate. The rate of development of the creep strains is also affected by the temperature, reaching a plateau at about 160°F. At the high temperatures encountered in fires, very large creep strains occur. The type of cement (i.e., normal or high-early-strength cement) and the water/cement ratio are important only in that they affect the strength at the time when the concrete is loaded.
For creep, as for shrinkage, several calculation procedures exist [3-6], [3-21], [3-48], and [3-49].For stresses less than creep is assumed to be linearly related to stress.
Beyond this stress, creep strains increase more rapidly and may lead to failure of the mem- ber at stresses greater than as shown in Fig. 3-2a. Similarly, creep increases signif- icantly at mean temperatures in excess of 90°F.
The total strain, at time tin a concrete member uniaxially loaded with a constant stress at time is
(3-32) where
The stress-dependent strain at time tis
(3-33) For a stress applied at time and remaining constant until time t, the creep strain between time and tis
(3-34) whereEc(28) is the modulus of elasticity at the age of 28 days, given by Eq. (3-17) or (3-18). Because creep strains involve the entire member, the value for the elastic modulus should be based on the average concrete strength for the full member. It is recommended that the value of mean concrete strength for a member, fcm, be taken as 1.2 fc⬘.
Pcc1t,t02 = sc1t02 Ec1282Ct t0
Pcc
t0 sc
Pcs1t2 = Pci1t02 + Pcc1t2 Ec1t02 = modulus of elasticity at the age of loading
PcT1t2 = thermal strain at time t Pcs1t2 = shrinkage strain at time t
Pcc1t2 = creep strain at time t where t is greater than t0 Pci1t02 = initial strain at loading = sc1t02/Ec1t02
Pc1t2 = Pci1t02 + Pcc1t2 + Pcs1t2 + PcT1t2 t0
sc1t02 Pc1t2, 0.75fœc,
0.40fœc, f.
Pc/Pi, Pc,
From reference [3-21], the creep coefficient as a function of time since load applica- tion,Ct, is given as:
(3-35) wheretis the number of days after application of the load and Cuis the ultimate creep co- efficient, which is defined below in Eq. (3-36). The constant, 10, may vary for different concretes and curing conditions, but this value is commonly used for steam-cured concrete and normal concrete that is moist-cured for 7 days.
As with the coefficient for ultimate shrinkage strain, the coefficient Cuconsists of a constant multiplied by correction factors.
(3-36) The constant in this equation can range from 1.30 to 4.15, but the value of 2.35 is com- monly recommended. The coefficients lrhandlvsaccount for the ambient relative humid- ity and the volume/surface ratio, respectively. As with shrinkage strains, a higher value of relative humidity and a larger volume/surface ratio (can also be expressed as a larger effective thickness), will tend to reduce the magnitude of creep strains. For an ambient rel- ative humidity (RH) greater than 40 percent, the modifier for relative humidity is:
(3-37) The modifier to account for the volume/surface ratio is:
(3-38) whereV/Sis the volume/surface area ratio in inches for the member in question.
The coefficient ltoin Eq. (3-36) is used to account for the age of the concrete when load is applied to the member. Early loading of a concrete member will result in higher shrinkage strains, as shown in Fig. 3-25 from reference [3-6], in which tois the time of initial loading in days, heis the member effective thickness, and f(t, to) is the symbol used for the creep coefficient in reference [3-6]. Values forltofrom ACI Committee 209 [3-21] are
for moist-cured concretes: lto=1.25*to-0.118 (3-39a) for steam-cured concretes: lto=1.13*to-0.094 (3-39b) wheretois the time in days at initial loading of the member.
lns = 0.67c1 + 1.13-0.54V/Sd lrh = 1.27 - 0.0067 * RH Cu = 2.35 * lrh * lns * lto
Ct = t0.6
10 + t0.6 * Cu
10 20 30 4.0
5.0
3.0
2.0
1.0
0.0
1 7 10 30 60 100 days 1 year 2 3
Age of concrete he=4 in.
t0=7 days
he= 4 in. t0= 1 year he= 24 in.
t0= 7 days
he= 24 in. t0= 1 year f⬘c= 3000 psi
RH = 50%
f(t,t0)
Fig. 3-25
Effect of effective thickness, and of age at loading, on creep coefficient.
t0, he,
The expressions given here for creep strains are intended for general use and do not consider significant variations in curing conditions and the types and amounts of aggregates used in the mix design. If creep deflections are anticipated to be a seri- ousproblem for a particular structure, consideration should be given to carrying out creep tests on the concrete to be used. Further, a more sophisticated approach is rec- ommended for applications where an accurate calculation of deflection versus time after initial loading is required, such as in segmentally constructed post-tension con- crete bridges.
Example 3-3 Calculation of Unrestrained Creep Strains
A plain concrete pedestal high is subjected to an average stress of 1000 psi. Compute the total shortening in 5 years if the load is applied 2 weeks after the concrete is cast. The properties of the concrete and the exposure are the same as in Example 3-2.
1. Compute the ultimate shrinkage strain coefficient, Cu. From Eq. (3-36), the ultimate creep coefficient is,
For a relative humidity of 50 percent, Eq. (3-37) is used to calculate the modification factor, lrh.
The load on the pedestal was applied at to= 14 days, and it is assumed that the pedestal was moist-cured. Thus, from Eq. (3-39a),
The volume of concrete in the pedestal is 2 ft *2 ft *10 ft. Assuming that only the sides of the pedestal are exposed to the atmosphere, the exposed surface area is 4 *2 ft *10 ft.
Thus, the volume/surface ratio is,
From Eq. (3-38), the modification factor for volume/surface ratio is,
Putting these coefficients into Eq. (3-36) results in,
Cu = 2.35 * 0.94 * 0.92 * 1.12 = 2.28
= 0.67
C
1 + 1.13-3.24D
= 0.67 [1 + 0.673] = 1.12lns = 0.67
C
1 + 1.13-0.54V/SD
V
S = 2 ft * 2 ft * 10 ft 4 * 2 ft * 10 ft = 2 ft
4 = 6 in.
= 1.25 * 14-0.118 = 1.25 * 0.732 0.92 lto = 1.25 * to-0.118
= 1.27 - 0.0067 * 50 = 1.27 - 0.33 = 0.94 lrh = 1.27 - 0.0067 * RH
Cu = 2.35 * lrh * lns * lto 24 in. * 24 in. * 10 ft