Figure 6.4 represents a general concrete section containing prestressed and non-prestressed steel. At time t0. the section is subjected to a normal force N at reference point O, combined with a bending moment M. The values of N and M include the effect of the prestressing, which is assumed to be introduced at the same instant t0. The normal force and the moment produced by other loads applied at t0 are included in the values of N and M. Thus, N and M represent resultants of the normal stress σ(t0) introduced at t0. The meaning of the symbols N and M will be further explained in and demonstrated by Examples 6.1 and 6.2. In tanks or silos, the general cross-section in Figure 6.4 may represent a horizontal section in a vertical strip from a cylindrical wall; it may also represent a cross-section of a member from the cover or the base of a tank.
The following four analysis steps give distributions of strain and stress changes at time t0 and time t after the occurrence of creep and shrinkage of concrete and relaxation of prestressed steel. Initially, before introducing N and M, the section may have a stress, whose distribution is defined by the parameters {σo, γ}initial; the values of these parameters are assumed to be known. Thus, when the loading or prestressing is applied in multi-stages, the four analysis steps presented below can be used to find the immediate and the time-dependent changes in stress at t0 and in the period (t−t0) with t0 and t representing, respectively, an instant at which a load or prestressing is applied, and an instant at which stress or strain distribution is required.
1 Apply equations (6.14) and (6.15) to determine the instantaneous stress and strain parameter changes: σo(t0), γ(t0), εo(t0) and ψ(t0). In these calculations, use properties of the transformed section existing at the instant t0. When post-tensioning is employed, exclude Aps and the cross-sectional area of the duct for each tendon prestressed at t0. Tendons stressed and ducts grouted at earlier stages should be included.
2 Determine the hypothetical changes in strain parameters that would occur if concrete can deform freely. The
hypothetical strain parameter changes are:
(6.20) (6.21)
3 Calculate the artificial concrete stress that, when gradually introduced on the concrete in the period (t0 to t), will prevent the occurrence of strain calculated in step 2. The restraining stress at any fibre is (equation 6.5):
(6.22) where is an age-adjusted elasticity modulus of concrete (equation 6.7); and εcs=εcs(t,t0) is the free shrinkage between t0 and t. The stress parameters that define σrestraint are:
(6.23) (6.24)
4 Substitute {σo, γ}restraint in equation (6.13) to determine values of forces restraining creep and shrinkage. Prevent the change in concrete strain due to relaxation of prestressed steel, by application of an artificial restraining force
at the level of prestressed steel. Replace this force by a force of the same magnitude at O plus a couple equal to Sum up to obtain {∆N, ∆M}restraint, the restraining force at O and the couple required to prevent artificially the strain change due to creep, shrinkage and relaxation. Eliminate the artificial forces by the
Figure 6.5 Flow chart for calculation of stress and strain changes in a section due to normal force N and bending moment M introduced at time t0. and sustained to time t.
corresponding changes in strain and stress parameters by equations (6.15) and (6.14).
The combined instantaneous plus time-dependent change in strain parameters is the sum of strain parameters determined in steps 1 and 4; the corresponding change in stress parameters is the sum of stress parameters determined in steps 1, 3 and 4.
There are a number of comments that need to be made, as follows:
The flow chart in Figure 6.5 indicates the application sequence of the four steps. If, after step 1, the stress at an extreme fibre exceeds the tensile strength of concrete, the calculation in step 1 must be repeated, using A, B and I of the
cracked section, in which concrete in tension is ignored. The compression zone depth c must be determined prior to applying steps 2, 3 and 4 to the cracked section in steps 2, 3 and 4. The flow chart also outlines the sequence of analysis steps in a less common case, in which cracking occurs during the period t0 to t; this will be detected only at the end of step 4. Although the four analysis steps presented above apply to non-cracked or cracked sections5, the time-dependent analysis examples discussed in this book will ignore cracking.
The four steps give stress and strain at time t, including the effect of loss in prestress force due to creep,
shrinkage and relaxation. Thus, estimation of this prestress loss before the stress analysis is not needed. Only the loss in prestress force due to friction should be included in calculation of N and M of the input data, when post-tensioned tendons are employed.
The analysis satisfies the requirements of compatibility and equilibrium: the strain changes in any reinforcement layer and adjacent concrete are equal; the time-dependent effects change the partitioning of forces between the concrete and the reinforcements, but do not change the stress resultants.
Figure 6.6 Time-dependent prestress loss in a cross-section for which the equations in section 6.5.1 apply.
The same four steps apply to a reinforced concrete section without prestressing, simply by setting Aps=0.
6.5.1 Special case: section subjected to axial force only
Figure 6.6 represents a section prestressed at time t0; it is required to determine stress in concrete and
reinforcements at time t, after the occurrence of creep and shrinkage of concrete and relaxation of prestressed steel. It is assumed that the centroids of areas of concrete (Ac), the prestressed reinforcement (Aps) and the non-prestressed reinforcement (Ans) coincide. In this special case, ∆Pc, ∆Pns and ∆Pps are situated at the same
centroid; the symbol ∆P represents change in force in the period t0 to t; the subscripts c, ns and ps refer to concrete, non-prestressed reinforcement and prestressed reinforcement, respectively. The four steps of analysis discussed above lead to equation (6.25), which gives the time-dependent change in resultant force in concrete:
(6.25) (6.26)
where As=Ans+Aps; Es is modulus of elasticity of reinforcement (assumed to be the same for the prestressed and the non-prestressed reinforcements); and εcs=εcs(t, t0); (equation 6.7); and σc(t0) is the stress in concrete at t0, immediately after prestressing. When post-tensioning is used, σc(t0) is equal to (−P/A), where P is the absolute value of the prestress force, and A is the area of the transformed section composed of the area of concrete (Ac) excluding the prestress duct plus (αAns), with α=Ens/Ec(t0).
The time-dependent change in concrete stress is
σc(t,t0)=∆Pc/Ac. (6.27)
This value, commonly positive, represents the loss in compression in concrete. The change in axial strain between t0 and t is:
(6.28) The changes in stress in the reinforcements in the same period are:
σns(t,t0)=Esεo(t,t0), (6.29) (6.30) The corresponding changes in force in the reinforcements are:
∆Pns=Ansσns(t,t0), (6.31)
∆Pps=Apsσps(t,t0). (6.32)
It can be shown that the above equations satisfy the equilibrium requirement:
∆Pc+∆Pns+∆Pps=0, (6.33)
which means that the resultant force on the section (composed of the three materials) does not change due to creep, shrinkage and relaxation. Commonly ∆Pns and ∆Pps are negative quantities; the first represents
compression picked up by the non-prestressed steel as concrete shortens due to creep and shrinkage; the second quantity represents the time-dependent loss in tension in the prestressed reinforcement. The absolute value |∆Pps| is equal to ∆Pc only in the absence of Ans; the difference between these two values is ∆Pns. Thus, ignoring the presence of non-prestressed reinforcement results in underestimation of σc(t,t0), which represents the loss in the precompression in concrete. It is this value that the designer should be concerned with to determine whether a section is cracked or not due to forces applied on the section after the occurrence of prestress loss.
The equations presented in this section apply when, at any instant between t0. and t, the stress is uniform on the section. This is the case when Aps,Ans and Ac have the same centroid. This will also be the case when the section in Figure 6.6 represents a vertical section of circular-cylindrical wall subjected to axisymmetrical loads. Because of such loading, or circumferential prestressing, the stress is uniform on all fibres and the equations presented above apply, even when the centroids of Aps, Ans and Ac do not coincide; this will be discussed further in section 6.6.1.
Example 6.1 Time-dependent stresses in a prestressed section: effect of presence of non-prestressed steel
The section in Figure 6.6 is prestressed at time t0, such that the stress in concrete immediately after prestressing is σc(t0)=−5.00MPa (0.725ksi). Determine the changes in stress in concrete and in the tension in the prestressed steel due to creep, shrinkage and relaxation between t0 and a later time t. Study the effect on the results of varying the non-prestressed reinforcement ratio ρns=Ans/Ac from zero to 1.0 per cent. The following data are given:
ρps=Aps/Ac=0.004; εcs(t,t0)=−300×10−6; Ec(t0)=30GPa (4350ksi);
Es=200GPa (29000ksi); χ(t,t0)=0.8.
The age-adjusted elasticity modulus of concrete (equation 6.7):
(6.34)
As=Ac (0.004+ρns). (6.35)
Substitution in equations (6.26), (6.25), (6.27), (6.28), (6.30) and (6.32) gives:
β=1.0693+17.33ρns, (6.36)
∆Pc=−β−1(0.7067+126.7ρns)106Ac, σc(t,t0)=−β−1(0.7067+126.7ρns)106,
εo(t,t0)=−10−6(633)+(11.54×109)−1σc(t,t0) (N, m units), σps(t,t0)=200×109εo(t,t0)−50×106 (N, m units).
Substitution of variable values of ρns in the above equations gives the results in Table 6.1.
Table 6.1
ρns (per cent) 0.0 0.2 0.4 0.6 0.8 1.0
σc(t, t0) (MPa) 0.661 0.870 1.066 1.250 1.424 1.588
σc(t, t0) (ksi) 0.096 0.126 0.155 0.181 0.206 0.230
[σc(t,t0)/σc(t0)]100 13.2 17.4 21.3 25.0 28.5 31.8
σps(t, t0) (MPa) −165 −161 −158 −155 −152 −149
σps(t, t0) (ksi) −23.9 −23.3 −22.9 −22.5 −22.0 −21.6
The results indicate that the presence of non-prestressed reinforcement significantly increases the loss in the precompression in concrete (from 13.2 to 31.8 per cent). This is accompanied by a relatively small reduction in the loss in tension in the prestressed reinforcement