Axially Loaded Members
5.4 Service Load Behavior of Compression Members
5.4.1 Linear Elastic Response
The linear-elastic solution for an axially loaded column follows from the assumptions of the preceding section. We first simplify the problem by assuming confinement is inactive for low strains, consistent with the findings of Chapter 4. Thus, fcc = fc in Figure 5.3. For a given strain ε within the linear range, stresses in concrete and reinforcement are fc = Ecε and fs = Esε, and corresponding stress resultants are Cc = Ac fc = (Ag – Ast′)fc, Cs1 = As1 fs, Cs2 = As2fs, and Cs3 = As3 fs. Note that we have taken Cc to represent the compression acting on the entire concrete section, with Ccc = 0 because confinement is inactive at this stage of loading. From equilibrium of the free-body diagram in Figure 5.3d, P – Cc – Cs1 – Cs2 – Cs3 = 0.
Combining terms, the relation between axial load and internal stresses can be written as
in which n = Es/Ec is the modular ratio. Solving Eq. (5.1) for concrete stress fc leads to
Because steel and concrete strains are equal at any point in the cross section, we can write
The results of Eq. (5.2) can be obtained using the transformed area method, which is applicable to linear materials. Recognizing that steel has elastic modulus Es = nEc, we can achieve equivalent cross-sectional properties by replacing steel area Ast by concrete having area nAst located at the centroid of the steel. As illustrated in Figure 5.4, when steel area Ast is transformed to concrete area nAst, the original concrete section is left with holes having area Ast. To simplify the analysis, those holes are filled, leaving (n – 1)Ast for the transformed steel sections. As shown in Figure 5.4, the transformed area is Ag + (n – 1)Ast; hence, the stress in concrete is given by Eq. (5.2).
FIGURE 5.4 Transformed section for compression member.
5.4.2 Effects of Drying Shrinkage and Creep
When freshly cast concrete is exposed to ambient temperature and humidity conditions, it generally undergoes volume changes associated with cooling (temperature shrinkage) and moisture loss (drying shrinkage). These occur in the absence of externally applied stresses. When stresses are applied, additional instantaneous strains occur. Under sustained stress, concrete also experiences a gradual increase in strain known as creep strain. Finally, concrete that is subjected to a given level of sustained strain (e.g., due to shrinkage) experiences a gradual decrease in stress due to creep known as stress relaxation.
Drying shrinkage and creep strains in concrete are associated mainly with the removal of adsorbed water from the hydrated cement paste. The main difference is that the driving force in drying shrinkage is the difference in relative humidity, whereas in the case of creep the driving force is sustained applied stress. Both are affected by concrete composition, initial curing, environment, geometry, and time. Additionally, creep is affected by the intensity of applied stress and the age of concrete when stresses are applied.1
Ultimate unrestrained shrinkage strain of concrete typically ranges from 0.0002 to 0.0008. Under normal conditions and for moderate-sized members, roughly 90% of drying shrinkage is completed by one year after casting and exposure to ambient conditions. Under long-term sustained load, creep strain, that is, the strain additional to the initial strain, can range from about 1.0 to 4.0 times the initial strain depending on materials and age at loading, with typical value around 2.0. Creep appears to continue indefinitely, but for most practical applications may be assumed to have reached its ultimate value after about five years.
We can estimate the effect of drying shrinkage on internal stresses of an otherwise unloaded reinforced concrete cross section using linear-elastic analysis methods. Figure 5.5 illustrates the procedure for a symmetric cross section. As a first step, the unrestrained shrinkage strain εsh is imposed on the section (Figure 5.5b). This results in compressive forces Cs1 and Cs2 in the longitudinal reinforcement, and a compressive force imbalance equal to P = Cs1 + Cs2. To restore equilibrium of the unloaded cross section, superimpose an equal and opposite tension force T = P (Figure 5.5c) to arrive at the equilibrium state (Figure 5.5d). Note that we are ignoring stress relaxation that would occur due to creep of the concrete under these internal stresses.
FIGURE 5.5 Internal stresses in symmetric cross section subject to longitudinal shrinkage strain.
For the problem at hand, we can show that the stresses in concrete and steel are given by
For concrete modulus of elasticity and tensile strength are approximately 4000 ksi (30,000 MPa) and 400 psi (3 MPa), resulting in n = 7. For typical values of εsh = 0.0004 and ρl = 0.01, Eqs.
(5.4) and (5.5) result in concrete tensile stress of 100 psi (0.7 MPa) and steel compressive stress of 10 ksi (70 MPa), which neither cracks the concrete nor yields the reinforcement. For greater values of εsh and ρ l, the tensile stress can be sufficient to crack the concrete.
Creep also affects internal stress distributions under service loads. In an axially loaded column, creep causes an increase in the axial compressive strain and, hence, an increase in the steel compressive stress. As the steel stress increases, the concrete stress must decrease, which, in turn, reduces the ultimate creep. This behavior is known as restrained creep because the longitudinal reinforcement is restraining creep in the concrete. The final steel stress can be two or more times the initial stress. The effect on steel stresses is greater for smaller steel ratios, which is one reason some building codes set a lower limit on the volume ratio of longitudinal reinforcement in compression members. For example, ACI 318 (2014) sets the minimum area of longitudinal reinforcement equal to 0.01Ag.
Restrained creep effects can be calculated using incremental analysis methods in which the loading period is broken into small time increments,
with material properties and stresses tracked for each increment.
Alternatively, the effective modulus method provides a closed-form solution to approximate the effects (Dilger, 1982).
The preceding analysis suggests that creep and shrinkage can appreciably alter the internal stresses of compression members under long- term loading. This is a considerable drawback for allowable stress design methods. Strength and deformation capacity of reinforced concrete members are much less susceptible to creep and shrinkage effects. This is one of the reasons why the load and resistance factor design (LRFD) method is generally preferred for reinforced concrete.