Columns with Rectilinear Hoop Reinforcement

Confined Concrete

4.3 Mechanism of Concrete Confinement

4.3.3 Columns with Rectilinear Hoop Reinforcement

Solution

Assuming concrete cover of 1.5 in (38 mm) over the spiral, the core diameter is 21 in (533 mm), resulting in spiral reinforcement ratio ρ_s = (4) (0.11 in²)/(21 in)(1.75 in) = 0.012. Confinement effectiveness factor is k_e = (1 – 1.75 in/21 in) = 0.92. Expected yield strength of A706 Grade 60 reinforcement is 1.15 × 60 ksi = 69 ksi (476 MPa). Using Eq. (4.8),

confined concrete strength is

Alternatively, we can use Figure 4.6. By this approach, from Eq. (4.5) we have f_2e = (0.92) (0.012)(69,000 ksi)/2 = 380 psi (2.6 MPa). Thus, From Figure 4.6, we can read Therefore, by this method,

FIGURE 4.7 Concrete column confined by rectilinear hoop reinforcement: (a) elevation; (b) section A-A; (c) free body diagram of a slice of the column core of thickness s.

Following the derivations in Section 4.3.2, force equilibrium requires

Where legs of hoops are inclined at an angle α relative to a normal to the section cut, a component of the area equal to A_shi cosα is used rather than A_shi. The ratio of volume of confinement reinforcement in the 3 direction to total volume of core confined by the hoops, ρ_s3, is defined as

Solving Eq. (4.9) for f₃ and using Eq. (4.10), we can write

Similarly, confinement stress in the orthogonal direction is given by

Stress f_s in the confinement reinforcement can be determined using Eq.

(4.6).

Arching Action and Confinement Effectiveness

Equations (4.11) and (4.12) define average confinement stresses f₂ and f₃ in the 2 and 3 directions, respectively. Unlike the case of spiral reinforcement, these stresses are not necessarily equal nor are they uniformly applied around the perimeter. Figure 4.8a illustrates this for a square core confined by a single perimeter hoop. As the core dilates under axial load, lateral expansion is effectively resisted by axial rigidity of the hoop reinforcement at the corners. Away from the corners, expansion must be resisted by the flexural rigidity of the hoop reinforcement. Tests have shown that, for practical hoop sizes, the flexural rigidity is insufficient to produce appreciable pressure away from the corners (Burdette and Hilsdorf, 1971).

Consequently, there is high confinement pressure at the corners of the core with little confinement pressure along the sides.

FIGURE 4.8 Development of confinement pressure for square column with perimeter hoop.

To improve the confinement of rectangular cross sections, additional hoop or crosstie legs can be added (Figure 4.8b). Each of these added legs should engage a longitudinal bar, serving to anchor the tie and spread the confining action longitudinally.

So far, we have considered only the confining pressure acting at the perimeter of the cross section, noting the concentrated forces associated

with each hoop or crosstie leg. These concentrated forces will spread laterally into the concrete core, reaching a more nearly uniform pressure distribution further within the core (Figure 4.9a and b). Likewise, the stresses must spread longitudinally between hoop sets (Figure 4.9c). This arching action is similar to that described for cores confined by circular hoops (Section 4.3.2), except in the case of rectilinear confinement the arching action occurs in three dimensions.

FIGURE 4.9 Confinement of concrete by reinforcement: (a) perimeter hoop with crossties; (b) perimeter hoop with added crossties to improve confinement effectiveness (note 90° hooks may not be fully effective under large core compressive strains); (c) arching action in column elevation.

Figure 4.10 illustrates three-dimensional arching in compression members confined by rectilinear hoop reinforcement. Where only perimeter hoops are provided at large spacing (Figure 4.10a), the effectively confined core is reduced considerably from the gross cross section. Additional hoop sets or crossties at reduced spacing can increase the effective cross section (Figure 4.10b).

FIGURE 4.10 Three-dimensional arching action in rectangular prismatic column core. (After Paultre and Légeron, 2008, used with permission from ASCE.)

Several analytical models for confined concrete include arching action and associated confinement effectiveness (Sheikh and Uzumeri, 1982;

Mander et al., 1988a; Razvi and Saatcioglu, 1999; Paultre and Légeron, 2008). Here we adopt a hybrid approach of these that is simple to implement, can be physically interpreted, and applies to a wide range of materials and core geometries.

A confinement effectiveness coefficient is defined according to the configuration and longitudinal spacing of the hoops (Figure 4.11). The values of k_e vary linearly from a maximum for zero longitudinal spacing to zero for spacing equal to the core dimension. The lines were positioned to represent an average of the above-cited models in the typical range of interest for seismic designs. The line for circular hoops and spirals is from Eq. (4.4). The lines for rectilinear confinement were calculated for square cross sections only. Confinement effectiveness for other rectangular sections can be adequately estimated by averaging k_e values for the two orthogonal directions.

FIGURE 4.11 Confinement effectiveness for various confinement configurations and hoop spacings.

Confinement effectiveness as shown in Figure 4.11 can be expressed algebraically as

in which n_l = number of longitudinal bars restrained by corners of hoops or legs of crossties around the column perimeter. For rectangular sections, for which s/b_c is different in the two orthogonal directions, the average of the s/b_c values in the two directions can be used. The expression was first presented in Paultre and Légeron (2008).

Long rectangular sections, such as those that occur in structural walls, present a special case (Figure 4.12). In the long direction, the confinement stresses are nearly uniform for most of the wall length (only the end sections have arching action). Therefore, the effective confinement stress in that direction, f_e2, can be calculated assuming k_e = 1. In contrast, arching action in both the horizontal and vertical directions will affect confinement of the wall through its thickness. The model by Mander et al. (1988a) can be used to estimate the confinement effectiveness in the 3 direction (through the thickness). Figure 4.12 presents results for a range of variables.

FIGURE 4.12 Confinement effectiveness for long rectangular sections.

Having determined the confinement effectiveness coefficient, the effective confinement stresses in the 2 and 3 directions are

Strength of the Confined Core

Knowing f_2e and f_3e from Eq. (4.14), we can use the strength relation given by Figure 4.6 to determine the confined concrete strength . The axial strength of a confined concrete core is thus in which A_ch is the cross- sectional area of the core measured to the outside of the hoops. The accuracy of the estimated confined concrete strength is presented in the next section following introduction of the loading rate effect.

Example 4.2. The rectangular cross section shown in Figure 4.13 has psi (34 MPa) in companion cylinders and A706 Grade 60 (420) reinforcement. Calculate the strength of the core concrete. Use expected material strengths. The solution is provided in tabular form.

FIGURE 4.13 Column cross section.