Axially Loaded Members
5.5 Inelastic Behavior of Compression Members
5.5.2 Longitudinal Reinforcement
Longitudinal reinforcement behaves differently in compression and tension for two reasons. First, under compression the Poisson effect increases the cross-sectional area, whereas in tension it decreases the area, so that engineering stresses in compression exceed those measured in tension tests (see Chapter 2). This effect is ignored in routine analyses. A second effect is reinforcement instability, which can occur if longitudinal reinforcement is inadequately supported or strain ranges are relatively large. Reinforcement buckling should not be overlooked, as it can have an appreciable effect on the axial load resistance of the cross section.
Buckling of longitudinal reinforcement in reinforced concrete members is complicated by interactions with the surrounding concrete. In a compression member, dilation of the confined core exerts outward pressure on the longitudinal bars, which in combination with axial compression increases the tendency for buckling. The cover concrete initially provides
some restraint against buckling, but this restraint diminishes as the cover itself becomes damaged at large strains. Transverse reinforcement thus plays a dominant role in restraining longitudinal bar buckling at large compressive strains. As will be discussed subsequently, for most practical problems, the transverse reinforcement spacing will be small enough that elastic buckling of longitudinal reinforcement is avoided. Thus, analysis of the buckling problem requires consideration of both geometric and material nonlinearities.
Bar buckling is further complicated in that different buckling modes can occur depending on the degree of restraint provided by the transverse reinforcement. The simplest mode involves buckling between two hoop sets or spiral turns (Figure 5.8b). In this case, the bar might be idealized as a fixed-ended column of length s, though the fixity at the ends is questionable.
A second mode involves buckling over two or more hoop sets or spiral turns (Figure 5.8d). In this case, the bar might be idealized as a column of length ns restrained by discrete hoops. A third mode (not shown) involves the bar buckling sideways in the direction parallel to the perimeter hoop or spiral.
FIGURE 5.8 Restraint of longitudinal bars and idealized buckling modes.
The buckling restraint provided by the hoop reinforcement depends on the configuration of the hoop set. For rectilinear transverse reinforcement, longitudinal bars located in the corner of a tie (e.g., bars a and c in Figure 5.8a) are effectively supported even if the tie has diameter as small as 15%
of the longitudinal bar diameter (Bresler and Gilbert, 1961). In contrast, longitudinal bars not supported in a corner of a tie (e.g., bar b in Figure 5.8a) must rely on the flexural stiffness of the hoop for support, which is
insufficient for practical hoop bar diameters. It can be argued that the unsupported bars gain indirect support from the adjacent restrained core and longitudinal bars (Pantazopoulou, 1998), but generally the unsupported bars weaken the cross section. For circular hoops or spirals, lateral restraint is provided by the radial component of the hoop tension (Figure 5.8c).
Whether longitudinal bars buckle between hoop sets or span multiple sets (Figure 5.8d) depends on the stiffness of the hoop tension action. Papia and colleagues (1988, 1989) and Brown et al. (2008) provide additional insight into the problem of buckling that spans multiple hoop sets.
Building codes include provisions for minimum size of rectilinear hoop reinforcement intended to prevent buckling over several hoop sets. ACI 318 requires that the hoop bar size be at least No. 3 (No. 10 metric) for longitudinal bars up to No. 10 (No. 32), and No. 4 (No. 13) for larger longitudinal bars. ACI 318 also requires that every corner and alternate longitudinal bar be supported by the corner of a tie with no unsupported bar more than 6 in (150 mm) clear from a supported bar. For columns of special moment frames with high concrete compressive strength or high axial load, ACI 318 requires that every longitudinal bar be supported. NZS 3101 (2006), referring to plastic hinges of beams and columns, requires that every longitudinal bar be restrained by ties, except that tie legs need not be placed closer than 8 in (200 mm) apart on centers (this permits unrestrained bars between the restrained bars). Furthermore, the diameter of ties is not to be less than 0.2 in (5 mm) and the area of a tie in the direction of potential buckling is not to be less than
in which ΣAbl is the sum of areas of all longitudinal bars to be restrained by the tie leg, including tributary portions of adjacent unrestrained bars. This expression was developed based on the assumption that, when the ties having spacing s = 6dbl, the provided strength should not be less than 1/16th the strength of the restrained bars. For spiral or circular hoop confinement of column plastic hinges, NZS 3101 (2006) requires
In Eqs. (5.6) and (5.7), fyt is not to be taken greater than 116 ksi (800 MPa).
Where transverse reinforcement provides sufficient stiffness and strength to prevent buckling spanning several hoop sets, the main design parameter is the spacing of transverse reinforcement to limit bar buckling between hoops. We can use the Euler equation for buckling of slender columns to analyze this problem. Accordingly, the critical stress at buckling is given by
For longitudinal bars with circular cross section of diameter db, r = db/4.
Recognizing that l = s, we can solve Eq. (5.8) for the critical hoop spacing as
To use Eq. (5.9), we must define the effective length factor k and the effective modulus of elasticity E. The value of k lies somewhere between 0.5 (fixed ends) and 1.0 (pinned ends). Recognizing that typical construction quality and damage after cover spalling result in less than fully fixed conditions, Bresler and Gilbert (1961) suggested a value of k = 0.7. For stresses up to the yield point, the elastic modulus of steel can be substituted for E. Using k = 0.7, E = 29,000 ksi (200,000 MPa), and yield stress fy = fcr
= 69 ksi (480 MPa), Eq. (5.9) results in scritical = 23db. This compares with the commonly recommended maximum hoop spacing of s = 16db for non- seismic design of columns (e.g., ACI 318).
We can extend the Euler model beyond the yield point using either the tangent modulus or the double (or reduced) modulus. In the tangent modulus approach, the instantaneous tangent modulus Et (Figure 5.9a) is substituted for E in Eq. (5.9). For example, to estimate requirements for a longitudinal bar just past the yield point, we might set E equal to the initial strain- hardening modulus, that is, E = Esh ≈ 1000 ksi (7000 MPa), along with k = 0.7 and fcr = 69 ksi (480 MPa), resulting in scritical = 4db. More generally, for the stress–strain relation shown in Figure 5.9a, which is characteristic
of A706 Grade 60 (420) reinforcement, the tangent modulus predicts the scritical/db versus strain relations shown in Figure 5.9b. According to this model, achieving large compressive strain without buckling requires very close spacing of hoop sets.
FIGURE 5.9 (a) Stress–strain relation and (b) calculated relation between scritical/db and maximum strain based on tangent modulus approach.
If the reinforcement has a yield plateau, Et = 0 along the plateau, and Eq.
(5.9) can be interpreted as predicting buckling immediately upon reaching the yield stress. Tests, however, demonstrate that reinforcement can remain stable well past the yield plateau if the unsupported length is not large. Two factors enable this to occur. First, the yield plateau represents the integration along a finite gauge length of slipbands occurring at discrete locations, and does not accurately represent the behavior of the entire length of bar at any strain along the plateau. Thus, the use of Et = 0 in Eq. (5.9) would be inappropriate. Equally important is the effect that buckling has on curvature and internal stresses of the bar, adding stability, as explained below.
When a reinforcing bar is restrained by closely spaced hoops, buckling involves considerable curvature of the bar, leading to increased compression on one side and unloading on the other side (Figure 5.10a).
Considering the stress–strain relation in Figure 5.10b, if the bar is at point A just prior to buckling, then the flexural compression side loads from A to B with initial modulus equal to the tangent modulus Et, whereas the flexural tension side of the bar unloads from A to C with modulus equal to Es. Thus, the bar behavior is determined by two moduli. This behavior is accounted for in the double (or reduced) modulus approach. Papia and Russo (1989)
and Pantazopoulou (1998) describe applications of this approach to reinforced concrete construction.
FIGURE 5.10 Double modulus buckling model: (a) buckled reinforcing bar; (b) loading and unloading stress–strain relation; (c) bar cross section showing change in stress from point A where buckling initiates.
Figure 5.11 compares the exact relation between tangent (Et) and double modulus (Er) for a circular cross section. In the range of interest, Er ≈ 2.5Et. Thus, the double modulus approach results in values of scritical approximately times greater than obtained by the tangent modulus approach.
FIGURE 5.11 Relation between tangent modulus and reduced (or double) modulus. (After Pantazopoulou, 1998, used with permission from ASCE.)
Several researchers have reported tests on restrained reinforcing bars loaded in compression (e.g., Monti and Nuti, 1992; Rodriguez et al., 1999;
Bae et al., 2005). Figure 5.12 shows sample results for bars with fixed ends loaded monotonically. According to the results shown, a compressed bar
can achieve compressive stress–strain response essentially equivalent to the tensile stress–strain relation if the unrestrained length is s = 5db.
FIGURE 5.12 Stress–strain response versus slenderness ratio: (a) test setup idealization; (b) test results. (After Monti and Nuti, 1992, used with permission from ASCE.)
The onset of buckling can be defined by any of several measures. In laboratory tests, visual observations commonly are used, but such observations are inherently subjective and inaccurate. Preferred approaches include measurement of lateral displacements or measurement of strains on opposing faces of the bar. Using the latter approach, Rodriguez et al. (1999) obtained relations between compressive strain and scritical/db for reinforcing bars satisfying A706 characteristics. Figure 5.13 compares measured data with results calculated using the tangent and double modulus approaches. Both the tangent modulus approach with k = 0.5 and the double modulus approach with k = 0.75 produce results that correlate well with the observed data.
FIGURE 5.13 Longitudinal strain versus critical hoop spacing for monotonic compression. (Data after Rodriguez et al., 1999.)
In most reported axial compression tests on confined concrete columns (see Chapter 4), hoops or crossties support all of the longitudinal bars in the cross section. Current building codes (e.g., ACI 318, NZS 3101), however, permit unsupported longitudinal bars provided that at least alternate bars are supported and no unsupported bar is more than a small distance from supported bars. Axial compression tests on such members show that the unsupported longitudinal bars can buckle over several hoop sets, deforming the perimeter hoop and enabling spalling to progress into the core (Figure 5.14). A conservative approach is to assume that unsupported bars lose all of their compressive strength following cover spalling.
FIGURE 5.14 Failure of cross sections with unsupported longitudinal reinforcement. (After Arteta and Moehle, 2014.)