Plastic Hinge Length of Reinforced Concrete Columns

INTRODUCTION

A reasonable estimation of the plastic hinge length is the key to successfully modeling the lateral load-drift response and conducting a proper seismic performance evaluation of the precast seismic force-resisting systems (SFRS). Plastic hinges form at the maximum moment region of reinforced concrete columns. Determining the plastic hinge length is a critical step in predicting the lateral load-drift response of columns. As it is difficult to calculate the plastic hinge length using sophisticated computer programs, it is often calculated based on experimental data or using empirical expressions. Several factors influence the plastic hinge length, such as (1) the diameter (db), (2) yield strength of longitudinal reinforcement (fy), (3) the section size (h), (4) level of axial load; (5) moment gradient; (6) the value of shear stress in the plastic hinge region; (7) the amount and mechanical properties of longitudinal and transverse reinforcement; (6) strength of concrete; and (7) level of confinement provided in the potential plastic hinge zone.

?Currently Available Expressions for Plastic Hinge Length

?Some of the widely adopted empirical expressions are shown in Table 1. The simplified expressions available in the literature do not contain all or most of the aforementioned factors. Hence, large variations exist in the value of plastic hinge length calculated using these empirical equations (Bae and Bayrak, 2008), It may be of interest to note that the plastic hinge length (Lp) usually ranges from 0.25h to 1.0h. Paulay and Priestley (1992) reported that, for all practical columns the value of plastic hinge length will be approximately 0.5h.

In above Table 1, Ag = Gross-area of concrete section, As = Area of longitudinal reinforcement, d = Effective depth, db = Diameter of longitudinal reinforcement, Ec = Modulus of elasticity of concrete, ?= Cylinder compressive strength of concrete, fy = yield strength of reinforcement, Gf = Fracture energy of concrete under compression, h = Overall depth of member section, P = Applied axial force, P0 = Nominal axial strength,? z = distance from critical section to point of contraflexure, L = Distance from column tip to foundation, ?= Peak compressive strain and ?= compressive strain at 20% of concrete compressive strength.

The equations suggested by Subramanian (2009) compare favorably with the measured values reported in the paper by Bae and Bayrak (2008). Bae and Bayrak (2008) identified (based on their experiments) the fact that the required length lo at the bottom of the column (over which closely-spaced reinforcement has to be provided) has to be increased from 1.0h (as per ACI 318-05) to a minimum of 1.5h. As per Clause R17.10.2 ?of current ACI 318-19, the plastic hinge zones are considered to extend a distance equal to twice the member depth from any column or beam face and also include any other sections in walls, frames, and slabs where yielding of reinforcement is likely to occur as a result of lateral displacements. It is interesting to note that the plastic hinge length for structural walls (Eq. (A.10.2a) and (A.10.2b) of ACI 318-19) have been adopted from Paulay and Priestley (1992).

?PRECAST CONSTRUCTION

?One of the precast column connections, which has been recently used in connection with accelerated bridge construction (ABC), utilizes mechanical bar splices (commonly referred to as couplers). For ABC, couplers have to be generally used in the plastic hinge region of precast columns. This would violate the current bridge seismic design codes, which do not allow the use of couplers in plastic hinge regions of ductile members.

?The existing empirical expressions of plastic hinge length need to be modified to reflect the effects of relatively large diameters and correspondingly high axial rigidity of couplers used to connect two adjacent reinforcements. These couplers barely deform under tension due to their relatively higher local axial stiffness, resulting in reduced curvatures in the splice region, as shown in Fig. 1(a). The reduced curvature could lead to a shorter plastic hinge length (Zhang and Lee, 2024). In addition, the effects of couplers on lateral drift behavior and plastic hinges are more complex due to the residual slip induced between couplers and connecting reinforcements. Tazarv and Saiidi (2016) recommended a reduced plastic hinge length (Lpsp) for the precast connection with couplers as below (see Figure at top):

?Where the term (1 – Hsp/Lp) reflects the location effect of couplers (Hsp and Lsp are shown in Fig. 1(a)).?The value of β has to be determined based on experiments- the rigid length factor (β) is quite low for the test specimens reported in Guan et al., 2021(see Fig. 2(b)). The effect of coupler location is explained in Fig. 1(b) to 1(e) for better understanding. When there is no splice, the plastic hinge will form at the base of the column, as shown in Fig. 1(b). If the reinforcements are mechanically spliced using couplers, as shown in Fig. 1(c), strain concentration occurs, leading to reduced strain above the couplers. As a result, the plastic hinge length above the couplers is necessarily reduced. If the location of couplers is moved up to some distance from the base of the column, but still within the range of base plastic hinge length (Lp), as shown in Fig. 1(d), full plastic hinge length can’t develop. Due to this, the plastic hinge length with couplers (Lpsp) will be lower than the base length of the plastic hinge (Lp). If the couplers are located beyond the base length of the plastic hinge (Lp), then the plastic hinge can be fully developed. In this case, the plastic hinge length with couplers (Lpsp) will be the same as that of the base length plastic hinge (Lp), as shown in Fig. 1(e).

?Recently, Zhang and Lee (2024) recommended that the couplers should be kept away from the base of the column at a distance equal to the plastic hinge length so that full development of the plastic hinge is achieved also in mechanically spliced precast connections. If the couplers are designed immediately next to the critical section, small-size couplers should be used to mitigate the detrimental effect of locally higher stiffness. In addition, the relative slip of couplers with rebars should be strictly limited to 0.3 mm. Zhang and Lee (2024) also found that the expression for plastic hinge length by Paulay and Priestley (1992) matched with the test and modeling results of their study.

SUMMARY AND CONCLUSIONS

Plastic hinges form at the maximum moment region of reinforced concrete columns. Determining the length of the plastic hinge length is a critical step in predicting the lateral load-drift response of columns. As it is difficult to calculate the plastic hinge length using sophisticated computer programs, it is often calculated based on experimental data or using empirical equations. Different authors have suggested expressions for calculating the plastic hinge length. Several factors influence the length of plastic hinges. The expressions proposed by Bae and Bayrak (2008) and Subramanian (2009) consider many of these parameters. As per ACI 318-19 the plastic hinge zones are considered to extend a distance equal to twice the member depth from any column or beam face. ACI 318-19 adopted the expression of Paulay and Priestley (1992) for the plastic hinge length for structural walls.

However, most of the existing expressions are not directly applicable to estimate the plastic hinge length of precast connections with couplers. Precast column connections may utilize couplers in the plastic hinge region of precast columns, thus, violating the current bridge seismic design codes. These couplers barely deform under tension due to their relatively higher local axial stiffness, resulting in reduced curvatures in the splice region-leading to shorter plastic hinge length. An expression has been suggested in the literature for calculating this shorter plastic hinge length. Anyhow, it is recommended that the mechanical splice should be kept away from the base of the column at a distance equal to the plastic hinge length (as in Fig.1(e)) so that full development of plastic hinge is achieved in mechanically spliced precast connections also.

References

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15.? Tazarv, M., and Saiidi, M. S. (2016) “Seismic Design of Bridge Columns Incorporating Mechanical Bar Splices in Plastic Hinge Regions,” Engineering Structures, V. 124, 2016, pp. 507-520. doi: 10.1016/j.engstruct.2016.06.041

16.? Guan, D.; Chen, Z.; Liu, J.; Lin, Z.; and Guo, Z. (2021) “Seismic Performance of Precast Concrete Columns with Prefabricated UHPC Jackets in Plastic Hinge Zone,” Engineering Structures, V. 245,? p. 112776 doi:10.1016/j.engstruct.2021.112776

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