Seismic performance of ductile corrosion-free reinforced concrete frames

1.   Introduction

Glass-fiber-reinforced-polymer (GFRP) rebars can be utilized as a replacement for steel rebars to prevent deterioration of structures due to corrosion. Additionally, they have the benefits of availability, high strength, lightweight, and resistance to electrical fields ^[1,2,3]. However, the use of GFRP rebars in concrete structures is limited, because of their poor seismic performance ^[4]. Concrete structures with GFRP reinforcement have significantly low energy dissipation and ductility, as compared to steel reinforced concrete (RC) structures. Additionally, their low modulus of elasticity leads to unacceptable deformations.

Many researchers have examined the behaviour of FRP-RC elements ^{[5,6,7,8,9,10]}. Based on experimental tests, the capacity of GFRP-RC columns was found similar to or higher than steel RC columns ^[11,12,13]. Additionally, the axial behavior of circular GFRP RC concrete cores was found similar to steel-confined concrete cores ^[14,15,16]. The experimental results by Li et al. ^[17] showed significant improvement in the performance of GFRP-concrete double tube composite columns compared to the corresponding normal GFRP-confined HSC columns. Halvonik et al. ^[18] reported that beams with GFRP longitudinal bars have a better shear performance per unit increase of the axial stiffness compared to beams with steel longitudinal bars. The numerical investigation by Kueres et al. ^[19] revealed that hybrid steel-GFRP RC sections were more efficient than traditionally steel RC frames in predetermining potential plastic hinges.

To control corrosion in concrete structures, researchers have also investigated the use of stainless steel (SS) as a replacement of regular steel. Other benefits for using SS include better resistance to fire, shock, and impact loads as compared to regular steel ^{[20,21,22,23]}. However, it is not practical to replace all regular steel rebars with SS rebars due to their high cost. Columns with SS dissipated more hysteretic energy at ultimate drift and showed better cyclic performance than the columns with traditional steel ^[24]. Rabi et al. ^[25] used a comprehensive numerical model to develop analytical models to predict the capacity of concrete beams with SS.

The excellent corrosion resistance of SS is due to composing of more than 10.5% of chromium. The physical properties and corrosion resistance of SS surpasses those of traditional steel due to including other metals, such as titanium, manganese, copper, silicon, and vanadium. The chromium forms an invisible corrosion resisting film on the SS steel surface. According to Gardner ^[26], the most common grades for structural applications are the austenitic and duplex grades. The duplex stainless steels offer higher strength, wear resistance and generally corrosion resistance than the austenitic, but at greater expense.

The seismic response of concrete columns, reinforced with SS at the plastic-hinge zones and FRP elsewhere, revealed that SS-RC columns had higher ductility than FRP-SS-RC columns ^[27]. It should be mentioned that the comparison is based on the same amount of reinforcement and concrete dimensions in both FRP-SS-RC and SS-RC columns. The resulting strength and stiffness for the FRP-SS RC columns were significantly less than those of SS-RC columns ^[27]. Youssef et al. ^[28] developed a set of design equations to design SS-GFRP reinforced elements that are equivalent to steel RC elements in terms of capacity, ductility, and stiffness. The developed equations were utilized to design concrete frames and investigate their lateral pushover performance ^[28].

One of the materials that was investigated as an alternative to steel rebars to eliminate corrosion problems and attain self-centering behavior is shape memory alloy (SMA). SMA rebars were found to be a perfect replacement for steel rebars because of their ability to experience inelastic deformations, and then restore their original shape when loads are removed ^[29]. However, such a replacement is not practical due to the high cost of SMA rebars. To overcome this problem, the hybrid use of SMA and steel rebars was investigated ^{[29,30,31,32,33]}. A concrete frame, reinforced with SMA rebars at plastic-hinge regions and steel rebars elsewhere, was found to recover most of its inelastic seismic deformations ^[29]. Youssef and Elfeki ^[33] further reduced the amount of SMA rebars by utilizing them only at the critical sections of the beams adjacent to the critical columns and obtained acceptable seismic performance.

Concrete joints reinforced with SMA rebars at the plastic zones and GFRP rebars elsewhere were found to have high-energy dissipation and insignificant seismic residual deformations ^[34]. Additionally, SMA-GFRP concrete columns were found to have adequate energy dissipation characteristics ^[27]. Youssef et al. ^[35] developed a set of design equations to design SMA-GFRP reinforced elements that are equivalent to steel RC elements in terms of capacity, ductility, and stiffness. The developed equations were utilized to design concrete frames and investigate their lateral pushover performance ^[35].

In the present study, the previously developed design equations ^[28,35] are used to design SS-GFRP and SS-SMA-GFRP RC frames. The seismic behavior of these frames is compared with that of a steel RC frame. The following sections provide details about the design of the frames, material models, finite element model, and the seismic response of the studied frames.

2.   Frame design

Figure 1 shows a steel RC frame (Frame 1) designed by Youssef and Elfeki ^[33]. Reinforcement detailing and concrete dimensions of the beams and columns, which satisfy the strong-column/weak-beam concept, are depicted in Figure 2. In the current study, a redesign of the same frame is conducted using austenitic SS at the beam plastic hinge zones and GFRP bars elsewhere (Frame 2). Frame 3 represents another design option that utilizes superelastic SMA at the plastic hinge regions of the first and fourth floors to minimize seismic residual deformations. These locations were recommended by Youssef and Elfeki ^[33] after examining the seismic performance of RC frames with different configurations of SMA bars. The assumed SMA bars have 55.0% nickel and 45.0% titanium by weight. For the other floors of Frame 3, austenitic SS was utilized at the beam plastic-hinge zones, while GFRP was used elsewhere. The columns of Frames 2 and 3 were reinforced with GFRP rebars, as plastic hinges were designed to develop in the beams. Equation 1 was utilized to estimate the length of SS and SMA rebars ^[33,36,37], which were assumed to be connected to the GFRP rebars using mechanical couplers ^[30,31].

where L is half the span of the beam (in mm), d_b is the diameter of longitudinal bars in the beam (in mm), and f_y is the yield strength of the utilized bars (in MPa).

The design of the SMA-GFRP, SS-GFRP, and GFRP elements focused on having the capacity, ductility, and stiffness of these elements equivalent to steel RC elements, as previously recommended by Youssef et al. ^[28,35]. This was achieved by utilizing the equations proposed by Youssef et al. ^[28,35] (Eqs. 2-7) to estimate the modified section height and modified reinforcement ratio. The height and reinforcement modification factors (F_h and F_r) are given by Eqs. 2 and 3 for GFRP-SS RC beams and Eqs 4 and 5 for GFRP-SMA RC beams. However, the height and reinforcement modification factors for GFRP RC columns are obtained using Eqs 6 and Eq 7. L_SS and L_SMA are lengths of the SS and SMA rebars, respectively. $\rho$ and ${\rho }^{\text{'}}$ are reinforcement ratios of the tensile and compressive reinforcing bars, respectively. E_f, P, and P_max are the GFRP modulus of elasticity, axial load, and axial load capacity. Although, interpolation can be used in case of having values outside the limits of Eqs 2-7. Extrapolation should not be used.

Plastic hinge zones of Beams 1 and 2 in Frame 1 (steel RC frame) have a steel reinforcement ratio (𝜌𝑠𝑡𝑒𝑒𝑙) of 0.77%. The zones outside the plastic hinges are reinforced with 𝜌𝑠𝑡𝑒𝑒𝑙 of 0.44% (Beam 1) and 0.52% (Beam 2). Columns Col 1, Col 2, Col 3, Col 4, and Col 5 have 𝜌𝑠𝑡𝑒𝑒𝑙 of 2.30%, 2.50%, 2.33%, 2.70%, and 1.60%, respectively. The interior columns of the 1^st, 2^nd, 3^rd, 4^th, 5^th, and 6^th stories are subjected to an axial load equal to 34%, 29%, 27%, 21%, 16%, and 12% of their axial load capacities, respectively. The exterior columns are subjected to an axial load representing 30%, 26%, 24%, 20%, 14%, and 9% of their axial capacities. The section heights and reinforcement areas for Frames 2 and 3 are adjusted using Eqs 2-7. The revised heights and reinforcement areas are presented in Table 1. Dimensions and reinforcement detailing of the three frames are shown in Figure 2.

3.   Materials and methods

3.1.   Material models

A uniaxial nonlinear constant confinement model is utilized to model the concrete ^[38,39]. The concrete compressive strength ($f{c}^{\text{'}})$ is 28 MPa. Young's modulus, yield strength, and strain hardening parameter are equal to 200,000 MPa, 400 MPa, and 0.02 for of reinforcing steel rebars, and 190,000 MPa, 400 MPa, and 0.025 for the simplified SS rebars. Figure 3 shows the assumed stress-strain curve for the steel and simplified SS rebars under cyclic loading ^[28]. A detailed stress-strain curve by Ramberg and Osgood ^[40] for SS is also utilized, as shown in Figure 4. This SS model in the most widely used model to simulate the nonlinear stress-strain behaviour for stainless steel ^[41,42]. The yield stress and proof strain are 400 and 0.002, respectively. Due to having a wide range of Ramberg and Osgood parameter (n) from the literature (5 ~ 11), three different values (6, 8, and 10) are adopted to estimate its effect on the results.

The modulus of elasticity (E_f), tensile strength (f_ftu), and ultimate tensile strain (ε_fu) for the transverse GFRP reinforcement are assumed equal to 44,000 MPa, 640 MPa, and 1.45%, respectively, as recommended by Tobbi et al. ^[11]. The tensile strength and strain of the GFRP rebars corresponding to each bar size are shown in Table 2. The modulus of elasticity (E_f) of the longitudinal GFRP reinforcement is 65,000 MPa.

Figure 5 shows the typical stress-strain model of 1D-superelastic SMA ^[43]. The model is defined by the following parameters: austenite to martensite starting stress (f_y-SMA) of 400 MPa, austenite to martensite finishing stress (f_P1) of 510 MPa, martensite to austenite starting stress (f_T1) of 370 MPa, martensite to austenite finishing stress (f_T2) of 130 MPa, superelastic plateau strain length (ε_l) of 6%, and modulus of elasticity (E_a) of 62.5 MPa.

3.2.   Modeling

SeismoStruct ^[46] was used for modelling and analysis of the investigated frames. Each Beam and each column were modeled using six and three displacement-based elements, respectively ^[46]. The spread of inelasticity is captured by dividing the cross section to 200 fibers. This number was selected based on a sensitivity analysis. The fundamental period for the steel-reinforced frame was estimated by dividing the cross section to 100,150,200, and 250 fibers. No significant differences in the results were noted between the 200 and 250 fibers. Therefore, the use of 200 fibers was adopted in the rest of the study. The fundamental periods of Frames 1, 2, and 3 are 0.498 s, 0.529 s, and 0.525 s, respectively.

The nonlinear uniaxial stress-strain response of the individual fibers is integrated to obtain the stress-strain state. Rigid connections between beams and columns are assumed, as the designs were following the current standards. The structural masses are assumed to be lumped at the beam column joints. A time step of 0.001 s is used for the dynamic analysis. The effect of geometrical non-linearity is considered. The investigated frames were analyzed under the selected earthquakes using incremental dynamic analysis with increment of 0.01 g. However, results are only reported at failure at increments of 0.25 g.

Twenty records from three earthquakes (1979 Imperial Valley, 1987 Superstition Hills, and 1989 Loma Prieta) were utilized as per the recommendation of Vamvatsikos and Cornell ^[47]. The characteristics of the 20 records are summarized in Table 3. They cover a wide range of frequencies and durations. Scaled versions of the twenty records were applied to the three frames to define the peak ground acceleration (PGA) causing failure. All records belong to a bin of relatively large magnitudes of 6.5-6.9 and moderate distances. These records also cover a wide range of ground motion frequencies as represented by the ratio between the peak ground acceleration and the peak ground velocity (A/v ratio).

To validate the adequacy of the simplified SS stress-strain curve, Frame 2 has been re-analyzed under the adopted 20 earthquake records using a detailed SS stress-strain by Ramberg and Osgood ^[40]. The response parameters considered in the evaluation are the damage mechanism, the maximum roof drift ratio, the maximum storey drift ratio, the residual roof drift ratio, and the residual storey drift ratio.

3.3.   Failure criteria

In the developed FEM, a failure criterion for confined concrete was applied in the concrete fibers inside the stirrups. While another failure criterion for unconfined concrete outside the stirrups was applied. Additionally, other failure criteria were set on traditional steel, SS, and FRP. The ultimate strain of unconfined concrete, ${\varepsilon }_{u\left(unconfined\right)}^{s}$, is assumed equal to 0.004. The core concrete ultimate strain, ${\varepsilon }_{u\left(confined\right)}^{s}$, for steel confined RC columns is obtained from Eq 8 ^[36]. Ultimate strain, ${\varepsilon }_{u}^{f}$, for columns laterally confined with GFRP bars is obtained from Eq 9, as recommended by Afifi et al. ^[48]. Stress f_P1 is used to define SMA failure. Failure of any frame is assumed to occur when the concrete ultimate strain is reached in all of the same storey columns.

where 𝜌𝑠 is the ratio between volumes of transverse steel reinforcement and concrete core, 𝜀𝑠𝑚 is the strain corresponding to the maximum tensile stress of the confining steel bars, and ${k}_{h}^{s}$ is a confinement factor ^[49].

where 𝑓𝑐𝑜′ is the unconfined concrete strength and 𝑓𝑙𝑒 is the lateral effective pressure. ${k}_{h}^{f}$ is GFRP stirrups confinement factor, which is estimated using Eq 10 ^[13].

4.   Results and discussions

4.1.   Damage mechanism

For each of the considered records, Frames 1, 2, and 3 failed at the same PGA value, shown in Table 4. All elements experienced concrete failure and the strains in the steel, SS, and GFRP bars did not reach their ultimate capacities. Considering Record #16, the locations of core concrete crushing at failure are shown in Figure 6. Similar damage was observed for the remaining records. The number of damaged beam and column sections for the steel RC frame is 25 beam sections and 4 column sections. This number was increased for the SS-GFRP RC frame (30 beam sections and 6 column sections) and was further increased for the SMA-SS-GFRP RC Frame (34 beam sections and 10 column sections). SS did not contribute to this observation as its properties are similar to steel. However, the observed increase in spread of plasticity when using GFRP (Frame 2) and GFRP-SMA (Frame 3) can be attributed to the lower Young's modulus of GFRP and SMA, which required increasing the stiffness of the elements (Table 1). This increase in stiffness resulted in redistributing the bending moments. The increase of the stiffness for elements of Frames 2 and 3 occurred due to using height factors, which have values higher than one. This increase in the stiffness is not equal for all members, as shown in Table 1. Thus, the distribution of moments in the elements of the corrosion-free frames was different than that of the steel-reinforced frame. Such increase in plastic hinge locations is desirable as it means reduced-damage state per hinge.

One can observe that PGA at failure for Frames 1, 2, and 3 were relatively similar for each earthquake record (maximum potential difference is less than 0.01 g, which is the used increment for incremental dynamic analysis). This have occurred because the developed procedures by Youssef et al. ^[28,35] provided cross sections and reinforcement for beams and columns of Frames 2 and 3 with approximately similar ductility, stiffness, and strength of the steel-reinforced elements of Frame 1.

4.2.   Seismic drifts

An Incremental dynamic analysis (IDA) is performed by obtaining the PGA at failure for each frame under the time history of each earthquake record. Then, the analysis was repeated for each frame under 0.25, 0.5, and 0.75 of the PGA at failure under each earthquake history. This procedure is followed to observe the effect of increasing the earthquake PGA on the behavior of the investigated frames. Figure 7 presents the mean maximum roof drift ratio (MRDR) and maximum storey drift ratio (MSDR) obtained from IDA at 0.25, 0.5, 0.75, and 1.0 of PGA at failure. The maximum difference between the values of the MRDR and MSDR for Frames 1, 2, and 3 at a specific PGA does not exceed 10%. The "mean" and "mean-plus-twice the standard deviation" MRDR and MSDR for Frames 1, 2, and 3 at failure are presented in Table 5. It is clear that the three frames experienced similar roof drifts. This observation proves the adequacy of using the equations proposed by Youssef et al. ^[28,35] to achieve ductile and corrosion-free frames. The utilization of these equations has clearly eliminated the problem of high MRDR and MSDR, which was associated with the low young's modulus for the GFRP and SMA bars, as observed by many researchers (Billah and Alam ^[27], Alam et al. ^[29], Youssef and Elfeki ^[33]).

Using the detailed SS stress-strain curve by Ramberg and Osgood ^[40] instead of the simplified SS curve resulted in a change in the MRDR and MSDR at failure depending on the Ramberg and Osgood parameter, as shown in Table 6. The MRDR changed with −1.95 ~ +2.02% while the MSDR changed with −8.85 ~ +8.96%.

It is worth mentioning that MSDR at failure for the three frames occurred either in Storey 1 or Storey 4 under the 20 earthquake records. This observation proves the adequacy of using the equations proposed by Youssef et al. ^[28,35] to achieve locations of maximum drifts similar to those obtained from traditionally steel reinforced frames.

The shown seismic performance is not intended to comply with the standards. Rather, the drift limits were pushed higher to fully understand the behaviour of the frames. All frames including the steel RC frame experienced high drift ratios. The expected high MRDR and MSDR for Frame 3 have been eliminated as they were similar to the steel RC frame regardless of the low modulus of elasticity for SMA and FRP.

4.3.   Seismic residual drifts

Figure 8 presents the residual roof drift ratio (RRDR) and residual storey drift ratio (RSDR) obtained from IDA at 0.25, 0.5, 0.75, and 1.0 of PGA at failure. The variations of the RRDR and RSDR for Frames 1 and 2 are found following the same trend, with a maximum difference in RRDR and RSDR values at a specific PGA of about 30%. However, the RRDR and RSDR values of Frame 3 are observed to be significantly lower than those of Frames 2 and 3 at all PGA values, with a reduction reaching about 90% in RRDR and 60% in RSDR at failure. The "mean" and "mean-plus-twice the standard deviation" RRDR and RSDR values for Frames 1, 2, and 3 at failure are shown in Table 7. The significant reductions in RRDR for Frame 3 are clearly due to the re-centering characteristic of the SMA rebars.

Using the detailed SS stress-strain curve by Ramberg and Osgood ^[40] instead of the simplified SS curve resulted in a change in the RRDR and RSDR at failure depending on the Ramberg and Osgood parameter, as shown in Table 8 The RRDR changed with −6.98 ~ +7.11% while the RSDR changed with −9.12 ~ +8.89%.

5.   Conclusions

Three frames are seismically analyzed using twenty different earthquake records. Frame 1 represents a typical steel reinforced concrete frame. In Frame 2, the steel rebars at the plastic hinge regions of the beams were replaced with stainless steel rebars. In Frame 3, the stainless steel rebars of the first and fourth stories were replaced with SMA rebars. The remaining length of the beams and the full height of the columns of Frames 2 and 3 were reinforced with GFRP rebars. The design of Frames 2 and 3 utilized the equations, previously proposed by the authors ^[28,35], which guarantee achieving elements having the same stiffness, ductility, and strength as steel RC elements. The damage mechanism, maximum roof drift ratio, residual roof drift ratio as well as maximum storey drift ratio of Frame 1 are compared to those of Frames 2 and 3. The following conclusions can be drawn from the conduct analyses.

1. Failure PGA values for the three frames were the same for the considered 20 ground motions.

2. Maximum and residual drift ratios for the steel RC frame and SS-GFRP RC frame were similar, indicating that the use of the equations previously proposed by the authors (Youssef et al. ^[28,35]) has eliminated the high drift ratios associated with the use of GFRP bars.

3. Maximum drift ratios for the steel RC frame and SS-SMA-GFRP RC frame were similar. However, the SS-SMA-GFRP RC frame has much reduced residual deformations. The equations previously proposed by the authors (Youssef et al. ^[28,35]) has led to a self-centering ductile corrosion-free reinforced concrete frame.

It should be noted that the mentioned conclusions are limited to the studied frame where the final response was evaluated using PGA at failure. Additional investigations for the response in terms of fragility curves are required to generalize these conclusions. It is worth mentioning that the current study aimed at examining the 2D behaviour of the designed frames and can be extended to consider 3D analysis.

Acknowledgments

The authors are grateful for the financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.