Plastic hinge length of shape memory alloy (SMA) reinforced concrete bridge pier

Engineering Structures 117 (2016) 321–331

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Plastic hinge length of shape memory alloy (SMA) reinforced concrete bridge pier A.H.M. Muntasir Billah, M. Shahria Alam ⇑ School of Engineering, The University of British Columbia, Kelowna, BC V1V1V7, Canada

a r t i c l e

i n f o

Article history: Received 13 June 2015 Revised 26 February 2016 Accepted 28 February 2016

Keywords: Shape memory alloy (SMA) Bridge pier Plastic hinge length Curvature Deformation capacity Strain

a b s t r a c t It is often assumed that the maximum seismic damage in a bridge pier will concentrate in the regions subjected to maximum inelastic curvature known as its plastic hinge length. Predicting the plastic hinge length accurately is an important part of seismic design of bridge piers. This study focuses on deriving an analytical expression for the plastic hinge length of shape memory alloy (SMA) reinforced concrete (RC) bridge pier based on the results from well calibrated nonlinear finite element models. A parametric study was performed to investigate the effect of different parameters on the plastic hinge length, including axial load ratio, aspect ratio, concrete strength, SMA properties, longitudinal and transverse reinforcement ratio. Multivariate regression analysis was performed to develop an expression to estimate the plastic hinge length in SMA-RC bridge pier. The results are compared with the existing plastic hinge length equations. The proposed equation was validated against test results which showed reasonable accuracy. Crown Copyright Ó 2016 Published by Elsevier Ltd. All rights reserved.

1. Introduction Shape memory alloys (SMAs) have been emerging as an alternative to conventional steel reinforcement in concrete structures due to its distinct shape recovery and superelastic properties. Numerous applications of SMAs in bridges such as dampers [1], isolators [2,3], restrainers [4,5], expansion joints [6] and reinforcement [7,8] have been investigated by different researchers. Considering the importance of bridge pier, it is necessary to predict the displacement capacity of bridge piers during earthquakes. Past research works have shown that SMA could significantly improve the seismic performance of bridge piers through recentering thereby significantly reducing the permanent damage. Previous researchers mostly used the Paulay and Priestely [9] equation for calculating the plastic hinge length in SMA-RC bridge pier and reported that this equation provides a reasonable estimate of the plastic hinge of SMA-RC pier. However, for seismic design of SMA-RC pier, it is necessary to identify the plastic hinge length of the pier which can be used for calculating the flexural displacement capacity. Plastic hinging regions indicate the area of concentrated damage for bridge piers that experience inelastic deformations [10]. Therefore, a consistent prediction of a plastic hinge length is also necessary to evaluate the length of the pier that needs to be rein-

⇑ Corresponding author. Tel.: +1 250 807 9397; fax: +1 250 807 9850. E-mail addresses: [email protected] (A.H.M.M Billah), shahria. [email protected] (M. Shahria Alam). http://dx.doi.org/10.1016/j.engstruct.2016.02.050 0141-0296/Crown Copyright Ó 2016 Published by Elsevier Ltd. All rights reserved.

forced with SMA and confined along the critical section. Most of the previous studies on plastic hinge length focused on beams and columns [9–13] where only a few studies were conducted for bridge piers [14,15]. A review of existing plastic hinge equations showed that the plastic hinge length of a bridge pier depends on many factors such as mechanical properties of longitudinal and transverse reinforcement, concrete strength, level of axial load, aspect ratio, reinforcement ratio, and level of confinement. Since the mechanical properties of SMA and its behavior under lateral load are significantly different from conventional RC piers, it warrants a specific plastic hinge expression for SMA-RC bridge pier. Although researchers have investigated the seismic performance of bridge piers considering different types of SMA [16–18], only one study [19] has been conducted so far to estimate the plastic hinge length in SMA reinforced concrete (RC) bridge pier. However, their proposed equation does not consider the effect of different parameters and could estimate the plastic hinge length with 11.6% error. This study adopted a numerical method to develop a plastic hinge length expression for SMA-RC bridge pier due to the absence of adequate experimental results and limitations in conducting experiments due to high cost of SMA. Using a well-calibrated finite element model, this study developed a plastic hinge length expression for SMA-RC bridge pier by investigating the distribution of curvature and strain in the longitudinal rebar (both steel and SMA rebar) along the height of the pier. Considering different parameters such as the level of axial load, aspect ratio, concrete

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strength, SMA properties and ratio of longitudinal and transverse reinforcement, a parametric study was conducted to derive a plastic hinge length expression for SMA-RC bridge pier. Finally, the proposed equation was used to estimate the drift capacity of SMA-RC bridge pier and compared with test results. Compared to conventional bridge pier, behavior of SMA-RC bridge pier is significantly different and governed by the distinct superelastic and thermo-mechanical properties of SMA. Estimating the plastic hinge length is a major step in predicting the load-drift response of a bridge pier. In order to limit the use of SMA rebar only in the plastic hinge region (i.e. to confine damages within the region that will eventually recover), the proposed equation will help determine the amount of SMA reinforcement to be used in the SMA-RC bridge pier.

2. Design and geometry of bridge pier This section briefly describes the design and configurations of different SMA-RC bridge piers used in this study. Since SMA is a costly material it is only used in the bottom plastic hinge region of the bridge pier. The bridge pier is assumed to be located in Vancouver, BC and was seismically designed following Canadian Highway Bridge Design Code [20]. Fig. 1 shows the cross section of the column. The diameter of all the columns was fixed to be 1.524 m. Several parameters govern the design and behavior of bridge piers. These parameters also affect the spread of plasticity along the length of the pier. The primary variables of the parametric study were selected as aspect ratio (L/d), axial load ratio (P/fc0 Ag), longitudinal reinforcement ratio (ql), transverse reinforcement ratio (qs), yield strength of SMA rebar (Fy-SMA) and concrete compressive strength (fc0 ). These parameters were selected based on existing literature on plastic hinge length of reinforced concrete elements [9,14,15,21–23]. Table 1 shows the list of considered parameters and their associated values. For each parameter three different values were considered. Table 2 shows the summary of the SMA-RC pier specimens analyzed in this study. A total of 18 piers were designed. In order to investigate the effect of different parameters on the plastic hinge length of SMA-RC pier, one parameter at a time was varied and others were kept constant. Apart from the investigated parameter, the plastic hinge length of the piers was also varied and three different plastic hinge lengths were considered: 0.5 LP/d, 0.75 LP/d and 1LP/d. These three lengths were selected as previous studies on SMA-RC bridge piers [19,24] showed that the plastic hinge length varies from 0.5 LP/d to 1.1 LP/d. All the bridge piers

Table 1 Details of variable parameters. Parameters

Values

Axial load (%) ql (%) Aspect ratio (L/d) fc0 (MPa) qs (%) fy-SMA (MPa)

5 1 3 35 0.8 210

10 2 5 50 1 450

20 3 7 60 1.2 750

were designed following Canadian Highway Bridge Design Code [20] meeting current seismic design criteria. The diameter and number of longitudinal reinforcement of different bridge piers were varied for different reinforcement percentages and 15 M (16 mm) spirals were used at different spacing as lateral reinforcement. In this study, in order to ensure flexure dominated behavior and avoid shear failure, three different aspect ratios (3, 5, 7) were considered. 3. Numerical modeling One of the main objectives of this study was to develop a fiberbased numerical model capable of predicting the nonlinear behavior in terms of strain and curvature distribution of SMA-RC bridge piers. The modeling and nonlinear analyses of SMA-RC bridge piers were conducted using fiber element based nonlinear analysis program SeismoStruct [25]. Using force based inelastic beam-column element, the circular bridge piers were modeled. In this study force based element type is selected since it is the most accurate among the four inelastic frame element types available in SeismoStruct and is capable of capturing the inelastic behavior along the entire length of a structural member. However, the mesh refinement issue (number of integration section and number of elements) was considered in order to ensure an optimum balance between accuracy, numerical stability and analysis’ run times. It was found that 5 integration sections and 250 fibers across the section provided reasonable accuracy with experimental results, computational stability and convenient computational time. The elements cross sections were represented by assemblages of longitudinally oriented, unidirectional steel and concrete fibers. Each fiber is associated with a prescribed stress-strain relationship that can be specified to represent the unconfined concrete, confined concrete and longitudinal reinforcement. The sectional stress-strain state of the inelastic beam-column elements is then obtained through the

Fig. 1. Geometry of SMA-RC bridge pier (a) Cross section, (b) Elevation and (c) Finite element modeling.

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A.H.M.M. Billah, M. Shahria Alam / Engineering Structures 117 (2016) 321–331 Table 2 Details of SMA-RC bridge piers. Variable

Pier

P/fc0 Ag

H (m)

fc0 (MPa)

ql (%)

fy-SMA (MPa)

Lp (m)

qs (%)

Axial load

P1-1 P1-2 P1-3 P2-1 P2-2 P2-3 P3-1 P3-2 P3-3 P4-1 P4-2 P4-3 P5-1 P5-2 P5-3 P6-1 P6-2 P6-3

0.05 0.1 0.2 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

7.62 7.62 7.62 4.572 7.62 10.668 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62

35 35 35 35 35 35 35 35 35 35 35 35 35 50 60 35 35 35

1 1 1 1 1 1 1 1 1 1 2 3 1 1 1 1 1 1

401 401 401 401 401 401 210 401 750 401 401 401 401 401 401 401 401 401

0.762 1.143 1.524 0.762 1.143 1.524 0.762 1.143 1.524 0.762 1.143 1.524 0.762 1.143 1.524 0.762 1.143 1.524

1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 0.8 1 1.2

Aspect ratio

SMA fy

ql (%) fc 0

qs (%)

integration of the nonlinear uniaxial stress–strain of the individual fibers in which the section has been subdivided. Hilber–Hughes– Taylor integration scheme [26] was utilized to acquire the global flexibility matrix for the Force-based (FB) elements; considering the potential strain softening or localized deformations phenomena. A total of 11 elements were used to model the SMA-RC bridge piers of which bottom three elements represent the plastic hinge length (SMA-RC section) and the remaining eight elements represent the steel-RC section. Fig. 2 shows the idealized bridge pier model with element number and the fiber discretization of the column section. The Mander et al. [27] concrete constitutive model was used to describe the confined and unconfined concrete and the steel reinforcement was represented using the Menegotto–Pinto [28] steel

model. This steel model takes into account the Bauschinger effect, which is relevant for the representation of the columns’ stiffness degradation under cyclic loading. The details of the parameters used to define the concrete and steel constitutive models are presented in Table 3. The superelastic SMA was modeled following the constitutive relation developed by Auricchio and Sacco [29] using the parameters provided in Table 4. Mechanical couplers were used to connect SMA with steel rebars [30] which is represented by introducing a zero length rotational spring at the bottom of the column section (Fig. 1c). The rotational spring was modeled by defining a link element in SeismoStruct [25]. The link element connects two initially coincident structural nodes, and requires the definition of an independent force-displacement (or momentrotation) response curve for each of its local six degrees-

Cover Concrete

Reinforcement

Core Concrete 250 Fibers and 5 Integration Sections

Fig. 2. Idealized numerical bridge pier model and fiber discretized cross section.

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Table 3 Modeling parameters used for concrete and steel constitutive model. Material

Property

Concrete

Compressive strength (MPa) Corresponding strain Tensile strength (MPa) Elastic modulus (GPa) Elastic modulus (GPa) Yield stress (MPa) Ultimate stress (MPa) Ultimate strain Plateau strain Strain Hardening Parameter Transition curve initial shape parameter Transition curve shape calibrating coefficient (A1) Transition curve shape calibrating coefficient (A2)

Steel

35 0.0029 3.5 23.1 200 475 692 0.14 0.016 0.0125 19.5 18.5 0.15

of-freedom (F1, F2, F3, M1, M2, M3). In order to define the link element, modified Takeda hysteresis curve was adopted in this study to define the moment-rotation relationship of link element. Five parameters are necessary to characterise the behavior of response curve which are, the yielding strength Fy (150 kN); the initial stiffness Ky (42,750 kN m); the post yielding to initial stiffness ratio a (0.0197); the outer loop stiffness degradation factor b0; and the inner loop stiffness degradation factor b1. Different parameters required for describing the response curve were obtained from the moment-rotation relationship of SMA-RC pier, and the stiffness degradation factors b0 and b1 were specified as 0.1 and 0.9, respectively, as to make the unloading stiffness close to the initial stiffness during the cyclic loading. Moment-rotation relationship of the bar-slip rotational spring is derived based on the calibration procedures described in [31]. More details about the stress-slip relationship of bars inside the coupler and the details of the splicing can be found elsewhere [31]. The rotational spring was provided at the bottom of the column since couplers are provided at the column—foundation interface (beginning of plastic hinge) and at the end of plastic hinge length. The rotational spring used in the model is an idealized condition that represents the slippage of SMA rebar inside the couplers. The authors believe that the modeling technique adopted in this study can be extended to other

cases. Similar modeling approach has also been used by other researchers and the authors have also validated their models with experimental results which proved the accuracy of adopted modeling technique.

4. Model validation The accuracy of the adopted finite element modeling program in predicting the seismic response of bridge structures has been demonstrated by several researchers through comparisons with experimental results [32,33]. However, in order to investigate the accuracy of the modeling technique in predicting the strain and curvature distribution, comparisons were made with experimental results of rectangular and circular SMA-RC bridge piers. Nakashoji and Saiidi [19] conducted experimental investigation on SMA-RC bridge piers and extensive measurements of rebar strains were made along the height of the pier. Specimen SR-99 LSE was a square column having an 18 in [457 mm] square cross section and a height of 62 in. [1575 mm]. The plastic hinge length (18 in or 457 mm) of the specimen was reinforced with 16–0.5 in [12.7 mm] diameter Ni–Ti SMA rebar and the remaining portion was reinforced with 16-#5 steel rebar. The vertical strains measured over a 20 inch [508 mm] gauge length from the base of Specimen SR-99 LSE are shown in Fig. 3a at two drift levels: 1% and 2% for strain gauges 2, 8, 18, 28, and 38. The predicted SMA rebar strains at 1% and 2% drift are also shown in Fig. 3a. Observation from Fig. 3a shows that, there is good agreement between the measured and predicted strains. From Fig. 3a it is evident that the analytical model was also able to predict the nonlinear strain profile observed from the experiment. This comparison shows that the local response of SMA-RC bridge pier can be determined satisfactorily with the adopted nonlinear finite-element modeling technique. Since this study used both rebar strain and curvature profile to predict the plastic hinge length of SMA-RC bridge pier, the ability of the adopted modeling technique in accurately predicting the curvature distribution was also investigated. O’Brien et al. [24] investigated the performance of a 1/5-scale circular SMA-RC bridge

Table 4 Properties of different types of SMA. Alloy

es (%)

E (GPa)

fy (MPa)

fp1 (MPa)

fT1 (MPa)

fT2 (MPa)

fy/E

Reference

NiTi45 FeNCATB CuAlMn

6 13.5 9

62.5 46.9 28

401.0 750 210.0

510 1200 275.0

370 300 200

130 200 150

0.0065 0.0159 0.0075

Alam et al. [32] Tanaka et al. [40] Shrestha et al. [41]

fy (austenite to martensite starting stress); fP1(austenite to martensite finishing stress); fT1 (martensite to austenite starting stress); fT2 (martensite to austenite finishing stress), es (superelastic plateau strain length); and E (modulus of elasticity).

(a)

1% drift (experiment) 2% drift (experiment) 1% drift (predicted) 2% drift (predicted)

Height (inch)

20 15 10

16

(b)

14

Height (inch)

25

1.5 % drift (experiment) 3% drift (experiment) 1.5% drift (predicted) 3% drift (predicted)

12 10 8 6 4

5

2 0

0 0

4000

8000

Strain (μ)

12000

0

0.002

0.004

0.006

Curvature (rad/inch)

Fig. 3. (a) Comparison of predicted and measured strain on SMA rebar [19] and (b) comparison of predicted and measured curvature [24].

A.H.M.M. Billah, M. Shahria Alam / Engineering Structures 117 (2016) 321–331

pier having a diameter of 254 mm and the height of the pier was 1143 mm. The bridge pier was reinforced with 15.9 mm diameter Ni–Ti SMA in the plastic hinge region. They tested the pier under reverse cyclic loading and measured the curvature distribution over a 355.6 mm gauge length from the base of Specimen RNC. Fig. 3b shows the comparison of the measured and predicted curvature at two different drift levels: 1.5% and 3% over the height of the specimen. From Fig. 3b it can be observed that, the profile of the curvature distribution predicted along the length of the pier not only matches closely to the measured response, but also mimics the trend in the curvature profile along the section. In the physical experiment (O’Brien et al. [24]), curvature was measured using displacement transducer. The transducer provided strain in the concrete by measuring the changes in length over the gauge length of 102 mm. The measured concrete strains on opposite sides of the column were added together and then divided by the horizontal distance between the transducers to obtain the curvature at the column cross section. This calculation was done for each layer of displacement transducers. The curvatures in the bottom two layers (the approximate plastic hinge length) were combined to better represent the curvature in the plastic hinge area. O’Brien et al. [24] represented the curvature distribution as a vertex of a broken line. In order to compare the results the curvature distribution are presented in the same pattern in this paper. In the numerical model, the curvature profile along the height of the pier was obtained directly from the numerical model. In the numerical model developed in Seismostruct, it is possible to define the elements for which the curvature output is necessary. In the preprocessor, different element groups were defined for generating the curvature output along the height of the pier. After the analysis, the curvature distribution at different integration sections were obtained and plotted against the height of the pier.

5. Approach for predicting plastic hinge length Accurate estimation of plastic hinge lengths in RC bridge piers using analytical approach can be complicated. Typically plastic hinge lengths are calculated using experimental results. However, several researchers [21,23] derived plastic hinge lengths of RC elements using analytical approach based on strain and curvature. This study adopted a numerical approach for deriving an expression for plastic hinge length of SMA-RC pier as there is lack of adequate test results. Two different measures, namely, the longitudinal rebar compressive strain profile and the curvature profile along the height of the pier were used to calculate the plastic hinge length of SMA-RC bridge pier. During an earthquake, bridge piers are subjected to lateral displacements while supporting gravity loads and the plastic hinges usually form at the maximum moment region. This inelastic portion causes a significant increase in inelastic curvature near the base of the bridge pier and forms the plastic hinge zone. As the curvature increases, the compression side of the member experiences increased strain and subsequently reaches a critical value when the concrete cover spalls off. After that, the longitudinal bars on the compression side experience yielding and subsequently core concrete starts to crush. Under increasing compressive strain, damage starts to accumulate and forms plastic hinges. The compressive strain in the longitudinal rebar is equal to the compressive strain in the outer core concrete fiber. Therefore, a rebar compressive strain profile along the height should give a clear indication on the formation of the plastic hinge. Similar approach was used by Bae and Bayrak [21] in predicting the plastic hinge length of reinforced concrete columns. In this study, the SMA-RC bridge piers were analyzed under reverse cyclic loading and the compressive strain profiles in the longitudinal rebar were plotted. By tracking

325

the yielding of longitudinal rebar in compression, the most damaged area i.e. the plastic hinge was identified. This study also used the curvature profile along the height of the pier to determine the plastic hinge length. After analyzing the bridge pier under reverse cyclic loading, the curvature profile of the piers were plotted to identify the zone where inelastic curvatures are localized. By tracking the yield curvature in the curvature profile, the plastic hinge was identified. The following section describes the effect of different parameters on the plastic hinge length of SMA-RC bridge pier. 5.1. Effect of axial load Several researchers [21,23] have considered axial load level as an important parameter for plastic hinge estimation of RC columns. However, researchers have reported contradictory conclusions regarding the effect of axial load. Mendis [34] and Park et al. [35] reported that the level of axial load does not have any influence on plastic hinge lengths. However, Tanaka and Park [36] and Légeron and Paultre [37] found that as the axial load increases the plastic hinge length increase. Except Berry et al. [38], most of the researchers considered very high levels of axial load which are unusual for bridge piers and most of them were for columns in a frame structure. In this study, three different axial load levels were considered to study the effect of axial load on the plastic hinge length. The ranges of axial loads (5%, 10% and 20%) were selected based on design codes or common practices. Keeping the other parameters constant, the piers were analyzed under reverse cyclic loading. Fig. 4 shows the variation of rebar compressive strain and curvature profile along the height of the pier. From Fig. 4a it is evident that the curvature profiles do not show any effect of axial load on the plastic hinge length. However, the compressive strain profile, as shown in Fig. 4b, clearly depicts the effect of increasing axial load on the compressive strain in the longitudinal reinforcement. It is evident from Fig. 4b that with the increase in axial load, the plastic hinge length increases. The strain profile in the significantly damaged zone drastically changes with the axial load as identified in the plastic hinge region. Yield strain of longitudinal rebar was used to determine the plastic hinge length. For different level of axial load the plastic hinge length varied between 0.78d to 1.18d where d is the diameter of the pier. 5.2. Effect of aspect ratio Previous researchers [11–13,34] identified that the plastic hinge length of a RC member is influenced by the aspect ratio (L/d). However, the widely used plastic hinge length equation proposed by Paulay and Priestley [9] does not account for the effect of aspect ratio. In order to investigate the influence of the aspect ratio on the plastic hinge length, circular SMA-RC piers with varying aspect ratios (3, 5, and 7) were considered keeping other parameters constant. The results of the analyses are summarized in Fig. 5. As can be observed in the curvature profile (Fig. 5a), plastic hinge length is independent of the aspect ratio of the pier. However, the plastic hinge length increases with the increasing aspect ratios as evident from the strain profile (Fig. 5b). As the aspect ratio increased from 3 to 7, the plastic hinge lengths were found to increase from 0.82d to 1.25d. Bae and Bayrak [21] and Alemdar [15] also reported that the lp increases with the increasing L/d for a given axial load level. Bae and Bayrak [21] found that the effect of change in aspect ratio is less pronounced in columns with small aspect ratio (2 < l/d < 3) as compared to columns having larger aspect ratio. They also concluded that the changes in plastic hinge length with increasing aspect ratio are insignificant for columns under low axial load. However, in this study it was found that aspect ratio contributes to the plastic hinge zone in SMA-RC bridge pier.

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9

(a)

8

Distance from base (m)

Distance from base (m)

9

0.2 Po 0.1 Po 0.05 Po

7 6 5 4 3 2 1 0

(b)

8

0.2 Po 0.1 Po 0.05 Po

7 6 5 4 3

εy-sma=0.0064

2 1 0

0

0.02

0.04

0.06

0.08

0.1

0

0.005

Curvature (1/m)

0.01

0.015

0.02

0.025

0.03

Longitudinal rebar strain (εs)

Fig. 4. Effect of axial load on (a) curvature profile and (b) longitudinal rebar strain profile.

12

(a)

AR-3

10

Distance from base (m)

Distance from base (m)

12

AR-5 AR-7

8 6 4 2

(b)

AR-3

10

AR-5 8

AR-7

6 4

εy-sma =0.0064 2 0

0 0

0.02

0.04

0.06

0.08

0

0.1

0.005

0.01

0.015

0.02

0.025

0.03

Longitudinal rebar strain (εs)

Curvature (1/m)

Fig. 5. Effect of axial load on (a) curvature profile and (b) longitudinal rebar strain profile.

5.3. Effect of SMA properties Since SMA possesses significantly different mechanical properties than conventional steel, it might affect the plastic hinge formation in the SMA reinforced bridge pier. In addition, several compositions of SMAs have been developed which have potential for application in bridge pier such as Ni–Ti, Fe-based and Cu-based. Most of the applications have been focusing on the use of Ni–Ti alloy while very few focused on the application of the alloys such as Cu-based SMAs [16,39], Fe- based SMAs [3]. This study employed three different types of SMA’s having different composition, yield strength, and superelastic strain to investigate the effect of SMA properties on the plastic hinge length. In this study, one nickel–titanium, one Cu-based, and one Fe- based shape memory alloys have been selected for the use in bridge piers. The selected SMAs along with their mechanical properties such as the

elastic modulus (E), austenite to martensite starting stress (fy); austenite to martensite finishing stress (fP1); martensite to austenite starting stress (fT1); martensite to austenite finishing stress (fT2); superelastic strain (es) are listed in Table 4. These three SMA’s were selected because of their variation in yield strength. Moreover, the selected SMA’s have very large superelastic strain, which allows recentering of bridge piers after a seismic event. As three different types of SMAs were used, the bridge piers were designed in such a way that they have comparable moment capacities. Fig. 6 shows the effect of different types of SMA on the curvature and rebar compressive strain profile. From Fig. 6a it can be observed that different types of SMA affects the curvature profile thereby affecting the plastic hinge length. Fig. 6b depicts that as the yield strength of SMA rebar increases the plastic hinge length increases. As the yield strength of SMA increased from 210 MPa to 750 MPa, the plastic hinge length increases from 0.8d to 1.06d. Previous researchers

9

(a)

8

SMA-210 SMA-450 SMA-750

7 6

Distance from base (m)

Distance from base (m)

9

5 4 3 2 1

(b)

8

SMA-210

7

SMA-450

6

SMA-750

5 4 3 2 1 0

0 0

0.02

0.04

0.06

Curvature (1/m)

0.08

0.1

0

0.005

0.01

0.015

0.02

0.025

Longitudinal rebar strain (εs)

Fig. 6. Effect of fy-SMA on (a) curvature profile and (b) longitudinal rebar strain profile.

0.03

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[15,38] also concluded that the plastic hinge length of concrete bridge pier increases as the yield strength of the reinforcement increases. 5.4. Effect of longitudinal reinforcement ratio The effect of longitudinal reinforcement ratio (ql) on the plastic hinge length has been ignored by many researchers. However, several researchers investigated the effect of ql on the plastic hinge length and reported contradictory conclusions. Mattock [11] concluded that, as the net tension reinforcement increases, the plastic hinge length also increases. On the contrary, Mendis [34] found that the plastic hinge length increases with increasing amount of tension reinforcement. These conclusions were based on beam test results. However, Bae and Bayrak [21] concluded that the plastic hinge length of column tend to increase with increasing longitudinal reinforcement ratio (ql). To study the effect of ql on the plastic hinge length of SMA-RC pier, three different reinforcement ratios (1%, 2% and 3%) consistent with current seismic design guidelines were selected. Fig. 7 shows the effect of longitudinal reinforcement ratio (ql) on the curvature and strain profile. As evident from both curvature and strain profile, the plastic hinge length tends to decrease with increasing longitudinal reinforcement ratio (ql). The change in plastic hinge length is more pronounced from longitudinal rebar strain profile (Fig. 7b) as compared to the curvature profile (Fig. 7a). 5.5. Effect of transverse reinforcement Most of the available plastic hinge equations do not consider the effect of transverse reinforcement ratio (qs). Corley [12] and Kazaz [23] did not consider qs in their proposed plastic hinge

5.6. Effect of concrete strength Several researchers have considered the effect of concrete strength on the plastic hinge length of RC members. Plastic hinge equations provided in the most of the design codes and proposed by different researchers do not consider the effect of concrete strength (Table A1, Appendix). However, only the plastic hinge expression proposed by Berry et al. [38] and Alemdar [15] consider the effect of concrete strength. They found that the plastic hinge length decreases as the concrete compressive strength increase as evident from their plastic hinge equations. This study also considered three different concrete strengths (35, 50 and 60 MPa) to investigate the variation in plastic hinge length of SMA-RC pier with varying concrete strength. Fig. 9 shows the changes in curvature and strain profile as the compressive strength varied from 35 to 60 MPa. The curvature profile depicts that (Fig. 9a) the change in

9

(a)

8

1% 2% 3%

7

Distance from base (m)

9

Distance from base (m)

expression. Only few researchers [14,34] considered the effect of qs on the plastic hinge length. Mendis [34] and Hines et al. [14] have concluded that as the qs increases the plastic hinge length decreases as evident from their proposed plastic hinge equation. Fig. 8 shows the variation in curvature and strain profile with changes in the transverse reinforcement ratio (qs). The change in plastic hinge length is more pronounced from strain profile as compared to the curvature profile. From the curvature profile (Fig. 8a) the plastic hinge length varied from 0.84d to 0.88d. However, from longitudinal rebar strain profile (Fig. 8b) the plastic hinge length varied from 0.76d to 1.02d. This can be attributed to the fact as the amount of transverse reinforcement increases, the core concrete experiences less damage, thereby, reduces the plastic hinge length.

6 5 4 3 2 1

(b)

8

1% 2% 3%

7 6 5 4 3 2 1 0

0 0

0.02

0.04

0.06

0.08

0

0.1

0.005

0.01

0.015

0.02

0.025

0.03

Longitudinal rebar strain (εs)

Curvature (1/m)

Fig. 7. Effect of longitudinal reinforcement ratio on (a) curvature profile and (b) longitudinal rebar strain profile.

Distance from base (m)

(a)

8

Distance from base (m)

9

9

0.8% 1% 1.2%

7 6 5 4 3 2 1 0

(b)

8

0.8% 1% 1.2%

7 6 5 4 3 2 1 0

0

0.02

0.04

0.06

Curvature (1/m)

0.08

0.1

0

0.005

0.01

0.015

0.02

0.025

Longitudinal rebar strain (εs)

Fig. 8. Effect of transverse reinforcement ratio on (a) curvature profile and (b) longitudinal rebar strain profile.

0.03

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9

(a)

8

Distance from base (m)

Distance from base (m)

9 35 MPa 50 MPa 60 MPa

7 6 5 4 3 2 1 0

(b)

8

35 MPa 50 MPa 60 MPa

7 6 5 4 3 2 1 0

0

0.02

0.04

0.06

0.08

0.1

0

Curvature (1/m)

0.005

0.01

0.015

0.02

0.025

0.03

Longitudinal rebar strain (εs)

Fig. 9. Effect of concrete compressive strength on (a) curvature profile and (b) longitudinal rebar strain profile.

plastic hinge length is independent of concrete strength as the plastic hinge length varied between 0.75d to 0.78d. On the other hand, the strain profile shows that as the concrete strength increased from 35 to 60 MPa, the plastic hinge length decreased from 1.08d to 0.68d. 6. Plastic hinge length expression for SMA-RC bridge pier The results presented in previous sections showed that the compressive strain profile of the longitudinal rebar facilitates a clearer observation of the plastic hinge length as compared to the curvature profile. As a result, this study utilized the compressive strain profile of the longitudinal rebar to develop the plastic hinge length expression for SMA-RC bridge pier. The discussions presented in preceding sections showed that several factors influence the length of plastic hinge in SMA-RC pier such as, level of axial load, aspect ratio, yield strength of SMA rebar, concrete compressive strength, longitudinal and transverse reinforcement ratio. Considering the effect of different parameters, a new expression for calculating the plastic hinge length of SMA-RC pier was derived by regression analysis. In this study, multivariate regression was used as it allows simultaneous testing and modeling of multiple independent variables. In this study, different forms of regression equations were tested to find the ‘‘best fit” line or curve for a series of data points. Table 5 shows a list of different equations tested for defining the plastic hinge expression for SMA-RC bridge pier. The criterion for selecting the suitable equation type was the minimum square of the error between the original data and the values predicted by the equation. Although technique may not be the most statistically robust method of fitting a function to a data set, it has the advantage of being relatively simple. Using the data obtained from numerical investigations as described in previous sections, the multivariate regression analysis technique was used for deriving the following expression (Eq. (1)) for estimating the plastic hinge length of SMA-RC pier:

Table 5 List of equations tested. Equation category

Equation name

Sample equation

Standard curves

Linear Quadratic

y ¼ y0 þ ax

Logarithm

2 parameter 3 parameter Linear Quadratic

Polynomial Power

3 parameter Modified pareto function

y ¼ y0 y ¼ y0 y ¼ y0 y ¼ y0

2

þ ax þ bx þ a ln x þ a lnðx  x0 Þ þ ax

y ¼ y0 þ ax þ bx y ¼ y0 þ axb 1 y ¼ 1  ð1þax bÞ

2

LP P ¼ 1:05 þ 0:25 0 d f c Ag

!

  L þ ð0:0002f ySMA Þ þ 0:08 d 0

 ð0:16ql Þ  ð0:019f c Þ  ð0:24qs Þ

ð1Þ

In Eq. (1) Lp is the plastic hinge length in mm, d is the diameter of circular column in mm. Except for the two parameters fy-SMA and f/c, the regression coefficients of all other parameters are dimensionless. However, the regression coefficients of these two parameters are MPa1. In this equation both ql and qs are expressed as percentage (i.e., 1% = 0.01, 2% = 0.02, etc.). From the proposed equation it can be observed that the plastic hinge length of SMARC pier is mostly influenced by the level of axial load, longitudinal and transverse reinforcement ratio and less sensitive to the aspect ratio. Although, the regression coefficients associated with the yield strength of SMA and concrete compressive strength look insignificant, a small change in fy-SMA or f/c will result in a significant change in the plastic hinge length. This can be attributed to the fact that only these two parameters (fy-SMA and f/c) are one dimensional whereas the other parameters are non-dimensional. Although Eq. (1) can be used for predicting the plastic hinge length of SMA-RC bridge pier, the applicability of the proposed equation is limited by the following conditions (Eqs. (2)–(6)):

0:05 6

P 6 0:20 0 f c Ag

0:8 6 ql 6 3:0 200 MPa 6 f ySMA 6 750 MPa 0

ð2Þ ð3Þ ð4Þ

30 MPa 6 f c 6 75 MPa

ð5Þ

3 6 L=d 6 15

ð6Þ

7. Validation of the proposed equation To verify the accuracy of the numerically derived expression for plastic hinge length of SMA-RC bridge pier, comparisons were made with plastic hinge length measured from experimental investigations. Since very limited number of test results are available on SMA-RC bridge piers which measured the plastic hinge length, a database composed of four SMA-RC pier test results were compiled. Table 6 shows the comparison of the measured and calculated plastic hinge length which illustrates that the use of proposed equation results in good estimates of plastic hinge length for all test specimens. From Table 6 it can be observed that the maximum variation was observed in Specimen SR99-LSE [19] which was 7.84%. This can be attributed to the fact that all other piers had circular section while SR99-LSE was a square column. Moreover, the proposed equation was derived based on the

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A.H.M.M. Billah, M. Shahria Alam / Engineering Structures 117 (2016) 321–331 Table 6 Comparison of experimental and predicted plastic hinge length. Parameter

Specimens

Axial load ratio (P/fc0 Ag) Aspect Ratio (L/d) Fy-SMA (MPa)

ql

fc’ (MPa)

qs Lp/d (measured) Lp/d (calculated) Lp (measured) (mm) Lp (calculated) (mm) Error (%)

Lp/d (Predicted)

1

RNC O’Brien et al. [24]

RNE O’Brien et al. [24]

SR-99-LSE Nakashoji and Saiidi [19]

SMAC-1 Saiidi and Wang [43]

0.1 4.5 413.7 0.02 31.03 0.024 0.98 0.92 249.9 233.48 6.57

0.1 4.5 413.7 0.02 35.8 0.024 0.84 0.83 212.3 210.46 0.87

0.0864 3.44 352 0.01 49.6 0.015 0.44 0.47 199 214.61 7.84

0.25 4.5 379.2 0.026 43.8 0.0068 0.75 0.71 229 216.66 5.39

y/x: μ=0.98 σ= 0.059 COV= 6%

0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

Lp/d (Experiment) Fig. 10. Comparison of measured and predicted plastic hinge lengths.

analyses on circular columns. Best match was observed for Specimen RNE [24] where the measured and predicted value differed by only 0.87%. Fig. 10 compares the Lp/d values measured from experimental results with those predicted using Eq. (1). Statistical parameters (mean, standard deviation and COV) displaying the degree of correlation between the measured and predicted values is also shown in the same figure. From Fig. 10 it is evident that the proposed equation provides a reasonable estimate of the plastic hinge length of SMA-RC bridge pier. The proposed plastic hinge equation was also used to calculate the maximum drift of a SMA-RC bridge pier (RNE) tested by O’Brien et al. [24]. Using plastic hinge length and the yield and ultimate curvature, the ultimate drift of a cantilever bridge pier can be calculated using the following equation:

1 Du ¼ /y L2 þ ð/u  /y ÞLp ðL  0:5Lp Þ 3

ð7Þ

Table 7 Comparison of measured and calculated ultimate drift. Reference RNC, O’Brien et al. [24]- Test Data Paulay and Priestley [9] Alemdar [15] Nakashoji and Saiidi [19] Berry et al. [38] Mander [42] Proposed equation

Lp (mm)

Ultimate displacement (mm)

–

137.4

207.10 141.45 232.20 151.51 182.60 233.48

123.05 95.70 133.02 100.01 113.06 133.52

% difference – 10.44 30.50 3.20 29.25 17.71 5.09

In order to predict the accuracy of the proposed plastic hinge expression in predicting the ultimate drift capacity of SMA-RC bridge pier, comparisons were made with experimental results and other plastic hinge expression available in literature. Table 7 shows a comparison of the measured ultimate drift value and ultimate drift calculated with different plastic hinge equations. From Table 7, it is evident that the proposed plastic hinge equation provides a reasonable estimate of the drift capacity of SMA-RC pier. The proposed Lp equation could predict the ultimate drift of the specimen RNE with only 5.09% error, which was the second most accurate among all the compared equations. The plastic hinge equation proposed by Nakashoji and Saiidi [19] predicted the drift capacity with higher accuracy where the difference was only 3.20%. This can be attributed to the fact that, the equation proposed by Nakashoji and Saiidi [19] back calculates the plastic hinge length using the results from the average measured moment curvature and force displacement idealizations. Moreover, this equation was derived for rectangular columns and they [19] recommended to use Paulay and Priestley [9] equation over their proposed equation for predicting the plastic hinge length of SMA-RC bridge pier. The plastic hinge equation proposed by Paulay and Priestley [9] also predicted the ultimate drift with only 10.4% error. The other equations differed by a large margin where the largest difference was 30.5% as predicted by the equation proposed by Alemdar [15]. 8. Conclusions This study proposed a new expression for estimating the plastic hinge length in SMA-RC bridge pier. The finite element model was validated with different experimental results to ensure the accuracy of the adopted modeling technique. A parametric study was conducted to evaluate the effect of different factors on plastic hinge length of SMA-RC bridge pier. A multivariate regression analysis was performed to develop the proposed plastic hinge length expression. The proposed equation was verified against test results of SMA-RC pier to check its accuracy. The accuracy of the proposed equation in predicting the drift capacity of SMA-RC pier was also validated against test results and compared with other plastic hinge expressions. Based on the analysis results, the following conclusions are drawn: 1. The effect of different parameters are more pronounced when plastic hinge length is estimated in terms of longitudinal rebar strain profile as opposed to the curvature profile. 2. Compressive strain profile of longitudinal rebar provides a better estimate of plastic hinge length as compared to the curvature profile of SMA-RC bridge pier.

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3. Plastic hinge length of SMA-RC pier increases as the axial load, aspect ratio and the yield strength of SMA rebar increases. On the contrary, plastic hinge length decreases with an increase in concrete compressive strength and the ratio of longitudinal and transverse reinforcement. 4. The proposed equation showed reasonable accuracy in predicting the plastic hinge length measured from experimental investigations. The proposed equation predicted the experimental plastic hinge lengths with a COV of only 6%. 5. The ultimate drift capacity of SMA-RC bridge pier can be predicted with reasonable accuracy using the proposed plastic hinge length equation as compared to other existing expressions.

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The scope of the present study was limited to flexure dominated circular bridge piers. Further research needs to be carried out considering shear dominated piers along with square and rectangular geometry.

Acknowledgements The financial contributions of Natural Sciences and Engineering Research Council of Canada (NSERC) through Discovery Grant and Industrial Postgraduate Scholarship Program were critical to conduct this study and are gratefully acknowledged.

Appendix A

Table A1

Author/Code Mattock [11]

Expressions   0 qffiffiffiffiffiffiffi   pffiffi d lp ¼ d=2 1 þ 1:14 dz  1 1  qq 16:2 q

Unit in

b

Corley [12]

lp ¼ 2d þ 0:2 pzffiffid

in

Park et al. [35] Priestley and Park [13] Paulay and Priestley [9]

lp ¼ 0:4d lp ¼ 0:08l þ 6db lp ¼ 0:08l þ 0:022db f y pffiffiffiffiffi lp ¼ 32 db þ 0:06l n   o     0:3 PPo þ 3 AAgs  0:1 hl þ 0:25 h lp ¼

in In mm, MPa

Mander [42] Bae [21] Berry et al. [38]

mm

mm, MPa

f d

lp ¼ 0:05l þ 0:1 py ffiffiffib0 fc

Alemdar [15] Caltrans 2010 [44] JTG/T B02-01 [45] Eurocode 8 [46] JRA 2002 [47] NZS-3101:2006 [48] Eurocode 8, Part-3 [49]

lp ¼ 14 þ

mm, MPa

3f y db 10000

l pffiffiffi0 þ 25000 fc

0.08L + 0.022fydb Minimum (0.08L + 0.022fydb P 0.044 fydb or 2/3h 0.1L + 0.015 fydb 0.2L-0.1h; 0.1h 6 Lp 6 0.5h Minimum (0.5he or 0.2 M/V) P 0.5 he f d

lp ¼ 0:1clv þ 0:17h þ 0:24 py ffiffiffib0 fc

d = h = Section depth; l = L = Member length. z = Distance of critical section to point of contra-flexure. db = The diameter of the longitudinal reinforcement. fy = The yield strength of longitudinal steel. P/Po = Axial load capacity of the section. As = Area of the longitudinal reinforcement. Ag = Area of the column. fc0 = Compressive strength of the concrete. c = Concrete cover to reinforcement. lv = Ratio of moment/shear at the end section.

mm, mm, mm, mm mm mm,

MPa MPa MPa

MPa

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