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Seismic response of steel braced frames with shape memory alloy braces
Journal of Constructional Steel Research 67 (2011) 65–74
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Journal of Constructional Steel Research journal homepage: www.elsevier.com/locate/jcsr
Seismic response of steel braced frames with shape memory alloy braces B. Asgarian, S. Moradi ∗ Civil Engineering Faculty, K.N. Toosi University of Technology, Tehran, Iran
article
info
Article history: Received 30 March 2010 Accepted 14 June 2010 Keywords: Shape memory alloys Superelasticity Braced frames Residual roof displacement
abstract In this paper, the seismic performance of steel frames equipped with superelastic SMA braces was investigated. To do so, buildings with various stories and different bracing configurations including diagonal, split X, chevron (V and inverted V) bracings were considered. Nonlinear time history analyses of steel braced frames equipped with SMA subjected to three ground motion records have been performed using OpenSees software. To evaluate the possibility of adopting this innovative bracing system and its efficiency, the dynamic responses of frames with SMA braces were compared to the ones with buckling restrained braces. After comparing the results, one can conclude that using an SMA element is an effective way to improve the dynamic response of structures subjected to earthquake excitations. Implementing the SMA braces can lead to a reduction in residual roof displacement and peak inter-story drift compare to the buckling restrained braced frames. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Buildings in seismically active regions are prone to severe damage and even collapse during earthquakes due to large lateral deformations. The flexibility of steel moment-resisting frames may result in great lateral drift induced nonstructural damage under strong ground motion. The weak performance of pre Northridge earthquake steel moment resisting frames is related to geometric nonlinearities and brittle failure of beam-to-column connections [1,2]. Therefore, the inter-story drift ratio should be limited in design [3] and hence compared with braced frames these systems generally require larger member sizes than those required for strength alone. Although extensive damage was reported in concentrically braced frames following recent earthquakes [4–7]. Several reasons for the weak performance of ordinary steel braced frames in recent earthquakes can be noted as limited ductility and low energy dissipation capacity due to braces buckling, failure of connections and asymmetric behavior of the braces in tension and compression. Observing the deficiencies for ordinary concentrically braced frames, seismic design requirements for braced frames were changed and the concept of special concentric braced frames was introduced [1]. Despite the improved performance of the special concentric braced frames over ordinary braced frames, research works have been done to search for more cost-effective and better-performance concentric braced frames [8]. As a result, Buckling Restrained Braced Frames (BRBFs)
were suggested. BRBFs are a relatively new type of concentrically braced system characterized by the use of braces that yield both in tension and compression. Unlike ordinary braced frames, BRBFs are able to achieve a stable, balanced hysteretic behavior and substantial ductility by accommodating compression yielding before the onset of buckling [9]. The results of some studies have shown that the behavior of BRBFs is comparable and often better than that of conventional ordinary braced frames and moment frames [1,10]. However, large building displacement and residual drift is a factor that may limit the adoption of buckling restrained bracing systems [8]. Since the inelastic behavior of braced frames subjected to lateral loads is strongly dependent on the behavior of bracing members, an alternative strategy can be pursued by using superelastic Shape Memory Alloys (SMAs) in bracing systems. As a result of the recentering and supplemental energy dissipation capabilities of SMA materials, inter-story drift and permanent displacement of the structure could be decreased compared to ordinary and even buckling restrained braced frames. The objective of this paper is to study the possibility and effectiveness of utilizing such materials in steel braced frames, as innovative seismic devices for the protection of buildings. A literature survey reveals that there is little reported work on the implementation of SMA braces in steel frames of varying heights and bracing types. To obviate this issue, the seismic performance of 24 different braced frames equipped with SMA braces is compared to the same frames with buckling restrained braces through nonlinear time history analyses. 1.1. Shape memory alloys
∗
Corresponding author. Tel.: +98 21 8877 9474x218; fax: +98 21 8877 9476. E-mail addresses: [email protected] (B. Asgarian), [email protected], [email protected] (S. Moradi). 0143-974X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2010.06.006
SMAs are materials capable of undergoing large recoverable strains of about 10% without exhibiting plasticity. No lifetime limits
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Fig. 1. Three dimensional stress, strain temperature relationship for the mechanical behavior of SMAs [13].
(no problem of maintenance or substitution, even after several strong earthquakes) high damping capacity, good control of forces, recentering capability, high fatigue resistance, and the recovery of strains are among the characteristics that have made SMAs an effective material for seismic applications [11–13]. SMAs were discovered in the 1932 and it was after 1962 that both in-depth research and practical applications of SMAs emerged. SMAs have more than one crystal structure. The prevailing crystal structure or phase in a polycrystalline metal depends on both temperature and external stress. The high temperature phase is called austenite, whereas the low temperature phase is referred to martensite [14,15]. The distinctive macroscopic behavior of SMA is closely linked to transformations between the two phases. The shape memory effect (SME) and superelastic effect (SE) are known as two unique properties of SMAs. The former property is the capacity to regain the original shape by heating and the latter is related to the ability of recovering large deformations after removal of the external load [16]. Fig. 1 represents the mechanical behavior of SMA as a function of temperature, strain and stress. As can be seen, below the martensite finish temperature, Mf , the SMA exhibits a shape memory effect. Deformations due to an applied stress are recovered by heating the material above the austenite finish temperature, Af . At a temperature above Af , the SMA is in its parent phase, austenite. Upon loading, stress-induced martensite is formed. Upon unloading, however, the material reverts to austenite at a lower stress, thereby resulting in the superelastic behavior. At a yet higher temperature (above Md , the temperature at which the stress to induce the martensitic transformation lies above the critical slip stress of the SMA resulting in permanent plastic deformation due to the thermo-elastic nature of the material), the SMA undergoes ordinary plastic deformation with much higher strength [13]. It should be noted that As in Fig. 1, indicates the temperature at which the transformation from the martensite into the austenite starts. Up to date many applications have been reported for SMA materials due to their unique properties. A number of the past studies has presented a review of the SMA properties and the applications of SMA technology in civil engineering [14,15,17]. Researchers are essaying to develop the seismic implementations of SMA materials in bridges as both base isolation systems [18] and passive seismic control devices [19]. Moreover, there have been several studies aimed at seismic hazard mitigation of buildings. Several researchers have investigated the performance of buildings implementing the SMA energy dissipation and recentering devices in the forms of braces for framed structures [20–24], dampers [25,26], connection elements [27,28] and retrofitting of existence structures [29]. However, further experimental and numerical studies are needed to have an appropriate implementation of SMA devices practically and to develop seismic design criteria for these innovative materials.
Table 1 Three cases of braced frames. Bracing system
Frame case
SMA bar segments connected to the frame through rigid elements SMA bar segments connected to the frame through buckling restrained elements Buckling restrained braces
A B C
2. Frame characteristics To evaluate the performance of the SMA bracing system in braced steel frames and compare their behavior to the same frames using buckling restrained braces, twenty-four buckling restrained braced frames designed by Asgarian and Shokrgozar [30] were used. All of the structures considered in this study have been designed according to the Iranian Earthquake Resistance Design Code [31] and Iranian National Building Code, steel structure design [32]. Fig. 2 shows the typical configuration of the models used. The 4, 6, 8, 10, 12 and 14 story buildings with four different bracing types (split X, chevron V, chevron-inverted V and diagonal types) were used. All these frames have spans of 6 m with story heights of 3.2 m. The dead and live loads of 6 and 2 kN/m2 have been considered for gravity loads respectively. Further details corresponding to the building design can be found in the paper by Asgarian and Shokrgozar [30]. In this work, in order to have a comparative study more comprehensively, the responses of three groups of braced frame were compared. Two groups of these braced frames have SMA bracing systems (namely case A and B) and the other group has buckling restrained braces (case C). Cases A and B have the same beam and column design as the buckling restrained braced frames (case C). In cases A and B, The buckling restrained braces are replaced with superelastic SMA bar segments connected to the frame through connective segments in order to reduce the length of SMA required. This connective segment is a rigid one in case A and it is a buckling restrained element in case B. Table 1 and Fig. 3 show these frame cases with more detail. 3. Modeling the structures The computational model of the structures was developed using the modeling capability of the software framework of OpenSees [33]. This software is finite element software which has been specifically designed in performance systems of soil and structure under earthquake. For modeling members in the nonlinear range of deformation, the following assumptions were assumed. To model the St-37 steel’s behavior, ‘steel01’ bilinear kinematic stress–strain curve was assigned to the elements from the library of materials introduced in OpenSees. Considering the idealized elastoplastic behavior of steel materials, compressive
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(a) Plane of the structures.
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(b) Brace configurations. Fig. 2. Configuration of model structures.
Fig. 3. Bracing system in cases A and B.
and tensional yield stresses were considered equal to steel’s yield stress. The strain hardening of 2% and the maximum ductility of 15 were considered for the member’s behavior in the inelastic range of deformation (Fig. 4). This was done, in part, for consistency purposes since this is the same approach taken by Asgarian and Shokrgozar [30]. All frame members, i.e. beams, columns and braces were considered as pin-ended. Therefore, beams will behave elastically under gravity loads and are not parts of the lateral resisting system. In this way, the earthquake lateral forces were carried only by the vertical braces (the braces have been designed for 100% of the lateral load). For the dynamic analysis, story masses were placed in the story levels considering rigid diaphragms action. A damping coefficient of 5% was assumed. A force-based beam–column element, consisting of fiber elements, was used for beams, columns and braces. Using a stress–strain curve, stress is calculated in parts of the element’s section according to the imposed strain. Integrating on the section area, total stress, force–deformation and moment curvature curves are calculated. Finally, taking all integration points into account, forces and deformations are calculated.
Fig. 4. Steel01 material.
The column was supposed to yield or buckle during severe earthquakes. Hence, both geometric and material nonlinearity were taken into account while modeling the structures. Using a finite element method based on uniaxial elements is a way to face the matter. Both lumped and distributed plasticity can be used for modeling the material nonlinearity [34]. The latter is used in the present study. For prediction of linear or nonlinear buckling of columns, both the element’s usual stiffness matrix and the element’s geometric stiffness matrix were considered. An initial mid span imperfection up to 1/1000 for all columns was considered and a fiber cross section element was considered for plastification of the element over the member’s length and cross section for linear and nonlinear buckling prediction. To consider the geometric nonlinear effect in columns (uniaxial elements), ‘corotational method’ was used. As noted before, braces were modeled as moment-released elements at both ends and are suppose to behave as axial members. It was also assumed that the braces are able to undergo
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(a) Stress–strain relationship: parameters needed for the model.
(b) Force–displacement relationship of SMA member.
Fig. 5. Superelastic behavior of SMA member. Table 2 Mechanical properties adopted for the numerical simulations [13]. Quantity steel
Fig. 6. Verification: uniaxial tension stress–strain response. Experimental data [39] and numerical simulation.
compressive loads without buckling. The following section is dedicated to the modeling of SMA braces and its assumptions. 3.1. Modeling and design of superelastic SMA braces Over the past decades, modeling of SMAs has attracted a large amount of attention and several researchers have proposed constitutive models for capturing the behavior of the SMAs. In this paper, for representing the superelastic behavior of the SMA braces a constitutive model proposed by Fugazza [35] was chosen. Only a few parameters are needed to construct the model (Fig. 5(a)). These parameters include the austenite to martensite starting stress (σsAS ), the austenite to martensite finishing stress (σfAS ), the martensite to austenite starting stress (σsSA ), the martensite to austenite finishing stress (σfSA ), modulus of elasticity for austenite
and martensite phases (E SMA ) and the superelastic plateau strain length (εL ). The main advantages of this model are robustness and simplicity of implementation, easiness to obtain the needed material parameters (from typical uniaxial tests conducted on either wires or bars) and the ability to simulate partial (i.e. subloops) and complete transformation patterns (i.e. from fully austenite to fully martensite) in both tension and compression. Considering such abilities and for the sake of simplicity, this model was adopted. This one-dimensional model can describe the material’s behavior under arbitrary loadings such as those involved in seismic excitations. The basis of its formulation, developed under the hypothesis of small deformation regime, is the assumption that the relationship between strains and stresses is represented by a series of linear curves whose form is determined by the extent of the transformation experienced.
Value
E E SMA
200 000 MPa 27 579 MPa
σsAS
414 MPa
σfAS
550 MPa
σsSA
390 MPa
σfSA
200 MPa
εL
3.5%
Fig. 5 shows the model schematically. Further assumptions that no strength degradation occurs during cycling and that the austenite and martensite branches have the same modulus of elasticity, in agreement with previous studies [16,36] are made. About the model formulation, it has been assumed to work with one scalar internal variable, representing the martensite fraction and with two conversion processes (from austenite to martensite and vice versa), which may produce variation of martensite fraction. Moreover, for describing the evolution in time of martensite fraction, linear kinetic rules have been used. More details of the model’s formulation and the integration technique can be found in the work by Fugazza [35]. The mechanical properties of the superelastic SMA braces, provided in Table 2, were selected based on the uniaxial cyclic tests carried out by DesRoches et al. [13]. In order to correctly consider the reduced energy dissipation capability of such materials at high frequency loads, those obtained from the dynamic tests of the 12.7 mm diameter NiTi bar were chosen. Although the model has been verified by Fugazza [35], the behavior of the SMA segment should be studied to make sure that the results of the numerical simulation using the model is true. For this purpose, the ability of the model was considered in comparison with experimental data or verified models. Figs. 6 and 7 compare the results of experimental and analytical models. In addition for the load patterns shown in Figs. 8 and 9, the responses of the superelastic model were compared with experimental data [37] and a verified model [38], as can be seen in Figs. 10 and 11, respectively. The results of comparisons in Figs. 6, 7, 10 and 11 demonstrate that the adopted model predicts the expected stress–strain behavior for SMA material accurately. About designing of the SMA braces, cross section and length of the SMA elements should be determined in such a way that two frame cases A and C be comparable. For this purpose (comparison between frame cases A and C), superelastic SMA braces were
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Table 3 Ground motion data. Earthquake
Year
PGA (g)
Duration (s)
ElCentro Kobe Tabas
1940 1995 1978
0.348 0.599 0.934
53.74 48 35
natural period of the corresponding C-case frames (note that, since the mass of the brace is small compared to the entire structure, the mass of the corresponding frame cases A and C were assumed equal). Based on these considerations, the following relations were achieved: Fig. 7. Verification: uniaxial tension stress–strain response. Experimental data [40] and numerical simulation.
ASMA = LSMA =
0.06
Fy
σsAS
=
2400 × Asteel
E SMA ASMA
4140
=
K = 0.08 × LSteel .
0.05
= 0.58 × Asteel
275790 × 0.58ASteel 2e6 × ASteel
(1)
× LSteel (2)
Strain
0.04 0.03 0.02 0.01 0 0
500
1000
1500
2000
2500
3000
3500
Since the required geometric properties (i.e. cross-sectional area, ASMA and element length, LSMA ) of the superelastic SMA braces can be calculated easily. It was also assumed that the SMA elements are made of a number of large diameter superelastic bars able to undergo compressive loads without buckling.
Time [s]
4. Nonlinear time history analyses and results Fig. 8. Strain paths applied in the experiment [37]. 0.08 0.06 0.04 Strain
0.02 0 -0.02
0
5
10
15
20
25
30
-0.04 -0.06 -0.08 Time [s]
Fig. 9. Strain paths applied in the Ref. [38].
designed to provide the same yielding strength, Fy and the same axial stiffness, K , as steel braces (buckling restrained braces in C-case frames). As a result, the A-case frames will have the same
As previously mentioned, in order to study in detail the dynamic behavior of SMA braced system, which acts in both tension and compression, nonlinear time history analyses were performed. Note that all the frame models were analyzed as two-dimensional (2D) models. 4, 6, 8, 10, 12 and 14 story frames with SMA bracing systems were considered. In addition, different bracing configurations (i.e. diagonal, split X, chevron V and inverted-V) were investigated. For conducting time history analyses on the frames implementing SMAs, the same three ground motion (GM) records that were used for analyses of BRBFs [30] were used again here, the details of these ground motion records can be seen in Table 3 and Fig. 12. Since the frames with buckling restrained braces have the same period as the corresponding frames with superelastic SMA braces, a good comparison can be made on the effectiveness of using the proposed SMA bracing system. To inspect the results of time history analyses, we first focus on some response results in regards to a case study and then an overall discussion is presented.
Fig. 10. Verification: stress–strain hysteresis loops for the strain path shown in Fig. 8. Numerical results for the adopted model (left) and experimental results [37] (right).
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750
0.75
600
0.60
450
0.45
300
0.30
150
0.15
0 -0.08
-0.06
-0.04
-0.02 0 -150
0.02
0.04
0.06
0.08
Stress
Stress MPa
70
-0.08
-0.06
-0.04
0.00 -0.02 0.00 -0.15
-300
-0.30
-450
-0.45
-600
-0.60
-750
-0.75 Strain
Strain
0.02
0.04
0.06
0.08
Fig. 11. Verification: stress–strain hysteresis loops for the strain path shown in Fig. 9. Numerical results for the adopted model (left) and the Ikeda model in Ref. [38] (right).
1.6 1.4
Sa =B×A Sa (ξ=5%)
1.2 1 0.8 0.6 0.4 0.2 0 1
2 Period (sec)
3
4
Fig. 12. Variation of spectral acceleration with period of structure.
4.1. Case study: ten-story braced frame The ten-story V-braced frame undergoing the ElCentro ground motion is chosen to be considered in detail. The force–displacement plots for one of the first floor braces in each braced system are shown in Fig. 13. Deformations occur in the buckling restrained braces, 16 mm, while the SMA members have lower displacement, 6.9 mm, due to the recentering capability guaranteed by the superelastic effect. The plots obviously indicate the ability of superelastic SMA bracing system to provide recentering, undergo compressive loadings, and stiffen at large strains as compared to the buckling restrained braces (BRBs). Further, a comparison of the hysteretic loops of the two types of braces suggests that BRBs allow for larger ductility demand with respect to superelastic SMA braces, while dissipating a significant amount of seismic energy due to their wider hysteretic loops. On the other hand, the innovative bracing system shows higher values of axial force due to hardening of the material, which may limit the displacements for unexpected strong earthquakes but at the same time, may result in larger shear forces transmitted in beams and columns. The time history for the top floor displacement of both frames is provided in Fig. 14. The peak roof displacements for the BRBF and SMA braced frame are approximately 233 mm and 193 mm, respectively. Since the beam and column members are the same in both of the frames and the yield force and initial stiffness of both types of bracing systems are constant, the difference in response must be attributed to the different behavior of the buckling restrained braces as compared to the superelastic SMA braces. As another important result, we can compare the values of permanent (residual) roof displacement in Fig. 14. The use of SMA braces results in a permanent roof displacement of approximately 1.21 mm while the buckling restrained braced frame undergoes
Fig. 13. First floor bracing force–displacement history for buckling restrained and SMA (Case A) braced frame (ten-story chevron V braced frame subjected to the ElCentro ground motion).
20 15 10 Displacement (mm)
0
5 0 -5 -10 -15 -20 -25 -30 0
5
10
15
20
25
30
35
40
45
50
55
Time (sec) Fig. 14. Roof displacement history of buckling restrained and SMA (Case A) braced frame (ten-story chevron V braced frame subjected to the ElCentro ground motion).
145.4 mm of residual displacement. These results reveal the excellent performance of SMA braces in reducing the residual displacement of the top floor. Besides, the maximum inter-story drift in the case of frame with buckling restrained braces is 1.17% whereas it is decreased to 0.82% using SMA braces (approximately 30% reduction). 4.2. Overall study: results and discussion The results from the case study show that the SMA braces work well at reducing both inter-story drifts and residual displacements.
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Fig. 15. Mean of the maximum inter-story drift for the chevron V (left) and inverted-V (right) braced frames (braces in case A, B and C).
Fig. 16. Mean of the maximum inter-story drift for the split X (left) and diagonal (right) braced frames (braces in case A, B and C).
Fig. 17. Permanent roof displacement for the chevron V (left) and inverted-V (right) braced frames subjected to ElCentro ground motion (braces in case A, B and C).
However, it is important to verify that the findings of the case study are consistent over the ground motion records. In addition, in order to have confidence in the predicted behavior of SMA bracing systems in real world structures, it is necessary to have a more comprehensive study of braced frame structures with various story numbers and different bracing configurations. Figs. 15 and 16 provide the mean values of maximum interstory drift for 4, 6, 8, 10, 12 and 14 story frames with four different bracing configurations and three varied bracing systems (for more brevity, mean values for the three GM records are presented). As noted earlier, the two frame cases A and C are comparable and the B-case frames represent the frames that braces include the SMA elements connected to the buckling restrained segments. Apparently, the average value of maximum inter-story drift is smaller for all the A-case frames as compared to the buckling restrained braced frames (case C). The results suggest that the ability of the SMA brace hysteresis to pass through the origin during cycling while undergoing large deformations reduces the maximum inter-story drift (on average 16% up to 60%, the least and peak difference) and subsequently the demand on the column members at each floor level.
As well as the maximum inter-story drift, the residual roof displacement can be considered to study the seismic performance of structures subjecting to dynamic loads. Figs. 17–22 show the values of residual roof displacement for the considered frames and the three aforementioned GM records. In almost all the results, the residual roof displacement for the frames equipped with the SMA braces (case A) is less than that for BRBFs (case C). These results suggest that the recentering capability and stiffening at large strains associated with the SMAs has a major influence in reducing the permanent deformation in the structures. It is important to notice that in some instances like the 12 story frame with chevron V, inverted-V and/or split X braces undergoing the Tabas GM record using the SMA bracing systems in place of the buckling restrained braces results in higher value of permanent roof displacement unexpectedly. This can be attributed to post-inelastic behavior (i.e. branch observed at the end of the upper plateau) of the SMA braces, which in their fully martensitic phase transmit high values of forces to columns and/or beams with consequent structural problems caused by yielding. In fact, this is a limitation of the model since the true behavior of a SMA material would show a third softening branch after yielding
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Fig. 18. Permanent roof displacement for the split X (left) and diagonal (right) braced frames subjected to ElCentro ground motion (braces in case A, B and C).
Fig. 19. Permanent roof displacement for the chevron V (left) and inverted-V (right) braced frames subjected to Kobe ground motion (braces in case A, B and C).
Fig. 20. Permanent roof displacement for the split X (left) and diagonal (right) braced frames subjected to Kobe ground motion (braces in case A, B and C).
Fig. 21. Permanent roof displacement for the chevron V (left) and inverted-V (right) braced frames subjected to Tabas ground motion (braces in case A, B and C).
of the martensite [22]. However, in structural design of SMA braces the force transferring to the other structural members as a result of the phase transformation (after martensitic yielding)
should be limited. This issue can be guaranteed by changing the mechanical properties of the SMA segments and shifting the phase transformation stress level to a lower one.
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Fig. 22. Permanent roof displacement for the split X (left) and diagonal (right) braced frames subjected to Tabas ground motion (braces in case A, B and C).
Although the B-case frames (i.e. the frames that have SMA bar segments connected to the frame through buckling restrained elements as bracing system) are not comparable to the other frame cases, it can be claimed that using a buckling restrained segment connected to the SMA element in bracing can provide the structure with a greater energy dissipating capability. Because of this ability, the response of the B-case frames can be even less than that for the A-case frames. To the contrary, in some instances, the response is higher as compared to BRBFs. Therefore energy dissipation capability of the structure can be increased through using the buckling restrained segment in SMA-based bracing system. Fig. 23 represents the added energy dissipation associate to the buckling restrained segments in the B-case frame. As shown, the BRBFs (i.e. case C) can exhibit higher amount of energy dissipation compared to the other frame cases. However, larger drifts occurred in respect to SMA-braced frames (case A).
Fig. 23. Base shear-first story drift for the 8 story inverted-V braced frame subjected to Tabas ground motion.
5. Conclusion
References
In this paper, the dynamic responses of twenty-four steel braced frames with various story numbers and types of bracing were evaluated aimed at investigating the SMA efficiency in seismic response reduction of structures. After verifying the model of superelastic behavior of shape-memory alloys, nonlinear time history analyses were carried out using the OpenSees computational platform. The seismic performance of the frame implementing superelastic SMAs was compared to the same frame with buckling restrained braces. The results of this comparative study showed that shapememory alloys could be effectively utilized for seismic design of structures. SMA braces can provide a means of minimizing losses associated with damage to structural systems during an earthquake, regardless of both the brace configuration and height of the structure. Although, buckling restrained braces possess a bigger energy dissipation capacity, some abilities of SMA elements such as recentering and undergoing strain hardening, make these alloys desirable for vibration response reduction. As observed in this study, implementing the SMA braces can lead to a reduction in residual roof displacement and peak inter-story drift with respect to the buckling restrained braced frames. In addition, the energy dissipation capability of the structure can be increased through using the buckling restrained segment in SMA-based bracing systems. Although the results of this article suggest promise for implementing the SMAs in braced frames, further works still are necessary to confirm the numerical simulations and the analytical results experimentally. In addition, it must be pointed out that there should be a methodology for seismic design of the braced steel frames endowed with SMA bracing systems.
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[15] Janke L, Czaderski C, Motavalli M, Ruth J. Applications of shape memory alloys in civil engineering structures — overview, limits and new ideas. Materials and Structures 2005;38:578–92. [16] Bruno S, Valente C. Comparative response analysis of conventional and innovative seismic protection strategies. Earthquake Engineering and Structural Dynamics 2002;31:1067–92. [17] Alam MS, Youssef MS, Nehdi M. Utilizing shape memory alloys to enhance the performance and safety of civil infrastructure: a review. Canadian Journal of Civil Engineering 2007;34:1075–86. [18] Ozbulut OE, Hurlebaus S. Evaluation of the performance of a sliding-type base isolation system with a NiTi shape memory alloy device considering temperature effects. Engineering Structures 2010;32:238–49. [19] Sharabash AM, Andrawes BO. Application of shape memory alloy dampers in the seismic control of cable-stayed bridges. Engineering Structures 2009;31: 607–16. [20] Yang W, DesRoches R, Leon RT. Design and analysis of braced frames with shape memory alloy and energy-absorbing hybrid devices. Engineering Structures 2010;32:498–507. [21] Speicher M, Hodgson DE, DesRoches R, Leon RT. Shape memory alloy tension/compression device for seismic retrofit of buildings. Journal of Materials Engineering and Performance 2009;18:746–53. [22] Auricchio F, Fugazza D, DesRoches R. Earthquake performance of steel frames with nitinol braces. Journal of Earthquake Engineering 2006;10(1):1–22. [Special issue]. [23] McCormick J, DesRoches R, Fugazza D, Auricchio F. Seismic vibration control using superelastic shape memory alloys. Journal of Engineering Materials and Technology 2006;128:294–301. [24] McCormick J, DesRoches R, Fugazza D, Auricchio F. Seismic assessment of concentrically braced steel frames with shape memory alloy braces. Journal of Structural Engineering 2007;133(6):862–70. [25] Motahari SA, Ghassemieh M, Abolmaali SA. Implementation of shape memory alloy dampers for passive control of structures subjected to seismic excitations. Journal of Constructional Steel Research 2007;63:1570–9. [26] Pan Q, C.h.o C. The investigation of a shape memory alloy micro-damper for MEMS applications. Sensors 2007;7:1887–900. [27] Ocel J, DesRoches R, Leon RT, Hess WG, Krumme R, Hayes JR, et al. Steel beam–column connections using shape memory alloys. Journal of the Structural Engineering 2004;130(5):732–40.
[28] Alam MS, Youssef MS, Nehdi M. Analytical prediction of the seismic behavior of superelastic shape memory alloy reinforced concrete elements. Engineering Structures 2008; doi:10.1016/j.engstruct.2008.05.025. [29] Cardone D, Dolce M, Ponzo FC, E. Coelho. Experimental behavior of R/C frames retrofitted with dissipating and re-centering braces. Journal of Earthquake Engineering 2004;8:361–96. [30] Asgarian B, Shokrgozar HR. BRBF response modification factor. Journal of Constructional Steel Research 2009;65:290–8. [31] BHRC. Iranian code of practice for seismic resistance design of buildings: standard no. 2800. 3rd edition. Building and housing research center; 2005. [32] MHUD. Iranian national building code, part 10, steel structure design. Tehran (Iran): Ministry of housing and Urban development; 2006. [33] Mazzoni S, McKenna F. OpenSees command language manual, version 2.1.0. Pacific earthquake engineering research center. http://opensees.berkeley.edu. 2009. [34] Crisfield MA. Nonlinear finite element analysis of solids and structures, Vol. 1. John Wiley & Sons; 1991. [35] Fugazza D. Shape memory alloy devices in earthquake engineering: mechanical properties, constitutive modeling and numerical simulations. Master’s thesis. Pavia (Italy): European school for advanced studies in reduction of seismic risk; 2003. [36] Andrawes B, McCormick J, DesRoches R. Effect of cyclic modeling parameters on the behavior of shape memory alloys for seismic applications. Proceedings of SPIE 2004;5390:324–34. [37] Ikeda T, Nae FA, Naito H. Constitutive model of shape memory alloys for unidirectional loading considering inner hysteresis loops. Smart Materials and Structures 2004;13:916–25. [38] Mishra SK. Modeling shape memory alloys with applications to seismic design of structural systems. M.Sc. dissertation. Department of civil, structural and environmental engineering state university of New York at Buffalo; 2006. [39] Fugazza D. Use of shape-memory alloy devices in earthquake engineering: mechanical properties. Advanced constitutive modeling and structural applications. Ph.D. thesis. Italy: University of Pavia; 2005. [40] Auricchio F. Shape memory alloys: applications, micromechanics, macromodeling and numerical simulations. Ph.D. thesis. Department of civil and environmental engineering, University of California at Berkeley; 1995.
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Journal of Constructional Steel Research journal homepage: www.elsevier.com/locate/jcsr
Seismic response of steel braced frames with shape memory alloy braces B. Asgarian, S. Moradi ∗ Civil Engineering Faculty, K.N. Toosi University of Technology, Tehran, Iran
article
info
Article history: Received 30 March 2010 Accepted 14 June 2010 Keywords: Shape memory alloys Superelasticity Braced frames Residual roof displacement
abstract In this paper, the seismic performance of steel frames equipped with superelastic SMA braces was investigated. To do so, buildings with various stories and different bracing configurations including diagonal, split X, chevron (V and inverted V) bracings were considered. Nonlinear time history analyses of steel braced frames equipped with SMA subjected to three ground motion records have been performed using OpenSees software. To evaluate the possibility of adopting this innovative bracing system and its efficiency, the dynamic responses of frames with SMA braces were compared to the ones with buckling restrained braces. After comparing the results, one can conclude that using an SMA element is an effective way to improve the dynamic response of structures subjected to earthquake excitations. Implementing the SMA braces can lead to a reduction in residual roof displacement and peak inter-story drift compare to the buckling restrained braced frames. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Buildings in seismically active regions are prone to severe damage and even collapse during earthquakes due to large lateral deformations. The flexibility of steel moment-resisting frames may result in great lateral drift induced nonstructural damage under strong ground motion. The weak performance of pre Northridge earthquake steel moment resisting frames is related to geometric nonlinearities and brittle failure of beam-to-column connections [1,2]. Therefore, the inter-story drift ratio should be limited in design [3] and hence compared with braced frames these systems generally require larger member sizes than those required for strength alone. Although extensive damage was reported in concentrically braced frames following recent earthquakes [4–7]. Several reasons for the weak performance of ordinary steel braced frames in recent earthquakes can be noted as limited ductility and low energy dissipation capacity due to braces buckling, failure of connections and asymmetric behavior of the braces in tension and compression. Observing the deficiencies for ordinary concentrically braced frames, seismic design requirements for braced frames were changed and the concept of special concentric braced frames was introduced [1]. Despite the improved performance of the special concentric braced frames over ordinary braced frames, research works have been done to search for more cost-effective and better-performance concentric braced frames [8]. As a result, Buckling Restrained Braced Frames (BRBFs)
were suggested. BRBFs are a relatively new type of concentrically braced system characterized by the use of braces that yield both in tension and compression. Unlike ordinary braced frames, BRBFs are able to achieve a stable, balanced hysteretic behavior and substantial ductility by accommodating compression yielding before the onset of buckling [9]. The results of some studies have shown that the behavior of BRBFs is comparable and often better than that of conventional ordinary braced frames and moment frames [1,10]. However, large building displacement and residual drift is a factor that may limit the adoption of buckling restrained bracing systems [8]. Since the inelastic behavior of braced frames subjected to lateral loads is strongly dependent on the behavior of bracing members, an alternative strategy can be pursued by using superelastic Shape Memory Alloys (SMAs) in bracing systems. As a result of the recentering and supplemental energy dissipation capabilities of SMA materials, inter-story drift and permanent displacement of the structure could be decreased compared to ordinary and even buckling restrained braced frames. The objective of this paper is to study the possibility and effectiveness of utilizing such materials in steel braced frames, as innovative seismic devices for the protection of buildings. A literature survey reveals that there is little reported work on the implementation of SMA braces in steel frames of varying heights and bracing types. To obviate this issue, the seismic performance of 24 different braced frames equipped with SMA braces is compared to the same frames with buckling restrained braces through nonlinear time history analyses. 1.1. Shape memory alloys
∗
Corresponding author. Tel.: +98 21 8877 9474x218; fax: +98 21 8877 9476. E-mail addresses: [email protected] (B. Asgarian), [email protected], [email protected] (S. Moradi). 0143-974X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2010.06.006
SMAs are materials capable of undergoing large recoverable strains of about 10% without exhibiting plasticity. No lifetime limits
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Fig. 1. Three dimensional stress, strain temperature relationship for the mechanical behavior of SMAs [13].
(no problem of maintenance or substitution, even after several strong earthquakes) high damping capacity, good control of forces, recentering capability, high fatigue resistance, and the recovery of strains are among the characteristics that have made SMAs an effective material for seismic applications [11–13]. SMAs were discovered in the 1932 and it was after 1962 that both in-depth research and practical applications of SMAs emerged. SMAs have more than one crystal structure. The prevailing crystal structure or phase in a polycrystalline metal depends on both temperature and external stress. The high temperature phase is called austenite, whereas the low temperature phase is referred to martensite [14,15]. The distinctive macroscopic behavior of SMA is closely linked to transformations between the two phases. The shape memory effect (SME) and superelastic effect (SE) are known as two unique properties of SMAs. The former property is the capacity to regain the original shape by heating and the latter is related to the ability of recovering large deformations after removal of the external load [16]. Fig. 1 represents the mechanical behavior of SMA as a function of temperature, strain and stress. As can be seen, below the martensite finish temperature, Mf , the SMA exhibits a shape memory effect. Deformations due to an applied stress are recovered by heating the material above the austenite finish temperature, Af . At a temperature above Af , the SMA is in its parent phase, austenite. Upon loading, stress-induced martensite is formed. Upon unloading, however, the material reverts to austenite at a lower stress, thereby resulting in the superelastic behavior. At a yet higher temperature (above Md , the temperature at which the stress to induce the martensitic transformation lies above the critical slip stress of the SMA resulting in permanent plastic deformation due to the thermo-elastic nature of the material), the SMA undergoes ordinary plastic deformation with much higher strength [13]. It should be noted that As in Fig. 1, indicates the temperature at which the transformation from the martensite into the austenite starts. Up to date many applications have been reported for SMA materials due to their unique properties. A number of the past studies has presented a review of the SMA properties and the applications of SMA technology in civil engineering [14,15,17]. Researchers are essaying to develop the seismic implementations of SMA materials in bridges as both base isolation systems [18] and passive seismic control devices [19]. Moreover, there have been several studies aimed at seismic hazard mitigation of buildings. Several researchers have investigated the performance of buildings implementing the SMA energy dissipation and recentering devices in the forms of braces for framed structures [20–24], dampers [25,26], connection elements [27,28] and retrofitting of existence structures [29]. However, further experimental and numerical studies are needed to have an appropriate implementation of SMA devices practically and to develop seismic design criteria for these innovative materials.
Table 1 Three cases of braced frames. Bracing system
Frame case
SMA bar segments connected to the frame through rigid elements SMA bar segments connected to the frame through buckling restrained elements Buckling restrained braces
A B C
2. Frame characteristics To evaluate the performance of the SMA bracing system in braced steel frames and compare their behavior to the same frames using buckling restrained braces, twenty-four buckling restrained braced frames designed by Asgarian and Shokrgozar [30] were used. All of the structures considered in this study have been designed according to the Iranian Earthquake Resistance Design Code [31] and Iranian National Building Code, steel structure design [32]. Fig. 2 shows the typical configuration of the models used. The 4, 6, 8, 10, 12 and 14 story buildings with four different bracing types (split X, chevron V, chevron-inverted V and diagonal types) were used. All these frames have spans of 6 m with story heights of 3.2 m. The dead and live loads of 6 and 2 kN/m2 have been considered for gravity loads respectively. Further details corresponding to the building design can be found in the paper by Asgarian and Shokrgozar [30]. In this work, in order to have a comparative study more comprehensively, the responses of three groups of braced frame were compared. Two groups of these braced frames have SMA bracing systems (namely case A and B) and the other group has buckling restrained braces (case C). Cases A and B have the same beam and column design as the buckling restrained braced frames (case C). In cases A and B, The buckling restrained braces are replaced with superelastic SMA bar segments connected to the frame through connective segments in order to reduce the length of SMA required. This connective segment is a rigid one in case A and it is a buckling restrained element in case B. Table 1 and Fig. 3 show these frame cases with more detail. 3. Modeling the structures The computational model of the structures was developed using the modeling capability of the software framework of OpenSees [33]. This software is finite element software which has been specifically designed in performance systems of soil and structure under earthquake. For modeling members in the nonlinear range of deformation, the following assumptions were assumed. To model the St-37 steel’s behavior, ‘steel01’ bilinear kinematic stress–strain curve was assigned to the elements from the library of materials introduced in OpenSees. Considering the idealized elastoplastic behavior of steel materials, compressive
B. Asgarian, S. Moradi / Journal of Constructional Steel Research 67 (2011) 65–74
(a) Plane of the structures.
67
(b) Brace configurations. Fig. 2. Configuration of model structures.
Fig. 3. Bracing system in cases A and B.
and tensional yield stresses were considered equal to steel’s yield stress. The strain hardening of 2% and the maximum ductility of 15 were considered for the member’s behavior in the inelastic range of deformation (Fig. 4). This was done, in part, for consistency purposes since this is the same approach taken by Asgarian and Shokrgozar [30]. All frame members, i.e. beams, columns and braces were considered as pin-ended. Therefore, beams will behave elastically under gravity loads and are not parts of the lateral resisting system. In this way, the earthquake lateral forces were carried only by the vertical braces (the braces have been designed for 100% of the lateral load). For the dynamic analysis, story masses were placed in the story levels considering rigid diaphragms action. A damping coefficient of 5% was assumed. A force-based beam–column element, consisting of fiber elements, was used for beams, columns and braces. Using a stress–strain curve, stress is calculated in parts of the element’s section according to the imposed strain. Integrating on the section area, total stress, force–deformation and moment curvature curves are calculated. Finally, taking all integration points into account, forces and deformations are calculated.
Fig. 4. Steel01 material.
The column was supposed to yield or buckle during severe earthquakes. Hence, both geometric and material nonlinearity were taken into account while modeling the structures. Using a finite element method based on uniaxial elements is a way to face the matter. Both lumped and distributed plasticity can be used for modeling the material nonlinearity [34]. The latter is used in the present study. For prediction of linear or nonlinear buckling of columns, both the element’s usual stiffness matrix and the element’s geometric stiffness matrix were considered. An initial mid span imperfection up to 1/1000 for all columns was considered and a fiber cross section element was considered for plastification of the element over the member’s length and cross section for linear and nonlinear buckling prediction. To consider the geometric nonlinear effect in columns (uniaxial elements), ‘corotational method’ was used. As noted before, braces were modeled as moment-released elements at both ends and are suppose to behave as axial members. It was also assumed that the braces are able to undergo
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(a) Stress–strain relationship: parameters needed for the model.
(b) Force–displacement relationship of SMA member.
Fig. 5. Superelastic behavior of SMA member. Table 2 Mechanical properties adopted for the numerical simulations [13]. Quantity steel
Fig. 6. Verification: uniaxial tension stress–strain response. Experimental data [39] and numerical simulation.
compressive loads without buckling. The following section is dedicated to the modeling of SMA braces and its assumptions. 3.1. Modeling and design of superelastic SMA braces Over the past decades, modeling of SMAs has attracted a large amount of attention and several researchers have proposed constitutive models for capturing the behavior of the SMAs. In this paper, for representing the superelastic behavior of the SMA braces a constitutive model proposed by Fugazza [35] was chosen. Only a few parameters are needed to construct the model (Fig. 5(a)). These parameters include the austenite to martensite starting stress (σsAS ), the austenite to martensite finishing stress (σfAS ), the martensite to austenite starting stress (σsSA ), the martensite to austenite finishing stress (σfSA ), modulus of elasticity for austenite
and martensite phases (E SMA ) and the superelastic plateau strain length (εL ). The main advantages of this model are robustness and simplicity of implementation, easiness to obtain the needed material parameters (from typical uniaxial tests conducted on either wires or bars) and the ability to simulate partial (i.e. subloops) and complete transformation patterns (i.e. from fully austenite to fully martensite) in both tension and compression. Considering such abilities and for the sake of simplicity, this model was adopted. This one-dimensional model can describe the material’s behavior under arbitrary loadings such as those involved in seismic excitations. The basis of its formulation, developed under the hypothesis of small deformation regime, is the assumption that the relationship between strains and stresses is represented by a series of linear curves whose form is determined by the extent of the transformation experienced.
Value
E E SMA
200 000 MPa 27 579 MPa
σsAS
414 MPa
σfAS
550 MPa
σsSA
390 MPa
σfSA
200 MPa
εL
3.5%
Fig. 5 shows the model schematically. Further assumptions that no strength degradation occurs during cycling and that the austenite and martensite branches have the same modulus of elasticity, in agreement with previous studies [16,36] are made. About the model formulation, it has been assumed to work with one scalar internal variable, representing the martensite fraction and with two conversion processes (from austenite to martensite and vice versa), which may produce variation of martensite fraction. Moreover, for describing the evolution in time of martensite fraction, linear kinetic rules have been used. More details of the model’s formulation and the integration technique can be found in the work by Fugazza [35]. The mechanical properties of the superelastic SMA braces, provided in Table 2, were selected based on the uniaxial cyclic tests carried out by DesRoches et al. [13]. In order to correctly consider the reduced energy dissipation capability of such materials at high frequency loads, those obtained from the dynamic tests of the 12.7 mm diameter NiTi bar were chosen. Although the model has been verified by Fugazza [35], the behavior of the SMA segment should be studied to make sure that the results of the numerical simulation using the model is true. For this purpose, the ability of the model was considered in comparison with experimental data or verified models. Figs. 6 and 7 compare the results of experimental and analytical models. In addition for the load patterns shown in Figs. 8 and 9, the responses of the superelastic model were compared with experimental data [37] and a verified model [38], as can be seen in Figs. 10 and 11, respectively. The results of comparisons in Figs. 6, 7, 10 and 11 demonstrate that the adopted model predicts the expected stress–strain behavior for SMA material accurately. About designing of the SMA braces, cross section and length of the SMA elements should be determined in such a way that two frame cases A and C be comparable. For this purpose (comparison between frame cases A and C), superelastic SMA braces were
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Table 3 Ground motion data. Earthquake
Year
PGA (g)
Duration (s)
ElCentro Kobe Tabas
1940 1995 1978
0.348 0.599 0.934
53.74 48 35
natural period of the corresponding C-case frames (note that, since the mass of the brace is small compared to the entire structure, the mass of the corresponding frame cases A and C were assumed equal). Based on these considerations, the following relations were achieved: Fig. 7. Verification: uniaxial tension stress–strain response. Experimental data [40] and numerical simulation.
ASMA = LSMA =
0.06
Fy
σsAS
=
2400 × Asteel
E SMA ASMA
4140
=
K = 0.08 × LSteel .
0.05
= 0.58 × Asteel
275790 × 0.58ASteel 2e6 × ASteel
(1)
× LSteel (2)
Strain
0.04 0.03 0.02 0.01 0 0
500
1000
1500
2000
2500
3000
3500
Since the required geometric properties (i.e. cross-sectional area, ASMA and element length, LSMA ) of the superelastic SMA braces can be calculated easily. It was also assumed that the SMA elements are made of a number of large diameter superelastic bars able to undergo compressive loads without buckling.
Time [s]
4. Nonlinear time history analyses and results Fig. 8. Strain paths applied in the experiment [37]. 0.08 0.06 0.04 Strain
0.02 0 -0.02
0
5
10
15
20
25
30
-0.04 -0.06 -0.08 Time [s]
Fig. 9. Strain paths applied in the Ref. [38].
designed to provide the same yielding strength, Fy and the same axial stiffness, K , as steel braces (buckling restrained braces in C-case frames). As a result, the A-case frames will have the same
As previously mentioned, in order to study in detail the dynamic behavior of SMA braced system, which acts in both tension and compression, nonlinear time history analyses were performed. Note that all the frame models were analyzed as two-dimensional (2D) models. 4, 6, 8, 10, 12 and 14 story frames with SMA bracing systems were considered. In addition, different bracing configurations (i.e. diagonal, split X, chevron V and inverted-V) were investigated. For conducting time history analyses on the frames implementing SMAs, the same three ground motion (GM) records that were used for analyses of BRBFs [30] were used again here, the details of these ground motion records can be seen in Table 3 and Fig. 12. Since the frames with buckling restrained braces have the same period as the corresponding frames with superelastic SMA braces, a good comparison can be made on the effectiveness of using the proposed SMA bracing system. To inspect the results of time history analyses, we first focus on some response results in regards to a case study and then an overall discussion is presented.
Fig. 10. Verification: stress–strain hysteresis loops for the strain path shown in Fig. 8. Numerical results for the adopted model (left) and experimental results [37] (right).
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750
0.75
600
0.60
450
0.45
300
0.30
150
0.15
0 -0.08
-0.06
-0.04
-0.02 0 -150
0.02
0.04
0.06
0.08
Stress
Stress MPa
70
-0.08
-0.06
-0.04
0.00 -0.02 0.00 -0.15
-300
-0.30
-450
-0.45
-600
-0.60
-750
-0.75 Strain
Strain
0.02
0.04
0.06
0.08
Fig. 11. Verification: stress–strain hysteresis loops for the strain path shown in Fig. 9. Numerical results for the adopted model (left) and the Ikeda model in Ref. [38] (right).
1.6 1.4
Sa =B×A Sa (ξ=5%)
1.2 1 0.8 0.6 0.4 0.2 0 1
2 Period (sec)
3
4
Fig. 12. Variation of spectral acceleration with period of structure.
4.1. Case study: ten-story braced frame The ten-story V-braced frame undergoing the ElCentro ground motion is chosen to be considered in detail. The force–displacement plots for one of the first floor braces in each braced system are shown in Fig. 13. Deformations occur in the buckling restrained braces, 16 mm, while the SMA members have lower displacement, 6.9 mm, due to the recentering capability guaranteed by the superelastic effect. The plots obviously indicate the ability of superelastic SMA bracing system to provide recentering, undergo compressive loadings, and stiffen at large strains as compared to the buckling restrained braces (BRBs). Further, a comparison of the hysteretic loops of the two types of braces suggests that BRBs allow for larger ductility demand with respect to superelastic SMA braces, while dissipating a significant amount of seismic energy due to their wider hysteretic loops. On the other hand, the innovative bracing system shows higher values of axial force due to hardening of the material, which may limit the displacements for unexpected strong earthquakes but at the same time, may result in larger shear forces transmitted in beams and columns. The time history for the top floor displacement of both frames is provided in Fig. 14. The peak roof displacements for the BRBF and SMA braced frame are approximately 233 mm and 193 mm, respectively. Since the beam and column members are the same in both of the frames and the yield force and initial stiffness of both types of bracing systems are constant, the difference in response must be attributed to the different behavior of the buckling restrained braces as compared to the superelastic SMA braces. As another important result, we can compare the values of permanent (residual) roof displacement in Fig. 14. The use of SMA braces results in a permanent roof displacement of approximately 1.21 mm while the buckling restrained braced frame undergoes
Fig. 13. First floor bracing force–displacement history for buckling restrained and SMA (Case A) braced frame (ten-story chevron V braced frame subjected to the ElCentro ground motion).
20 15 10 Displacement (mm)
0
5 0 -5 -10 -15 -20 -25 -30 0
5
10
15
20
25
30
35
40
45
50
55
Time (sec) Fig. 14. Roof displacement history of buckling restrained and SMA (Case A) braced frame (ten-story chevron V braced frame subjected to the ElCentro ground motion).
145.4 mm of residual displacement. These results reveal the excellent performance of SMA braces in reducing the residual displacement of the top floor. Besides, the maximum inter-story drift in the case of frame with buckling restrained braces is 1.17% whereas it is decreased to 0.82% using SMA braces (approximately 30% reduction). 4.2. Overall study: results and discussion The results from the case study show that the SMA braces work well at reducing both inter-story drifts and residual displacements.
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Fig. 15. Mean of the maximum inter-story drift for the chevron V (left) and inverted-V (right) braced frames (braces in case A, B and C).
Fig. 16. Mean of the maximum inter-story drift for the split X (left) and diagonal (right) braced frames (braces in case A, B and C).
Fig. 17. Permanent roof displacement for the chevron V (left) and inverted-V (right) braced frames subjected to ElCentro ground motion (braces in case A, B and C).
However, it is important to verify that the findings of the case study are consistent over the ground motion records. In addition, in order to have confidence in the predicted behavior of SMA bracing systems in real world structures, it is necessary to have a more comprehensive study of braced frame structures with various story numbers and different bracing configurations. Figs. 15 and 16 provide the mean values of maximum interstory drift for 4, 6, 8, 10, 12 and 14 story frames with four different bracing configurations and three varied bracing systems (for more brevity, mean values for the three GM records are presented). As noted earlier, the two frame cases A and C are comparable and the B-case frames represent the frames that braces include the SMA elements connected to the buckling restrained segments. Apparently, the average value of maximum inter-story drift is smaller for all the A-case frames as compared to the buckling restrained braced frames (case C). The results suggest that the ability of the SMA brace hysteresis to pass through the origin during cycling while undergoing large deformations reduces the maximum inter-story drift (on average 16% up to 60%, the least and peak difference) and subsequently the demand on the column members at each floor level.
As well as the maximum inter-story drift, the residual roof displacement can be considered to study the seismic performance of structures subjecting to dynamic loads. Figs. 17–22 show the values of residual roof displacement for the considered frames and the three aforementioned GM records. In almost all the results, the residual roof displacement for the frames equipped with the SMA braces (case A) is less than that for BRBFs (case C). These results suggest that the recentering capability and stiffening at large strains associated with the SMAs has a major influence in reducing the permanent deformation in the structures. It is important to notice that in some instances like the 12 story frame with chevron V, inverted-V and/or split X braces undergoing the Tabas GM record using the SMA bracing systems in place of the buckling restrained braces results in higher value of permanent roof displacement unexpectedly. This can be attributed to post-inelastic behavior (i.e. branch observed at the end of the upper plateau) of the SMA braces, which in their fully martensitic phase transmit high values of forces to columns and/or beams with consequent structural problems caused by yielding. In fact, this is a limitation of the model since the true behavior of a SMA material would show a third softening branch after yielding
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Fig. 18. Permanent roof displacement for the split X (left) and diagonal (right) braced frames subjected to ElCentro ground motion (braces in case A, B and C).
Fig. 19. Permanent roof displacement for the chevron V (left) and inverted-V (right) braced frames subjected to Kobe ground motion (braces in case A, B and C).
Fig. 20. Permanent roof displacement for the split X (left) and diagonal (right) braced frames subjected to Kobe ground motion (braces in case A, B and C).
Fig. 21. Permanent roof displacement for the chevron V (left) and inverted-V (right) braced frames subjected to Tabas ground motion (braces in case A, B and C).
of the martensite [22]. However, in structural design of SMA braces the force transferring to the other structural members as a result of the phase transformation (after martensitic yielding)
should be limited. This issue can be guaranteed by changing the mechanical properties of the SMA segments and shifting the phase transformation stress level to a lower one.
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Fig. 22. Permanent roof displacement for the split X (left) and diagonal (right) braced frames subjected to Tabas ground motion (braces in case A, B and C).
Although the B-case frames (i.e. the frames that have SMA bar segments connected to the frame through buckling restrained elements as bracing system) are not comparable to the other frame cases, it can be claimed that using a buckling restrained segment connected to the SMA element in bracing can provide the structure with a greater energy dissipating capability. Because of this ability, the response of the B-case frames can be even less than that for the A-case frames. To the contrary, in some instances, the response is higher as compared to BRBFs. Therefore energy dissipation capability of the structure can be increased through using the buckling restrained segment in SMA-based bracing system. Fig. 23 represents the added energy dissipation associate to the buckling restrained segments in the B-case frame. As shown, the BRBFs (i.e. case C) can exhibit higher amount of energy dissipation compared to the other frame cases. However, larger drifts occurred in respect to SMA-braced frames (case A).
Fig. 23. Base shear-first story drift for the 8 story inverted-V braced frame subjected to Tabas ground motion.
5. Conclusion
References
In this paper, the dynamic responses of twenty-four steel braced frames with various story numbers and types of bracing were evaluated aimed at investigating the SMA efficiency in seismic response reduction of structures. After verifying the model of superelastic behavior of shape-memory alloys, nonlinear time history analyses were carried out using the OpenSees computational platform. The seismic performance of the frame implementing superelastic SMAs was compared to the same frame with buckling restrained braces. The results of this comparative study showed that shapememory alloys could be effectively utilized for seismic design of structures. SMA braces can provide a means of minimizing losses associated with damage to structural systems during an earthquake, regardless of both the brace configuration and height of the structure. Although, buckling restrained braces possess a bigger energy dissipation capacity, some abilities of SMA elements such as recentering and undergoing strain hardening, make these alloys desirable for vibration response reduction. As observed in this study, implementing the SMA braces can lead to a reduction in residual roof displacement and peak inter-story drift with respect to the buckling restrained braced frames. In addition, the energy dissipation capability of the structure can be increased through using the buckling restrained segment in SMA-based bracing systems. Although the results of this article suggest promise for implementing the SMAs in braced frames, further works still are necessary to confirm the numerical simulations and the analytical results experimentally. In addition, it must be pointed out that there should be a methodology for seismic design of the braced steel frames endowed with SMA bracing systems.
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