Investigation of the Seismic Performance of a Multi-Story, Multi-Bay Special Truss Moment Steel Frame with X-Diagonal Shape Memory Alloy Bars

Abstract

In this work, the seismic response of a multi-story, multi-bay special truss moment frame (STMF) with Ni-Ti shape memory alloys (SMAs) incorporated in the form of X-diagonal braces in the special segment is investigated. The diameter of the SMAs per diagonal in each floor was initially determined, considering the expected ultimate strength of the special segment, developed when the frame reaches its target drift and the desirable collapse mechanism, i.e., the formation of plastic hinges, according to the performance-based plastic design procedure. To further investigate the response of the structure with the SMAs incorporated, half the calculated SMA diameters were introduced. Continuing, three more cases were investigated: the mean value of the SMA diameter was introduced at each floor (case DC1), half the SMA diameter of case DC1 (case DC2), and twice the SMA diameter of case DC1 (case CD3). Dynamic time history analyses under seven benchmark earthquakes were conducted using commercial nonlinear Finite Element software (SeismoStruct 2024). Results were presented in the form of top-displacement time histories, the SMAs force–displacement curves, and maximum inter-story drifts, calculating also maximum SMA displacements. The analysis outcomes highlight the potential of the SMAs to be considered as a novel material in the seismic retrofit of steel structures. Both design approaches presented exhibit a certain amount of effectiveness, depending on the distribution, with the placement of the SMA bars and the seismic excitation considered. Further research is suggested to fully understand the capabilities of the use of SMAs as dissipation devices in steel structures.

1. Introduction

Shape memory alloys (SMAs) are metallic smart materials capable of sustaining large inelastic strains that can be recovered by heating or by unloading [1], exhibiting the shape memory effect and the superelastic effect, respectively. The origin of this behavior is their ability to form two crystal structures through the rearrangement of atoms within the crystal lattice, the austenitic, and the martensitic [2]. SMAs also exhibit some other characteristics which are the following: ductility, low elastic anisotropy, superb corrosion and abrasion resistance [3], excellent fatigue performance, and their thermo-mechanical properties can be controlled by the selection of the material composition and various heat treatments [4]. In structural engineering applications, systems with SMA-based elements can be used as dissipation devices exploiting their energy dissipation capacity and their recentering capability, enhancing the stability and reliability of steel framing structures and preserving their structural integrity and functionality. Numerous other publications concerning the application of SMAs in civil structures may also be cited. An excellent relevant work is Song et al. [5], which provides valuable information about the subject, while the work by Habieb et al. [6] may also be quoted.

In the following paragraphs, recent advances in scientific research on SMA applications in framing systems are summarized, highlighting the potential regarding their deployment as diagonal braces.

Lafortune et al. [7] investigated experimentally (via reduced scale tests) as well as analytically the possible benefits of the use of SMA braces for the control of the seismic response of a single-bay frame, and its advantage over the use of steel braces of the same stiffness and yield force characteristics. At the same time, an analytical verification study was also presented, based on the FE modeling of the cyclic behavior of SMAs. Both approaches were focused on the effect of pre-straining the SMA braces, and the results obtained revealed an optimum level of pre-strain range from 1 to 1.5%. A consistent reduction in the response (due mainly to recentering) in the presence of SMA braces was reported. The need for further studies regarding SMA models under pre-strain conditions was also pointed out.

Furthermore, McCormick et al. [8], studied three- and six-story concentrically braced frames equipped with SMA braces, to establish their seismic response compared to traditional relevant structures. The authors developed an analytical model of the former type of frames in detail, and two suits of ground excitations were introduced to evaluate their performance, in terms of inter-story and residual drift. The results of the investigation showed that the use of SMA braces satisfactorily addresses the lack of ductility and energy dissipation issues reported in traditional solutions. The development of more precise simulations of SMA systems was also suggested.

Ghassemieh and Kargarmoakhar [9] evaluated the seismic response of steel-braced frames employing shape memory alloy braces. Various buildings of several heights and different bracing systems were selected to examine the overstrength, ductility, and response modification factors of braced frames utilizing SMAs.

Static pushover analysis, incremental nonlinear dynamic analysis, and linear dynamic analysis were carried out to predict the behavior and calculate overstrength, ductility, and response modification factors. The effects of building height, number of spans, and different types of bracings including diagonal X and Chevron are considered. Comparing the response modification factor of varying system configurations, the study indicated that the seismic response of Chevron bracing is preferable when compared with other types of bracing.

The first objective of the work presented by Vafaei and Eskandari [10] was to compare the dynamic behavior of mega SMA braced frames subjected to far-fault and near-fault seismic excitations.

A significant number of frames in the vicinity of either of these fault types with varying geometries were studied via nonlinear dynamic analyses. The investigation results clearly showed the differences in response between low- and high-rise frames, depending on the location of the faults. Moreover, the second objective aimed to demonstrate the excellent performance of SMA braces compared to BRBs, a fact especially valid for frames under near-fault ground motions.

Qiu and Zhu [11] investigated the seismic design of SC steel frames equipped with SMABs. A performance-based plastic approach, along with nonlinear time history analyses, validated the proposed PBSD. The SMABs were assumed to function in an indoor environment; hence, the authors suggested developing design methods suitable for accounting for temperature changes in future studies.

In another study, Qiu et al. [12] investigated analytically the seismic performance of four frames incorporating SMA braces (SMABs) with various story numbers by subjecting them to ground motion records corresponding to design-based earthquakes. SMABs were chosen since they can exhibit large ductility and various hysteretic parameters, depending on the specific configuration, working mechanism, and SMA types. Compared with buckling-restrained braces, the SMABs used showed a high ‘post-yield’ stiffness ratio and a low energy dissipation capacity. The analysis results showed that all the resulting frames were capable of coinciding well with the prescribed targets of peak inter-story drift ratios and ductility demands and the structures returned to at-rest positions without producing residual deformation after earthquake events.

Sultana and Youssef [13] investigated the potential of using SMA braces to reduce seismic residual deformation and improve the seismic performance of typical modular steel braced frames (MSBFs). A six-story modular steel building (MSB) located in Vancouver, Canada was selected as a case study. Incremental dynamic analysis (IDA) was utilized to judge the benefits of such a system. Five different schemes of SMA braces were investigated using rigid elements with the SMA truss element to make sure that all the deformations would happen in the SMA segments of braces. It was observed that utilizing superelastic SMA braces can reduce inter-story residual drift. The study also concluded that instead of replacing all steel braces with SMA braces, the best seismic performance can be achieved using SMA braces at some desired locations, highlighting that Frame 6 configuration can be considered the most suitable solution.

Shi et al. [14] dealt with the influence of the properties of SMA design on the performance and collapse characteristics of SMA-braced frames, under seismic actions. The authors explored the SMA failure using pushover and IDA analyses, and quite interesting results for structural design purposes were obtained, as given in the conclusions of the full-length paper.

Mirzay and Attarnejad [15] performed nonlinear dynamic time history and pushover analysis for nine-story steel frame buildings designed for a seismically active site (Tehran), incorporating SMA links to evaluate the system performance.

SMAs were selected to address the limitation of conventional eccentrically braced frames (EBFs) with vertical links to achieve better residual and maximum inter-story drifts. Their study presents a vibration control system equipped with SMAs acting as a rapid repair fuse to achieve improved seismic performance. Compared to conventional EBFs, the proposed system named recentering damping device (RDD) is easy to fabricate and implement and allows for the redesign of fuse members. Results of the time history analysis demonstrate the better performance of the proposed system as compared to EBF, as it effectively reduced the maximum inter-story drift ratio by 57% and the residual drift by up to 86.99% and produced minimal residual deformation.

The subject of a recent publication by Qiu et al. [16] was the comparative study of the seismic behavior of knee-braced frames having either steel or NiTi buckling-restrained braces (BRBs). It was found that NiTi braces are advantageous over steel ones since they are capable of eliminating residual drift and hence possess a higher seismic resilience.

Moreover, Qui and collaborators [17] extended in the same year the previous work by analyzing multi-story steel frames with iron-based SMA BRBs (FeSMA) incorporated. The authors concluded (once more) that this type of BRBs controls in a better manner than the steel ones’ residual inter-story drift ratios.

Thaer and Asheena [18] conducted finite element analysis in a three-bay portal frame with and without braced systems by performing nonlinear static analysis, considering different configurations of bracings with SMAs. The results showed that using SMAs can improve seismic performance in steel frames by dissipating energy and reducing stresses.

Hassanzadeh and Moradi [19] studied the seismic topology optimization of steel-braced frames with SMA NiTi or Fe-based braces. An optimization algorithm was implemented to optimize SMA-BFs in a performance-based context following ASCE 41-13. The optimization study found eight optimal SMA-BFs with either NiTi or Fe-based braces. Incremental dynamic analysis was performed on the optimal frames to extract fragility curves. The results demonstrated that Fe-based SMA braces exhibited higher collapse capacity and a more uniform distribution of lateral displacement over the frame height while being more cost-effective than NiTi SMAs and reducing material usage. In frames with Fe-based SMA braces, the material usage was reduced up to 80%.

Within the above context, the scope of the present work is to evaluate the efficiency and highlight the potential of X-braced SMA configurations incorporated in the special segment of multi-story special truss moment frames (STMFs) considering five different dimensioning cases, as innovative high-performance materials for vibration control exhibiting the superelastic effect through which their initial geometry can be recovered after mechanical distortion by unloading through the phase transformation (martensitic-austenitic phase). The initial energy dissipation mechanism of STMFs, which consisted of the formation of four plastic hinges in a special segment located in the middle of the truss, is replaced with the input energy dissipation through the SMA deformation in a pin-ended special segment. Nonlinear dynamic time history analysis is conducted, applying finite element methodology under selected ground motions, and the proposed schemes are compared in terms of maximum displacement and maximum inter-story drifts with the ones of the conventional frame.

Two design approaches are proposed, the one based on the distribution of the SMAs according to the performance-based calculation findings (different diameters on each floor) and the alternative one on the same diameter (three cases) on each floor. It was found that both approaches contain a certain amount of effectiveness, depending on the case considered and the seismic event. The whole subject is still open, but the use of SMAs is indeed encouraging.

2. Dynamic Response of a Multi-Story STMF with SMAs

2.1. Description

A benchmark nine-story, five-bay structure with STMFs of open Vierendeel type, studied by Chao and Goel [20], was selected to examine its behavior incorporating STMFs with X-diagonal Ni-Ti SMA bars as dissipation devices in a pin-ended special segment under seismic loading. Standard profiles of the US steel industry were used for the members of the frame, which are given in detail in the publication mentioned above. The structure is 45.70 m in length and 39.90 m in elevation and it consists of five 9.14 m wide bays on each floor. The levels of the structure are numbered concerning the ground level. The 9th level is the roof and under the ground level, there is a basement. The floor-to-floor height of the basement is 4.30 m, for the 1st floor is 5.50 m, and for the 2nd through 8th floors is 4.30 m. The height of the truss is 1.20 m and the length of the special segment is 2.43 m. Columns are made of 355 MPa steel, placed in strong axis bending. Column bases are pinned and secured to the ground. Concrete foundation walls and the surrounding soil are assumed to restrain the structure at the ground level from horizontal displacement.

The original structure was designed according to the performance based design procedure (PBDP), aiming to achieve a predictable and controlled yield mechanism through the formation of plastic hinges at the four corners of the chord members of each special segment. A thorough description of PBDP is given (among others) in the work by Goel et al. [21] and the review article of Ghobarah [22]. Members outside of the special segment need to remain elastic and are designed to resist the gravity loads and the maximum expected vertical shear strength ($V_{ne}$), developed at the midpoint of special segments when the frame reaches its target drift and the desirable collapse mechanism, i.e., the formation of plastic hinges. The original structure is modified by placing SMA bars as dissipation devices in pin-ended chord members in the special segment. All members’ sections are kept the same as derived from the design of the conventional structure. Both structures, the conventional and the proposed one, are modeled using SeismoStruct 2024 software [23]. Columns of both structures were modeled as elastic elements. Chord and truss members outside of the special segment of the conventional structure were modeled as elastic elements except for the chord members in the special segment, which were modeled as inelastic elements with concentrated inelasticity within a fixed length. On the contrary, chord and truss members outside of the special segment of the proposed structure were modeled as inelastic elements while chord members in the special segment were modeled as elastic elements with moment releases at both ends. Chord members support gravity loads as well. Simple connection details were considered for girder-to-column moment connections. Double channels were used for all truss members. The undeformed shape of the conventional structure without SMAs and without incorporating a pin-ended special segment is depicted in Figure 1.

Regarding the inelasticity of selected members, the FE procedure adopted was the more recent forced-based (FB) formulation, a brief description of which can be found in the software’s user manual. Moreover, the damping ratio used in the analyses was equal to 5%, a value typical for steel structures.

The SMA rate-independent constitutive law se_sma2 (of Abdulridha et al. 2013 [24]) embedded in SeismoStruct was used. SMAs were modeled via SeismoStruct as truss elastic elements since they undergo only axial deformation. They were considered to be made of a circular solid section, to have a length of 2.71 m, a Young’s modulus of 35 GPa, 300 kPa forward and reverse transformation stress, 373 kPa forward transformation finishing stress, and 150 kPa reverse transformation finishing stress. Although in practical engineering, the same diameter of SMA X-bars in each special segment on all frame floors is incorporated (for convenient construction), in this work, an alternative and novel approach is first encountered in the relevant literature. Namely, the dimensioning of SMAs on each floor was performed by taking into account the maximum expected vertical shear strength developed in the midpoint of the special segment ($V_{ne}$). For this structure, one SMA per diagonal was placed to resist the $V_{ne}$ developed in the special segment. The calculated SMA diameters on each floor are given in

Table 1. SMA diameter/diagonal at each floor.

Floor	SMA Diameter (m)	Vne (kN)
9	0.035	598
8	0.046	1019
7	0.052	1308
6	0.057	1581
5	0.061	1821
4	0.061	1821
3	0.061	1821
2	0.066	2126
1	0.066	2126

.

The SMA bars used are assumed to be equipped with buckling-restrained devices, or to be large enough in diameter (which is the case of the ones in Table 1) to prevent buckling in all stages of dynamic deformation.

2.2. Dynamic Time History Analysis

To verify the effectiveness of the SMA incorporation in the STMF configuration, the dynamic response of the conventional STMF (abbreviated as Fr1) is compared with the corresponding one of the proposed configurations with the full diameter of SMAs in the special segment (Fr2) and with the one with half the diameter (Fr3) listed in Table 1.

In our work, the systems/structures dynamics were analyzed until the ground motion seizes. Thereafter (free motion), the frames’ recentering (due to the SMAs) capability will surely exhibit its benefits. At the end of the Conclusion section, the continuation of the analysis is suggested for future work.

However, in the very recent literature, other types of self-centering dissipative braces/devices were reported. These can be purposely designed to ensure the absence (or at least minimize) of residual inter-story drifts at the end of the earthquake shaking. In this regard, one must quote the pertinent works of De Domenico et al. [25] and Yan et al. [26].

Before discussing the dynamic history analysis results that follow, and to gain a first insight of the dynamics of the frames considered, the outcome of free vibration in terms of the two first eigen periods is in order. These are contained in

Table 2. First and second eigen periods of the frames dealt with.

Frame	T1 (s)	T2 (s)
Fr1	3.45	1.24
Fr2	3.07	1.09
Fr3	4.52	1.55

.

The presence of the full diameter of SMAs decreases the fundamental period to about 11% and the second one to about 12.1%. Contrary to this, for Fr2, an increase in T₁ and T₂ to about 31% and 25%, respectively, is observed.

Seven benchmark earthquake records were selected with varying characteristics of PGA and magnitude, as shown in

Table 3. Characteristics of benchmark earthquakes.

Earthquake/Year/Location	PGA (g)	Magnitude (Mw)	Horizontal Component
Loma Prieta/1989/California	0.37	6.93	ESE
Northridge/1994/California	0.57	6.69	NNW
Kocaeli/1999/Izmit, Turkey	0.35	7.51	WSW
Imperial Valley/1997/Mexico	0.32	6.53	NNW
Kobe/1995/Japan	0.35	6.90	NW-WNW
Chichi/1999/Taiwan	0.36	7.62	N-NNW
Landers/1992/California	0.78	7.28	NNE

. These can be considered sufficient in both number and salient features for the dynamic time history analysis given in this section. Their vast response spectrum records, as well as the numerous average response spectra, not only vary but also differ significantly, but the two first periods of the structures dealt with are contained in the period of each spectrum. The direct integration of the equations of motion is accomplished using the numerically dissipative α-integration algorithm [Hilber et al., 1977] or a special case of the former, the well-known Newmark scheme [Newmark, 1959], with automatic time–step adjustment for optimum accuracy and efficiency; modeling of seismic action is achieved by introducing acceleration loading curves (accelerograms) at the supports.

For the first earthquake record selected, the Loma Prieta one (LPE), using SMAs as in Table 1 per diagonal, seems to lead to a symmetrical response, as shown by the displacement time history (Figure 2). Recentering is encountered, due to the presence of the SMAs, which is quite important and advantageous for the size of the structure dealt with. SMAs respond linearly at lower floors up to the 5th floor, and at the 6th, 7th, and 8th floors, they exhibit a small loop, and at the 9th floor, a loop of 0.057 m (2.1% strain) maximum displacement (

Table 4. Maximum SMA displacements for LPE.

	Table 1 SMAs		Half the Diameter of Table 1
Floor	Displacement (m)	Corresponding Strain	Displacement (m)	Corresponding Strain
9	0.057	2.1%	0.26	9.6%
8	0.045	1.6%	0.17	6.3%
7	0.040	1.5%	0.10	3.7%
6	0.035	1.3%	0.06	2.2%
5	Linear response	Negligible	0.05	1.8%
4	Linear response	Negligible	0.05	1.8%
3	Linear response	Negligible	0.05	1.8%
2	Linear response	Negligible	0.04	1.5%
1	Linear response	Negligible	0.032	1.2%

).

The incorporation of half the calculated diameter of SMAs per diagonal leads to a more flexible structure (Figure 3). SMAs at lower floors up to the 7th floor undergo deformations up to 0.1 m (3.7% strain). At upper floors (8th, 9th), they undergo deformations up to 0.26 m. This corresponds to a 9.6% strain. For this study, to consider that the SMA deforms in the range of superelasticity, the strain that it undergoes must be under 6%. Consequently, this is a large strain for the SMAs to accommodate and can provoke plastic deformation by slip. As far as inter-story drifts are concerned (Figure 4a), the incorporation of SMAs according to $V_{ne}$ results in an even distribution with larger values compared to those of the structure with the conventional STMF at the lower floors, and smaller at the upper floors. This can be attributed to the fact that the structure with SMAs has pin-ended special segments and to the fact that SMAs respond linearly at the lower floors, while at the upper floors, they exhibit a loop. The structure with the conventional STMF and the structure with half the calculated diameter of SMAs presents large drift values at upper floors indicating damages. Using a target drift of 2% for a ground motion hazard level, with a 10% probability of exceedance in 50 years as a criterion, as Chao and Goel also used in their 2006 report (see [2]), it is observed that only the structure with the SMAs according to $V_{ne}$ has drifts below the drift limit.

For the Northridge earthquake (NE), SMAs deform in the limits of superelasticity up to the 8th floor (

Table 5. Maximum SMA displacements for NE.

	Table 1 SMAs		Half the Diameter of Table 1
Floor	Displacement (m)	Corresponding Strain	Displacement (m)	Corresponding Strain
9	0.17	6.3%	0.28	10.3%
8	0.065	2.4%	0.1	3.6%
7	0.025	0.9%	0.09	3.3%
6	0.02	0.74%	0.06	2.2%
5	0.03	1.1%	0.06	2.2%
4	0.025	0.9%	0.06	2.2%
3	Linear response	Negligible	0.05	1.8%
2	Linear response	Negligible	0.05	1.8%
1	Linear response	Negligible	0.04	1.4%

). On the 9th floor, the maximum SMA displacement of 0.15 m slightly exceeds the 6% strain limit of superelasticity. The incorporation of half the calculated diameter leads to larger displacements but within the limits of superelasticity, except for the 9th floor where the SMA displacement grows dramatically up to 0.28 m, corresponding to a 10.3% strain. The performance of all three structures is revealed via the maximum inter-story drift diagram (Figure 4b), where the upper floors exhibit values greater than the target drift limit of 2%. However, the incorporation of SMAs according to Table 1 makes the inter-story drift distribution more uniform with larger values on the lower floors and smaller on the upper floors, compared to those of the conventional structure.

For the Kocaeli (K1E), the Imperial Valley (IVE), and the Kobe (K2E) earthquakes, SMAs deform in the limits of superelasticity at all floors (

Table 6. Maximum SMA displacements for K1E.

	Table 1 SMAs		Half the Diameter of Table 1
Floor	Displacement (m)	Corresponding Strain	Displacement (m)	Corresponding Strain
9	0.096	3.5%	0.46	16.9%
8	0.05	1.8%	0.36	13.2%
7	0.07	2.6%	0.25	9.2%
6	0.08	3.0%	0.17	6.3%
5	0.12	4.4%	0.12	4.4%
4	0.15	5.5%	0.16	5.9%
3	0.16	5.9%	0.15	5.5%
2	0.14	5.2%	0.12	4.4%
1	0.10	3.6%	0.10	3.6%

,

Table 7. Maximum SMA displacements for IVE.

	Table 1 SMAs		Half the Diameter of Table 1
Floor	Displacement (m)	Corresponding Strain	Displacement (m)	Corresponding Strain
9	0.05	1.8%	0.17	6.3%
8	0.03	1.1%	0.1	3.6%
7	Linear response	Negligible	0.065	2.4%
6	Linear response	Negligible	0.045	1.6%
5	Linear response	Negligible	0.04	1.4%
4	Linear response	Negligible	0.045	1.6%
3	0.02	0.74%	0.04	1.4%
2	0.02	0.74%	0.035	1.3%
1	0.02	0.74%	0.03	1.1%

and

Table 8. Maximum SMA displacements for K2E.

	Table 1 SMAs		Half the Diameter of Table 1
Floor	Displacement (m)	Corresponding Strain	Displacement (m)	Corresponding Strain
9	0.026	0.95%	0.20	7.4%
8	0.03	1.1%	0.08	3.0%
7	0.02	0.74%	0.04	1.4%
6	0.016	0.59%	0.05	1.8%
5	0.01	0.36%	0.04	1.4%
4	0.008	0.3%	0.04	1.4%
3	0.01	0.36%	0.044	1.6%
2	0.01	0.36%	0.04	1.4%
1	0.01	0.36%	0.03	1.1%

). The incorporation of half the calculated diameter of SMAs per diagonal leads to a more flexible structure with SMA displacements exceeding the 6% limit. The maximum inter-story drift diagram for the Kocaeli earthquake (Figure 5a) reveals that the structure with the conventional STMF presents a more even distribution with lower values compared to those of the other two configurations. All inter-story drift values are equal or greater to the target drift limit of 2%, indicating that the structures undergo damage, while for the Imperial Valley (Figure 5b) and the Kobe earthquakes (Figure 6a), a more uniform drift distribution of the structure with the SMAs according to $V_{ne}$ compared to the conventional STMF is revealed, with drifts below the 2% drift limit; the incorporation of half the calculated diameter of SMAs causes drifts larger than the 2% limit at upper floors and damage is caused.

For the Chichi earthquake (CE), SMAs respond to the limits of superelasticity at all floors (

Table 9. Maximum SMA displacements for CE.

	Table 1 SMAs		Half the Diameter of Table 1
Floor	Displacement (m)	Corresponding Strain	Displacement (m)	Corresponding Strain
9	0.025	0.90%	0.10	3.6%
8	0.02	0.74%	0.065	2.4%
7	0.016	0.59%	0.05	1.8%
6	0.016	0.59%	0.045	1.6%
5	0.01	0.36%	0.045	1.6%
4	0.01	0.36%	0.05	1.8%
3	0.01	0.36%	0.045	1.6%
2	0.01	0.36%	0.042	1.55%
1	0.01	0.36%	0.035	1.3%

). The incorporation of half the calculated diameter of SMAs per diagonal still gives SMA responses in the limits of superelasticity, unlike the previously examined earthquakes. The maximum inter-story drift diagram (Figure 6b) shows more uniform drifts for the incorporation of SMAs according to $V_{ne}$ compared to the ones of the conventional structure.

For the Landers earthquake (LE), SMAs deform within the limits of superelasticity at all floors (

Table 10. Maximum SMA displacements for LE.

	Table 1 SMAs		Half the Diameter of Table 1
Floor	Displacement (m)	Corresponding Strain	Displacement (m)	Corresponding Strain
9	0.03	1.1%	0.12	4.4%
8	0.03	1.1%	0.09	3.3%
7	0.03	1.1%	0.09	3.3%
6	0.035	1.3%	0.08	3.0%
5	0.035	1.3%	0.06	2.2%
4	0.03	1.1%	0.042	1.55%
3	0.03	1.1%	0.04	1.4%
2	Linear	Negligible	0.03	1.1%
1	Linear	Negligible	0.03	1.1%

). The maximum inter-story drift diagram (Figure 7) shows that inter-story drifts, unlike the previously examined earthquakes, are not uniformly distributed along the height of the structure neither for the conventional nor for the incorporation of SMAs according to $V_{ne}$.

Taking into consideration the above remarks for each earthquake record, the following assumptions can be observed: For all earthquake records, except the Landers earthquake, the incorporation of SMAs according to the maximum expected vertical shear strength ($V_{ne}$) decreased the drift at upper floors and especially of the 9th floor, compared to the corresponding drift of the conventional structure. SMAs responded to the limits of superelasticity for all earthquakes on all floors except for the SMAs on the 9th floor for the Northridge earthquake, which slightly exceeded the 6% strain limit. The incorporation of SMAs with half the calculated diameter of Table 1 led to larger drifts and larger SMA deformations that, in some cases, led to SMA failure and more damages, as expected.

It should be noted that an integrated in-depth mechanism analysis of the dynamic response for all cases considered is not feasible, since it depends on the following varying parameters:

• The distribution, placement, and dimensioning of SMA X-bars in each special segment and on each floor, with a direct effect also on the free vibration characteristics of each frame configuration, i.e., on the eigenfrequencies and eigenmodes;
• The seismic excitation features (duration and PGA);
• The energy dissipated by each floor component, starting from ground one and ending at the roof.

All the above remarks are also valid for the alternative design approach that follows. However, the foregoing difficult task has been extensively investigated by the second author in her Ph.D. thesis [2].

2.3. An Alternative Design Approach

Instead of distributing the SMA X-diagonals in the special segment of every floor according to Table 1 and the half diameter of these, we introduce new proposed SMA distributions, based on incorporating the same diameter of SMA X-bars in each special segment in all floors of the frame. Namely, we introduce three new cases: the first, designated as DC1, is related to the incorporation of the mean Table 1 diameter of SMA bars to every floor i.e., bars of the diameter of 0.056 m at each diagonal; the second, DC2, the half of this number, i.e., 0.028 m; and the third, DC3, twice the mean number, i.e., 0.112 m.

As in Section 2.1, and for the same reasons mentioned therein, modal analysis results are also presented at this point and given in

Table 11. First and second eigen periods of the frames dealt with in the foregoing approach.

Frame	T1 (s)	T2 (s)
Fr1	3.45	1.24
DC1	3.23	1.08
DC2	4.75	1.50
DC3	2.64	0.92

.

DC1 and DC3 are related to a decrease in both eigen periods. Namely, for DC1, this amounts to 6.38% for T₁ (small but not negligible) and 12.9% for T₂. DC3 leads to a significant decrease in T₁ and T₂ up to 23.48% and 25.80%, respectively. However, for DC2, both T₁ and T₂ exhibit a significant increase (37.68% and 20.97%).

For the Loma Prieta earthquake, case DC1 SMAs respond to the limits of superelasticity (

Table 12. Maximum SMA displacements for LPE for the configurations investigated.

	DC1		DC2		DC3
Floor	Displacement (m)	Strain	Displacement (m)	Strain	Displacement (m)	Strain
9	0.015	0.55%	0.1	3.7%	Linear response	Negligible
8	0.02	0.7%	0.08	3.0%	Linear response	Negligible
7	0.03	1.1%	0.2	7.4%	Linear response	Negligible
6	0.04	1.4%	0.08	3.0%	Linear response	Negligible
5	0.03	1.1%	0.06	2.2%	Linear response	Negligible
4	0.03	1.1%	0.045	1.6%	Linear response	Negligible
3	0.03	1.1%	0.05	1.8%	Linear response	Negligible
2	0.02	0.7%	0.05	1.8%	Linear response	Negligible
1	0.03	1.1%	0.05	1.8%	Linear response	Negligible

). The maximum inter-story drift diagram (Figure 8a) shows uniform drift distribution. By comparing case DC1 drifts to those of the structure incorporating $V_{ne}$ dimensioning (Figure 4a), it can be observed that case DC1 drifts are larger on lower floors and smaller on upper floors. This is justified as the diameters obtained from $V_{ne}$ dimensioning (Table 1) are smaller than those of the DC1 case at upper floors (7th, 8th, 9th floors) and larger at lower floors (1st to 5th floors). The incorporation of half the case DC1 diameter of SMAs (case DC2) per diagonal leads to a more flexible structure with the SMAs exhibiting larger strains than the DC1 case, but still within the limits of superelasticity. Despite this, drifts at the upper floors are larger than the 2% target drift limit. Case DC3 SMAs respond elastically at all floors, and this indicates that the input energy is dissipated through the formation of plastic hinges at regions outside the special segment. Even though the special segment is pin-ended, the free rotation of the chord members inside the special segment relative to the trusses on either side is not allowed, due to the stiffness introduced by the large SMA diameter.

For the Northridge earthquake, for case DC1, SMAs respond in the limits of superelasticity (

Table 13. Maximum SMA displacements for NE for the configurations investigated.

	DC1		DC2		DC3
Floor	Displacement (m)	Strain	Displacement (m)	Strain	Displacement (m)	Strain
9	Linear response	Negligible	0.1	3.7%	Linear response	Negligible
8	0.025	0.9%	0.08	3.0%	Linear response	Negligible
7	0.025	0.9%	0.08	3.0%	Linear response	Negligible
6	Linear response	Negligible	0.075	2.8%	Linear response	Negligible
5	0.05	1.8%	0.055	2.0%	Linear response	Negligible
4	0.045	1.6%	0.065	2.4%	Linear response	Negligible
3	0.04	1.4%	0.07	2.6%	Linear response	Negligible
2	Linear response	Negligible	0.06	2.2%	Linear response	Negligible
1	0.04	1.4%	0.055	2.0%	Linear response	Negligible

) as well. Compared to SMA displacements dimensioned according to $V_{ne}$, a decrease of 62% is observed at the 8th floor’s maximum strain and a tremendous decrease in strain at the 9th floor. The maximum inter-story drift diagram (Figure 8b) shows uniform drifts for case DC1, which are smaller at upper floors compared to those of the conventional structure and the structure with SMAs according to $V_{ne}$ drifts. Case DC2 gives larger SMA strains than case DC1 but is still within the limits of superelasticity. However, the drift diagram shows an uneven distribution with drifts on the 9th floor approaching the 2% target drift limit, indicating that plastic hinges may have been formed at members outside the special segment. Case DC3 leads to a very stiff special segment; plastic hinges may form again at the members that are supposed to remain elastic, and SMAs on every floor exhibit negligible strains.

For the Kocaeli earthquake, SMA strains for the case DC1 (

Table 14. Maximum SMA displacements for K1E for the configurations investigated.

	DC1		DC2		DC3
Floor	Displacement (m)	Strain	Displacement (m)	Strain	Displacement (m)	Strain
9	Linear response	Negligible	0.27	9.9%	Linear response	Negligible
8	Linear response	Negligible	0.27	9.9%	Linear response	Negligible
7	0.05	1.8%	0.2	7.4%	Linear response	Negligible
6	0.1	3.6%	0.18	6.6%	Linear response	Negligible
5	0.15	5.5%	0.17	6.3%	Linear response	Negligible
4	0.18	6.6%	0.16	5.9%	Linear response	Negligible
3	0.18	6.6%	0.12	4.4%	Linear response	Negligible
2	0.15	5.5%	0.12	4.4%	Linear response	Negligible
1	0.12	4.4%	0.1	3.6%	Linear response	Negligible

) are larger at lower floors and negligible at the 8th and 9th floors due to the increased strength provided in the special segments. Case DC2 leads to a more flexible structure, with SMA strains exceeding the limit of superelasticity. The above remarks are also depicted in the maximum inter-story drifts diagram (Figure 9a), as inter-story drifts for the DC1 case at the upper floors are not only smaller than the corresponding ones of the conventional structure but also lower than the 2% target drift limit.

For the Imperial Valley earthquake, SMA strains for the case DC1 (

Table 15. Maximum SMA displacements for IVE for the configurations investigated.

	DC1		DC2		DC3
Floor	Displacement (m)	Strain	Displacement (m)	Strain	Displacement (m)	Strain
9	0.013	0.48%	0.08	3.0%	Linear response	Negligible
8	0.019	0.70%	0.08	3.0%	Linear response	Negligible
7	0.019	0.70%	0.075	2.8%	Linear response	Negligible
6	0.02	0.74%	0.06	2.2%	Linear response	Negligible
5	0.024	0.86%	0.05	1.8%	Linear response	Negligible
4	0.02	0.74%	0.05	1.8%	Linear response	Negligible
3	0.023	0.84%	0.047	1.73%	Linear response	Negligible
2	0.03	1.1%	0.045	1.6%	Linear response	Negligible
1	0.036	1.33%	0.04	1.4%	Linear response	Negligible

) are close to each other except for the bottom and the 9th floors. By comparing to the corresponding values of the distribution of SMAs according to $V_{ne}$, it can be perceived that placing smaller diameters (0.056 m) at lower floors than the one derived from $V_{ne}$ dimensioning provokes larger strains to SMAs. Case DC2 leads to a more flexible structure, but SMA strains do not exceed superelasticity limits. The maximum inter-story drifts diagram (Figure 9b) shows a rather uniform distribution for the DC1 case with reduced drifts at upper floors and an uneven one for the DC2 case. Case DC3 leads, once again, to a special segment with increased strength that cannot rotate freely and lets the SMAs incorporated deform, resulting in the uneven drift distribution depicted in Figure 9, which is an unwanted behavior.

For the Kobe and Chichi earthquakes, the incorporation of case DC1 SMAs (

Table 16. Maximum SMA displacements for K2E for the configurations investigated.

	DC1		DC2		DC3
Floor	Displacement (m)	Strain	Displacement (m)	Strain	Displacement (m)	Strain
9	0.015	0.55%	0.06	2.2%	Linear response	Negligible
8	0.015	0.55%	0.08	3.0%	Linear response	Negligible
7	0.016	0.59%	0.08	3.0%	Linear response	Negligible
6	0.02	0.74%	0.046	1.70%	Linear response	Negligible
5	0.017	0.63%	0.043	1.59%	Linear response	Negligible
4	0.017	0.63%	0.04	1.4%	Linear response	Negligible
3	0.016	0.59%	0.045	1.6%	Linear response	Negligible
2	0.015	0.55%	0.05	1.8%	Linear response	Negligible
1	0.015	0.55%	0.04	1.4%	Linear response	Negligible

and

Table 17. Maximum SMA displacements for CE for the configurations investigated.

	DC1		DC2		DC3
Floor	Displacement (m)	Strain	Displacement (m)	Strain	Displacement (m)	Strain
9	0.01	0.36%	Linear response	Negligible	Linear response	Negligible
8	0.014	0.52%	0.02	0.74%	Linear response	Negligible
7	0.016	0.59%	0.045	1.6%	Linear response	Negligible
6	0.017	0.63%	0.05	1.8%	Linear response	Negligible
5	0.018	0.66%	0.05	1.8%	Linear response	Negligible
4	0.017	0.63%	0.06	2.2%	Linear response	Negligible
3	0.015	0.55%	0.06	2.2%	Linear response	Negligible
2	0.016	0.59%	0.05	1.8%	Linear response	Negligible
1	0.018	0.66%	0.04	1.4%	Linear response	Negligible

) shows their activation at all floors. Compared to the corresponding values of $V_{ne}$ dimensioning (Table 8 and Table 9), it can be observed that SMA strains at upper floors decreased and at lower floors increased. Case DC2 SMA strains are also within the limits of superelasticity. Maximum inter-story drift diagrams (Figure 10a,b) show a uniform distribution for the DC1 case, with reduced drifts at upper floors compared to the conventional structure and the structure with SMAs according to $V_{ne}$ drifts, and an uneven one for the DC2 case. Case DC3 shows an uneven drift distribution for the Kobe earthquake while for the Chichi earthquake, the distribution is uniform.

For the Landers earthquake, the incorporation of case DC1 SMAs (

Table 18. Maximum SMA displacements for LE for the configurations investigated.

	DC1		DC2		DC3
Floor	Displacement (m)	Strain	Displacement (m)	Strain	Displacement (m)	Strain
9	Linear	Negligible	0.04	1.4%	Linear response	Negligible
8	Linear	Negligible	0.055	2.03%	Linear response	Negligible
7	Linear	Negligible	0.04	1.4%	Linear response	Negligible
6	0.03	1.1%	0.04	1.4%	Linear response	Negligible
5	0.035	1.3%	0.06	2.2%	Linear response	Negligible
4	0.045	1.6%	0.06	2.2%	Linear response	Negligible
3	0.04	1.4%	0.05	1.8%	Linear response	Negligible
2	0.05	1.8%	0.04	1.4%	Linear response	Negligible
1	0.04	1.4%	0.045	1.6%	Linear response	Negligible

) shows that at upper floors (7th, 8th, and 9th floors), they are activated in their elastic range as the strains that they exhibit are negligible, while at lower floors, they exhibit strains ranging from 1.1% to 1.8%. Case DC2 SMA strains are within the limits of superelasticity. The maximum inter-story drift diagram (Figure 11) depicts again a highly uneven distribution not only when SMAs are incorporated but also for the conventional structure.

As it can be deduced from the above, for case DC1 SMAs responded, in general, in the limits of superelasticity, exhibiting strains that decreased at upper floors and increased at lower floors compared to the corresponding ones of the structure with SMAs according to $V_{ne}$. This is also depicted on the maximum inter-story drifts as case DC1 drifts are larger at lower floors and smaller at upper floors. In addition, maximum inter-story drift diagrams show uniform drift distribution below the 2% target drift limit except in the excitation of the structure with the Kocaeli and Landers earthquakes. For the DC2 case, SMAs respond generally in the limits of superelasticity except for the Kocaeli earthquake. However, larger inter-story drifts occur at upper floors, exceeding, for some earthquakes, the 2% target drift limit, and the drift distribution is uneven for all earthquake records examined. The last case studied, designated as DC3, creates a very stiff special segment. For all earthquakes studied, SMAs respond elastically on all floors, and the input energy is dissipated through the formation of plastic hinges at regions outside the special segment, which is an undesirable performance. This STMF configuration fails to serve the desired behavior, which is the confinement of inelastic activity in the special segment through SMA deformation.

3. Conclusions

This work investigates the seismic response of a multi-story, multi-bay special truss moment frame (STMF) with shape memory alloys (SMAs) incorporated as X-diagonal braces in the special segment. The diameter of SMAs per diagonal on each floor was initially determined considering the expected ultimate strength of the special segment. To further investigate the structure’s response with the SMAs incorporated, half the SMA diameter of each floor of the previous case were introduced. Continuing, the mean value of the SMA diameter was introduced at each floor (case DC1), half of it (case DC2), and twice of it (case DC3). The outcome of this investigation can be summarized as follows:

• The analyses of the nine-story, five-bay structure’s displacement time histories for the examined earthquakes along with the maximum inter-story drift diagrams show that in the case of the conventional structure, the energy dissipation is achieved through the formation of plastic hinges. Moreover, the desired yield mechanism requires that the distribution of yielding is even along with the height of the structure to prevent extensive yielding from concentrating at a few floors and to achieve uniform behavior. Maximum inter-story drift diagrams present a uniform distribution, in general, revealing that the structure responds as it was designed. Drifts exceeded the 2% target drift limit for three earthquakes, either on the two top floors or on all floors.
• In the analyses of the first SMA distribution for all but one earthquake, the structure exhibits uniform drift distributions and a decrease in the drift at upper floors, especially on the 9th floor, compared to the corresponding drift of the conventional structure. All drifts were below the 2% target drift except for the case of one earthquake. SMAs responded to the limits of superelasticity for all earthquakes on all floors except the SMAs on the 9th floor for the Northridge earthquake, which slightly exceeded the 6% strain limit.
• The second distribution (half the diameter of the previous case at each floor) showed larger drifts, larger SMA deformations (leading to failure of the SMAs), and more damage was encountered.
• The third SMA bar distribution, which is related to the placement of the mean of the diameters of SMAs in all floors, as calculated from the original distribution, leads to the generation of a uniform drift distribution within the limits of superelasticity. The SMAs’ maximum displacements at each floor are close to each other along the height of the structure for almost every ground motion. Compared to the structure incorporating SMAs according to the maximum expected vertical shear strength, the maximum SMA strains decrease on upper floors and increase on lower floors. This is also deduced from the maximum inter-story drifts, as case DC1 drifts are larger on lower floors and smaller on the upper floors. This scheme seems to introduce more strength than needed at special segments on the upper floors. Consequently, the relative rotation of chord members in the special segment to the members outside the special segment may be obstructed. In that case, input energy may be dissipated through the formation of plastic hinges in members outside of the special segment, an unwanted behavior as the concept of the incorporation of the SMAs was to absorb the input energy through their deformation.
• DC2, the fourth SMA bar distribution, incorporates half the SMA diameter of case DC1 at each bay. SMAs respond to the limits of superelasticity except for one earthquake. However, larger inter-story drifts occur at upper floors, exceeding, for some earthquakes, the 2% target drift limit, and the drift distribution is uneven for all earthquake records examined.
• Finally, using twice the same diameters of SMAs along the structure’s height leads to a very stiff special segment. For all earthquakes studied, SMAs respond elastically on all floors, and the input energy is dissipated via the formation of plastic hinges at regions outside the special segment, which is an undesirable performance. This STMF configuration fails to serve the desired behavior, which is the confinement of inelastic activity in the special segment through the SMA deformation.
• Given the above, it can be concluded that the structure with the SMAs dimensioned according to ($V_{ne}$) exhibited an interesting performance showing uniform drift distributions with decreased drifts at upper floors compared to the conventional structure. The design methodology proposed is that of the SMA distribution according to $V_{ne}$, as inter-story drifts are more successfully controlled.
• For future work, the following are suggested:
• Free vibration analysis of the SMA-equipped frames after the end of earthquake shaking.
• The use of a rate-dependent SMA model in the analyses would most probably produce more reliable and accurate results due to the dynamic nature of the problem.
• The theoretical and experimental investigation of the buckling of SMA bars and the establishment of relevant constitutive models will allow the use of these bars as dissipation elements without the need for buckling prevention mechanisms and devices.
• The study of the potential use of SMA bars as dissipation devices in other types of steel frames.
• The use of 3D analyses via more sophisticated software.