Nitinol Belleville elastic nonlinear (Ni-Belle-E-N) structural assembly

A structural assembly and a structural bracing system including the structural assembly are presented. The structural assembly includes Belleville disks and shape memory alloy (SMA) rods with nonlinear elastic behavior (e.g., Nitinol rods) to resist lateral force. Stacked Belleville disks are placed between plates resulting in nonlinear elastic behavior in compression. Shape memory alloy rods are placed at the corners of the plates and held by nuts at the exterior faces of the plates. The rods are loose when a compression load is applied to the plates and will work in tension when a tensile load is applied to the plates. A shaft (e.g., steel tube) is placed at the center of the Belleville disks to stabilize the assembly. Addition of the assembly with nonlinear elastic behavior in both tension and compression loadings to a structural bracing system improves the structural behavior of the bracing system.

TECHNICAL FIELD

This application relates generally to structural resiliency, and, more particularly, to seismic control technologies and methodologies.

BACKGROUND

Due to the growing need of minimizing the economic loss against natural hazards, structural resiliency has emerged as an important research field. In the U.S., a 2014 statement jointly issued by 44 professional associations in the built-environment industry (design and construction stakeholders, owners, and operators) formally recognized that natural and manmade hazards are posing an increasing threat to the safety of the public; and agreed on a resolution to promote resilience in contemporary planning, building materials, design, construction, and operational techniques (AIA (2012), Industry Statement on Resilience. USA, The American Institute of Architects). Collapse prevention limit state, which is a critical component of seismic hazard mitigation, has been advocated by modern design methodologies since the 1960s and been incorporated in most seismic design codes worldwide since the 1970s. This design philosophy is now prevalent worldwide, including the most recent versions of the U.S. codes (ASCE/SEI7-16 (2016), Minimum Design Loads and Associated Criteria for Buildings and Other Structures, American Society of Civil Engineers; IBC-18 (2017), International Building Code, International Code Council, Inc.). However, the recent strong earthquakes worldwide have demonstrated that buildings designed for the collapse prevention limit state experienced extensive nonstructural and structural damage, even when the intensity of the ground shaking was reasonably within the range of the corresponding design spectrum for the site of interest (Kurth, M. H., Keenan, J. M., Sasani, M. and Linkov, I. (2019), “Defining Resilience for the Us Building 612 Industry.” Building Research & Information, Vol. 47, No. 4, pp. 480-492; Tena-Colunga, A., Hernandez-Ramirez, H. and de Jesus Nangullasmil-Hernandez, H. (2019), Resilient Design of Buildings with Hysteretic Energy Dissipation Devices as Seismic Fuses. Resilient Structures and Infrastructure, Springer, pp. 77-103).

In order to achieve an efficient and resilient seismic design, the primary structural system, which carries the gravitational loads, should remain essentially elastic after strong earthquakes. At the same time, one typically has a secondary system that would be activated when an earthquake strikes. Furthermore, the structural system should be able to withstand the earthquake ground motions in a stable manner, protecting the primary system to remain essentially elastic or with at most minor damage. This secondary system should be easily replaceable after a strong earthquake, in the event that the developed damage within this system interferes with the functionality of the structure. Therefore, this secondary system acts as a structural “fuse” during the earthquake; it absorbs most of the earthquake energy and, if damaged, it could be easily replaced to minimize interruptions to the building usage (Liu, Y., Guo, Z. Liu, X., Chicchi, R. and Shahrooz, B. (2019), “An Innovative Resilient Rocking Column with Replaceable Steel Slit Dampers: Experimental Program on Seismic Performance.” Engineering Structures, Vol. 183, pp. 830-840; Tena-Colunga et al., 2019). To achieve such behavior, novel approaches based on implementing seismic control technologies such as passive, semi-active, and active damping systems have been used. Among the available seismic control methodologies, passive devices represent an advantage over semi-active and active devices, because passive devices do not need an external energy source for their operation. Moreover, they can be easily repaired or replaced after a major seismic event (Soong, T. T. and Costantinou, M. C. (2014), Passive and Active Structural Vibration Control in Civil Engineering, Springer; Aydin, E., Farsangi, E. N., Ozturk, B., Bogdanovic, A. and Dutkiewicz, M. (2019), Improvement of Building Resilience by Viscous Dampers. Resilient Structures and Infrastructure, Springer, pp. 105-127).

A new self-centering device was constructed by Dong et al. (Dong, H., Du, X., Han, Q., Hao, H., Bi, K. Iltang, X. (2017), “Performance of an Innovative Self-Centering Buckling Restrained Brace for Mitigating Seismic Responses of Bridge Structures with Double-Column Piers.” Engineering Structures, Vol. 148, pp. 47-62) based on inserting one buckling restrained brace inside the holes of disc springs. The test results indicated the energy dissipation due to friction in compression disks was quite random and it was very difficult to be accurately predicted in the tests. Thus, Dong et al. (2017) did not consider any energy dissipation by compression disks in their physical model for the self-centering device. Xu et al. (Xu, L., Fan, X. and Li, Z. (2017), “Experimental Behavior and Analysis of Self-Centering Steel Brace with Pre-Pressed Disc Springs.” Journal of Constructional Steel Research, Vol. 139, pp. 363-373) used pre-pressed disk springs to provide restoring force and added special friction devices to the self-centering steel brace to provide energy dissipation in the bracing system. A non-asbestos-organic friction pad was clamped between stainless steel plates by a high-strength bolt in the special friction device. They applied lubricating oil to the disk springs to reduce the contact friction and suggested to use fewer pieces of disk springs to improve self-centering capability. Although the above procedures simplify the physical model by reducing the frictional action in disc springs, they reduce the total energy dissipation capacity of overall brace. Ding and Liu (Ding, Y. and Liu, Y. (2020), “Cyclic Tests of Assemble Self-Centering Buckling-Restrained Braces with Pre-Compressed Disc Springs.” Journal of Constructional Steel Research, Vol. 172, pp. 106229) included frictional actions on overlapped surfaces of stacked pre-compressed disk springs and used them with steel bars and stud bolts to propose a new self-centered buckling restrained brace.

Innovative devices based on superelastic shape memory alloy have been studied by a large number of researchers in base isolators (Dezfuli, F. H. and Alam, M. S. (2013), “Shape Memory Alloy Wire-Based Smart Natural Rubber Bearing.” Smart Materials and Structures, Vol. 22, No. 4, pp. 045013), dampers (Parulekar, Y., Kiran, A. R., Reddy, G., Singh, R. and Vaze, K. (2014), “Shake Table Tests and Analytic Simulations of a Steel Structure with Shape Memory Alloy Dampers.” Smart materials and structures, Vol. 23, No. 12, pp. 125002.) and bracing systems (Yang, C.-S. W., DesRoches, R. and Leon, R. T. (2010), “Design and Analysis of Braced Frames with Shape Memory Alloy and Energy-Absorbing Hybrid Devices.” Engineering Structures, Vol. 32, No. 2, pp. 498-507). According to Gur et al. (Gur, S., Xie, Y. and DesRoches, R. (2019), “Seismic Fragility Analyses of Steel Building Frames Installed with Superelastic Shape Memory Alloy Dampers: Comparison with Yielding Dampers.” Journal of Intelligent Material Systems and Structures, Vol. 30, No. 18-19, pp. 2670-2687) these devices outperform other passive devices because they (1) dissipate significant input energy through their flag-shaped hysteresis loops and (2) are able to recover their original shapes after experiencing large tensile strains, which leaves negligible permanent deformation in the structure. Large diameter threaded shape memory alloy bolts were used in connections to achieve a reasonable level of load resistance (Fang, C., Yam, M. C., Lam, A. C. and Xie, L. (2014), “Cyclic Performance of Extended End-Plate Connections Equipped with Shape Memory Alloy Bolts.” Journal of Constructional Steel Research, Vol. 94, pp. 122-136; Fang, C., Yam, M. C., Ma, H. and Chung, K. (2015), “Tests on Superelastic Ni—Ti Sma Bars under Cyclic Tension and Direct-Shear: Towards Practical Recentering Connections.” Materials and Structures, Vol. 48, No. 4, pp. 1013-1030). Due to fracture susceptibility of such bolts over the threaded area, Fang et al. (Fang, C., Zhou, X., Osofero, A. I., Shu, Z. and Corradi, M. (2016), “Superelastic Sma Belleville Washers for Seismic Resisting Applications: Experimental Study and Modelling Strategy.” Smart Materials and Structures; Vol. 25, No. 10, pp. 105013) suggested the net threaded-to-shank area ratio (which decreases the stress demand over the threaded area) be increased to improve the ductility of the bolts.

SUMMARY

In one implementation of the invention, a structural assembly is provided. The structural assembly includes an upper plate. The upper plate includes at least one aperture. The structural assembly further includes a lower plate. The lower plate includes at least one aperture, and the lower plate is located a distance below the upper plate. The structural assembly also includes a plurality of Belleville disks. The plurality of Belleville disks is stacked one on top of another to form a stack of Belleville disks. The stack of Belleville disks is located between the upper plate and the lower plate. The structural assembly further includes at least one shape memory alloy rod. An upper end of the at least one shape memory alloy rod is inserted into the at least one aperture in the upper plate. A lower end of the at least one shape memory alloy rod is inserted into the at least one aperture in the lower plate. The at least one shape memory alloy rod spans the distance between the upper plate and the lower plate.

In one embodiment, the upper plate is a rectangular steel plate having four corners. The at least one aperture of the upper plate includes four apertures. Each of the four corners of the upper plate include an aperture. Further, the lower plate is a rectangular steel plate having four corners. The at least one aperture of the lower plate includes four apertures. Each of the four corners of the lower plate include an aperture.

In another embodiment, the stack of Belleville disks is arranged into groups of Belleville disks. The Belleville disks within a group of the groups are arranged in parallel. The groups of Belleville disks may be arranged in series. Specifically, the stack of Belleville disks may include two groups of Belleville disks arranged in series. Each group of the two groups comprising 11 Belleville disks are arranged in parallel, for example. Alternatively, the stack of Belleville disks may include one group of Belleville disks arranged in series. This one group may include 21 Belleville disks arranged in parallel. Further alternatively, the stack of Belleville disks may include two groups of Belleville disks arranged in series. Each group of the two groups may include six Belleville disks arranged in parallel. Even further alternatively, the stack of Belleville disks may include one group of Belleville disks arranged in series. This one group may include 18 Belleville disks arranged in parallel, for example.

In yet another embodiment, the upper end of the at least one shape memory alloy rod extends through the at least one aperture of the upper plate and the lower end of the at least one shape memory alloy rod extends through the at least one aperture of the lower plate. The upper end and the lower end of the at least one shape memory alloy rod may be threaded. An upper nut may be removably fastened to the upper end of the at least one shape memory alloy rod and a lower nut may be removably fastened to the lower end of the at least one shape memory alloy rod.

In a further embodiment, the at least one shape memory alloy rod may include four shape memory alloy rods. The at least one shape memory alloy rod may be a Nitinol rod. Further, the at least one shape memory alloy rod does not carry a load when the structural assembly is in compression and the at least one shape memory alloy rod carries the load when the structural assembly is in tension.

In one embodiment, the assembly further includes a shaft located at a center of the stack of Belleville disks. The stack of Belleville disks surrounds the shaft between the upper plate and the lower plate. The shaft stabilizes the structural assembly. Additionally, the at least one aperture of the upper plate may include a central aperture. The central aperture is located at or near a center of the upper plate. An upper end of the shaft is inserted into the central aperture of the upper plate such that the shaft spans the distance between the upper plate and the lower plate. The shaft may be a hollow steel tube.

In another implementation of the invention, a structural bracing system is provided. The structural bracing system includes a rectilinear frame. The frame includes beam elements operatively connected to column elements. The structural bracing system also includes a plurality of brace elements. The brace elements may be arranged in a chevron orientation and operatively connected to the frame. The structural bracing system may further include at least one structural assembly operatively connected to an end of each of the brace elements. The structural assembly may include an upper plate. The upper plate may include at least four apertures. The structural assembly may also include a lower plate. The lower plate may include at least four apertures and the lower plate is located a distance below the upper plate. The structural assembly then may further include a stack of Belleville disks located between the upper plate and the lower plate. The structural assembly may also include a plurality of Nitinol rods (such as at least four Nitinol rods). An upper end of each Nitinol rod is inserted into an aperture in the upper plate and a lower end of each Nitinol rod is inserted into an aperture in the lower plate. Thus, in a specific embodiment having four Nitinol rods, an upper end of each Nitinol rod is inserted into an aperture of the at least four apertures in the upper plate and a lower end of each Nitinol rod is inserted into an aperture of the at least four apertures in the lower plate.

In one embodiment, the plurality of brace elements may include a compression brace and a tension brace. Both the compression brace and the tension brace may include at least one structural assembly attached to an end thereof.

In another embodiment, the structural assemblies may be operatively connected to both ends of the brace elements. Further, at least one structural assembly may be operatively connected to a gusset plate of the frame via steel C-shaped sections. Additionally, at least one structural assembly may be welded to the brace element.

In yet another embodiment, the structural assembly further includes a shaft located at a center of the stack of Belleville disks. The stack of Belleville disks surrounds the shaft between the upper plate and the lower plate. The shaft stabilizes the structural assembly.

DETAILED DESCRIPTION OF THE INVENTION

The exemplary embodiments described herein are provided for illustrative purposes and are not limiting. Other exemplary embodiments are possible, and modifications may be made to the exemplary embodiments within the scope of the present disclosure. Therefore, this Detailed Description is not meant to limit the scope of the present disclosure.

Referring to the Figures, various aspects and embodiments of the present invention are shown. Referring at least toFIGS.6A and10A, these include a structural assembly10and structural bracing system12that are based on the assemblage of Belleville disks14(also referred to as disk springs, compression disks, or similar) and shape memory alloy rods16with nonlinear elastic behavior to achieve a resilient lateral force resistance. In one embodiment, Belleville disks14are arranged in a stack18and are placed between plates20,22(e.g., steel plates) of the structural assembly10to achieve nonlinear elastic behavior in compression. Shape memory alloy rods16(e.g., Nitinol rods) with proper threads at ends24,26thereof are placed at the corners28of the plates20,22and held by nuts30,32at the exterior faces of the plates20,22. The rods16are loose when a compression load is applied to the plates20,22and will work in tension when a tensile load is applied to the plates20,22. A shaft34(e.g., steel tube) is placed at the center of the disks14to stabilize the assembly10. Addition of the structural assembly10with nonlinear elastic behavior in both tension and compression loadings to a structural bracing system12improves the structural behavior of the bracing system12. In an embodiment, a structural assembly10may be operatively connected on one end to a frame36(e.g., a gusset plate38of a frame36) of a structural bracing system12through use of two C-shape sections40(e.g., made of steel). On an opposing end, the structural assembly10may be operatively connected to a brace element42of a structural bracing system12through welding, for example.

Belleville Disk

Belleville disks, also known as Belleville springs, disk springs, or similar, are conically formed from washers and have a rectangular cross section. The disk spring concept was invented by a Frenchman Louis Belleville in 1865. His springs were relatively thick and had a small amount of cone height or “dish”, which determined axial deflection. Coned disc springs are widely used in mechanical systems such as clamping systems, internal combustion engines, clutch and brake systems, and aerospace applications to resist large loads under small deflections. They are compact and can be used for demonstrating various nonlinear load—deflection curves by controlling their dimensions and stacking them in series or parallel. Large coned disc springs are also increasingly being used in base-isolated systems for protection against earthquakes (Ozaki, S., Tsuda, K. and Tominaga, J. (2012), “Analyses of Static and Dynamic Behavior of Coned Disk Springs: Effects of Friction Boundaries.” Thin-walled structures, Vol. 59, pp. 132-143).

Almen (Almen, J. O. (1136). The Uniform-Section Disk Spring. ASME) derived an equation for the relationship between compressive load F and deflection d of a coned disc spring having thickness t, inner diameter Di, outer diameter Do, and free height h. Almen's equation is shown in Equation (1) in which E is the modulus of elasticity and μ is the Poisson's ratio of the disk material. Mechanical properties of an exemplary disk material are shown below in Table 1.FIGS.1A(physical) and1B (simulated) show the configuration of an embodiment of a compression disk14with Do=180 mm, Di=92 mm, t=10 mm, h=4 mm. Such a disk14was modeled in ABAQUS (Simulia, D. S. (2013), “Abaqus 6.13 User's Manual.” Dassault Systems, Providence, RI, Vol. 305, pp. 306) (FIG.1B) to compare simulation results to those obtained using Almen's equation (Equation (1)).

The results of the comparison between the simulation results and the results obtained by Almen's equation (Equation (1)) are shown inFIG.1C. Almen's equation is widely known for overestimating the compression load for a certain amount of deformation since it is based on the assumption that the loads are concentrically distributed and that the radial stresses are negligible (Ozaki et al., 2012; Zheng, E., Jia, F. and Zhou, X. (2014), “Energy-Based Method for Nonlinear Characteristics Analysis of Belleville Springs.” Thin-Walled Structures, Vol. 79, pp. 52-61; Patangtalo, W., Aimmanee, S. and Chutima, S. (2016), “A Unified Analysis of Isotropic and Composite Belleville Springs.” Thin-Walled Structures, Vol. 109, pp. 285-295). This trend can be seen in the force-deformation response shown inFIG.1C.FIG.1Cshows that the difference between the force-deformation response in the simulated results and the results obtained from Almen's equation becomes larger as the applied load increases. It is suggested by some industrial catalogues (Schnorr, A. (2003), “Handbook for Disk Springs.”) to use 75% of the disk capacity in design applications due to the better correlation between Almen's equation and experimental results in the mentioned domain. Based on the results, there is approximately a 15% difference between the results when a disk with h/t<1 is loaded up to 65% of its capacity. Thus, only 65% of the disk capacity is used in the design of the structural assembly.

Compression disks can be stacked in parallel or series to obtain a certain amount of strength or stiffness. Equations (3) to (6)—below—are used to estimate the force-deformation response of stacked disks. Curti and Montanini (Curti, G. and Montanini, R. (1999), “On the Influence of Friction in the Calculation of Conical Disk Springs”) studied the influence of friction on the loading branch of the force-deformation response of compression disks, made out of the same material as is presented in Table 1, both numerically and experimentally. They determined that the average friction coefficient factor on commercial conical disks is equal to 0.14 and observed that the maximum error obtained by using Almen's equation in the evaluation of the disk response is in the range of 2-5% when the friction coefficient factor is 0.14. For n disks arranged in parallel, the following equations are applicable:
Ftotal=F×n(3)
dtotal=d(4)

F and d are the force and deformation of one disk. For n disks arranged in series the following equations are applicable:
Ftotal=F(5)
dtotal=n×d(6)

According to Oberg (Oberg, E. (2012), Machinery's Handbook 29th Edition-Full Book, Industrial Press), the parallel and series theory (Equations (3) to (6)) provides accurate results for compression disks with the following ratios: Do/Di=1.3 to 2.5 and h/t≤1.5.

FIG.2Ashows an embodiment of a stack18of Belleville disks14. The depicted embodiment includes 10 disks14—a group44of 5 parallel disks14in series with another group44of 5 parallel disks. It is to be understood, and will be described later, that the stack18of disks14can take on several forms and include more or less than 10 disks14and that the groups44of disks14can take on several forms and may be arranged differently than is depicted inFIG.2A.FIG.2Bcompares the finite element simulation (ABAQUS) results to the theoretical results (Almen's equation) for the depicted stack18of disks14. Due to the large amount of contact between adjacent stacked disks14, the explicit analysis method with the general contact properties were used in ABAQUS simulations (Simulia, 2013). The penalty function algorithm showed acceptable results in earlier studies of force-deformation response of compression disks (Zhu, D., Ding, F., Liu, H., Zhao, S. and Liu, G. (2018), “Mechanical Property Analysis of Disc Spring.” Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 40, No. 4, pp. 230). The friction coefficient was taken as 0.14 based on Curti and Montanini (1999). The simulation results of stacked disks (FIG.2B) are similar to those for single-disk results (FIG.1C)—with one difference. As the loading reaches the maximum value, stacked disks begin to behave “stiffer” than a single disk. This is attributed to the enhanced stiffening influence of friction between the contact surfaces of adjacent disks as the axial load increases, which does not exist in the case of a single disk.

Shape Memory Alloy

Shape Memory Alloy (SMA) is a smart material discovered in 1932 by Olander (Olander, A. (1932), “An Electrochemical Investigation of Solid Cadmium-Gold Alloys.” Journal of the American Chemical Society, Vol. 54, No. 10, pp. 3819-3833). SMAs possess an interesting property by which the metal “remembers” its original size or shape and reverts to it at a characteristic transformation temperature (Funakubo, H. and Kennedy, J. (1987), “Shape Memory Alloys.” Gordon and Breach, xii+275, 15×22 cm, Illustrated). This feature is known as the shape memory effect (SME) and can be used to make sensors and actuators for smart civil structures (Xu, Y.-L. and He, J. (2017), Smart Civil Structures, CRC Press). Another important feature of SMAs is super elastic or pseudo-elastic behavior. Super elasticity for a generic SMA is shown inFIG.3. The material microstructure of a SMA is initially austenitic (point 1). During loading, the critical stress for phase transition is reached (point 2) and the material transforms directly into detwinned martensite (plateau from points 2 to 3). Once the phase transition is complete, further loading causes elastic deformation of the detwinned martensite (slope from points 3 to 4). Austenite is the only stable phase at high temperature and no stress. Therefore, the critical stress for the reverse phase transition is reached during unloading and the macroscopic deformation is recovered (plateau from points 5 to 6). Because of the presence of a small hysteresis loop between the loading and unloading plateau, this property is often referred to as super elastic behavior (Lecce, L. (2014), Shape Memory Alloy Engineering: For Aerospace, Structural and Biomedical Applications, Elsevier).

In 1965, Buehler and Wiley of the U.S. Naval Ordnance Laboratory (NOL) received a United States patent for a series of Ni—Ti alloys, whose generic name is 55-Nitinol, having shape memory behavior (Buehler, W. J., Gilfrich, J. and Wiley, R. (1963), “Effect of Low-Temperature Phase Changes on the Mechanical Properties of Alloys near Composition Tini.” Journal of applied physics, Vol. 34, No. 5, pp. 1475-1477). In honor of the Naval Ordnance Laboratory where the material was first discovered, NiTi SMA is also widely known as Nitinol. NiTi alloy is still one of the most successful alloys of SMA in practice (Fang, C. and Wang, W. (2020), Shape Memory Alloys for Seismic Resilience, Springer). Typical applications of NiTi in civil structures require larger diameter elements because of the magnitudes of the loads, particularly those associated with a seismic event, along with ease of implementation. DesRoches et al. (DesRoches, R., McCormick, J. and Delemont, M. (2004), “Cyclic Properties of Superelastic Shape Memory Alloy Wires and Bars.” Journal of Structural Engineering, Vol. 130, No. 1, pp. 38-46) studied the cyclic properties of super elastic, large diameter Nitinol rods (the largest diameter was 25.4 mm) and observed the residual strain gradually increases from an average of 0.15% following 3% strain to an average of 0.65% strain following four cycles at 6% strain. They also noticed continued loading beyond 6% strain typically resulted in unacceptably large increases in residual strains.

One quite simple and effective model for super elastic behavior was introduced by (Auricchio, F., Taylor, R. L. and Lubliner, J. (1997), “Shape-Memory Alloys: Macromodelling and Numerical Simulations of the Superelastic Behavior.” Computer methods in applied mechanics and engineering, Vol. 146, No. 3-4, pp. 281-312) based on a Drucker—Prager-type loading function. Due to the high computational efficiency of the model, it has been adopted as a built-in user-defined material model by many commercial finite element software packages such as ABAQUS.FIG.4and Table 2 illustrate parameters that can be input in ABAQUS to model super elastic behavior.

TABLE 2ABAQUS input values for super elastic behavior (Lecce, 2014)VariableValueEa:Young's modulus of austenite40GPavA: Poisson's ratio of austenite0.33Em: Young's modulus of martensite32GPavM: Poisson's ratio of martensite0.33εL: Maximum transformation strain0.041σtLS: Forward transformation start stress440MPaσtLE: Forward transformation finish stress540MPaσtUS: Reverse transformation start stress250MPaσtUE: Reverse transformation finish stress140MPa

ASTM-F2516-18 (ASTM-F2516-18 (2018), Standard Test Method for Tension Testing of Nickel-Titanium Superelastic Materials. West Conshohocken, PA, ASTM International) specifies the tension testing method to determine some stress-strain related properties of Nickel-Titanium super elastic materials. Upper plateau strength (UPS) is the stress at 3% strain during loading of the sample and is specified as 500 MPa on average for 2.5 mm diameter specimens (the largest tested in the standard).FIG.5Acompares the results obtained by finite element simulation (ABAQUS) of a tensile rod versus the specified UPS by ASTM. An exemplary rod16is shown inFIG.5B.

Structural Assembly

A Bellville disk14has a nonlinear elastic behavior in compression and SMA (e.g., Nitinol) rods16have super elastic behavior in tension. Combining the aforementioned two parts in one assembly10(as shown inFIG.6A) results in a structural assembly10that has nonlinear elastic behavior in both tension and compression. Such is illustrated byFIG.6Bin which Kstackis the stiffness of the disk stack18in compression and KNiTiis the stiffness of Nitinol rods16in tension. The compression disks14are arranged in a stack18and are inserted between two plates20,22(e.g., steel plates)—one at the at the top of the stack18and one at the bottom of the stack18. It is to be understood that the plates20,22may be made of a suitable material other than steel. A shaft34(e.g., made out of steel) is placed at the center, inside the disks14, to stabilize the assembly10. It is to be understood that the shaft34may be made out of a suitable material other than steel. The disks14surround the shaft34. In an embodiment, an upper end46of the shaft34extends through a central aperture48(hole) in at least one of the two plates20,22—the top plate20, for example. The stack18of disks14will carry the load when the assembly10is in compression.

In an embodiment, the upper and lower plates20,22include further apertures50(holes) in each of the corners28of the plates20,22. In the case of a square or rectangular plate20,22, each plate20,22will have at least four holes or apertures50at or near the corners28of the plate20,22. Shape memory alloy rods16(e.g., Nitinol rods) with sufficient threads at the ends24,26are inserted through the holes50at the corners28of the plates20,22and are held by nuts30,32on the exterior faces of the plates20,22. It is to be understood that the rods16may be connected to the plates20,22in another suitable manner—using a different mechanical fastener, for example. With such configuration, the rods16are loose when the assembly10is in compression and will carry the load when the assembly10is in tension. Based on a target stiffness for the assembly10(Kasm), and for a given load (Fuit), Equations (7) to (18) are used to design the assembly10. In these equations, nDis the number of parallel disks14in a group44and nGis the number of disk14groups44arranged in series. Therefore, the total number of disks14in a stack18is nG×nD.

In order to accurately estimate the disks' behavior and also leave some margin of safety in the design of stacked disks, 65% of the disk capacity is used under the given load. This consideration is shown in Equation (7) and incorporated in the Almen's equation as shown in Equation (8). By choosing a compression disk from a manufacturer's catalog and including its geometrical and material parameters (Do, Di, h, t, E, μ) in Equation (8), the number of disks per group (nD) can be found using Equation (9). Note that the disk deformation (d) is assumed to be 65% of the disk capacity (0.65×h) when the ultimate load is applied on the stack.

After finding the number of parallel disks in one group (nD), Equations (10) and (11) are used to obtain the number of groups in a stack (nG) in order to reach the target stiffness (Kstack).

Since the tensile stiffness should be the same as the compression stiffness, Equations (12) to (16) are used to obtain the Nitinol rods diameter. According to Equation (14) the length of each Nitinol rod is the total height of the disk stack plus the thickness of the bottom and top plates (tpit). By substituting the length of each rod into the axial stiffness equation (Equation (15)), the diameter of each Nitinol rod (DNiTi) is found through Equation (16).

To ensure the assembly has enough strength in tension while the tensile strain in rods is lower than 6% (to avoid any residual deformation), the condition mentioned in Equation (17) needs to be satisfied. Otherwise, either the diameter of Nitinol rods obtained from Equation (16) has to be increased and the plate thickness be tuned accordingly to satisfy Equation (15), or another compression disk with different dimensions (Do, Di, h, t) needs to be selected and the assembly be redesigned to satisfy all the aforementioned conditions and equations.
Check: 4×ANiTi×(σNiTi)εNiTi=6%≥Fult(17)
Cyclic Behavior of the Structural Assembly

In order to check the cyclic behavior of the assembly10, five embodiments (e.g., assemblies10) are presented based on a target stiffness (Kasm) and an assumed applied load (Fuit) and were then simulated in ABAQUS. A572-GR50 steel is used for the steel plates20,22, central shaft34, and the steel nuts30,32. Though it is to be understood that other suitable materials could be used. The loading protocol for the simulation is shown inFIG.7. The loading protocol is according to loading histories for quasi-static cyclic testing suggested by FEMA-461 (FEMA-461 (2007), Interim Testing Protocols for Determining the Seismic Performance Characteristics of Structural and Nonstructural Components. Washington, DC.). The details of the embodiments of the assemblies10are presented in Table 3 and the drawings of the embodiments of the assemblies10with their cyclic response (determined by finite element analyses) are shown inFIGS.27A-31D.

The stiffness and load values are based on the force and stiffness of braces in a 5-story special concentrically braced frame (SCBF) building. The target stiffness (Kasm) is determined based on brace stiffness by applying equivalent lateral force (ELF) load (ASCE/SEI7-16, 2016) and the load (Fult) is determined based on brace load by using maximum considered event (MCE) loading. Final equations to find Fultand Kasmare developed based on the cyclic behavior results and are presented in Equations (18) and (24). As an example, the drawings of the first embodiment of an assembly10(e.g., assembly1) are shown inFIGS.8A and8Bwith the 3-dimensional (perspective) view of this assembly10shown inFIG.6A.

TABLE 3Details of designed assemblies for studying cyclic behaviorLoad andRodTarget stiffnessDisk geometryStackdiameterFultKasmDoDithinformationDNiTiAssembly(kN)(kN/mm)(mm)(mm)(mm)(mm)nDnG(mm)1191033420010212.04.1911231.82122034315071.06.004.8021125.4342314410051.07.002.216214.74138035920082.08.006.2018124.2595629415071.06.004.8018122.6

Cyclic responses of the embodiments of the assemblies10listed in Table 3 were obtained through finite element simulations in ABAQUS and the responses are shown inFIGS.9A-9EandFIGS.27D,28D,29D,30D, and31D.FIGS.9A and27Dcorrespond to a first embodiment of the assembly10(assembly1),FIGS.9B and28Dcorrespond to a second embodiment of the assembly10(assembly2), and so on. Further,FIGS.27D,28D,29D,30D, and31Dshow some amount of residual deformation in the behavior of structural assembly10. Since the residual deformation is negligible in comparison to the total deformation under the loading, the assemblies10are deemed to behave in a nonlinear elastic manner.

The compression disks14are capable of some energy dissipation due to the different loading and unloading paths in their response; however, the amount of dissipated energy is small in comparison to the energy dissipated by Nitinol rods16(the amount of area enclosed by the cycles in the negative region compared to the area inside the cycles in the positive region of the force-deformation responses). The energy dissipation in compression disks14is a function of friction between the disks14which is highly unpredictable as discussed. The results indicate the stacked compression disks14exhibit some energy dissipation capability due the differences between loading and unloading paths as evident from the negative domain of cyclic responses.

This type of cyclic response has also been observed in other studies focused on behavior of compression disks (Ozaki et al., 2012; Mastricola, N. P., Dreyer, J. T. and Singh, R. (2017), “Analytical and Experimental Characterization of Nonlinear Coned Disk Springs with Focus on Edge Friction Contribution to Force-Deflection Hysteresis.” Mechanical Systems and Signal Processing, Vol. 91, pp. 215-232). Cycles with the same amplitude on the positive (tensile) domain of responses almost completely cover each other. The amplitude of the last two cycles have force almost equal to the point where the flag shape behavior ends, and Nitinol gains more stiffness (see σtlEinFIG.4). That is the reason for having the extreme points of the last two cycles located at the higher force values. The maximum load in the cyclic behavior (Fult) uses 65% of disk stack capacity (see Equation (7)) in compression and causes 6% strain in Nitinol rods (see Equation (17)) in tension. Thus, the assembly10shows nonlinear elastic behavior when brace force is at MCE demand-level.

Structural Bracing System

In an embodiment, the structural bracing system12is based on a chevron configured bracing system in which at least one structural assembly10is added to each of the braces42. It is to be understood that the structural bracing system12could take on other forms besides a chevron configured bracing system. In order to obtain symmetric cyclic behavior in the depicted structural bracing system12, the structural assembly10is attached to both the tension and compression braces52,54. It is to be understood that the same may not be true in other embodiments of the structural bracing system12.FIGS.10A-10Dshow an embodiment of a braced frame36with structural assemblies10added to both of the braces42(e.g., compression54and tension52), which is referred to as the structural bracing system12. The frame36includes beam elements56operatively connected to column elements58, at right angles, for example. The structural assembly10can be added to the brace42, which may be a hollow structural section (HSS) section, at one end or both ends of the brace42depending on the demands of a particular situation. In an embodiment, a structural assembly10may be operatively connected on one end to a frame36(e.g., a gusset plate38of a frame36) of a structural bracing system12through use of two C-shape sections40(e.g., made of steel). On an opposing end, the structural assembly10may be operatively connected to a brace section42of a structural bracing system12through welding, for example. It is to be understood that the structural assembly10may be connected to the frame36and to the braces42in other manners.

Due to the usage of the structural assembly10in both the tension and compression braces52,54in the frame36, the system12possesses a symmetric nonlinear elastic behavior. While the nonlinearity of the behavior dissipates the earthquake energy, the elasticity of the system12impedes the formation of residual deformations in the system12, which leads to a reduction in the structural repairs after ground motions. The aforementioned behavior plays a key role in improving the seismic resiliency of structures.

Cyclic Behavior of Structural Bracing System

Cyclic behavior of a frame using the structural bracing system12, similar to the one shown inFIG.10A, was studied analytically by using RUAUMOKO-2D (Carr, A. J. (2008), “Ruaumoko-lnelastic Dynamic Analysis Program.” Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand) which is a comprehensive dynamic analysis program. The RUAUMOKO model is shown inFIG.11. The columns58, beam56, and the HSS brace sections42were modelled with frame elements, as shown inFIG.11. Elastic-perfectly-plastic behavior was assigned to the frame elements. The behavior of the structural assembly10was modelled with a spring element having the cyclic behavior shown inFIG.12A. This model is available in the RUAMOKO library for super elastic cyclic behavior. The values of the parameters used to define the model were found by matching the RUAUMOKO hysteretic response to the five cyclic behavior responses obtained from finite element analysis and shown inFIGS.9A-9E.FIG.12Bshows a good match between the hysteretic responses for the first embodiment of the structural assembly10(e.g., assembly1), for example. The same is true for the second (e.g., assembly2), third (e.g., assembly3), fourth (e.g., assembly4), and fifth (e.g., assembly5) embodiments of the structural assembly10(FIGS.12C-12F). The values of the modeling parameters are listed in Table 4. The first parameter in the table, a (ALPHA), is the fraction of yield point at unloading path intersection. The second parameter, β (BETA), is the stiffness factor at stiffening path after reaching the end of the flag shape. The third parameter, Δ (DELTA), is the multiplier of yield deformation where stiffening starts. The final parameter, r, is the secondary stiffness factor in the unloading path. These parameters are illustrated inFIG.12A.

TABLE 4Cyclic behavior modeling parameters in RUAUMOKOαβΔr0.3010.5785.060.016

Cyclic behavior of the structural bracing system12modelled in RUAUMAKO-2D is shown inFIG.13. The lateral loading applied to the frame36was based on the loading protocol shown inFIG.7. Since the structural bracing system12showed a flag-shape behavior in both tension and compression regions, the system12is capable of carrying seismic loads without experiencing permanent deformations. Determination of the structural bracing system12ultimate force (Fult) and assembly stiffness (Kasm) of structural assemblies10is the key in achieving the desirable behavior (i.e., nonlinear elastic) for the structural bracing system12.

Determination of Ultimate Force (Fult) and Stiffness (Kasm)

The ultimate force and stiffness of a structural assembly10is determined based on the force and stiffness of SCBF braces when ELF load is applied to the SCBF building. The ultimate force in the structural assembly10(Fult) is determined from Equation (18). FSbrcis the force in SCBF brace when ELF loading is applied. The value of ΩSXBFis 2 according to ASCE/SEI7-16 (2016).
Fult=ΩSCBF×(FSbrc)ELF(18)

Equation 19 was used to determine KSbrc, the stiffness of SCBF brace. Since the assembly10and the HSS steel section in the structural bracing system12function as two springs in series (seeFIG.10A), Equation (20) was used to define the total stiffness of the brace42including structural assemblies10(KNibellen) The factor of 2 in Equation (20) is based on the assumption that two assemblies10have been used (similar to that shown inFIG.10A). For cases with one assembly10operatively connected to in the brace42, the factor of 2 becomes a factor of 1.

Since seismic design parameters and drift ratio limits suggested by ASCE/SEI7-16 (2016) are not available for the structural bracing system12yet, the total stiffness of brace42including one or more structural assemblies10(KNibellen) is considered equal to the stiffness of SCBF brace (KSbrc) to pass the drift limits suggested by ASCE/SEI7-16 (2016) for SCBF system. This is shown in Equation (21).
KNibellen=KSbrc(21)

Trial designs showed the stiffness of a steel HSS section (KHSS) in a brace42including one or more structural assemblies10(seeFIG.10A) should be approximately twice the stiffness of SCBF brace in order to have the total stiffness of the brace42including one or more structural assemblies10(KNibellen) be equal to the stiffness of the SCBF brace. Thus, Equation (22) is considered in the derivations.
KHSS=2×KSbrc(22)

Substituting Equations (21) and (22) into Equation (20), Equation (23) is obtained to calculate the stiffness of the assembly10or assemblies10as a function of the stiffness of a SCBF brace.

The cyclic behavior of the assemblies10obtained from finite element analyses indicate that the stiffness of the assemblies10is, on average, 87% of the target stiffness determined from the design equations. This difference is attributed to the additional deformations in the top and bottom plates20,22and also the nuts30,32, which are not included in the design calculations. To include this effect, a K factor is included in Equation (23) to obtain a practical design value for the stiffness of assembly(s)10. Such is shown in Equation (24).

Since the structural assembly10is the “fuse” in the structural bracing system12, the maximum force generated from the stack18of disks14in the assembly10is the critical compression force for design of the HSS section. The maximum disk force was obtained by using Equation (8) and setting the value of the disk deformation (d) equal to the disk free height (h), as shown in Equation (25) in which (σcre)HSSis the compressive strength of the HSS section.

The stiffness of the HSS section should satisfy Equation (22). Thus, Equation (26) is used to check the axial stiffness of the HSS section in the structural bracing system12. Note that the length of the HSS section (LHSS) is calculated in Equation (27) based on the assumption that part of the brace length will be allocated to the bracing connections and gusset plates (Lc). The value of Lcwas taken as 1.5 m (5 ft.). Height of the structural assembly10(Hasm) is multiplied by 2 in cases where two assemblies10are used with a brace42. This factor is replaced by 1 when only one assembly10is used with the brace42.

Chevron beams56, columns58, and the connections (e.g., gusset plate38) have to be able to carry the maximum force that can be generated by the structural bracing system12. In the compressive brace54, this force is the stack force when disks14are fully deformed (d=h). In the tensile52brace, it is the force generated by the Nitinol rods16when the tensile deformation of the brace42is equal to its compressive deformation when disks14are fully deformed. These forces are calculated by Equations (28) and (29), in which σtLSis the stress at which Nitinol transforms from austenite phase to martensite phase (see Table 2) and εtLSis the strain corresponding to σtLS. The “Chevron Effect” method introduced by Fortney and Thornton (Fortney, P. J. and Thornton, W. A. (2015), “The Chevron Effect—Not an Isolated Problem.” Engineering Journal, Vol. 52, No. 2, pp. 125-163; Fortney, P. J. and Thornton, W. A. (2017), “The Chevron Effect and Analysis of Chevron Beams-a Paradigm Shift.” Engineering Journal-American Institute of Steel Construction, Vol. 54, No. 4, pp. 263-296) was used to analyze the chevron frame when the mentioned forces are applied. Hadad and Fortney (Hadad, A. A. and Fortney, P. J. (2019). Studying the Ductility Factor for Middle Gusset Connections in Chevron Braced Frame Configurations. Structures Congress 2019: Buildings and Natural Disasters, American Society of Civil Engineers Reston, VA; Hadad, A. A. and Fortney, P. J. (2020), “Investigation on the Performance of a Mathematical Model to Analyze Concentrically Braced Frame Beams with V-Type Bracing Configurations.” Engineering Journal, Vol. 57, No. 2, pp. 91-108) have shown a better accuracy of the “Chevron Effect” method in comparison to the Net Vertical Force (NVF) method through studying several cases of beam-gusset assemblies.

EXAMPLES

Example 1: Seismic Assessment of the Structural Bracing System (5-Story)

A 5-story office building was selected to study the seismic behavior of an embodiment of the structural bracing system12and to compare its performance with a SCBF system. The plan of the building is shown inFIG.14. The dashed spans in this figure are braced with the structural bracing system12. Each story has a height of 4.3 m (14 ft.). The loading information is given in Table 5. The building was designed initially based on a SCBF system, and subsequently a structural bracing system12. The member sizes and dimensions are listed in Tables 6 to 10. The sections were selected according to Steel Construction Manual (AISC (2017), Steel Construction Manual. Chicago, Ill., American Institute of Steel Construction).

FIG.15compares the construction cost of the designed members in both of the buildings. Construction cost data (Gordian (2018), Building Construction Costs with Rsmeans Data 2018, RS Means Company, Incorporated) were used to estimate the cost of common structural members. Table 11 shows the costs of a number of different diameter Belleville disks14and Nitinol rods16. A compression disk manufacturer (Schnorr, 2003) provided disk costs per item, and a Nitinol manufacturer (SEAS-Group (2018), Shape Memory Alloy Costs (Quotation), Memry corporation) provided the cost of Nitinol rod per weight for the diameter available in stock. The cost of Nitinol rods16and Belleville disks14used in the studied buildings were estimated based on the cost provided in Table 11. It is worth noting that Bellville disk diameters (Do, Di) and thickness (t) are the key parameters in determining the price for a given disk size, according to the data provided by Schnorr (2003).

Although the brace cost has increased 108% by using new material in the structural bracing system12, the total construction cost has increased only 4% due to the reduction in the costs of beams and columns. This reduction is possible because of smaller demands in the beams56and columns58of the chevron braces in the structural bracing system12in which the difference between the maximum tension and compression forces (mechanism forces) is less in comparison to SCBF brace mechanism forces. Accounting for the cost of nonstructural elements, the difference between the total costs of the systems will be less noticeable.

To ensure a broad representation of different recorded earthquakes, seven far-field ground motion records were selected from the list of large-magnitude earthquakes provided by FEMA-P695 (FEMA-P695 (2009), Quantification of Building Seismic Performance Factors. Washington, DC.). The response of the 5-story buildings was studied under three ground motion demand levels: (1) maximum base shear of each ground motion is equal to the ELF base shear, (2) the mean spectral acceleration of ground motions matches design-base-event (DBE) spectrum, and (3) the mean spectral acceleration of ground motions matches maximum-considered-event (MCE) spectrum.

Each demand-level was obtained by scaling the ground motion records. The scale factor for the first demand-level was obtained by performing linear response history analysis for each ground motion record, finding the maximum base shear of each record, and calculating the ratio of the maximum base shear over the ELF base shear.

PEER ground motion data base in conjunction with PEER record-scaling-tool were used to find the scale factor for the second demand-level. Target DBE spectrum was generated based on (ASCE/SEI7-16, 2016) for the site of interest (Northern California Region) and then introduced to the tool to match the average ground motion spectrum to the target spectrum.FIG.16shows the response spectrum for scaled ground motions versus the DBE spectrum. Since the MCE spectrum can be obtained by multiplying the DBE spectrum by 3/2 (ASCE/SEI7-16, 2016), the scale factors used in the second demand-level were multiplied by 3/2 to obtain the third demand level.

The horizontal components of each ground motion acceleration were applied to the buildings concurrently per (ASCE/SEI4—(2014), Seismic Evaluation and Retrofit of Existing Buildings, ASCE; NITS (2017), Guidelines for Nonlinear Structural Analysis for Design of Buildings Part I—General, Applied Technology Council) to perform response history analyses on the 3-dimensional building models. The aforementioned codes do not require concurrent analysis for regular buildings; however, such analysis was performed to study the behavior of the structural bracing system12when used in both building directions. RUAUMOKO-3D (Carr, 2008) was used to perform the analyses. Columns58, beams56, and braces42were modelled to have an elastic-perfectly-plastic behavior. Structural assemblies10were modelled based on the cyclic behavior discussed above with respect toFIGS.12A-12F. A concrete slab with 10 cm (4 in.) thickness was used as the floor system in buildings. Plane-stress quadrilateral elements with elastic material behavior were used to model the floor system. The stiffness for elastic diaphragms, which were modeled by quadrilateral elements, was determined according to the approach recommended by NIST (2017): (a) 30 percent of concrete Young's modulus to reflect an effective stiffness based on the expected deformation and cracking, and (b) 100 percent concrete Poisson's ratio.

FIGS.17A-17Cshows representative roof displacement histories for Landers ground motion at different demand-levels. Both systems could resist the earthquake load without residual roof displacement at the first demand level in which the maximum base shear of buildings is equal to the ELF base shear (seeFIG.17A). The SCBF system showed residual roof displacements at the DBE demand level, and a much larger residual value at the MCE demand level (FIGS.17B and17C, respectively). The structural bracing system12, however, did not show any roof residual displacement at both of the DBE and MCE demand levels (FIGS.17B and17C, respectively).FIG.17Bshows that ground motion spectral acceleration matches DBE.FIG.17Cshows that ground motion spectral acceleration matches MCE. A similar trend is seen for the other ground motions as evident from Table 12 in which the maximum values of residual roof drifts in the X- and Y-directions are provided in Table 12. The structure using the structural bracing system12did not show any residual roof drift at any of the demand levels for any of the ground motions.

The average of the maximum inter-story drift is shown inFIGS.18A-18Cfor each of the three ground motion demand levels.FIG.18Ashows max base shear equals ELF base shear.FIG.18Bshows mean spectral acceleration matches DBE.FIG.18Cshows mean spectral acceleration matches MCE. Comparing the drift values shows the stiffness of the structural bracing system12is close to the SCBF system, as intended in Equation (21). The reason for the larger difference between the drift values at MCE demand level, is the plastic deformations in the SCBF system while the structural bracing system12could return to its initial form without plastic deformations.

Resiliency of the System

Seismic resiliency was defined analytically by Bruneau et al. (Bruneau, M., Chang, S. E., Eguchi, R. T., Lee, G. C., O'Rourke, T. D., Reinhorn, A. M., Shinozuka, M., Tierney, K., Wallace, W. A. and Von Winterfeldt, D. (2003), “A Framework to Quantitatively Assess and Enhance the Seismic Resilience of Communities.” Earthquake spectra, Vol. 19, No. 4, pp. 733-752) through the measurement of quality degradation in the infrastructure. Equation (30) was used to determine the resiliency. R is the resiliency of the system, Q(t) is the functionality term defined in Equation (31), H is the Heaviside step function, t0Eis the event occurrence time, TLCis the control time of the system, TREis the recovery (repair) time for the event, and LS(I) is the system's structural loss occurring instantaneously after the seismic event. The structural loss was calculated based on Equation (32) in which CS,jis the damage cost due to damage state j, Pjis the probability of exceeding performance limit state j, and ISis the total replacement cost.

Performance Assessment Calculation Tool (PACT) provided by FEMA-P58 (FEMA-P58 (2018), Seismic Performance Assessment of Buildings. Washington, DC.) was used to estimate the damage and loss costs. In PACT, each building component and content is associated with a fragility curve that correlates engineering demand parameters to the probability of that item reaching a particular damage state. Story drift ratio was used as the demand parameter. The component's damage was related to a loss (e.g., repair cost or repair time) utilizing consequence functions. The total loss at a certain hazard level was then estimated by integrating losses over all components of a system. To account for the many uncertainties affecting calculation of seismic performance, the FEMA P-58 methodology uses a Monte Carlo procedure to perform loss calculations (Cimellaro, G. P. (2016), “Urban Resilience for Emergency Response and Recovery.” Geotechnical, Geological and Earthquake Engineering, Springer International Publishing).

Different types of recovery function (ƒrec) can be selected depending on the system and society preparedness response. The simplest form is the linear recovery function written in Equation (33) which is generally used when there is no information regarding the preparedness, resources available, and societal response (Kafali, C. and Grigoriu, M. (2005). Rehabilitation Decision Analysis. Proceedings of the Ninth International Conference on Structural Safety and Reliability (ICOSSAR'05)).

Fragility curves were defined as the probability of reaching or exceeding a specific damage state under earthquake excitation. Several fragility functions have been introduced (Nazri, F. M. (2018), Seismic Fragility Assessment for Buildings Due to Earthquake Excitation, Springer), but Equation (34) is the most common equation for fragility, which is based on a research conducted by Yamaguchi and Yamazaki (Yamaguchi, N. and Yamazaki F. (2000). Fragility Curves for Buildings in Japan Based on Damage Surveys after the 1995 Kobe Earthquake. Proceedings of the 12th conference on earthquake engineering, Auckland, New Zealand), and is suitable for all structural types. In this equation, F is the probability of reaching a certain damage state at drift D, φ is the standard normal cumulative distribution function, θ is the median value probability distribution, and β is the logarithmic standard deviation of probability distribution (dispersion).

Incremental dynamic analysis (IDA) was performed to understand the mean value and the dispersion value for each damage state in the systems.FIGS.19A and19Bshow the results of IDA on the 5-story SCBF and structural bracing system12buildings, respectively. The vertical axis shows the spectral acceleration of the ground motions at the first-mode vibration period of the building, and the horizontal axis shows the maximum inter-story drift. The ground motion records were scaled up incrementally until the systems reached collapse. Collapse state was defined as the numerical instability of response history analysis due to the yielding and plastic deformations of gravity members leading to excessive deformations in the building.

In order to build the fragility curves, two damage states were considered for a SCBF brace: (1) brace compression buckling, and (2) brace tensile yielding. Three damage states were defined for a brace42including one or more structural assemblies10: (1) structural assembly10reaching the tensile force corresponds to 6% strain in Nitinols, (2) structural assembly10reaching a stack's maximum compression force, and (3) HSS section in the brace42including one or more structural assemblies10reaching compression buckling force.FIG.20shows the fragility curves based on the defined damage states in the systems. The mean and dispersion values for each damage state were obtained by tracking the member's state (force) versus maximum inter-story drift at different increments of the dynamic analysis and saving the value once it reached the damage state limit.

The beams56and columns58remained elastic under the hazard levels (DBE and MCE) utilized to assess resiliency of the structural bracing system12. For example,FIGS.21A and21Bshow the axial force-bending moment elastic interaction diagram for an interior first floor column (which carries the maximum gravity loads) of the structural bracing system12and SCBF buildings, respectively, subjected to Chi-Chi ground motion (which among other ground motions caused the maximum demands in the columns) at MCE hazard level. The elastic interaction diagram was obtained from Equation (15) in which MEis the bending moment strength calculated by Equation (16) and PEis the tensile and compressive axial strength determined by Equation (17) and Equation (18), respectively. The steel yield strength (ƒy) was taken as 345 MPa (50 ksi). The critical compressive stress (ƒcre) of the steel sections in compression was obtained from Table 4-22 of AISC (2017). Table 1-1 of AISC (2017) was used to obtain the necessary cross-sectional parameters: area (As), moment of inertia (Is), and depth (ds).FIGS.22A and22Bshow the axial force-bending moment response of a first-floor chevron beam of the structural bracing system12and SCBF buildings, respectively, subjected to Chi-Chi ground motion at MCE hazard level. Equation (15) was also used to obtain the beam interaction diagrams. Beam buckling is prevented by continuous lateral support of the floor diaphragm; hence, PEis the same for compression and tension. Due to the elastic behavior of beams and columns under the hazard levels, the fragility curves of beams and columns were not included.

The repair cost and time of the systems were developed based on PACT supporting material and digital libraries (FEMA-P58, 2018), construction costs data (Gordian, 2018), and engineering judgment. This data is input into PACT through consequence functions. Consequence functions are relationships that indicate the potential distribution of losses as a function of the damage state, and they translate damage into potential repair and replacement costs and repair time.FIGS.23A and23Bshow the consequence functions defined for the SCBF and structural bracing system12. To repair the first damage state in a SCBF brace, only the braces in a chevron frame need be replaced since they have buckled. Repairing the second damage state in the SCBF includes further actions since the braces have yielded in tension, and all other members in the system have experienced the maximum force (mechanism force) that can be generated by the braces. Thus, the repair includes replacing the braces, replacing the connection gusset plates, and some local repairs on the beams and columns.

Regarding the repair of the structural bracing system12, only the Nitinol rods16have to be replaced in the first damage state. The structural assembly10has to be replaced in the second damage state. Repair of the third damage state requires replacement of structural assembly10, the HSS brace42section, and the connection gusset plates38in addition to some local repairs of the beams56and columns58. Some generic costs with respect to removal of nonstructural elements (such as removal of architectural and mechanical, electrical, and plumbing systems), obtaining access to damaged elements, and temporary activities (such as removal or protection of the contents adjacent to the damaged area and protection of the surrounding area against dust and noise with temporary enclosures) were also included in all the repair costs based on FEMA P-58 provided data.

The structural loss, the repair cost, and the repair time were determined by inputting the defined fragility functions (FIG.20) and consequence functions (FIGS.23A and23B) and also the maximum inter-story drift values (FIGS.18A-18C) into the PACT software.FIGS.24-25Bcompare the total repair cost and repair time of each floor, respectively at DBE and MCE demand levels. All the floors were assumed to be repaired concurrently. The complete repair of the SCBF system took 19 days at the DBE demand level, and 29 days at the MCE demand level. These numbers are 0 and 5 days, respectively, in the structural bracing system12.

The building system using bracing42including one or more structural assemblies10did not reach any damage state under DBE demand level and, accordingly,FIGS.24-25Bdo not show any repair cost and repair time. On the other hand, the SCBF building experienced some damage, resulting in appreciable repair cost and time. While both systems experience some damage at MCE, the extent of damage is higher in the SCBF system. Although the construction cost was increased by 4% in the structural bracing system12(seeFIG.15), the repair cost has been reduced by 100% at DBE demand level and by 95% at MCE demand level in comparison to the SCBF building, hence, negating the construction cost increase of 4%.

Functionality curves and resiliency were derived based on Equations (30) to (33).FIGS.26A-26Dshow the results. Initially the functionality was 100% in all systems because there is no initial damage. The ground motion was arbitrarily assumed to occur after 5 days once monitoring the functionality of buildings began (i.e., t0Eis 5). Selecting different values for t0Eshifts the functionality curve forward or backward without changing the final results. Functionality of the SCBF system was decreased to 91% at the DBE demand level (FIG.26A) and 83% at the MCE demand level (FIG.26B) due to the damages experienced by the system. Since the structural bracing system12did not reach any damage state at the DBE demand level, the functionality remained 100% without any decrease (FIG.26C). Due the small amount of damages experience by the structural bracing system12at the MCE demand level, the functionality was decreased slightly to 99% (FIG.26D) in this system. The functionality returns to 100% in all systems after passing the repair time and following the linear recovery function.

The resiliency of the systems (R) was obtained by calculating the area under the functionality curve according to Equation (30). These values are presented inFIGS.26A-26D. Comparing the repair costs, time and the resiliencies shows the better seismic performance of the structural bracing system12versus the SCBF system.

Conclusion

Utilizing Belleville disks14and shape memory alloy16, a resilient bracing system with nonlinear-elastic behavior was developed. This system is referred to as the structural bracing system12. Detailed equations for proportioning the various components of the structural bracing system12were derived and used to design a 5-story building. A companion special concentrically braced frame (SCBF) was also designed as a benchmark to evaluate the performance of the structural bracing system.

Fragility curves were generated for both systems by performing incremental nonlinear dynamic analyses of three-dimensional models of the designed buildings. A suite of ground motions was selected for the analyses Consequence functions were also generated for the systems in order to estimate the loss in the system after seismic events. In an effort to determine the resiliency of the structural bracing system12, the level of damage and the associated repair costs were quantified.

While the SCBF building had residual deformations at design base event (DBE) and maximum considered event (MCE) demand levels, the structural bracing system12could resist the ground motions with less drift and no residual deformation. The cost of new materials in the structural bracing system12results in a 4% construction cost increase in comparison to the SCBF system. However, this increase is rather negligible in comparison to the repair costs, 100% reduction at the DBE and 95% at MCE for the structural bracing system12because of its improved performance.

The stiffness and strength of a brace42including one or more structural assemblies10were determined based on SCBF bracing system since the seismic design parameters listed in ASCE7-16 are for common structural systems and not yet available for the proposed novel system.

Considering that the beams and columns remained elastic for the reported case studies, the focus was on evaluating the resiliency and performance of the braces—the structural bracing system12versus conventional braces.

Example 2: Further Seismic Assessments of the Structural Bracing System (5-, 10-, and 15-Story)

The structural bracing system12is a resilient bracing system based on the application of Bellville disks14and Nitinol rods16. The cyclic behavior of the structural assembly10was obtained, and the design equations were developed based on the available literature. Seismic performance of the system was studied analytically. Two groups of buildings with different lateral force resisting systems were designed and studied: one group with the structural bracing system12, and the other with the special concentrically braced frame (SCBF) system. Each building group consisted of 5-, 10-, and 15-story buildings. The Design-Base-Event (DBE) and Maximum Considered Event (MCE) were considered as the seismic hazard, and a suite of seven ground motions were scaled accordingly for response history analyses. Finally, the resiliency of the buildings was studied by obtaining the functionality curve of the buildings before and after the seismic event. The construction cost of the 5-story building with the structural bracing system12increased, but the post-earthquake cost decreased significantly. The application of structural bracing system12in the 10- and 15-story buildings reduced both the construction and repair costs, considerably. Resiliency of all the buildings was improved when structural bracing system12was used as the lateral force resisting system.

Archetype Buildings

Two groups of buildings were studied. The lateral force resisting system in one group was the structural bracing system12, and in the other was the SCBF. Each building group consisted of 5-, 10-, and 15-story buildings. All buildings were office buildings with the same floor plan. The three-dimensional view of 10-story buildings are shown inFIGS.32A and32B. The structural bracing system12used in the buildings was designed based on having one assembly10in each of the chevron braces42. The assemblies10are shown inFIG.32B. Each story has a height of 4.3 m (14 ft.). The buildings were assumed to be in Northern California region in a site classified as “D” per ASCE 7-16 (2016). The floors carried 340 (kg/m2) dead load and 245 (kg/m2) live load. The roof of the buildings carried 220 (kg/m2) dead load and 100 (kg/m2) live load. Structural sections for the buildings were selected according to Steel Construction Manual (AISC, 2017). Tables 13 to 17 list the structural member sections for the 10-story buildings. The chevron effect analysis method was used to analyze and design the chevron beams in the systems.

Tables 16 and 17 list the designed assemblies for the 10-story structural bracing system building12. The assemblies10were designed according to the equations obtained by Hadad et al. (Hadad, A. A., B. M. Shahrooz and P. J. Fortney (2021). “Innovative resilient steel braced frame with Belleville disk and shape memory alloy assemblies,” Engineering Structures, Vol. 237, No. pp. 112166). In the aforementioned tables, the number of assemblies10used in each brace42is presented by nasm, the outside diameter of the selected Bellville disk14for the assembly10is D0, the inside diameter of the selected Bellville disk is Di, the thickness of the disk is t, the free height (rise) of the disk is h, the number of disks stacked together in parallel is nD, the number of disk groups in the assembly is nG, and the dimeter of the selected Nitinol rod for the assembly is DNiTi.

Construction costs of the buildings were estimated based on the designed sections and according to the available construction cost data (Gordian, 2018) and industrial quotes for Nitinol rods and Bellville disks (Schnorr, 2003; SEAS-Group, 2018).FIGS.33A-33Ccompares the construction cost of the buildings. Due to the usage of new material and design procedure in a brace42including one or more structural assembly10, the cost of the brace42including one or more structural assembly10is almost twice the cost of regular SCBF brace. However, the total construction cost of 5-story building with the structural bracing system12is only 4% greater than the SCBF building. The total construction cost of 10- and 15-story buildings with the structural bracing system12are 25% and 23% lower than the buildings with SCBF system, respectively. As discussed earlier, the value of maximum tensile force in the tensile brace52is close to the value of maximum compressive force in the compressive brace54(i.e., mechanism forces) in the structural bracing system12. Thus, they apply a lower demand on braced frame beams56and columns58when compared to the SCBF brace. Such lower demand in the structural bracing system12leads to lighter beam56and column58sections in the braced frames36. Due to the greater number of beams56and columns58in the 10- and 15-story buildings, the amount of cost reduction in the beams56and columns58exceeds the expense increase because of the addition of one or more structural assemblies10in the braces42, resulting in the lower total construction cost of structural bracing system12buildings in comparison to the SCBF buildings.

Seismic Ground Motions

PEER ground motion database and record scaling tool (PEER (2013). PEER ground motion database. University of California, Berkeley, CA, Pacific Earthquake Engineering Research Center) were used to obtain seven ground motion records (Table 18) and the necessary scale factors to match the ground motion intensity to the design base earthquake (DBE) hazard level. The maximum considered earthquake (MCE) and design base earthquake (DBE) acceleration response spectra (Sa) were obtained for the buildings' location according to ASCE/SEI7-16 (2016) provided guidelines and seismic design factors.

FIG.34shows the acceleration spectra of scaled ground motions versus the target DBE for the 5-story building. The scale factors obtained by the PEER tool to match ground motions to DBE spectrum were then multiplied by 1.5 to increase the hazard level of the ground motions to the MCE level.

Three dimensional models of the buildings were created in RUAUMOKO-3D (Carr, 2008), and the horizontal components of each ground motion acceleration were applied to the buildings concurrently per ASCE/SEI41-13 (2014); NIST (2017) to perform response history analyses on the building models. The aforementioned codes do not require concurrent analysis for regular buildings; however, such analyses were performed to study the behavior of the structural bracing system12when used in both building directions.

Columns58, beams56, and braces42were modelled to have an elastic-perfectly-plastic behavior. Structural assemblies10were modelled based on the cyclic behavior discussed in Hadad et al. (2021). Concrete slab with 10 cm (4 in.) thickness was used as the floor system. Plane-stress quadrilateral elements with elastic material behavior were used to model the floor system. According 178 to NIST (2017), stiffness for elastic diaphragms was calculated using expected material properties and reduced to reflect an effective stiffness based on the expected deformations and cracking. Thus, 30 percent of concrete Young's modulus was input as the in-plane stiffness, and concrete Poisson's ratio was input as the Poisson's ratio of the quadrilateral elements.

The maximum inter-story drift ratio for each building was obtained for each ground motion, and then the average of the seven values was calculated and reported as the average of maximum inters-story ratio. The results are shown inFIGS.35A to37B. Based onFIGS.35A and35B, the 5-story building experienced more drift in comparison to the structural bracing system12building at both DBE and MCE hazard levels. This figure also shows the drift difference between the SCBF system and the structural bracing system12was increased as the SCBF system experienced more plastification at the MCE level.FIGS.36A-37Bshow the drift values of the two systems were close to each other in the 10- and 15-story buildings at both DBE and MCE hazard levels. Since the SCBF system did not experience significant plastic deformations in the 10- and 15-story buildings, the drift values of the two systems in the selected mid-rise and high-rise buildings did not show a great difference. Comparison ofFIGS.35A to37Balso reveals that the maximum drift occurred at lower levels in the 5-story buildings while the 10- and 15-story buildings experienced the maximum drift at the higher levels.

FIGS.38A to40Bshow representative roof displacement histories for Friuli ground motion at different hazard levels. According toFIGS.38A and38B, the structural bracing system did not experience any permanent deformation while the SCBF systems experienced a considerable amount of residual deformation.FIGS.39A-40Bshow the amount of residual deformation was decreased as the building height was increased.

Incremental Dynamic Analysis (IDA) and Fragility Curves

In order to evaluate the performance of seismic systems, probable damage states were identified for each brace type. Also, the inter-story drift at which each damage states occurred was obtained by performing Incremental Dynamic Analysis (IDA) (Vamvatsikos, D. and C. A. Cornell (2002). “Incremental dynamic analysis,” Earthquake engineering & structural dynamics, Vol. 31, No. 3, pp. 491-514) on the buildings.FIGS.41A-42Cshow the results of IDA for the buildings. The vertical axis shows the spectral acceleration of the ground motions, and the horizontal axis shows the maximum inter-story drift. The ground motion records were scaled up incrementally until the systems reached collapse. Collapse state was defined as the numerical instability of response history analysis due to the yielding and plastic deformations of gravity members leading to excessive deformations in the building.

Fragility function measures the probability of reaching a specific damage state under earthquake excitation at a certain inter-story drift. Equation 34 (above) was used to obtain the fragility function of the systems. According to FEMA-P58 (2018), fragility estimates obtained with as few as seven ground motions per intensity level are likely to be of comparable quality to those obtained using a greater number of ground motions (order of thirty).

In order to build the fragility curves, two damage states were considered for a SCBF brace: (1) brace compression buckling, and (2) brace tensile yielding. Three damage states were defined for a brace42including one or more structural assemblies10: (1) structural assembly10reaching the tensile force corresponding to 6% strain in Nitinols, (2) structural assembly10reaching stack's maximum compression force, and (3) HSS section in the brace42including one or more structural assemblies10reaching compression buckling force. The strain of 6% was selected for Nitinol because this material experiences permanent deformations when deformed beyond 6% strain in tension (DesRoches et al., 2004). Table 19 lists the drift ratio median (θ) and dispersion (β) values for all the selected damage states in all three buildings for the SCBF system and the structural bracing system12. The mean and dispersion values for each damage state were obtained by tracking the member's state (force) versus maximum inter-story drift at different increments of the dynamic analysis and saving the value once it reached the damage state limit. Since larger brace sections experienced each damage states at larger drifts, the brace sections in the buildings were categorized into three groups based on the brace unit weight. The 5-story building had one brace group (Group 1), the 10-story building had two brace groups (Group 2 braces were used in levels 1 to 5, and Group 1 braces were used in levels 6 to 10), and the 15-story building had three brace groups (Group 3 braces were used in levels 1 to 5, Group 2 braces were used in levels 6 to 10, Group 1 braces were used in levels 11 to 15).

As seen from Table 19, the mean drift ratio (θ) causing damage state i (i=1, 2) in brace group n (n=1, 2, 3) of SCBF brace is lower than the corresponding value for the brace42including one or more structural assemblies10, i.e., the SCBF brace experiences damage state i at lower drift values when compared to the structural bracing system12. The dispersion of the drift ratios (β) in each damage state determines the extent to which the fragility curve is stretched.FIGS.43A-43Cshow all the fragility curves based on the defined damage states in the systems. Considering a certain group of braces (e.g., Group 1) for a certain damage state (e.g., Damage state 1), the fragility curve for the brace42including one or more structural assemblies10is lower than the SCBF brace curve (FIG.43A) which means the brace42including one or more structural assemblies10is less likely to experience the damage state at a given drift value when compared to the SCBF brace. This observation matches the median drift values presented in Table 19: the median drift value for damage state 1 in SCBF Group 1 braces (θ=0.29) is lower than the corresponding value in the structural bracing system12(θ=0.58) indicating SCBF system experiences this damage earlier in comparison to the structural bracing system12.

Equation (30), which was defined by Bruneau et al. (2003), was used to measure the resiliency of systems. In this equation, R is the resiliency of the system, Q (t) is the functionality of the system under consideration which is defined in Equation (31). The structural loss was calculated based on Equation (32). Linear recovery function (ƒrec) was shown in Equation (33).

Consequence functions are relationships that indicate the potential distribution of losses as a function of damage state and translate damage into potential repair and replacement costs as well as repair time. The consequence functions for the bracing systems were developed based on PACT supporting material and digital libraries (FEMA-P58, 2018), construction costs data (Gordian, 2018), and estimated construction costs shown inFIGS.33A-33C. This data was input into PACT.

The structural loss, the repair cost, and the repair time were determined by inputting the defined fragility functions, consequence functions, and the maximum inter-story drift values into PACT.FIGS.44A-44Ccompares the total repair cost of the buildings for each of the hazard levels. Although the application of the structural bracing system12in the 5-story building increased the total construction cost by 4%, it removed 100% and 95% of the repair costs at the DBE and MCE hazard level, respectively. The repair cost in the 10-story structural bracing system12building was lower than the SCBF building repair costs by 76% and 42% at the DBE and MCE hazard levels, respectively. Similarly for the 15-story building, the repair costs of structural bracing system12at DBE and MCE hazard levels were 66% and 41%, respectively lower than those in the SCBF system. Thus, the application of the structural bracing system12in high-rise buildings (10- and 15-story) reduces both the construction costs (seeFIGS.33A-33C) and repair costs (seeFIGS.44A-44C).

The structural bracing system12experienced some damage at all the applied hazard levels in the 10- and 15-story buildings; however, the repair costs for the damage were lower than those in the SCBF system. Such lower damage in the structural bracing system12is due to the application of different elements in the braces42including one more structural assemblies10in comparison to the SCBF brace. In the brace42including one or more structural assemblies10, Nitinol rods16are the first elements to experience damage corresponding to tensile strain values of 6% or more. At higher loads, the compression disks14experience full deformation that ultimately leads to buckling of the HSS section. Thus, for different load intensities, only some elements of the brace42might need to be replaced due to damage while the other elements remain undamaged. In the SCBF brace, however, the entire brace buckles once it reaches the compressive strength which requires replacement of the entire brace.

FIGS.45A to47Bcompare the repair time of the buildings required for each floor when different levels of ground motion hazard were applied to the buildings. The floors were assumed to be repaired concurrently. Since the 5-story structural bracing system building12did not experience any damage at the DBE hazard level, the repair times for this building are zero inFIGS.45A and45B.FIGS.45A and45Balso shows lower levels of the 5-story buildings experienced more damage in comparison to the upper levels. This behavior was reversed in the 10- and 15-story buildings (FIGS.46A-47B) in which the majority of damage occurred in the upper levels. Such observations are similar to the trend of the maximum inter-story drift ratios shown inFIGS.35A-37B.

FIGS.48A to50Bshow the functionality (Q) of the systems. Initially the functionality was 100% since there was no initial damage. The ground motion was arbitrarily assumed to occur after 5 days once monitoring the buildings' functionality began (i.e., t0Eis 5). As observed earlier through the repair cost and time diagrams, the 5-story structural bracing system12building did not experience any damage at the DBE hazard level. This is reflected in the functionality curve ofFIG.48A. The steady horizontal functionality curve means no disruption on the system performance. The graphs show a greater decrease in the functionality of systems when the intensity of ground motions increased. It should be noted that the functionality curves are only based on the performance of the braces in order the compare the brace42including one or more structural assemblies10versus SCBF brace. Inclusion of other structural and non-structural members leads to greater decrease in the functionality curves due to the damages incurred in such members.

Resiliency (R) of the systems was obtained by calculating the area under the functionality curves based on Equation (30). Table 20 shows the resiliency of the buildings at different hazard levels. According to these results, application of the structural bracing system12improved the resiliency of all buildings by 5% on average at all the considered hazard levels. Since only the damage in the braces was the only loss metric in the buildings (Ls), the resiliency values are relatively close. Inclusion of fragility curves and consequence functions for other structural and non-structural members is expected to result in a greater variation of the values.

TABLE 20Resiliency of buildings (%)SystemNumber of storiesHazard level 1Hazard level 2Structural Bracing510097System109996159897SCBF59391109693159391

CONCLUSION

In order to assess the seismic performance of the structural bracing system12, two groups of buildings were evaluated. The structural bracing system12was used as the lateral force resisting system in the first group, and the other group had SCBF system. Each group consisted of 5-, 10-, and 15-story buildings. Two levels of seismic hazard were considered and a suite of seven ground motions were applied to the buildings to evaluate seismic performance. The first hazard level had 10% probability of occurrence in 50 years (DBE), and the second had 2% probability of occurrence in 50 years (MCE).

While the SCBF buildings experienced residual deformations under some ground motions at the DBE and MCE hazard level, the structural bracing system12did not experience any residual deformation. The nonlinear elastic behavior of the structural assembly10enables the system to resist the earthquake load without permanent deformations. The maximum inter-story drift ratios showed the 5-story SCBF system experienced more drift in comparison to the structural bracing system12at lower levels. In the 10- and 15-story buildings, however, the drift values were close and both systems experienced the maximum drift at upper levels.

The estimated construction cost of the structural bracing system12was 4% higher than that in the SCBF system for 5-story buildings. In the 10- and 15-story buildings, however, the estimated construction cost of the structural bracing system12buildings was 25% and 35%, respectively, lower than that in the SCBF buildings. The brace mechanism forces in the structural bracing system12were closer leading to lower demands in the chevron beams56and the supporting columns58in the braced frame36. Thus, lighter beam and column section were required in the braced bays of the structural bracing system12. Inclusion of additional design equations in the structural bracing system12in order to have equal tension and compression mechanism forces in the brace42including one or more structural assemblies10will further reduce the demands on chevron beams56and columns58which results in lighter sections for those members. Lighter sections reduce construction costs in the structural bracing system12.

Fragility curves were generated for both systems by performing incremental nonlinear dynamic analyses of three-dimensional models. Consequence functions were also generated for the systems to estimate the loss in each system after seismic events. To determine the resiliency of the structural bracing system12, the level of damage and the associated repair costs were quantified. The structural bracing system12required less repair time and cost in comparison to the SCBF system in all buildings when the systems were evaluated under different hazard levels. According to the measured resiliency values, application of the structural bracing system12improved the resiliency of all buildings by 5%, on average for all the considered hazard levels.

While the present invention has been illustrated by the description of various embodiments and while these embodiments have been described in some detail, it is not the intention of the Applicant to restrict or in any way limit the scope of the invention to such detail. Additional advantages and modifications will readily appear to those skilled in the art. The invention in its broader aspects is therefore not limited to the specific details and illustrative examples shown and described. Accordingly, departures may be made from such details without departing from the scope of the general inventive concept.