Out‐of‐plane stability of buckling‐restrained braces including moment transfer capacity

Buckling‐restrained braces (BRBs) are widely used as ductile seismic‐resistant and energy‐dissipating structural members in seismic regions. Although BRBs are expected to exhibit stable hysteresis under cyclic axial loading, one of the key limit states is global flexural buckling, which can produce an undesirable response. Many prior studies have indicated the possibility of global buckling of a BRB before its core yields owing to connection failure. In this paper, BRB stability concepts are presented, including their bending‐moment transfer capacity at restrainer ends for various connection stiffness values with initial out‐of‐plane drifts, and a unified simple equation set for ensuring BRB stability is proposed. Moreover, a series of cyclic loading tests with initial out‐of‐plane drifts are conducted, and the results are compared with those of the proposed equations. © 2013 The Authors. Earthquake Engineering & Structural Dynamics published by John Wiley & Sons Ltd.     

An experimental study on steel‐encased buckling‐restrained brace hysteretic dampers

This paper describes a seismic test program performed on 12 steel‐encased buckling‐restrained braces (BRBs). The use of rolled or built‐up buckling‐restraining mechanisms with welded or bolted attachments was examined. In addition, the effects of bolt pretension, core‐to‐encasing attachment details, aspect ratio of core segment and imperfections due to manufacturing on the brace response were investigated. All specimens were subjected to a stepwise incremental quasi‐static testing protocol with a maximum axial strain amplitude of 2%. All specimens except one showed satisfactory performance with stable hysteretic response and sustained cumulative inelastic deformations in excess of 200 times the yield deformation. Based on the experimental results, the compression strength adjustment factor, and the strain hardening adjustment factor for each excursion were quantified. Test results revealed that these two factors are significantly influenced by the parameters investigated in the experimental program. BRB details were also found to influence the buckling and the yielding patterns of the core segment. Copyright © 2009 John Wiley & Sons, Ltd.     

Developing high‐performance aluminum alloy buckling‐restrained braces based on series of low‐cycle fatigue tests

In order to enhance the durability of high‐performance buckling‐restrained braces (BRBs) used in bridge engineering, which are expected to withstand severe earthquakes three times without being replaced, aluminum alloys were employed to manufacture BRBs. A series of low‐cycle fatigue tests, including 18 specimens, were conducted to address the low‐cycle fatigue strength of the aluminum alloy BRB. Test results of all specimens show that stable hysteretic curves were obtained without overall buckling occurrence. Failure mode of the welded aluminum alloy BRB is obviously affected by the ribs' welding under the variable or constant strain amplitude condition. Therefore, another type of aluminum alloy BRB, the bolt‐assembled BRB with or without spot‐welded stoppers, is proposed and tested. Results showed that the low‐cycle fatigue performance of bolt‐assembled BRBs with stoppers improved four to five times compared with welded BRBs. However, the stoppers' spot welding has an adverse effect on the failure mode because the crack, which induced the specimen's failure, initiated from the spot weld toes of the stoppers. Both bolt‐assembled BRBs with and without stoppers can meet the cumulative inelastic deformation requirement proposed for high‐performance BRBs under the constant strain amplitude, not larger than 2%. In addition, under the variable strain amplitude condition, only the bolt‐assembled BRB without stoppers has an excellent cumulative inelastic deformation capacity and sustains two cycles of 2.5% strain amplitude. Finally, recommended Manson–Coffin equations and preliminary cumulative damage formulae for welded and bolt‐assembled BRBs are given as the references of the strain‐based damage evaluation. Copyright © 2011 John Wiley & Sons, Ltd.     

Seismic design and testing of buckling‐restrained braces with a thin profile

A thin‐profile buckling‐restrained brace (thin‐BRB) consists of a rectangular steel casing and a flat steel core that is parallel to a gusset plate. A thin configuration reduces the width of the restraining member and thus saves usable space in buildings. However, deformable debonding layers, which cover the steel core plate in order to mitigate the difference between the peak tensile and compressive axial forces, provide a space for the steel core to form high mode buckling waves when the thin‐BRB is under compression. The wave crests squeeze the debonding layers and produce outward forces on the inner surface of the restraining member. If the restraining member is too weak in sustaining the outward forces, local bulging failure occurs and the thin‐BRB loses its compression capacity immediately. In order to investigate local bulging behavior, a total of 22 thin‐BRB specimens with a ratio of steel core plate to restraining steel tube depth ranging from 0.3 to 0.7 and axial yield force capacities ranging from 421 kN to 3036 kN were tested by applying either cyclically increasing, decreasing, or constant axial strains. The restraining steel tube widths of all the specimens were smaller than 200 mm and were infilled with mortar with a compressive strength of 97 MPa or 55 MPa. Thirteen of the 22 thin‐BRB specimens' restraining members bulged out when the compressive core strains exceeded 0.03. A seismic design method of the thin‐BRB in preventing local bulging failure is proposed in this study. Test and finite element model (FEM) analysis results suggest that the outward forces can be estimated according to the BRB compressive strength, steel core high mode buckling wavelength, and the debonding layer thickness. In addition, the capacity of the restraining member in resisting the outward forces can be estimated by using the upper bound theory in plastic analysis. Both the FEM analysis and test results indicate that the proposed method is effective in predicting the possibility of local bulging failure. Test results indicate that the proposed design method is conservative for thin‐BRB specimens with a large steel core plate to restraining steel tube depth ratio. This paper concludes with design recommendations for thin‐BRBs for severe seismic services. Copyright © 2015 John Wiley & Sons, Ltd.     

Seismic performance analysis of BRBs and gussets in a full‐scale 2‐story BRB‐RCF specimen

The implementation of buckling‐restrained braces (BRBs) for new reinforced concrete frame (RCF) constructions is limited. This study investigates the seismic forces and stability in the BRBs and gussets of a 2‐story full‐scale RCF specimen by using Abaqus models and a newly proposed stability evaluation method. The hybrid and cyclic loading test results are accurately predicted by the Abaqus analyses. Existing methods for computing the gusset interface forces for steel buildings from both the brace and the frame actions are compared with the Abaqus results. The applicability of these methods for the BRB‐RCF design is critically evaluated. It is confirmed that the Parallel‐2 method is suitable for estimating the BRB force demand imposed on the corner gusset and the generalized uniform force method is good for the corner gusset at the base. In addition, existing stability evaluation methods for BRBs and gussets are applied to investigate the out‐of‐plane (OOP) buckling of the first‐story BRB observed at the end of tests. The proposed stability model incorporates the BRB restrainer's flexural effects and 4 rotational springs in assessing the BRB's buckling. This model confirms that the BRB and the gusset's OOP buckling limit states could be coupled and must be evaluated together. By incorporating the flexural effects of the steel casing and the infilled grout, the proposed model satisfactorily predicts the OOP buckling of the first‐story BRB and gussets. These research results can be used for the implementation of BRBs in new RC frame constructions.     

Experimental and numerical studies on buckling restrained braces with posttensioned carbon fiber composite cables

There has been an increasing interest in using residual deformation as a seismic performance indicator for earthquake resistant building design. Self-centering braced structural systems are viable candidates for minimizing residual deformations following a major earthquake. Hence, this study proposes an alternative type of buckling restrained brace (BRB) with externally attached posttensioned (PT-BRB) carbon fiber composite cables (CFCCs). The steel core of the brace is used as an energy dissipator, whereas the CFCCs provide the self-centering force for minimizing residual story drifts. Three proof-of-concept specimens are designed, fabricated, and cyclically tested at different posttensioning force levels. The CFCC behavior to obtain cyclic response, including the anchorage system, is examined closely. A parametric study is also conducted to show the effect of the different configurations of PT-BRBs on the inelastic response. Furthermore, optimal brace parameters are discussed to realize design recommendations. The results indicated that the implementation of partially self-centering BRBs in building frames can lead to the target residual displacements. A stable behavior is obtained for the proposed PT-BRBs when subjected to the loading protocol specified in the American Institute of Steel Construction (AISC) 2016 Seismic Provisions.     

Welded end‐slot connection and debonding layers for buckling‐restrained braces

The debonding mechanism has a significant effect on the performance of a buckling‐restrained brace (BRB). In this paper, a method for estimating the compression strength adjustment factor for any given BRB core strain is presented. Experimental investigations were conducted on four BRBs to examine the efficiency of four different debonding materials in reducing the difference between the cyclic peak compression and tension. Test results indicate that chloroprene rubber is very easy to install and very effective in minimizing the difference between the compressive and tensile capacities. The excellent performance of 13 full‐scale welded end‐slot BRBs (WES‐BRBs) is illustrated through experiments. Cyclic loading test results of a 12.5‐m long jumbo WES‐BRB reveal that its peak compressive strength exceeds 16,800 kN and its maximum core strain reaches 0.035. All WES‐BRBs show satisfactory performance with a very stable hysteresis response, modest peak compressive to tensile strength ratio, and very predictable axial stiffness. These specimens sustain a cumulative plastic deformation of greater than 400 times the yield deformation. The hysteresis responses can be satisfactorily predicted by using a two‐surface plasticity analytical model. Advantages of the welded end‐slot connections are also presented through a discussion on the effects of the BRB yield region length ratio on the effective stiffness, the yield story drift, and the core strain level. Copyright © 2014 John Wiley & Sons, Ltd.     

Seismic design and hybrid tests of a full‐scale three‐story buckling‐restrained braced frame using welded end connections and thin profile

A series of hybrid and cyclic loading tests were conducted on a three‐story single‐bay full‐scale buckling‐restrained braced frame (BRBF) at the Taiwan National Center for Research on Earthquake Engineering in 2010. Six buckling‐restrained braces (BRBs) including two thin BRBs and four end‐slotted BRBs, all using welded end connection details, were installed in the frame specimen. The BRBF was designed to sustain a design basis earthquake in Los Angeles. In the first hybrid test, the maximum inter‐story drift reached nearly 0.030 rad in the second story and one of the thin BRBs in the first story locally bulged and fractured subsequently before the test ended. After replacing the BRBs in the first story with a new pair, a second hybrid test with the same but reversed direction ground motion was applied. The maximum inter‐story drifts reached more than 0.030 rad and some cracks were found on the gusset welds in the second story. The frame responses were satisfactorily predicted by both OpenSees and PISA3D analytical models. The cyclic loading test with triangular lateral force distribution was conducted right after the second hybrid test. The maximum inter‐story drift reached 0.032, 0.031, and 0.008 rad for the first to the third story, respectively. This paper then presents the findings on the local bulging failure of the steel casing by using cyclic test results of two thin BRB specimens. It is found that the steel casing bulging resistance can be computed from an equivalent beam model constructed from the steel core plate width and restraining concrete thickness. This paper concludes with the recommendations on the seismic design of thin BRB steel casings against local bulging failure. Copyright © 2011 John Wiley & Sons, Ltd.     

Effect of non‐moment braced frame seismic deformations on buckling‐restrained brace end connection behavior: Theoretical analysis and subassemblage tests

In‐plane buckling‐restrained brace (BRB) end rotation induced by frame action is a commonly observed phenomenon in buckling‐restrained braced frames (BRBFs). However, its effect on BRB end connection behavior has not yet been clear. In this study, four BRB end deformation modes for quick determination of end rotational demand are proposed for non‐moment BRBF considering different BRB arrangements, installing story of BRBs, and boundary condition of corner gussets connected with column base. Key factors affecting BRB end rotation and flexural moments are examined theoretically by parametric analysis. Subassemblage tests of seven BRB specimens under horizontal cyclic loading were conducted by adopting two loading frames to impose the expected BRB end deformations. It shows that BRB end rotation subjected BRB ends to significant flexural moments, leading to premature yielding of BRB ends or even tendency of end zone buckling. The deformation modes, the flexural rigidity of BRB ends, and the initial geometric imperfections of BRBs were found to have significant influence on BRB end connection behavior. The triggering moment induced by BRB end rotation was the main contributor to end flexural moment. However, the moment amplification effect induced by flexure of BRB end zones became prominent especially for small flexural rigidity of BRB ends. Implications and future research needs for design of BRB end connections are provided finally based on the theoretical and experimental results. Copyright © 2015 John Wiley & Sons, Ltd.     

Seismic performance evaluation of single damped‐outrigger system incorporating buckling‐restrained braces

The outrigger system is an effective means of controlling the seismic response of core‐tube type tall buildings by mobilizing the axial stiffness of the perimeter columns. This study investigates the damped‐outrigger, incorporating the buckling‐restrained brace (BRB) as energy dissipation device (BRB‐outrigger system). The building's seismic responses are expected to be effectively reduced because of the high BRB elastic stiffness during minor earthquakes and through the stable energy dissipation mechanism of the BRB during large earthquakes. The seismic behavior of the BRB‐outrigger system was investigated by performing a spectral analysis considering the equivalent damping to incorporate the effects of BRB inelastic deformation. Nonlinear response history analyses were performed to verify the spectral analysis results. The analytical models with building heights of 64, 128, and 256 m were utilized to investigate the optimal outrigger elevation and the relationships between the outrigger truss flexural stiffness, BRB axial stiffness, and perimeter column axial stiffness to achieve the minimum roof drift and acceleration responses. The method of determining the BRB yield deformation and its effect on overall seismic performance were also investigated. The study concludes with a design recommendation for the single BRB‐outrigger system.     

Hybrid experimental performance of a full‐scale two‐story buckling‐restrained braced RC frame

This paper proposes a novel implementation of buckling‐restrained braces (BRB) in new reinforced concrete (RC) frame construction. Seismic design and analysis methods for using a proposed steel cast‐in anchor bracket (CAB) to transfer normal and shear forces between the BRB and RC members are investigated. A full‐scale two‐story RC frame with BRBs (BRB‐RCF) is tested using hybrid and cyclic loading test procedures. The BRBs were arranged in a zigzag configuration and designed to resist 70% of the story shear. The gusset design incorporates the BRB axial and RCF actions, while the beam and column members comply with ACI 318‐14 seismic design provisions. Test results confirm that the BRBs enhanced the RCF stiffness, strength, and ductility. The hysteresis energy dissipation ratios in the four hybrid tests range from 60% to 94% in the two stories, indicating that BRBs can effectively dissipate seismic input energy. When the inter‐story drift ratio for both stories reached 3.5% in the cyclic loading test, the overall lateral force versus deformation response was still very stable. No failure of the proposed steel CABs and RC discontinuity regions was observed. This study demonstrates that the proposed design and construction methods for the CABs are effective and practical for real applications. Copyright © 2016 John Wiley & Sons, Ltd.     

Steel buckling‐restrained braced frames with single and dual corner gusset connections: seismic tests and analyses

The design of a three‐story buckling‐restrained braced frame (BRBF) with a single‐diagonal sandwiched BRB and corner gusset was evaluated in cyclic tests of a one‐story, one‐bay BRBF subassembly and dynamic analyses of the frame subjected to earthquakes. The test focused on evaluating (1) the seismic performance of a sandwiched BRB installed in a frame, (2) the effects of free‐edge stiffeners and dual gusset configurations on the corner gusset behavior, (3) the frame and brace action forces in the corner gusset, and (4) the failure mode of the BRBF under the maximum considerable earthquake level. The subassembly frame performed well up to a drift of 2.5% with a maximum axial strain of 1.7% in the BRB. Without free‐edge stiffeners, the single corner gusset plate buckled at a significantly lower strength than that predicted by the specificationof American Institute of Steel Construction (2005). The buckling could be eliminated by using dual corner gusset plates similar in size to the single gusset plate. At low drifts, the frame action force on the corner gusset was of the same magnitude as the brace force. At high drifts, however, the frame action force significantly increased and caused weld fractures at column‐to‐gusset edges. Nonlinear time history analyses were performed on the three‐story BRBF to obtain seismic demands under both design and maximum considerable levels of earthquake loading. The analytical results confirmed that the BRB and corner gusset plate achieved peak drift under cyclic loading test. Copyright © 2011 John Wiley & Sons, Ltd.     

High‐mode buckling responses of buckling‐restrained brace core plates

Cyclic loading tests and finite element analyses on six novel all‐steel buckling‐restrained braces (BRBs) are conducted using different loading patterns to investigate the core plate high‐mode buckling phenomenon. The proposed BRB is composed of a core member and a pair of identical restraining members, which restrains the core member by using bolted shim spacers. The design of the proposed BRB allows the core plate to be visually inspected immediately following a major earthquake. If necessary, the pair of restraining members can be conveniently disassembled, and the damaged core plate can be replaced. Test results indicate that the proposed BRBs can sustain large cyclic strain reversals and cumulative plastic deformations in excess of 400 times the yield strain. Experimental and analytical results confirm that the high‐mode buckling wavelength is related to the core plate thickness and the applied loading patterns. The larger the axial compressive strain is applied, the shorter the high‐mode buckling wavelength would be developed. The buckling wavelength is about 12 times the core plate thickness when the high‐mode buckling shape is fully developed. However, it reduces to about 10 times the core plate thickness when a compressive core strain reaches greater than 0.03. The high‐mode bucking wavelength can be satisfactorily predicted using the proposed method or from the finite element analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Retrofit of reinforced concrete frames with buckling‐restrained braces

Damage to buildings observed in recent earthquakes suggests that many old reinforced concrete structures may be vulnerable to the effects of severe earthquakes. One suitable seismic retrofit solution is the installation of steel braces to increase the strength and ductility of a building. Steel bracings have some compelling advantages such as their comparatively low weight, their suitability for prefabrication, and the possibility of openings for utilities, access, and light. The braces are typically connected to steel frames that are fixed to the concrete structure using post‐installed concrete anchors along the perimeter. However, these framed steel braces are not without some disadvantages such as heavier steel usage and greater difficulties during the installation. Therefore, braces without steel frames appear to be an attractive alternative. In this study, braces were connected to gussets furnished with anchor brackets, which were fixed by means of a few post‐installed concrete anchors. The clear structural system and the increased utilization of the anchors allowed the anchorage to be designed precisely and economically. The use of buckling‐restrained braces (BRBs) provides additional benefits in comparison with conventional braces. BRBs improve the energy dissipation efficiency and allow the limitation of the brace force to be taken up by the highly stressed anchorage. Cyclic loading tests were conducted to investigate the seismic performance of BRBs connected with post‐installed anchors used to retrofit reinforced concrete frames. The tests showed that the proposed design method is feasible and increases strength as well as ductility to an adequate seismic performance level. Copyright © 2014 John Wiley & Sons, Ltd.     

Earthquake response analyses of a full‐scale five‐story steel frame equipped with two types of dampers

The seismic performance tests of a full‐scale five‐story passively controlled steel building were conducted on the E‐Defense shaking table in Japan in March 2009. Before the tests, a blind prediction contest was held to allow researchers and practitioners from all over the world to construct analytical models and predict the dynamic responses of the steel frame specimen equipped with buckling‐restrained braces (BRBs) or viscous dampers (VDs). This paper presents the details of two refined prediction models made and results obtained before the tests. When the proposed analytical modeling techniques are adopted as in the two refined prediction models, the overall prediction accuracy is about 90%. Sensitivity studies conducted after the tests are also presented in this paper. The effects of varying each modeling feature on the response simulation accuracy have been investigated. The analytical results suggest that considering concrete full‐composite actions for beam members could improve prediction accuracy by about 20% against using the simplified bare steel beam model. Adopting refined BRB stiffness computed from incorporating finite‐element gusset stiffness only improves the overall prediction accuracy by 0.9%. Considering the BRB dynamic loading test results for analytical BRB strength reduces the error by 1.9%. For the VD frame, incorporating the brace and VD stiffness could improve the overall prediction accuracy by about 15%. Copyright © 2012 John Wiley & Sons, Ltd.     

Out‐of‐plane stability assessment of buckling‐restrained braces including connections with chevron configuration

One of the key limit states of buckling‐restrained braces (BRBs) is global flexural buckling including the effects of the connections. The authors have previously proposed a unified explicit equation set for controlling the out‐of‐plane stability of BRBs based on bending‐moment transfer capacity at the restrainer ends. The proposed equation set is capable of estimating BRB stability for various connection stiffnesses, including initial out‐of‐plane drift effects. However, it is only valid for symmetrical end conditions, limiting application to the single diagonal configuration. In the chevron configuration, the out‐of‐plane stiffness in the two ends differs because of the rotation of the attached beam. In this study, the equation set is extended to BRBs with asymmetric end conditions, such as the chevron configuration. Cyclic loading tests of the chevron configuration with initial out‐of‐plane drifts are conducted, and the results are compared with the proposed equation set, which is formulated as a function of the normalized stiffness of the attached beam. © 2016 The Authors. Earthquake Engineering & Structural Dynamics published by John Wiley & Sons Ltd. © 2016 The Authors. Earthquake Engineering & Structural Dynamics published by John Wiley & Sons Ltd.     

Computational study of self‐centering buckling‐restrained braced frame seismic performance

Recent research developed and experimentally validated a self‐centering buckling‐restrained brace (SC‐BRB) that employs a restoring mechanism created using concentric tubes held flush with pretensioned shape memory alloy rods, in conjunction with a buckling‐restrained brace (BRB) that dissipates seismic energy. The present computational study investigated how the SC‐BRB can be implemented in real buildings to improve seismic performance. First, a computational brace model was developed and calibrated against experimental data, including the definition of a new cyclic material model for superelastic NiTi shape memory alloy. A parametric study were then conducted to explore the design space for SC‐BRBs. Finally, a set of prototype buildings was designed and computationally subjected to a suite of ground motions. The effect of the lateral resistance of gravity framing on self‐centering was also examined. From the component study, the SC‐BRB was found to dissipate sufficient energy even with large self‐centering ratios (as large as 4) based on criteria found in the literature for limiting peak drifts. From the prototype building study, a SC‐BRB self‐centering ratio of 0.5 was capable of reliably limiting residual drifts to negligible values, which is consistent with a dynamic form of self‐centering discussed in the literature. Because large self‐centering ratios can create significant overstrength, the most efficient SC‐BRB frame designs had a self‐centering ratio in the range of 0.5–1.5. Ambient building resistance (e.g., gravity framing) was found to reduce peak drifts, but had a negligible effect on residual drifts. Copyright © 2014 John Wiley & Sons, Ltd.     

Study of a new-type of steel buckling-restrained brace

The rectangle core plate of all-steel buckling-restrained braces(BRBs) usually exhibit obvious local buckling, due to the lack of longitudinal restraint from the encasing tube. To eliminate the undesirable effects, a novel steel BRB is proposed. In this new-type steel BRB, two T-shaped steels are adopted as the minor restraint elements to restrain the core plate instead of infilled concrete or mortar. Meanwhile, the ingot-iron material with low yielding strength and high elongation is applied to the steel core to study the mechanical properties of steel BRBs. To validate the theoretical requirements for the width-to-thickness ratio of the steel core and the thickness of angle steel, quasi-static tests of eight specimens were conducted. The tests focused on the energy dissipation capacity and failure modes of the proposed steel BRBs. Nonlinear finite element analysis was also carried out to validate the experimental results. Both the aforementioned results imply that appropriately designed steel BRBs can meet the performance requirements for BRB components.     

Seismic behaviour of self‐centring braced frame buildings with reusable hysteretic damping brace

This paper presents the seismic behaviour of a concentrically braced frame system with self‐centring capability, in which a special type of bracing element termed reusable hysteretic damping brace (RHDB) is used. The RHDB is a passive energy dissipation device with its core energy‐dissipating component made of superelastic Nitinol wires. Compared with conventional bracing in steel structures, RHDB has a few prominent performance characteristics: damage free under frequent and design basis earthquakes in earthquake prone areas; minimal residual drifts due to the self‐centring capability of RHDB frame; and ability to survive several strong earthquakes without the need for repair or replacement. This paper also includes a brief discussion of the RHDB's mechanical configuration and analytical model for RHDB. The seismic performance study of RHDB frame was carried out through a non‐linear time history analysis of 3‐storey and 6‐storey RHDB frame buildings subjected to two suites of 20 earthquake ground motions. The analysis results were compared with buckling‐restrained brace (BRB) frames. This study shows that RHDB frame has a potential to outperform BRB frames by eliminating the residual drift problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Development of a new rubber seismic isolator: ‘Ball Rubber Bearing (BRB)’

This paper presents an experimental research aimed at developing a new rubber‐based seismic isolator called ‘Ball Rubber Bearing (BRB)’. The BRB is composed of a conventional steel‐reinforced multi‐layered rubber bearing with its central hole filled with small diameter steel balls that are used to provide energy dissipation capacity through friction. A large set of BRBs with different geometrical and material properties are manufactured and tested under reversed cyclic horizontal loading at different vertical compressive load levels. Extensive test results indicate that steel balls do not only increase the energy dissipation capacity of the elastomeric bearing (EB), but also increase its horizontal and vertical stiffness. It is also observed that the energy dissipation capacity of a BRB does not degrade as the number of loading cycles increases. Copyright © 2011 John Wiley & Sons, Ltd.