Brace Stiffness Research Articles

Intermediate-point bracing for columns: New stiffness–buckling load formulae

According to the stiffness equation provided in the American Institute of Steel Construction (AISC) design specifications for intermediate-point braces, the required point brace stiffness increases with the number of point braces. This research aims to formulate a new stiffness design equation for intermediate-point braces that accounts for the decline in required stiffness with an increasing number of braces. The brace stiffness (β)–buckling load (Pcr) relationships for columns with point braces were investigated by performing a buckling analysis using the energy method with the Fourier series. A formula was derived, defining the β−Pcr relationships for each buckling mode of the columns, regardless of the number of point braces and whether the buckling is elastic or inelastic. The AISC stiffness equations for point braces were significantly conservative under most buckling loads. The proposed stiffness design equation requires lower brace stiffness with an increasing number of point braces, resulting in smaller brace members than the AISC design equations. Additionally, the proposed strength design equation can set relatively high strength requirements for the brace, reflecting that stiffness governs the design of the point brace. These new equations offer a more precise and efficient method for calculating required brace stiffness, providing valuable insights to improve structural design capabilities.

Pseudo-dynamic tests of steel frame with disc spring self-centering energy dissipation braces under doublet earthquake and mainshock-aftershock sequence

To provide valuable insights into the seismic performance and self-centering behavior of self-centering braced systems, a 1/2-scale 3-story 1-bay steel frame equipped with disc spring self-centering energy dissipation braces (DS-SCBs) was designed, fabricated, and subjected to pseudo-dynamic testing. This testing encompassed various earthquake hazard levels, a doublet earthquake, and a mainshock-aftershock sequence. The experimental findings revealed the stable hysteretic response of the DS-SCB, with no observed degradation in stiffness or activation forces. Additionally, the test frame exhibited outstanding seismic performance, sustaining only confined damage under very rare earthquakes. Due to the presence of the DS-SCBs, the test frame performed commendable self-centering behavior, characterized by minimal residual deformations. Peak story drifts and floor accelerations under two earthquakes of the same hazard level were regarded as equivalent. Furthermore, the subsequent aftershocks exhibited no discernible impact on the seismic performance of the test frame. Floor acceleration spikes were observed at locations where the brace stiffness underwent changes or where adjacent stories moved in opposite directions. The steel frame equipped with DS-SCBs proved to be an attractive, low-damage alternative to conventional steel systems for the development of resilient urban areas.

Seismic evaluation of a self-centering retrofit solution for modular steel structure connections

In this study, a novel plug-in modular steel structure (MSS) connection is proposed, in which the connector, cover plates, and bolts form a complete connecting system. To improve the seismic performance and post-earthquake functional recoverability of the MSS connection, a retrofit solution with diagonal self-centering (SC) haunch braces is introduced, and the self-centering modular steel structure (SC-MSS) connection is developed. Theoretical analysis of the mechanical properties of the MSS and SC-MSS connections are conducted, and the calculation formulas for the initial stiffness and bending bearing capacity are derived. A comparative experimental study of one MSS connection and five SC-MSS connections under cyclic loading was carried out to evaluate their seismic performance, and to explore the influences of the main haunch parameters on the behavior of the SC-MSS connection. The results demonstrate that the plug-in connector, cover plate, and bolts can realize reliable continuity and integrity between modules. The SC haunch braces have good collaborative work with the modules and provide sufficient restoring force, and the damage in all the SC-MSS connections was found to be effectively controlled and delayed as compared with the MSS connection. The MSS connection displays a full shuttle-shaped hysteretic response with adequate plastic development, while the SC-MSS connections successfully exhibit expected flag-shaped hysteretic responses with higher bearing capacity and lower residual displacement. The adjustment of the post-activation stiffness and pre-pressed force of the haunch brace is found to improve the connection performance, and a reasonable selection of the installation scheme could achieve optimal self-centering efficiency. The experimental results also verify the feasibility and correctness of the proposed theoretical formulas and mechanical assumptions.

Approximate closed-form solutions for free vibration of circular arches with discrete lateral braces

Discrete lateral braces are commonly employed to enhance the out-of-plane stability of arches. However, these braces generate coupling effects among vibration modes, which significantly impact the vibration characteristics of the arches. This paper presents an analytical solution for the free vibration analysis of two-hinged arches with equally-spaced lateral translational and/or rotational braces. A simplified method of evaluating the fundamental frequency for the out-of-plane vibration of arches with equally-spaced lateral braces is introduced. The method relies on the conventional approach of representing arch braces through elastic springs, but here the out-of-plane deflection of the braced arch is constructed by adding related coupled sinusoidal functions to that of the unbraced arch, satisfying both the boundary and kinematic conditions. Subsequently, the fundamental frequency and threshold bracing stiffness or stiffness requirement are derived using the Rayleigh–Ritz method. Finite element analysis (FEA) is applied to investigate the flexural-torsional vibration mode and fundamental frequency of arches with discrete braces, and the results from FEA closely match the theoretical predictions. The effects of number of lateral translational and/or rotational brace, out-of-plane slenderness, included angle, bracing stiffness and bracing position on the fundamental frequency are comprehensively analyzed. It is found that when the fundamental frequency of the braced arch increases due to changes in bracing stiffness, the mode shape displays more intricate behavior, rather than following a straightforward low-to-high mode growth pattern. Furthermore, it is observed that in some cases, the addition of rotational bracing stiffness can markedly enhance the maximum fundamental frequency of the arch with both translational and rotational braces.

Study on seismic behavior and collapse risk of super high-rise braced mega frame-core tube structural system

The braced mega frame-core tube (BMFCT) structure could provide an efficient lateral-force resisting (LFR) system for high-rise or super high-rise buildings, especially for those higher than 500 m. The BMFCT structure is a dual LFR system, including the outer braced mega frame and the inner core tube. The seismic behavior and design method of this structural system is not clear. In this study, a parametric study based on collapse risk-targeted analysis is conducted to clarify seismic mechanism and performance and to make clear means of matching reasonable stiffness and setting energy-dissipation components. Five comparable BMFCT structures with different mega brace stiffness proportions and frame-tube stiffness ratios are designed. An efficient elasto-plastic modeling method is introduced to establish the accurate FE models. The corresponding collapse risks are calculated through Incremental Dynamic Analysis and fragility theory. The results show that the structural collapse risk increases with the increased frame-tube stiffness ratio. Therefore，the stiffness and bearing capacity of peripheral frames should be strengthened to ensure the structural collapse margin. On the other hand, the collapse resistance capacity gradually decreases with the increasing of the mega brace stiffness. Thus, the stiffness proportion of mega braces should be recommended to meet the basic lateral stiffness requirements, which shall not exceed 35 % of the overall structural stiffness. Moreover, the distribution feature and ideal arrangement of the energy dissipation components are proposed. The seismic design recommendation based on the uniform collapse risk is also proposed for the BMFCT system, which serves a reference for the structural design of super high-rise buildings.

Bracing requirements and design for a single column considering semi-rigid connections and initial curvature

The use of bracing has been widely recognized as an effective technique to increase the strength of columns by reducing their effective length. This paper presents an assessment of the bracing requirements for a semi-rigidly connected column, which is laterally braced at the mid-height. By utilizing the proposed half-length column model, the effects of column stiffness and initial curvature on the brace strength and stiffness requirements are investigated. The results indicate that the column’s initial curvature coefficient increases with a higher applied load, but decreases with increased column end connection stiffness. The required brace stiffness requirement stipulated in AISC 360-22 should be increased when considering the effect of column’s initial curvature. Following the design concepts of AISC 360-22 and CSA S16-19, equations for assessing the bracing requirements for a semi-rigidly connected column are proposed. These equations are validated against the finite element analysis results, demonstrating their ability to accurately assess the bracing requirements for semi-rigidly connected columns. These equations can be adopted in current engineering practice.

Design method for determining appropriate stiffness for end point braces

The high dependence of end point braces on rotational restraint conditions differentiates them from intermediate point braces. Although the point brace stiffness equation of the American Institute of Steel Construction (AISC) design specifications applies to both braces, it restricts the stiffness to only the buckling load based on the unbraced length and an effective length factor of 1, which is too conservative for end point braces. This study investigated the relationship between end point brace stiffness and column buckling loads through buckling analyses using slope–deflection equations with stability functions, considering diverse supports, end rotation constraints, and column counts. Irrespective of the column count, the columns with end point braces exhibited a non-sway elastic buckling load of up to four times the Euler buckling load depending on the rotational restraint at the column ends. The proposed procedure enhances the AISC regulations by introducing a new equation that allows up to 90% utilization of the buckling load based on the unbraced length and an effective length factor of 1 or less, considering column end rotational constraints. The procedure was applied to six frames undergoing elastic or inelastic buckling and validated by assessing the buckling loads and related point brace stiffness. The proposed procedure significantly enlarged the load-bearing capacity of the column compared with the AISC design procedure, although both approaches obtained comparable point brace cross-sectional areas. It extends the buckling load limit for columns or frames with an end point brace, providing predictive insights into brace stiffness to improve structural design.

Experimental, mathematical model, and simulation of a dual-system self-centering energy dissipative brace equipped with SMA and variable friction device

The application of shape memory alloy (SMA) in seismic-resistant devices has received extensive attention due to its unique superelastic and shape memory effect. The poor energy dissipative capacity, insufficient reliability, and small bearing capacity are the main challenges that hinder the application of the developed SMA braces and dampers. The new system of self-centering energy dissipative (SCED) brace was proposed, fabricated, and tested, and the numerical model was defined and verified. The dates manifest that the load-displacement curve of the new dual-system SCED brace displays a typical flag shape. The new dual-system SCED brace exhibits the desirable energy dissipative ability and self-centering capacity and the theoretical and FE model established in this research can preferably express the hysteretic performance. The FE parametric analysis shows that the SMA screw preload has a small effect on the hysteretic curve of the new system brace, but it plays a decisive role in the shape of the hysteretic curve under small deformation. Furthermore, the bearing capacity and stiffness of the dual-system SCED brace can be adjusted by designing key parameters. The research shows that the new dual-system SCED device compensates for the shortcomings of the existing SMA braces and dampers, and it is a very promising and recommended self-centering energy dissipative device to use the wedge block system as the main energy dissipative system and SMA system as the main self-centering system.

Bracing requirements for a semi‐rigidly connected column considering column lateral stiffness and initial curvature

AbstractThis paper theoretically assesses the required brace stiffness and strength for a semi‐rigidly connected column with a lateral brace at the mid‐height, by introducing a half‐length column model and adopting the concept of storey‐based buckling. Unlike that in Winter's model, column lateral stiffness and initial curvature associated with column initial out‐of‐straightness are accounted for. A coefficient, which is a function of applied load and column end connection stiffness, is introduced to characterize the effect of column initial curvature on bracing requirements. By comparing the results obtained from the analytical method and current structural steel and cold‐formed steel design standards, it has been found that the required brace strength stipulated in these standards are generally conservative in various degrees for design loads; however, the required brace stiffness should be increased if the effect of column initial curvature is considered. The results obtained from the proposed approach are verified against the finite element analysis, indicating that the proposed approach provides an accurate evaluation of required brace strengths for semi‐rigid columns and is recommended to be adopted for engineering practice.

A comparison of stiffness of six knee braces with application for posterolateral corner reconstructions.

Posterolateral corner knee injuries are clinically significant, and often require surgical reconstruction. The optimal knee brace following posterolateral corner reconstructions has not yet been determined via clinical nor biomechanical study. We sought to evaluate the stiffness of six types of knee braces to determine the ideal brace type for reducing varus forces, which may have clinical utility for posterolateral corner knee reconstruction rehabilitation. Six different groups of knee braces underwent mechanical testing: cruciate braces, cruciate braces with a valgus bend, medial unloaders, articulating sleeves, hinged braces, and tri-panel immobilizers. Each brace was fitted to the same fiberglass leg model and was secured to the testing apparatus. Force was applied under four-point bending to generate a varus moment about the artificial knee. The stiffness in Newtons per millimeter (N/mm) of each brace was calculated from the slope of the force-displacement curve. The cruciate brace with a valgus bend had the highest average stiffness at 192.61 N/mm (SD 28.53). The articulating sleeve was the least stiff with an average stiffness of 49.86 N/mm (SD 8.99). Stiffness of the cruciate brace was not statistically different compared to cruciate valgus (p = 0.083) or medial unloader (p = 0.098). In this experimental design, a cruciate brace with a valgus bend was shown to have the highest overall stiffness, while an articulating sleeve had the lowest stiffness. Future work will investigate whether this translates into clinical performance.

Analytically optimal parameters of supplemental brace-damper system in single-degree-of-freedom structure based on stability maximization criterion

In this study, optimal parameters (i.e., optimal additional stiffness ratio α and damping ratio λ) for supplemental brace-damper system in a single-degree-of-freedom (SDOF) structure are analytically derived. Different from the design formulations for supplemental brace-damper system proposed by Londoño, which are essentially fitting formulae through extensive parametric study, analytically optimal parameters for supplemental brace-damper system are derived by combining the research work by Londoño and the stability maximization criterion (SMC). Similar to Optimum Brace Stiffness (OBS) criterion and Optimum Damper Size (ODS) criterion proposed by Londoño, SMC-based OBS criterion and SMC-based ODS criterion are respectively proposed to assure a desired structural performance represented by the targeted overall damping ratio of the considered system. Through numerical study, it is demonstrated that the seismic response of the proposed SMC-based OBS criterion and SMC-based ODS criterion excellently agree with those of the existing OBS criterion and ODS criterion, respectively.

Development of small-scale buckling-restrained brace analog for physical experiments of structural systems

Experimental testing is critical for evaluating the performance of novel structural systems. However, the cost of full-scale testing can be limiting. Small-scale testing provides a lower cost alternative that can produce reliable data if each component of the scaled model is properly developed. Given their stable hysteretic behavior, buckling-restrained braces (BRBs) are featured in several seismic force-resisting systems. The design of typical BRBs involves an axially yielding steel core that is prevented from buckling by a concrete casing. Designs for adaptable and easy-to-implement small-scale BRBs that can be included in structural testing have not been established. In this work, a small-scale BRB analog is developed that utilizes a core steel yielding plate that is prevented from buckling by a pair of steel casing plates. A flexural yielding mechanism is employed in the small-scale BRB analog, which allows for practical dimensions and the ability to independently tune brace stiffness and strength. Prototype braces were fabricated and validated through quasi-static cyclic testing in which they exhibited the full and stable hysteretic behavior characteristic of BRBs. To aid in the use of this BRB analog by others, numerical results were utilized to generate empirical equations that relate the strength and stiffness of the scaled BRB to the archetypical geometry of the core plate. Given its ability to deliver a stable hysteretic performance, the wide range of strength and stiffnesses that it can provide, and its easy reusability, the developed device is a good choice for experiments where a small-scale BRB analog is needed.

Evaluation of Criteria for Out-of-Plane Stability of Steel Arch Bridges in Major Design Codes by FE Analysis

The provisions for out-of-plane stability of steel arch bridges in three major design codes are presented in this paper. By employing an existing steel arch bridge as a model, the influence of bridge type, arch rib to lateral bracing stiffness ratio, rise-to-span ratio, arch rib spacing, and range of lateral bracing arrangements on the out-of-plane critical axial force of the arch rib is studied using FE analysis. The accuracy of the critical axial force provisions is then evaluated against the FE analysis. The results show that the influence of the rise-to-span ratio on critical axial force is generally small. The critical axial force decreases with increasing arch rib spacing when the stiffness ratio is relatively large. A smaller ratio of arch rib length provided with lateral bracing (γ-value) significantly reduces the critical axial force and normalized critical axial force decreases with increasing stiffness ratio. The critical axial force of half-through type arch bridges is lowest when the stiffness ratio is relatively small. A deck-type bridge has a larger critical axial force than a through-type bridge when the stiffness ratio is relatively large, while the results are the opposite when the ratio is small. The different assumptions made in the provisions result in the various parameters having different impacts on the out-of-plane critical axial force in each code, thus affecting code accuracy. Considering the influence of the rise-to-span ratio, ratio of lateral bracing, and arch rib spacing with different stiffness ratios, factors to improve the accuracy of the critical axial force obtained by the three codes are proposed for a practical design process.

Bracing requirements for multiple parallel nonidentical columns with initial curvature

For a system consisting of multiple columns, horizontal discrete braces are often used to increase the system’s strength by decreasing the effective length of columns. Since the bracing requirements for such systems in current design standards were developed based on the assumption of identical columns with pin-ends, an analytical method is proposed to evaluate the ideal brace stiffness and brace forces for multiple columns by accounting for the stiffness interaction among columns and braces. For systems with uniform column lateral stiffness, the explicit solution for the maximum brace force is derived, and a simple formula is subsequently proposed by fitting the analytical results. The proposed formula explicitly accounts for the influences of column initial curvature and semi-rigid end connections on the maximum brace force. The method is comprehensive and applicable to multi-column systems with nonuniform column lateral stiffness, which can result from differences in column sizes, end connections, or applied loads. The results obtained from the proposed formulae are verified against those of finite element analyses. The ideal brace stiffness and brace forces for a system containing a distinctive column with a greater moment of inertia than the other columns are investigated. The results show that the location of the distinctive column affects the magnitudes of the ideal brace stiffness and the maximum brace force, which is yet to be considered in current design standards.

Calculation Method of Bracing Stiffness in Foundation Pit: A Review

In the excavation engineering, the bracing stiffness is an important parameter reflecting the supporting performance of the inner bracing system. Bracing stiffness has great influence on the deformation of the retaining structure and the foundation pit, and its calculation method is the theoretical basis for the design and deformation analysis of the foundation pit. The limit equilibrium method, subgrade reaction method and finite element method used in the design, stability and deformation analysis are introduced in detail. Different calculation models and calculation methods of bracing stiffness are summarized in detail in each analysis method. The method of providing foundation pit retaining design standard and the optimization method proposed by scholars are introduced and analyzed in depth. It is suggested that the subsequent research should further establish a support stiffness calculation method taking influence of important parameters such as the difference of support stiffness and uneven support spacing.

Seismic performance of a shear-type energy dissipation brace: Experimental investigation and numerical analysis

In this study, a novel shear-type energy dissipation brace (SEDB) was proposed, which integrated multiple pieces of steel shear plates on a single brace. The brace stiffness and yield load can be adjusted by changing the number of pieces and layers of the steel shear plates, which is more flexible than the traditional shear-type dampers. Three configurations of SEDBs were developed, including the flat steel plate-SEDB (F-SEDB), F-SEDB without core tube and corrugated steel plate-SEDB (C-SEDB). The quasi-static cyclic tests were conducted on three SEDBs to investigate the seismic performance. Experimental results showed that the proposed SEDB can achieve a stable hysteretic behavior, especially by adopting the core tube to prevent global out-of-plane deformation. Meanwhile, the C-SEDB has a more stable energy dissipation performance due to the high out-of-plane stiffness of the corrugated steel plates. The design procedure of the SEDBs was developed. Meanwhile, a finite element model was established and verified by the experimental results. The influences of the height-thickness and height-width ratio of steel plates were systematically investigated, which benefited the application and offered suggestions in engineering design.

Numerical Study on the Seismic Behavior of Eccentrically Braced Composite Frames with a Vertical Low-Yield-Point Steel Shear Link

An eccentrically braced composite frame with a low-yield-point (LYP) steel shear link is an efficient energy dissipation system that exhibits good mechanical properties. However, existing experimental studies have not fully demonstrated the superiority and applicability of the structural system. We present a structural mechanics and finite element model analysis of an eccentrically braced composite frame with a vertical shear link. The effect of the design parameters on the seismic performance of the structure is analyzed. First, a theoretical model of the mechanics of the structural system is established to provide a comprehensive description of the key parameters. Then, a finite element model is developed using the computer program ABAQUS to analyze the mechanical and energy dissipation mechanisms. Finally, the beam-to-column stiffness ratio, shear link web thickness, shear link web width and length, and diagonal brace stiffness are analyzed to determine their effects on the mechanical properties of the structural system. Furthermore, some design parameter values are suggested.

Lateral Bracing Stiffness Requirements for Systems of Parallel Compression Members

AbstractPractical models have been developed to determine the threshold stiffness for lateral bracing components used in systems of parallel compression members or sub‐assemblages. Model formulation is based on the analysis of n parallel compression members with intermediate tie‐bracing and flexible anchorages. Intermediate bracing options include both continuous lateral tie‐bracing with flexible shear connectors, and discrete tie‐braces (blocking) directly linking adjacent members without connectors. Anchor bracing cases include both rigid and flexible braces and connectors, and both equal and unequal stiffnesses. Analytical solutions for the threshold stiffness were derived using equilibrium, compatibility, and constitutive relationships for a system that modeled bracing with linear‐elastic springs. The resulting eigenvalue problem is solved with computational efficiency and speed that far exceeds that of a root‐finding solution scheme reliant on symbolic computations. Two approximate expressions for the ideal tie‐brace stiffness have also been developed. Comparisons with the exact solutions indicate satisfactory accuracy for both approximations, with mean percent errors of 1% ‐ 2% for realistic variable values. A design procedure using these approximations is also proposed.

A Novel Low-Cost Three-Dimensional Printed Brace Design Method for Early Onset Scoliosis

Abstract Early onset scoliosis (EOS) is a type of spine deformity that presents before 10 years of age. The biomechanical properties in scoliosis have been found to be different, especially in the case of the concave and convex paravertebral muscles. Based on this fact, a novel three-dimensional (3D) printed patient-specific asymmetric stiffness brace design method is proposed in this paper, aiming to provide asymmetric stiffness to match “imbalanced” biomechanical properties of the concave and convex paravertebral muscles, respectively, and treat EOS by applying the block-structure brace. A 3D computer aided design draft model of the brace contour was implemented from 3D scanning. The asymmetric stiffness block-structure brace was designed in Rhinoceros and the finite element (FE) model was imported into abaqus. FE simulation was employed to study the mechanical characteristics of the brace, which provided a quantitative index for the imbalanced property of brace stiffness. The results of the FE simulation showed that the stiffnesses of the concave and convex sides were 145.88 N/mm and 35.95 N/mm, respectively. The block-structure brace was fabricated using 3D printing. Asymmetric stiffness was evaluated by corrective force measurements, which were obtained from a thin-film pressure sensor equipped on the brace. The patient-specific asymmetric stiffness brace was applied to clinical practice in a one-year-old EOS patient. A novel low-cost 3D printed brace design method for EOS was proposed in this study that could potentially be useful in patient treatment acceptance.

Seismic performance of steel frame structure adopting parallel spine frames with elastic braces

Strong structural spines increase structural integrity and enhance the seismic performance of multistory buildings. However, they usually do not provide a resilient force, which may lead to large residual drifts after major earthquakes. To address this problem, an earlier study by the authors proposed a super-elastic bracing system comprising parallel spine frames with buckling-restrained dampers and steel elastic braces. This study further investigated the seismic performance of the proposed system using a five-story building prototype. Evaluations of the lateral stiffness and strength of the elastic braces and dampers were first formulated and validated using pushover analysis. Extensive seismic response analyses were then performed for models with different brace configurations and sections, damper yield strengths, and ground motion intensities. The analyses showed that adding elastic braces efficiently reduced the peak and residual drifts and had no significant influence on the story drift distribution. The demand for brace stiffness to reduce the residual drift to a particular level has been suggested.