Experimental and theoretical study on a novel dual-functional replaceable stiffening angle steel component
Xinzheng Lu a,*, Lei Zhang b, Yao Cui c, Yi Li d, Lieping Ye a
Soil Dynamics and Earthquake Engineering, 2018, 114: 378-391.
a Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Department of Civil Engineering, Tsinghua University, Beijing 100084, China
b Beijing Engineering Research Center of Steel and Concrete Composite Structures, Tsinghua University, Beijing 100084, China
c State Key Laboratory of Costal and Offshore Engineering, Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, China
d Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, Beijing 100124, China
Abstract: Recently, seismic resilience has become a research frontier in civil engineering. The self-centering steel frame can effectively control structural damage and reduce structural residual deformation, which ensures rapid repair after an earthquake. Therefore, such a structural system has attracted extensive attention from researchers. One of the important research directions on self-centering steel frames is the development of high-performance energy-dissipating components. A new type of dual-functional replaceable stiffening angle steel (SAS) component is proposed here. It can effectively improve the stiffness and strength of beam-column connections and has sufficient energy-dissipating performance and ductility. Seven different energy-dissipating components were tested, including one angle steel component and six SAS components. The strength and deformation capacity of the components were compared based on monotonic loading tests. The SAS component with the highest out-of-plane stability and sufficient strength and initial stiffness was selected and subsequently tested under hysteretic loading to investigate its energy-dissipating performance. The theoretical analysis methods of the initial stiffness and the yield moment provided by the SAS components were proposed and validated by the finite-element (FE) models calibrated using experimental data.
Key words: stiffening angle steel (SAS); dual-function; monotonic loading test; hysteretic test; theoretical analysis; numerical simulation.
* Corresponding author. Tel.: +86 10 62795364.
E-mail address: firstname.lastname@example.org (Xinzheng Lu)
Owing to their advantages of lightweight, high strength, and excellent ductility, steel structures have been widely used in tall buildings. However, since the 1994 Northridge earthquake, many steel joints have been severely damaged when subjected to earthquakes. Such damages are difficult to repair. Therefore, ways to improve structural seismic resilience have become the focus of many researchers in recent years. The main ways to improve structural resilience include the following: (1) avoiding the damage of key components; (2) using replaceable or easy-repair energy-dissipating components; and (3) reducing the residual deformation after an earthquake. Based on these approaches, many resilient structural systems have been proposed [1-6]. Among them, self-centering steel frames [7-15] can well meet structural resilience requirements, and consequently they are widely studied.
One of the key issues in the development of self-centering steel frames is the improvement of the performance of energy-dissipating components. Four types of energy-dissipating devices are commonly used in a self-centering steel frame: (a) energy-dissipating bar, (b) friction devices, (c) angle steels, and (d) shape memory alloy rods. For example, Christopoulos et al.  proposed a post-tensioned energy-dissipating connection with energy-dissipating bars, which can undergo large deformations with energy-dissipating characteristics while retaining the beam and column undamaged and without residual drift. Vasdravellis et al. [17, 18] adapted web hourglass-shaped pin devices in a self-centering beam-to-column connection as a reliable method for increasing the energy-dissipating capacity of structures under seismic loading. The above energy-dissipating components exhibit remarkable energy-dissipating performance. However, it is inconvenient to replace them. Flange friction devices [13, 14] and web friction devices [19, 20] were used in self-centering frames to dissipate seismic energy. The test results reveal that the energy-dissipating performance of the friction devices is satisfactory. However, the friction coefficient of the friction devices are likely to change after long-term static pressure, and this will influence the seismic-energy-dissipating performance. Angle steels, which are convenient to replace and capable of dissipating seismic energy, were widely used to connect the beams and columns in self-centering steel frames. For example, Garlock et al. [21, 22] experimentally investigated the behavior of angles in a bolted angle beam-to-column connection and discussed the influence of the design parameters on the seismic response of post-tensioned steel moment resisting frame systems; Deng et al.  tested and simulated full-scale self-centering beam-to-column connection, and the result revealed that this type of connection exhibits a reasonably resilient performance with only a straightforward replacement of the angles. Moradi and Alam  developed detailed three-dimensional finite-element (FE) models of steel beam-column connections with post-tensioned strands and validated these models with respect to the results of prior experiments on interior post-tensioned connections with top-and-seat angles; furthermore, the performance of these models under cyclic loading was analyzed. Moradi et al. [25, 26] identified the parameters that significantly influence the lateral load-drift response and seismic response of steel post-tensioned connections, through FE analysis. L¨®pez-Barraza et al.  and Shiravand and Mahboubi  used post-tensioned steel moment resisting frames with semi-rigid connections to regulate the hysteretic energy demands, reduce the maximum inter-story drift, eliminate structural damage, and minimize residual drifts. The results of the above studies on semi-rigid connection demonstrated that this type of connection when adopted in self-centering steel frames exhibits a reasonably resilient performance. The disadvantage of this type of semi-rigid connection is that it exhibits lower stiffness and strength compared to traditional bolt-welded connections or full-welded connections. Wolshi et al.  adopted angle steel in conjunction with bottom flange friction device as beam-to-column connection. The test results indicate that this type of energy-dissipating device can provide reliable energy dissipation. In addition, the connection remains damage-free under the design earthquake. Wang et al. [30, 31] used shape memory alloy rods or a combination of shape memory alloy rods and angle steels, as beam-column connections. Shape memory alloys can effectively improve the energy-dissipating performance and self-centering capability of beam-to-column connections. However, the shape memory alloy is expensive, which hinders its popularization and application. Therefore, this study attempted to develop a new type of energy-dissipating device, which can be adopted in self-centering steel frames with convenient reparability and higher economic benefit and can simultaneously guarantee adequate stiffness, strength, and energy-dissipating performance.
In this work, a new type of dual-functional replaceable stiffening angle steel (SAS) component (comprising an angle steel and two rib stiffeners) was proposed. It can be adopted as the beam-column connections in self-centering steel frames. Rib stiffeners formed by cold bending, combined with angle steels can improve the strength and initial stiffness of beam-to-column connections in comparison with angle steel connections. Additionally, the energy dissipation of the beam-to-column connections can be increased by tensile and compressive deformation of rib stiffeners caused by the rotation at the beam-column interface. Moreover, the SAS component is connected with the beam and column through bolts and can be replaced rapidly after an earthquake. Therefore, the proposed SAS components can promote the structural seismic resilience. The connection between the proposed SAS components and floor slab is not considered in this work, which will be implemented in the future. In this work, seven different energy-dissipating components are examined, including one angle steel component and six SAS components. The strength and deformation performance of each component were compared based on a monotonic loading test. The SAS component with the highest out-of-plane stability and sufficient strength and initial stiffness was selected. Subsequently, the hysteretic test was carried out to investigate the energy-dissipating performance. Theoretical models of the initial stiffness and the yield moment provided by SAS components are proposed, and they were validated by FE models calibrated using experimental data.
2. Principle of dual-functional replaceable SAS components
The construction of the self-centering steel joint discussed in this work is shown in Fig. 1. The SAS components have two functions: to resist the bending moment and to dissipate seismic energy. Meanwhile, the shear panels can resist the shear force, and the pre-stressed steel strands can provide self-centering capacity after earthquakes. A number of researchers have studied this type of self-centering steel joint [26-31] and verified that the energy-dissipating component is the key to ensuring energy-dissipating performance, and the convenient reparability of the energy-dissipating devices is the key to ensuring seismic resilience. Therefore, to improve the performance of the traditional angle steel connection, a new type of dual-functional replaceable SAS component is proposed in this work, as shown in Fig. 2, in which SAS-IN refers to the rib stiffener installed inside the angle steel, while SAS-OUT refers to the rib stiffener installed outside the angle steel. By combining the angle steels with the rib stiffeners, the strength and energy-dissipating performance are improved. Additionally, the rib stiffeners are manufactured through cold forming to avoid the negative effect due to welding and other thermal processing, which may reduce the deformation and energy-dissipating performance.
Fig. 1. Principle of SAS in self-centering steel frames
Fig. 2. Assembly drawing of the beam-column connection with SAS
3. Experimental program
3.1. Design of specimens and material properties
The specimens were designed based on the prototype of the web-bolted flange-welded beam-column composite joints of a high-rise building . The prototype beam-column joints were designed following the Code for Seismic Design of Buildings (GB 50011-2010) , in which the specified limitation of nonlinear story drift is 0.02 rad and seismic performance is defined as the state wherein the structure can be repaired after the design basis earthquakes, whose 50-y probability of exceedance is 10%. The specimens were designed according to the yield strength specified in the design code . In order to balance the energy dissipation and self-centering performance of the connection, the yield moment provided by the energy-dissipating device SAS was designed to be half of the yield moment provided by the web-bolted and flange-welded connection or the welded connection, which was the target performance for the preliminary exploration. The objective of this test was to investigate the contribution of the SAS component to the seismic performance of the angle steel bolted beam-column connection. Therefore, the pretension strand was not included in the specimens. The replicability of the SAS components is also investigated in this test. An additional objective of these tests was to verify whether replaceable energy-dissipating components can attain the required plastic rotation without significant damage and strength deterioration and thus achieve the performance levels equivalent to the SMFs designed according to AISC .
Seven different types of energy-dissipating components (including one angle steel component and six SAS components) were designed and compared. The proposed SAS components are shown in Fig. 3: Specimens AR-IN-M (moon-shaped SAS), AR-IN-T (triangle-shaped SAS), AR-IN-S (triangle-shaped with an edge stiffener SAS), AR-OUT-M (moon-shaped SAS), AR-OUT-T (triangle-shaped SAS), and AR-OUT-S (triangle-shaped with an edge stiffener SAS). The difference between Specimen SAS-IN series and Specimen SAS-OUT series is the restraint condition of rib stiffeners. The rib stiffeners in Specimen SAS-IN series are constrained by bolts, while the rib stiffeners in Specimen SAS-OUT series are constrained by angle steels (Fig. 3).
Fig. 3. Construction of SAS
The details of the proposed SAS components are shown in Figs. 4 and 5. L200 ´ 200 ´ 14 (mm) angle steels are used in this work. The design thickness of the rib stiffeners was 3 mm, whereas the actual thickness of the specimens was 3.5 mm. The material properties of the steel plates are listed in Table 1.
Fig. 4. Dimensions of Specimen SAS-IN series (unit: mm)
Fig. 5. Dimensions of Specimen SAS-OUT series (unit: mm)
Material properties of the steel plates
3.2. Loading setup and loading history
To explore the performances of different components, the loading setup shown in Fig. 6 was used for all tests in this work including monotonic loading tests and hysteretic test. An H300 ¡Á 150 ¡Á 6.5 ¡Á 9 (mm) steel beam was connected to the support beam through the proposed energy-dissipating components and friction-type high-strength bolts (M24, Level 10.9). Note that the research objects were the energy-dissipating components in this work. Therefore, the H-shaped steel beam and support beam were strengthened to remain elastic during the tests. The distance between the loading point and support beam was L = 1.4 m. The beam-column rotation angle (q) could be obtained by measuring the displacement of the loading point (denoted by D), i.e., q = D / L.
Fig. 6. Loading setup
The strength and deformation performance of each energy-dissipating component were compared through the monotonic loading test. The SAS component with the highest out-of-plane stability and sufficient strength and initial stiffness was selected. Subsequently, the hysteretic test was carried out to investigate its energy-dissipating performance. The maximal displacement of the loading actuator was 120 mm (q = 0.086 rad), which was sufficient to meet the ductility requirement of special moment frames (SMFs) proposed by AISC , i.e., q ¡Ý 0.04 rad. The loading in the hysteretic test was performed under displacement control, and two cycles were imposed at each displacement level (Fig. 7). The first and the second displacement levels are 2.5 mm and 5 mm, respectively. The incremental displacement of each subsequent level is 5 mm. The hysteretic test stopped when the strength decreased to 80% of the peak load.
Fig. 7. Loading history in hysteretic test
3.4.1. Monotonic loading test
The objective of the monotonic loading test was to investigate the contribution of the SAS component on the initial stiffness and strength and to identify the SAS component with the highest stability for the hysteretic test. The load-rotation curves of all the monotonic loaded specimens are shown in Fig. 8. The final phenomena of all the specimens are illustrated in Fig. 9. The three aspects¡ªinitial stiffness, ultimate strength, and stability¡ªwere discussed, and the following conclusions were formed from Figs. 8 and 9.
Fig. 8. Load-rotation curves of all specimens
Both Specimen SAS-IN series and Specimen SAS-OUT series exhibited larger initial stiffness and strength than that of Specimen A. Moreover, compared with Specimen SAS-IN series, Specimen SAS-OUT series exhibited larger initial stiffness and strength (Fig. 8). This is because the rib stiffeners could be effectively constrained by the angle steel in Specimen SAS-OUT series to ensure that the capacity of the rib stiffeners could be utilized completely. In contrast, the rib stiffeners in Specimen SAS-IN series were constrained only by the bolts, which limited the performance of the rib stiffeners. As shown in Fig. 8(a), Specimen SAS-IN series exhibited approximately 25% differences in the initial stiffnesses with various details of the SAS components. In contrast, the initial stiffnesses of Specimen SAS-OUT series were similar (Fig. 8(b)). A similar behavior was observed in the case of strength. It indicates that the constraints between the SAS component and the beam flange impacted the specimen¡¯s behavior. The final phenomena of the monotonic loading tests demonstrated an absence of local buckling in Specimen AR-IN-S and Specimen AR-OUT-S (Fig. 9(d) and (g)), whereas different degrees of local buckling occurred in the other SAS specimens (Fig. 9(b), (c), (e) and (f)). Therefore, Specimen AR-IN-S and Specimen AR-OUT-S exhibited higher stability. The edge stiffener significantly improved the buckling resistance of the triangle-shaped rib stiffener, which contributed to the stable energy dissipation during the hysteretic test.
Fig. 9. Final phenomena of all specimens
In conclusion, the peak bending moments of Specimen AR-OUT-S were two times that of Specimen A with a 41% increase in the steel consumption. The initial stiffness of Specimen AR-OUT-S was increased by 1.73 times. The ductility of Specimen AR-OUT-S was not reduced compared with that of Specimen A. Therefore, Specimen AR-OUT-S, in which the strengthened rib stiffeners were incorporated in the SAS components, was adopted for the cyclic loading test to further investigate the hysteresis behavior.
3.4.2. Hysteretic test
The objective of the hysteretic test was to investigate the seismic performance, failure mode, and ductility of the selected specimen (Specimen AR-OUT-S). The hysteretic curve of Specimen AR-OUT-S is shown in Fig. 10. In order to explore the hysteretic characteristics of the component, the hysteretic loops at q = 0.01 rad, 0.02 rad, 0.03 rad, 0.04 rad, and 0.05 rad are summarized in Fig. 11. The experimental phenomena of Specimen AR-OUT-S are summarized in Fig. 12. Based on the hysteretic characteristics and experimental phenomena, the failure process of Specimen AR-OUT-S can be divided into three stages: elastic stage (q ¡Ü 0.01 rad), elastic-plastic stage (0.01 rad < q ¡Ü 0.04 rad), and damage stage (q ¡Ý 0.043 rad) (Fig. 13). In the elastic stage, no pinching behavior of the hysteresis curves was observed before the beam-to-column rotation angle reached 0.01 rad (Fig. 11). This is because the angle steel was in the local bending state and the rib stiffener exhibited certain plane deformations, as shown in Fig. 13(a). In the elastic-plastic stage (0.01 rad < q ¡Ü 0.04 rad), the hysteretic loops exhibited pinching behavior (Fig. 11) owing to the change of the bearing-force state of the angle steel from local bending to local tension (Fig. 13(b)), which caused the increase in the stiffness and strength of the connection. Therefore, a pinching behavior is shown on the hysteretic curves. However, owing to the presence of the rib stiffeners, the pinching behavior was alleviated. No damage on the specimen was observed during the test before the beam-to-column rotation angle reached 0.043 rad. In the damage stage (q ¡Ý 0.043 rad), the rib stiffeners began to rupture as a result of the excessive tensile stress (Figs. 12 and 13(c)). Consequently, the strength and reloading stiffness declined, resulting in a more significant pinching behavior (Fig. 11) and the decline of the ultimate strength (Fig. 10).
Fig. 10. Hysteretic curve of Specimen AR-OUT-S
Fig. 11. Typical hysteretic loops of Specimen AR-OUT-S
Fig. 12. Experimental phenomena of Specimen AR-OUT-S in hysteretic test
Fig. 13. Failure process of Specimen AR-OUT-S during hysteretic test
In summary, the rib stiffeners of Specimen AR-OUT-S did not exhibit apparent buckling or rupture when q ¡Ü 0.04 rad; this property can impart stable strength and stiffness and simultaneously reduce the pinching caused by the plastic hinge of the angle steel. When q = 0.054 rad, the strength decreased to 80% of the peak load. In addition, when the energy-dissipating component on one side of the beam was completely damaged, the beam-to-column rotation angle became 0.061 rad. Therefore, considering the cumulative damage caused by the hysteresis loading, the fracture capacity of the SAS component can be achieved through the hysteretic test rather than the monotonic loading test. In summary, Specimen AR-OUT-S can completely satisfy the ductility requirement of SMFs proposed by AISC , i.e., q ¡Ý 0.04 rad.
4. FE model
4.1. General description
To study the failure mechanism of the SAS components and provide the basis for the following theoretical analysis, the FE models of Specimens A and AR-OUT-S were developed using the general-purpose FE software, MSC. Marc . The solid elements were used, and the FE models of Specimens A and AR-OUT-S are shown in Fig. 14. The steel was modeled as an elastic-plastic material. The loading method and boundary conditions were consistent with the tests. Contact algorithm was adopted among different components of the FE models, which was consistent with the actual situation. The pre-stressed force in the bolts was modeled by the temperature stress. The displacement of symmetric surface was constrained in the x-direction (ux = 0), and the displacements of bottom of support beam were constrained in the x-, y-, and z-direction (ux = uy = uz = 0).
Fig. 14. FE models (half-model adopted owing to symmetry)
4.2. Validation of FE model
In practical engineering, the initial geometric imperfections of steel components are unavoidable. For the SAS components proposed in this work, the initial out-of-plane deformation is the main imperfection. The initial geometric imperfections of the rib stiffeners were simulated using the method proposed in Eurocode 3  and the Code for Design of Steel Structures (GB 50017-2003) . The comparisons between the predicted and experimental results of Specimens A and AR-OUT-S are shown in Figs. 15 and 16. It can be seen from the figures that the simulated curves are in good agreement with those of the tests. Additionally, Specimen AR-OUT-S can effectively restrain the buckling deformation of rib stiffeners; therefore, the initial geometric imperfections of the rib stiffeners have little effect on the structural strength, which will not be considered in the following study. In conclusion, the FE models developed in this work can accurately simulate the response of the specimens under monotonic loading and, consequently, were used for the following parametric analysis.
Fig. 15. Comparison between predicted and experimental results of Specimen A
Fig. 16. Comparison between predicted and experimental results of Specimen AR-OUT-S
5. Theoretical analysis
The SAS component is a new type of energy-dissipating component. Because Specimen AR-OUT-S exhibited the highest stability and sufficient strength and initial stiffness, the SAS component discussed hereafter is similar to Specimen AR-OUT-S. The yield bending moment of the SAS connection can be calculated by the superposition of the bending moments provided by the angle steel and the rib stiffener, respectively. The initial stiffness of the SAS connection can be calculated through the same superposition method. Note that the design methods of the angle steel connection have been widely studied by many researchers [38-41]. Therefore, these research outcomes can be used in this work, while the design method of rib stiffeners needs to be developed.
5.1. Parameter calibration for the model of angle steel connection
Many studies have shown that the three-parameter power model proposed by Kishi and Chen  can accurately describe the behavior of angle steel connections [38-40]. Therefore, the Kishi¨CChen power model was adopted to predict the moment¨Crotation relations of angle steel connection in this work. The Kishi¨CChen power model contains three key parameters: initial stiffness , ultimate moment , and shape parameter . The equation is as follows:
in which , M A and q are moment and relative rotation in the connection, respectively.
Kishi and Chen  proposed the computational equations of the initial stiffness and the ultimate moment . The shape parameter nA is determined through the least-squares fitting between the predicted moments and the experimental ultimate moments. Note that this method can effectively predict the moment¨Crotation relations when the moment approaches the ultimate moment. However, the ultimate moment is significantly larger than the yield moment. The test of Specimen A shows that the moment¨Crotation relations determined by the shape parameter nA cannot predict the yield moment well (Fig. 17). Therefore, to predict the yield moment of the angle steel connection accurately, the shape parameter was recalibrated in this work by ensuring that the predicted and experimental moment¨Crotation relations have same energy within the specified rotation qp, and the new shape parameter is denoted as .
Fig. 17. Differences between nA and (neergy with Specimen A)
qp should be greater than the yield rotation, but it should not be excessively large. Otherwise, it will affect the accuracy of the predicted yield moment. The maximum acceptable story drift of the steel structure specified in the Code for Seismic Design of Buildings (GB 50011-2010)  is 0.02 rad; thus, in this work, qp is set to 0.02 rad. Based on the data from literature listed in Table 2, can be determined by least-squares curve fitting (shown in Eq. (2) and Fig. 18).
The yield moment of the angle steel connection MyA is defined as follows:
in which is the yield rotation.
Fig. 18. Least-squares curve fitting of
Geometrical and mechanical characteristics of monotonic loading tests
Taking Specimen A as an example, the comparisons of the predicted moment¨Crotation curves (q ¡Ü 0.02 rad) determined by nA and are shown in Fig. 17. The results show that can better determine moment¨Crotation relations of angle steel connection when q ¡Ü 0.02 rad. In other words, the yield moment of angle steel connection can be predicted more accurately by .
5.2. Theoretical calculating method of rib stiffeners
5.2.1. Initial stiffness provided by rib stiffeners
Based on the FE results, the effective stress area of rib stiffeners can be simplified as a set of strips with a specific angle a. Fig. 19(a) shows the equivalent stress area of rib stiffeners for the calculation of initial stiffness, in which the red area represents tension and the blue area represents compression. Fig. 19(b) shows the deformation mode of rib stiffeners with a connection rotation q. Fig. 19(c) shows the deformation of simplified strips with a connection rotation q. Fig. 19(d) shows the equivalent force model for the calculation of initial stiffness provided by rib stiffeners.
Fig. 19. Schematics for the calculation of the initial stiffness provided by the rib stiffeners
The strain of tensile strips is
where is the coefficient accounting for the nonuniform stress of tensile strips when calculating the initial stiffness, h is the height of beam, x1 is a variable which is the distance between the strip and the beam end, a is the inclination angle of strips, and the recommended value is 45º.
The cross-sectional area of tensile strips is
where tr is the thickness of the rib stiffeners.
The arm of force of tensile strips to Point O2 is
The moment provided by a tensile rib stiffener to Point O2 is
Consequently, the initial stiffness provided by a tensile rib stiffener to Point O2 is
(2) For compression rib stiffeners:
The strain of compression strips is
where is the coefficient accounting for the nonuniform stress of compression strips when calculating the initial stiffness, and x2 is a variable that is the distance between the strip and the beam end.
The cross-sectional area of the compression strips is
The arm of force of the compression strips to Point O2 is
The moment provided by a compression rib stiffener to Point O2 is
where l2 is the distance between the beam end and the strip closest to the beam end (Fig. 19).
The initial stiffness provided by a compression rib stiffener to Point O2 is
Take a = 45º, the initial stiffness provided by the rib stiffeners is determined as follows:
5.2.2. Yield moment provided by rib stiffeners
The yield moment provided by the rib stiffeners can be defined as the sum of the yield moments provided by all the strips. The corresponding equivalent area, deformation mode and equivalent force model for the calculation of the yield moment provided by the rib stiffeners can be determined in Fig. 20(a-d).
Fig. 20. Schematics for calculation of yield moment provided by rib stiffeners
(1) For tensile rib stiffeners:
The resultant tensile force is
where b and c are the dimensions of the rib stiffeners shown in Fig. 20, is the yield strength of the rib stiffeners, and is the coefficient accounting for the nonuniform stress of tensile strips when calculating the yield moment.
The arm of force of the tensile strips to Point O2 is
The yield moment provided by a tensile rib stiffener to Point O2 is
(2) For compression rib stiffeners:
The resultant compression force is
where is the coefficient accounting for the nonuniform stress of compression strips when calculating the yield moment.
The arm of force of compression strips to Point O2 is
The yield moment provided by a compression rib stiffener to Point O2 is
Take a = 45º, the yield moment provided by rib stiffeners is determined as follows:
5.2.3. Coefficients accounting for nonuniform stress determined by FE models
Based on the above analysis, the initial stiffness and the yield moment provided by the rib stiffeners can be calculated, if the coefficients accounting for nonuniform stress ( , , , and ) are determined. Due to the limited number of experiments available, the FE model calibrated by experimental data (shown in Section 4) is adopted for the parametric analysis herein. To calculate the initial stiffness and yield moment provided by the rib stiffeners and eliminate the influence of angle steel, the specific elements at the corner of the angle steel in the FE model are deleted (Fig. 21).
Fig. 21. FE models with different height of rib stiffeners
The variable parameters in the FE model include the height of beam h (0.3, 0.35, and 0.4 m), the thickness of rib stiffeners tr (0.001, 0.002, 0.003, 0.004, 0.005, 0.006, and 0.007 m), the height of rib stiffeners lr (0.1, 0.15, and 0.2 m), and the yield strength of rib stiffeners (235, 345, and 368 MPa). The definition of the variable parameters (h, tr, and lr) are shown in Fig. 19(a). The FE models with different heights of rib stiffeners are shown in Fig. 21. The simulation results show that if tr/ta (ta is the thickness of the angle steel) is too large, it will result in the deformation of the angle steel, which will reduce the energy-dissipating capacity. In contrast, if tr/ta is too small, the rib stiffeners have little contribution. Consequently, after a series of parametric discussions, tr/ta should meet the following criterion: .
With the comparisons of initial stiffness and yield moment between the simulation results and the computational results from Eqs. (14) and (21), , , , and can be determined as follows by data fitting:
Substituting , , , and into Eqs. (14) and (21) results in
The comparisons of the initial stiffness and yield moment between the FE results and the calculation results from Eqs. (25) and (26) are shown in Figs. 22 and 23, respectively, and the results show good agreement.
Fig. 22. Initial stiffness provided by rib stiffeners
Fig. 23. Yield moment provided by rib stiffeners
Additionally, the moment-rotation relations provided by rib stiffeners can be defined as follows, similar to Eq. (1):
Because the strength provided by some rib stiffeners will decrease when , is taken as a basic parameter in this work. Based on the former mentioned FE results (Figs. 24 and 25) and Eqs. (25) and (26), and can be determined as follows:
Fig. 24. Data fitting for
Fig. 25. Theoretical calculation of
Consequently, can be determined as follows:
5.3. Evaluation of initial stiffness and yield moment provided by SAS
Based on the above analysis, the calculation method of the initial stiffness and the yield moment provided by SAS can be determined as follows:
The initial stiffness provided by SAS is the sum of that provided by the angle steels and the rib stiffeners:
Note that the stiffness of SAS will significantly decrease after the yield of the rib stiffeners. Therefore, it is assumed that SAS will reach the yield point when the rib stiffeners yield, and, consequently, the yield moment provided by SAS can be determined as follows: (1) determine the yield moment provided by the rib stiffeners through Eq. (26); (2) determine the yield rotation of the rib stiffeners through Eq. (30); and (3) determine the yield moment provided by the angle steel through Eq. (3). Subsequently, the yield moment provided by SAS can be determined as follows:
To validate the above equation, a series of FE models of SAS were established, and the variable parameters in the FE models included the following: the thickness of the angle steel ta = 14 mm; the other parameters of angle steels, the same as the FE model shown in Fig. 14(a); the height of beam h (0.3, 0.35, and 0.4 m); the thickness of rib stiffeners tr (0.002, 0.003, and 0.004 m); and the height of rib stiffeners lr (0.1, 0.15, and 0.2 m). The comparisons of the initial stiffness and the yield moment provided by SAS between the simulation results and the calculation results from Eqs. (31) and (32) are shown in Fig. 26 and Table 3, with a good agreement, which proves the rationality of the calculation method proposed in this work.
Comparisons of initial stiffness and yield moment provided by SAS between the FE results and the proposed model
In this work, a new type of dual-functional replaceable SAS component was proposed for self-centering frames, and seven replaceable energy-dissipating components, namely, an angle steel component and six SAS components, were designed. Monotonic and hysteretic loading experiments, FE simulation, and theoretical analysis were conducted. The main conclusions are as follows:
(1) The results of the monotonic loaded specimens indicate that Specimen AR-OUT-S had higher stability compared with the other specimens and simultaneously exhibited sufficient strength and initial stiffness. The strength of Specimen AR-OUT-S was 100% higher than that of the angle steel component with a 41% increase in the steel consumption. In addition, Specimen AR-OUT-S did not exhibit damage when q ¡Ü 0.04 rad in the hysteretic test; this property can impart stable strength and stiffness and simultaneously improve the structural seismic resilience.
(2) The FE models of the specimens were developed, in which the contact algorithm among the different components were adopted. The FE models could accurately simulate the response of the angle steel component and Specimen AR-OUT-S under monotonic loading and consequently, could be used for the parametric analysis to study the effect of the dimension of the rib stiffeners on the initial stiffness and the yield moment.
(3) The theoretical calculation equations of the initial stiffness and the yield moment of the beam-column connection provided by the SAS component were proposed. In the proposed equations, the contribution of the angle steels and rib stiffeners were considered. The theoretical equations are reasonably consistent with the FE results and can be applied for designing the proposed SAS components.
In conclusion, the proposed SAS components exhibited higher strength and energy-dissipating performance than the angle steel components and simultaneously satisfied the ductility requirement of SMFs proposed in AISC . In addition, the SAS components can be replaced rapidly after an earthquake, which ensures the rapid repair of structures and satisfies seismic resilience requirements.
This work was supported by Beijing Natural Science Foundation .
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