Energy dissipation estimation of steel slit shear panel dampers with tapered links considering the effect of out-of-plane buckling

Abstract

The steel slit shear panel (SSSP) with tapered links is one type of efficient energy-dissipating devices, with tapered shape of the links accommodating the bending moment demand when subjected to lateral shear deformation. To dissipate the maximum amount of seismic energy, full plasticity is expected to be developed. However, out-of-plane buckling, which is almost inevitable for thin plates, significantly reduces its energy dissipation capacity. This work aims to quantify the amount of energy dissipation of SSSPs with tapered links considering the effect of out-of-plane buckling. Quasi-static cyclic loading tests on five SSSP specimens with different design parameters are conducted, based on which the refined numerical model is built and validated. In estimating the energy dissipation, the effect of out-of-plane buckling needs to be considered. Otherwise, the estimated energy dissipation can be as much as over two times the tested value. To consider the effect of out-of-plane buckling, the index of average pinching parameter is adopted. A new method is derived for estimating the energy dissipation of SSSPs with considering the effect of out-of-plane buckling, feasibility of which is validated using the obtained test results.

Keywords

steel slit shear panel tapered link out-of-plane buckling energy dissipation pinching

Introduction

For designing new buildings and strengthening of existing buildings, energy dissipation devices are more and more applied. By introducing energy dissipation devices, damage is concentrated thereby such that damage of the main structure can be mitigated. Metallic dampers use metal yielding in different forms, such as bending, shear, tension and compression, torsion and so on. The steel slit shear panel (SSSP) belongs to one type using the combined shear and bending mechanism. By introducing slits in the monolithic steel plate, the stress pattern of the plate changes from in-plane shear to primary flexural bending of individual plate links separated by slits. The most common SSSP is the type having equally-spaced vertical slits. Hitaka and Matsui (2003) are generally believed to be the pioneers conducting research on SSSPs. They focused on the essential performance of SSSPs, and proposed corresponding theoretical calculation methods of strength and stiffness. Following theirs, several other works investigating the seismic behavior of SSSPs were conducted (Cortés and Liu, 2011; Hebdon et al., 2012; Khatamirad and Shariatmadar, 2017). Also, some researches were conducted to study the capabilities of SSSPs with different link configurations, including links with variation of length and width (Jacobsen et al., 2010; Lu et al., 2018; Milad and Hashem, 2017). Influence of material property on the hysteretic behavior of SSSPs was investigated (Bu et al., 2021b; Hwang et al., 2022). Seismic performance under additional tensile load was also analyzed (Oh and Park, 2022). In the structural level, contributions of SSSPs on improving structural performance were analyzed for frame structures (Ahmadi et al., 2021a; Bu et al., 2021a; Maida et al., 2021). Self-centering mechanism was adopted in combination with SSSPs to achieve seismic resilience (Chu et al., 2021; Lu et al., 2021). To further improve the energy dissipation capacity of SSSPs, different restrainers were adopted, such as reinforced concrete plate (Zhao and Astaneh-Asl, 2004), mortar plate (Hitaka et al., 2007), wood plate (Ito et al., 2011), steel plate (Deng et al., 2015) or steel-concrete composite plate (Lin et al., 2019). Due to the large stress concentrating at slit ends, fracture was thereby generated (Ito et al., 2011), which surely reduced the energy dissipation capacity. To solve the fracture issue, one solution is to adjust the location of large stress away from link ends. Kobori et al. (1992) proposed the SSSP with tapered links, which demonstrated plump hysteresis. Later, Ma et al. (2010) conducted study on the energy dissipation performance of SSSP with tapered links in rocking system. Kurata et al. (2015) used the SSSPs with tapered links for the main purpose of monitoring inter-story deformation. To achieve even yielding, shape optimization of the links was also investigated (Ahmadi et al., 2021b; Garivani et al., 2016; Ghabraie et al., 2010; Shen et al., 2022) and stable hysteresis curves were obtained. In fact, the tapered shape of links in this study is very close to the optimized shape but using straight edges for simplicity. Overall, SSSPs with tapered links showed largely improved ductility than those with rectangular links. Another solution to solve the fracture issue is to use more ductile steel (He et al., 2016). The significant ductility of low yield steel eliminated the potential fracture and plump hysteresis was obtained without the need of out-of-plane constrainer. For the two solutions to solve the fracture issue, the first solution by using tapered links made by common steel is more preferred to the authors, considering the less common and more expensive low yield steel.

To achieve an economical design and meanwhile consider the inevitable buckling of SSSPs under rare earthquakes, this study focuses on investigating their pinched hysteretic behavior with the occurrence of out-of-plane buckling. For SSSPs with tapered links, the width-thickness ratio of tapered links dominates the occurrence of out-of-plane bucking. The SSSP with wide links (having relatively large width-thickness ratio) buckles easily and produces pinched hysteresis. To estimate their energy dissipation capacity, pinching in the hysteresis apparently needs to be considered. With considering estimation of energy dissipation, Ke and Chen (2012) proposed basic method for the designing of SSSPs with rectangular links. In their work, the effect of pinching in the hysteresis was not considered in the estimation of energy dissipation. In a recent work by the authors (Ma et al., 2022), the effect of pinching in the hysteresis was considered in estimating the energy dissipation of SSSPs with rectangular links. Different from SSSPs with rectangular links, adoption of tapered links gives different shear strength, stiffness and accordingly energy dissipation capacity. For SSSPs with tapered links, to the best of our knowledge, there is no work conducted yet on estimating their energy dissipation, especially taking account of the influence of pinching in the hysteresis.

With the backgrounds mentioned above, this work investigates the method to estimate the energy dissipation of SSSPs with tapered links considering the effect of out-of-plane buckling occurrence. To this end, quasi-static cyclic loading tests were conducted on five specimens. Different design parameters of width-thickness ratio and aspect ratio of tapered links were used to investigate their effect on the energy dissipation of SSSPs and to provide reference data in validating the proposed method of energy dissipation estimation. Test results in terms of shear strength and stiffness, hysteretic curve and dissipated energy were discussed in detail. A refined numeral analysis was also conducted to study the influence of width-thickness ratio and aspect ratio of tapered links on the out-of-plane buckling and energy dissipation capacity.

SSSP with tapered links

Figure 1 illustrates a representative SSSP with tapered links. With the tapered shape of the link, the initiation of material yielding will not happen at the link ends when subjected to lateral shear deformation. In order to lead the initiation of plasticity at the ¼ and ¾ height section, the end width was set as 3 times the mid-section width (He et al., 2016). Thus, the major controlling parameters of the tapered link are the mid-section width a, link height h and the plate thickness t. The critical width at the ¼ and ¾ height section is 2a. Two parameters were defined, width-thickness ratio (λ = 2a/t) and the aspect ratio (β = h/2a). Bolt holes are reserved in the boundary zone for the connection. The SSSP with tapered links is connected to beams only, which gives a simple and clear interaction with the main lateral force resisting system.

Figure 1.

Steel slit shear panel with tapered links.

Elastic buckling of tapered link

For thin plates, elastic local buckling may occur. Under lateral loading, the tapered link deforms in a symmetrical manner, as shown in Figure 2. Half of the link is analyzed for simplicity. Due to the tapered shape of the link, it is difficult to apply the available plate buckling theory. Therefore, the tapered link is treated as rectangular type. When the links subjected to lateral deformation, tension and compression are the stress state of the two free edges. Compared to the edge in compression, the edge in tension is more rigid and unlikely to occur buckling. Therefore, the edge in tension is assumed as simply supported boundary. The lower edge is also assumed as simply supported, in view of no vertical displacement but certain freedom of rotation. To adopt the available buckling theory, boundary of the upper edge is also approximately equated to simply supported, considering that the upper edge will be connected by angels in practice and there is always certain flexibility thereby.

Figure 2.

Plate buckling analytical model for the tapered link.

According to the plate buckling theory (Timoshenko and Gere, 1961), the critical bending stress for a rectangular plate is governed by the following equation

σ_{c r} = k \frac{π^{2} E}{12 (1 - ν^{2}) {(b / t)}^{2}}

(1)

where k is a coefficient, influenced by dimension of the plate and boundary conditions; E is Young’s modulus; ν is Poisson’s ratio; b = 2a is the equivalent critical width and t is thickness.

With the boundary conditions illustrated in Figure 2(b), the lower boundary of k is chosen as 0.85. Then, according to equation (1), the critical width-thickness ratio of the rectangular plate link at which the bending stress equals the yield strength can be estimated. For instance, with ν = 0.3, E = 205 GPa and a yield strength of f_y = 374 MPa, the critical width-thickness ratio of rectangular plate links is λ = 20.5. For energy dissipation, development of plasticity prior to the occurrence of elastic buckling is preferred. In another word, SSSPs with these links having λ no larger than 20.5 will yield first before the occurrence of elastic buckling.

According to the plane cross-section assumption, the buckling strength Q_cr can be obtained when the bending stress reaches the critical buckling stress according to equation (1)

Q_{c r} = \frac{n t b^{2} σ_{c r}}{3 h}

(2)

where h is link length; n is the number of links; σ_cr is critical buckling stress.

For relatively long plate links, overall lateral torsional buckling may also occur. The lateral torsional buckling strength Q_LTB can be obtained by

Q_{L T B} = n \frac{4.013}{{(1 - \sqrt{\frac{E I_{W}}{G J {(h / 2)}^{2}}})}^{2}} \frac{\sqrt{E I G J}}{{(h / 2)}^{2}}

(3)

where EI is the bending stiffness, EI_w is the warping stiffness and GJ is the torsional constant.

Maximum shear strength and stiffness

As shown in Figure 1, width at end section is designed as 3 times that at the middle section, which produces plasticity at the ¼ and ¾ height sections of the link. The yield strength Q_{y, link}, initiation of steel yielding at the edge fiber at the ¼ and ¾ height sections, is estimated as follows (He et al., 2016)

Q_{y, l i n k} = \frac{8 a^{2} t}{3 h} f_{y}

(4)

where a is width at middle section, h is the link length, and f_y is the steel yield strength.

With the yielding expands, plasticity development at the ¼ and ¾ height sections gives its plastic strength Q_p,link

Q_{p, l i n k} = \frac{4 a^{2} t}{h} f_{y}

(5)

Its elastic stiffness K_link is estimated by summating the contribution from both flexural and shear deformation (He et al., 2016)

K_{l i n k} = \frac{1}{\frac{1}{12.7 E t {(h / a)}^{3}} + \frac{0.55 h}{G a t}}

(6)

Based on equations (4)–(6), summation of all links gives the wall’s overall yield strength Q_y, plastic strength Q_p and elastic stiffness K

Q_{y,} = n Q_{y, l i n k}

(7a)

Q_{p,} = n Q_{p, l i n k}

(7b)

K = n K_{l i n k}

(7c)

Energy dissipation assuming full plasticity

In the work by Ke and Chen (2012), energy dissipation of SSSPs with rectangular links at a certain lateral deformation was estimated according to the principle of work. In their work, full plasticity was assumed to occur at link ends. Inspired by their work, this paper proposes the following equation to estimate the energy dissipation of a single tapered link E_p,_link

E_{p, l i n k} = M_{p, l i n k} θ_{p, l i n k} = f_{y} a^{2} t (\frac{7 δ}{2 h} - \frac{f_{y} h}{E a})

(8)

where M_{p, link} is full plastic moment at link ends, θ_{p, link} is plastic hinge rotation, δ is lateral displacement, f_y is yield stress, E is Young’s modulus.

For a SSSP with multiple links, a larger number of individual links certainly gives larger energy dissipation. The plastic energy dissipation of the wall E_p, _wall at a certain lateral displacement can be estimated by summing up all individual links as follows

E_{p, w a l l} = n f_{y} a^{2} t (\frac{7 δ}{2 h} - \frac{f_{y} h}{E a}) = f_{y} V_{L} (\frac{7}{8} \frac{Δ}{h} \frac{1}{β} - \frac{f_{y}}{2 E})

(9)

where n is number of links, Δ is lateral displacement, V_L = 2naht is total volume of tapered links.

Equation (9) tells that with given steel type, total volume of steel and link height, energy dissipation is inversely proportional to aspect ratio. For validation, three common carbon steel tapered links, Link 1 with λ = 8.9 and β = 9, Link 2 with λ = 13.3 and β = 6, and Link 3 with λ = 17.8 and β = 4.5 respectively, were numerically analyzed under monotonic loading. Considering the overall structure of the paper, information regarding the numerical modeling is not given here but in a later section. The dissipated energy obtained from numerical analysis was compared with the calculation according to equation (8). As shown in Figure 3, the proposed method gives a fair agreement with the simulation.

Figure 3.

Comparison of energy dissipation between prediction and simulation.

Note that out-of-plane buckling of the tapered links is constrained in simulation and thus they can behave only in plane, to allow the sufficient development of plasticity at quarter-height sections. While, buckling out of plane is almost inevitable. Its effect on the energy dissipation will be investigated through both experimental tests and numerical analysis described later.

Test preparation

Specimen design

To maximize the energy dissipation, majority of the lateral story drift should be input to the installed SSSP. To realize this, a shorter SSSP relative to story height is recommended, for instance, by using stiff members between story beam and the middle installed SSSP. Considering the available test setup, the height of SSSP specimens was designed as 480 mm. Five scaled SSSP specimens were designed, with detailed dimensions shown in Figure 4. The key dimension parameters of the specimens are summarized in Table 1 (geometric notations defined in Figure 1). Openings between tapered links were manufactured by laser cutting. The height of steel panel and link are L = 480 mm and h = 300 mm respectively. The plate portion beyond the links was 90 mm for all specimens, which was treated with circular holes for bolt connection. By controlling the mid-section width and link thickness, four different width-thickness ratios and aspect ratios were adopted as shown in Table 1. Common carbon mild steel was used in manufacturing the specimens. By standard uniaxial tensile coupon tests, the actual yield strength of the steel was obtained as 374 MPa.

Figure 4.

Specimen dimensions (unit: mm).

Table 1.

Specimen details.

Specimen	a (mm)	h (mm)	t (mm)	n	λ	β
1	25	300	2.2	3	22.7	6.0
2	15	300	2.2	10	13.6	10.0
3	30	300	4.3	5	14.0	5.0
4	25	300	4.3	6	11.6	6.0
5	20	300	4.3	7	9.3	7.5

Note that except for Specimen 4, the rest test specimens can also be found in literature (He et al., 2016), which studied the quantification of out-of-plane buckling deformation amplitude. Herein, the focus is on investigating the influence of out-of-plane buckling deformation on the hysteresis and quantifying the resulted energy dissipation.

Loading protocol and test setup

Quasi-static cyclic loading was conducted using displacement control. Figure 5 shows the adopted incremental two-cycle loading protocol (FEMA 461, 2007). The ordinate is drift ratio, which is defined as the ratio between experienced lateral deformation and link height.

Figure 5.

Loading protocol.

For the convenience of loading, the specimens were rotated 90° and installed in a loading setup as shown in Figure 6. The steel frame was made of wide-flange H-shaped steel, with two side columns anchored to the base and the center column attached to the vertical jack. Moving of the center column gives specimen’s lateral deformation. The vertical movement of the center column was recorded by displacement transducer and carefully adjusted to meet the required drift ratio. To balance the vertical load, two identical specimens were installed for each loading. The shear force was determined by half of the force obtained from the jack.

Figure 6.

Test setup.

Test results and discussions

Shear strength and stiffness

As discussed in earlier, both elastic local buckling and lateral torsional buckling of individual plate links may occur. The calculated buckling strengths are listed in Table 2. The calculated plate local buckling strength Q_cr of Specimen 1 is the smallest, which indicates the first occurrence of plate local buckling prior to steel yielding. This is consistent with its largest width-thickness ratio as shown in Table 1. With the largest aspect ratio, lateral torsional buckling strength Q_LTB of Specimen 2 was smaller than both yielding strength Q_y and local buckling strength Q_cr, indicating the dominant pattern of lateral torsional buckling. With the further decrease of width-thickness ratio, steel yielding happens first, which is desirable for seismic energy dissipation.

Table 2.

Elastic buckling strength.

Specimen	Q_cr (kN)	Q_LTB (kN)	Q_y (kN)	Q_cr/Qy	Q_LTB/Qy
1	5.59	7.09	13.71	0.41	0.52
2	18.63	13.28	16.46	1.13	0.81
3	69.56	109.60	64.33	1.08	1.70
4	83.48	105.94	53.61	1.56	1.98
5	97.39	95.64	40.03	2.43	2.39

The obtained maximum shear strength and elastic lateral stiffness are summarized in Table 3. For all specimens, out-of-plane buckling occurred in the tapered links and thus the maximum strength in test (Q_max,t) was smaller than the plastic strength (Q_p). With the largest width-thickness ratio, Specimen 1 had the most obvious torsional deformation, which gave Q_max,t less than 80% of Q_p. With the decrease of width-thickness ratio, Q_max,t became closer to Q_p. The obtained average K_t was 88% of the predicted K₀. Potential cause for this difference was the non-rigid connection, which gave certain flexibility.

Table 3.

Maximum shear strength and elastic stiffness.

Specimen	Q_p (kN)	Q_max.t (kN)	Q_max.t/Q_p	K₀(kN/mm)	K_t (kN/mm)	K _t /K ₀
1	20.6	16.0	0.78	7.9	7.1	0.89
2	24.7	23.2	0.94	6.1	6.0	0.98
3	96.5	79.3	0.82	40.3	31.7	0.79
4	80.4	69.6	0.87	29.6	24.8	0.84
5	60.0	55.5	0.92	18.9	16.6	0.88

Hysteretic behavior

Hysteretic curves of all specimens were shown in Figure 7. The ordinate is the actual shear force divided by the plastic strength (Q_y) and abscissa is lateral drift ratio. Tapered links in Specimen 1 buckled out of plane beyond 1% drift ratio, which caused obvious pinching. For Specimen 5, notable out-of-plane buckling of tapered links did not occur until the drift ratio of 3% and thus plumper hysteretic behavior resulted. With the same width-thickness ratio, a larger aspect ratio made the shear strength continue to increase. For instance, the increased shear strength of Specimen 2 was more obvious than the corresponding Specimen 3. The increased shear strength was caused by the more obvious tension field effect at large drift ratio. With the increase of width-thickness ratio, the pinching phenomenon became more notable, which significantly degraded the energy dissipating capacity. Considering the pinching level in the obtained hysteresis, the width-thickness ratio no larger than 10 is recommended for good energy dissipation.

Figure 7.

Hysteretic curves of Specimens 1–5 respectively.

Observed out-of-plane deformation

Out-of-plane deformation of specimens was recorded by digital cameras during the test. For demonstration, Figures 8–10 show the deformed shapes of Specimen 1, Specimen 3 and Specimen 5 respectively. With the increase of lateral displacement, out-of-plane buckling of tapered links became obvious. Compared with Specimen 3 and Specimen 5 with smaller width-thickness ratios and larger aspect ratios, more notable out-of-plane buckling was observed in Specimen 1. With the smallest width-thickness ratio, Specimen 5 exhibited the least amplitude of out-of-plane deformation. This comparison shows that the width-thickness ratio of tapered links was the controlling parameter for out-of-plane buckling. After the completion of loading with a maximum drift ratio of 6%, all the specimens remained intact and no visible fracture was observed. The advantage of the tapered shape of the links in eliminating fracture was demonstrated.

Figure 8.

Deformed shape of Specimen 1 at drift ratios of 2%, 4% and 6% respectively.

Figure 9.

Deformed shape of Specimen 3 at drift ratios of 2%, 4% and 6% respectively.

Figure 10.

Deformed shape of Specimen 5 at drift ratios of 2%, 4% and 6% respectively.

Finite element analysis

Analysis models

The occurrence of out-of-plane buckling produces pinching in the hysteresis and obviously affects the energy dissipation performance. For estimating energy dissipation, the effect of pinching needs to be quantified. As a compromise of limited test data, supplemental numerical simulation was carried out.

In the numerical simulation, finite element analysis software ABAQUS is used for modelling SSSPs with tapered links. Three-dimensional four-node shell element with reduced integration (S4R) is used. For the material model of steel, the commonly adopted bilinear elastic-plastic model is used. The post-elastic stiffness is set as 0.5% of the initial stiffness. For the boundary condition, bottom end of the wall is fixed and lateral displacement history is applied on the top end to impose cyclic loading.

The tested specimens were simulated, with the obtained hysteretic curves plotted in dashed lines in Figure 7. As the adopted material model was the simple elastic perfectly plastic material model, degradation of unloading stiffness was not able to be considered. Except for the unloading part, simulation agreed well with test and thus the simulation can basically capture the hysteretic behavior of SSSPs. Difference between the simulation and test data will be further discussed in next section.

To save the time in analysis, individual tapered links instead of a SSSP with multiple links were built for parametric analysis. A total of 24 tapered links with 8 different width-thickness ratios (λ = 5–25) and three aspect ratios (β = 5, 7.5 and 10) were modelled. Details of 24 models are listed in Table 4.

Table 4.

Model parameters.

Model	h (mm)	a (mm)	t (mm)	λ	β
1	300	15	1.2	25.0	10.0
2			1.5	20.0
3			2.0	15.0
4			2.5	12.0
5			3.0	10.0
6			4.0	7.5
7			5.0	6.0
8			6.0	5.0
9	300	20	1.6	25.0	7.5
10			2.0	20.0
11			3.0	13.3
12			4.0	10.0
13			5.0	8.0
14			6.0	6.7
15			7.0	5.7
16			8.0	5.0
17	300	30	2.4	25.0	5.0
18			3.0	20.0
19			4.0	15.0
20			5.0	12.0
21			6.0	10.0
22			8.0	7.5
23			10.0	6.0
24			12.0	5.0

Pinching parameter

Both test and simulation results showed that significant pinching occurred. To quantify the effect of pinching on energy dissipation, the average pinching parameter was proposed herein. For a typical hysteretic curve shown in Figure 11, the pinching parameter is defined as follows

η = \sum_{i = 1}^{N} S_{(A B C D A), i} / \sum_{i = 1}^{N} S_{(A b C d A), i}

(10a)

S_{(A b C d A), i} = (V_{i +} - V_{i -}) (Δ_{i +} - Δ_{i -})

(10b)

where S_(ABCDA), _i is the area of each hysteresis loop, S_(AbCdA), _i is the area of each rectangle AbCd, V_{i +}, V_i- and Δ_{i +}, Δ_i- are the maximum force and displacement at each direction of displacement respectively.

Figure 11.

Determination method of pinching parameter η.

As discussed earlier, there is difference between the simulated and tested hysteresis, especially in the unloading branch (Figure 7). Comparison of pinching parameter between the simulation and test is shown in Figure 12. Difference of pinching parameter between the simulation and test exists because of the incapability of capturing unloading stiffness degradation in simulation. And the difference is relatively large at large drift ratios, which is natural considering the more pinched hysteresis. Considering the simplicity of the adopted elastic perfectly plastic material model and current difficulty in developing a more suitable model, this study adopts an indirect approach: adjusting the simulated pinching parameter. For instance, to reflect the more serious pinching obtained in the test, an overall adjusting factor of 0.8 for the simulation is adopted. The adjusted pinching parameter obtained in simulation is also plotted in Figure 12. The adjusted simulation results agree well with the test data. To compensate the inaccurate simulation of unloading stiffness degradation, the indirect approach by adjusting the simulated pinching parameter is found feasible.

Figure 12.

Comparison of pinching parameter.

To quantify the pinching of hysteresis in simulation, the average pinching parameter η obtained from the parametric analysis of tapered links is shown in Figure 13. The tendency is clear that η decreases with the increase of width-thickness ratio. It is natural that links with larger width-thickness ratio buckle earlier and larger and thus pinching is more serious. And the reduction rate slows down at large width-thickness ratio. Compared to width-thickness ratio, the influence of aspect ratio is limited. Similarly, η decreases with the increase of aspect ratio. For links with larger aspect ratio, flexural deformation is more dominant than shear deformation, which compromises the energy dissipation capacity due to the less developed plasticity at quarter-height sections.

Figure 13.

Average pinching parameter η.

By fitting the obtained data, equation (11) can be used to calculate the average pinching parameter as a function of the width-thickness ratio and the aspect ratio of links. In this equation, the width-thickness ratio is much more influential than aspect ratio, to reflect the controlling effect of width-thickness ratio on the pinching behavior of SSSPs. By incorporating both parameters, pinching level of SSSPs can be easily estimated. In a more intuitive way, equation (11) together with simulation results are plotted in Figure 13. Equation (11) gives a reasonably fair estimation of the average pinching parameter. It is worth mentioning that equation (11) is at most a tentative solution as there lacks sufficient physical interpretations. Further study is still needed to quantify the pinching level more rationally. To consider the deficiency of current numerical model and quantify the pinching of hysteresis obtained in test, an overall adjustment of average pinching parameter obtained in simulation by multiplying 0.8 can be used. In seismic design of SSSPs, the acceptable level of pinching can be determined flexibly considering different levels of earthquake intensity.

η = - 0.025 β + 0.32 + \frac{0.58}{1 + {(\frac{λ}{14.7 - 0.5 β})}^{3.54 + 0.003 β}}

(11)

Proposed energy dissipation estimation

Equation (9) gives the energy dissipation at a certain lateral deformation under pure in-plane monotonic loading. Under cyclic loading, assuming no occurrence of out-of-plane buckling and cyclic deterioration, energy dissipation in one cycle of a lateral amplitude can be estimated by four times the value calculated by equation (9), yielding the new equation (12).

E_{p, w a l l, c y c l i c} = 4 f_{y} V_{L} (\frac{7}{8} \frac{Δ}{h} \frac{1}{β} - \frac{f_{y}}{2 E})

(12)

For the test specimens, dissipated energy at each drift ratio is calculated by summing the hysteretic loop area in the first cycle of each drift ratio. The comparison between the predicted value according to equation (12) and the tested value is shown in Figure 14. Prediction by equation (12) is much larger than the tested value, over two times. As described earlier, the calculation method of energy dissipation of SSSP with tapered links by equation (9) assumes that plasticity occurs fully at quarter-height sections. However, once buckling of links happens, the resulting out-of-plane deformation makes the occurrence of full plasticity rather difficult. Consequently, pinching appears in the hysteretic curves. Therefore, pinching needs to be considered in the estimation of energy dissipation.

Figure 14.

Dissipated energy in one cycle.

Considering the effect of out-of-plane buckling, estimation of energy dissipation under cyclic loading is proposed in equation (13). Compared to equation (12), the difference is the inclusion of two multiplication factors, the average pinching parameter η and adjusting factor of 0.8 to compensate deficiency of current numerical model.

E_{p, p r o p o s e d} = 0.8 η * 4 f_{y} V_{L} (\frac{7}{8} \frac{Δ}{h} \frac{1}{β} - \frac{f_{y}}{2 E}) = 3.2 η f_{y} V_{L} (\frac{7}{8} \frac{Δ}{h} \frac{1}{β} - \frac{f_{y}}{2 E})

(13)

For validation of the proposed new method, dissipated energy E_p,proposed calculated by equation (13) is compared with the test results at each amplitude (first cycle), as shown in Figure 14. Compared to equation (12), the new method by equation (13) with considering the effect of out-of-plane buckling estimates dissipated energy much closer to the test results, especially at relatively large drift ratios. With the ideal assumption of in-plane behavior without out-of-plane buckling behind equation (12), naturally it overestimates the energy dissipation. With considering the effect of out-of-plane buckling, the proposed new method gives the closest prediction over the whole loading process. Certainly, there is still considerable error of the proposed method. The closeness between calculations by the proposed method and test results varies at different drift ratios, which is reasonable considering the adoption of average pinching parameter in the proposed method. Influence of out-of-plane buckling development on the pinching parameter is different at different drift ratios. As shown in Figure 12, the pinching parameter increases with the increase of drift ratios before the occurrence of out-of-plane buckling, and beyond that the pinching parameter decreases. The average pinching parameter clearly could not reflect the dynamic variation of pinching parameter. However, within the interested range of drift ratios, adoption of average pinching parameter makes the estimation of energy dissipation simple.

Cumulative energy dissipation can also be calculated by summing up all the loading cycles, as shown in Figure 15. In the calculation, both two cycles at each amplitude were calculated. Obviously, the new method gives better agreement with the tested value. Compared with Figure 14, the new method predicts the cumulative energy dissipation better than that in each one cycle.

Figure 15.

Cumulative dissipated energy.

Equation (13) indicates that the energy dissipation capacity of SSSP is dependent on both steel volume of the shear panel and aspect ratio of the link. For the convenience of testing, the specimens are designed as small in this particular study. In practice, the SSSP is suggested to be designed as wide as possible, which first is beneficial for energy dissipation considering a large steel volume and second avoids the potential lateral torsional buckling of beam with concentrated force from the installed SSSP. In addition, to concentrate the majority of story drift into the SSSP for sufficient development of plasticity, link height is suggested to be short relative to story height. Besides energy dissipation, as a structural member, contributions of shear strength and lateral stiffness of SSSP also need to be carefully estimated. For instance, the influences of link buckling on the maximum shear strength and flexibility in connection on lateral stiffness need to be taken into account in designing SSSPs.

Conclusions

The present work studies how to estimate the energy dissipation of SSSPs with tapered links considering the influence of out-of-plane buckling occurrence. Through both experimental and numerical study, the following major conclusions can be drawn:

(1) Width-thickness ratio dominantly controls the out-of-plane buckling of tapered links. Links with a larger width-thickness ratio buckle out of plane earlier and produce more serious pinching in the hysteretic curve.

(2) For estimating energy dissipation of SSSPs with tapered links, the influence of out-of-plane buckling needs to be considered. Otherwise, the estimation obviously overestimates the amount of dissipated energy, as much as over two times.

(3) To consider the effect of out-of-plane buckling on the energy dissipation, the average pinching parameter is introduced in the equation for estimating the dissipated energy. The prediction of dissipated energy in both one cycle and cumulative values agrees well with the obtained test data.

(4) Future work is still required to develop a more refined model for SSSPs with tapered links capable of considering the material fracture damage and stiffness degradation especially in the unloading part under cyclic loading. To facilitate the practical design, development of a general macro model in common design software is also needed.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Financial support from National Key Research and Development Program of China under (Grant No. 2022YFC3803003) is greatly appreciated.

ORCID iD

Liusheng He

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