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Hourglass-shaped strip damper subjected to monotonic and cyclic loadings
Engineering Structures 119 (2016) 122–134
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Hourglass-shaped strip damper subjected to monotonic and cyclic loadings Chang-Hwan Lee a, Seung-Hee Lho b, Do-Hyun Kim c, Jintak Oh d, Young K. Ju d,⇑ a
Research Institute of Structural Engineering & System, DongYang Structural Engineers Co., Ltd., Seoul 05836, Republic of Korea Urban Architecture Division, Korea Agency for Infrastructure Technology Advancement, Anyang-si 14066, Republic of Korea c Department of Architecture and Interior Design, Gyeonggi College of Science and Technology, Siheung-si, Gyeonggi 15073, Republic of Korea d School of Civil, Environmental and Architectural Engineering, Korea University, Seoul 02841, Republic of Korea b
a r t i c l e
i n f o
Article history: Received 14 January 2015 Revised 8 April 2016 Accepted 11 April 2016
Keywords: Metallic damper Passive damping device Seismic retrofitting Energy dissipation Backbone curve Membrane effect Hysteretic model Combined isotropic–kinematic hardening
a b s t r a c t An hourglass-shaped strip damper (HSD) was proposed to improve on the conventional slit damper. The damper has non-uniform strips which have a smaller cross-sectional area close to the middle height. To find the structural capacities of HSD subjected to monotonic and cyclic loadings, experimental tests were carried out in this study. Test parameters were loading rate, material strength, and the number of damper plates. The results showed substantial load–resistance capacity under monotonic loadings, and excellent ductility and energy dissipation were exhibited under cyclic loadings, with even distribution of damage over the entire height of strips. Based on the test results, a simple hysteretic model using a combined isotropic–kinematic hardening rule was also proposed. The comparison demonstrated that it represents the tested cyclic load–displacement hysteresis well. It is expected that the proposed model can be successfully used to predict the behavior of HSD in real-world applications. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Many older low- to mid-rise buildings were not properly designed for resisting seismic loads. Since these buildings have insufficient strength and stiffness, as well as non-ductile detailing, they are vulnerable to serious damage or collapse during a strong seismic event [1]. Therefore, in order to use them safely, seismic strengthening strategies should be employed in existing buildings which lack good seismic performance. Many types of passive control systems have been studied over the past 40 years, and these are very promising for seismic retrofitting. Such systems do not require an external energy supply, but rather are activated by the movement of the main structure [2]. Therefore, they continue to function even during a power outage. A metallic damper is one type of passive damping device. It dissipates earthquake energy through the plastic deformation of metal. As a displacement-dependent damper, the metallic damper is less sensitive to changes in the external environment such as velocity, frequency, and temperature. Because it is fabricated and installed using conventional construction methods, a structure ⇑ Corresponding author. Tel.: +82 2 3290 3327; fax: +82 2 921 2439. E-mail address: [email protected] (Y.K. Ju). http://dx.doi.org/10.1016/j.engstruct.2016.04.019 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.
can be strengthened at a relatively low-cost. For seismic application of a metallic damper, the following characteristics are required according to the applied lateral loadings: (a) the damper should provide adequate stiffness to the structure under service loads, such as wind; (b) seismic energy dissipation by the damper, not by the main structural system, must be maximized by designing the damper to yield at a low load level. By equipping a structure with metallic dampers, stiffness is usually added as well as increasing damping [3]. Added stiffness is affected by damping systems which incorporate damping devices and connection elements such as bracings and wall-type columns. However, in terms of the device’s stiffness itself, metallic dampers which resist the applied load through in-plane behavior [4,5] are more efficient than those which resist load through outof-plane bending [3,6,7]. Among dampers which display in-plane behavior, a shear panel damper made of low-yield-point steel has excellent elongation capacity and considerable strain hardening [5,8]. On the other hand, a slit damper uses a normal strength steel and has slits in a direction perpendicular to the horizontal load so that it can be activated at relatively low load level [4,9]. Due to the unvarying width and thickness of strips in conventional slit dampers, stress is concentrated at the ends when subjected to external loads. In order to improve ductility and fatigue
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performance, recent studies attempting to optimize strip shapes have been carried out. In the study by Ghabraie et al. [10], strips were optimized with diamond-shaped holes derived through the application of a bi-directional evolutionary structural optimization algorithm, and performance was evaluated experimentally. Several types of non-uniform strip shapes (a dumbbell-shaped strip, a tapered strip, and an hourglass-shaped strip) were proposed by Woo et al. [11] and Lee et al. [12,13] for the purpose of simplifying the design and improving performance. The proposed dampers were tested under monotonic and cyclic loadings. They showed superior ductility and energy dissipation capacity compared to a conventional slit damper. This paper concentrates on a detailed investigation of the structural performance of an hourglass-shaped strip damper (HSD) proposed by the authors. With this purpose in mind, various conditions focusing on real-world applications of the damper were established as variables, specifically loading type, loading rate (v), material strength of the steel, and the number of damper plates used. In order to evaluate the effect of each variable, a total of eight full-scale specimens were tested. From the experimental results, the structural characteristics are compared and discussed in terms of failure mode, strength, stiffness, deformation and energy dissipation. In addition, a simple hysteretic model defined by the combined isotropic–kinematic hardening rule is also proposed for predicting the behavior of HSD. Finally, the validity of the proposed model is evaluated based on the test results.
The equations for the characteristics of non-uniform strip dampers are presented in Eqs. (1)–(3), which were derived from pure flexural behavior [12]. Plastic strength (Pp) denotes the horizontal force when the entire cross sections of the strip reach Mp, and Eq. (1) was formulated based on the cross section of the ends. 2
Pp ¼ 2F y t1 b1 =4h Ke ¼ c
dy ¼
24
R h=2 0
ð1Þ
E
ð2Þ
3
ðx2 =bx tx Þdx
2 1 2F y t 1 b1 24 c 6h
R h=2 0
3
ðx2 =bx t x Þdx E
ð3Þ
where Fy is the yield stress, t1 and b1 are the thickness and width of the strip at the ends, tx and bx are the thickness and width at location x, h is the height of the strip, c is the stiffness coefficient, and E is Young’s modulus. To calculate elastic stiffness (Ke) and yield displacement (dy) of the strip in Eqs. (2) and (3), semi-fixed end conditions were assumed and c was incorporated. The stiffness coefficient (c) is expressed as a ratio compared to the stiffness for fixed–fixed ends. The theoretical process for deriving these equations was explained by Lee et al. [12]. From the prior study, it was verified experimentally that the structural characteristics of the damper are predicted well by the equations. 3. Test program
2. Hourglass-shaped strip damper
3.1. General
The typical configuration and geometry of an hourglass-shaped strip damper (HSD) is shown in Fig. 1. The strips each have identical dimensions. After removing unnecessary volume to optimize the shape, the strips have an hourglass shape in which the crosssection decreases from the ends to the middle. The shape was designed so that the plastic bending moment (Mp) can be reached at all cross sections simultaneously, and the area of the central part of the strip was decided in order to safely resist shear force [12] (detailed dimensions are given in Section 3.2). The damper material is KS SS400 steel which has a relatively low nominal yield stress (Fyn = 235 MPa) and is widely used in Korea.
A series of experimental tests for HSD were performed previously, the results of which are discussed in Refs. [11,12], including the fundamental behavior under monotonic and cyclic loadings. The previous research showed that the shape of HSD was well designed to improve ductility and fatigue performance. Valuable insight was gained from these efforts, but a limited number of conditions was covered by those tests. Forcing velocity and the tensile property of the damper plate may vary in real-life situations, so the impact of these variables should be evaluated. Additionally, behavioral characteristics need to be investigated with the number of damper plates as a variable in order to extensively design and
Fig. 1. Configuration and geometry of HSD.
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apply HSD according to the performance required of the device [14]. Therefore, in order to study the behavior of HSD in greater detail, experimental tests for eight specimens were carried out. The objective of the test was to evaluate the performance of HSD under the various conditions discussed above (higher loading rate, different material strength of the damper plate, and extended application of damper plates placed in parallel) which may be encountered upon actual application. 3.2. Specimens The configuration of specimens is presented in Fig. 1. The width at the ends (b1) and the height (h) of the strip were 36 mm and 180 mm, respectively, making a strip with an aspect ratio of five. The thickness of the strip decreased linearly from 17 mm at the ends to 10 mm at the middle. As can be seen from the list of specimens in Table 1, two specimens were subjected to monotonic loadings. Cyclic behavior was then evaluated with six specimens. Higher loading rates (v) were applied to specimens C3 and C4, Table 1 List of specimens. Specimena
nb
Materialc
Loading
v (mm/s)
M1 M2 C1 C2 C3 C4 2-C1 2-C2
1 1 1 1 1 1 2 2
(1) (1) (1) (1) (1) (2) (1) (2)
Monotonic Monotonic Cyclic Cyclic Cyclic Cyclic Cyclic Cyclic
0.1 0.1 0.1–0.5 0.1–0.5 5.0 10.0 0.1–0.5 0.1–0.5
Note: a M1 is the identical sample as specimen ‘‘HSD” in Ref. [11], and C1 is the identical sample as specimens ‘‘HSD” and ‘‘B1” in Refs. [12] and [13], respectively. b n: the number of damper plates used. c Material: the mechanical properties of test pieces are presented in Table 2.
and specimens 2-C1 and 2-C2, unlike the others, consisted of two damper plates. To investigate the effect of material strength, specimens C4 and 2-C2 were fabricated using a different steel plate (though it was still SS400 steel). 3.3. Test setup All specimens were tested quasi-statically using a 500 kN hydraulic actuator with a stroke of ±200 mm. Fig. 2 shows the test setup and instrumentation apparatus. The testing frame was designed to allow the actual behavior of the damper to be assessed. Specifically, in order that strips be able to deform in a double curvature when subjected to horizontal loadings, rotation at the upper and lower ends of the specimen must be restrained. To this end, the upper loading block was connected to a strong floor by pinconnected fixing rods. Lateral support frames were also installed to restrict the loading to an in-plane direction. The horizontal load (P) resisted by the damper was obtained from a load cell connected to the actuator. Pushing from the actuator toward the specimen was defined as a positive direction, and displacement-controlled loading was applied to the specimens through the built-in displacement transducer. A set of linear variable differential transformers (LVDT) were used to measure displacements. The relative displacement of the damper (d) was obtained from the difference between the measurements in LVDT 2 and 3 which were installed at the top and bottom of the 180 mm-height strips. To observe the load–resistance characteristics, strains were also measured. All the tests were conducted at room temperature. In specimens M1 and M2, monotonically increasing loads (v = 0.1 mm/s) were applied in the positive direction. The tests continued until strips finally fractured. Other specimens were subjected to cyclic loadings where the displacement amplitude (a) increased gradually; the loading protocol is illustrated in Fig. 3 [15]. In cyclic loading tests of specimens C1, C2, 2-C1, and 2-C2,
Fig. 2. Test setup and instrumentation apparatus.
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Fig. 3. Cyclic loading protocol.
Table 2 Mechanical properties of coupons (KS SS400). Test piece
Yield stress (MPa)
Tensile stress (MPa)
Yield ratio
Elongation (%)
(1) (2)
308.5 265.0
446.5 431.0
0.69 0.61
29.5 30.7
a stepwise increasing v ranging from 0.1 to 0.5 mm/s was used in order to control the total loading time. Unlike these specimens, higher v fixed at 5 mm/s and 10 mm/s were applied to specimens C3 and C4, respectively. Each v corresponded to the strain rates of 0.01 s1 and 0.02 s1 in extreme fibers at strip ends. The values were chosen to investigate the impact of different v within a reasonable range where the impact of temperature increase would be negligible [16]. The number of cycles (N) was three in steps 1 and 2, and the remaining steps had two cycles for each amplitude. 3.4. Material properties Coupons of the KS SS400 steel [17] used in the specimens were prepared in accordance with KS D ISO 377 [18], and tensile tests were conducted at room temperature as specified in KS B 0802 [19]. Table 2 shows the mechanical properties. The results satisfied the requirements for strength and material properties. 4. Test results 4.1. Load–displacement relationships The load–displacement relationships for specimens are presented in Fig. 4. The curves of two specimens are overlapped, and the applied v and type of steel plate used are indicated on each graph. Specimens M1 and M2 (Fig. 4(a)) and C1 and C2 (Fig. 4(b)) each had identical conditions, and they showed similar hysteretic behavior. Monotonically-loaded specimens (MLS) displayed much higher maximum strength and larger displacement compared to cyclically-loaded specimens (CLS), whereas the load-bearing capacity of MLS was suddenly lost at a displacement of more than 80 mm. On the contrary, CLS represented gradual degradation of strength as damage accumulated. Even though different v and steel plates were applied in specimens C3 and C4, as seen in Fig. 4(c), the
hystereses showed no significant difference. Fig. 4(d) presents the hysteresis curves for specimens 2-C1 and 2-C2 where two damper plates were connected in parallel. During the loading of specimen 2-C2, there was a malfunction in the hydraulic system. As a result, an unexpected large displacement at step 15 was caused in the positive direction. Specimens 2-C1 and 2-C2 had different material strengths, though their envelope lines were quite similar. As described earlier, the test setup was designed so that rotation could be restrained at the upper and lower ends of specimens. This also prevented vertical displacements at strip ends. Because the damper plates will be connected to upper and lower support members as slip-critical high-strength bolted connections in real applications, the test setup used in this study faithfully reflected actual conditions. With this setup, the strips experience longitudinal elongation when subjected to horizontal displacement (i.e., the membrane effect) [3,12,20] which subsequently causes axial tension forces to occur as well as double curvature bending moments and constant shear forces in the strips. The contribution of the membrane effect becomes greater as the applied horizontal displacement increases, achieving higher load–resistance efficiency. Prior to failure, MLS were able to withstand more than twice the displacements of CLS (see Fig. 4). As a result, considerable membrane effect contributed to MLS exhibiting a significantly larger load-bearing capacity, as seen in Fig. 4(a). Essentially, flexural fatigue crack propagation was insignificant in MLS because they did not undergo repeated cyclic excursions, decreasing a loss of the cross sectional area which resists axial force. This allowed the strips to resist larger displacements, and in turn, caused an appreciable membrane effect in MLS. The distinct behavioral differences between MLS and CLS are further discussed later.
4.2. Failure modes Typical failure modes for MLS and CLS are shown in Fig. 5. There was a clearly observable difference based on the loading type. MLS displayed large maximum displacements (dmax), and fractures finally occurred at the middle of the strips after experiencing a considerable degree of plastic deformation (Fig. 5(a)). The fact that strips fractured at the middle (in which bending moment was negligible) suggested the failure mechanism of MLS was derived from a combination of shear and tension caused by the membrane
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Fig. 4. Load–displacement relationships.
Fig. 5. Failure modes.
Fig. 6. Crack patterns; arrows indicate major cracks.
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Fig. 7. Fracture surfaces.
effect. CLS only reached approximately half the dmax of MLS, but they tended to fail through a gradual propagation of flexural fatigue cracks. The strips finally fractured at the ends in a ductile manner (Fig. 5(b)). Fig. 6 shows the crack patterns of specimens. Regardless of loading type, flexural cracks were initiated at the strip ends. After that, major cracks occurred at the middle of strips (Fig. 6(a)), and the necking phenomenon was also displayed just before fracture in MLS. However, in CLS, damage was evenly distributed over the entire height of the strips (Fig. 6(b) and (c)) since the first cracks appeared at the ends. Failure characteristics can also be found from the fracture surfaces. As seen in Fig. 7(a), MLS experienced ductile fracture leaving a rough surface (cup- and cone-like appearance) even though they fractured so rapidly that crack propagation was not clearly observable. In CLS, progressive ductile fractures developed on the inside of the cross-section after cracks were initiated at the sides of strips. Following a sufficient reduction of the cross section, the middle part finally fractured abruptly (Fig. 7(b) and (c)). 4.3. Backbone curves The differences in strength, stiffness, and deformation capacity among specimens can be easily compared through backbone curves. Two types of backbone curves were plotted (i.e., a skeleton curve and a cyclic envelope curve), and Fig. 8 illustrates the concept of the backbone curves. A skeleton curve is obtained by successively connecting each of the skeleton parts which exceeds the load level attained at the preceding loading step [21,22]. Additionally, a cyclic envelope curve is defined by directly connecting the points of the maximum load (Pmax) at each loading step [23]. The two backbone curves are shown in Fig. 9; only curves for the positive direction were plotted for clarity. In order to make a fair comparison of the load resisted by one damper plate, the loads of specimens 2-C1 and 2-C2 were reduced by half in the graph. M0 curve represents the finite element (FE) analysis result
[24]. It predicted the behavior of specimens reasonably well for monotonically increasing loading. Up until a skeletal displacement of 20 mm, skeleton curves matched the curve of monotonic behavior accurately. Skeleton curves, however, began to diverge from monotonic behavior thereafter due to the membrane effect and fatigue crack propagation in CLS (Fig. 9(a)). As shown by the cyclic envelope curves (Fig. 9(b)), specimens exhibited cycle-dependent hardening in an approximate range of displacement less than 30 mm. The yield ratios of the applied steel plates were 0.69 for coupon (1) and 0.61 for coupon (2) (Table 2), and the strain hardening ratios (the ratios of yield stress to tensile stress) corresponded to 1.45 and 1.63, respectively. The hardening behavior of cyclic specimens was consistent with predictions made in prior research, specifically that a metal is expected to exhibit cyclic hardening if a strain hardening ratio is greater than 1.4 [16]. In Fig. 9, two kinds of plastic strengths calculated by Eq. (1) (i.e., Ppn and Ppe) are also expressed. The nominal plastic strength (Ppn) was calculated using a nominal yield stress (Fyn) of 235 MPa as the material yield stress. However, the overstrength factor in material yield stress (Ry = 1.3) [25] was applied in the calculation of the expected plastic strength (Ppe); this is mentioned again in Section 5. It can be seen on the backbone curves that Ppn was located at the upper limit of linear behavior and Ppe was in the transition region between the initial and post-yield behavior. Table 3 summarizes the experimental results for positive and negative directions. Structural characteristics were determined based on cyclic envelope curves, using the same rule as in Ref. [12]. Pmax denotes the maximum load resisted by the specimen, and dmax is defined as the maximum displacement at the loading step where Pmax was reached. dy,exp and Ke,exp represent the yield displacement and the elastic stiffness which were derived from the experiments. 4.4. Ultimate load and stiffness The bar graphs for Pmax and Ke,exp are represented in Figs. 10 and 11. To compare the performance for one damper plate, the load
Fig. 8. Construction of backbone curves.
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Fig. 9. Backbone curves.
Table 3 Structural responses. Specimen
M1 M2 C1 C2 C3 C4 2-C1 2-C2
Pmax (kN)
dmax (mm)
dy,exp (mm)
Ke,exp (kN/mm)
(+)
()
(+)
()
(+)
()
Mean
(+)
()
Mean
356.21 355.06 195.99 202.62 205.89 197.99 404.11 461.70
– – 191.18 200.84 205.51 191.79 392.17 397.31
81.32 82.20 31.74 32.20 32.85 33.39 35.38 52.82
– – 31.37 31.52 33.44 33.84 33.70 31.97
1.000 1.025 1.101 1.038 1.231 1.482 1.026 1.111
– – 1.053 1.046 1.277 1.448 1.057 1.135
1.000 1.025 1.077 1.042 1.254 1.465 1.042 1.123
75.49 73.67 72.95 75.14 69.79 58.43 144.44 128.17
– – 73.63 74.86 71.47 59.50 145.96 130.47
75.49 73.67 73.29 75.00 70.63 58.97 145.19 129.32
Fig. 10. Comparison of the maximum load (Pmax).
Fig. 11. Comparison of the elastic stiffness (Ke,exp).
and stiffness of specimens 2-C1 and 2-C2 were reduced by half from the values obtained in the tests (Table 3). When subjected to monotonic loadings (specimens M1 and M2), significantly higher Pmax was achieved, which was due primarily to an appreciable membrane effect resulting from negligible flexural crack growth (see Section 4.1). As a consequence of strain-hardening as well as the membrane effect, all specimens displayed load–resistance capacities larger than Ppe (more than 3 times in MLS and approximately 1.8 times in CLS). In CLS, specimen 2-C2 showed an unusually high load–resistance capacity; this was caused by an unintended large displacement, as stated before. Except for this specimen, Pmax was not significantly affected by the variables of v, material strength, nor the number of damper plates used among CLS, representing a variation range of less than 2.4% compared to the average value (200.91 kN).
The theoretical elastic stiffness (Ke,cal) was calculated as 106.97 kN/mm when both ends of the strips were fixed (c = 1), but the actual elastic stiffness (Ke,exp) was lower than Ke,cal (Fig. 11). This was attributed to the softening effect of end fixity [12]; the stiffness coefficient (c) ranged from 0.551 to 0.706. Showing almost constant values, Ke,exp was not affected by the loading type (M1 (or M2) vs. C1 (or C2)) nor the number of damper plates used (C1 (or C2) vs. 2-C1); the average value of c was 0.695 for these five specimens. However, unlike the prediction derived from the theory of statics (Eq. (2)), Ke,exp was associated with material strength of the damper plate. The stiffness of specimen 2-C2 decreased by 10.9% compared to specimen 2-C1, and it was observed that the lower yield stress (and/or tensile stress) of the damper plate caused the lower stiffness. On the other hand, the higher v also resulted in lower stiffness between specimens C1
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4.6. Energy dissipation
Fig. 12. Deformation capacity.
(or C2) and C3; i.e., Ke,exp of specimen C3 (v = 5 mm/s) was 4.7% lower than the average value for specimens C1 and C2 (v = 0.1– 0.5 mm/s). Since v may be higher during an earthquake than those experienced under the considered experimental conditions, dynamic effect needs to be adequately considered in the estimation of the initial stiffness of the damper. The lowest Ke,exp value was found in specimen C4 owing to the combined impact of both factors: lower material strength and higher v. The increased dy,exp due to the reduction of stiffness can be seen in Table 3. 4.5. Deformation capacity Fig. 12 shows bar plots of total cumulative peak displacement (dt) and total cumulative ductility (gt). Each of the values of dt and gt accumulates the peak displacement (dpeak) and ductility (l) for positive and negative directions of each cycle until the final failure. These represent the deformation capacity of the damper. CLS had a range of dt from 458 mm to 542 mm. They experienced five times larger dt even though their dmax values were much smaller than MLS (Table 3). Given that the number of cycles to failure (Nf) differed slightly and there was a malfunction during the test of specimen 2-C2, dt did not differ greatly among CLS. However, the difference became bigger in gt, and it showed a tendency that higher v decreases ductility capacity like the general characteristic of an engineering metal [26].
Fig. 13 plots the variation of dissipated energy with cumulative peak displacement (again, the values of specimens 2-C1 and 2-C2 were converted to the equivalent of one damper plate). CLS dissipated energy six times greater as compared to MLS. In the graph, the slope of the curves denotes the efficiency of energy dissipation. The portion of elastic strain energy was higher when the displacement was relatively small, thus it can be observed that the slopes increased gradually up to 100 mm (up until step 10 or 11). Thereafter, linear slopes were formed as a result of the fat hysteresis loop. The pattern of the slope showed great variance based on the loading type (M1 (or M2) vs. C1 (or C2)), but v (C1 (or C2) vs. C3), and the number of damper plates used (C1 (or C2) vs. 2-C1) had almost no effect on the slope. Lower material strength tended to decrease the slope slightly (C4 and 2-C2). Besides specimen 2-C2 where the extensive damage induced was more than intended, the amount of the total dissipated energy (Et) did not show a significant difference depending on variables under cyclic loadings; it was within a range of 8.8% deviation from the average value (117.2 kJ) in CLS. In order to further investigate the energy dissipation characteristics of specimens tested under cyclic loading, an extended energy analysis was carried out. Following the approach presented by Kato et al. [21], the cyclic load–displacement curves obtained from the tests were decomposed into a skeleton part and a Bauschinger part. The detailed procedure is not provided here; it can be found in references [21,27,28]. Based on the decomposed curves, the energy dissipated by each part and their sum (Eq. (4)) were computed and are presented in Table 4.
Wt ¼ Ws þ WB
ð4Þ
where Wt is the total plastic strain energy dissipated by the damper, and Ws and WB are the plastic strain energy dissipated by the skeleton part and the Bauschinger part, respectively. Ws and WB include values for both positive and negative directions. The loading of each step was repeated twice in cyclic tests, except for steps 1 and 2 which corresponded to the elastic regime. The values in Table 4 were obtained with consideration for this repetition effect. In the decomposition method, the unloading stiffness (Ku) of each Bauschinger segment was approximated using the initial elastic stiffness (Ke,exp). Nevertheless, the values of Wt computed on the basis of a number of split segments were almost iden-
Fig. 13. Energy dissipation.
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Table 4 Decomposition of energy contribution. Specimen
M1 M2 C1 C2 C3 C4 2-C1 2-C2
Skeleton part (kJ) Wþ s
W s
19.15 18.93 11.45 10.87 11.84 10.84 11.52 14.32
– – 11.98 10.99 11.81 10.53 11.00 10.80
Bauschinger part (kJ)
Total energy (kJ)
Ratio
Ws
Wþ B
W B
WB
Wt
Et
Wt/Et
19.15 18.93 23.44 21.85 23.65 21.38 22.52 25.12
– – 47.51 44.40 52.15 47.97 47.70 35.96
– – 47.58 42.64 52.10 46.83 47.13 38.64
– – 95.08 87.04 104.25 94.80 94.83 74.61
19.15 18.93 118.52 108.90 127.90 116.18 117.35 99.73
19.15 18.93 115.95 108.39 127.53 118.35 115.84 98.60
1.000 1.000 1.022 1.005 1.003 0.982 1.013 1.011
þ Note: The energy values of specimens 2-C1 and 2-C2 were reduced by half. The items that have a superscript (W þ s , W s , W B , and W B ) denote the values for the positive or negative direction.
tical to the actual Et obtained from the cyclic hysteresis curves, with differences of less than 2%. WB was found to be much bigger than Wt, and the average contribution of the Bauschinger part accounted for 80 percent of the total energy in CLS. This was probably because CLS displayed fully ductile behavior where the plastic plateau extended over a wider region and the load increased slowly thereafter, as can be seen in Fig. 4. In the case of specimen 2-C2, the contribution of WB was the lowest at 74.8%, affected by an unintended extensive damage at step 15. Excluding this specimen, Ws incorporating the repetition effect did not significantly differ between MLS and CLS, with a variation of 15.6% on average, even though the skeleton curves were clearly distinct from the load–displacement curves of MLS (Fig. 9). 4.7. Strain distributions Fig. 14 shows the strain distributions of MLS (specimen M2). The strains increased linearly with the increase of displacement at the ends of strips, as seen in Fig. 14(b). For both tensile and compressive strains, the same distributions were observed from the internal and external strips (SG1 vs. SG5 and SG3 vs. SG6) and also from the top and bottom ends (SG1 vs. SG4 and SG2 vs. SG3). At the same displacements, larger strains were measured in the tension zone (SG2, SG3, and SG6) than in the compression zone (SG1, SG4, and SG5). This indicates that the membrane effect affected strain distributions even at strip ends where bending strain was the most predominant. Strain variations at the mid-height of the internal and external strips (SG7 and SG8) are plotted in Fig. 14 (c). These strain gauges were placed where normal strains would not be caused by bending, and therefore the initial response
displayed almost pure flexural behavior. On the other hand, tensile strains began to rapidly increase at a displacement of 20 mm, resulting from the membrane effect. This was consistent with the point where monotonic behavior started to deviate from skeleton curves (see Section 4.3). The strain distributions of CLS (specimen 2-C1) are shown in Fig. 15. Strains were measured only on one internal strip. SG1 to SG4 and SG5 to SG8 were each installed on the front plate and the rear plate. From the strain distributions obtained, it can be found that the front and rear plates resisted the applied load equally. Even under cyclic loadings, the same strains were measured both at the top and bottom of strips, while strains began to change into asymmetrical distributions as cracks caused by damage occurred. From the above analysis, it was observed that the strains were induced identically in each strip regardless of the locations of not only strips (internal or external, top or bottom) but also the damper plates (front or rear); namely, the external load was uniformly distributed on each strip. Consequently, it is expected that the number of strips and the arrangement of damper plates can be applied in various ways according to the strength requirements for the damper. 5. Hysteretic model Based on the experimental results, a nonlinear hysteretic model is proposed in this section. For ease of computation, it is preferable that a hysteretic model be defined by simple rules. However, it is essential that the model be able to represent the realistic cyclic behavior, specifically: (a) variables that can affect the strength
Fig. 14. Strain distributions of a monotonically-loaded specimen (MLS).
C.-H. Lee et al. / Engineering Structures 119 (2016) 122–134
and stiffness should be properly taken into account; (b) the predicted hysteresis loop should display a similar shape to the actual test; and (c) the model must predict dissipated energy accurately. 5.1. Bilinear idealization In this study, the relationship between the load and displacement of HSD was approximated by a bilinear curve with an initial stiffness (Ki) and a positive post-yield stiffness (Kp) [23]. Additionally, the load value at the intersection point of the two lines was defined as the plastic strength of the damper (Pp). To calculate each of the stiffness values, a least square method was used. It was previously mentioned that the nominal plastic strength (Ppn) represented an approximate upper limit of initial behavior. Thus, the experimental data points with loads less than Ppn (86.29 kN) were used only in the calculation of Ki (Fig. 16(a)). Referring back to Eq. (1), it can be seen that Pp is proportional to the material yield stress. The specified minimum yield stress, i.e., nominal yield stress (Fyn), of KS SS400 steel is given as 235 MPa (16 mm < t 6 40 mm) in KS D 3503 [17], but material yield stress is generally larger than Fyn. The average yield stress of the two steel plates used in the specimens was also 1.22 times Fyn (Table 2). In practice, the material test results may not be provided, and therefore the performance of the damper can be more precisely predicted by considering an overstrength in material [29]. With
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this background, during the calculation process of the expected plastic strength (Ppe), Fy was replaced by the expected yield stress (RyFyn) in Eq. (1), where Ry denotes the ratio of the expected material yield stress to Fyn. Following Korean Building Code [25], Ry took the value of 1.3. The concept of bilinear approximation is illustrated in Fig. 16. It can be seen that the bilinear approximation curve represents the actual behavior of CLS closely. Table 5 summarizes the stiffness characteristics. In the table, the 1st set shows the result calculated using the method presented above. With a range of 55.53– 68.44 kN/mm, the ratios of Ki to Ke,cal (106.97 kN/mm) were between 0.52 and 0.64. In addition, HSD displayed significant stiffness in the post-yield region with a post-yield slope (a), which is the ratio of Kp to Ki, ranging from 0.0485 to 0.0574. On the other hand, the stiffness characteristics in the 2nd set were calculated by applying 0.5 as the value of c (Ki = 53.48 kN/mm). In the 2nd set, a had a range of 0.0538–0.0656. From the comparison of the 1st set and the 2nd set, it was found that a decrease of Ki led to an increase of Kp. In actual applications, various conditions may not be clearly defined, such as unknown material properties and dynamic effects due to high v. Therefore, in the bilinear idealization of HSD, Ki was simplified to 53.48 kN/mm, 50% of Ke,cal (c = 0.5) as in the 2nd set in Table 5. Using the average a value of 0.06, the corresponding Kp had a value of 3.21 kN/mm. The comparison between the tested
Fig. 15. Strain distributions of a cyclically-loaded specimen (CLS).
Fig. 16. Concept of bilinear approximation.
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Table 5 Summary of stiffness characteristics. Specimen
1st set
2nd set
Ki
Kp
a
Ki
Kp
a
C1 C2 C3 C4 2-C1 2-C2
66.23 67.95 68.44 55.53 63.65 55.72
3.22 3.44 3.39 3.19 3.09 2.87
0.0486 0.0506 0.0495 0.0574 0.0485 0.0515
53.48 53.48 53.48 53.48 53.48 53.48
3.29 3.51 3.46 3.20 3.14 2.88
0.0615 0.0656 0.0647 0.0598 0.0587 0.0538
Total (+) Total ()
62.37 61.62
3.15 3.22
0.0505 0.0523
53.48 53.48
3.19 3.26
0.0596 0.0610
Note: The stiffness values of specimens 2-C1 and 2-C2 were reduced by half. The unit for Ki and Kp is kN/mm.
Fig. 19. Variation of scaled secant stiffness.
Fig. 17. Comparison between tested and predicted loads.
Fig. 18. Illustration of combined hardening model.
load and the predicted load is presented in Fig. 17. It can be seen that the bilinear idealization successfully predicted the behavior of specimens with different variables, showing values of the normalized loads (the ratios of the predicted load to the tested load) close to 1. 5.2. Hysteretic model for cyclic behavior The cyclic envelope of specimens was represented well by the above bilinear idealization with one post-yield stiffness. However,
in cyclic hysteresis curves (Fig. 4), as the experienced displacement became larger, the load value where hysteresis met the vertical axis increased continuously while the post-yield stiffness decreased gradually. Therefore, in the hysteretic model for cyclic behavior, (a) kinematic hardening and isotropic hardening rules should be combined, and (b) the degradation of post-yield stiffness should be adequately considered. Fig. 18 shows an illustration of the combined hardening model proposed in this study. As in the bilinear idealization, the model has a slope of Ki (=0.5Ke,cal) at the first loading segment (Path 0– 1), and it has a reduced slope when Ppe is reached (Path 1–2). The unloading stiffness (Ku) and the reloading stiffness (Kr) are equal to Ki (Paths 2–3 and 4–5). Considering combined isotropic– kinematic hardening, load change (DP) from the unloading (or reloading) point to the reversed yielding is defined as the sum of Ppe and the peak load (Ppeak) reached previously. Consequently, the load values at yielding points are Ppe during loading, unloading, and reloading (Points 1, 3, and 5). The slope from a yield point to the next dpeak (i.e., the post-yield stiffness at the ith half cycle) is denoted by Kpi. In order to consider stiffness degradation in the post-yield region, this model uses the secant stiffness (Ksec) of the curve obtained by the bilinear idealization. Fig. 19 plots the variation of scaled secant stiffness, and Kpi takes the value of the scaled secant stiffness which corresponds to the largest absolute value of the experienced displacement prior to the ith half cycle. Based on the hysteresis curve obtained from the test, a scale ratio of 1:5 (Kpi = Ksec/5) was adopted. The comparison of load–displacement hysteresis curves for specimen C1 is presented in Fig. 20. The kinematic hardening model resulted in a different shape and area of the hysteresis loop. On the other hand, it can be seen that numerical simulation using the combined hardening model showed good agreement with the test result. Fig. 21 plots the variation of normalized energy with cumulative peak displacement, where normalized energy denotes the ratio of dissipated energy calculated by the combined hardening model to the tested value. After 100 mm of cumulative peak displacement, the normalized energy showed values close to 1. It ranged from 0.946 to 0.989 at the final stage. Because the model was defined simply to reach plastic strength in a linear relationship without a transition region, it underestimated the dissipated energy in the range less than 50 mm of cumulative peak displacement. In actuality, elastic behavior predominated there, so the dissipated energy was only less than 1 kJ different between the prediction and the test. Given this, it may be said that the dissipated energy of cyclic specimens throughout the entire history is well predicted by the proposed hysteretic model considering the combined hardening rule.
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Fig. 20. Comparison of hysteresis curves (specimen C1).
Fig. 21. Variation of normalized energy with cumulative peak displacement.
6. Conclusions In order to clarify the detailed behavior of the hourglass-shaped strip damper (HSD), experimental tests were carried out under various conditions which may be encountered in real-life applications. In addition, a hysteretic model defined by simple rules was proposed and also compared with the test results. The main findings of this study can be summarized as follows: (1) The monotonically-loaded specimens (MLS) and cyclicallyloaded specimens (CLS) showed noticeably different behavior. MLS displayed significantly larger maximum loads (Pmax) and displacements (dmax) which were greater by 1.8 and 2.3 times compared to CLS, respectively, and ductile fractures ultimately occurred at the middle of the strips after undergoing considerable plastic deformation. On the other hand, CLS represented superior deformation and energy dissipation capacities; each was 5 and 6 times greater compared to MLS. Furthermore, as damage accumulated, the strips fractured progressively at the ends in a ductile manner. (2) In CLS, Pmax and the total dissipated energy (Et) were within the ranges of 2.4% and 8.8% variation, respectively, compared
to the average values; principal structural characteristics such as Pmax and Et displayed no significant dependence upon the variables of loading rate (v), material strength, nor the number of damper plates used (n). However, lower material strength decreased the elastic stiffness of the damper by 10.9% in specimen 2-C2 and higher v (5 mm/s) decreased the stiffness by 4.7% in specimen C3. This led to a decrease in cumulative ductility even though specimens had similar cumulative displacement and dissipated energy. (3) The distribution of measured strains demonstrated that the external load was evenly distributed among all internal and external strips, irrespective of the number of damper plates used. This feature will allow for flexibility of composition; that is, the number of strips and the arrangement of damper plates can be extended in various ways without limitation according to the required performance when designing the damper. (4) The cyclic envelope of specimens was represented well by the idealized bilinear curve where a value of 0.5 was used as the stiffness coefficient (c) based on the experimental results. It was also found that HSD had significant postyield stiffness, with a range of 0.0538–0.0656 as the value of the post-yield slope (a). (5) The hysteretic model for cyclic behavior was proposed using the combined isotropic–kinematic hardening in order to consider the increase of strength and the degradation of post-yield stiffness. The shape of the hysteresis loop predicted by the model matched well with the test results, and the model predicted the dissipated energy well not only at the final stage but also throughout the entire loading history. It is expected that the proposed model can be practically used to predict the behavior of a building during an earthquake where HSD has been installed.
Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2013R1A1A2013578). The authors would also like to express their gratitude for the financial support of UNISON eTech Co., Ltd., Republic of Korea.
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[15] FEMA 461. Interim testing protocol for determining the seismic performance characteristics of structural and nonstructural components. Washington, DC (USA): Federal Emergency Management Agency; 2007. [16] Benson DK, Hancock JR. The effect of strain rate on the cyclic response of metals. Metall Trans 1974;5:1711–5. [17] Korean Agency for Technology and Standards (KATS). KS D 3503: rolled steels for general structure. Seoul (Korea): KATS; 2008. [18] Korean Agency for Technology and Standards (KATS). KS D ISO 377: steel and steel products – location and preparation of samples and test pieces for mechanical testing. Seoul (Korea): KATS; 2012. [19] Korean Agency for Technology and Standards (KATS). KS B 0802: method of tensile test for metallic materials. Seoul (Korea): KATS; 2008. [20] de la Llera JC, Esguerra C, Almazan JL. Earthquake behavior of structures with copper energy dissipaters. Earthq Eng Struct Dyn 2004;33:329–58. [21] Kato B, Akiyama H, Yamanouchi H. Predictable properties of material under incremental cyclic loading. In: IABSE symposium on resistance and ultimate deformability of structures acted on by well defined repeated loads, Lisbon, Portugal; 1973. p. 119–24. [22] Kim YJ, Kim MH, Jung IY, Ju YK, Kim SD. Experimental investigation of the cyclic behavior of nodes in diagrid structures. Eng Struct 2011;33:2134–44. [23] FEMA P440A. Effects of strength and stiffness degradation on seismic response. Washington, DC (USA): Federal Emergency Management Agency; 2009. [24] Hibbitt H, Karlsson B, Sorensen P. Abaqus analysis user’s manual. Version 6.10. Providence, RI (USA): Dassault Systemes Simulia Corp; 2010. [25] Architectural Institute of Korea (AIK). Korean Building Code (KBC). Seoul (Korea): AIK; 2009. [26] Dowling NE. Mechanical behavior of material: engineering methods for deformation, fracture, and fatigue. 3rd ed. Upper Saddle River, NJ (USA): Prentice-Hall; 2007. [27] Benavent-Climent A. An energy-based damage model for seismic response of steel structures. Earthq Eng Struct Dyn 2007;36:1049–64. [28] Kim YJ, Oh SH, Moon TS. Seismic behavior and retrofit of steel moment connections considering slab effects. Eng Struct 2004;26:1993–2005. [29] Lee CH, Kim J, Kim DH, Ryu J, Ju YK. Numerical and experimental analysis of combined behavior of shear-type friction damper and non-uniform strip damper for multi-level seismic protection. Eng Struct 2016;114:75–92.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Hourglass-shaped strip damper subjected to monotonic and cyclic loadings Chang-Hwan Lee a, Seung-Hee Lho b, Do-Hyun Kim c, Jintak Oh d, Young K. Ju d,⇑ a
Research Institute of Structural Engineering & System, DongYang Structural Engineers Co., Ltd., Seoul 05836, Republic of Korea Urban Architecture Division, Korea Agency for Infrastructure Technology Advancement, Anyang-si 14066, Republic of Korea c Department of Architecture and Interior Design, Gyeonggi College of Science and Technology, Siheung-si, Gyeonggi 15073, Republic of Korea d School of Civil, Environmental and Architectural Engineering, Korea University, Seoul 02841, Republic of Korea b
a r t i c l e
i n f o
Article history: Received 14 January 2015 Revised 8 April 2016 Accepted 11 April 2016
Keywords: Metallic damper Passive damping device Seismic retrofitting Energy dissipation Backbone curve Membrane effect Hysteretic model Combined isotropic–kinematic hardening
a b s t r a c t An hourglass-shaped strip damper (HSD) was proposed to improve on the conventional slit damper. The damper has non-uniform strips which have a smaller cross-sectional area close to the middle height. To find the structural capacities of HSD subjected to monotonic and cyclic loadings, experimental tests were carried out in this study. Test parameters were loading rate, material strength, and the number of damper plates. The results showed substantial load–resistance capacity under monotonic loadings, and excellent ductility and energy dissipation were exhibited under cyclic loadings, with even distribution of damage over the entire height of strips. Based on the test results, a simple hysteretic model using a combined isotropic–kinematic hardening rule was also proposed. The comparison demonstrated that it represents the tested cyclic load–displacement hysteresis well. It is expected that the proposed model can be successfully used to predict the behavior of HSD in real-world applications. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Many older low- to mid-rise buildings were not properly designed for resisting seismic loads. Since these buildings have insufficient strength and stiffness, as well as non-ductile detailing, they are vulnerable to serious damage or collapse during a strong seismic event [1]. Therefore, in order to use them safely, seismic strengthening strategies should be employed in existing buildings which lack good seismic performance. Many types of passive control systems have been studied over the past 40 years, and these are very promising for seismic retrofitting. Such systems do not require an external energy supply, but rather are activated by the movement of the main structure [2]. Therefore, they continue to function even during a power outage. A metallic damper is one type of passive damping device. It dissipates earthquake energy through the plastic deformation of metal. As a displacement-dependent damper, the metallic damper is less sensitive to changes in the external environment such as velocity, frequency, and temperature. Because it is fabricated and installed using conventional construction methods, a structure ⇑ Corresponding author. Tel.: +82 2 3290 3327; fax: +82 2 921 2439. E-mail address: [email protected] (Y.K. Ju). http://dx.doi.org/10.1016/j.engstruct.2016.04.019 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.
can be strengthened at a relatively low-cost. For seismic application of a metallic damper, the following characteristics are required according to the applied lateral loadings: (a) the damper should provide adequate stiffness to the structure under service loads, such as wind; (b) seismic energy dissipation by the damper, not by the main structural system, must be maximized by designing the damper to yield at a low load level. By equipping a structure with metallic dampers, stiffness is usually added as well as increasing damping [3]. Added stiffness is affected by damping systems which incorporate damping devices and connection elements such as bracings and wall-type columns. However, in terms of the device’s stiffness itself, metallic dampers which resist the applied load through in-plane behavior [4,5] are more efficient than those which resist load through outof-plane bending [3,6,7]. Among dampers which display in-plane behavior, a shear panel damper made of low-yield-point steel has excellent elongation capacity and considerable strain hardening [5,8]. On the other hand, a slit damper uses a normal strength steel and has slits in a direction perpendicular to the horizontal load so that it can be activated at relatively low load level [4,9]. Due to the unvarying width and thickness of strips in conventional slit dampers, stress is concentrated at the ends when subjected to external loads. In order to improve ductility and fatigue
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performance, recent studies attempting to optimize strip shapes have been carried out. In the study by Ghabraie et al. [10], strips were optimized with diamond-shaped holes derived through the application of a bi-directional evolutionary structural optimization algorithm, and performance was evaluated experimentally. Several types of non-uniform strip shapes (a dumbbell-shaped strip, a tapered strip, and an hourglass-shaped strip) were proposed by Woo et al. [11] and Lee et al. [12,13] for the purpose of simplifying the design and improving performance. The proposed dampers were tested under monotonic and cyclic loadings. They showed superior ductility and energy dissipation capacity compared to a conventional slit damper. This paper concentrates on a detailed investigation of the structural performance of an hourglass-shaped strip damper (HSD) proposed by the authors. With this purpose in mind, various conditions focusing on real-world applications of the damper were established as variables, specifically loading type, loading rate (v), material strength of the steel, and the number of damper plates used. In order to evaluate the effect of each variable, a total of eight full-scale specimens were tested. From the experimental results, the structural characteristics are compared and discussed in terms of failure mode, strength, stiffness, deformation and energy dissipation. In addition, a simple hysteretic model defined by the combined isotropic–kinematic hardening rule is also proposed for predicting the behavior of HSD. Finally, the validity of the proposed model is evaluated based on the test results.
The equations for the characteristics of non-uniform strip dampers are presented in Eqs. (1)–(3), which were derived from pure flexural behavior [12]. Plastic strength (Pp) denotes the horizontal force when the entire cross sections of the strip reach Mp, and Eq. (1) was formulated based on the cross section of the ends. 2
Pp ¼ 2F y t1 b1 =4h Ke ¼ c
dy ¼
24
R h=2 0
ð1Þ
E
ð2Þ
3
ðx2 =bx tx Þdx
2 1 2F y t 1 b1 24 c 6h
R h=2 0
3
ðx2 =bx t x Þdx E
ð3Þ
where Fy is the yield stress, t1 and b1 are the thickness and width of the strip at the ends, tx and bx are the thickness and width at location x, h is the height of the strip, c is the stiffness coefficient, and E is Young’s modulus. To calculate elastic stiffness (Ke) and yield displacement (dy) of the strip in Eqs. (2) and (3), semi-fixed end conditions were assumed and c was incorporated. The stiffness coefficient (c) is expressed as a ratio compared to the stiffness for fixed–fixed ends. The theoretical process for deriving these equations was explained by Lee et al. [12]. From the prior study, it was verified experimentally that the structural characteristics of the damper are predicted well by the equations. 3. Test program
2. Hourglass-shaped strip damper
3.1. General
The typical configuration and geometry of an hourglass-shaped strip damper (HSD) is shown in Fig. 1. The strips each have identical dimensions. After removing unnecessary volume to optimize the shape, the strips have an hourglass shape in which the crosssection decreases from the ends to the middle. The shape was designed so that the plastic bending moment (Mp) can be reached at all cross sections simultaneously, and the area of the central part of the strip was decided in order to safely resist shear force [12] (detailed dimensions are given in Section 3.2). The damper material is KS SS400 steel which has a relatively low nominal yield stress (Fyn = 235 MPa) and is widely used in Korea.
A series of experimental tests for HSD were performed previously, the results of which are discussed in Refs. [11,12], including the fundamental behavior under monotonic and cyclic loadings. The previous research showed that the shape of HSD was well designed to improve ductility and fatigue performance. Valuable insight was gained from these efforts, but a limited number of conditions was covered by those tests. Forcing velocity and the tensile property of the damper plate may vary in real-life situations, so the impact of these variables should be evaluated. Additionally, behavioral characteristics need to be investigated with the number of damper plates as a variable in order to extensively design and
Fig. 1. Configuration and geometry of HSD.
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apply HSD according to the performance required of the device [14]. Therefore, in order to study the behavior of HSD in greater detail, experimental tests for eight specimens were carried out. The objective of the test was to evaluate the performance of HSD under the various conditions discussed above (higher loading rate, different material strength of the damper plate, and extended application of damper plates placed in parallel) which may be encountered upon actual application. 3.2. Specimens The configuration of specimens is presented in Fig. 1. The width at the ends (b1) and the height (h) of the strip were 36 mm and 180 mm, respectively, making a strip with an aspect ratio of five. The thickness of the strip decreased linearly from 17 mm at the ends to 10 mm at the middle. As can be seen from the list of specimens in Table 1, two specimens were subjected to monotonic loadings. Cyclic behavior was then evaluated with six specimens. Higher loading rates (v) were applied to specimens C3 and C4, Table 1 List of specimens. Specimena
nb
Materialc
Loading
v (mm/s)
M1 M2 C1 C2 C3 C4 2-C1 2-C2
1 1 1 1 1 1 2 2
(1) (1) (1) (1) (1) (2) (1) (2)
Monotonic Monotonic Cyclic Cyclic Cyclic Cyclic Cyclic Cyclic
0.1 0.1 0.1–0.5 0.1–0.5 5.0 10.0 0.1–0.5 0.1–0.5
Note: a M1 is the identical sample as specimen ‘‘HSD” in Ref. [11], and C1 is the identical sample as specimens ‘‘HSD” and ‘‘B1” in Refs. [12] and [13], respectively. b n: the number of damper plates used. c Material: the mechanical properties of test pieces are presented in Table 2.
and specimens 2-C1 and 2-C2, unlike the others, consisted of two damper plates. To investigate the effect of material strength, specimens C4 and 2-C2 were fabricated using a different steel plate (though it was still SS400 steel). 3.3. Test setup All specimens were tested quasi-statically using a 500 kN hydraulic actuator with a stroke of ±200 mm. Fig. 2 shows the test setup and instrumentation apparatus. The testing frame was designed to allow the actual behavior of the damper to be assessed. Specifically, in order that strips be able to deform in a double curvature when subjected to horizontal loadings, rotation at the upper and lower ends of the specimen must be restrained. To this end, the upper loading block was connected to a strong floor by pinconnected fixing rods. Lateral support frames were also installed to restrict the loading to an in-plane direction. The horizontal load (P) resisted by the damper was obtained from a load cell connected to the actuator. Pushing from the actuator toward the specimen was defined as a positive direction, and displacement-controlled loading was applied to the specimens through the built-in displacement transducer. A set of linear variable differential transformers (LVDT) were used to measure displacements. The relative displacement of the damper (d) was obtained from the difference between the measurements in LVDT 2 and 3 which were installed at the top and bottom of the 180 mm-height strips. To observe the load–resistance characteristics, strains were also measured. All the tests were conducted at room temperature. In specimens M1 and M2, monotonically increasing loads (v = 0.1 mm/s) were applied in the positive direction. The tests continued until strips finally fractured. Other specimens were subjected to cyclic loadings where the displacement amplitude (a) increased gradually; the loading protocol is illustrated in Fig. 3 [15]. In cyclic loading tests of specimens C1, C2, 2-C1, and 2-C2,
Fig. 2. Test setup and instrumentation apparatus.
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Fig. 3. Cyclic loading protocol.
Table 2 Mechanical properties of coupons (KS SS400). Test piece
Yield stress (MPa)
Tensile stress (MPa)
Yield ratio
Elongation (%)
(1) (2)
308.5 265.0
446.5 431.0
0.69 0.61
29.5 30.7
a stepwise increasing v ranging from 0.1 to 0.5 mm/s was used in order to control the total loading time. Unlike these specimens, higher v fixed at 5 mm/s and 10 mm/s were applied to specimens C3 and C4, respectively. Each v corresponded to the strain rates of 0.01 s1 and 0.02 s1 in extreme fibers at strip ends. The values were chosen to investigate the impact of different v within a reasonable range where the impact of temperature increase would be negligible [16]. The number of cycles (N) was three in steps 1 and 2, and the remaining steps had two cycles for each amplitude. 3.4. Material properties Coupons of the KS SS400 steel [17] used in the specimens were prepared in accordance with KS D ISO 377 [18], and tensile tests were conducted at room temperature as specified in KS B 0802 [19]. Table 2 shows the mechanical properties. The results satisfied the requirements for strength and material properties. 4. Test results 4.1. Load–displacement relationships The load–displacement relationships for specimens are presented in Fig. 4. The curves of two specimens are overlapped, and the applied v and type of steel plate used are indicated on each graph. Specimens M1 and M2 (Fig. 4(a)) and C1 and C2 (Fig. 4(b)) each had identical conditions, and they showed similar hysteretic behavior. Monotonically-loaded specimens (MLS) displayed much higher maximum strength and larger displacement compared to cyclically-loaded specimens (CLS), whereas the load-bearing capacity of MLS was suddenly lost at a displacement of more than 80 mm. On the contrary, CLS represented gradual degradation of strength as damage accumulated. Even though different v and steel plates were applied in specimens C3 and C4, as seen in Fig. 4(c), the
hystereses showed no significant difference. Fig. 4(d) presents the hysteresis curves for specimens 2-C1 and 2-C2 where two damper plates were connected in parallel. During the loading of specimen 2-C2, there was a malfunction in the hydraulic system. As a result, an unexpected large displacement at step 15 was caused in the positive direction. Specimens 2-C1 and 2-C2 had different material strengths, though their envelope lines were quite similar. As described earlier, the test setup was designed so that rotation could be restrained at the upper and lower ends of specimens. This also prevented vertical displacements at strip ends. Because the damper plates will be connected to upper and lower support members as slip-critical high-strength bolted connections in real applications, the test setup used in this study faithfully reflected actual conditions. With this setup, the strips experience longitudinal elongation when subjected to horizontal displacement (i.e., the membrane effect) [3,12,20] which subsequently causes axial tension forces to occur as well as double curvature bending moments and constant shear forces in the strips. The contribution of the membrane effect becomes greater as the applied horizontal displacement increases, achieving higher load–resistance efficiency. Prior to failure, MLS were able to withstand more than twice the displacements of CLS (see Fig. 4). As a result, considerable membrane effect contributed to MLS exhibiting a significantly larger load-bearing capacity, as seen in Fig. 4(a). Essentially, flexural fatigue crack propagation was insignificant in MLS because they did not undergo repeated cyclic excursions, decreasing a loss of the cross sectional area which resists axial force. This allowed the strips to resist larger displacements, and in turn, caused an appreciable membrane effect in MLS. The distinct behavioral differences between MLS and CLS are further discussed later.
4.2. Failure modes Typical failure modes for MLS and CLS are shown in Fig. 5. There was a clearly observable difference based on the loading type. MLS displayed large maximum displacements (dmax), and fractures finally occurred at the middle of the strips after experiencing a considerable degree of plastic deformation (Fig. 5(a)). The fact that strips fractured at the middle (in which bending moment was negligible) suggested the failure mechanism of MLS was derived from a combination of shear and tension caused by the membrane
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Fig. 4. Load–displacement relationships.
Fig. 5. Failure modes.
Fig. 6. Crack patterns; arrows indicate major cracks.
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Fig. 7. Fracture surfaces.
effect. CLS only reached approximately half the dmax of MLS, but they tended to fail through a gradual propagation of flexural fatigue cracks. The strips finally fractured at the ends in a ductile manner (Fig. 5(b)). Fig. 6 shows the crack patterns of specimens. Regardless of loading type, flexural cracks were initiated at the strip ends. After that, major cracks occurred at the middle of strips (Fig. 6(a)), and the necking phenomenon was also displayed just before fracture in MLS. However, in CLS, damage was evenly distributed over the entire height of the strips (Fig. 6(b) and (c)) since the first cracks appeared at the ends. Failure characteristics can also be found from the fracture surfaces. As seen in Fig. 7(a), MLS experienced ductile fracture leaving a rough surface (cup- and cone-like appearance) even though they fractured so rapidly that crack propagation was not clearly observable. In CLS, progressive ductile fractures developed on the inside of the cross-section after cracks were initiated at the sides of strips. Following a sufficient reduction of the cross section, the middle part finally fractured abruptly (Fig. 7(b) and (c)). 4.3. Backbone curves The differences in strength, stiffness, and deformation capacity among specimens can be easily compared through backbone curves. Two types of backbone curves were plotted (i.e., a skeleton curve and a cyclic envelope curve), and Fig. 8 illustrates the concept of the backbone curves. A skeleton curve is obtained by successively connecting each of the skeleton parts which exceeds the load level attained at the preceding loading step [21,22]. Additionally, a cyclic envelope curve is defined by directly connecting the points of the maximum load (Pmax) at each loading step [23]. The two backbone curves are shown in Fig. 9; only curves for the positive direction were plotted for clarity. In order to make a fair comparison of the load resisted by one damper plate, the loads of specimens 2-C1 and 2-C2 were reduced by half in the graph. M0 curve represents the finite element (FE) analysis result
[24]. It predicted the behavior of specimens reasonably well for monotonically increasing loading. Up until a skeletal displacement of 20 mm, skeleton curves matched the curve of monotonic behavior accurately. Skeleton curves, however, began to diverge from monotonic behavior thereafter due to the membrane effect and fatigue crack propagation in CLS (Fig. 9(a)). As shown by the cyclic envelope curves (Fig. 9(b)), specimens exhibited cycle-dependent hardening in an approximate range of displacement less than 30 mm. The yield ratios of the applied steel plates were 0.69 for coupon (1) and 0.61 for coupon (2) (Table 2), and the strain hardening ratios (the ratios of yield stress to tensile stress) corresponded to 1.45 and 1.63, respectively. The hardening behavior of cyclic specimens was consistent with predictions made in prior research, specifically that a metal is expected to exhibit cyclic hardening if a strain hardening ratio is greater than 1.4 [16]. In Fig. 9, two kinds of plastic strengths calculated by Eq. (1) (i.e., Ppn and Ppe) are also expressed. The nominal plastic strength (Ppn) was calculated using a nominal yield stress (Fyn) of 235 MPa as the material yield stress. However, the overstrength factor in material yield stress (Ry = 1.3) [25] was applied in the calculation of the expected plastic strength (Ppe); this is mentioned again in Section 5. It can be seen on the backbone curves that Ppn was located at the upper limit of linear behavior and Ppe was in the transition region between the initial and post-yield behavior. Table 3 summarizes the experimental results for positive and negative directions. Structural characteristics were determined based on cyclic envelope curves, using the same rule as in Ref. [12]. Pmax denotes the maximum load resisted by the specimen, and dmax is defined as the maximum displacement at the loading step where Pmax was reached. dy,exp and Ke,exp represent the yield displacement and the elastic stiffness which were derived from the experiments. 4.4. Ultimate load and stiffness The bar graphs for Pmax and Ke,exp are represented in Figs. 10 and 11. To compare the performance for one damper plate, the load
Fig. 8. Construction of backbone curves.
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Fig. 9. Backbone curves.
Table 3 Structural responses. Specimen
M1 M2 C1 C2 C3 C4 2-C1 2-C2
Pmax (kN)
dmax (mm)
dy,exp (mm)
Ke,exp (kN/mm)
(+)
()
(+)
()
(+)
()
Mean
(+)
()
Mean
356.21 355.06 195.99 202.62 205.89 197.99 404.11 461.70
– – 191.18 200.84 205.51 191.79 392.17 397.31
81.32 82.20 31.74 32.20 32.85 33.39 35.38 52.82
– – 31.37 31.52 33.44 33.84 33.70 31.97
1.000 1.025 1.101 1.038 1.231 1.482 1.026 1.111
– – 1.053 1.046 1.277 1.448 1.057 1.135
1.000 1.025 1.077 1.042 1.254 1.465 1.042 1.123
75.49 73.67 72.95 75.14 69.79 58.43 144.44 128.17
– – 73.63 74.86 71.47 59.50 145.96 130.47
75.49 73.67 73.29 75.00 70.63 58.97 145.19 129.32
Fig. 10. Comparison of the maximum load (Pmax).
Fig. 11. Comparison of the elastic stiffness (Ke,exp).
and stiffness of specimens 2-C1 and 2-C2 were reduced by half from the values obtained in the tests (Table 3). When subjected to monotonic loadings (specimens M1 and M2), significantly higher Pmax was achieved, which was due primarily to an appreciable membrane effect resulting from negligible flexural crack growth (see Section 4.1). As a consequence of strain-hardening as well as the membrane effect, all specimens displayed load–resistance capacities larger than Ppe (more than 3 times in MLS and approximately 1.8 times in CLS). In CLS, specimen 2-C2 showed an unusually high load–resistance capacity; this was caused by an unintended large displacement, as stated before. Except for this specimen, Pmax was not significantly affected by the variables of v, material strength, nor the number of damper plates used among CLS, representing a variation range of less than 2.4% compared to the average value (200.91 kN).
The theoretical elastic stiffness (Ke,cal) was calculated as 106.97 kN/mm when both ends of the strips were fixed (c = 1), but the actual elastic stiffness (Ke,exp) was lower than Ke,cal (Fig. 11). This was attributed to the softening effect of end fixity [12]; the stiffness coefficient (c) ranged from 0.551 to 0.706. Showing almost constant values, Ke,exp was not affected by the loading type (M1 (or M2) vs. C1 (or C2)) nor the number of damper plates used (C1 (or C2) vs. 2-C1); the average value of c was 0.695 for these five specimens. However, unlike the prediction derived from the theory of statics (Eq. (2)), Ke,exp was associated with material strength of the damper plate. The stiffness of specimen 2-C2 decreased by 10.9% compared to specimen 2-C1, and it was observed that the lower yield stress (and/or tensile stress) of the damper plate caused the lower stiffness. On the other hand, the higher v also resulted in lower stiffness between specimens C1
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4.6. Energy dissipation
Fig. 12. Deformation capacity.
(or C2) and C3; i.e., Ke,exp of specimen C3 (v = 5 mm/s) was 4.7% lower than the average value for specimens C1 and C2 (v = 0.1– 0.5 mm/s). Since v may be higher during an earthquake than those experienced under the considered experimental conditions, dynamic effect needs to be adequately considered in the estimation of the initial stiffness of the damper. The lowest Ke,exp value was found in specimen C4 owing to the combined impact of both factors: lower material strength and higher v. The increased dy,exp due to the reduction of stiffness can be seen in Table 3. 4.5. Deformation capacity Fig. 12 shows bar plots of total cumulative peak displacement (dt) and total cumulative ductility (gt). Each of the values of dt and gt accumulates the peak displacement (dpeak) and ductility (l) for positive and negative directions of each cycle until the final failure. These represent the deformation capacity of the damper. CLS had a range of dt from 458 mm to 542 mm. They experienced five times larger dt even though their dmax values were much smaller than MLS (Table 3). Given that the number of cycles to failure (Nf) differed slightly and there was a malfunction during the test of specimen 2-C2, dt did not differ greatly among CLS. However, the difference became bigger in gt, and it showed a tendency that higher v decreases ductility capacity like the general characteristic of an engineering metal [26].
Fig. 13 plots the variation of dissipated energy with cumulative peak displacement (again, the values of specimens 2-C1 and 2-C2 were converted to the equivalent of one damper plate). CLS dissipated energy six times greater as compared to MLS. In the graph, the slope of the curves denotes the efficiency of energy dissipation. The portion of elastic strain energy was higher when the displacement was relatively small, thus it can be observed that the slopes increased gradually up to 100 mm (up until step 10 or 11). Thereafter, linear slopes were formed as a result of the fat hysteresis loop. The pattern of the slope showed great variance based on the loading type (M1 (or M2) vs. C1 (or C2)), but v (C1 (or C2) vs. C3), and the number of damper plates used (C1 (or C2) vs. 2-C1) had almost no effect on the slope. Lower material strength tended to decrease the slope slightly (C4 and 2-C2). Besides specimen 2-C2 where the extensive damage induced was more than intended, the amount of the total dissipated energy (Et) did not show a significant difference depending on variables under cyclic loadings; it was within a range of 8.8% deviation from the average value (117.2 kJ) in CLS. In order to further investigate the energy dissipation characteristics of specimens tested under cyclic loading, an extended energy analysis was carried out. Following the approach presented by Kato et al. [21], the cyclic load–displacement curves obtained from the tests were decomposed into a skeleton part and a Bauschinger part. The detailed procedure is not provided here; it can be found in references [21,27,28]. Based on the decomposed curves, the energy dissipated by each part and their sum (Eq. (4)) were computed and are presented in Table 4.
Wt ¼ Ws þ WB
ð4Þ
where Wt is the total plastic strain energy dissipated by the damper, and Ws and WB are the plastic strain energy dissipated by the skeleton part and the Bauschinger part, respectively. Ws and WB include values for both positive and negative directions. The loading of each step was repeated twice in cyclic tests, except for steps 1 and 2 which corresponded to the elastic regime. The values in Table 4 were obtained with consideration for this repetition effect. In the decomposition method, the unloading stiffness (Ku) of each Bauschinger segment was approximated using the initial elastic stiffness (Ke,exp). Nevertheless, the values of Wt computed on the basis of a number of split segments were almost iden-
Fig. 13. Energy dissipation.
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Table 4 Decomposition of energy contribution. Specimen
M1 M2 C1 C2 C3 C4 2-C1 2-C2
Skeleton part (kJ) Wþ s
W s
19.15 18.93 11.45 10.87 11.84 10.84 11.52 14.32
– – 11.98 10.99 11.81 10.53 11.00 10.80
Bauschinger part (kJ)
Total energy (kJ)
Ratio
Ws
Wþ B
W B
WB
Wt
Et
Wt/Et
19.15 18.93 23.44 21.85 23.65 21.38 22.52 25.12
– – 47.51 44.40 52.15 47.97 47.70 35.96
– – 47.58 42.64 52.10 46.83 47.13 38.64
– – 95.08 87.04 104.25 94.80 94.83 74.61
19.15 18.93 118.52 108.90 127.90 116.18 117.35 99.73
19.15 18.93 115.95 108.39 127.53 118.35 115.84 98.60
1.000 1.000 1.022 1.005 1.003 0.982 1.013 1.011
þ Note: The energy values of specimens 2-C1 and 2-C2 were reduced by half. The items that have a superscript (W þ s , W s , W B , and W B ) denote the values for the positive or negative direction.
tical to the actual Et obtained from the cyclic hysteresis curves, with differences of less than 2%. WB was found to be much bigger than Wt, and the average contribution of the Bauschinger part accounted for 80 percent of the total energy in CLS. This was probably because CLS displayed fully ductile behavior where the plastic plateau extended over a wider region and the load increased slowly thereafter, as can be seen in Fig. 4. In the case of specimen 2-C2, the contribution of WB was the lowest at 74.8%, affected by an unintended extensive damage at step 15. Excluding this specimen, Ws incorporating the repetition effect did not significantly differ between MLS and CLS, with a variation of 15.6% on average, even though the skeleton curves were clearly distinct from the load–displacement curves of MLS (Fig. 9). 4.7. Strain distributions Fig. 14 shows the strain distributions of MLS (specimen M2). The strains increased linearly with the increase of displacement at the ends of strips, as seen in Fig. 14(b). For both tensile and compressive strains, the same distributions were observed from the internal and external strips (SG1 vs. SG5 and SG3 vs. SG6) and also from the top and bottom ends (SG1 vs. SG4 and SG2 vs. SG3). At the same displacements, larger strains were measured in the tension zone (SG2, SG3, and SG6) than in the compression zone (SG1, SG4, and SG5). This indicates that the membrane effect affected strain distributions even at strip ends where bending strain was the most predominant. Strain variations at the mid-height of the internal and external strips (SG7 and SG8) are plotted in Fig. 14 (c). These strain gauges were placed where normal strains would not be caused by bending, and therefore the initial response
displayed almost pure flexural behavior. On the other hand, tensile strains began to rapidly increase at a displacement of 20 mm, resulting from the membrane effect. This was consistent with the point where monotonic behavior started to deviate from skeleton curves (see Section 4.3). The strain distributions of CLS (specimen 2-C1) are shown in Fig. 15. Strains were measured only on one internal strip. SG1 to SG4 and SG5 to SG8 were each installed on the front plate and the rear plate. From the strain distributions obtained, it can be found that the front and rear plates resisted the applied load equally. Even under cyclic loadings, the same strains were measured both at the top and bottom of strips, while strains began to change into asymmetrical distributions as cracks caused by damage occurred. From the above analysis, it was observed that the strains were induced identically in each strip regardless of the locations of not only strips (internal or external, top or bottom) but also the damper plates (front or rear); namely, the external load was uniformly distributed on each strip. Consequently, it is expected that the number of strips and the arrangement of damper plates can be applied in various ways according to the strength requirements for the damper. 5. Hysteretic model Based on the experimental results, a nonlinear hysteretic model is proposed in this section. For ease of computation, it is preferable that a hysteretic model be defined by simple rules. However, it is essential that the model be able to represent the realistic cyclic behavior, specifically: (a) variables that can affect the strength
Fig. 14. Strain distributions of a monotonically-loaded specimen (MLS).
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and stiffness should be properly taken into account; (b) the predicted hysteresis loop should display a similar shape to the actual test; and (c) the model must predict dissipated energy accurately. 5.1. Bilinear idealization In this study, the relationship between the load and displacement of HSD was approximated by a bilinear curve with an initial stiffness (Ki) and a positive post-yield stiffness (Kp) [23]. Additionally, the load value at the intersection point of the two lines was defined as the plastic strength of the damper (Pp). To calculate each of the stiffness values, a least square method was used. It was previously mentioned that the nominal plastic strength (Ppn) represented an approximate upper limit of initial behavior. Thus, the experimental data points with loads less than Ppn (86.29 kN) were used only in the calculation of Ki (Fig. 16(a)). Referring back to Eq. (1), it can be seen that Pp is proportional to the material yield stress. The specified minimum yield stress, i.e., nominal yield stress (Fyn), of KS SS400 steel is given as 235 MPa (16 mm < t 6 40 mm) in KS D 3503 [17], but material yield stress is generally larger than Fyn. The average yield stress of the two steel plates used in the specimens was also 1.22 times Fyn (Table 2). In practice, the material test results may not be provided, and therefore the performance of the damper can be more precisely predicted by considering an overstrength in material [29]. With
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this background, during the calculation process of the expected plastic strength (Ppe), Fy was replaced by the expected yield stress (RyFyn) in Eq. (1), where Ry denotes the ratio of the expected material yield stress to Fyn. Following Korean Building Code [25], Ry took the value of 1.3. The concept of bilinear approximation is illustrated in Fig. 16. It can be seen that the bilinear approximation curve represents the actual behavior of CLS closely. Table 5 summarizes the stiffness characteristics. In the table, the 1st set shows the result calculated using the method presented above. With a range of 55.53– 68.44 kN/mm, the ratios of Ki to Ke,cal (106.97 kN/mm) were between 0.52 and 0.64. In addition, HSD displayed significant stiffness in the post-yield region with a post-yield slope (a), which is the ratio of Kp to Ki, ranging from 0.0485 to 0.0574. On the other hand, the stiffness characteristics in the 2nd set were calculated by applying 0.5 as the value of c (Ki = 53.48 kN/mm). In the 2nd set, a had a range of 0.0538–0.0656. From the comparison of the 1st set and the 2nd set, it was found that a decrease of Ki led to an increase of Kp. In actual applications, various conditions may not be clearly defined, such as unknown material properties and dynamic effects due to high v. Therefore, in the bilinear idealization of HSD, Ki was simplified to 53.48 kN/mm, 50% of Ke,cal (c = 0.5) as in the 2nd set in Table 5. Using the average a value of 0.06, the corresponding Kp had a value of 3.21 kN/mm. The comparison between the tested
Fig. 15. Strain distributions of a cyclically-loaded specimen (CLS).
Fig. 16. Concept of bilinear approximation.
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Table 5 Summary of stiffness characteristics. Specimen
1st set
2nd set
Ki
Kp
a
Ki
Kp
a
C1 C2 C3 C4 2-C1 2-C2
66.23 67.95 68.44 55.53 63.65 55.72
3.22 3.44 3.39 3.19 3.09 2.87
0.0486 0.0506 0.0495 0.0574 0.0485 0.0515
53.48 53.48 53.48 53.48 53.48 53.48
3.29 3.51 3.46 3.20 3.14 2.88
0.0615 0.0656 0.0647 0.0598 0.0587 0.0538
Total (+) Total ()
62.37 61.62
3.15 3.22
0.0505 0.0523
53.48 53.48
3.19 3.26
0.0596 0.0610
Note: The stiffness values of specimens 2-C1 and 2-C2 were reduced by half. The unit for Ki and Kp is kN/mm.
Fig. 19. Variation of scaled secant stiffness.
Fig. 17. Comparison between tested and predicted loads.
Fig. 18. Illustration of combined hardening model.
load and the predicted load is presented in Fig. 17. It can be seen that the bilinear idealization successfully predicted the behavior of specimens with different variables, showing values of the normalized loads (the ratios of the predicted load to the tested load) close to 1. 5.2. Hysteretic model for cyclic behavior The cyclic envelope of specimens was represented well by the above bilinear idealization with one post-yield stiffness. However,
in cyclic hysteresis curves (Fig. 4), as the experienced displacement became larger, the load value where hysteresis met the vertical axis increased continuously while the post-yield stiffness decreased gradually. Therefore, in the hysteretic model for cyclic behavior, (a) kinematic hardening and isotropic hardening rules should be combined, and (b) the degradation of post-yield stiffness should be adequately considered. Fig. 18 shows an illustration of the combined hardening model proposed in this study. As in the bilinear idealization, the model has a slope of Ki (=0.5Ke,cal) at the first loading segment (Path 0– 1), and it has a reduced slope when Ppe is reached (Path 1–2). The unloading stiffness (Ku) and the reloading stiffness (Kr) are equal to Ki (Paths 2–3 and 4–5). Considering combined isotropic– kinematic hardening, load change (DP) from the unloading (or reloading) point to the reversed yielding is defined as the sum of Ppe and the peak load (Ppeak) reached previously. Consequently, the load values at yielding points are Ppe during loading, unloading, and reloading (Points 1, 3, and 5). The slope from a yield point to the next dpeak (i.e., the post-yield stiffness at the ith half cycle) is denoted by Kpi. In order to consider stiffness degradation in the post-yield region, this model uses the secant stiffness (Ksec) of the curve obtained by the bilinear idealization. Fig. 19 plots the variation of scaled secant stiffness, and Kpi takes the value of the scaled secant stiffness which corresponds to the largest absolute value of the experienced displacement prior to the ith half cycle. Based on the hysteresis curve obtained from the test, a scale ratio of 1:5 (Kpi = Ksec/5) was adopted. The comparison of load–displacement hysteresis curves for specimen C1 is presented in Fig. 20. The kinematic hardening model resulted in a different shape and area of the hysteresis loop. On the other hand, it can be seen that numerical simulation using the combined hardening model showed good agreement with the test result. Fig. 21 plots the variation of normalized energy with cumulative peak displacement, where normalized energy denotes the ratio of dissipated energy calculated by the combined hardening model to the tested value. After 100 mm of cumulative peak displacement, the normalized energy showed values close to 1. It ranged from 0.946 to 0.989 at the final stage. Because the model was defined simply to reach plastic strength in a linear relationship without a transition region, it underestimated the dissipated energy in the range less than 50 mm of cumulative peak displacement. In actuality, elastic behavior predominated there, so the dissipated energy was only less than 1 kJ different between the prediction and the test. Given this, it may be said that the dissipated energy of cyclic specimens throughout the entire history is well predicted by the proposed hysteretic model considering the combined hardening rule.
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Fig. 20. Comparison of hysteresis curves (specimen C1).
Fig. 21. Variation of normalized energy with cumulative peak displacement.
6. Conclusions In order to clarify the detailed behavior of the hourglass-shaped strip damper (HSD), experimental tests were carried out under various conditions which may be encountered in real-life applications. In addition, a hysteretic model defined by simple rules was proposed and also compared with the test results. The main findings of this study can be summarized as follows: (1) The monotonically-loaded specimens (MLS) and cyclicallyloaded specimens (CLS) showed noticeably different behavior. MLS displayed significantly larger maximum loads (Pmax) and displacements (dmax) which were greater by 1.8 and 2.3 times compared to CLS, respectively, and ductile fractures ultimately occurred at the middle of the strips after undergoing considerable plastic deformation. On the other hand, CLS represented superior deformation and energy dissipation capacities; each was 5 and 6 times greater compared to MLS. Furthermore, as damage accumulated, the strips fractured progressively at the ends in a ductile manner. (2) In CLS, Pmax and the total dissipated energy (Et) were within the ranges of 2.4% and 8.8% variation, respectively, compared
to the average values; principal structural characteristics such as Pmax and Et displayed no significant dependence upon the variables of loading rate (v), material strength, nor the number of damper plates used (n). However, lower material strength decreased the elastic stiffness of the damper by 10.9% in specimen 2-C2 and higher v (5 mm/s) decreased the stiffness by 4.7% in specimen C3. This led to a decrease in cumulative ductility even though specimens had similar cumulative displacement and dissipated energy. (3) The distribution of measured strains demonstrated that the external load was evenly distributed among all internal and external strips, irrespective of the number of damper plates used. This feature will allow for flexibility of composition; that is, the number of strips and the arrangement of damper plates can be extended in various ways without limitation according to the required performance when designing the damper. (4) The cyclic envelope of specimens was represented well by the idealized bilinear curve where a value of 0.5 was used as the stiffness coefficient (c) based on the experimental results. It was also found that HSD had significant postyield stiffness, with a range of 0.0538–0.0656 as the value of the post-yield slope (a). (5) The hysteretic model for cyclic behavior was proposed using the combined isotropic–kinematic hardening in order to consider the increase of strength and the degradation of post-yield stiffness. The shape of the hysteresis loop predicted by the model matched well with the test results, and the model predicted the dissipated energy well not only at the final stage but also throughout the entire loading history. It is expected that the proposed model can be practically used to predict the behavior of a building during an earthquake where HSD has been installed.
Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2013R1A1A2013578). The authors would also like to express their gratitude for the financial support of UNISON eTech Co., Ltd., Republic of Korea.
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