Evaluation of Structural Behavior of Hysteretic Steel Dampers under Cyclic Loading

: This study proposes a relatively simple steel damper with high energy dissipation capacity. Three types of steel dampers were evaluated for structural performance. The ﬁrst damper with U-shape had two vertical members and a semicircular connecting member for energy dissipation. The second damper with an angled U-shape replaced the connecting member with a horizontal steel member. The last damper with D-shape had a horizontal member added to the U-shaped damper. All the dampers were designed with steel plates on both sides that transmitted external shear force to the energy-dissipating members. To evaluate the structural performance of the dampers, an in-plane cyclic shear force was applied to the specimens. The D-shaped damper showed ductile behavior with excellent energy dissipation capacity after yielding without decreasing in strength during cyclic load. In other words, the D-shaped specimen showed excellent performance, with about 3.5 times the strength of the U-shaped specimen and about 3.8 times the energy dissipation capacity due to the additional horizontal member. Furthermore, the e ﬃ cient energy dissipation of the proposed D-shaped steel damper was conﬁrmed from the ﬁnite element (FE) analytical and experimental results.


Introduction
One of most important purposes of building structures is to protect humans from nature, and design methods that reduce structural damage caused by external loads are constantly being developed. There are even methods being developed to mitigate damage caused by natural phenomena that are difficult to predict, such as earthquakes.
Seismic design includes seismic resistance, seismic isolation, and seismic damping to design structures that are safe from earthquakes. Seismic resistance allows a structure to directly absorb seismic energy, and it can protect human life during a strong earthquake, but severe residual deformation due to the inelastic behavior of the structure results in enormous social costs and environmental pollution due to demolition. Seismic isolation is the best way to avoid earthquake damage by separating a structure from the ground, thereby rendering it unaffected by earthquakes. However, seismic isolation is difficult to apply to many buildings because the cost of the seismic isolator can be prohibitive. Finally, seismic damping allows the damping device, instead of the structure, to absorb the seismic energy. Damping devices can be divided into active and passive control systems. The active control system is relatively expensive and complex because it requires several mechanical devices. On the other hand, the passive control system is relatively simple and economical, and it can easily be replaced when retrofitting structures after an earthquake. For these reasons, passive control systems are widely used in building structures. As shown in Figure 1, all the steel dampers were divided into energy dissipation members and shear plates that transferred external shear force. The size of each shear plate was 75 mm × 110 mm and the thickness was 20 mm. The shear plates were located on each side, and the distance between them was 90 mm. The steel dampers were designed in opened U-and angled U-shapes and a closed D-shape according to the shape of the energy dissipation parts. The thickness of all dampers was 20 mm, and the width of the member for energy dissipation was planned to be 15 mm. The spacing between the shear plates, the thickness of the dampers, and the width and height of the energy dissipation members were designed in consideration of the strength and deformation capacities of the specimens. The corners of the steel dampers were rounded to prevent stress concentration.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 12 dissipation members were designed in consideration of the strength and deformation capacities of the specimens. The corners of the steel dampers were rounded to prevent stress concentration. As shown in Figure 1, the U-shaped energy dissipation part of the SD1 damper specimen had a height of 90 mm. The energy dissipation part had two vertical members, each with a height of 30 mm, and a semicircular member with an inner diameter of 90 mm. SD2 consisted of horizontal and vertical steel members in order to observe the change in structural behavior when the connecting member was changed from semicircular to horizontal. The vertical members were planned to be 45 mm high, and other details of the specimen were the same as for SD1. The SD3 damper had an additional horizontal member added to the lower part of SD1 to increase the load resisting capacity. The lower horizontal member had a cross-sectional area of 15 mm × 20 mm. The dimensions of the vertical and connecting members of SD3 were the same as those of SD1.

Test Setup and Loading Procedure
As shown in Figure 2, four identical dampers were installed on each specimen to prevent eccentric deformation of the damper device and to distribute the force acting on the bolt fastened to the shear plates. The dampers were connected to the tops and bottoms of the fronts and rears of Tshaped loading plates, which were connected to L-shaped steel loading frames. As shown in Figure  2, two L-shaped loading frames were used in this experiment: one fastened to the bottom steel frame and the other to a loading actuator of a universal testing machine (UTM) with a capacity of 2000 kN. When a load was applied to the top of the L-shaped loading frame, the loading frame moved vertically, and the shear force was transmitted to the dampers via loading plates.  As shown in Figure 1, the U-shaped energy dissipation part of the SD1 damper specimen had a height of 90 mm. The energy dissipation part had two vertical members, each with a height of 30 mm, and a semicircular member with an inner diameter of 90 mm. SD2 consisted of horizontal and vertical steel members in order to observe the change in structural behavior when the connecting member was changed from semicircular to horizontal. The vertical members were planned to be 45 mm high, and other details of the specimen were the same as for SD1. The SD3 damper had an additional horizontal member added to the lower part of SD1 to increase the load resisting capacity. The lower horizontal member had a cross-sectional area of 15 mm × 20 mm. The dimensions of the vertical and connecting members of SD3 were the same as those of SD1.

Test Setup and Loading Procedure
As shown in Figure 2, four identical dampers were installed on each specimen to prevent eccentric deformation of the damper device and to distribute the force acting on the bolt fastened to the shear plates. The dampers were connected to the tops and bottoms of the fronts and rears of T-shaped loading plates, which were connected to L-shaped steel loading frames. As shown in Figure 2, two L-shaped loading frames were used in this experiment: one fastened to the bottom steel frame and the other to a loading actuator of a universal testing machine (UTM) with a capacity of 2000 kN. When a load was applied to the top of the L-shaped loading frame, the loading frame moved vertically, and the shear force was transmitted to the dampers via loading plates.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 12 dissipation members were designed in consideration of the strength and deformation capacities of the specimens. The corners of the steel dampers were rounded to prevent stress concentration. As shown in Figure 1, the U-shaped energy dissipation part of the SD1 damper specimen had a height of 90 mm. The energy dissipation part had two vertical members, each with a height of 30 mm, and a semicircular member with an inner diameter of 90 mm. SD2 consisted of horizontal and vertical steel members in order to observe the change in structural behavior when the connecting member was changed from semicircular to horizontal. The vertical members were planned to be 45 mm high, and other details of the specimen were the same as for SD1. The SD3 damper had an additional horizontal member added to the lower part of SD1 to increase the load resisting capacity. The lower horizontal member had a cross-sectional area of 15 mm × 20 mm. The dimensions of the vertical and connecting members of SD3 were the same as those of SD1.

Test Setup and Loading Procedure
As shown in Figure 2, four identical dampers were installed on each specimen to prevent eccentric deformation of the damper device and to distribute the force acting on the bolt fastened to the shear plates. The dampers were connected to the tops and bottoms of the fronts and rears of Tshaped loading plates, which were connected to L-shaped steel loading frames. As shown in Figure  2, two L-shaped loading frames were used in this experiment: one fastened to the bottom steel frame and the other to a loading actuator of a universal testing machine (UTM) with a capacity of 2000 kN. When a load was applied to the top of the L-shaped loading frame, the loading frame moved vertically, and the shear force was transmitted to the dampers via loading plates.  To maintain a constant distance between the two loading frames, each roller with a capacity of 300 kN was installed on the upper and lower portions of the L-shaped steel loading frames. As shown in Figure 3, the repeated load with the target displacement of ±15mm was applied to the specimens at a speed of 0.06 mm/s to evaluate whether there were changes in the strength, stiffness, and energy dissipation capacity of the specimens during the test.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 12 To maintain a constant distance between the two loading frames, each roller with a capacity of 300 kN was installed on the upper and lower portions of the L-shaped steel loading frames. As shown in Figure 3, the repeated load with the target displacement of ±15mm was applied to the specimens at a speed of 0.06 mm/s to evaluate whether there were changes in the strength, stiffness, and energy dissipation capacity of the specimens during the test. To measure the relative displacement of the specimens, each LVDT (linear variable differential transducer) was installed on the front and rear surfaces of the loading plates. To check the yielding and deformation characteristics of the specimens, strain gauges were attached to the ends of vertical and horizontal members where the maximum moment occurred. In addition, a strain gauge was attached to the top of each semicircular member to evaluate the deformation characteristics of the connecting members of SD1 and SD3.

Shear Force Versus Displacement Relationships
The shear force versus displacement relationships of the specimens are shown in Figure 4, and the experimental results for the first yielding and maximum load of the specimens are shown in Table  2. As shown in Figure 4, the stiffness of all specimens decreased significantly after first yielding, and gradually increased until the target displacement. For all specimens during the repeated load cycles, the peak load slightly increased or almost remained steady, and no decrease in structural performance of the specimens was found. The experiment of Specimen SD1 was terminated after performing only 2.5 repetitions due to an error with the test equipment, but it showed sufficient behavioral characteristics to allow for a comparison with the hysteresis behavior of the Specimens SD2 and SD3.   To measure the relative displacement of the specimens, each LVDT (linear variable differential transducer) was installed on the front and rear surfaces of the loading plates. To check the yielding and deformation characteristics of the specimens, strain gauges were attached to the ends of vertical and horizontal members where the maximum moment occurred. In addition, a strain gauge was attached to the top of each semicircular member to evaluate the deformation characteristics of the connecting members of SD1 and SD3.

Shear Force Versus Displacement Relationships
The shear force versus displacement relationships of the specimens are shown in Figure 4, and the experimental results for the first yielding and maximum load of the specimens are shown in Table 2. As shown in Figure 4, the stiffness of all specimens decreased significantly after first yielding, and gradually increased until the target displacement. For all specimens during the repeated load cycles, the peak load slightly increased or almost remained steady, and no decrease in structural performance of the specimens was found. The experiment of Specimen SD1 was terminated after performing only 2.5 repetitions due to an error with the test equipment, but it showed sufficient behavioral characteristics to allow for a comparison with the hysteresis behavior of the Specimens SD2 and SD3.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 12 To maintain a constant distance between the two loading frames, each roller with a capacity of 300 kN was installed on the upper and lower portions of the L-shaped steel loading frames. As shown in Figure 3, the repeated load with the target displacement of ±15mm was applied to the specimens at a speed of 0.06 mm/s to evaluate whether there were changes in the strength, stiffness, and energy dissipation capacity of the specimens during the test. To measure the relative displacement of the specimens, each LVDT (linear variable differential transducer) was installed on the front and rear surfaces of the loading plates. To check the yielding and deformation characteristics of the specimens, strain gauges were attached to the ends of vertical and horizontal members where the maximum moment occurred. In addition, a strain gauge was attached to the top of each semicircular member to evaluate the deformation characteristics of the connecting members of SD1 and SD3.

Shear Force Versus Displacement Relationships
The shear force versus displacement relationships of the specimens are shown in Figure 4, and the experimental results for the first yielding and maximum load of the specimens are shown in Table  2. As shown in Figure 4, the stiffness of all specimens decreased significantly after first yielding, and gradually increased until the target displacement. For all specimens during the repeated load cycles, the peak load slightly increased or almost remained steady, and no decrease in structural performance of the specimens was found. The experiment of Specimen SD1 was terminated after performing only 2.5 repetitions due to an error with the test equipment, but it showed sufficient behavioral characteristics to allow for a comparison with the hysteresis behavior of the Specimens SD2 and SD3.     As shown in Figure 4, SD1 with U-shaped damper and SD2 with the upper horizontal connecting member showed similar loading history curves. On the other hand, as shown in Tables 2 and 3, SD2 had a similar load at first yielding as SD1 but showed some differences in maximum load and elastic and plastic stiffness. This is due to the height of the vertical members and the shape of the connecting member. However, the increase in load-resisting capacity of SD2 was not higher than that of SD3. In other words, SD3 with additional horizontal connecting member between shear plates showed a greater load-resisting capacity than SD1 and SD2 did. As shown in Table 2, with the addition of the lower horizontal connecting member, the yield strength of SD3 was about 2.0 times higher than that of SD1. In addition, the maximum load of SD3 was up to 3.9 times higher in the first cycle and 3.5 times higher in the second cycle than that of SD1. Furthermore, as shown in Figure 4, the overall behavior of SD3 was similar to that of SD1, which means SD3 was ductile despite its high load-resisting capacity. Both SD2 and SD3 showed no decrease in strength at the target displacement during five repetitions. Table 3 indicates the experimental results for the elastic and plastic stiffness of the specimens. SD2 was 1.29 times and 1.97 times greater than SD1 in the elastic and plastic moduli, respectively, during the first positive loading phase. On the other hand, after showing the Bauschinger effect, the plastic modulus of SD2 was 1.13 times that of SD1, meaning there was no significant difference. SD3 showed a 5.9 times greater elastic modulus compared to U-shaped SD1. A high elastic modulus means the damper reacts more quickly to an external seismic load. In particular, as shown in Table 2, closed-type SD3 had a faster yielding point and higher initial rigidity than open-type SD1 and SD2, which indicates excellent energy dissipation capability. The plastic modulus of SD3 was 9.1 times and 2.4 times greater than that of SD1. In this experiment, no significant decrease in strength, stiffness, and energy dissipation capacity was observed in SD3, which is similar to the experimental results of SD1 and SD2.

Deformation of Specimens
None of the specimens fractured or buckled until the end of the experiment. Figure 5 shows the deformation of each specimen at the target displacement. As shown in the figure, when the target displacement was reached, the deformation of each specimen was very stable. In all the specimens, Appl. Sci. 2020, 10, 8264 6 of 11 the upper connecting member served to induce deformation so the vertical members could dissipate energy as much as possible.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 12 the upper connecting member served to induce deformation so the vertical members could dissipate energy as much as possible.
(a) (b) (c) Since both ends of the horizontal member of SD3 were fixed to both shear plates, it received an anti-symmetric moment as vertical displacement occurred. Adding the horizontal connecting member to the lower part of the shear plates contributed to the increase in load resisting capacity while maintaining ductile behavior characteristics. As shown in Figure 5, the semicircular connecting members of SD1 and SD3 helped to make a relatively small lateral deformation at the maximum displacement, which is due to the geometric shape of the circle.

Strain Distribution
The strains of the specimens as measured by the strain gauges are shown in Figure 6. SD1 showed a maximum strain of about 0.023 after the vertical member reached its first yield strain at 17.9 kN in the forward direction for first cycle. On the other hand, no yielding was observed in the semicircular connecting member. As shown in Figure 6, energy dissipation of SD1 was concentrated in the vertical member. Since both ends of the horizontal member of SD3 were fixed to both shear plates, it received an anti-symmetric moment as vertical displacement occurred. Adding the horizontal connecting member to the lower part of the shear plates contributed to the increase in load resisting capacity while maintaining ductile behavior characteristics. As shown in Figure 5, the semicircular connecting members of SD1 and SD3 helped to make a relatively small lateral deformation at the maximum displacement, which is due to the geometric shape of the circle.

Strain Distribution
The strains of the specimens as measured by the strain gauges are shown in Figure 6. SD1 showed a maximum strain of about 0.023 after the vertical member reached its first yield strain at 17.9 kN in the forward direction for first cycle. On the other hand, no yielding was observed in the semicircular connecting member. As shown in Figure 6, energy dissipation of SD1 was concentrated in the vertical member. Since both ends of the horizontal member of SD3 were fixed to both shear plates, it received an anti-symmetric moment as vertical displacement occurred. Adding the horizontal connecting member to the lower part of the shear plates contributed to the increase in load resisting capacity while maintaining ductile behavior characteristics. As shown in Figure 5, the semicircular connecting members of SD1 and SD3 helped to make a relatively small lateral deformation at the maximum displacement, which is due to the geometric shape of the circle.

Strain Distribution
The strains of the specimens as measured by the strain gauges are shown in Figure 6. SD1 showed a maximum strain of about 0.023 after the vertical member reached its first yield strain at 17.9 kN in the forward direction for first cycle. On the other hand, no yielding was observed in the semicircular connecting member. As shown in Figure 6, energy dissipation of SD1 was concentrated in the vertical member. Similar to SD1, SD3 did not yield in the semicircular connecting member. On the other hand, both the vertical and the lower horizontal members reached yield strain in first cycle. The first yielding was observed at 36.2 kN in the lower horizontal member, and then at 55.5 kN in the vertical member. The vertical member of SD3 showed a maximum strain of about 0.027, showing similar strain characteristics to SD1. It can be seen from Figure 6 that the lower horizontal member, which is unique to SD3, exhibited a maximum strain of about 0.066, and both the vertical and the lower horizontal members greatly contributed to energy dissipation. It is noted that the maximum strain range of the damper can be controlled by planning the geometric dimensions of the energy dissipation members of the damper. Figure 7 shows the energy dissipation of the specimens for each cycle. In this study, the energy dissipation was the inner area of the load-displacement relationship curve. In the first cycle, since the elastic and plastic phases existed together, the value was slightly lower than those of the other cycles. The cumulative energy dissipation of Specimens SD2 and SD3 showed almost proportional increase, as shown in Figure 7a.

Energy Dissipation
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 12 member. The vertical member of SD3 showed a maximum strain of about 0.027, showing similar strain characteristics to SD1. It can be seen from Figure 6 that the lower horizontal member, which is unique to SD3, exhibited a maximum strain of about 0.066, and both the vertical and the lower horizontal members greatly contributed to energy dissipation. It is noted that the maximum strain range of the damper can be controlled by planning the geometric dimensions of the energy dissipation members of the damper. Figure 7 shows the energy dissipation of the specimens for each cycle. In this study, the energy dissipation was the inner area of the load-displacement relationship curve. In the first cycle, since the elastic and plastic phases existed together, the value was slightly lower than those of the other cycles. The cumulative energy dissipation of Specimens SD2 and SD3 showed almost proportional increase, as shown in Figure 7a. SD1 showed an energy dissipation capacity of 1154.5 kN•mm in the first cycle, as shown in Figure 7b. Afterward, the energy dissipation area increased by 11.4% to 1286.3 kN•mm in the second cycle, and the total accumulated energy dissipation capacity was 2440.8 kN•mm. As shown in Figure  7b, SD2 had an energy dissipation capacity of 1386.6 kN•mm, 20.1% higher than that of SD1, in the first cycle. In the second cycle, the energy dissipation capacity of SD2 increased by 10.5% from the first cycle, a similar result to that obtained from SD1. The final fifth cycle ended with a 12.5% increase from the first cycle. The experimental results confirmed that the energy dissipation capacity of SD2 remained almost unchanged after the second cycle.

Energy Dissipation
SD3 had an energy dissipation capacity of 4417.4 kN•mm in the first cycle. This was 3.8 times higher than that of SD1, which meant that even though only the lower horizontal member was added, the energy dissipation capacity was greatly improved. SD3 showed 12.8% and 14.9% higher energy dissipation capacities from the first cycle in the second and fifth cycles, respectively, as shown in Figure 7b. In other words, there was no decrease in energy dissipation capacity during five cycles, such as SD2. From the experimental results, it was confirmed that SD3 has excellent energy dissipation capacities.

Equivalent Damping Ratio
The damping capacity of the specimens can be determined from the energy dissipated by damping, D E , which is the area enclosed by the hysteresis loop, and the maximum strain energy SD1 showed an energy dissipation capacity of 1154.5 kN·mm in the first cycle, as shown in Figure 7b. Afterward, the energy dissipation area increased by 11.4% to 1286.3 kN·mm in the second cycle, and the total accumulated energy dissipation capacity was 2440.8 kN·mm. As shown in Figure 7b, SD2 had an energy dissipation capacity of 1386.6 kN·mm, 20.1% higher than that of SD1, in the first cycle. In the second cycle, the energy dissipation capacity of SD2 increased by 10.5% from the first cycle, a similar result to that obtained from SD1. The final fifth cycle ended with a 12.5% increase from the first cycle. The experimental results confirmed that the energy dissipation capacity of SD2 remained almost unchanged after the second cycle.
SD3 had an energy dissipation capacity of 4417.4 kN·mm in the first cycle. This was 3.8 times higher than that of SD1, which meant that even though only the lower horizontal member was added, the energy dissipation capacity was greatly improved. SD3 showed 12.8% and 14.9% higher energy dissipation capacities from the first cycle in the second and fifth cycles, respectively, as shown in Figure 7b. In other words, there was no decrease in energy dissipation capacity during five cycles, such as SD2. From the experimental results, it was confirmed that SD3 has excellent energy dissipation capacities.

Equivalent Damping Ratio
The damping capacity of the specimens can be determined from the energy dissipated by damping, E D , which is the area enclosed by the hysteresis loop, and the maximum strain energy corresponding to the shaded area, E So , as shown in Figure 8 [24]. This study derived the equivalent damping ratio ξ as follows: corresponding to the shaded area, So E , as shown in Figure 8 [24]. This study derived the equivalent damping ratio ξ as follows: Figure 8. Estimation of equivalent damping ratio. Figure 9 shows the equivalent damping ratios of the specimens for each cycle. As shown in the figure, both open and closed specimens showed high equivalent damping ratios. That is, the equivalent damping ratios of from 48.7% to 50.4% for SD1 and SD2 and from 45.7% to 49.1% for SD3 were experimentally evaluated. SD1 showed an equivalent damping ratio of 49.1% in the first cycle. The second cycle showed an equivalent damping ratio of 45.9%, which was 3.2%p lower than the ratio obtained in the first cycle. For SD2, the equivalent damping ratios of the first and second cycles were 48.2% and 46.5%, respectively, showing a difference of 1.7%p, a similar result to that obtained from SD1. SD2 showed an equivalent damping ratio of 45.7% in the fifth cycle, which was 2.5%p lower than the ratio obtained in the first cycle. SD3 showed an equivalent damping ratio of 48.7% in the first cycle and then showed a higher equivalent damping ratio of about 1.5%p in the second cycle, which it maintained until the final cycle. From the experimental results, the excellent damping capacity of the proposed damper SD3 was confirmed.

FE Modeling
This study performed a FE analysis to analytically evaluate the stress distributions and energy dissipation zones of the dampers used in the specimens and to verify the accuracy of the experiment.  Figure 9 shows the equivalent damping ratios of the specimens for each cycle. As shown in the figure, both open and closed specimens showed high equivalent damping ratios. That is, the equivalent damping ratios of from 48.7% to 50.4% for SD1 and SD2 and from 45.7% to 49.1% for SD3 were experimentally evaluated. SD1 showed an equivalent damping ratio of 49.1% in the first cycle. The second cycle showed an equivalent damping ratio of 45.9%, which was 3.2%p lower than the ratio obtained in the first cycle. For SD2, the equivalent damping ratios of the first and second cycles were 48.2% and 46.5%, respectively, showing a difference of 1.7%p, a similar result to that obtained from SD1. SD2 showed an equivalent damping ratio of 45.7% in the fifth cycle, which was 2.5%p lower than the ratio obtained in the first cycle. SD3 showed an equivalent damping ratio of 48.7% in the first cycle and then showed a higher equivalent damping ratio of about 1.5%p in the second cycle, which it maintained until the final cycle. From the experimental results, the excellent damping capacity of the proposed damper SD3 was confirmed.
corresponding to the shaded area, So E , as shown in Figure 8 [24]. This study derived the equivalent damping ratio ξ as follows: Figure 8. Estimation of equivalent damping ratio. Figure 9 shows the equivalent damping ratios of the specimens for each cycle. As shown in the figure, both open and closed specimens showed high equivalent damping ratios. That is, the equivalent damping ratios of from 48.7% to 50.4% for SD1 and SD2 and from 45.7% to 49.1% for SD3 were experimentally evaluated. SD1 showed an equivalent damping ratio of 49.1% in the first cycle. The second cycle showed an equivalent damping ratio of 45.9%, which was 3.2%p lower than the ratio obtained in the first cycle. For SD2, the equivalent damping ratios of the first and second cycles were 48.2% and 46.5%, respectively, showing a difference of 1.7%p, a similar result to that obtained from SD1. SD2 showed an equivalent damping ratio of 45.7% in the fifth cycle, which was 2.5%p lower than the ratio obtained in the first cycle. SD3 showed an equivalent damping ratio of 48.7% in the first cycle and then showed a higher equivalent damping ratio of about 1.5%p in the second cycle, which it maintained until the final cycle. From the experimental results, the excellent damping capacity of the proposed damper SD3 was confirmed.

FE Modeling
This study performed a FE analysis to analytically evaluate the stress distributions and energy dissipation zones of the dampers used in the specimens and to verify the accuracy of the experiment.

FE Modeling
This study performed a FE analysis to analytically evaluate the stress distributions and energy dissipation zones of the dampers used in the specimens and to verify the accuracy of the experiment. The FE analysis was performed with the 3D nonlinear FE analysis program MIDAS-NFX [25]. This program can take into account both material and geometric nonlinearities. The kinematic hardening model considering the Bauschinger effect was applied as the hysteresis model of steel subjected to cyclic loading. The elastoplastic tri-linear curve of the steel with strain hardening was used for stress versus strain relationship of the material. Geometric nonlinearity was applied to consider the large deformation of the element. Figure 10 shows the FE modeling of each steel damper of the specimens. A 3D solid hexahedron element was used for the FE modeling, and the size of the element was about 4 mm. To induce vertical displacement in each specimen, the bolted region of one shear plate was fixed and the other was modeled with a roller.
The FE analysis was performed with the 3D nonlinear FE analysis program MIDAS-NFX [25]. This program can take into account both material and geometric nonlinearities. The kinematic hardening model considering the Bauschinger effect was applied as the hysteresis model of steel subjected to cyclic loading. The elastoplastic tri-linear curve of the steel with strain hardening was used for stress versus strain relationship of the material. Geometric nonlinearity was applied to consider the large deformation of the element. Figure 10 shows the FE modeling of each steel damper of the specimens. A 3D solid hexahedron element was used for the FE modeling, and the size of the element was about 4 mm. To induce vertical displacement in each specimen, the bolted region of one shear plate was fixed and the other was modeled with a roller.  Figure 11 shows the stress distribution of each damper by FE analysis when the target displacement was reached. In the figure, the red color refers to the plastic region after yielding, and the stress decreases as the color turns blue. From Figure 11 it can be seen that the region reaching yield stress in each damper was almost the same as it was in the experimental result in Section 3.3. In the case of SD1, the yield and energy dissipation zones were concentrated in the vertical members. The analysis confirmed that some yielding occurred in the semicircular connecting member adjacent to the vertical member, which was similar to the experimental results obtained from the horizontal member adjacent to the vertical member in SD2. As shown in Figure 11c, in SD3 with an additional horizontal member, the yield region was concentrated on the vertical and lower horizontal members, the same as it was in the experimental results. In particular, most of the area of the lower horizontal member reached yield stress. In addition, by adding the lower horizontal member, a larger area of the damper engaged in stress transmission and energy dissipation than did the areas of SD1 and SD2 damper.  Figure 11 shows the stress distribution of each damper by FE analysis when the target displacement was reached. In the figure, the red color refers to the plastic region after yielding, and the stress decreases as the color turns blue. From Figure 11 it can be seen that the region reaching yield stress in each damper was almost the same as it was in the experimental result in Section 3.3. In the case of SD1, the yield and energy dissipation zones were concentrated in the vertical members. The analysis confirmed that some yielding occurred in the semicircular connecting member adjacent to the vertical member, which was similar to the experimental results obtained from the horizontal member adjacent to the vertical member in SD2.

Comparison of Experimental and Analytical Results
Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 11 Figure 10 shows the FE modeling of each steel damper of the specimens. A 3D solid hexahedron element was used for the FE modeling, and the size of the element was about 4 mm. To induce vertical displacement in each specimen, the bolted region of one shear plate was fixed and the other was modeled with a roller.  Figure 11 shows the stress distribution of each damper by FE analysis when the target displacement was reached. In the figure, the red color refers to the plastic region after yielding, and the stress decreases as the color turns blue. From Figure 11 it can be seen that the region reaching yield stress in each damper was almost the same as it was in the experimental result in Section 3.3. In the case of SD1, the yield and energy dissipation zones were concentrated in the vertical members. The analysis confirmed that some yielding occurred in the semicircular connecting member adjacent to the vertical member, which was similar to the experimental results obtained from the horizontal member adjacent to the vertical member in SD2. As shown in Figure 11c, in SD3 with an additional horizontal member, the yield region was concentrated on the vertical and lower horizontal members, the same as it was in the experimental results. In particular, most of the area of the lower horizontal member reached yield stress. In addition, by adding the lower horizontal member, a larger area of the damper engaged in stress transmission and energy dissipation than did the areas of SD1 and SD2 damper.

Conclusions
This study experimentally and analytically evaluated the structural performance of the proposed D-shaped steel damper subjected to cyclic loading, and the following results were obtained.
(1) As a result of the experiment, SD1 and SD2 in which the vertical members were connected by semicircular and horizontal members, respectively, had similar load history characteristics, energy dissipation capacities, and equivalent damping ratios. Therefore, even if the shape of a connecting member is changed from the semicircular type to the relatively simple horizontal type, there will be no significant degradation in structural performance. As shown in Figure 11c, in SD3 with an additional horizontal member, the yield region was concentrated on the vertical and lower horizontal members, the same as it was in the experimental results. In particular, most of the area of the lower horizontal member reached yield stress. In addition, by adding the lower horizontal member, a larger area of the damper engaged in stress transmission and energy dissipation than did the areas of SD1 and SD2 damper.

Conclusions
This study experimentally and analytically evaluated the structural performance of the proposed D-shaped steel damper subjected to cyclic loading, and the following results were obtained.
(1) As a result of the experiment, SD1 and SD2 in which the vertical members were connected by semicircular and horizontal members, respectively, had similar load history characteristics, energy dissipation capacities, and equivalent damping ratios. Therefore, even if the shape of a connecting member is changed from the semicircular type to the relatively simple horizontal type, there will be no significant degradation in structural performance. (2) SD3, which added the horizontal member to U-shaped SD1, showed about 3.5 times more load-resisting capacity and about 3.8 times greater energy dissipation capacity than did SD1 in the experiment. In addition, SD3 had higher elastic stiffness and lower displacement at first yielding compared to those of SD1. (3) From the experiment, both SD2 and SD3 showed no decrease in stiffness and strength during five loading cycles, and they showed equivalent damping ratios of about 50%, confirming excellent damping capacity. In addition, SD2 and SD3 almost maintained their original shapes, even after five reversed cyclic loads, and no in-plane or out-of-plane buckling was found. (4) The FE analysis results for the energy dissipation zone of each damper were almost consistent with the experimental results. In the case of SD3, the yielding zone was concentrated in the vertical member and the lower horizontal member. Furthermore, the semicircular connecting member of SD3 showed similar stress distribution as SD1. This means the additional horizontal member is an important factor in determining the behavioral characteristics of the proposed D-shaped steel damper.