**1. Introduction**

Earthquakes are sudden natural disasters endangering people's lives and property. The may not only cause housing damage, tra ffic interruption, water disaster, fire, disease, and other secondary disasters, but can also endanger human life and safety.

To minimize casualties and economic losses, preventing structural collapse and serious damage to the civil infrastructure against seismic hazard has become a challenging task that researchers need to address. Improving the seismic capacity of engineering structures through novel techniques and technical measures is the most e ffective way to mitigate seismic hazards. The traditional seismic method takes the "anti-seismic" approach as an important way to resist earthquakes by enlarging the section of structures and adding more reinforcements. The result is that the larger the section of structural components is, the greater the sti ffness becomes, and the greater the impact of earthquakes. In this vicious circle, it is not only di fficult to ensure safety, but construction costs required for earthquake resistance also increase greatly.

An e ffective way to overcome increased inertia properties is structural vibration control through energy dissipation devices. Energy dissipation devices absorb and dissipate seismic energy, thereby reducing the dynamic response of the main structure under an earthquake. Based on numerous

practical applications of energy dissipation devices, the vibration control technique can be used to mitigate seismic hazard and achieve better seismic performance of the structure under vibrations.

The passive energy dissipation method has been widely used in the seismic control of structures since the 1980s [1,2]. Dampers can be classified into metal, viscoelastic [3], viscous [4], tuned absorption dampers [5], smart materials based isolation and suspension system [6,7], recently developed smart materials based damping or negative stiffness structure [8–10], actively tuned damper and driver system [11,12], according to the energy dissipation methods, control force characteristics [13] as well as materials used. The metal damper has a certain stiffness, and its energy dissipation mechanism depends on the elastic-plastic deformation. It is cheap, easy to install, and less affected by temperature. Therefore, it is suitable for all kinds of building structures and has a good economic value and effectiveness for the reinforcement and reconstruction of existing and new buildings. Metallic dampers can further be categorized into axial yield [14,15], shear yield [16], flexural yield [17,18], and combined yield devices [19].

Existing forms of flexural yield dampers include the added damping and stiffness (ADAS) device [20–22], triangular added damping and stiffness (TADAS) device [23,24], knee brace device [25,26], steel-composited wall dampers and B-C-W members [27,28], non-uniform steel strip damper [29], rhombic low yield strength steel plate [30], J-shaped steel hysteresis damper [31,32], U-shaped damper [33–35], pipe damper [36], dual-pipe damper [37], bar-fuse damper [38], accordion metallic damper [39], pipe-fuse damper [40], pure bending yielding dissipater [41], crawler steel damper [42], and hourglass-shaped strip damper [43].

Because the yield load of single-sheet steel is small, some bending plate dampers need to be combined with multiple sheets of the steel plate. In order to facilitate the energy consumption and reduce the stress concentration, it is necessary to optimize the shape of the steel plate, which makes the processing difficult and cannot eliminate the influence of the vertical force on the damper. In view of the above problems, this paper proposes the improvement of a curved plate damper based on the principle of the U-shaped damper. The mechanical characteristics and energy dissipation capacity of the curved plate damper were investigated using theoretical and experimental methods.

#### **2. Performance Parameters of Curved Plate Damper**

Three views with dimensioning is shown in Figure 1. The parameters of the curved plate damper mainly include thickness (*t*), width (*b*), and radius (*R*, *R* = *R* − *t*/2). The performance of the damper will vary with the parameters. The AB section and CD section of the damper are straight sections connected with other components, which constitutes the non-energy-consuming part of the damper, as shown in Figure 1a. The half-arc AC and BD segments are the main work parts of the damper.

**Figure 1.** Front (**a**), lateral (**b**), and vertical (**c**) views of the curved plate damper.

#### *2.1. Elastic Sti*ff*ness Calculation Formula*

A single semi-circular arc steel plate was taken as the research object, which can be regarded as a curved beam with two fixed ends [44], as shown in Figure 2a. Figure 2b shows the deformation sketch and internal force analysis diagram of the central axis of the semi-circular arc steel plate when displacement (Δ = 1) occurs at the support.

**Figure 2.** Diagram of a single semi-circular arc steel plate (**a**) and computing model (**b**).

According to the elastic center method, using the symmetry of the structure, *M* and *Q* are unknown symmetric forces, *F* is an unknown anti-symmetric force, and the force method equation can be simplified as follows:

$$\begin{cases} \delta\_{11}M + \delta\_{12}Q + \Delta\_{1C} = 0\\ \delta\_{21}M + \delta\_{22}Q + \Delta\_{2C} = 0\\ \delta\_{33}F + \Delta\_{3C} = 0 \end{cases} \tag{1}$$

Using the rigid arm, the sum of δ12 and δ21 equals zero. Therefore, the simplified form of the equation is expressed as follows:

$$\begin{cases} \delta\_{11}\mathcal{M} + \Delta\_{1\mathcal{C}} = 0\\ \delta\_{22}\mathcal{Q} + \Delta\_{2\mathcal{C}} = 0\\ \delta\_{33}\mathcal{F} + \Delta\_{3\mathcal{C}} = 0 \end{cases} \tag{2}$$

where Δ1*C* and Δ2*C* are zero due to the horizontal displacement of the support. In addition, the bending moment (*M*) and vertical force (*Q*) at the elastic center are both zero. Thus, the above simplified equation can be written as follows:

$$
\Delta \delta\_{33} F + \Delta\_{3\mathcal{C}} = 0 \tag{3}
$$

where δ33 and Δ3*C* can be written as

$$\begin{array}{ll} \delta\_{33} &= \int \frac{\overline{M}\_3^2}{EI} d\_s + \int \frac{\overline{P}\_3^2}{EA} d\_s + \int \frac{\overline{Q}\_3^2}{GA} d\_s\\ &= \int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{R^2 \sin^2 \alpha}{EI} R d\alpha + \int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\sin^2 \alpha}{EA} R d\alpha + \int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\cos^2 \alpha}{GI} R d\alpha\\ &= \frac{R\pi}{2} \left( \frac{R^2}{EI} + \frac{1}{EA} + \frac{1}{CA} \right) \end{array} \tag{4}$$

$$
\Delta\_{\text{\%C}} = -1 \tag{5}
$$

Introducing Equations (4) and (5) into (3), the force can be obtained as

$$F = -\frac{\Delta\_{3\overline{C}}}{\delta\_{3\overline{3}}} = \frac{2}{R\pi \left(\frac{R^2}{EI} + \frac{1}{EA} + \frac{1}{GA}\right)}\tag{6}$$

where *F* is the force at the end of the damper with unit displacement, which is the elastic stiffness of the damper with a semi-curved steel plate, and *G* is the shear modulus of elasticity (*G* = 0.4*E*). The elastic stiffness of the damper can be simplified as follows:

$$K = \frac{Ebt^3}{R\pi(6R^2 + 7t^2/4)}\tag{7}$$

where *E* is the elastic modulus of the material, *b* is the width of the curved plate damper, and *t* is the thickness of the curved plate damper.

#### *2.2. Formula for Calculating Yield Strength*

From the balance mechanism of the forces, the expression of elastic ultimate strength can be obtained as

$$F\_{\mathfrak{k}} \cdot \mathfrak{2}R = \mathfrak{2M} \tag{8}$$

The elastic ultimate strength can also be written as

$$F\_{\varepsilon} = \frac{M\_{\varepsilon}}{R} = \frac{f\_{y}W\_{\varepsilon}}{R} = \frac{f\_{y}bt^{2}}{6R} \tag{9}$$

where *fy* is the yield load, and *We* is the elastic moment of resistance.

The method for determining the yield displacement in this paper is shown in Figure 3. The horizontal and vertical coordinates at the intersection of tangent OA and AB are the yield displacement and yield strength, respectively. Because the yield strength is greater than the elastic ultimate strength, the adjustment coefficient β is introduced.

**Figure 3.** Schematic diagram of yield displacement.

The adjustment coefficient of the yield load can be determined by relevant tests and numerical analysis. The yield load can be written as

$$F\_y = \frac{\beta f\_y b t^2}{6R} \tag{10}$$

#### *2.3. Formula for Calculating Yield Displacement*

The yield displacement can be expressed as

$$
\Delta\_y = \frac{F\_y}{K} = \frac{\beta f\_y \pi (6R^2 + 7t^2/4)}{6Et} \tag{11}
$$

From Equations (7), (9), and (11), the elastic stiffness, yield strength, and yield displacement of curved plate dampers can be calculated. By adjusting the relevant parameters, a damper was designed to meet the needs of the project.

#### **3. Mechanical Properties of Curved Plate Damper**

The curved plate damper was made of Q235B steel. Four groups of specimens were designed for the material properties test. The thickness of steel plates for specimens 1 and 2 was 10 mm, and that for specimens 3 and 4 was 6 mm. The material property test was loaded with a servo actuator (maximum force capacity: 500 kN) in Civil Test Center of Southeast University (Nanjing, China) with a loading rate of 1.2 mm/min. Based on the analysis of stress and strain data, the yield point was taken as the yield strength of the steel. The mechanical properties of steel are shown in Table 1.


**Table 1.** Mechanical properties of steel.

The four groups of curved plate dampers were named as CSPD-1, CSPD-2, CSPD-3, and CSPD-4, respectively. The length of the straight section of the four groups was the same. Their geometric dimensions are shown in Table 2.


**Table 2.** Specimen sizes.

#### *3.1. Loading Device and Test Scheme*

The test was carried out in Civil Test Center of Southeast University (Nanjing, China). The loading equipment was a MTS 50-ton fatigue testing machine. The experiment was divided into four groups with eight specimens, with each group having two identical specimens. In order to connect with the servo actuator, a fixture was designed as shown in Figure 4. The inverted T-splint and T-base are equipped with triangular sti ffening ribs, and the components of the fixture are connected by the bolts.

**Figure 4.** Schematic diagram of the fixture (**a**) and photograph (**b**) of the testing device.

The test was divided into standard loading and fatigue loading. According to the multiple of the yield displacement of the damper ( ), the target displacements of the CSPD-1 specimens were 2 mm (0.5 ), 4 mm (1 ), 8 mm (2 ), 12 mm (3 ), 16 mm (4 ), 20 mm (5 ), 24 mm (6 ), 28 mm (7 ), 32 mm (8 ), 36 mm (9 ), and 40 mm (10 ), respectively. According to the multiple of yield displacement, all specimens in this test were loaded in 11 stages. The target displacements of CSPD-2 were 1.2, 2.4, 4.8, 7.2, 9.6, 12, 14.4, 16.8, 19.2, 21.6, and 24 mm, respectively. The target displacements of CSPD-3 were 0.9, 1.8, 3.6, 5.4, 7.2, 9, 10.8, 12.6, 14.4, 16.2, and 18 mm, respectively. The target displacements of CSPD-4 were 2.1, 4.2, 8.4, 12.6, 16.8, 21, 25.2, 29.4, 33.6, 37.8, and 42 mm, and the target displacement circulated three times. For fatigue loading, the Code for Seismic Design of Buildings (GB 50011-2010) stipulates that the energy dissipator should circulate 30 times [13] under the designed displacement, and there should be no obvious low-cycle fatigue phenomenon. The fatigue loading displacements of CSPD-1, CSPD-2, CSPD-3, and CSPD-4 were 40, 24, 18, and 42 mm, respectively.

During the test, force and displacement data were automatically recorded by the servo actuator. In order to study the mechanical properties of the damper, strain gauges were attached to the damper. The locations of the measuring points are shown in Figure 5. Two strain gauges were attached to each group of specimens at positions 1, 2, and 3.

**Figure 5.** Gauge position.

#### *3.2. Experiment Result Analysis*

The described four groups of specimens underwent standard loading. It was observed that when the displacement was small, the damper showed no obvious change. However, with the increase in displacement, the steel oxide layer of the damper's semi-circular arc energy dissipation section appeared to warp and spall with a relatively higher spall at the end of the semi-circular arc. The damper produced plastic deformation during the cyclic loading process. The damper's deformation when it was loaded to the maximum displacement is shown in Figure 6. After 30 cycles of fatigue loading, the damper had obvious deformations but no cracks. No obvious damage was observed and the damper had good integrity.

**Figure 6.** *Cont.*

**Figure 6.** Deformation of CSPD-1 (**a**), CSPD-2 (**b**), CSPD-3 (**c**), and COPD-4 (**d**) specimens at maximum displacement.

## 3.2.1. Hysteretic Curve

From the data collected by MTS equipment, the hysteretic curves of curved steel plate dampers are plotted in Figure 7. It can be seen that the hysteretic curves of dampers are full, the elastic stiffness is large, the stiffness of dampers after yielding decreases, and the maximum and minimum loads of dampers are not exactly the same. The reason may be that there were errors in the installation. The four groups of specimens circulated three times at each target displacement, and the three curves basically coincide, which indicates the stability of the damper.

Hysteretic curves under fatigue loading are shown in Figure 8. After 30 cycles of the cyclic loading, the load indices of the specimens CSPD-1 and CSPD-4 decreased, but the attenuation was far less than 15%. However, the load attenuation of the specimens CSPD-2 was higher, about 13.51%. The load attenuation of the specimens CSPD-3 presented positive and negative asymmetric state, which may be due to an installation error. Therefore, the performance of CSPD-2 was slightly worse than that of the other three specimens, but still met the relevant requirements of the code for dampers. It can be concluded that overall, the curved plate damper had good hysteretic performance.

**Figure 7.** Hysteretic curves of CSPD-1 (**a**), CSPD-2 (**b**), CSPD-3 (**c**), andCSPD-4 (**d**) under standard loading.

**Figure 8.** Hysteretic curve of CSPD-1 (**a**), CSPD-2 (**b**), CSPD-3 (**c**), and CSPD-4 (**d**) under fatigue loading.

#### 3.2.2. Ductility Coefficient and Energy Dissipation Coefficient

According to the relevant provisions of test data processing in the code for seismic test methods of buildings (JGJ-96), the ductility coefficient is the ratio of ultimate displacement to yield displacement. The energy dissipation coefficient is measured by the envelope area of the hysteresis curve. The diagram of the hysteresis curve is shown in Figure 9. The ductility coefficient can be written as

$$E = \frac{\mathcal{S}\_{\text{(ABC} + \text{CDA)}}}{\mathcal{S}\_{\text{(OBE} + \text{ODF})}} \tag{12}$$

**Figure 9.** Diagram of the hysteresis curve.

The ductility coefficient and energy dissipation coefficient of the curved plate damper are shown in Table 3. The ductility and energy dissipation capacity of the four groups of specimens were good with few exceptions. Among them, CSPD-2 had the smallest ductility coefficient, and CSPD-3 had the smallest energy dissipation coefficient.


**Table 3.** Mechanical properties of curved plate damper.
