Hysteretic behavior of cambered surface steel tube damper: Theoretical and experimental research

Abstract

A novel cambered surface steel tube damper (CSTD) with a cambered surface steel tube and two concave connecting plates is proposed herein. The steel tube is the main energy dissipation component and comprises a weakened segment in the middle, a transition segment, and an embedded segment. It is believed that during an earthquake, the middle weakened segment of the CSTD will be damaged, whereas the reliability of the end connection is ensured. Theoretical and experimental studies are conducted to verify the effectiveness of the proposed CSTD. Formulas for the initial stiffness and yield force of the CSTD are proposed. Subsequently, two CSTD specimens with different steel tube thicknesses are fabricated and tested under cyclic quasi-static loads. The result shows that the CSTD yields a stable hysteretic response and affords excellent energy dissipation. A parametric study is conducted to investigate the effects of the steel tube height, diameter, and thickness on the seismic performance of the CSTD. Compared with equal-stiffness design steel tube dampers, the CSTD exhibits better energy dissipation performance, more stable hysteretic response, and better uniformity in plastic deformation distributions.

Graphical abstract

Keywords

cambered surface steel tube damper / energy dissipation capacity / finite element model / hysteretic performance / parametric study

1 Introduction

Passive dampers have been widely adopted in building structures and bridge engineering to reduce earthquake hazards and risks [1,2]. When an earthquake occurs, the damper absorbs a significant amount of seismic energy, which can effectively reduce the seismic response of the structures [3]. Currently developed dampers can be classified into displacement- and velocity-dependent dampers based on their operating mechanisms [4]. The metallic damper is a type of displacement-dependent damper that is widely used owing to its simple structure, low cost, and ease of manufacture [5–8].

Additional damping and stiffness devices are typically used in metallic dampers, which dissipate seismic energy via the flexural yield deformation of steel plates [9–12]. In these devices, a series of rhombic, triangular, or holed steel plates is placed in parallel to increase the energy dissipation of the metal damper and to provide stiffness [13–15]. By contrast, the shear yield damper depends on the inelastic shear deformation of the metal plate under an in-plane load. Under cyclic loading, the damper exhibits stable and excellent hysteretic energy dissipation [16–19]. Low-yield-strength steel [20,21] and aluminum plates [22–24] were used as shear yield dampers to achieve high effectiveness. A damping device that combines shear and flexure yielding has been proposed recently [25] in addition to a series of metal dampers, such as slit dampers [26], that relies on a configurational arrangement to realize energy dissipation [27–29].

However, in contrast to metal dampers configured using metal plates, steel-tube dampers have been investigated less. Maleki and Bagheri [30,31] investigated a pipe damper with a low stiffness and strength. Dual-pipe [32] and infilled-pipe dampers [33] were used to improve the performance of the pipe damper. Guo et al. proposed an X-shaped pipe damper based on a dual-pipe damper [34,35]. However, these pipe dampers do not reflect the isotopic characteristics of the steel pipes. Abebe et al. [36,37] investigated circular hollow-section dampers both experimentally and numerically. The dampers exhibited uniform structural performance in all directions owing to their shapes, as well as excellent hysteretic performance. Park et al. [38] directly compared the performances of H-shaped shear panels and circular hollow section dampers; both types of dampers exhibited stable shuttle-shaped hysteresis characteristics as well as sufficient plastic deformation and energy absorption. Results showed that the steel tube damper at the end of the steel tube was damaged. Hence, because the maximum equivalent plastic strain occurred therein, the energy consumption of the steel tube could not be fully utilized.

Hence, a novel cambered surface steel tube damper (CSTD) is proposed herein to fully exploit the material strength and improve the seismic performance of a conventional steel tube damper. The CSTD comprises a cambered surface steel tube and two concave connecting plates. The steel tube is the main energy dissipation component and comprises a weakened segment in the middle, a transition segment, and an embedded segment. It is believed that during an earthquake, the middle weakened segment of the CSTD will be damaged, whereas the reliability of the end connection is ensured. Thus, material energy dissipation can be fully utilized. Theoretical and experimental studies are performed to investigate the seismic behavior of the newly proposed CSTD. First, theoretical formulas are proposed to calculate the initial stiffness and yield force of the CSTD based on the virtual work principle and von Mises yield criterion, respectively. Subsequently, two CSTD specimens with different thicknesses of the cambered surface steel tube are fabricated and tested under quasi-static cyclic loads. The hysteresis and skeleton curves as well as the failure modes of the specimens are investigated. In addition, a finite element model that considers the material nonlinearity, large deformation, and contact is established using the ABAQUS software. The accuracy of the finite element model is verified by comparing the predicted and experimental hysteresis curves and the failure modes. Subsequently, a parametric study is conducted to investigate the effects of the steel tube height, diameter, and thickness on the seismic performance of the CSTD, and the optimum size is proposed. Finally, a comparison of the seismic performance of the CSTD and that of a conventional steel tube dampers with equal stiffness show the superior energy dissipation of the proposed CSTD.

2 Cambered surface steel tube damper

Fig.1 shows the processing of the CSTD. The CSTD comprises a cambered surface steel tube and two concave connecting plates (bottom and top). The cambered surface steel tube was formed using a circular arc that rotated around the central axis of the steel tube. A concave connecting plate was formed by laser cutting the steel plate. Finally, a cambered surface steel tube and concave connecting plates were welded to form the CSTD.

The cambered surface steel tube comprises three main components: a weakened segment, a transition segment, and an embedded segment, as shown in Fig.2. The weakened segment with height H_w is the main energy-consuming component of the damper. The embedded and weakened segments are connected by the transition segment. The height of the transition segment is H_b. The embedded segment enters the concave connecting plate to ensure a reliable connection of the ends, which is beneficial to the steel tube. The depth of the steel tube embedded in the connecting plate is t_q. The embedded and transition segments in these three components are straight, and the weakened segment features a cambered surface. This ensures the reliability of the end connection and controls the energy-dissipation component of the damper in the cambered surface segment.

Here, H is the total height of the damper; t₀ is the wall thickness of the thinnest section of the weakened segment; t₁ is the wall thickness of the thickest section of the weakened segment; D is the inner diameter of the steel tube; and L and t_b are the length and thickness of the connecting plate, respectively. Groove welding was performed at a groove angle of 45°. The CSTD offers (1) a consistent in-plane and out-of-plane stiffness and yield force; (2) an accurate control of the damper damage location and a complete utilization of the material energy dissipation; (3) a simple structure, convenient manufacturing, and low cost; and (4) easy installation and replacement.

The CSTD dissipates energy primarily through shear deformation and bending deformation; therefore, the damper can be installed in structures that exhibit relative deformation. Fig.3 shows application scenarios of the CSTD. Generally, a CSTD can be installed between the upper and lower beams of a frame structure through wall piers or chevron bracing, as shown in Fig.3(a) and Fig.3(b). Using multiple CSTDs in parallel can provide a large force and excellent energy dissipation. Other potential application scenarios include eccentrically braced structural systems and coupling beams of reinforced concrete shear walls, as shown in Fig.3(c) and Fig.3(d), respectively.

3 Theoretical calculation of cambered surface steel tube damper

The total deformation (∆) of the CSTD includes shear deformation (∆1) and bending deformation (∆2). The deformation mechanism is illustrated in Fig.4. The CSTD is designed to remain in an elastic state and to increase the structural stiffness of buildings when they are not excited. When encountering earthquake excitations, the CSTD enters yield and plastic deformation states to dissipate energy. The initial stiffness and yield force of the metal damper, which are important indicators of the CSTD, are presented in this section.

3.1 Initial stiffness calculation

The following assumptions are introduced in this study: (1) the connecting plate always remains elastic, and the weld connections between the cambered surface steel tube and concave connecting plate are rigid constraints during the elastic deformation period; (2) the plane section is satisfied during the steel tube deformation, and the section perpendicular to the deformation axis of the steel tube remains unchanged as a plane after being subjected to tension, compression, bending, or shearing.

Fig.5(a) and Fig.5(b) show the computational diagram and positioning coordinates of the damper, respectively. The calculated coordinate O is located at the geometric center of the steel tube. The loading direction is defined as the x-direction; the direction perpendicular to the loading direction, the y-direction; and the direction along the height of the steel tube, the z-direction. The cambered surface steel tube can be described for two different sections: the outer radius of the steel tube R_1\left(\mathit{\text{z}}\right) for the weakened segment, and the outer radius of the steel tube R_2\left(\mathit{\text{z}}\right) for the transition segment. The outer radius of the steel tube R\left(\mathit{\text{z}}\right) can be expressed as follows:

(1)R(\mathit{\text{z}})=\{\begin{array}{c}{R}_1(\mathit{\text{z}})=\sqrt{R^2-{\mathit{\text{z}}}^2}-a,\mathit{\text{z}}\in \left[-\frac{H_{\text{W}}}{2},\frac{H_{\text{W}}}{2}\right],\\ R_2(\mathit{\text{z}})=\frac{D}{2}+t_1,\mathit{\text{z}}\in \left[-\frac{h}{2},-\frac{H_{\text{W}}}{2}\right)\cup \left(\frac{H_{\text{W}}}{2},\frac{h}{2}\right],\end{array}

(2)\begin{array}{rl}& a=\frac{H_{\text{W}}^2}{8(t_1-t_0)}+\frac{D+t_0+t_1}{2},\\ & R=\frac{H_{\text{W}}^2}{8(t_1-t_0)}+\frac{t_1-t_0}{2},\end{array}

where a is a constant (a > 0); R and a can be calculated using H_w, D, t₀, and t₁; and the height of the damper is calculated as h=H_{\mathrm{W}}+H_{\mathrm{b}}.

Based on the two assumptions mentioned above, the shear force (the letter “F”) and bending moment (the letter “M”) of the CSTD are illustrated as shown in Fig.6. Using the virtual work principle, the virtual work balance is expressed as follows:

(3)1\cdot \mathrm{\Delta }={\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2={\displaystyle {\int }_{-\frac{h}{2}}^{\frac{h}{2}}\frac{k{\overline{F}}_{\text{Q}}{F}_{\text{QP}}}{G_{\text{s}}{A}_{\text{s}}}}\text{d}\mathit{\text{z}}+{\displaystyle {\int }_{-\frac{h}{2}}^{\frac{h}{2}}\frac{\overline{M}{M}_{\text{P}}}{E_{\text{s}}{I}_y}}\text{d}\mathit{\text{z}},

where {\overline{F}}_{\mathrm{Q}} and \overline{M} are the internal forces due to the dummy unit load; F_{\mathrm{Q}\mathrm{P}} and M_{\mathrm{P}} are the internal forces generated by actual loads; k is the correction coefficient, which is related to the section form (e.g., k = 2 for the thin-walled ring section); G_{\mathrm{s}} is the shear modulus of the steel tube; and E_{\mathrm{s}} is the elastic modulus of the steel tube. The shear modulus and elastic modulus are correlated as follows:

(4)G_{\text{s}}=\frac{E_{\text{s}}}{2(1+{\upsilon }_{\text{s}})},

where {\upsilon }_{\text{s}} is the Poisson’s ratio of the steel tube, which is specified as 0.3 in this study based on the literature [39–41]. The steel tube section area A_{\mathrm{s}} can be expressed as

(5)A_{\text{s}}=\text{π}\left(R(\mathit{\text{z}}{)}^2-{\left(\frac{D}{2}\right)}^2\right).

Thus, the moment of inertia of the steel tube section, I_y, can be expressed as

(6)I_y={\int }_{A_{\text{s}}}{x}^2\text{d}{A}_{\text{s}}=\frac{\text{π}}{4}\left(R(\mathit{\text{z}}{)}^4-{\left(\frac{D}{2}\right)}^4\right).

Substituting Eqs. (4)−(6) into Eq. (3) yields

(7)\begin{array}{rl}1\cdot \mathrm{\Delta }=& {\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2={\displaystyle {\int }_{-\frac{h}{2}}^{\frac{h}{2}}\frac{4(1+{\upsilon }_{\text{s}})}{\text{π}{E}_{\text{s}}\left[R{(\mathit{\text{z}})}^2-{\left({\displaystyle \frac{D}{2}}\right)}^2\right]}}\text{d}\mathit{\text{z}}\\ & +{\displaystyle {\int }_{-\frac{h}{2}}^{\frac{h}{2}}\frac{4{\mathit{\text{z}}}^2+2h\mathit{\text{z}}}{\text{π}{E}_{\text{s}}\left[R{(\mathit{\text{z}})}^4-{\left({\displaystyle \frac{D}{2}}\right)}^4\right]}}\text{d}\mathit{\text{z}}.\end{array}

The initial stiffness provided by the steel tube is expressed as

(8)\begin{array}{rl}{K}_1=& \frac{1}{\mathrm{\Delta }}\\ =& \frac{1}{{\displaystyle {\int }_{-\frac{h}{2}}^{\frac{h}{2}}\frac{4(1+{\upsilon }_{\text{s}})}{\text{π}{E}_{\text{s}}\left[R{(\mathit{\text{z}})}^2-{\left({\displaystyle \frac{D}{2}}\right)}^2\right]}\text{d}\mathit{\text{z}}+{\displaystyle {\int }_{-\frac{h}{2}}^{\frac{h}{2}}\frac{4{\mathit{\text{z}}}^2+2h\mathit{\text{z}}}{\text{π}{E}_{\text{s}}\left[R{(\mathit{\text{z}})}^4-{\left({\displaystyle \frac{D}{2}}\right)}^4\right]}\text{d}\mathit{\text{z}}}}}.\end{array}

3.2 Yield force calculation

The yield force F_y represents the bearing capacity of the damper when the elastic phase of the damper terminates and the damper begins to enter the plastic phase. The von Mises shape change energy density theory was adopted as the basis for judging the yield of steel tubes. This theory assumes that the shape change energy density causes the material to yield regardless of the stress state and that provided the shape changes, the energy density at a certain point in the component reaches the limit value of the material, at which time material plastic yielding occurs.

The yield condition is expressed as [42–45]

(9)\sqrt{\frac{1}{2}[{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_1-{\sigma }_3)}^2+{({\sigma }_2-{\sigma }_3)}^2]}=\sigma ,

where {\sigma }_1, {\sigma }_2, and {\sigma }_3 denote the three principal stresses at certain points in the steel tube. Because the out-of-plane force of the damper is zero, only the plane stress state is considered; the stress state of the microelement is shown in Fig.5(c).

(10)\{\begin{array}{c}{\sigma }_1=\frac{{\sigma }_x+{\sigma }_{\mathit{\text{z}}}}{2}+\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_{\mathit{\text{z}}}}{2}\right)}^2+{\tau }_{x\mathit{\text{z}}}^2},\\ {\sigma }_2=0,\\ {\sigma }_3=\frac{{\sigma }_x+{\sigma }_{\mathit{\text{z}}}}{2}-\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_{\mathit{\text{z}}}}{2}\right)}^2+{\tau }_{x\mathit{\text{z}}}^2}.\end{array}

The direction angle \theta of the main plane is expressed as

(11)\tan (2\theta )=-\frac{2{\tau }_{x\mathit{\text{z}}}}{{\sigma }_x-{\sigma }_{\mathit{\text{z}}}}.

Because the damper is not subjected to an axial force, {\sigma }_{\mathit{\text{z}}}=0. Therefore, Eq. (9) becomes

(12)\begin{array}{rl}\sigma =& \sigma (x,\mathit{\text{z}})=\sqrt{\frac{1}{2}[{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_1-{\sigma }_3)}^2+{({\sigma }_2-{\sigma }_3)}^2]}\\ =& \sqrt{{\sigma }_x^2+3{\tau }_{x\mathit{\text{z}}}^2}.\end{array}

Material mechanics is considered for deriving the normal stress and shear stress of the steel tube, which are expressed as follows:

(13){\sigma }_x=\frac{M\cdot x}{I_y},

(14){\tau }_{x\mathit{\text{z}}}=\frac{V\cdot S_y^{\ast }}{I_y\cdot b},

where b is the section width of the steel tube in the y-direction, as shown in Fig.5(d), and is calculated as follows.

(15)

S_y^{\ast } is the first moment of the cross-sectional area of the steel tube to the neutral axis and is calculated as follows.

(16)

Based on Eq. (12), the von Mises stress is calculated as follows.

(17){\sigma }_{max}=max(\sigma (x,\mathit{\text{z}})).

The yield force F_{\mathrm{y}} can be calculated as follows:

(18)F_{\text{y}}={\sigma }_{\text{y}}\cdot \frac{1}{{\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}}}=\frac{{\sigma }_{\text{y}}}{\mathrm{m}\mathrm{a}\mathrm{x}(\sigma (x,\mathit{\text{z}}))},

where {\sigma }_{\text{y}} is the yield stress of the steel tube material obtained from material property tests.

4 Experimental test preparation

In practical applications, dampers are designed to enter the plastic state and participate in energy dissipation through plasticity development. However, the plastic performance is difficult to describe using the aforementioned elastic derivations. Hence, quasi-static tests were conducted to demonstrate the cyclic performance and energy-dissipation ability of the CSTD. The accuracy of the design formula for the initial stiffness and yield force is examined.

4.1 Specimen design and fabrication

Tab.1 provides information regarding the test specimens. The heights of the specimen, weakened segment, embedded segment, and transition segment were 330, 240, 15, and 15 mm, respectively. The connecting endplate was a square, and its length and thickness were 300 and 30 mm, respectively. Two steel tubes of diameter D and wall thickness t₀ – t₁, namely, 100 mm (5–12 mm) and 101.2 mm (4.4–11.4 mm), were selected. The energy dissipation performance, deformation capacity, failure mode, and effect of steel tube wall thickness on the mechanical properties of the CSTD were investigated experimentally.

The tubes were prepared using of 20# seamless steel tubes based on Chinese standards [46], whereas the connecting end plates were fabricated using ordinary structural steel, Q345. The material properties of the steel tube listed in Tab.2, including the yield stress, tensile stress, and elongation at break, were obtained via uniaxial tensile testing [47–50].

4.2 Loading protocol and measurement plan

The cyclic loading test system used in this study, as shown in Fig.7, includes an actuator, a four-link loading device, and a reaction wall (floor). The four-link loading device comprises two rigid beams and columns connected by four hinged supports using high-strength bolts. The damper is connected to the top and bottom piers using high-strength bolts, and the pier is fixed to the loading device. The bottom beam is securely fixed on a firm reaction floor. In addition, one end of the actuator is connected to the top beam and the other end is fixed to the reaction wall. The displacement range of the actuator is ±250 mm, and the output force is −1000 to 3000 kN.

Because the CSTD absorbs seismic energy cyclically, quasi-static cyclic loading was adopted in the test. According to China’s JGJ 297-2013 [51] and JGJ/T 101-2015 [52], the target displacement should be set as the integral time of the yield displacement, and each loading level after the yielding of the damper should be performed for three cycles. Therefore, the yield displacement Δy of the CSTD was calculated as 1 mm, as shown in Fig.8. The loading history was repeated with further increments until the specimen bearing capacity force decreased to 85% of the maximum value.

Fig.9 shows the measurement plan for the test. The ejector-rod displacement meter (D) was fixed to the lower connecting plate of the damper. The ejector rod was in contact with the upper connecting plate used to measure the horizontal displacement of the damper. Fig.9(a) and 9(c) show the detailed strain gauge distributions used to monitor strain development on the damper surface.

5 Test result and discussion

5.1 Hysteretic performance

The hysteretic curves of the two specimens are shown in Fig.10. The two specimens exhibited stable pump hysteretic loops after yielding. The hysteresis curves of CSTD-1 and CSTD-2 were parallelograms, and the CSTD-1 specimen exhibited clear strain-hardening behavior. Compared with CSTD-2, CSTD-1 indicated significantly higher bearing capacity and stiffness. The maximum bearing capacity of CSTD-2 began to decrease when the loading displacement was 12 mm; however, the damper failed when the loading displacement reached 16 mm. Meanwhile, the maximum bearing capacity of CSTD-1 did not decrease until a loading displacement of 16 mm; however, the damper failed when the loading displacement reached 20 mm. The ultimate displacement capacity was only 16 mm because the elongation of the tube material was small and the CSTD size in the test was not the final size. The ultimate displacement of the CSTD can be increased by selecting a tube material with a larger elongation and by optimizing the CSTD size. The results show that CSTD-1 offers better cyclic performance and energy dissipation capacity than CSTD-2.

5.2 Skeleton curves and strength calculation comparisons

The skeleton curves of the two specimens, as shown in Fig.11(a), and were obtained by connecting the points at the maximum force during each amplitude cycle [53]. The stiffness curves of the two specimens are shown in Fig.11(b), and the stiffness values can be obtained as follows:

(19)K_{\text{eqv}}=F_i^{+}/X_i^{+}\mathrm{o}\mathrm{r}{K}_{\text{eqv}}=F_i^{-}/X_i^{-},

where K_{\mathrm{e}\mathrm{q}\mathrm{v}} is the stiffness at loading level i, F_i is the peak load of loading stage i, and X_i is the corresponding maximum displacement of the ith level loading.

Specimens CSTD-1 and CSTD-2 exhibited different stiffness values after the yielding and ultimate bearing capacities were reached. Additionally, the maximum bearing capacity and initial stiffness of CSTD-1 were 75.35% and 13.32% higher than those of CSTD-2, respectively.

A comparison of the calculated initial stiffness and yielding force with the test data are shown in Tab.3. The ratios of the initial stiffness and yield force calculation results to the test results were within 10%. The comparison shows that the theoretical stiffness and yield force calculations are similar the test results. Therefore, the theoretical derivations based on the assumptions are reasonable.

5.3 Energy dissipation ability and strain distribution

The energy-dissipation capacity can be measured using the equivalent viscous damping ratio \xi and cumulative energy dissipation, which can be calculated from the hysteresis curves of the specimens. The calculation of the equivalent viscous damping ratio is illustrated in Fig.12.

The equivalent viscous damping ratio is expressed as

(20)\xi =\frac{S_{\mathrm{A}\mathrm{B}\mathrm{C}+\mathrm{C}\mathrm{D}\mathrm{A}}}{2\text{π}\cdot S_{\mathrm{O}\mathrm{B}\mathrm{E}+\mathrm{O}\mathrm{D}\mathrm{F}}}.

The cumulative energy dissipation of the specimens is calculated as the area within the load–displacement curve at the specified displacement. As each load case is repeated thrice, the total energy dissipated by the specimen for all displacement cycles represents the cumulative energy dissipation.

Fig.13(a) shows the equivalent viscous damping ratio, and Fig.13(b) shows the cumulative energy dissipation vs. the cumulative displacement. The results show that the loading displacement was 1 mm, the equivalent viscous damping ratio exceeded 0%, and the equivalent viscous damping ratios at a displacement of 16 mm for specimens CSTD-1 and CSTD-2 were 43.98% and 46.75%, respectively. This indicates that the displacement of the specimens for entering the dissipation energy state was small. The equivalent viscous damping ratio exceeded 40% under a large displacement, which indicates excellent energy consumption. The cumulative energy consumption and cumulative displacement of CSTD-1 were significantly higher than those of CSTD-2, i.e., approximately 56% higher than those of CSTD-2 at the maximum displacement level. Hence, CSTD-2 exhibited better energy consumption.

During the tests, the strain development was monitored using strain gauges. Results obtained using the strain gauge results are shown in Fig.14, where the two dashed black lines indicate that the strain was in the elastic phase. Strain gauge F2 first indicated the yield strain, and the corresponding displacement was approximately 1 mm. The yield strains indicated by end strain gauges F1, F3, S1, and S3 appeared later than that of F2, indicating that the cambered surface structure can ensure the reliability of the end connection and accurately control the energy-dissipation position of the damper. Under the same level of lateral displacement, the strains indicated by F3, S2, and S3 of CSTD-2 appeared one direction and exhibited larger values compared with those of CSTD-1. This is because the buckling deformation of CSTD-2 was greater than that of CSTD-1, which will be discussed in more detail in Section 5.4.

5.4 Deformed state and failure mode

The specimens were loaded under the standard cycling protocol until fracture failure; the experimental results are shown in Fig.15 and Fig.16. All specimens exhibited similar behaviors, and the failure process of the specimens can be categorized into three stages.

1) Elastic stage: The cambered surface steel tube remains elastic without any specific behavior and is characterized by a rapidly increasing damper force and a constant stiffness.

2) Buckling stage: The slight buckling becomes more severe as the horizontal displacement increases. All the local buckling of the specimen occurs in the middle of the cambered surface steel tube. Buckling first occurs on the sides and then extends gradually to the entire middle of the steel tube. The buckling displacement of CSTD-1 occurred later than that of CSTD-2.

3) Failure stage: The lateral displacement continues to increase during this stage, and small cracks appear in the severely buckled section. Subsequently, cracks penetrate the middle of the steel tube. The experiment is terminated when the bearing capacity is reduced to 85% of the ultimate bearing capacity.

The failure positions of the two specimens were located in the middle of the specimen. The CSTD-1 specimen presented a “Z”-shaped crack, whereas the CSTD-2 specimen presented a “-”-shaped crack. This is due to the severe local buckling of CSTD-2 and that fracture did not develop at either end. No visible peaks were observed at the ends, indicating that the cambered surface effectively ensured the reliability of the end connection.

6 Finite element analysis

6.1 Numerical modeling

As shown in Fig.17, a three-dimensional finite element model of the CSTD was established. A numerical model was developed and analyzed using the ABAQUS software. An eight-node linear hexahedron (C3D8R) was used to simplify the integral solid element for the connecting plate and cambered surface steel tube, which can effectively avoid the shear self-locking problem and offers high good calculation accuracy. Considering the balance between computational accuracy and efficiency, the grid sizes of the steel tube and connecting plate were set as 8 and 10 mm, respectively.

The steel tube and end plates were welded together. Therefore, the contact between the steel tube and connecting plate is defined by the “tie”. The contact method was face-to-face, and the contact surfaces of the connecting plate and steel tube were the master and slave surfaces, respectively. To simulate an actual situation, the rotations and displacements of the bottom-connecting plate were fixed. The upper connection plate was constrained in all directions except for the x-axis loading to realize cyclic reciprocating displacement control loading. The analysis method was a static-general method.

A combined model combining kinematic hardening and isotropic hardening was established in ABAQUS to model the steel tube. The model was used to simulate the nonlinear characteristics of the steel tube under cyclic loading [54,55]. The hardening behavior and Bauschinger effect of metals can be simulated using the combined model, and the conditions for the numerical model are as follows:

(21)\begin{array}{rl}& \alpha =\sum _k^n{\alpha }_k=\sum _k^n\frac{C_k}{{\gamma }_k}(1-{\text{e}}^{-{\gamma }_k{\overline{\epsilon }}^{\text{pl}}}),\\ & {\sigma }^0={\sigma |}_0+Q_{\mathrm{\infty }}(1-{\text{e}}^{-b{\overline{\epsilon }}^{\text{pl}}}),\end{array}

where {\overline{\epsilon }}^{\text{pl}} is the equivalent plastic strain; C_k and {\gamma }_k are the kinematic hardening modulus and rate factor, respectively; Q_{\mathrm{\infty }} is the maximum value of the yield surface change; and b represents the rate at which the size of the yield surface changes with plasticity, which can be calibrated via material property experiments. The material for the connecting plate was Q345 steel, and a bilinear constitutive model was adopted. All steels used in this model had a Young’s modulus E of 206 GPa. The parameters of the combined hardening model are presented in Tab.4.

6.2 Model verification and analysis

Based on the settings above, the force–displacement curves and deformation modes of the numerical results were compared with those of the experiments, as shown in Fig.18(a) and Fig.18(b), respectively. The hysteresis curve and deformation modes obtained from the simulation were consistent with those of the test specimens, thus confirming the reliability of the model and its settings. Hence, the proposed finite element model is well-calibrated and can thus be used for the subsequent analysis and parametric studies (detailed in Subsection 6.3).

Fig.19 shows the stress and maximum buckling displacement of the steel tube at a loading displacement of 16 mm, which could not be monitored during the test. Based on the figure, the maximum stress of CSTD-1 is 1.17 times that of CSTD-2, and the maximum buckling displacement of CSTD-1 is 0.33 times that of CSTD-2. This implies that the reasonable size of the cambered surface steel tube can reduce the buckling displacement and improve the energy dissipation of the steel tube.

6.3 Parametric study

A systematic parametric analysis of the behavior of the CSTD under cyclic loading was conducted using the finite element model established. In particular, the weakening ratio 1 − t₀/t₁, height-to-diameter ratio H_w/D, and diameter-to-thickness ratio D/t₀ were investigated in the same order to discuss their effects on the energy dissipation and mechanical characteristics of the damper.

Tab.5 presents the dimensions of the 21 CSTD models used for the analysis. To account for engineering applications, the maximum inner diameter of all model steel tubes used in the parametric study did not exceed 200 mm.

In the parametric analysis, the equivalent plastic strain (PEEQ) was used as an indicator to examine the strain distribution of the damper. The PEEQ is expressed as a cumulatively increasing variable governed by the initial equivalent plastic strain {\overline{\epsilon }}^{\text{pl}} and the plastic strain rate tensor {\dot{\epsilon }}^{\text{pl}}, as shown in Eq. (22).

(22)PEEQ={{\overline{\epsilon }}^{\text{pl}}|}_0+{\int }_0^t\sqrt{\frac{2}{3}{\dot{\epsilon }}^{\text{pl}}:{\dot{\epsilon }}^{\text{pl}}}\text{d}t.

6.3.1 Effect of 1 − t₀/t₁

Seven CSTD models with different 1 − t₀/t₁ values (0.2 to 0.8) were established for analysis, and the results are shown in Fig.20. A comparison of the CSTD envelope curves with different 1 − t₀/t₁ values (see Fig.20(a)) indicates that the force of the specimens increased significantly with 1 − t₀/t₁. The ultimate force of the specimens with 1 − t₀/t₁ = 0.8 was 64.08% lower than that of the specimens with 1 − t₀/t₁ = 0.2.

Fig.20(b) shows the PEEQ curve, which presents the envelope values of the black and red lines on the surface of the cambered surface steel tube. The results show that the middle maximum PEEQ of the specimens improved significantly, whereas the end maximum PEEQ of the specimens decreased as 1 − t₀/t₁ increased. The maximum PEEQ of the specimens with 1 − t₀/t₁ = 0.2 was 63.33% lower than those with 1 − t₀/t₁ = 0.2. At 1 − t₀/t₁ = 0.2, the PEEQ value at the end was greater than that in the middle, indicating that the damper was broken at the end, which defeated the intended purpose; therefore, 1 − t₀/t₁ = 0.2 should not be used.

Based on the change in the mechanical properties and the PEEQ values for different 1 − t₀/t₁ values, the 1 − t₀/t₁ value should be set between 0.3 and 0.5 during design to ensure desirable mechanical properties and a uniform plastic distribution.

6.3.2 Effect of H_w/D

Based on the recommended optimal range of 1 − t₀/t₁, 1 − t₀/t₁ = 0.3 was selected to analyze the effect of H_w/D. The responses of the CSTD specimen with various H_w/D values (1.8, 2.1, 2.4, 2.7, 3.0, 3.3, and 3.6) under cyclic loading were analyzed. Fig.21 shows the envelope and PEEQ curves. Based on a comparison of the CSTD envelopes for different H_w/D values in Fig.21(a), the ultimate forces of the CSTD with H_w/D values of 1.8 and 2.1 were 14.39% and 6.45% higher than those of the CSTD with a H_w/D value of 2.4, respectively. As H_w/D incraseed from 2.4 to 2.7, 3.0, 3.3, and 3.6, the ultimate force decreased by 7.19%, 14.74%, 20.84%, and 27.20%, respectively.

Based on a comparison of the PEEQ curves of different H_w/D values (see Fig.21(b)), the PEEQ value in the middle of the damper decreased gradually as H_w/D increased. In the middle of the steel tube, the maximum PEEQ values of the CSTD with H_w/D values of 1.8 and 2.1 were 84.62% and 40.68% higher than those of the CSTD with a H_w/D value of 2.4, respectively. As the maximum PEEQ increased from 2.4 to 2.7, 3.0, 3.3, and 3.6, the maximum PEEQ decreased by 29.79%, 50.55%, 68.82%, and 79.67%, respectively. The results indicate that the ultimate force and maximum PEEQ of the CSTD decreased as H_w/D increased.

Based on the mechanical properties and PEEQ value changes of the CSTD under different H_w/D values, the H_w/D value should be between 2.1 and 2.7 when designing the CSTD.

6.3.3 Effect of D/t₀

Based on the findings obtained thus far as well as 1 − t₀/t₁ = 0.3 and H_w/D = 2.4, an analysis was performed based on D/t₀ values of 9.5, 11.9, 14.3, 16.7, 19.0, 21.4, and 23.8. Based on a comparison of different D/t₀ envelope curves (see Fig.22(a)), the ultimate forces of the CSTD with D/t₀ values of 9.5 and 11.9 were 39.42% and 19.37% lower than that of the CSTD with a D/t₀ value of 14.3, respectively. The ultimate forces of the CSTD with D/t₀ values of 16.7, 19.0, 21.4, and 23.8 were 18.54%, 36.63%, 54.59%, and 74.17%, respectively, which were higher than those of the model with a D/t₀ value of 14.3.

The PEEQ curves for different D/t₀ values (see Fig.22(b)) show that as D/t₀ increased, the middle maximum PEEQ value initially increased rapidly and then stabilized gradually. The maximum PEEQ of the CSTD with D/t₀ values of 9.5 and 11.9 were 39.17% and 20.08%, respectively, which were lower than that of the CSTD with a D/t₀ value of 14.3. The maximum PEEQ of the CSTD with D/t₀ values of 16.7, 19.0, 21.4, and 23.8 were 13.59%, 22.61%, 28.80%, and 30.17%, respectively, which were higher than that of the model with a D/t₀ value of 14.3. However, at D/t₀ = 9.5, the maximum PEEQ was recorded at the end of the steel tube, which indicates that damage occurred therein; hence, D/t₀ = 9.5 should not be used in the design.

Based on the envelope and PEEQ curves of the CSTD under different D/t₀, D/t₀ should be in the range of 11.9–14.3. The analysis above indicates that the weakening ratio 1 − t₀/t₁ exerts the most significant effect on the CSTD, in particular on its bearing capacity, stress distribution, diameter-to-thickness ratio D/t₀, and height-to-diameter ratio H_w/D, in that order.

6.3.4 Calculation comparisons

The initial stiffness and yield force obtained from the finite element analysis and theoretical calculations are presented in Tab.6 and Fig.23, respectively. As shown in Tab.6, the differences in the initial stiffness and yield force between the simulation results and theoretical calculation are within 15%. This indicates that the design method can yield accurate values of the initial stiffness and yield force and that the theoretical calculation formula can be used in the subsequent analysis and design.

6.4 Comparison with steel tube damper

The aforementioned parameter analysis of the CSTD provides the recommended value range of the key parameters. The theoretical calculation results were compared with the results obtained using the finite element method to verify the feasibility of the design method. Specimen CSTD indicated in this section was designed within the range of the parameter values, whereas the steel tube damper (STD) with the same initial stiffness as the damper was designed using the theoretical design method. The CSTDs were further compared and investigated. The dimensions and mechanical parameters of the specimens are presented in Tab.7.

Fig.24 shows the different positions of the PEEQ curve of the steel tube at the end of loading. The middle maximum PEEQ of the CSTD was 38.24% higher than that of the STD. The end maximum PEEQ of the CSTD was 31.94% lower than that of the STD. The PEEQ of the STD specimen was concentrated at the end of the steel tube, which significantly affected its low-cycle fatigue performance and caused the STD to stop functioning prematurely. When the CSTD was optimized, the energy dissipation area was transferred to the middle and the distribution was more uniform, which improved the low-cycle fatigue performance.

The von Mises stress clouds of the two dampers at the end of loading are shown in Fig.25. The yield area of the CSTD steel tube was significantly larger than that of the STD steel tube. Compared with the STD specimen, the CSTD specimen did not indicate any stress concentration at the end of the steel tube. Hence, the stress distribution and material utilization rate of the steel tube improved significantly.

Fig.26 shows the buckling displacement amplitudes of the two dampers at the end of loading. Compared with the result shown in Fig.19, the buckling displacement of the CSTD decreased significantly. Meanwhile, the buckling displacement of the optimized CSTD was smaller than that the STD, thus indicating that the optimized cambered surface steel tube exhibited better deformation.

7 Conclusions

A CSTD fabricated from a typical steel tube to provide lateral stiffness and energy dissipation was presented herein. Formulas for the initial stiffness and yield force were derived. The hysteretic curves, stiffness degradation, energy dissipation, and failure modes based on test observations and results were analyzed. Subsequently, a detailed finite element model was established and validated against the test results; parametric analyses were performed the analyze the effects of the weakening ratio 1 − t₀/t₁, height to diameter ratio H_w/D, and diameter-to-thickness ratio D/t₀; and the optimal size of the CSTD was determined. The results of this study can be summarized as follows.

1) The cambered surface steel tube ensured the reliability of the end connection, realized the controllability of the yield and failure positions, and fully utilized the energy dissipation of the steel tube.

2) The CSTD exhibited stable pump hysteretic loops, high ductility, and excellent energy dissipation capacity. It began to dissipate energy only when loaded to 1 mm. The failure modes of the CSTD under cyclic loading primarily included local buckling and steel tube fractures in the middle section.

3) The design formulae for the initial stiffness and yield force agreed well with the test results. The maximum error between the test result and theoretical calculation was within 15%.

4) A systematic parametric analysis of the CSTD under cyclic loading was conducted. The 1 − t₀/t₁ value is recommended to be 0.3–0.5; H_w/D, 2.1–2.7; and D/t₀, 11.9–14.3. In these ranges, the CSTD exhibited excellent energy dissipation performance, small buckling displacements, uniform strain distributions, and improved energy dissipation efficiency.

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