A simplified analytical model for U-shaped steel dampers considering horizontal bidirectional deformation

Abstract

Hysteresis steel dampers are widely used in earthquake-resistant structures, where some of them are anisotropic and capable of sustaining earthquake-induced bidirectional deformation. In this paper, a simplified analytical model is proposed for simulating the hysteretic behavior of U-shaped steel dampers with horizontal bidirectional deformation. The proposed model is composed of a series of shear springs with different nonlinear characteristics in a radial configuration, and the Menegotto–Pinto hysteresis model is employed to represent the hysteretic characteristics of the springs. The mechanical and shape-related parameters of the hysteresis model are set according to the multi-directional deformation characteristics of steel dampers. With the aim of validating the effectiveness and applicability of the analytical model, a U-shaped steel damper was used as an example. The pseudo-static hysteretic characteristics of the steel damping element were analyzed and the elasto-plastic seismic response of a curved bridge featuring a steel hysteresis device was investigated. The results showed that the proposed model is sufficiently accurate to simulate the hysteretic behavior of U-shaped steel dampers, and thus provides a practical method to assess U-shaped steel dampers through seismic response analysis.

Appendices

Appendix 1: Effectiveness of the FE model

Finite element method (FEM) is widely recognized as a mainstream for simulating mechanical behavior of structures. With particular regards to structural components made of steel with well-defined boundary conditions, FEM can be regarded as an alternative to experiments. For instance, by means of numerical simulations, Kato and Kim (2006) and Deng et al. (2015) investigated the influence of geometric parameters on hysteretic behavior of J-shaped and U-shaped steel dampers, respectively.

In this paper, the hysteretic behavior of a U-shaped steel damper selected as the example were assessed by a fine large-deformation elastic–plastic finite element (FE) model. To validate the effectiveness of the numerical model, analysis results obtained using the FE model were compared against experimental results reported in literature (Jiao et al. 2015), and then the sensitivity study of the FE model was carried out. A general commercial software ABAQUS 6.13 was employed in the above analyses.

The FE model of the investigated U-shaped steel specimen, geometric and material parameters of which were obtained from extant literature, is shown in Fig. 22. Note that the width of the damping element gradually changes from 45 mm at the end of the bent part to 60 mm at one end of the connecting part. At the same time, however, widths of the above two parts themselves remain constant. The model was constructed with an eight-node solid element (C3D8R) and a mesh layout divided into 6 and 10 layers along the thickness and width directions, respectively. The total number of mesh-elements was 13,260. SN490B (a Japanese high-ductile steel), whose yield strength and elastic modulus measure 378 and 1.989 × 10⁵ MPa, respectively, was used during testing, and a nonlinear mixed hardening model was selected as the constitutive hysteresis model.

Fig. 22

Finite element model of investigated U-shaped steel specimen

The comparative results of hysteretic curves between the experimental specimen and the FE model of a single U-shaped damping element subjected to unidirectional cyclic loading are shown in Fig. 23. As can be seen from the figure, the resulting hysteretic loops of the FE model demonstrate good agreement with experimental data, thereby verifying the effectiveness of the numerical model.

Fig. 23

Comparison of hysteretic curves between the experimental specimen and the FE model under unidirectional cyclic loading. a x direction (in-plane direction). b y direction (out-of-plane direction)

In addition, hysteretic curves obtained from three models employing different grid-partition criteria were compared to perform sensitivity analysis of the FE model. Table 3 lists details of the grid partition, wherein “model A” “model B” and “model C” represent the models with low, medium, and high meshing densities, respectively. Comparison of results are shown in Fig. 24, wherein it can be seen that the three curves tend to almost coincide with each other. This illustrates the high computational convergence of the FE model as well as the fact that the mesh density has negligible effect on the elasto-plastic hysteretic behavior of damping elements.

Table 3 Griding partition details of the FE model

Fig. 24

Influence of meshing size on the results of the FE model. a x direction. b y direction

Appendix 2: Effectiveness of the MMSS model

To confirm the effectiveness of the MMSS model more persuasively, hysteretic loops of the proposed analytical model were compared against those obtained experimentally in this section.

As mentioned in the main text of the manuscript, four directional deformation performances of a damping element are required to determine related parameters of the MMSS model. However, only two directional (in-plane and out-of-plane direction) experimental results were provided in the literature (Jiao et al. 2015), which are insufficient for identifying related parameters. In accordance with the conclusion of Appendix 1 that the numerical model can accurately simulate hysteretic behavior of the damping element, related parameters of the proposed MMSS model, as listed in Table 4, were determined using calculation results of the FE model. Simulation results obtained for cases of loading along the x, y, 45° and − 45° directions are shown in Fig. 25.

Table 4 Nonlinear parameters of springs in the MMSS model

Fig. 25

Simulation results of the MMSS model. a x direction. b y direction. c 45° direction. d − 45° direction

Based on the MMSS model obtained above which represent one damping element, hysteretic performances of a U-shaped damper unit consisting of two orthogonal damping elements with geometric parameters identical to those in Fig. 22 were analyzed. Figure 26 shows the relationship between the element and the corresponding proposed model. With the aim of simulating overall hysteretic behavior of the damper unit, two analytical models with the identical parameters but different orientations were computed simultaneously. In the figure, x and y represent local directions for a single damping element; 0° and 90° represent global directions of the U-damper unit.

Fig. 26

Diagram of the U-shaped damper unit

Figure 27 shows the comparative results of hysteretic curves for the damper unit subjected to unidirectional cyclic loading along the 0° direction. The experimental data were got from the drawing line in the literature (Jiao et al. 2015) by a shareware try-before-you-buy software called GetData Graph Digitizer. Note that only the second hysteresis loop was compared owing to limited data availability. As can be seen from the figure, both the FE model and the MMSS model tend to coincide with experimental results.

Fig. 27

Comparison of hysteretic curves among the experimental specimen, the FE model and the MMSS model under unidirectional cyclic loading

To further testify the effectiveness of the MMSS model, hysteretic curves for the first two cycles of the damper unit subjected to circular and elliptical loading were compared. Note that experimental data for bidirectional loadings were, once again, obtained from literature (Ene et al. 2016a) using GetData Graph Digitizer. Patterns of the bidirectional loading are shown in Fig. 28. The loads were first implemented to the target radius along the 90° direction of the unit, then conducted in the clockwise direction along the specified path. From the comparative results shown in Fig. 29, it can be seen that the analytical results of the fine FE model can still basically coincide with the experimental curves under conditions of bidirectional loading. Also, resulting curves obtained using the proposed analytical model, which is a simplified method, demonstrate good agreement with both the numerical and experimentally derived curves.

Fig. 28

Patterns of the bidirectional loading. a Circular loading. b Elliptical loading

Fig. 29

Comparison of hysteretic curves among the experimental specimen, the FE model and the MMSS model under bidirectional loading. a Circular loading. b Elliptical loading

Based on all of the above analyses, it can be inferred that the proposed MMSS model can be utilized as a simplified analytical method demonstrating good simulation accuracy.