Damage Evaluation Method for Steel Beams Subjected to Cyclic Loading

Abstract

Evaluating the cumulative damage of seismic members is an important issue in seismic design. Steel beams are the primary components of seismic moment-resisting frames, wherein the story drift angle and ductility factor with respect to the number of cycles are used as indices of the degree of damage. The former is widely used in the United States and several other countries, whereas the latter is used in Japan. Additionally, these indices are used to express the plastic deformation capacity up to the ultimate state determined by fracture or other failure modes. In this study, an appropriate evaluation method was developed based on the numerical analysis of sub-assemblies for determining the cumulative damage and plastic deformation capacity of steel beams. The analytical results indicate that a unified evaluation is possible based on the relationship between the ductility factor of the beam of each steel grade and the number of cycles to fracture. The proposed method can be used regardless of geometric conditions, except in the case of short-span beams.

Appendices

Appendix 1

1.1 Hysteresis Model of Steel (Yamada & Jiao, 2016; Yamada et al., 2020)

Previous studies have reported the details of the hysteresis model of steel (Yamada & Jiao, 2016; Yamada et al., 2020), which can be summarized as follows.

• The hysteresis curve under cyclic loading of steel was decomposed into the skeleton curve; Fig. 

  Fig. 13

  

  Explanation of the hysteresis model

  13a depicts the Bauschinger and the elastic unloading parts.

• The skeleton curve was formed by sequentially connecting the paths of the loads that exceeded the maximum load attained in the preceding cycle. This was adopted in the hysteresis model as the true stress–true strain relationship derived from the nominal stress–nominal strain relationship of the coupon test using Equations (3) and (4).

  $${}_{t}^{{}} \sigma = \left( {1 + {}_{n}^{{}} \varepsilon } \right)\Delta {}_{n}^{{}} \sigma$$

  (3)
  $${}_{t}^{{}} \varepsilon = {\text{ln}}\left( {1 + {}_{n}^{{}} \varepsilon } \right)$$

  (4)

  where ${}_{t}^{{}} \sigma$ denotes the true stress, ${}_{n}^{{}} \sigma$ indicates the nominal stress, ${}_{t}^{{}} \varepsilon$ represents the true strain, and ${}_{n}^{{}} \varepsilon$ denotes the nominal strain.

• The Bauschinger parts were softened owing to the Bauschinger effect and modeled as bilinear (Fig. 13b). The model involves stress when entering the skeleton curve (${}_{t}^{{}} \sigma {}_{Bs}^{{}}$), elastic stiffness (E), plastic strain increments in each cycle of the Bauschinger part ($\Delta {}_{t}^{{}} \varepsilon {}_{B}^{{}}$), and stress of the stiffness changing point (${}_{t}^{{}} \sigma {}_{E}^{{}}$). Among them, ${}_{t}^{{}} \sigma {}_{Bs}^{{}}$ was set as the maximum stress from the preceding cycle, and $\Delta^{{}} \varepsilon_{B}^{{}}$ can be calculated using Equations (5) and (6).

In the case of 400 and 490 N/mm² class structural steel

$$\Delta_{t} \varepsilon_{B} = 0.33\sum \Delta_{t} \varepsilon_{s}$$

(5)

For 590 N/mm² class structural steel,

$$\Delta_{t} \varepsilon_{B} = 0.72\sum \Delta_{t} \varepsilon_{s}$$

(6)

Here, $\sum \Delta \varepsilon$ denotes the cumulative plastic strain of the skeleton curve.

The stress of the stiffness changing point of the Bauschinger part ($_{t} \sigma_{E}$) can be calculated using Equation (5).

For 400, 490, and 590 N/mm² class structural steel,

$$_{t} \sigma_{E} = 0.67\Delta_{t} \sigma_{Bs}$$

Figure 13c depicts an example (tensile start) of the hysteresis to understand the model.

• The compressive side of the first cycle was considered as the skeleton curve owing to the initial compressive stress experienced by the steel material (loop 3–6 in Fig. 13c). The softening caused by the plastic strain in the tensile side was considered by adopting the bilinear model of the Bauschinger part (loop 4–5 in Fig. 13c).

• The entering and unloading points in the skeleton curve were reset each time when the hysteresis loop entered a skeleton curve and remained unchanged until the loop entered the subsequent skeleton curve fragment. (loops 0–2, 3–6, 7–10, 11–14, and 15–18 in Fig. 13c).

• In the case of unloading within the plastic region of the Bauschinger part, the plastic region moved to the subsequent entering point of the skeleton curve before unloading (point 23 in Fig. 13c, which moves toward point 18).

Appendix 2

2.1 I Comparison of Analytical and Experimental Results of Beams Subjected to Cyclic Loading (Jiao & Yamada, 2020)

The analytical method proposed in this study can reproduce the behavior of the steel beam under cyclic loading, including strain history, by adopting the steel hysteresis model described in Sect. 2.3 and Appendix 1. The analytical and experimental results are compared in this Appendix. The experimental data used for the comparison were obtained from the cyclic loading tests of H-section steel beams connected to thick end plates (Tenderan et al., 2019). Figure

Fig. 14

Comparison of analytical and experimental results

14a depicts the shape of the specimen. The details of the beam end connection are identical to those of the typical Japanese shop-welding type incorporated in the analytical model in this study. The steel grade of the beam was 490 N/mm² (SN490; JIS, 2012). The experiment was performed under constant deformation amplitude conditions. Figure 14b illustrates a comparison of the analytical and experimental results of the load–deformation relationships. Additionally, the measurements of the plastic strain gauges on each flange were compared with the nominal analytical strain history results at the same position, as indicated in Fig. 14c. The experimental strain value of the flange was the average value of the gauges attached to the flange section. In addition to the load–deformation relationships and strain histories of the beam flange, the analytical results concurred with the experimental results.