The secant stiffness method is applied to the equivalent linearization of nonlinear system. Two types of hysteretic models and a set of 72 ground motions were considered. A new type of flag-shape hysteretic model was introduced which would be compared with the benchmark model.
The two hysteretic models which were used in the analyses were calibrated by the experimental data results to get a set of parameter α, β, and γ. The nonlinear behaviors of the two models were evaluated with Jacobsen’s equivalent damping approach which also adopted the secant stiffness method for the linearization process. The evaluation revealed that Jacobsen’s approach overestimated damping in long period range on other hand underestimated damping in short period range. Two equations to modify the area-based viscous damping by Jacobsen’s approach for each type of hysteretic model were proposed. The two hysteretic models have also been verified by comparing the pseudo-dynamics loading result with the analyses result.

The secant stiffness method is applied to the equivalent linearization of nonlinear system. Two types of hysteretic models and a set of 72 ground motions were considered. A new type of flag-shape hysteretic model was introduced which would be compared with the benchmark model.
The two hysteretic models which were used in the analyses were calibrated by the experimental data results to get a set of parameter α, β, and γ. The nonlinear behaviors of the two models were evaluated with Jacobsen’s equivalent damping approach which also adopted the secant stiffness method for the linearization process. The evaluation revealed that Jacobsen’s approach overestimated damping in long period range on other hand underestimated damping in short period range. Two equations to modify the area-based viscous damping by Jacobsen’s approach for each type of hysteretic model were proposed. The two hysteretic models have also been verified by comparing the pseudo-dynamics loading result with the analyses result.

[1] AASHTO. (2004). LRFD Bridge Design Specifications, American Association of State Highway and Transportation Officials, 3rd Ed., Washington, D.C.
[2] Blandon C.A. (2004). “Equivalent viscous damping equations for direct displacement based design.” A Dissertation Submitted in Partial of the Requirements for the Master Degree in Earthquake Engineering, Rose School, Pavia, Italy.
[3] Chopra A. K. (1995). Dynamics of structures, Prentice-Hall, New Jersey.
[4] Chou, C. C. and Hsu, C.P. (2008). “Hysteretic model development and seismic response of unbonded post-tensioned precast CFT segmental bridge columns.” Earthquake Engineering and Structural Dynamics, 37(6), 919–934.
[5] Dwairi, H. M., Kowalsky, M. J., and Nau, J. M. (2007). “Equivalent Damping in Support of Direct Displacement-Based Design.” Journal of Earthquake Engineering, 11(4), 1-19.
[6] FEMA-356. (2000). Prestandard and Commentary for the Seismic Rehabilitation of Buildings, Federal Emergency Management Agency, Washington, D.C.
[7] Herwanto, A. C. (2013). “Displacement-Based Design for precast segmental columns.” M.Sc. Thesis, Dept. of Civil and Construction Eng., National Taiwan University of Science and Technology (NTUST), Taipei, Taiwan.
[8] Ibarra L. F. and Krawinkler H. (2005). Global collapse of frame structures under seismic excitations, Report No. PEER 2005/06, Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA.
[9] Kwan W. P. and Bilington S. L. (2003). “Influence of Hysteretic Behavior on Equivalent Period and Damping of Structural Systems.” Journal of Structural Engineering, 129(5), 576-585.
[10] Lin, J. C., and Mo, Y. L. (2000). “The shear transfer behavior of precast prestressed hollow rectangular bridge columns.” M.Sc. Thesis, Dept. of Civil Engineering, National Cheng Kung University, Tainan, Taiwan (in Chinese).
[11] Otani, S. (1981). “Hysteretic Models of Reinforced Concrete for Earthquake Response Analysis.” Journal of Faculty of Engineering, University of Tokyo, 36(2), 407-441.
[12] Ou, Y. C. (2007). “Precast segmental post-tensioned concrete bridge columns for seismic regions.” A Dissertation Submitted in partial fulfillment for of the requirements of the degree of Doctor Philosophy in Faculty of the Graduate School of State University of New York at Buffalo.
[13] Ou, Y. C., Chiewanichakorn, M., Aref A. J., and Lee, G. C. (2007). “Seismic performance of segmental precast unbonded posttensioned concrete bridge columns.” Journal of Structural Engineering, ASCE, 133(11), 1636-1647.
[14] Ou, Y. C., Song, J. Wang, P., Adidharma, L., Chang, K., and Lee, G. (2014). ”Ground motion duration effects on hysteretic behavior of reinforced concrete bridge columns.” Journal of Structure Eng., 140(3), 04013065.
[15] Ou, Y. C., Tsai, M. S., Chang, K. C., and Lee, G.C. (2010). “Cyclic behavior of precast segmental concrete bridge columns with high performance or conventional steel reinforcing bars as energy dissipation bars.” Journal of Earthquake Engineering and Structural Dynamics, 39(11), 1181-1198.
[16] Ou, Y. C., Wang, P. H., Tsai, M. S., Chang, K. C., and Lee, G.C. (2009). “Large-scale experimental study of precast segmental unbonded post-tensioned concrete bridge columns for seismic regions.” Journal of Structural Engineering, ASCE, 136(3), 255-264.
[17] Priestley, M. J. N. and Grant, D. N. (2005). “Viscous Damping in Seismic Design and Analysis.” Journal of Earthquake Engineering, 9(SP2), 229-255.
[18] Priestley, M. J. N., Calvi, G. M. and Kowalsky, M. J. (2007). Displacement Based Seismic Design of Structures. IUSS Press, Pavia, Italy.
[19] Seo, C. Y., and Sause, R. (2005). “Ductility demands on self-centering systems under earthquake loading.” Journal of ACI Structure, 102(2), 275-285.
[20] Sullivan, T. J., Calvi, G. M., and Priestley, M. J. N. (2004). “Initial stiffness versus secant stiffness in displacement based design.” 13th World Conference of Earthquake Engineering (WCEE), No. 2888.
[21] Wang, P. H., Ou, Y. C., Chang, K. C., and Lee, G.C. (2008). “Large-Scale Seismic Tests of Tall Concrete Bridge Columns with Precast Segmental Construction.” Journal of Earthquake Engineering and Structural Dynamics, 37(12), 1449–1465.