Summaries

The obtained results may be summarized as follows:

• 1. The multiple impulse input of equal time interval can be a good substitute for the long-duration earthquake ground motion which is expressed in terms of harmonic waves. A closed-form expression can be derived for the maximum response of an elastic-perfectly plastic (EPP) single-degree-of-freedom (SDOF) model under the critical multiple impulse input.
• 2. While the critical set of input velocity amplitude and input frequency (timing of impulse) has to be computed iteratively for the multi-cycle sinusoidal wave, it can be obtained directly without iteration for the multiple impulse input by introducing a modified version (only the timing between the first and second impulses is modified so that the second impulse is given at the zero restoring force). The resonance was proved by using energy investigation. It was made clear that the critical timing of the multiple impulses can be characterized by the time attaining the zero restoring force in the unloading process. This decomposition of input amplitude and input frequency overcame the difficulty in finding the resonant frequency without repetition. This is one of the most original contributions in this chapter.
• 3. Since only free-vibration is induced under such multiple impulse input, a simple energy approach plays an important role in the derivation of the closed-form expression for a complicated elastic-plastic critical response. The energy approach enabled the derivation of the maximum critical elastic-plastic response without resorting to solving the equation of motion. The maximum elastic-plastic response after each impulse can be obtained by equating the initial kinetic energy given by the initial velocity to the sum of hysteretic dissipation and elastic strain energies. The critical inelastic deformation and the corresponding critical input frequency can be captured by the substituted multiple impulse input depending on the input velocity level. This is the second one of the most original contributions in this chapter.
• 4. The validity and accuracy of the theory explained in this chapter were assured through the comparison with the time-history response to the corresponding equivalent multi-cycle sinusoidal input. It was found that if the adjustment of both inputs is made by using the equivalence of the maximum Fourier amplitude and a modification based on the response correspondence at some points with different input levels for better correspondence in a wider range of input level, the maximum elastic-plastic deformation to the multiple impulse exhibits a good correspondence with that to the multi-cycle sinusoidal wave.
• 5. While the conventional methods (Caughey 1960a, b, Iwan 1961, 1965a, b) are aimed at constructing an equivalent linear structural model or solving transcendental equations iteratively to an unchanged input (sinusoidal input) to enable the simple approximate computation of complicated elastic-plastic responses, the method explained in the present chapter is aimed at finding an equivalent input model for an unchanged exact elastic-plastic model. The most significant difference between two approaches is that while the conventional methods require the repetition both in the computation of equivalent model parameters or the solution of transcendental equations for one input frequency and the computation of the resonant frequency giving the maximum response, the approach explained in this chapter does not require any repetition in the computation of the critical input timing (resonant frequency) and the critical response. The approach explained in this chapter also enabled the computation of the steady-state response for an EPP model that cannot be treated by the conventional approaches.

The approach for an EPP model can be extended to a bilinear hysteretic model with some modifications (see Chapter 6).

References

Abbas, A. M., and Manohar, C. S. (2002). Investigations into critical earthquake load models within deterministic and probabilistic frameworks, Earthquake Eug. Struct. Dyn., 31, 813-832.

Akehashi, H., Kojima, K., Noroozinejad Farsangi, E., and Takewaki, I. (2018). Critical response evaluation of damped bilinear hysteretic SDOF model under long duration ground motion simulated by multi impulse motion, Int. ]. of Earthquake and Impact Eng, 2(4), 298-321.

Akehashi, H., and Takewaki, I. (2020). Critical response of nonlinear base-isolated building considering soil-structure interaction, Proc. of the 17th World Conference on Earthquake Engineering (17WCEE), Sendai, Japan, September 13-18 (postponed).

Alavi, B., and Krawinkler, H. (2004). Behaviour of moment resisting frame structures subjected to near-fault ground motions, Earthquake Eng. Struct. Dyn., 33(6), 687-706.

Bertero, V. V., Mahin, S. A., and Herrera, R. A. (1978). Aseismic design implications of near-fault San Fernando earthquake records, Earthquake Eng. Struct. Dyn. 6(1), 31^12.

Bozorgnia, Y., and Campbell, K. W. (2004). Engineering characterization of ground motion, Chapter 5 in Earthquake engineering from engineering seismology to performance-based engineering, (eds.) Bozorgnia, Y. and Bertero, V. V., CRC Press, Boca Raton, FL.

Bray, J. D., and Rodriguez-Marek, A. (2004). Characterization of forward-directivity ground motions in the near-fault region, Soil Dyn. Earthquake Eng., 24(11), 815-828.

Caughey, T. K. (1960a). Sinusoidal excitation of a system with bilinear hysteresis,/. AppiMech., 27(4), 640-643.

Caughey, T. K. (1960b). Random excitation of a system with bilinear hysteresis,/. AppiMech., 27(4), 649-652.

Drenick, R. F. (1970). Model-free design of aseismic structures,/. Eng. Mech. Div., ASCE, 96(EM4), 483-493.

Hall, J. F, Heaton, T. H., Hailing, M. W., and Wald, D. J. (1995). Near-source ground motion and its effects on flexible buildings, Earthquake Spectra, 11(4), 569-605.

Hayashi, K., Fujita, K.,Tsuji, M., and Takewaki, I. (2018). A simple response evaluation method for base-isolation building-connection hybrid structural system under long-period and long-duration ground motion, Frontiers in Built Environment (Specialty Section: Earthquake Engineering), Volume 4: Article 2.

Hayden, C. P., Bray, J. D., and Abrahamson, N. A. (2014). Selection of near-fault pulse motions,/. Geotechnical Geoenvironmental Eng., ASCE, 140(7).

Iwan, W. D. (1961). The dynamic response of bilinear hysteretic systems, Ph.D. Thesis, California Institute of Technology.

Iwan, W. D. (1965a). The dynamic response of the one-degree-of-freedom bilinear hysteretic system, Proceedings of the Third World Conference on Earthquake Engineering, 1965.

Iwan, W. D. (1965b). The steady-state response of a two-degree-of-freedom bilinear hysteretic system, Journal of Applied Mechanics, 32(1), 151-156.

Kalkan, E., and Kunnath, S. K. (2006). Effects of fling step and forward directivity on seismic response of buildings, Earthquake Spectra, 22(2), 367-390.

Kalkan, E., and Kunnath, S. K. (2007). Effective cyclic energy as a measure of seismic demand,/. Earthquake Eng. 11, 725-751.

Kawai, A., Maeda, T, and Takewaki, I. (2020). Smart seismic control system for high-rise buildings using large-stroke viscous dampers through connection to strong-back core frame, Frontiers in Built Environment (Specialty Section: Earthquake Engineering), Volume 6, Article 29.

Khaloo, A. R., Khosravi, H., and Hamidi Jamnani, H. (2015). Nonlinear interstory drift contours for idealized forward directivity pulses using “Modified Fish-Bone” models; Advances in Structural Eng. 18(5), 603-627.

Kojima, K., Fujita, K., and Takewaki, I. (2015). Critical double impulse input and bound of earthquake input energy to building structure, Frontiers in Built Environment, Volume 1, Article 5.

Kojima, K., and Takewaki, I. (2015a). Critical earthquake response of elastic-plastic structures under near-fault ground motions (Part 1: Fling-step input), Frontiers in Built Environment, Volume 1, Article 12.

Kojima, K., and Takewaki, I. (2015b). Critical earthquake response of elastic-plastic structures under near-fault ground motions (Part 2: Forward-directivity input), Frontiers in Built Environment, Volume 1, Article 13.

Kojima, K., and Takewaki, I. (2015c). Critical input and response of elastic-plastic structures under long-duration earthquake ground motions, Frontiers in Built Environment (Specialty Section: Earthquake Engineering), Volume 1, Article 15.

Kojima, K., and Takewaki, I. (2016). Closed-form critical earthquake response of elastic-plastic structures on compliant ground under near-fault ground motions, Frontiers in Built Environment (Specialty Section: Earthquake Engineering), Volume 2, Article 1.

Kojima, K., and Takewaki, I. (2017). Critical steady-state response of SDOF bilinear hysteretic system under multi impulse as substitute of long-duration ground motions, Frontiers in Built Environment (Specialty Section: Earthquake Engineering), Volume 3: Article 41.

Liu, C.-S. (2000). The steady loops of SDOF perfectly elastoplastic structures under sinusoidal loadings,/. Marine Science and Technology, 8(1), 50-60.

Luco, J. E. (2014). Effects of soil-structure interaction on seismic base isolation, Soil Dyn. Earthq. Eng., 66, 67-177. doi: 10.1016/j.soildyn.2014.05.007

Mavroeidis, G. P., and Papageorgiou, A. S. (2003). A mathematical representation of near-fault ground motions, Bull. Seism. Soc. Am., 93(3), 1099-1131.

Mavroeidis, G. R, Dong, G., and Papageorgiou, A. S. (2004). Near-fault ground motions, and the response of elastic and inelastic single-degree-freedom (SDOF) systems, Earthquake Eng. Struct. Dyn., 33,1023-1049.

Moustafa, A., Ueno, K., and Takewaki, I. (2010). Critical earthquake loads for SDOF inelastic structures considering evolution of seismic waves, Earthquakes and Structures, 1(2), 147-162.

Mukhopadhyay, S., and Gupta, V. K. (2013a). Directivity pulses in near-fault ground motions—I: Identification, extraction and modeling, Soil Dyn. Earthquake Eng., 50,1-15.

Mukhopadhyay, S., and Gupta, V. K. (2013b). Directivity pulses in near-fault ground motions—II: Estimation of pulse parameters, Soil Dyn. Earthquake Eng., 50, 38-52.

Roberts, J. B., and Spanos, P. D. (1990). Random vibration and statistical linearization. Wiley, New York.

Rupakhery, R., and Sigbjornsson, R. (2011). Can simple pulses adequately represent near-fault ground motions?,/. Earthquake Eng., 15,1260-1272.

Sasani, M., and Bertero, V. V. (2000). Importance of severe pulse-type ground motions in performance-based engineering: historical and critical review, in Proceedings of the Twelfth World Conference on Earthquake Engineering, Auckland, New Zealand.

Takewaki, I. (1996). Design-oriented approximate bound of inelastic responses of a structure under seismic loading, Computers and Structures, 61(3), 431-440.

Takewaki, I. (1997). Design-oriented ductility bound of a plane frame under seismic loading,/. Vibration and Control, 3(4), 411^134.

Takewaki, I. (2002). Robust building stiffness design for variable critical excitations, /. Struct. Eng., ASCE, 128(12), 1565-1574.

Takewaki, I. (2004). Bound of earthquake input energy,/. Struct. Eng., ASCE, 130(9), 1289-1297.

Takewaki, I. (2007). Critical excitation methods in earthquake engineering, Oxford, Elsevier (Second edition in 2013).

Takewaki, I., and Fujita, K. (2009). Earthquake input energy to tall and base-isolated buildings in time and frequency dual domains, /. of The Structural Design of Tall and Special Buildings, 18(6), 589-606.

Takewaki, I., Murakami, S., Fujita, K., Yoshitomi, S., and Tsuji, M. (2011). The 2011 off the Pacific coast of Tohoku earthquake and response of high-rise buildings under long-period ground motions, Soil Dyn. Earthquake Eng., 31(11), 1511-1528.

Takewaki, I., andTsujimoto, H. (2011). Scaling of design earthquake ground motions for tall buildings based on drift and input energy demands, Earthquakes Struct., 2(2), 171-187.

Takewaki, I., Moustafa, A., and Fujita, K. (2012). Improving the earthquake resilience of buildings: The worst case approach, Springer, London.

Tamura, G., Kojima, K., and Takewaki, I. (2019). Critical response of elastic-plastic SDOF systems with nonlinear viscous damping under simulated earthquake ground motions, Heliyon, 5, e01221. doi:10.1016/j.heliyon.2019.e01221.

Thomson, W. T. (1965). Vibration theory and applications, Prentice-Hall, Englewood Cliffs, NJ.

Vafaei, D., and Eskandari, R. (2015). Seismic response of mega buckling-restrained braces subjected to fling-step and forward-directivity near-fault ground motions, Struct. Design Tall Spec. Build., 24, 672-686.

Xu, Z., Agrawal, A. K., He, W.-L., and Tan, P. (2007). Performance of passive energy dissipation systems during near-field ground motion type pulses, Eng. Struct., 29, 224-236.

Yamamoto, K., Fujita, K., and Takewaki, I. (2011). Instantaneous earthquake input energy and sensitivity' in base-isolated building, Struct. Design Tall Spec. Build., 20(6), 631-648.

Yang, D., and Zhou, J. (2014). A stochastic model and synthesis for near-fault impulsive ground motions, Earthquake Eng. Struct. Dyn., 44, 243-264.

Zhai, C., Chang, Z., Li, S., Chen, Z.-Q., and Xie, L. (2013). Quantitative identification of near-fault pulse-like ground motions based on energy, Bull. Seism. Soc. Am., 103(5), 2591-2603.

A. Appendix 1: Multi-impulse and correspondingmulti-cycle sine wave with thesame frequency and samemaximum fourier amplitude

The velocity amplitude V of the multi-impulse is related to the maximum velocity of the corresponding multi-cycle sine wave with the same frequency (the period is twice the interval of the multi impulse) so that the maximum Fourier amplitudes of both inputs coincide. The detail is explained in this section.

The multi-impulse is expressed by

where V is the velocity amplitude of the multi-impulse and S(t) is the Dirac delta function. The Fourier transform of Eq. (4.A1) can be obtained as

Let A_h Т/, and w,= 2л/Т, denote the acceleration amplitude, the period, and the circular frequency of the corresponding multi-cycle sine wave, respectively. The acceleration wave of the corresponding multi-cycle sine wave is expressed by

The time interval t₀ of consecutive two impulses in the multi-impulse is related to the period T, of the corresponding multi-cycle sine wave by T; = 21₀. Although the starting points of both inputs differ by tj2, the starting time of multi-cycle sine wave does not affect the Fourier amplitude. For this reason, the starting time of multi-cycle sine wave will be adjusted so that the responses of both inputs correspond well. In this section, the relation of the velocity amplitude V of the multi-impulse with the acceleration amplitude A, of the corresponding multi-cycle sine wave is derived. The ratio a_M^h(t₀,N) of A, to V is introduced by

The Fourier transform of ^ug in Eq.(4.A3) is computed by

From Eqs. (4. A2), (4. A5), the Fourier amplitudes of both inputs are expressed

by

The coefficient a_M^b(t₀,N) can be derived from Eqs. (4.A4), (4.A6), (4.A7) and the equivalence of the maximum Fourier amplitude ILfco) =

ргнI •

N-l

In the numerator of Eq. (4.A8), max 0.5 + ^(-1)" е~^шпи> + 0.5B,v,<’ = N

»=1

holds. The denominator of Eq. (4.A8) will be evaluated next.

Let us define the function f_M^b(x,N) given by

When we substitute x = wt₀ in Eq. (4.A9), Eq. (4.A7) can be transformed into

Eq. (4.A9) means that once the maximum value of f_M^b(x = wt₀,N) is obtained, the denominator of Eq. (4.A8) is evaluated. After some manipulation on the maximization of Eq. (4.A10), the following relation is derived.

where /_Mmax*(N) is the maximum value of f_M^b(x,N) for the variable x. Finally, for a sufficiently large number N(>20), the function f_M^b(x,N) becomes maximum at x = к and f_M^b(x = n,N) = N/(2л). Then, Eq. (4.A11) is reduced to

Eq. (4.A12) provides the relation between V and A, in terms of t₀(=T,/2) together with the relation between V and V;.