Closed-Form Dynamic Stability Criterion for Elastic–Plastic Structures under Near-Fault Ground Motions

Abstract

A dynamic stability criterion for elastic-plastic structures under near-fault ground motions is derived in closed form. A negative post-yield stiffness is treated in order to consider the P-delta effect. The double impulse is used as a substitute of the fling-step near-fault ground motion. Since only the free vibration appears under such double impulse, the energy approach plays a critical role in the derivation of the closed-form solution of a complicated elastic-plastic response of structures with the P-delta effect. It is remarkable that no iteration is needed in the derivation of the closed-form dynamic stability criterion on the critical elastic-plastic response. It is shown via the closed-form expression that several patterns of unstable behaviors exist depending on the ratio of the input level of the double impulse to the structural strength and on the ratio of the negative post-yield stiffness to the initial elastic stiffness. The validity of the proposed dynamic stability criterion is investigated by the numerical response analysis for structures under double impulses with stable or unstable parameters. Furthermore, the reliability of the proposed theory is tested through the comparison with the response analysis to the corresponding onecycle sinusoidal input as a representative of the fling-step near-fault ground motion. The applicability of the proposed theory to actual recorded pulse-type ground motions is also discussed.

Collapse-Limit Input Level of Critical Double Impulse for Damped Bilinear Hysteretic SDOF System With Negative Post-yield Stiffness

“…Dynamic instability induced by collapse is one of the most important and challenging problems in the field of earthquake-resistant design of building structures and infrastructures, and such phenomena have been investigated extensively from the theoretical and numerical viewpoints (Herrmann, 1965;Jennings and Husid, 1968;Sun et al, 1973;Tanabashi et al, 1973;Bertero et al, 1978;Takizawa and Jennings, 1980;Bernal, 1987Bernal, , 1992Bernal, , 1998Nakajima et al, 1990;Ger et al, 1993;Challa and Hall, 1994;Hall, 1998;Hjelmstad and Williamson, 1998;Uetani and Tagawa, 1998;Araki and Hjelmstad, 2000;Sasani and Bertero, 2000;Williamson and Hjelmstad, 2001;Miranda and Akkar, 2003;Ibarra andKrawinkler, 2005, Sivaselvan et al, 2009;Adam and Jager, 2012;Khoshnoudian et al, 2014;Kojima and Takewaki, 2016a). Jennings and Husid (1968) defined the statically stable limit for an elastic-plastic single-degree-of-freedom (SDOF) system with a rotational spring as the zero restoring moment in the plastic range.…”

“…However, previous studies provide only the stability or instability condition, e.g., "the zero-restoring-force point in the post-yield stiffness range" or "the negative eigenvalue by the tangent stiffness matrix." On the other hand, Kojima and Takewaki (2016a) derived the collapse-limit input level of the double impulse for an undamped elastic-plastic SDOF system with the P-delta effect in the closed-form. The double impulse can represent the main part of the fling-step near-fault ground motion and the response to the double impulse can be expressed by only free vibration with the initial velocity.…”

“…The collapse-limit level of the critical double impulse can be obtained by using the energy balance law where the kinetic energy by the initial velocity of mass is transformed into the sum of the elastic strain energy and the energy dissipated by the plastic deformation at the maximum displacement. Kojima and Takewaki (2016a) adopted the zerorestoring-force point in the post-yield stiffness range as the collapse point.…”

A Simple Evaluation Method of Seismic Resistance of Residential House under Two Consecutive Severe Ground Motions with Intensity 7

“…Consider first an undamped SDOF model of normal bilinear hysteresis with negative second slope (steel structures) as shown in Figure 3A (Kojima and Takewaki, 2016). The negative slope of the first model can be understood as a modeling result of P-delta effect and structural degradation.…”

“…Following a procedure similar to that in the reference (Kojima and Takewaki, 2016), the limit input velocity for one impulse can be derived as follows by equating the kinetic energy (1/2)mV 2 provided by one impulse with the dissipated energy (triangle in Figure 4). Figure 4).…”