Kotaro Kojima

Critical Earthquake Response of Elastic–Plastic Structures Under Near-Fault Ground Motions (Part 1: Fling-Step Input)

The double impulse input is introduced as a substitute of the fling-step near-fault ground motion and a closed-form solution of the elastic-plastic response of a structure by the ‘critical double impulse input’ is derived. Since only the free-vibration appears under such double impulse input, the energy approach plays an important role in the derivation of the closed-form solution of a complicated elastic-plastic response. It is shown that the maximum inelastic deformation can occur either after the first impulse or after the second impulse depending on the input level. The validity and accuracy of the proposed theory are investigated through the comparison with the response analysis to the corresponding one-cycle sinusoidal input as a representative of the fling-step near-fault ground motion.

Critical Input and Response of Elastic–Plastic Structures Under Long-Duration Earthquake Ground Motions

The multiple impulse input is introduced as a substitute of the long-duration earthquake ground motion, mostly expressed in terms of harmonic waves, and a closed-form solution is derived of the elastic-plastic response of a single-degree-of-freedom structure under the ‘critical multiple impulse input’. Since only the free-vibration appears under such multiple impulse input, the energy approach plays an important role in the derivation of the closed-form solution of a complicated elastic-plastic response. It is shown that the critical inelastic deformation and the corresponding critical input frequency can be captured depending on the input level by the substituted multiple impulse input in the form of original and modified input sequence. The validity and accuracy of the proposed theory are investigated through the comparison with the response analysis to the corresponding sinusoidal input as a representative of the long-duration earthquake ground motion.

Critical Earthquake Response of Elastic–Plastic Structures Under Near-Fault Ground Motions (Part 2: Forward-Directivity Input)

The triple impulse input is used as a simplified version of the forward-directivity near-fault ground motion and a closed-form solution of the elastic-plastic response of a structure by this triple input is obtained. It is noteworthy that only the free-vibration appears under such triple impulse input. An almost critical excitation is defined and its response is derived. The energy approach plays an important role in the derivation of the closed-form solution of a complicated elastic-plastic response. It is shown that the maximum inelastic deformation can occur after the second impulse or the third impulse depending on the input level. The validity and accuracy of the proposed theory are discussed through the comparison with the response analysis result to the corresponding three wavelets of sinusoidal waves as a representative of the forward-directivity near-fault ground motion.

Closed-Form Critical Earthquake Response of Elastic–Plastic Structures on Compliant Ground under Near-Fault Ground Motions

The double impulse is introduced as a substitute of the fling-step near-fault ground motion. A closed-form solution of the elastic-plastic response of a structure on compliant (flexible) ground by the ‘critical double impulse’ is derived for the first time based on the solution for the corresponding structure with fixed base. As in the case of fixed-base model, only the free-vibration appears under such double impulse and the energy approach plays an important role in the derivation of the closed-form solution of a complicated elastic-plastic response on compliant ground. It is remarkable that no iteration is needed in the derivation of the critical elastic-plastic response. It is shown via the closed-form expression that, in the case of a smaller input level of double impulse to the structural strength, as the ground stiffness becomes larger, the maximum plastic deformation becomes larger. On the other hand, in the case of a larger input level of double impulse to the structural strength, as the ground stiffness becomes smaller, the maximum plastic deformation becomes larger. The criticality and validity of the proposed theory are investigated through the comparison with the response analysis to the corresponding one-cycle 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.

Closed-Form Overturning Limit of Rigid Block under Critical Near-Fault Ground Motions

A closed-form limit on the input level of the double impulse as a substitute of a near-fault ground motion is derived for the overturning of a rigid block. The rocking vibration of the rigid block is formulated by using the conservation law of angular momentum and the conservation law of mechanical energy. The initial rotational velocity after the first impulse and the rotational velocity after the impact are determined by the conservation law of angular momentum. The velocity change after the second impulse is also characterized by the conservation law of angular momentum. The maximum angles of rotation of the rigid block in both the clockwise and anti-clockwise directions, which are needed for the computation of the overturning limit, are derived by the conservation law of mechanical energy. This enables us to avoid the computation of complicated non-linear time-history responses. The critical timing of the second impulse to the first impulse is characterized by the time of impact after the first impulse. It is clarified that the action of the second impulse just after the impact corresponds to the critical timing. It is derived from the closed-form expression of the critical velocity amplitude limit of the double impulse that its limit is proportional to the square root of size, i.e. the scale effect.

Critical Response of 2DOF Elastic–Plastic Building Structures under Double Impulse as Substitute of Near-Fault Ground Motion

The double impulse is introduced as a substitute of the fling-step near-fault ground motion and a critical elastic-plastic response of a 2DOF (two-degree-of-freedom) building structure under the ‘critical double impulse’ is evaluated. Since only the free-vibration appears under such double impulse, the energy balance approach plays an important and essential role in the derivation of the solution of a complicated elastic-plastic critical response. It is shown that the critical timing of the double impulse is characterized by the timing of the second impulse at the zero story shear force in the first story. This timing guarantees the maximum energy input by the second impulse which causes the maximum plastic deformation after the second impulse. Because the response of 2DOF elastic-plastic building structures is quite complicated due to the phase difference between two masses compared to SDOF models for which a closed-form critical response can be derived, the upper bound of the critical response is introduced by using the convex model.

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

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 one-cycle 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.

Critical double impulse input and bound of earthquake input energy to building structure

A theory of earthquake input energy to building structures under single impulse is useful for disclosing the property of energy transfer function. This property shows that the area of the energy transfer function is constant irrespective of natural period and damping of building structures. However single impulse may be unrealistic from a certain viewpoint because the frequency characteristic of input cannot be expressed by this input. In order to resolve such issue, a double impulse is introduced in this paper. The frequency characteristic of the Fourier amplitude of the double impulse is found in an explicit manner and a critical excitation problem is formulated with an interval of two impulses as a variable. The solution to that critical excitation problem is derived. An upper bound of the earthquake input energy is then derived by taking full advantage of the property of the energy transfer function that the area of the energy transfer function is constant. The relation of the double impulse to the corresponding one-cycle sinusoidal wave as a representative of near-fault pulse-type waves is also investigated.

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

In the 2016 Kumamoto earthquake in Japan, two severe ground shakings with the seismic intensity 7 (the highest level in Japan Metheorological Agency (JMA) scale; approximately X-XII in Mercalli scale) occurred consecutively on April 14 and April 16. In the seismic regulations of most countries, it is usually prescribed that such severe earthquake ground motion occurs once in the working period of buildings. In this paper, a simple evaluation method is presented on the seismic resistance of residential houses under two consecutive severe ground motions with intensity 7. Therefore the proposed method can be used for the design of buildings under two consecutive severe ground motions. The present paper adopts an impulse as a representative of near-fault ground motion and two separated impulses are used as the repetition of intensive ground shakings with the seismic intensity 7. Two scenarios to building collapse (collapse limit in terms of zero restoring force with P-delta effect and collapse limit in terms of maximum deformation) under two repeated severe ground shakings are provided and energy consideration is devised for the response evaluation. The validity and accuracy of the proposed theories are discussed through numerical analysis using recorded ground motions.

Simplified Analysis of the Effect of Soil Liquefaction on the Earthquake Pile Response

The time-history response of a structure-pile system during soil liquefaction is highly complicated and several analytical methods have been proposed through the accuracy verification based on the comparison with the experimental works. However, the analytical methods with higher accuracy often require large computational loads and are not necessarily preferred in the actual design practice. On the other hand, while the response spectrum method is not accurate compared to the aforementioned methods, it can provide useful design guidelines in the preliminary stage for structure-pile systems under soil liquefaction with acceptable accuracy. In this paper, the previously proposed response spectrum method for a structure-pile-soil system is used where the effect of soil liquefaction is taken into account by introducing the so-called p-multiplier method. It is shown that, while in the case of inner partial liquefaction with a non-liquefied layer at the top, the demand on the pile moment is large due to the inertial effect of that non-liquefied layer at the top, in the case of overall liquefaction near the ground surface, the demand is smaller than the case of inner partial liquefaction.