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Edited by: Nikos D. Lagaros, National Technical University of Athens, Greece

Reviewed by: Paolo Castaldo, Polytechnic University of Turin, Italy; Claudia Casapulla, University of Naples Federico II, Italy; Aristotelis E. Charalampakis, National Technical University of Athens, Greece

This article was submitted to Earthquake Engineering, a section of the journal Frontiers in Built Environment

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The collapse-limit input velocity level of the critical double impulse simulating the principal part of near-fault ground motions is derived for an elastic-plastic structure with viscous damping and P-delta effect. The structural system is modeled by a bilinear hysteretic SDOF system with negative post-yield stiffness reflecting the P-delta effect which plays a key role in the collapse behavior. Since the critical timing of the second impulse in the double impulse has been proven as the zero-restoring force timing after the first impulse for the elastic-plastic SDOF system with viscous damping, that property is used again in this paper. It is shown that the collapse-limit input level of the critical double impulse can be obtained as a function of the post-yield stiffness and the damping ratio by using the energy balance law and the quadratic-function approximation of the damping force-deformation relation. The applicability of the collapse-limit level to actual recorded ground motions is investigated through the time-history response analysis for the stable models and the collapse models under two actual earthquake ground motions.

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,

Jennings and Husid (

The dynamic stability has also been investigated for MDOF systems (Maier and Perego,

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 (

The phenomenon caused by the P-delta effect, which is represented by the negative second slope, may be related to the phenomenon of the rocking of a rigid block. Regarding the resonance and overturning phenomenon, some interesting researches have been conducted (Chatzis and Smyth,

In this paper, the collapse-limit input velocity level of the critical double impulse is derived approximately for an elastic-plastic structure with viscous damping and P-delta effect. The system is modeled by a damped bilinear hysteretic SDOF system with negative post-yield stiffness. The critical timing of the second impulse was proven as the zero-restoring force timing after the first impulse for the elastic-plastic SDOF system with viscous damping in the previous investigations (Kojima et al.,

The double impulse input and the model used in this study are explained in section Double Impulse and Damped Bilinear Hysteretic SDOF System With Negative Post-yield Stiffness. The collapse-limit input velocity level of the critical double impulse is derived for 4 collapse patterns in section Collapse Limit Input Level for Damped Bilinear Hysteretic SDOF System With Negative Post-yield Stiffness. Accuracy of the proposed approximate closed-form solution for the collapse-limit level is investigated through the time-history response analysis for stable and unstable models in section Accuracy Check for Approximate Collapse-Limit Input Velocity Level of Critical Double Impulse. The effect of the damping ratio on the collapse-limit input level is clarified in section Transition of Collapse-Limit Input Velocity Level With Respect to Damping Ratio. Applicability of the proposed solution of the collapse level to the one-cycle sinusoidal wave is investigated through the comparison of the proposed level of the double impulse and that of the one-cycle sinusoidal wave in section Applicability of the Proposed Collapse-Limit Input Level to the Corresponding One-Cycle Sinusoidal Wave. Further applicability of the proposed theory to actual near-fault ground motions is discussed in section Applicability of the Proposed Collapse-Limit Input Level to Actual Recorded Ground Motions. The conclusions are summarized in section Conclusions.

A ground acceleration ü_{g}(

where _{0} denotes the time interval between two impulses and δ(

Input motion and damped bilinear hysteretic SDOF system,

Consider a damped bilinear hysteretic SDOF system with negative post-yield stiffness as shown in _{1} = 2π/ω_{1} and _{R} and _{D}, respectively. The yield deformation is given by _{y} and the yield force is by _{y} = _{y}. The bilinear hysteretic restoring-force characteristic with the negative post-yield stiffness is shown in _{y} = ω_{1}_{y} is the input velocity level of the single impulse at which the maximum deformation of the undamped SDOF system just attains the yield deformation _{y} and _{y} is used to normalize the input velocity level

The collapse-limit input velocity level of the double impulse is derived for the bilinear hysteretic SDOF system with the negative post-yield stiffness ratio α and the damping ratio _{max 1}, _{p1}, _{max 2}, _{p2} denote the maximum deformation and the plastic deformation after the first and second impulses, respectively. Note that _{max 1}, _{max 2} are the absolute values. The critical timing of the second impulse is the zero-restoring force timing after the first impulse for the elastic-plastic SDOF system with viscous damping (Kojima et al., _{c}.

Restoring force-deformation relation and damping force-deformation relation of damped bilinear hysteretic SDOF system with negative post-yield stiffness under critical double impulse,

The collapse limit is characterized by the zero-restoring force point in the negative post-yield stiffness range and four collapse patterns, where the maximum deformation under the critical double impulse just attains the collapse limit (stability limit), are assumed as shown later. The collapse-limit input velocity level of the critical double impulse is derived via the energy balance law and the quadratic-function approximation of the damping force-deformation relation and the normalized collapse-limit input level _{y} is obtained as a function of the post-yield stiffness ratio α and the damping ratio

The four collapse patterns can be categorized as follows.

Collapse Pattern 1: Collapse limit after the second impulse without plastic deformation after the first impulse

Collapse Pattern 2: Collapse limit after the second impulse with plastic deformation after the first impulse

Collapse Pattern 3: Collapse limit after the second impulse with closed-loop in the restoring force-deformation relation

Collapse Pattern 4: Collapse limit after the first impulse.

The collapse-limit input levels in Collapse Patterns 1–4 are derived in the following sections.

The first collapse pattern represents the pattern where the SDOF system just attains the zero restoring force in the second stiffness range after the second impulse without plastic deformation after the first impulse. _{y} in Collapse Pattern 1 has to satisfy the following equation since the plastic deformation is allowed only after the second impulse (Akehashi et al.,

Elastic-plastic response corresponding to Collapse Pattern 1,

The left-hand side of the above inequality indicates the input velocity level at which the damped bilinear hysteretic SDOF system just attains the yield deformation after the second impulse and the right-hand side corresponds to the input velocity level at which the SDOF system just attains the yield deformation after the first impulse.

From

The left-hand side of Equation (3) indicates the kinetic energy for the velocity (_{c} + _{max 2} = _{y} + _{p2}.

It can also be understood from _{p2} after the second impulse can be obtained from _{y} + α_{p2} = 0. Then _{p2} can be derived as

Let _{c} denote the velocity of the state when the restoring force becomes zero in the unloading process. It can be obtained by solving the equation of motion in the unloading process (point A to point O in

By substituting Equations (5) and (6) into Equation (4), the following equation is obtained.

From Equation (7), the input velocity level _{y} of the critical double impulse in Collapse Pattern 1 can be derived by characterizing that the SDOF system just attains the zero-restoring force in the post-yield stiffness range after the second impulse without the plastic deformation after the first impulse.

where Inequality (2) should be satisfied.

The second collapse pattern expresses the pattern such that the SDOF system just attains the collapse limit (the zero-restoring force in the second stiffness range) after the second impulse with the plastic deformation after the first impulse. _{y} in Collapse Pattern 2 has to satisfy the following equation since the plastic deformation is allowed even after the first impulse (Akehashi et al.,

From

The left-hand side of Equation (10) represents the kinetic energy for the velocity (_{c} +

Elastic-plastic response corresponding to Collapse Pattern 2,

It can also be understood from _{p2} after the second impulse is calculated from _{y} − α_{p1} + α_{p2} = 0. Then, _{p2} can be expressed by

By substituting Equation (11) into Equation (10) and dividing both side of the resulting equation by (

With the notation 1 − α_{p1}/_{y} =

From Equation (13), the normalized velocity (_{c} + _{y} just after the second impulse can be derived by

where

Note that _{c} denotes the velocity of the state when the restoring force becomes zero in the unloading process. It can be obtained by solving the equation of motion in the unloading process (point B to point C in

With the notation

The plastic deformation _{p1}/_{y} after the first impulse can be obtained from the following energy balance law after the first impulse (Akehashi et al.,

From Equation (17), _{p1}/_{y} can be obtained by

where _{p1}/_{y} =

From the definition of _{y}).

where

From Equation (20), the input velocity level _{y} of the critical double impulse in Collapse Pattern 2 can be derived by characterizing that the SDOF system just attains the zero-restoring force in the post-yield stiffness range after the second impulse with the plastic deformation after the first impulse.

where Inequality (9) should be satisfied.

The third collapse pattern is the pattern such that the SDOF system just attains the collapse limit after the second impulse with a closed loop in the restoring force-deformation relation. In this collapse pattern, the SDOF system yields even after the first impulse [the input velocity level _{y} must satisfy Inequality (9) as with Collapse Pattern 2] and the direction of the collapse limit is same as the maximum deformation after the first impulse.

Elastic-plastic response corresponding to Collapse Pattern 3,

The following equation can be obtained from the energy balance law between Point E and Point H in

Point E indicates the starting point in the unloading process after experiencing the maximum deformation after the second impulse and Point H is the point at which the maximum deformation after experiencing the closed loop after the second impulse attains the collapse limit in the same direction as the maximum deformation after the first impulse. Let ṽ denote the maximum velocity in the unloading process after the second impulse. Note that ṽ is the absolute value. The left-hand side of Equation (22) expresses the elastic strain energy at Point E and the right-hand side indicates the sum of the elastic strain energy, the energy dissipated by the plastic deformation and the work done by the damping force. The work done by the damping force between Point E and Point H is obtained by the quadratic-function approximation for the damping force-deformation relation (Akehashi et al.,

By substituting ṽ into Equation (22) and arranging the resulting equation, a quartic equation of _{y} can be derived. The detailed analysis of ṽ and the quartic equation is presented in Appendix. Then, the input velocity level _{y} in Collapse Pattern 3 can be computed by solving the quartic equation. The collapse-limit level has to be a real number and satisfy Inequality (9).

The fourth collapse pattern expresses the pattern in which the SDOF system just attains the collapse limit (the zero-restoring force in the post-yield stiffness range) after the first impulse. In Collapse Pattern 4, the input velocity level _{y} must satisfy Inequality (9) as in Collapse Patterns 2 and 3.

From

Elastic-plastic response corresponding to Collapse Pattern 4,

The left-hand side of Equation (23) indicates the kinetic energy calculated for the velocity

It can also be understood from _{p1} after the first impulse can be obtained from

By substituting _{p1} = −_{y}/α derived from Equation (24) into Equation (23) and arranging the equation, the following equation can be derived.

By solving Equation (25), the input velocity level in Collapse Pattern 4 can be obtained as follows.

where Inequality (9) should be satisfied.

The approximate collapse-limit input velocity levels of the critical double impulse in Collapse Patterns 1–4 were derived in section Collapse Limit Input Level for Damped Bilinear Hysteretic SDOF System With Negative Post-yield Stiffness. The collapse-limit level with respect to the negative post-yield stiffness ratio for damping ratio _{y} and the post-yield stiffness ratio α. Case 1 in _{y}, α, and

Collapse-limit input velocity level of critical double impulse with respect to post-yield stiffness ratio for specific damping ratio (

In this section, the accuracy of the proposed collapse-limit level is investigated through the comparison with the time-history response analysis result. _{y} and the negative post-yield stiffness ratio α. These 18 points express the slightly larger or smaller than the approximate collapse-limit level with α = −0.20, −0.50, −0.65, −0.80. The restoring force-deformation relations of 18 points are shown in

Restoring force-deformation relation of 18 stability or collapse cases (damping ratio

In order to investigate the collapse-limit level in detail, the input velocity level of the critical double impulse at which the maximum deformation just attains the collapse limit is evaluated by the time-history response analysis. The elastic-plastic response to the critical double impulse can be evaluated by changing the time interval in a parametric manner in the time-history response analysis.

Comparison of proposed collapse-limit input velocity level and minimum collapse-limit input velocity level by time-history response analysis,

The effect of damping ratio on the collapse-limit level of the critical double impulse is investigated here.

Collapse–limit input velocity level of critical double impulse with respect to post-yield stiffness ratio for specific damping ratio,

Input velocity level of Collapse Patterns 1–4 for specific damping ratio,

The applicability of the proposed collapse input level of the critical double impulse is investigated to the one-cycle sinusoidal wave which can represent the main-part of near-fault ground motions through the comparison with the collapse velocity level of the one-cycle sinusoidal wave. The following relation between the input velocity level _{p} of the corresponding one-cycle sinusoidal wave has been proposed based on the equivalence of the maximum Fourier amplitude (Kojima and Takewaki,

The maximum deformation to the critical one-cycle sinusoidal wave is evaluated by the time-history response analysis by changing the input wave period for the constant maximum velocity _{p} and the collapse-limit input level _{p}/1.222) of the one-cycle sinusoidal wave is evaluated where the maximum deformation attains the collapse limit (the zero-restoring force point in the second stiffness range). The critical one-cycle sinusoidal wave indicates the one-cycle sinusoidal wave with the input wave period which maximizes the maximum deformation for the constant maximum velocity _{p}.

Comparison of proposed collapse-limit input velocity level of critical double impulse and collapse level of one-cycle sinusoidal wave,

In order to investigate the validity of the double impulse as a substitute for near-fault ground motions and the applicability of the proposed theory to actual recorded ground motions, the time-history response analysis is conducted to actual recorded ground motions. Then the collapse level of actual recorded ground motions is investigated. In this paper, the Rinaldi station FN component during the 1994 Northridge earthquake and the Kobe University NS component during the 1995 Hyogoken-Nanbu earthquake are used as the representative near-fault ground motions. _{p}(= π_{p}/_{p}) and the period _{p} of the one-cycle sinusoidal wave equivalent to the Rinaldi station FN component are _{p} = 7.85[m/sec^{2}] and _{p} = 0.8[sec]. On the other hand, the acceleration amplitude and the period of the one-cycle sinusoidal wave equivalent to the Kobe University NS component are _{p} = 2.60[m/sec^{2}] and _{p} = 1.0[sec]. The input velocity level of the double impulse corresponding to the Rinaldi station FN component is

Actual recorded ground motions and equivalent one-cycle sine wave,

In above sections, the critical double impulse or the critical one-cycle sinusoidal wave has been determined for a certain input velocity level (or a certain maximum velocity). On the other hand, the critical elastic-plastic response under a given actual earthquake ground motion (fixed) for a certain structural parameter _{y}(= ω_{1}_{y}) is evaluated here by changing the natural circular frequency and the yield deformation (Kojima and Takewaki, _{y}. From _{y} = 1.080 for _{y} = 1.080 of the Rinaldi station FN component is close to the collapse-limit level _{y} = 1.058 evaluated by the proposed method in Collapse Pattern 1. _{y} = 1.079) and the collapse case (_{y} = 1.080). On the other hand, _{y}. From _{y} = 0.949 for _{y} = 0.949 of the Kobe University NS component is close to the collapse-limit level _{y} = 0.981 evaluated by the proposed method in Collapse Pattern 1. _{y} = 0.948) and the collapse case (_{y} = 0.949). It can be observed that the proposed theory provides a reasonably accurate collapse-limit velocity level.

Elastic-plastic response under Rinaldi station FN component of system with _{y} = 1.079) and collapse case (_{y} = 1.080),

Elastic-plastic response under Kobe University NS component of system with _{y} = 0.948) and collapse case (_{y} = 0.949),

The double impulse has been introduced as a substitute for the fling-step near-fault ground motion and the approximate closed-form solution for the collapse-limit input velocity level of the critical double impulse has been derived for a damped bilinear hysteretic SDOF system with negative post-yield stiffness. The conclusions can be summarized as follows.

The collapse-limit input velocity level of the critical double impulse can be derived approximately by introducing the quadratic-function approximation of the damping force-deformation relation and the energy balance law. Since the critical timing of the second impulse had been proved to be the zero-restoring-force timing in the unloading process in the previous study (Kojima et al.,

The applicability of the approximate solution for the collapse-limit input velocity level to near-fault ground motions was investigated through the comparison with the collapse-limit input velocity level of the one-cycle sinusoidal wave. The proposed solution can provide the collapse-limit input velocity level of near-fault ground motions with reasonable accuracy.

The applicability of the collapse-limit input velocity level to actual recorded ground motions was investigated through the time-history response analysis for the stable and collapse models under the Rinaldi station FN component during the 1994 Northridge earthquake and the Kobe University NS component during the 1995 Hyogo-ken Nanbu earthquake. It was confirmed that the proposed theory can evaluate the collapse level of these two earthquake ground motions with reasonable accuracy.

The proposed method enables a closed-form expression useful for the judgement of a stable or collapse state of a structure under earthquake ground motions. However, the readers should keep in mind again that the present theory is based on the following three assumptions: (a) the principal part of a near-fault ground motion can be simulated by a critical double impulse, (b) the critical timing of the second impulse taken equal to zero-restoring force timing after the first impulse is an assumption following (Kojima et al.,

The datasets generated for this study are available on request to the corresponding author.

YS formulated the problem, conducted the computation, and wrote the paper. KK conducted the computation, discussed the results, and wrote the paper. IT supervised the research and wrote the paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Detailed derivation of the quartic equation of _{y} for Collapse Pattern 3 is explained here.

From _{p3} after experiencing the maximum deformation after the second impulse can be obtained from −_{y} − α_{p1} + α_{p2} − α_{p3} = 0.

This means that the maximum deformation after experiencing the closed loop just attains the collapse limit in the same direction as the maximum deformation after the first impulse. Then, _{p3} can be obtained by

where _{p1} in Equation (A1) can be obtained from Equation (18). _{p2} can be derived from the following energy balance law between the point at the second impulse (Point C in

From Equation (A2), _{p2}/_{y} can be derived by

where _{c} can be obtained by Equation (15).

The velocity ṽ in Equation (22) can be obtained by solving the equation of motion in the unloading process after experiencing the maximum deformation −_{max 2} (Point E in

The displacement, velocity, and acceleration responses can be computed by solving Equation (A4) and substituting

In Equations (A5a–c), _{v max} when _{v max} into Equation (A5b).

With the notation λ = (− α_{p1} + α_{p2})/_{y} and Equation (A6), Equation (22) can be transformed into the following equation.

where

By substituting _{p1} by Equation (18) and _{p2} by Equation (A3) into λ = (− α_{p1} + α_{p2})/_{y}, the following equation can be derived.

where 1 − α_{p1}/_{y} = _{c} + _{y} =

From Equation (15) and the notation

Equation (A10) can be transformed into

By substituting Equation (18) and _{p1}/_{y} into Equation (A11), the following equation can be obtained.

Define

By substituting Equations (A13a–d) into Equation (A12) and arranging the equation, the following equation can be obtained.

Here,

where ^{2} + (8/3)^{2}α − (λ + 1)^{2} − (16/3)

By substituting Equations (A13a,b), (A15a,b) into Equation (A14), the following quartic equation can be derived.

where

The input velocity level _{y} in Collapse Pattern 3 can be computed by solving the Equation (A16). Then, the collapse-limit level has to be a real number and satisfy Inequality (9).