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Edited by: Dario De Domenico, University of Messina, Italy

Reviewed by: Christian Málaga-Chuquitaype, Imperial College London, United Kingdom; Georgios Kampas, University of Greenwich, United Kingdom

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.

A dynamic collapse criterion for elastic–plastic structures under near-fault ground motions is derived analytically by approximately transforming near-fault ground motions into double impulse and using an energy balance law. A negative post-yield stiffness is introduced to treat the P-delta effect in the single-degree-of-freedom (SDOF) model. The principal part of fling-step near-fault ground motions is modeled by a one-cycle sine wave and then a double impulse. The double impulse enables the efficient use of the energy approach in the derivation of compact expressions of complicated elastic–plastic responses of structures with the negative post-yield stiffness. In contrast to the previous work (Kojima and Takewaki,

The dynamic collapse of structures is of permanent interest in the field of structural and earthquake engineering and applied mechanics. Historically, many significant works have been conducted (Jennings and Husid,

It seems that the first theoretical achievement on collapse of structures subjected to earthquake ground motions was made by Jennings and Husid (

The tangent stiffness has also attracted some researchers in the investigation of dynamic response of elastic–plastic structures in view of instability. If the tangent stiffness goes into the negative range, residual displacements are induced and accelerated. In addition, there were some discussions that a negative eigenvalue of the tangent stiffness matrix is strongly related to either the accumulation of deformation (Uetani and Tagawa,

However, it does not seem that a simple dynamic collapse criterion has been proposed even for a rather simple input except the recent work (Kojima and Takewaki,

The effects of near-fault ground motions on structural response have been studied extensively (Bertero et al.,

While the inelastic earthquake responses were analyzed for the steady-state response to a harmonic input or the non-stationary response to a simple sinusoidal input in the 1960s−1970s (Caughey,

In this paper, a dynamic collapse criterion for elastic–plastic structures under near-fault ground motions is derived analytically by approximately transforming near-fault ground motions into a double impulse and using an energy balance law. A negative post-yield stiffness is introduced to treat the P-delta effect in the single-degree-of-freedom (SDOF) model. The principal part of fling-step near-fault ground motions is modeled by a one-cycle sine wave and then a double impulse. The use of the double impulse enables the efficient use of the energy approach in the derivation of explicit expressions of a complicated elastic–plastic response of structures with negative post-yield stiffness. In contrast to the previous work (Kojima and Takewaki,

There exist two major advantages of the proposed method against the method using time-history response analysis: (1) if the collapse limit figure is prepared, structural designers can judge the state of collapse or non-collapse at once without time-history response analysis and know the safety factor (both for velocity level and input frequency) for the collapse without many time-history response analyses, and (2) while time-history response analysis of the structural model with negative post-yield stiffness exhibits the response results sensitive to the time increment of the numerical integration, the proposed method does not have such drawback (transcendental equation can be solved stably). Furthermore, since the proposed collapse limit figure is drawn in a normalized form with respect to input velocity level and input frequency, it can be used for various combinations of structural models and input properties.

As explained in the previous papers (Kojima and Takewaki,

Transformation of acceleration wavelets into a series of impulses.

Consider a simplified ground acceleration ü_{g}(

_{0} is the time interval of two impulses. The time derivative is denoted by an over-dot.

The Fourier transform of the acceleration _{g}(

Kojima and Takewaki (_{y} and _{y} denote the yielding force and the yield displacement, respectively. The natural period of this SDOF model is denoted by _{y} denotes the velocity level at which the SDOF model just attains the yield level after the first impulse. In this non-linear resonant case, the second impulse acts at the point of zero restoring force in the first slope range with a positive slope. They classified the collapse pattern into five patterns, i.e., pattern 1, pattern 2, pattern 3, additional pattern 1, and additional pattern 2. Pattern 1 is the collapse pattern such that the SDOF model collapses after the second impulse without plastic deformation after the first impulse. Pattern 2 is the collapse pattern such that the SDOF model collapses after the second impulse with plastic deformation after the first impulse. Pattern 3 is the collapse pattern such that the SDOF model collapses after the second impulse with plastic deformation after the first impulse and with closed loop after the second impulse. Additional pattern 1 is the collapse pattern such that the SDOF model collapses after the first impulse. Additional pattern 2 is the collapse pattern such that the SDOF model has an elastic limit after the second impulse.

Regions of collapse and non-collapse and several patterns of collapse limit (patterns of collapse).

CASE 1 indicates the input velocity range such that the SDOF model remains elastic even after the second impulse. CASE 2 expresses the input velocity range such that the SDOF model remains elastic after the first impulse and goes into the plastic range after the second impulse. CASE 3 presents the input velocity range such that the SDOF model goes into the plastic range after the first impulse.

In this section, a preparation for the next section to derive the collapse limit is made. Two classifications, one based on the input level of the double impulse and the other based on the timing of the second impulse, are introduced.

In the first classification based on the input level of the double impulse, three cases exist, i.e., CASE-A, B, C.

_{y} ≤ 1.

The parameter

In the second classification based on the timing of the second impulse, four cases exist, i.e., CASE-I, II, III, and IV.

Consider the principal points on the restoring-force characteristic, Point O: origin, Point A: initial yielding point, Point B: point of the maximum displacement after the first impulse, Point C: point of zero restoring force after the first impulse, Point D: point of collapse after the first impulse, Point E: point of zero velocity after Point C, Point F: point of zero restoring force after Point E, Point P: the maximum displacement in the negative direction (CASE-A), and Point Q: the maximum displacement in the positive direction (CASE-A). The time between two principal points (e.g., O and A) is indicated by _{OA}.

In CASE-A (

This is because the vibration after the first impulse is a free vibration of an elastic SDOF model.

Classification based on input level of double impulse and timing of second impulse.

In CASE-B (_{1} can be expressed by

In this case, the following condition must be satisfied.

The plastic deformation _{p1} after the first impulse can be expressed by using the energy balance law (Kojima and Takewaki,

The time history from Point A through Point B and the time _{AB} between Points A and B are derived next. The equation of motion from Point A through Point B is

The energy balance law between Points O and A yields

Equation (8) provides the velocity _{A} at Point A.

From Equation (7) and the initial condition (_{y},

The ratio of the time between Points A and B to _{1} can be expressed by

The time history after Point B and the times between two principal points are derived next. The equation of motion (free vibration) after Point B (maximum displacement point) can be described by

The solution of Equation (13) for the initial condition

The following results for the ratios of the times between the principal points to _{1} can be obtained.

Based on these results, the ratio of the critical interval of the double impulse to _{1} can be expressed by

Consider CASE-C (_{AD}) = −_{1} can be obtained as

_{OA} between two principal points (O and A) is indicated by a solid light green line. This line exists only in CASE-B and C. The normalized time _{OD} between two principal points (O and D) is indicated by a dotted black line. This line exists only in CASE-C. The normalized time _{OP} or _{OQ} between two principal points (O and P or O and Q) is indicated by a dotted light green line. This line exists only in CASE-A (elastic response after the first impulse). The normalized times _{OB}, _{OE} between two principal points (O and B, O and E) are indicated by a dashed light green line and a dashed–dotted light green line. These lines exist only in CASE-B (elastic response after the first impulse). The normalized time _{OF} between two principal points (O and F) is indicated by a solid black line. This line exists only in CASE-A and B. Finally, the normalized critical time interval

On the other hand,

Consider here several collapse patterns, Collapse patterns 1′-4′. This naming comes from the similarity to the previous formulation for the non-linear resonant case (Kojima and Takewaki,

The first collapse pattern is the case where the structure remains elastic after the first impulse and attains the collapse limit after the second impulse with arbitrary timing as shown in

Collapse pattern 1′ (CASE-A): _{0}/_{1}-input velocity relation.

Let O and A denote the point of the first impulse (origin of the restoring force characteristic) and the point of initial yielding in the negative direction. The interval of two impulses is denoted by _{0}, and the passing time between Points O and A is indicated by _{OA}. Since the structure does not yield after the first impulse, the following two cases exist.

Consider the respective cases shown in

_{y} ≤ 1 (CASE-A)]

_{y} ≤ 1), the structure does not collapse for the input 0 < _{y} ≤ 0.5. Therefore, the condition 0.5 < _{y} ≤ 1 is necessary to satisfy the collapse condition.

The displacement and velocity of the mass just before the second impulse can be expressed by

When the structure just attains a zero restoring force after the second impulse, the plastic deformation _{p2} after the second impulse can be obtained as

The energy balance law between the state just after the second impulse and the collapse Point H in

Substitution of Equations (21), (22), and (23) into Equation (24) leads to the collapse limit input velocity for CASE-A in collapse pattern 1′.

Since 0.5 < _{y} ≤ 1 is necessary, the following condition for α and _{0} must be satisfied.

_{0}/_{1} = 0.5 (Kojima and Takewaki,

_{y} (CASE-B, C) and 0 < _{0} ≤ _{OA}

After some manipulation, it was found that α ≤ −1 is required in this case to satisfy the collapse condition in collapse pattern 1′. Since a usual case corresponds to the model with α > −1, the detail of analysis is not shown here.

The second collapse pattern is the case where the structure exhibits plastic deformation after the first impulse and attains the collapse limit after the second impulse (see _{y} > 1 must be satisfied.

Collapse pattern 2′ (CASE-B),

Because the second impulse acts after the structure goes into a plastic region under the first impulse, the case is divided into the following two cases, CASE-II and CASE-III.

Since the second impulse acts before the attainment of the maximum displacement (Point B) after the yielding under the first impulse, the interval of two impulses has to satisfy

The displacement and velocity at time ^{*} just before the action of the second impulse can be obtained from Equations (10), (11).

_{OA} in Equations (29), (30) can be obtained from Equation (4). The plastic deformation after the first impulse can be expressed by

When the maximum displacement after the second impulse just attains a zero restoring force, the plastic deformation after the second impulse can be expressed by

The energy balance law between the point just after the second impulse and the point H where the maximum displacement after the second impulse just attains a zero restoring force can be expressed by

Substitution of Equations (4), (29)–(32) into Equation (33) provides the collapse input velocity level _{y} for collapse pattern 2′.

After some manipulation, it was found that α ≤ −1 is required in this case to satisfy the collapse condition in collapse pattern 2′. Since a usual case corresponds to the model with α > −1, the detail of analysis is not shown here.

Since the second impulse acts before the attainment of the collapse point D with a zero restoring force after the yielding under the first impulse, the following condition must be satisfied from Equations (4), (19).

After some manipulation, it was found that α ≤ −1 is required in this case to satisfy the collapse condition in collapse pattern 2′. Since a usual case corresponds to the model with α > −1, the detail of analysis is not shown here.

In this case, the displacement and velocity of the mass just before the action of the second impulse are described from Equations (14) and (15) as

_{OA} and _{AB} in Equations (36), (37) can be obtained from Equations (4), (12). The plastic deformation _{p1} after the first impulse can be derived by using the energy balance law between the point just after the first impulse and the point of the maximum deformation (Point B). The plastic deformation _{p2} after the second impulse can be obtained as Equation (32).

The energy balance law between the point just after the second impulse and the point of the maximum deformation _{max2} = _{y} − _{p1} + _{p2} after the second impulse (Point H) can be expressed as

Since substitution of Equations (4), (6), (12), (32), (36), (37) into Equation (38) provides the transcendental equation, it is difficult to derive a closed-form expression for the input velocity corresponding to the collapse. To determine the input velocity corresponding to the collapse, this transcendental equation can be computed for given α and _{0}.

The third collapse pattern is the case where the structure exhibits plastic deformation after the first impulse and attains the collapse limit with a closed loop after the second impulse (Kojima and Takewaki,

Since the structure exhibits plastic deformation after the first impulse in this case, _{y} > 1 must be satisfied. Because the second impulse acts after the structure goes into a plastic region under the first impulse, the case is divided into two cases, CASE-II and CASE-III, as shown in Equation (27). According to the classification shown in Equation (27), the collapse limit velocity corresponding to the collapse pattern 3′ is derived.

Collapse pattern 3′ (CASE-B and CASE-II). _{0}/_{1}-input velocity relation.

Since the second impulse acts before the structure goes into the unloading path naturally at Point B, the impulse interval _{0} must satisfy Equation (28).

In this case, the displacement and velocity of the mass just before the action of the second impulse are described from Equations (29) and (30). _{OA} in Equations (29), (30) can be obtained from Equation (4). As shown in _{p1} after the first impulse can be derived as in Equation (31).

The energy balance law between the point just after the second impulse and the point J in

The plastic deformation _{p2} after the second impulse can be obtained from Equation (39). In this case, the condition 0 < _{p2} < −(1/α)_{y} must be satisfied. By solving the quadratic equation, Equation (39), under the condition 0 < _{p2} < −(1/α)_{y}, _{p2} can be obtained in closed form.

Another energy balance law between Point J and Point H provides

The input velocity corresponding to the collapse can be obtained by solving the quartic equation transformed from Equation (41).

Since the input velocity level in this case is too large, the solution to satisfy the collapse condition does not exist in this case.

_{0} must satisfy Equation (35).

Collapse pattern 3′ (CASE-B and CASE-III). _{0}/_{1}-input velocity relation.

The displacement and velocity just before the action of the second impulse can be obtained from Equations (36), (37). _{OA} and _{AB} in Equations (36), (37) can be obtained from Equations (4), (12). The plastic deformation _{p1} after the first impulse can be derived as Equation (6) by using the energy balance law between the point just after the first impulse and the point of the maximum deformation (Point B) in

The energy balance law between the point just after the second impulse and the point of the maximum deformation _{max2} after the second impulse (Point J in

By solving the quadratic equation, Equation (42), under the condition 0 < _{p2} < −(1/α)_{y}, _{p2} can be obtained in closed form.

The collapse limit level in this pattern can be obtained by solving the quartic equation derived by substituting Equation (43) into Equation (41).

The fourth collapse pattern is the case where the structure exhibits plastic deformation after the first impulse and attains the collapse limit after experiencing unloading (positive direction) and reloading–reyielding (negative direction) for the second impulse.

Since the structure exhibits plastic deformation after the first impulse in this case, _{y} > 1 must be satisfied. Because the second impulse acts after the structure goes into a plastic region under the first impulse, the case is divided into two cases, CASE-II and CASE-III, as shown in Equation (27). According to the classification shown in Equation (27), the collapse limit velocity corresponding to the collapse pattern 4′ is derived.

_{0} must satisfy Equation (28).

Restoring-force characteristic corresponding to collapse pattern 4′,

In this case, the displacement and velocity of the mass just before the action of the second impulse are expressed by Equations (29) and (30). _{OA} in Equations (29), (30) can be obtained from Equation (4). As shown in _{p1} after the first impulse can be derived as in Equation (31). Since the structure does not go into a plastic region just after the second impulse, the following relation must hold.

The energy balance law between the point just after the second impulse and Point H in

Since the second impulse acts before the structure attains the collapse state, Point D, the impulse interval _{0} must satisfy Equation (34).

In this case, the displacement and velocity of the mass just before the action of the second impulse are described by Equations (29) and (30). _{OA} in Equations (29), (30) can be obtained from Equation (4). As shown in _{p1} after the first impulse can be derived as in Equation (31). Since the structure does not go into a plastic region just after the action of the second impulse, Equation (44) must be satisfied. The energy balance law between the point just after the second impulse and the point H in

Second impulse timing _{0}/_{1}-input velocity relation for CASE-B and CASE-II, CASE-C, CASE-II in collapse pattern 4′.

_{0} must satisfy Equation (35).

Collapse pattern 4′ (CASE-B and CASE-III). _{0}/_{1}-input velocity relation. _{0}/_{1}-input velocity relation.

When the structure goes into a plastic region after the first impulse and the second impulse acts after the structure attains the maximum deformation, Point B, the displacement and velocity of the mass just before the action of the second impulse are described by Equations (36) and (37). _{OA} and _{AB} in Equations (36), (37) can be obtained by Equations (4), (12). As shown in _{p1} after the first impulse can be derived as in Equation (6) by using the balance law between the point just after the first impulse and the point B of the maximum deformation.

Since the structure does not go into a plastic region just after the action of the second impulse, Equation (46) must be satisfied.

The energy balance law between the point just after the second impulse and the point H in

Since substitution of Equations (4), (6), (12), (32), (36), (37) into Equation (47) provides the transcendental equation, it is difficult to derive a closed-form expression for the input velocity corresponding to the collapse. To determine the input velocity corresponding to the collapse, this transcendental equation can be computed for given α and _{0}.

_{0}/_{1}-input velocity relation for Case-B and CASE-III in collapse pattern 4′.

Based on the collapse patterns explained above, a limit curve on the second impulse timing _{0}/_{1}-input velocity _{y} relation for the collapse and non-collapse states can be proposed. As an example, _{0}/_{1}-input velocity relation for the collapse and non-collapse states for α = −0.4. It should be remarked that the present SDOF model is an undamped model, and the states of _{0}/_{1}=0.5 and _{0}/_{1}=1.5 provide the same collapse limit. It can be observed that an isolated region of the collapse state exists around the level of _{0}/_{1}=0.5 (also 1.5) and the level of _{y}=1. The most important point to be remarked is that the critical state (Kojima and Takewaki,

Second impulse timing _{0}/_{1}-input velocity relation for collapse and non-collapse states (α = −0.4).

_{0}/_{1}=0.5 and _{0}/_{1}=1.5 certainly correspond to the critical state in the reference (Kojima and Takewaki,

Correspondence between arbitrary timing and critical timing of second impulse in collapse velocity level (α = −0.4).

_{0}/_{1}-input velocity relation for collapse and non-collapse states appear and α = −1/3 gives the boundary of the change of phases. When α is larger than −1/3, the non-linear resonance does not provide the minimum input level corresponding to collapse.

Collapse limit input velocity of double impulse with arbitrary interval for SDOF system with various negative post-yield slopes (α = −0.2, −1/3, −0.8).

Verification of the proposed collapse limit by time-history response analysis to double impulses with various input velocities and impulse timings.

Restoring-force characteristics corresponding to various combinations of impulse timing and input velocity level (blue marks indicate double impulse timings and red mark shows collapse state).

A dynamic collapse criterion for elastic–plastic structures under double impulse as a substitute of a near-fault ground motion has been derived. The conclusions may be summarized as follows:

The use of the double impulse enables the efficient use of the energy approach in the derivation of explicit expressions of a complicated elastic–plastic response of structures with the P-delta effect.

In contrast to the previous work (Kojima and Takewaki,

Discussions on several patterns of dynamic collapse behaviors introduced in the previous critical case are useful for deriving a boundary between the collapse and the non-collapse in the plane of the input velocity and the input frequency.

The most important point to be remarked is that the critical state (Kojima and Takewaki,

The validity of the proposed collapse criterion has been investigated by the numerical response analysis for structures under double impulses with collapse or non-collapse parameters. It has been confirmed that the proposed criterion has a reasonable accuracy.

The present paper dealt with an undamped system. This is because, if a damped system is treated, the formulation is too complicated even for the case of critical input (Saotome et al.,

All datasets generated for this study are included in the article/supplementary material.

SH formulated the problem, conducted the computation, and wrote the paper. KK conducted the computation and discussed the results. IT supervised the research and wrote the paper. All authors contributed to the article and approved the submitted version.

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.