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, 1968; Sun et al., 1973; Tanabashi et al., 1973; Bertero et al., 1978; Takizawa and Jennings, 1980; Bernal, 1987, 1998; Nakajima 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; Ibarra and Krawinkler, 2005; Adam and Jager, 2012). While earthquake and structural engineering made clear its significance in the real world, e.g., safety check of structures and infrastructures, applied mechanics contributed to the theoretical advancement in this field (Herrmann, 1965; Ishida and Morisako, 1985; Maier and Perego, 1992; Araki and Hjelmstad, 2000; Williamson and Hjelmstad, 2001).

It seems that the first theoretical achievement on collapse of structures subjected to earthquake ground motions was made by Jennings and Husid (1968). They dealt with a single-degree-of-freedom (SDOF) system with an elastic–plastic spring and demonstrated that the P-delta effect makes the natural period of the structure longer. Sun et al. (1973) focused on the free vibration of the SDOF system subjected to an initial impact and made clear the stability or collapse boundary. Extension to multi-degree-of-freedom (MDOF) systems has also been attempted (Takizawa and Jennings, 1980; Nakajima et al., 1990).

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, 1998) or the localization of deformation (Maier and Perego, 1992). Dynamic collapse behaviors and responses of actual and realistic frame models have been investigated by some researchers (Ger et al., 1993; Challa and Hall, 1994; Hall, 1998; Sivaselvan et al., 2009). Through such investigations, various effects, such as non-linear geometric effect, non-linear material behavior, and spread of the plastic zone, were incorporated in the numerical methods.

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, 2016b). In the paper of Kojima and Takewaki (2016b), a simple closed-form dynamic collapse criterion has been proposed for the double impulse as a simplification of the near-fault ground motion by taking full advantage of the energy balance law. They focused on the non-linear resonant situation (Drenick, 1970; Takewaki, 2002; Moustafa et al., 2010) and showed that several collapse patterns exist.

The effects of near-fault ground motions on structural response have been studied extensively (Bertero et al., 1978; Hall et al., 1995; Sasani and Bertero, 2000; Alavi and Krawinkler, 2004; Makris and Black, 2004; Mavroeidis et al., 2004; Kalkan and Kunnath, 2006; Xu et al., 2007; Rupakhety and Sigbjörnsson, 2011; Yamamoto et al., 2011; Jarernprasert et al., 2013; Minami and Hayashi, 2013; Khaloo et al., 2015; Vafaei and Eskandari, 2015). These many investigations classified the input characteristics into two ones, i.e., the fling-step input and the forward-directivity input (Mavroeidis and Papageorgiou, 2003; Bray and Rodriguez-Marek, 2004; Kalkan and Kunnath, 2006; Mukhopadhyay and Gupta, 2013a,b; Zhai et al., 2013; Hayden et al., 2014; Yang and Zhou, 2015).

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, 1960a,b; Iwan, 1961, 1965a,b; Kojima and Takewaki, 2015a,c, 2016a,b) developed a completely different innovative approach to the peak elastic–plastic response using an energy balance law without solving directly the equations of motion. Furthermore, the resonant and overturning phenomena have been investigated from various viewpoints (Chatzis and Smyth, 2012; Makris and Vassiliou, 2013; Casapulla, 2015; Nabeshima et al., 2016; Casapulla and Maione, 2017).

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, 2016b) for the resonant critical case, a general collapse criterion is provided for the velocity amplitude and the frequency of the double impulse. It is significant that no iteration is needed in the derivation of the dynamic collapse criterion except the solution of transcendental equations. It is shown that 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, 2016b) corresponding to the non-linear resonance does not necessarily provide the minimum input velocity level with respect to arbitrary impulse timing. The validity of the proposed dynamic collapse criterion is examined by the numerical response analysis for structures under double impulses with collapse or non-collapse parameters.

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, 2015a,b; Kojima et al., 2015), two types are distinctive in near-fault ground motions. The fault-parallel fling-step input can be expressed by a one-cycle sinusoidal wave (Mavroeidis and Papageorgiou, 2003; Kalkan and Kunnath, 2006) and the fault-normal forward-directivity input can be represented by three sinusoidal wavelets (see Figure 1). The fling step results from the permanent displacement of the ground induced by the fault dislocation and the forward directivity effect can be explained by the relation of the movement of the rupture front with the site. In this paper, a double impulse is used following the references (Kojima and Takewaki, 2015a, 2016a; Kojima et al., 2015). The double impulse enables the derivation of a straightforward expression of the elastic–plastic responses based on an energy approach by taking advantage of the properties of free vibrations. Another advantage of the double impulse is the ease of the derivation of the critical resonant timing of impulses which is not possible for the sinusoidal and other inputs without a repetitive procedure. While most of the conventional methods (Caughey, 1960a,b; Iwan, 1961) use the equivalent linearization techniques for the structural model under the original unchanged input, the method using the double impulse (Kojima and Takewaki, 2015a,b) employs the transformation of the input for the unchanged structural model. This property leads to an advantageous feature that the method using the double impulse is appropriate even for large plastic deformation.

Consider a simplified ground acceleration ü_g(t) as shown in Figure 1A (Kojima and Takewaki, 2015a) which is expressed by

V is the velocity amplitude in both positive and negative directions, and t₀ is the time interval of two impulses. The time derivative is denoted by an over-dot. Figure 1A also illustrates the comparison with the corresponding one-cycle sinusoidal wave. For reference, the velocity and displacement of both inputs are plotted in Figures 1A,B indicates the triple impulse as a substitute of a forward-directivity input. Good agreement can be observed even in the form of velocity and displacement. However, the correspondence in the response should be discussed carefully (see Kojima and Takewaki, 2016a).

The Fourier transform of the acceleration ü_g(t) of the double impulse can be derived as,

Kojima and Takewaki (2016b) treated the case of non-linear resonant input to derive the collapse limit of a bilinear hysteretic SDOF model with negative second slope under the double impulse. The terminology “nonlinear resonant input” in this paper means the case where the non-linear response of the SDOF model attains the maximum with respect to the interval of two impulses of the double impulse. When the restoring-force in the second stiffness range becomes zero, the model collapses. This is because the model cannot sustain the external force at this stage (Kojima and Takewaki, 2016b). Let f and u denote the restoring force of the spring of stiffness k and the displacement of the mass m, respectively. In addition, let f_y and d_y denote the yielding force and the yield displacement, respectively. The natural period of this SDOF model is denoted by T1=2π/ω1(ω1=k/m:natural circular frequency). The ratio of the second slope to the initial slope is expressed by α. The negative stiffness depends on the magnitude of elastic stiffness and the effect of the P–Δ effect. When the magnitude of the effect of the P–Δ effect against the elastic stiffness is large, the post-yield stiffness is apt to become negative. V_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.

Figure 2 summarizes the collapse limit of input velocity amplitude with respect to the second slope.

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.

The parameter 1-(1/α) is related to the coefficient such that the structure just attains the collapse under only the first impulse with the input velocity level V/Vy=1-(1/α) (Kojima and Takewaki, 2016b). This classification is different from CASE-1, 2, 3 in the previous section which are defined for resonant critical input. In non-resonant cases, a different kind of classification is necessary.

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 t_OA.

(CASE-A)

In CASE-A (Figure 3A), it is evident that

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

(CASE-B)

In CASE-B (1<V/Vy<1-(1/α)) (Figure 3A), the ratio of the time between Points O and A to T₁ can be expressed by

In this case, the following condition must be satisfied.

The plastic deformation u_p1 after the first impulse can be expressed by using the energy balance law (Kojima and Takewaki, 2015a, 2016b) between the point of the first impulse and the maximum deformation point.

The time history from Point A through Point B and the time t_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 v_A at Point A.

From Equation (7) and the initial condition (u(t = 0) = −d_y, u∙(t=0)=vA), the displacement and velocity between Points A and B are

The ratio of the time between Points A and B to T₁ 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 u(t=0)=-dy-up1,u∙(t=0)=0 and t = 0 at Point B leads to

The following results for the ratios of the times between the principal points to T₁ can be obtained.

Based on these results, the ratio of the critical interval of the double impulse to T₁ can be expressed by

(CASE-C)

Consider CASE-C (Figure 3A). As in CASE-B, the condition (5) is satisfied. The displacement and velocity after yielding (Point A) can be expressed by Equations (10), (11) as in CASE-B. Since the displacement at the collapse (Point D) is −dy{1−(1/α)} and u(t_AD) = −dy{1−(1/α)} for time t from Point A, the ratio of the time between Point A and Point D to T₁ can be obtained as

Figure 3B shows the classification based on the input level of the double impulse and the time between origin O and principal points for the case α =-0.4. In this case of α =-0.4, CASE-A, B, C exist. The normalized time t_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 t_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 t_OP or t_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 t_OB, t_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 t_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 t0c between two impulses is indicated by a solid blue line. This line exists only in CASE-A and B. These boundaries of arrival times are useful for the classification of regions.

On the other hand, Figure 3C presents the classification of regions based on the timing of the second impulse for the case α =-0.4. In this case of α =-0.4, CASE-I, II, III, IV exist. CASE-I exists in CASE-A, B, C and CASE-II exists in CASE-B and C. Furthermore, CASE-III exists in CASE-B and CASE-IV exists in CASE-C. Once such classification of regions based on the timing of the second impulse is conducted, the judgment of collapse or non-collapse is made efficiently by introducing the energy balance law (Kojima and Takewaki, 2015a, 2016b).

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, 2016b). Collapse pattern 4′ represents a new type.

Based on the collapse patterns explained above, a limit curve on the second impulse timing t₀/T₁-input velocity V/V_y relation for the collapse and non-collapse states can be proposed. As an example, Figure 11 shows the second impulse timing t₀/T₁-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 t₀/T₁=0.5 and t₀/T₁=1.5 provide the same collapse limit. It can be observed that an isolated region of the collapse state exists around the level of t₀/T₁=0.5 (also 1.5) and the level of V/V_y=1. The most important point to be remarked is that the critical state (Kojima and Takewaki, 2016b) corresponding to the non-linear resonance does not necessarily provide the minimum input velocity level with respect to arbitrary impulse timing.

Figure 12 presents the correspondence between arbitrary timing and the critical timing (Kojima and Takewaki, 2016b) of the second impulse in the collapse velocity level (α = −0.4). It can be confirmed that the states of t₀/T₁=0.5 and t₀/T₁=1.5 certainly correspond to the critical state in the reference (Kojima and Takewaki, 2016b).

Figure 13 indicates the collapse limit input velocity of the double impulse with arbitrary interval for the SDOF system with various negative post-yield slopes (α = −0.2, −1/3, −0.8). The correspondence between the arbitrary timing and critical timing (Kojima and Takewaki, 2016b) of the second impulse in the collapse velocity level is also shown again. It can be found that, as the parameter α changes, different phases of the limit curve in the second impulse timing t₀/T₁-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.

Figure 14A shows the verification of the proposed collapse limit by time-history response analysis to double impulses with various input velocities and impulse timings (α = −0.2, −0.4). Many combinations of the impulse timing and the input velocity level in the collapse and non-collapse states were selected in the case of α = −0.2, −0.4. For strict verification, many combinations of the impulse timing and the input velocity level were chosen near the boundary of the proposed limit curve. Figure 14B presents the displacement time histories of the model with α = −0.4 to five inputs (1), (2), (5), (6), (8) shown in Figure 14A. It can be observed that strict classification into the collapse state and the non-collapse state has been made with the proposed limit curve.

Figure 15 presents the 9 restoring-force characteristics corresponding to various combinations of the impulse timing and the input velocity level shown in Figure 14 (blue marks indicate the double impulse timings and red mark shows the collapse state). The cases (1), (2), (5), (6), (8) exhibit collapse behaviors. The response behaviors can be well-understood from these figures.

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.

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., 2019). The discussion on the damped system may be the future work.

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.