Since a long time ago, a half-cycle sine wave is used as a simple representative of impulsive ground motions, e.g., see Housner (1963). In this paper, a near-fault impulsive ground motion is simplified into a half-cycle sine wave and then into a single impulse (see Figure 1). The introduction of impulses enables a simple energy evaluation of elastic–plastic structures (see Figure 2).

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)mV² provided by one impulse with the dissipated energy (triangle in Figure 4).

In Eq. (1), u_p = − (1/α)d_y is used (see Figure 4). Therefore, the ratio of limit input velocity to the reference velocity V_y (input velocity just attaining the yield of the model after one impulse) may be expressed by (2)V/Vy=1−(1/α)

Then the reference velocity (strength indicator of the model) may be derived for a specified input velocity level V.

The corresponding model strength fy[1] can then be obtained as (4)fy[1]=k[1]dy=(m/dy)Vy[1]2=(mV2/dy)/1−(1/α) where k^[1] is the stiffness of this model.

The limit input velocity for two consecutive impulses can be derived from the following procedure. The maximum deformation and the residual deformation after the first impulse can be computed by equating the kinetic energy (1/2)mV² provided by the first impulse with the dissipated and strain energy (quadrangle in Figure 5). Then the limit input velocity for the two consecutive impulses inducing collapse after the second impulse can be derived by equating the kinetic energy (1/2)mV² provided by the second impulse with the dissipated energy (triangle in Figure 6).

Let up(1) and up(2) denote the plastic deformation after the first impulse and the second impulse, respectively. Since the plastic deformation of the model just attaining the collapse after the second impulse can be described by −(1/α)d_y, the following relation holds (see Figure 6).

First of all, find up(1) from A₁ = A₂ (see Figure 6) which is guaranteed by the same input energy (1/2)mV² by the first and second impulses. The condition A₁ = A₂ in terms of up(1),up(2) can be expressed by (6)(1/2)kdy2+fyup(1)+(1/2)αkup(1)2=(1/2)fy+αkup(1)dy+αup(1)+up(2)=(1/2)k1−(1/α)dy+αup(1)2 Rearrangement of Eq. (6) leads to (7)α(α−2)up(1)/dy2+2(α−2)up(1)/dy−(1/α)=0 From Eq. (7), up(1)/dy can be obtained as (8)up(1)/dy=(1/α)−1±(α−1)/(α−2) Since 0≤up(1)≤−(1/α)dy, the following expression is derived.

By using the energy balance, the input velocity level V of the single impulse can be related to A₁ (or A₂) and up(1). The energy balance after the first impulse can then be expressed as (10)(1/2)mV2=A1=(1/2)kdy2+fyup(1)+(1/2)αkup(1)2 Therefore, the limit input velocity corresponding to the collapse after two impulses may be derived as (11)V=ω1dy1+2up(1)/dy+αup(1)/dy2=Vy1+2up(1)/dy+αup(1)/dy2 By substituting Eq. (9) into Eq. (11), the ratio V/V_y is obtained as (12)V/Vy=1+{1/(α2−2α)}

The reference velocity (strength of the model) may then be derived for a specified input velocity level V.

(13)Vy[2]=V/1+{1/(α2−2α)} The corresponding model strength fy[2] can be obtained as (14)fy[2]=k[2]dy=(m/dy)Vy[2]2=(mV2/dy)/1+{1/(α2−2α)} where k[1] is the stiffness of this model.

In summary, the ratio of the reference velocity for collapse after two impulses to that for collapse after one impulse can be computed as (15)Vy[2]/Vy[1]=(α−2)/(α−1)

Finally, the ratio of the model strength for collapse after two impulses to that for collapse after one impulse can be expressed as (16)fy[2]/fy[1]=(α−2)/(α−1)

Figure 7 shows the plot of Vy[2]/Vy[1] with respect to α. Furthermore, Figure 8 presents the plot of fy[2]/fy[1] with respect to α. In Figure 8, numerical results using recorded ground motions (Kumamoto earthquake in 2016) are also plotted. It should be noted that, since the purpose of this section is to investigate the ratio fy[2]/fy[1], the same earthquake ground motion recorded on April 16, 2016 has been input twice.

Figure 9 shows the restoring-force characteristics of the models with different second slopes, under the earthquake ground motion of April 16, 2016, designed so that they just collapse after one single impulse. T₁ in Figure 9 is the natural period of the model. In the top left one, the plastic deformation proceeds toward the reverse direction different from the other three cases. On the other hand, Figure 10 illustrates the restoring-force characteristics of the models with different second slopes, under the twice repeated earthquake ground motion of April 16, 2016, designed so that they just collapse after two impulses.

The energy balance law for the single impulse can be expressed by (see Figure 11) (17)(1/2)mV2=(1/2)kdy2+fy(du−dy) where d_u indicates the ultimate deformation. Eq. (17) can be reduced to (18)V/Vy=2(du/dy)−1

For a specified input velocity level V, the reference velocity Vy[1] (strength indicator of the model) may be derived as (19)Vy[1]=V/2(du/dy)−1

Then the corresponding strength fy[1] of the model exhibiting collapse just after a single impulse can be obtained as (20)fy[1]=k[1]dy=(m/dy)Vy[1]2=(mV2/dy)/2(du/dy)−1 where k^[1] is the stiffness of this model.

Let up(1) and up(2) denote the plastic deformation after the first impulse and the second impulse, respectively. Since the plastic deformation of the model just attaining the collapse after the second impulse can be described by d_u − d_y, the following relation holds (see Figure 12).

First of all, find up(1) from A₁ = A₂ which is guaranteed by the same input energy (1/2)mV² by the first and second impulses. The condition A₁ = A₂ in terms of up(1),up(2) can be expressed by (22)(1/2)kdy2+fyup(1)=(1/2)kdy2+fyup(2) From Eq. (22), up(1)/dy can be obtained as (23)up(1)=up(2)=(1/2)(du/dy)−1dy

By using the energy balance, the input velocity level V of the single impulse can be related to A₁ (or A₂) and up(1). The energy balance after the first impulse can then be expressed as (24)(1/2)mV2=A1=(1/2)kdy2+fyup(1) Eq. (24) can be reduced to (25)V/Vy=du/dy Then the reference velocity (strength indicator of the model) may be derived for a specified input velocity level V.

(26)Vy[2]=V/du/dy The corresponding model strength fy[2] can be obtained as (27)fy[2]=k[2]dy=(m/dy)Vy[2]2=(mV2/dy)/(du/dy)

Therefore, the ratio of the reference velocity for collapse after two impulses to that for collapse after one impulse can be computed as (28)Vy[2]/Vy[1]=2−(du/dy)−1

Finally, the ratio of the model strength for collapse after two impulses to that for collapse after one impulse can be expressed as (29)fy[2]/fy[1]=2−(du/dy)−1

Vy[2]/Vy[1] in Eq. (28) and fy[2]/fy[1] in Eq. (29) are plotted in Figures 13 and 14, respectively. It can be found from Figure 14 that the strength ratio fy[2]/fy[1] exhibits the value between 1.5 and 1.8 for the ductility μ = d_u/d_y between 2 and 5.

The formulation presented in Section “Limit Input Velocity for Two Consecutive Impulses” is applied to wooden structures and reinforced-concrete structures with slip-type hysteresis. As explained in the beginning of Section “Case of Collapse Limit on Maximum Deformation,” the formulation in section “Limit Input Velocity for Two Consecutive Impulses” was made for an ideal model as shown in Figure 11 and the results may be applied approximately to wooden structures and reinforced-concrete structures.

Consider wooden structures here. Figure 15 shows the restoring-force characteristics of the models with different strength ratios under repeated 2016 Kumamoto earthquake ground motion (April 16) designed so that they just collapse after one impulse or two impulses [reference model: collapse just after two impulses/slip and bilinear model (wooden structures)]. This model is a combination of an elastic-perfectly plastic model and a slip model. The resistance ratios with which the respective models (elastic-perfectly plastic model and slip model) govern are 0.2 and 0.8. The mass is 60 × 10³ (kilogram), the yield deformation is d_y = 0.1 (m) and the structural damping ratio is 0.02. It can be observed that the strength ratio 1.5 is reasonable for wooden structures. In other words, the maximum deformation of the wooden structure, designed for two impulses, under two consecutive severe earthquake ground motions is almost equivalent to that of the wooden structure, designed for one impulse, under one severe earthquake ground motion and their strength ratio is about 1.5.

Consider next reinforced-concrete structures. Since reinforced-concrete structures have complex hysteresis rules, a new scenario different from Figures 11 and 12 may be necessary. However, a simple numerical investigation is conducted here in order to obtain the property on strength ratio between the structure designed for two impulses and that for one impulse.

Figure 16 shows the restoring-force characteristics of the models with different strength ratios under repeated 2016 Kumamoto earthquake ground motion (April 16) designed so that they just collapse after one impulse or two impulses [reference model: collapse just after two impulses/Takeda model (Takeda et al., 1970) (reinforced-concrete structures)]. The mass is 60 × 10³ kg, the crack deformation is d_c = 0.003 m, the yield deformation is d_y = 0.03 m, and the structural damping ratio is 0.02. It can be observed that the strength ratio 1.3–1.4 is reasonable for reinforced-concrete structures. In other words, the maximum deformation of the reinforced-concrete structure, designed for two impulses, under two consecutive severe earthquake ground motions is almost equivalent to that of the reinforced-concrete structure, designed for one impulse, under one severe earthquake ground motion and their strength ratio is about 1.3–1.4.

It may be useful to remind that the present paper introduced two scenarios on the collapse. For steel buildings considering the P-delta effect or strength-degradation effect, the zero restoring force represents the collapse limit. This was demonstrated in Figures 8–10. On the other hand, for wooden and reinforced-concrete structures, the maximum deformation defines the collapse limit. Regarding this collapse scenario, a simple slip-type hysteretic model as shown in Figure 12 has been introduced. Although a theoretical result has been obtained in Figure 14, wooden and reinforced-concrete structures have more complicated hysteretic models. Therefore, the comparison as shown in Figure 8 for steel buildings is difficult in wooden and reinforced-concrete structures and another investigation has been conducted in wooden and reinforced-concrete structures. In Figures 15 and 16, the strength ratio has been investigated in which a strengthened building exhibits the same maximum deformation under two consecutive ground motions as that of the corresponding building with the original strength under one ground motion. This provides an appropriate strength ratio for wooden and reinforced-concrete structures.