# SDOF system

Consider an undamped elastic-perfectly plastic (EPP) single-degree-of-free- dom (SDOF) system of mass *m* and stiffness *k.* The yield deformation and yield force are denoted by *d _{y}* and

*f*respectively. Let ta, =

_{y},*yjk/m*,

*u,*and

*f*denote the undamped natural circular frequency, the displacement of the mass relative to the ground (deformation of the system), and the restoring force of the model, respectively. The time derivative is denoted by an overdot as in the previous chapters. In Section 4.4, these parameters will be dealt with in a nondimensional or normalized form to derive the relation of permanent interest between the input and the elastic-plastic response. Flowever numerical parameters will be introduced partially in Sections 4.4 and 4.5.

# Maximum elastic-plastic deformation of SDOF system to multiple impulse

## Non-iterative determination of critical timing and critical plastic deformation by using modified input sequence

Firstly, consider Input Sequence 1 as shown in Figure 4.2(a). If the SDOF system is in an elastic range, the critical timing *t _{0}* is exactly a half of the natural period of the SDOF system. On the other hand, if the SDOF system goes into a plastic range, the critical set of input velocity amplitude and input frequency (interval of impulses) need to be computed iteratively. This situation is the same for the multi-cycle sinusoidal wave (Caughey 1960, Iwan 1961, 1965a, b).

To avoid and overcome this difficulty, consider Input Sequence 2, as shown in Figure 4.2(b), which introduces a modified input. In Input Sequence 2, only the time interval between the first and second impulses is modified so that the second impulse is given at the time corresponding to the zero restoring-force. More specifically, Input Sequence 2 was introduced based on the assumption that if the steady state exists in which the impulse is given at the time corresponding to the zero restoring-force, such impulse provides the maximum steady-state plastic deformation. This assumption is assured by giving the critical timing obtained from Input Sequence 2 to Input Sequence 1. In other words, if the critical timing obtained from Input Sequence 2 is given to Input Sequence 1, the timing of impulse converges to the time attaining the zero restoring-force. This fact is also supported by the one-to-one correspondence between the input velocity amplitude and its critical timing of impulses. In this case, impulses have to be given at the times corresponding to the zero restoring-force points. It may also be possible to derive the Input Sequence 3 and its response with zero residual displacement (see Figure 4.2(c)).

The elastic-plastic response of the undamped EPP SDOF model to the multiple impulse can be described by the continuation of free-vibrations after considering sudden change of velocity of mass.

Let n_{maxl} and *u _{max2}* denote the maximum deformations after the first and second impulses in Input Sequence 2, respectively, as shown in Figure 4.3, and let

*Up*denote the plastic deformation amplitude in the critical steady state, as shown in Figure 4.3. The input of each impulse is expressed by the instantaneous change of velocity of the structural mass. Such responses can be derived by an energy balance approach without resorting to solving the equation of motion directly. The kinetic energy given at the initial stage (the time of action of the first impulse) and at the time of action of the second impulse is transformed into the sum of the hysteretic dissipation energy and the strain energy corresponding to the yield deformation (see Figures 1.5

*Figure 4.3* Definition of response quantities and response transition: (a) schematic diagram of deformation quantities in force-deformation relation; (b) transition of response process and impulse timing (Input Sequence 2) (Kojima andTakewaki 2015c).

and 1.7). By using this rule, the maximum deformation and plastic deformation amplitude can be obtained in a simple manner.

It should be emphasized that while the resonant equivalent frequency can be computed for a specified input velocity level by moving the excitation frequency parametrically in the conventional methods dealing with the sinusoidal input (Caughey 1960a, b, Iwan 1961, 1965a, b, Roberts and Spanos 1990, Liu 2000, Moustafa et al. 2010), no iteration is required in the method for the multiple impulse explained in this chapter. This is because the resonant equivalent frequency (resonance can be proved by using energy investigation: see Appendix in Kojima and Takewaki 2015a and Chapter 2 in this book) can be obtained directly without the repetitive procedure including nonlinear time-history response analysis. As a result, the timing of the second impulse can be characterized as the time corresponding to the zero restoring force.

Only critical response is captured by the method explained in this chapter, and the critical resonant frequency can be obtained automatically for the increasing input velocity level of the multiple impulse. One of the original points in this chapter is the introduction of the concept of “critical excitation” in the direct elastic-plastic response (Drenick 1970, Abbas and Manohar 2002, Takewaki 2002, 2007, Moustafa et al. 2010). “Direct” means the evaluation without the use of an equivalent linear model for which an iteration is required for determination. Once the frequency and amplitude of the critical multiple impulse are found, the corresponding multi-cycle sinusoidal motion as a representative of the long-duration earthquake ground motion can be identified.

Ler us explain the evaluation method of и_{тах1}, и_{тах2}, and *u _{p}.* Let

*u*and

_{pX}*u*denote the plastic deformations after the first and second impulses, respectively. In this problem, three cases exist depending on the yielding stage (Kojima and Takewaki 2015a). Let denote the input level of

_{p2}velocity of the impulse at which the undamped EPP SDOF system at rest just attains the yield deformation after the single impulse of such velocity.

Consider the case where the undamped EPP SDOF model goes into the yielding stage even after the first impulse. This case corresponds to CASE 3 in the problem of the double impulse (Kojima and Takewaki 2015a and Chapter 2) (the input level of the first impulse is 0.5 V instead of V). Figure 4.3(a) shows the schematic diagram of the response in this case. The maximum deformation after the first impulse и_{тах1} can be obtained from the following energy balance law (see Eq. (3.20) for the triple impulse).

On the other hand, the maximum deformation after the second impulse м_{тах2} ^{can} be computed from another energy balance law (see Figure 2.2(b) for the double impulse).

where *v _{c}(=V)* is characterized by

*mu 42 = f*and

_{y}dJ2*u*is characterized by M

_{pl}_{max2}+ («maxi -

*+*

**dy)=**^{d}y*Upl*•

^{В}У using this relation,

*u*can be obtained from

_{m3x2}

From Eqs. (4.5a) and (4.5b), *u _{p}(=u_{p2})* and и

_{тах2}can be obtained as follows.

As in the above case, the velocity V induced by the second impulse is added to the velocity *v _{c}* introduced by the first impulse. This velocity

*v*indicates the maximum velocity in the unloading process. Although only CASE 3 in the double impulse (Kojima and Takewaki 2015a and Chapter 2) was considered here, CASE 2 (yielding only after the second impulse) can be treated by replacing

_{c}*v*in Eqs. (4.5a, b) by 0.5 V. It is noted that when the input velocity V is smaller than 2, the number of impulses more than two is

_{c}*Figure 4.4* Comparison of response among Input Sequence I (multiple impulse), Input Sequence I (sine wave) and Input Sequence 2 (multiple impulse): (a) plastic deformation amplitude; (b) residual deformation (Kojima andTakewaki 2015c).

necessary to attain the inelastic critical steady state in Input Sequence 2, but the plastic deformation amplitude can be calculated by Eq. (4.6a) in all input level.

Figure 4.4 shows the plastic deformation amplitude *u _{p}* (

*u*in this case) and the residual deformation

_{pl}*u*shown in Figure 4.3(a), with respect to the input level

_{r},*V/V*for Input Sequence 1 and 2. It is interesting to note that while the plastic deformation amplitude is the same for Sequence 1 and 2, the residual deformations are different. This results from the difference in the initial disturbances in Input Sequence 1 and 2.