## Abstract

The use of bounding scenarios is a common practice that greatly simplifies the design and qualification of structures. However, this approach implicitly assumes that the quantities of interest increase monotonically with the input to the structure, which is not necessarily true for nonlinear structures. This paper surveys the literature for observations of nonmonotonic behavior of nonlinear systems and finds such observations in both the earthquake engineering and applied mechanics literature. Numerical simulations of a single degree-of-freedom mass-spring system with an elastic–plastic spring subjected to a triangular base acceleration pulse are then presented, and it is shown that the relative acceleration of this system scales nonmonotonically with the input magnitude in some cases. The equation of motion for this system is solved symbolically and an approximate expression for the relative acceleration is developed, which qualitatively agrees with the nonmonotonic behavior seen in the numerical results. The nonmonotonicity is investigated and found to be a result of dynamics excited by the discontinuous derivative of the base acceleration pulse, the magnitude of which scales nonmonotonically with the input magnitude due to the fact that the first yield of the spring occurs earlier as the input magnitude is increased. The relevance of this finding within the context of defining bounding scenarios is discussed, and it is recommended that modeling be used to perform a survey of the full range of possible inputs prior to defining bounding scenarios.

## 1 Introduction

A common practice in mechanical engineering is the use of so-called bounding scenarios in the design and qualification of structures. For example, if a structure is required to withstand drops up to a particular height, test and analysis efforts might focus solely on the maximum drop height rather than the full range of possible drop heights. While this practice enormously simplifies the work of engineers, it implicitly assumes that all quantities of interest (QoI) (strains, accelerations, etc.) increase monotonically with the magnitude of the applied input. While this is guaranteed by definition for linear systems, it is not necessarily true for nonlinear systems. If QoI do not increase monotonically with applied input, it is possible that a “bounding scenario” (e.g., a maximum drop height) actually produces a less severe structural response than a lesser scenario (e.g., a lower drop height), which could potentially lead to nonconservative errors in which a structure is believed to be safe based upon test and analysis of a bounding scenario, when in reality it cannot withstand the full range of environments it might encounter. Therefore, it behooves us to understand the conditions under which nonlinear structures exhibit nonmonotonic scaling of QoI, as well as the prevalence and severity of this phenomenon. While there does not appear to be literature directly addressing the issue of nonmonotonicity within the context of defining bounding scenarios, the fact that nonlinear systems can exhibit nonmonotonicity has been previously observed in the literature. There appear to be two major veins of literature in which such behavior is observed. The first is in the earthquake engineering literature, where the response to earthquake excitation of nonlinear systems ranging from single degree-of-freedom (SDOF) mass-spring systems to full-scale building models is studied and found to sometimes vary nonmonotonically with excitation level. The second is in the applied mechanics literature, where the counterintuitive behavior of simple beams and plates is studied using analytical, numerical, and experimental approaches. While the term “counterintuitive behavior” is used in this context to mean final plastic deformation opposite the direction of loading, these investigations also indicate some degree of nonmonotonicity. The literature in both of these areas is discussed below.

Throughout the earthquake engineering literature, extensive use is made of SDOF mass-spring systems subjected to earthquake excitation. An early study of the effect of material nonlinearity on such systems is documented in Ref. [1], in which the responses of SDOF systems with elastic-perfectly plastic springs were compared to those of the corresponding elastic systems. The results of this study show the maximum displacement of these oscillators *decreasing* as the displacement (or load) capacity at yield is decreased in some cases, which is equivalent to increasing the load level for this simple system. This unexpected result is not discussed in Ref. [1], but similar studies were later carried out in Ref. [2], and the nonmonotonicity was explicitly noted in this work. Reference [2] also noted that nonmonotonicity occurs when the branches for the positive and negative peaks intersect. Subsequent work in Ref. [3] also examined earthquake response of elastic-perfectly plastic SDOF oscillators and found nonmonotonic scaling of the ductility response with the first-mode spectral acceleration, which is a commonly used metric of environmental severity in earthquake engineering defined as the acceleration response spectrum of the input evaluated at the structure’s first period. This nonmonotonicity was explained in Ref. [3] as being the result of early, weaker portions of the loading which do not cause yielding at lower input levels but do cause yielding at higher input levels, thus altering the structure for later cycles.

Building upon this SDOF analysis, Ref. [3] goes on to develop an approach referred to as Incremental Dynamic Analysis (IDA) which has become a widely used tool in the earthquake engineering community. The basic idea behind IDA is to perform a series of nonlinear dynamic analyses with the input earthquake acceleration record scaled to several different levels ranging from linear structural response to structural collapse. The results are presented by plotting an intensity measure (IM), which is basically a measure of environmental severity, versus a demand measure (DM), which is a QoI from the model. IDA curves for a five-story steel braced frame subjected to 30 different ground acceleration records are presented in Ref. [3] in terms of first-mode spectral acceleration and maximum interstorey drift, and several of these curves are nonmonotonic, with maximum interstorey drift decreasing as the spectral acceleration is increased. IDA has been discussed extensively in the earthquake engineering literature and applied to a wide variety of structures (e.g., Refs. [4–10]), and nonmonotonic IDA curves appear to be quite common in this work. However, little attention has been given in this work to *why* the IDA curves are nonmonotonic, aside from the explanation given above [3]. This is likely because the IDA curves are used within a probabilistic analysis framework known as performance-based earthquake engineering (PBEE), which analyzes the full range of potential earthquakes rather than a single bounding scenario, so the question of how nonmonotonicity impacts the validity of bounding scenarios is less relevant in this context. As a final note concerning IDA, it is observed in Ref. [3] that the nonmonotonicity of the IDA curves can sometimes be quite extreme, resulting in a phenomenon referred to as structural resurrection, where an intermediate range of IM results in global collapse of the structure, while larger values of the IM do not result in collapse. This has clear implications for the definition of bounding scenarios, since a bounding scenario defined in the higher input regime where collapse does not occur would overlook the intermediate collapse regime, resulting in a nonconservative and potentially catastrophic error.

The second relevant branch of literature is concerned with the “counterintuitive behavior” of simple structures, such as beams and plates, and investigates this behavior through analytical, numerical, and experimental methods. In this context, “counterintuitive behavior” refers to final plastic deformation that is opposite the direction of loading. The first investigations into this question appear to have been conducted in the Soviet Union in the 1970s and documented in Ref. [11] (in Russian) and much later described in English in Refs. [12–14]. These papers describe a series of tests in which an explosive charge was detonated in a tank of water, causing shock waves to impinge upon a thin metal plate covering a hole in the thick-walled tank. The experiments were performed for various levels of explosive charge and different materials and geometries for the plates. While in general the final plastic deformation of the plate was directed in the shock direction, it was found that for certain plates and charge sizes, the final plastic deformation was directed opposite the shock direction. This phenomenon was not investigated further in Ref. [11] but was the subject of a great deal of subsequent work. One of the earliest analytical investigations of counterintuitive behavior is presented in Ref. [15], where an Abaqus model of an elastic-perfectly plastic beam supported by pins on both ends and loaded by a uniformly distributed rectangular pulse was studied. For certain parameters and load levels, this system was shown to exhibit final midpoint plastic deflection opposite the loading direction. Another interesting example that exhibits counterintuitive behavior is the case of a fixed circular plate subjected to a rectangular pressure pulse, similar to the plates originally tested in Ref. [11]. Using both an Abaqus finite element model and a Galerkin approximation, Ref. [16] studies the response of three different plates at a range of input levels, and it is found that certain ranges of input produce a chaotic response which can lead to final plastic deformation opposite the direction of loading, accompanied by oscillations that take much longer to ring out than at higher input levels. For both systems, the counterintuitive behavior appears to be due to differences in the static loading–unloading curves at different load levels, with the intermediate load levels associated with the counterintuitive behavior allowing for cycles of buckling in which the structure snaps from one side to the other. Although Refs. [15,16] do not explicitly discuss nonmonotonicity, a few figures in these papers suggest that some QoI for these structures scale nonmonotonically with load level, and further study of these systems in Ref. [17] demonstrates that several QoI are highly nonmonotonic and that the nonmonotonicity and counterintuitive behavior frequently occur at the same load levels. The counterintuitive behavior of the beam and plate structures discussed above is an example of what is referred to in Refs. [18,19] as “reverse buckling and post-chaotic self-organization.” These references provide an excellent survey of the literature on this subject, beginning with the counterintuitive plate experiments from Ref. [11]. They describe a wide variety of experiments and analyses in which counterintuitive behavior similar to that of Ref. [11] was observed (Refs. [20–25]). References [18,19] explain that many structures exhibit a so-called anomalous region that lies between the better-understood regions of fully linear response and large plastic deformation. While one might intuitively expect a structure to transition smoothly from linear response to large plastic deformation as the applied load is increased, the reality is often much stranger, with this transition region showing extreme sensitivity to geometric, material, numerical, and loading parameters, as well as final deformation opposite the loading direction. For the example of the circular plate subjected to a uniform pressure pulse, Refs. [18,19] explain that if the loading is large enough, the resulting “dome” configuration is stable enough to resist unloading and so the structure oscillates about this deformed state. However, if the loading is smaller, the deformed configuration is not stable, and so the plate snaps back to the opposite side, which leads to all of the unexpected behavior of the anomalous region. The numerical simulations of Ref. [17] suggest that one additional unexpected behavior associated with the anomalous region is nonmonotonic scaling of QoI with load level. This is but one of many different ways in which such nonmonotonic scaling can occur, with the elastic–plastic SDOF example from Ref. [3] providing another example of how such nonmonotonicity can occur.

The present work studies a very simple system exhibiting nonmonotonicity of QoI, namely an SDOF mass-spring system with an elastic–plastic spring subjected to a triangular base acceleration pulse. Section 2 presents numerical simulations of this system and demonstrates that for certain parameters and loads, the peak relative acceleration scales nonmonotonically with the base acceleration magnitude. The equation of motion for this system is solved symbolically in Sec. 3, and an approximate symbolic expression for the peak acceleration as a function of input magnitude is developed in Sec. 4 and shown to agree qualitatively with the numerical results from Sec. 2. Section 5 investigates the source of the nonmonotonicity and shows that it is due to dynamics excited by the discontinuous derivative of the applied input, the magnitude of which is proportional to the velocity at the time of the discontinuity. Since the first yield of the spring occurs earlier as the input magnitude is increased, this introduces a nonmonotonic dependence of the velocity at the time of the discontinuity, and thus the magnitude of the subsequent dynamics, upon the applied input magnitude. The paper concludes in Sec. 6, where the relevance of this work within the original context of defining bounding scenarios is discussed.

## 2 Mass-Spring System Subjected to a Triangular Base Acceleration Pulse

*A*and duration

*T*

*f*

_{i}is the spring force on step

*i*. The spring force is stored as a fifth-state variable which is updated after the first four states as follows:

*k*is determined from the hardening rule as

*f*

_{y}represents the current yield force for the spring, which is initialized at

*f*

_{0}and updated on each step where the spring is yielding

Equations (2) through (5) were used to study the response of this system at a range of input magnitudes for a variety of input waveforms. For the triangular pulse given by Eq. (1), these simulations showed an unexpected nonmonotonic dependence of the peak relative acceleration $(x\xa8\u2212u\xa8)$ upon the input magnitude for certain pulse durations. For example, Fig. 2 plots the peak relative acceleration versus the input magnitude for *T* = 14 and 34 ms. These curves show multiple lobes, where the peak acceleration initially increases, then levels off, and later decreases with increasing input magnitude. As the pulse duration is lengthened, additional lobes appear in these curves. Figure 3 shows acceleration time histories at the top and bottom of one of these lobes for *T* = 14 ms (*A* = 3700 and 6900 g). While the numerical results reveal this unusual dependence of the peak acceleration upon the input magnitude, they do little to explain it. Therefore, a symbolic solution to the equation of motion for this system is developed in Sec. 3.

## 3 Solution to Equation of Motion

*z*≡

*x*−

*u*is the relative displacement across the spring and

*f*denotes the spring force, which evolves over time according to the bilinear hardening rule shown in Fig. 1. We define the natural frequencies when the spring is yielding and not yielding, respectively, as $\omega 2\u2261k2/m$ and $\omega 1\u2261k1/m$.

In the special case where the spring never yields *f* = *kz* and it is a trivial matter to solve Eq. (6) and show that the peak relative acceleration scales monotonically with *A*. Therefore, this paper focuses on the more interesting case in which the spring does yield. It is shown in Appendix A that, to a first-order approximation, the following statement is true: when *ω*_{1}*T* ≫ 1 (i.e., the applied loading is slow compared to the elastic natural frequency), the spring either yields during the initial increasing portion of the loading ([0 *T*/2]) or it does not yield at all. This enormously simplifies the solution of the equation of motion for this system because, as will be demonstrated in the remainder of this section, either the spring remains elastic for all time when *A* ≤ *f*_{0}/*m*, or it undergoes a single yielding phase beginning at *t*_{y} ∈ [0 *T*/2] and ending at *t*_{2} > *T*/2 after which it does not yield again. In contrast, for the general case where we do not assume *ω*_{1}*T* ≫ 1, initial yield can occur at any time during or after the applied load, after which arbitrary cycles of yielding and not yielding can occur, which greatly complicates attempts to solve the equation of motion. For this reason, we focus on the special case where *ω*_{1}*T* ≫ 1 here, for which we have *t*_{y} ∈ [0 *T*/2]. We also assume that *k*_{1} ≫ *k*_{2}, as this is typically true of elastic–plastic hardening, and it allows us to simplify our peak acceleration expression in Sec. 5.

*z*(0) = 0 and $z\u02d9(0)=0$, we obtain the solution:

*t*

_{y}satisfies the equation

*k*

_{1}

*z*(

*t*

_{y}) = ±

*f*

_{0}. Substituting Eq. (8) into this, we obtain

*ω*

_{1}

*t*

_{y}≫ 1 (in practice, greater than roughly 10 radians), the linear term dominates the sine term and

*t*

_{y}can be approximated by

*t*

_{y}, the spring yields and the equation of motion for this yielding phase can be written as

*z*

_{y}≡

*z*(

*t*

_{y}). Equation (11) has a solution of the form

*z*(

*t*

_{y}) and $z\u02d9(ty)$ with Eq. (8) and solving for

*A*

_{1}and

*A*

_{2}, we obtain

*T*/2 or until the spring stops yielding, whichever comes first. However, it is shown in Appendix B that the spring cannot stop yielding until after

*T*/2 so Eq. (12) is valid for

*t*∈ [

*t*

_{y}

*T*/2]. After

*T*/2, the spring continues to yield until a later time

*t*

_{2}at which the yielding ends. During this period, the equation of motion becomes

*t*∈ [

*T*/2 min(

*t*

_{2},

*T*)]. Enforcing continuity of

*z*(

*T*/2) and $z\u02d9(T/2)$ with Eq. (12) and solving for

*B*

_{1}and

*B*

_{2}, we obtain

*T*/2; i.e.,

*t*

_{2}is the first solution greater than

*T*/2 to $z\u02d9(t2)=0$. Differentiating Eq. (16) and rearranging, this becomes

*T*

_{1}satisfies Eq. (19)

*n*∈

*Z*, the set of all integers. If there are elements of

*T*

_{1}in the interval [

*T*/2

*T*], then

*t*

_{2}is the smallest such element. If there are no elements of

*T*

_{1}in the interval [

*T*/2

*T*], then Eq. (16) applies throughout [

*T*/2

*T*] and it is necessary to solve the equation of motion for the subsequent period [

*T t*

_{2}] to find an expression for

*t*

_{2}. This has been done in Ref. [17], but for the sake of brevity, we do not show this solution here and instead choose to focus strictly on the case where

*t*

_{2}<

*T*since this is sufficient to investigate the unexpected acceleration scaling shown in Sec. 2. By the same token, it is possible to continue the solution for the case where

*t*

_{2}<

*T*into the subsequent periods [

*t*

_{2}

*T*] and [

*t*

_{2}

*T*], and while this work has been done in Ref. [17], it is not shown here since the solution up through [

*T*/2

*t*

_{2}] is sufficient to understand the nonmonotonic scaling observed in Sec. 2. Finally, an important result proven in Appendix C is that $z\xa8(t2)>0$ when

*t*

_{2}<

*T*. Differentiating Eq. (16) twice, this can be stated as

## 4 Peak Acceleration Expression

*t*

_{2}<

*T*, the displacement during [

*T*/2

*t*

_{2}] is given by Eq. (16), which can be restated as

*ϕ*

_{B}= arctan(−

*B*

_{1}/

*B*

_{2}). This can be differentiated once to obtain the velocity and again to find the acceleration

*ω*

_{2}

*t*+

*ϕ*

_{B}=

*nπ*where

*n*∈

*Z*. Furthermore, Eq. (23) shows that at these times the velocity is $z\u02d9=2mA/Tk2>0$. In addition, it is proven in Appendices B and C that $z\u02d9\u22640$ during [

*t*

_{y}

*t*

_{2}]. Therefore, we conclude that $z\xa8$ does not reach a local peak during [

*T*/2

*t*

_{2}] and that the maximum acceleration must occur at one of the endpoints of this interval. The acceleration at the endpoints can be compared by first noting that since $z\u02d9(t2)=0$ by definition, Eq. (23) shows that

*ω*

_{2}

*T*/2 +

*ϕ*

_{B})| > |sin(

*ω*

_{2}

*t*

_{2}+

*ϕ*

_{B})|, which further implies that |cos(

*ω*

_{2}

*T*/2 +

*ϕ*

_{B})| < |cos(

*ω*

_{2}

*t*

_{2}+

*ϕ*

_{B})|. Returning to Eq. (24), we now see that $|z\xa8(t2)|>|z\xa8(T/2)|$. Thus, the peak acceleration in the interval [

*T*/2

*t*

_{2}] always occurs at

*t*

_{2}and can be written as

It is apparent from this expression that the peak acceleration during this period does not scale monotonically with *A*, but instead shows interesting dependence upon *f*_{0}/*mA*. This is investigated further in Sec. 5. We also note here (without derivation for the sake of brevity) that the peak acceleration in [*t*_{2} ∞] also follows Eq. (33) so that this expression can be generalized to apply to [*T*/2 ∞], and that a similar expression applies when *t*_{2} > *T*. The derivation of these results can be found in Ref. [17].

## 5 Explanation of Unexpected Peak Acceleration Scaling

*A*that one might expect, the second portion shows a nonmonotonic and much less intuitive dependence on

*A*. Examination of Eq. (33) shows that it has local maxima of $42A/T\omega 2$ when

*n*∈

*Z*. Note that since the system is elastic up to

*A*=

*f*

_{0}/

*m*, only values of

*A*=

*f*

_{0}/

*m*are meaningful since the solution underlying Eq. (33) assumes that the spring yields. Thus, the first maximum of Eq. (33) occurs at

*A*=

*f*

_{0}/(

*m*(1 − 2

*π*/

*ω*

_{2}

*T*)) and the first zero occurs at

*A*=

*f*

_{0}/(

*m*(1 − 4

*π*/

*ω*

_{2}

*T*)). In order for these to correspond to meaningful (positive) values of

*A*, we must have

*ω*

_{2}

*T*> 2

*π*for the maximum and

*ω*

_{2}

*T*> 4

*π*for the zero. This shows that the load duration must be at least as long as one full period of the system’s yielding dynamics in order for Eq. (33) to reach a maximum, and at least as long as two full periods to achieve a zero. If the load duration is less than one period of the yielding dynamics, Eq. (33) will increase monotonically with

*A*and the radical in this expression will only cause a reduction in the slope as

*A*is increased. On the other hand, if the load duration is significantly more than two periods of the yielding dynamics, Eq. (33) will exhibit multiple maxima and zeros as

*A*is increased (i.e., one for each positive integer value of

*n*that can be substituted into the expressions above without resulting in

*A*< 0). Figure 4 plots the actual peak acceleration from numerical simulations, the predicted peak acceleration from the symbolic solution of Sec. 4 (i.e., the maximum over time of the second derivatives of Eqs. (8), (12), and (16)), as well as Eq. (33) versus the input magnitude

*A*for a system with the parameters from Fig. 1 and

*T*= 3.1, 7.5, 13.8, and 20.8 ms. These durations correspond to roughly 1, 2.4, 4.4, and 6.6 periods of the yielding dynamics, respectively. This figure shows that the symbolic solution from Sec. 4 generally matches the numerical results quite well. Furthermore, this figure shows that Eq. (33) matches the numerical results near the maxima and diverges from them near the zeros. This is because Eq. (33) gives the maximum acceleration after

*T*/2, and when this value approaches zero, the overall maximum occurs before

*T*/2. This causes the behavior seen in Fig. 4 where the peak acceleration curve consists of lobes corresponding to peaks after

*T*/2 interspersed with linear regions corresponding to peaks before

*T*/2. Finally, we note that the process illustrated in Fig. 4, whereby the load duration is increased and additional lobes appear in the peak acceleration curve, can be continued indefinitely to produce any number of peaks.

*T*/2

*T*] can be shown to be a sum of sine and cosine terms with coefficients $\u2212\omega 12G1$ and $\u2212\omega 12G2$ by differentiating Eq. (A2) twice, where

*G*

_{1}and

*G*

_{2}are given by Eqs. (A3) and (A4). Thus, the peak acceleration (and the magnitude of the dynamic part of the displacement) during [

*T*/2

*T*] can be shown to have periodic dependence upon the number of elastic periods swept out in the first half of the load duration

*ω*

_{1}

*T*/2 = 10

*π*and 11

*π*, which illustrates that the dynamic part of the solution is much larger when

*ω*

_{1}

*T*/2 is an odd multiple of

*π*as opposed to an even one. While one might have expected that increasing the pulse duration from 9.9 to 10.9 ms in this case would decrease the dynamic response because the load is already quite slow compared to the natural frequency of 1000 Hz, the effect is the opposite. This perhaps unexpected result arises from the discontinuous slope of the applied load at

*T*/2. To see this, note that, in solving for

*G*

_{1}and

*G*

_{2}, we must match the displacement and velocity at

*T*/2 between the solutions for [0

*T*/2] and [

*T*/2

*T*] (Eqs. (8) and (A2)). From these equations, we have

*T*/2] is a sinusoidal term minus a positive constant, while the velocity in [

*T*/2

*T*] is a sinusoidal term plus the same positive constant. As shown in Fig. 6, when $z\u02d9(T/2)=0$, the velocity at

*T*/2 can be matched with $G12+G22=2mA/T\omega 1k1$, which is the minimum value of $G12+G22$. In contrast, as shown in Fig. 7, when $z\u02d9(T/2)$ is near its most negative, a much larger value of $G12+G22$ is required (near the maximum of 6

*mA*/

*Tω*

_{1}

*k*

_{1}). Thus, we see that the discontinuity in the slope of the applied input excites dynamic response in [

*T*/2

*T*] (even if there was minimal dynamic response in [0

*T*/2]) and that the magnitude of the this response is proportional to the magnitude of the velocity at

*T*/2, which is maximized when

*ω*

_{1}

*T*/2 is an odd multiple of

*π*.

*T*/2

*t*

_{2}] (for

*t*

_{2}<

*T*) is given by Eq. (16) and the acceleration during this period can be shown to be a sum of sine and cosine terms with coefficients $\u2212\omega 22B1$ and $\u2212\omega 22B2$. It is further shown in Eq. (31) that, analogous to Eq. (34), the magnitude of the dynamic part of the solution during [

*T*/2

*t*

_{2}] has periodic dependence upon the number of yielding periods swept out in [

*t*

_{y}

*T*/2]. Furthermore, it is shown in Eq. (10) that

*t*

_{y}is inversely proportional to

*A*, and this introduces a periodic dependence of $B12+B22$ upon 1/

*A*as shown in Eq. (32). Just as for the elastic system above, this can be explained in terms of the discontinuous slope of the applied input at

*T*/2. In solving for

*B*

_{1}and

*B*

_{2}, it is again necessary to match the displacement and velocity at

*T*/2, and the velocity during [

*t*

_{y}

*T*/2] and [

*T*/2

*t*

_{2}] can be found by differentiating Eqs. (12) and (16) and rearranging

*ϕ*

_{A}= arctan(−

*A*

_{1}/

*A*

_{2}). These expressions exhibit the same switch that was observed in the elastic system from being centered on a negative mean value in [

*t*

_{y}

*T*/2] to a positive mean value in [

*T*/2

*t*

_{2}]. Thus, when $z\u02d9(T/2)$ is near its most negative value, $B12+B22$ must assume a larger value than when $z\u02d9(T/2)$ is less negative in order to achieve continuity of the velocity between the two solutions at

*T*/2. Furthermore, it is demonstrated in Appendix D that when

*k*

_{1}≫

*k*

_{2}, the minima of $z\u02d9$ in [

*t*

_{y}

*T*/2] occur when

*ω*

_{2}(

*t*−

*t*

_{y}) ≈ (2

*n*− 1)

*π*, where

*n*∈

*Z*

^{+}. Thus, we see that the velocity at

*T*/2 is at its most negative, leading to the maximum value of $B12+B22$, when

*ω*

_{2}(

*T*/2 −

*t*

_{y}) ≈ (2

*n*− 1)

*π*. The nonmonotonic dependence of the peak acceleration upon

*A*is therefore explained by the dynamics excited by the discontinuous derivative of the applied input at

*T*/2, and the magnitude of which is dependent upon the velocity at

*T*/2. As

*A*is increased, the length of the period [

*t*

_{y}

*T*/2] is also increased, which can either increase or decrease the velocity at

*T*/2 and hence the magnitude of the subsequent dynamics $B12+B22$.

The behavior described above is illustrated in Figs. 8 and 9, which show the relative velocity and acceleration time histories for the system of Fig. 1 subjected to 7.5 ms pulses with 2200 and 4200 g magnitudes. At the lower input level of 2200 g, the velocity at *T*/2 is not far from its negative extreme as shown in Fig. 8. This excites a relatively large dynamic term after *T*/2, with peak accelerations of just over 800 g as shown in Fig. 9. In contrast, when the input level is increased to 4200 g, the velocity at *T*/2 is closer to zero and the peak acceleration is reduced to approximately 500 g.

## 6 Conclusion

This paper has studied nonmonotonic scaling of QoI for an SDOF mass-spring system with an elastic–plastic spring subjected to a triangular base acceleration pulse. After surveying the literature in Sec. 1 and noting that nonmonotonic scaling of QoI with applied input has been previously observed in both the earthquake engineering and applied mechanics literature, this paper presents numerical results for the SDOF system in Sec. 2 and notes that for certain parameters and loads, the peak relative acceleration scales nonmonotonically with the base acceleration magnitude. For this simple system, it is possible to solve the equation of motion under certain conditions, and this solution is presented in Sec. 3. An approximate symbolic expression for the peak acceleration as a function of input magnitude is then developed in Sec. 4 and shown to agree qualitatively with the numerical results from Sec. 2. Finally, Sec. 5 investigates the source of the nonmonotonicity and shows that it is due to dynamics excited by the discontinuous derivative of the applied input, the magnitude of which is proportional to the velocity at the time of the discontinuity. Since the first yield of the spring occurs earlier as the input magnitude is increased, this introduces a nonmonotonic dependence of the velocity at the time of the discontinuity, and thus the magnitude of the subsequent dynamics, upon the applied input magnitude.

While the system analyzed here is extremely simple compared to actual structures of interest, this problem provides a rare case where the equation of motion for a nonlinear system can be solved symbolically and shown to exhibit nonmonotonic scaling of QoI with load level. As such, it provides clear mathematical evidence that such nonmonotonicity can occur for even the simplest structures and explains the mechanism by which it occurs. Note that the cause of the nonmonotonicity in this case (i.e., dynamics excited by a load discontinuity) is distinct from the two causes of the nonmonotonicity previously observed in the literature and discussed in Sec. 1. We therefore observe that there are at least three mechanisms by which nonmonotonicity can occur, and it is hypothesized that in complex structures of practical interest involving contact, failure, and other complexities, there are likely to be many other ways in which nonmonotonicity could occur. This suggests that the use of bounding scenarios should be rethought to include some representation of the full spectrum of possible insults. In cases where practical considerations limit the number of scenarios that can be tested, modeling should be used to understand how the QoI vary with the applied input (similar to the IDA approach) in order to define these scenarios, rather than just testing the maximum input scenario.

## Acknowledgment

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

### First-Order Approximation Showing That, when *ω*_{1}*T* ≫ 1, *t*_{y} ∈ [0 *T*/2] If *t*_{y}, Exists

*ω*

_{1}

*T*≫ 1, the spring either yields during [0

*T*/2] or does not yield at all. To see this, it is helpful to first solve the equation of motion for the case in which the spring never yields. In this case, the solution of Eq. (8) applies throughout [0

*T*/2]. In [

*T*/2

*T*], the equation of motion becomes

*z*(

*T*/2) and $z\u02d9(T/2)$ with Eq. (8) and solving for

*G*

_{1}and

*G*

_{2}, we obtain

*T*∞], the equation of motion is

*z*(

*T*) and $z\u02d9(T)$ with Eq. (A2) and solving for

*H*

_{1}and

*H*

_{2}, we obtain

*T*/2. By Eq. (8), the spring force at this time is

*ω*

_{1}

*T*≫ 1, this can be approximated as

*f*

_{0}≥

*mA*. Using Eq. (A2) and rearranging, the spring force during [

*T*/2

*T*] can be written as

*ϕ*

_{G}= arctan (−

*G*

_{1}/

*G*

_{2}). Substituting Eqs. (A3) and (A4), this becomes

*ω*

_{1}

*T*≫ 1, this can be approximated as

*mA*at

*T*/2, and since

*f*

_{0}≥

*mA*, we conclude that the spring does not yield during [

*T*/2

*T*]. Similarly, the spring force during [

*T*∞] can be written from Eq. (A6)

*ϕ*

_{H}= arctan (−

*H*

_{1}/

*H*

_{2}). Substituting Eqs. (A7) and (A8) shows that

*T*∞]. We therefore conclude that for

*ω*

_{1}

*T*≫ 1, the spring either yields during [0

*T*/2] if

*A*≥

*f*

_{0}/

*m*, or else never yields. In reality, the expressions used in this section are approximate and it is possible for the spring to undergo first yield slightly after

*T*/2, and/or for inputs slightly less than

*f*

_{0}/

*m*. When

*ω*

_{1}

*T*is not particularly large and

*A*is slightly less than

*f*

_{0}/

*m*, this can cause significant differences between the numerical and symbolic solutions since the former yields and the latter does not. However, these errors are clustered about

*A*≈

*f*

_{0}/

*m*and decrease dramatically as

*A*is increased or decreased. Since the interesting acceleration scaling observed in Sec. 2 occurs at magnitudes well beyond

*f*

_{0}/

*m*, this approximation is deemed appropriate for this work.

### Proof That the Spring Does Not Stop Yielding During [*t*_{y}*T*/2]

*t*

_{y}

*T*/2], first note that from Eq. (8), the relative velocity at the time of the first yield is less than or equal to zero

*t*∈ [

*t*

_{y}

*T*/2] can be found by differentiating Eq. (12)

*x*such that cos

*x*− 1 = 0, as discussed after Eq. (B1), the first yield cannot occur at a time when the relative velocity is zero, so

*t*

_{y}must be such that cos

*ω*

_{1}

*t*

_{y}− 1 < 0 and the result above is a contradiction. Therefore, $z\u02d9<0$ for

*t*∈ [

*t*

_{y}

*T*/2] and since the velocity does not change sign during this period, the spring remains yielding and the solution of Eq. (12) remains valid throughout.

### Proof That Relative Acceleration at *t*_{2} is Positive (for *t*_{2} < *T*)

*t*

_{2}<

*T*, the relative velocity during [

*T*/2

*t*

_{2}] can be found by differentiating Eq. (16) and restating as

*t*

_{y}

*T*/2]. Since $z\u02d9$ is continuous and its first zero after

*T*/2 occurs at

*t*

_{2}, we can extend this to say that $z\u02d9<0$ during [

*t*

_{y}

*t*

_{2}). Furthermore, by definition, $z\u02d9(t2)=0$. Based on the form of Eq. (C1) (i.e., a cosine term plus a positive constant), we therefore conclude that $z\xa8(t2)>0$. This can be proven by contradiction by supposing that $z\xa8(t2)<0$. In this case, $z\u02d9(t2\u2212\epsilon )$ for any small

*ɛ*> 0 can be approximated by the following backward difference:

*t*∈ [

*t*

_{y}

*t*

_{2}). This proves that $z\xa8(t2)\u22650$. Furthermore, $z\xa8(t2)\u22600$ because this would imply that

*t*

_{2}is one of the local maxima of Eq. (C1), but this contradicts $z\u02d9(t2)=0$ since both terms of Eq. (C1) would be positive. Therefore, $z\xa8(t2)>0$ when

*t*

_{2}<

*T*.

### Proof That Minima of $z\u02d9$ in [*t*_{y}*T*/2] Occur When *ω*_{2}(*t* − *t*_{y}) ≈ (2*n* − 1)*π*

*ϕ*

_{X}= arctan(−

*X*

_{2}/

*X*

_{1}), $\varphi X\u2032=arctan(X1/X2)$, and it has been noted that sgn(

*X*

_{1}) = 1. From these definitions of

*ϕ*

_{X}and $\varphi X\u2032$, we note that when

*X*

_{2}< 0, we have $\varphi X=\varphi X\u2032+\pi /2$ (

*ϕ*

_{X}is in the first quadrant and $\varphi X\u2032$ is in the fourth quadrant). On the other hand, when

*X*

_{2}> 0, we have $\varphi X\u2032=\varphi X+\pi /2$ (

*ϕ*

_{X}is in the fourth quadrant and

*φ*is in the first quadrant). Thus, we have

_{X′}*ϕ*

_{A}= arctan(−

*A*

_{1}/

*A*

_{2})

*n*

_{1}is an integer chosen such that

*ϕ*

_{A}∈ [−

*π*/2

*π*/2]. Therefore,

*n*

_{1}is given by

*ω*

_{1}≫

*ω*

_{2}, we conclude from Eqs. (D1) and (D2) that

*X*

_{1}≫

*X*

_{2}and that $\varphi X\u2032\u2248\xb1\pi /2$. Thus, we see from Eq. (D11) that the minima and maxima of $z\u02d9$ in [

*t*

_{y}

*T*/2] occur when

*ω*

_{2}(

*t*−

*t*

_{y}) ≈

*nπ*. In addition, the sinusoidal term in Eq. (D11) can be shown to be positive at

*t*

_{y}, thus demonstrating that

*t*

_{y}is a maximum of $z\u02d9$. To see this, first denote this term by $z\u02d9$

Thus, when *n*_{1} is even, $\omega 2ty+\varphi X\u2032$ lies in the first or fourth quadrants so $sgn(cos(\omega 2ty+\varphi X\u2032))=1$. In addition, we have cos(*n*_{1}*π*) = 1 for *n*_{1} even. Therefore from Eq. (D15), $sgn(z\u02d9sin(ty))=1$ when *n*_{1} is even. On the other hand, when *n*_{1} is odd, $\omega 2ty+\varphi X\u2032$ lies in the second or third quadrants so $sgn(cos(\omega 2ty+\varphi X\u2032))=\u22121$. Likewise, we have cos(*n*_{1}*π*) = −1 for *n*_{1} odd. Therefore, from Eq. (D15), $sgn(z\u02d9sin(ty))=1$ when *n*_{1} is odd. Therefore, we conclude that $z\u02d9sin$ is positive at *t*_{y}. Since the local minima and maxima of $z\u02d9$ occur when *ω*_{2}(*t* − *t*_{y}) ≈ *nπ*, this implies that *t*_{y} is close to a local maximum of $z\u02d9$. Thus, we conclude that the maxima of $z\u02d9$ in [*t*_{y}*T*/2] occur when *ω*_{2}(*t* − *t*_{y}) ≈ 2*nπ* and that the minima occur when *ω*_{2}(*t* − *t*_{y}) ≈ (2*n* − 1)*π*, where *n* ∈ *Z*^{+}.