Unexpected Scaling of Peak Acceleration for a Yielding Mass-Spring System Subjected to a Triangular Base Acceleration Pulse

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.

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 $(x¨−u¨)$ 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.

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 ω₁T ≫ 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 ≤ f₀/m, or it undergoes a single yielding phase beginning at t_y ∈ [0 T/2] and ending at t₂ > T/2 after which it does not yield again. In contrast, for the general case where we do not assume ω₁T ≫ 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 ω₁T ≫ 1 here, for which we have t_y ∈ [0 T/2]. We also assume that k₁ ≫ k₂, as this is typically true of elastic–plastic hardening, and it allows us to simplify our peak acceleration expression in Sec. 5.

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 f₀/mA. This is investigated further in Sec. 5. We also note here (without derivation for the sake of brevity) that the peak acceleration in [t₂ ∞] also follows Eq. (33) so that this expression can be generalized to apply to [T/2 ∞], and that a similar expression applies when t₂ > T. The derivation of these results can be found in Ref. [17].

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.

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.

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.

There are no conflicts of interest.

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.